Physics Reports vol.346

J. Manjavidze, A. Sissakian / Physics Reports 346 (2001) 1}88

1

VERY HIGH MULTIPLICITY HADRON PROCESSES

J. MANJAVIDZEa, A. SISSAKIANb a

b

Joint Institute for Nuclear Research, Djelepov Lab. of Nucl. Problems, 141980 Dubna, Russia Joint Institute for Nuclear Research, Bogolyubov Lab. of Theor. Physics, 141980 Dubna, Russia

AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO

Physics Reports 346 (2001) 1}88

Very high multiplicity hadron processes J. Manjavidze *, A. Sissakian
Joint Institute for Nuclear Research, Djelepov Lab. of Nucl. Problems, 141980 Dubna, Russia
Joint Institute for Nuclear Research, Bogolyubov Lab. of Theor. Physics, 141980 Dubna, Russia Received July 2000; editor: J.V. Allaby

Contents 1. Introduction 2. Qualitative inside of the problem 2.1. Phenomenology of VHM processes 2.2. S-matrix interpretation of thermodynamics 2.3. Classi"cation of asymptotics over multiplicity 3. Model predictions 3.1. Peripheral interaction 3.2. Hard processes 3.3. Phase transition } condensation 4. Conclusion 4.1. Discussion of physical problems 4.2. Model predictions 4.3. Experimental perspectives Acknowledgements Appendix A. Matsubara formalism and the KMS boundary condition Appendix B. Constant temperature formalism B.1. Kubo}Martin}Schwinger boundary condition Appendix C. Local temperatures C.1. Wigner functions Appendix D. Multiperipheral kinematics D.1. Deep inelastic reactions D.2. Large angle production

4 6 6 10 19 27 29 34 38 41 41 42 43 44 45 46 50 53 56 58 60 60

Appendix E. Reggeon diagram technique for generating function Appendix F. Pomeron with
'0 Appendix G. Dual resonance model of VHM events G.1. Partition function G.2. Thermodynamical parameters G.3. Asymptotic solutions Appendix H. Correlation functions in DIS kinematics H.1. Generating function Appendix I. Solution of the jets evolution equation Appendix J. Condensation and type of asymptotics over multiplicity J.1. Unstable vacuum J.2. Stable vacuum Appendix K. New multiple production formalism and integrable systems K.1. S-matrix unitarity constraints K.2. Dirac measure K.3. Multiple production in sin-Gordon model References

 Permanent address: Institute of Physics, Tbilisi, Georgia. * Corresponding author. Tel.: #7-(09621)-6-35-17; fax: #7-(09621)-6-66-66. E-mail addresses: [email protected] (J. Manjavidze), [email protected] (A. Sissakian). 0370-1573/01/$ - see front matter  2001 Published by Elsevier Science B.V. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 0 2 - 2

61 62 63 64 66 67 68 71 72 74 74 76 76 76 78 82 85

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Abstract The paper contains a description of a "rst attempt to understand the extremely inelastic high-energy hadron collisions, when the multiplicity of produced hadrons considerably exceeds its mean value. Problems with existing model predictions are discussed. The real-time "nite-temperature S-matrix theory is built to have a possibility to "nd model-free predictions. This allows to take the statistical e!ects into consideration and build the phenomenology. The questions to experiment are formulated at the very end of the paper.  2001 Published by Elsevier Science B.V. PACS: 13.85.!t; 12.38.!t; 05.70.Ln; 05.30.!d Keywords: Multiple production; Stochastization; QCD; Integrability

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J. Manjavidze, A. Sissakian / Physics Reports 346 (2001) 1}88

1. Introduction The intuitive feeling that hadron matter should be maximally perturbed in the high-energy extremely inelastic collisions was the main reason of our e!ort to consider such processes. We had hoped to observe new dynamical phenomena, or new degrees of freedom, unattainable in other ordinary hadron reactions. This paper presents a "rst attempt to describe the particularity of considered processes, to give a review of existing models prediction and, at the end, we will o!er the "eld-theoretical formalism for hadron inelastic processes. Thus, considering hadrons mean multiplicity n
(s) as a natural scale of the produced hadrons multiplicity n at given CM energy (s, we would assume that n
(1.1)

At the same time we wish to have (1.2) n;n "(s/m ,

 where mK0.1 Gev is the characteristic hadron mass. The last restriction is introduced to weaken the unphysical constraints from the "nite, for given s, phase space volume. We should assume therefore that s is high enough. The multiple production is the process of colliding particles where kinetic energy is dissipated into the mass of produced particles [1]. Then one may validate that the entropy S accedes its maximum in the domain (1.1) since the multiplicity n characterizes the rate of stochastization, i.e. the level of incident energy dissipation over existing (free) degrees of freedom. There is also another quantitative de"nition of our reactions. Let  be the energy of the fastest

 particle in the given frame and let E be the total incident energy in the same frame. Then the di!erence (E! ) is the energy spent on production of the less energetic particles. It is useful to

 consider the inelasticity coe$cient  E!

 "1!  41 . " E E

(1.3)

It de"nes the portion of spent energy. Therefore, we wish to consider processes with 1!;1 .

(1.4)

So, the produced particles have comparatively small energies. This property may be used for experimental triggering of our processes. Indeed, using the energy conservation law, n(1!)'1 .

(1.5)

Following (1.2) we will assume that m 1!< . E

(1.6)

Therefore, the kinetic energy of the particles produced in our processes cannot be arbitrarily small.

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Using thermodynamical terminology, we wish to investigate the production and properties of comparatively &cold' multi-hadron (mostly of
-mesons) state. We would like to note from the very beginning that we have only a qualitative scenario of such states which may be produced and the review of the corresponding why's may be considered as the main purpose of this paper. At the end of this paper (Appendix K) we will describe principal features of possible "eld-theoretical solution of our problem. The absence of any authentic experimental information concerning discussed processes should be noted. Moreover, actually the hadron inelastic interactions with a set peculiar to them of unsolved theoretical problems will be considered. Nevertheless, we suggest to work in this "eld in spite of these di$culties because the system with extremal properties may be more transparent since the asymptotics always simplify a picture. We would demonstrate this idea and will try to put it in the basis of developed theoretical methods. The absence of experimental information about such high inelastic hadron processes is the consequence of the smallness of corresponding cross sections. Besides this, it was unclear for what purpose the experimental e!orts should be done. We would like to convince the reader that the discussed problem is interesting and important. For instance, we will discuss a possibility that asymptotics over n may replace in a de"nite sense the asymptotics over (s. A short address to experimentalists will be given in Section 4.1.2. We hope that the paper would be useful both for theorists and experimentalists. For this reason the main text of this paper will contain only the qualitative discussion of the problems and results. The quantitative proof, formulation of pure theoretical methods, etc. are added in the appendixes. Considered extremal problem is a good theoretical laboratory and is described in the appendixes and theoretical methods may be applied for other physical problems. We would like to point out that a special technique was built for the problem discussed, see Section 2. E Having the very high multiplicity (VHM) state it is natural to use the thermodynamical methods. We will o!er for this purpose the real-time S-matrix interpretation of thermodynamics. It can be shown in what quantitative conditions it will coincide with simpler canonical imaginary-time Matsubara formalism. We will give also the generalization of the real-time "nite-temperature perturbation theory in the case of local temperature ¹"1/ distribution, when "(x, t). This will allow to use the thermodynamical description if the system is far from equilibrium. E The particle spectrum in the VHM region is soft. It is just a situation when the collective phenomena should be important. To describe these phenomena, the decomposition on correlators will be adopted. The origin of this decomposition lies in Mayer's &group decomposition'. In multiple production physics this decomposition is known also as the &multi-component description' [2]. It is based on the idea that the multiple production process may include various mechanisms. In Section 3 we will investigate model predictions for VHM region. We would like to note two main conclusions: E Existing multiperipheral-type models are unable to describe the VHM region. E The infrared region of the pQCD becomes important even if constraint (1.2) is taken into account.

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2. Qualitative inside of the problem In Section 2.1 we will try to formulate the phenomenology of our problem, i.e. the way the VHM processes may be described and what type of phenomena one may expect. The importance of thermodynamical methods will become evident and we will o!er in Section 2.2 a general description of the corresponding formalism. It is important to note that we may classify the possible asymptotics over n. We will "nd that there exist only three classes of asymptotics. This will simplify consideration de"nitely restricting the possibilities. In Section 2.3 we will use the thermodynamical language to give a physical interpretation of these classes of asymptotics. We will see in result that in our choice of the VHM "nal state this should lead to reorganization of multiple production dynamics: we will get out of the habitual multiperipheral picture in the VHM domain. Moreover, one may assume that the semiclassical approximation becomes exact in the VHM domain. This naturally leads to the idea to search for such a scheme of calculation which depends on the choice of "nal state. Quantitative description of this idea may be realized as is described in Appendix K.

2.1. Phenomenology of VHM processes The VHM production phenomena include two sub-problems. First of all, it is the dynamical problem of incident energy degradation into the secondary particle energies and the second one is the description of the "nal state. We will start discussion in Section 2.1.1 from the second part of the problem to explain that the statistical methods are essential for us. In Section 2.1.2 we will try to outline at least qualitatively the main mechanisms of hadron production. The peculiarity of hadron production phenomena consists in the presence of hidden constraints, the consequence of local nonAbelian gauge symmetry. The constraints may prevent thermalization and the incident energy dissipation is confusing in this case. Just the &confusing' e!ect is dominant in the hadron multiple production processes if n&n
(s). The expected change of dominant mechanism of hadron production is discussed in Section 2.1.3. It is important that, in spite of hidden constraints, the system may freely evolve to de"ne the VHM state. Such a VHM state should be in equilibrium. A formal de"nition of the &equilibrium' notion will be given in Section 2.2.2. The problem described contains small parameters (n
(s)/n);1 and (1!);1. To have the possibility of estimation of contributions in accordance with these parameters one should include them into the formalism. This becomes possible if and only if the integral quantities are calculated. So, the multiplicity n is an index only if the multiple production amplitudes a (p ,2, p ) are considered. But the cross section  (s) is a nontrivial function of n. L  L L We will calculate by this logic mostly integrals of a , excluding from consideration the L amplitudes, see also Section 2.2, where the "rst realization of this idea is o!ered (a naive attempt to realize this idea one may "nd in [3]). This is a general methodological feature of our consideration.

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2.1.1. Formulation of the problem The multiple production cross section  (s) falls down rapidly in the discussed very high L multiplicity (VHM) domain (1.1) and for this reason the multiplicities n&n are not accessible

 experimentally. At the LHC energy n
(s)K100 is valid and we will assume that n&n
(s)K10 000 is just the discussed VHM region (n K100 000 at the LHC energy). We will explain later why

 n&n
(s) (2.1) is chosen for the de"nition of the VHM region. Generally speaking, having the state of a large number of particles, it is reasonable to depart from an exact de"nition of the number n of created particles, their individual energies  , momenta G q , etc. since they cannot be de"ned exactly by experiment. Indeed, for instance, full reconstruction G of kinematics is a practically impossible task because of neutral particles, neutrinos, the more so as n&10 000 is considered. We suppose that nothing will happen if n is measured with nO0 accuracy since (n/n);1 is easily attainable in the VHM region. Besides, it is practically impossible to deal with theory which operates by the N"3n!4 (&30 000!) variables. Arti"cial reduction of the set of the necessary variables may lead to a temporary success only. Indeed, the last 30 years of multiple production physics development was based on the inclusive approach [4], when the measured quantities (cross sections) depend on a few dynamical parameters only. But later on the experiment and its fractal analyses show that the situation is not so simple, also as, for instance, for the classical turbulence. So, the event-by-event experimental data show that the particle density #uctuation is unexpectedly large [5] and the fractal dimension D is  not equal to zero [6]. We know that if the fractal dimension is non-trivial, then the system is extremely &non-regular' [7]. So, D K0.3 for the perimeter of Great Britain and D K0.5 for Norway. The discrepancy   marks the fact that the shore of Norway is much more broken than that of Great Britain [8]. It is noticeable that the fractal dimension D crucially depends on the type of reaction, incident energy  and so on. It is evident that one may choose from N"3n!4 an arbitrary "nite set of variables to characterize the multiple production process. But the fractal analyses show that such an approach would lead to the same e!ect as if one may hear, for example, only the "rst violin of Mahler's music. So, it is important to understand when the restricted set of dynamical variables will allow to describe the process (state) completely. The same problem was solved in statistical physics, where the &rough' description by a restricted number of (thermodynamical) parameters is a basis of its success, see the discussion of rough variables description, e.g. in the review of Uhlenbeck [9]. We will search for the same solution desiring to build a complete theory of the VHM hadron reactions. We want to note that just VHM process may be in this sense &simple': from all evidence, the system becomes &quiet' in the VHM region and for this reason its &rough' thermodynamical description is available. It seems natural, therefore, to start investigation of multiple production phenomena from the (extremely rare) VHM processes. 2.1.2. Soft channel of hadron production The dominant inelastic hadron processes at n&n
(s) are saturated by production of low transverse momentum hadrons [10]. One of the approaches explains this phenomenon by the nonperturbative e!ect of quarks created from vacuum.

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The corresponding dynamics appears as follows. At the expense of transverse kinetic motion color charges may separate at large distances. Nevertheless, the transverse motion is suppressed since separation leads to increasing polarization of vacuum, because of con"nement phenomenon. Then, as in QED [11], the vacuum becomes unstable in regard to the tunneling creation of real fermions. Just on their creation, the transverse kinetic energy is spent and, as a result, particles cannot have, with exponential accuracy, high transverse energies. This picture is attractive being simple and transparent, but despite numerous e!orts [12], there is no quantitative description of this phenomenon till now. Brie#y, the problem is connected with the unknown mechanism of strong colored electric "elds formation among distant colored charges, see also [13]. One may use other terms. The soft channel of multiple production means the long-range correlation among hadron colored constituents. Under this special correlation the nonAbelian gauge "eld theory conservation laws constraints were implied. They are important in dynamics since each conservation law decreases the number of independent degrees of freedom at least on one unity (this may explain why hadrons n
(s);n (s)), i.e. it has a nonperturbative e!ect.

 Moreover, in the so-called integrable systems each independent integral of motion (in involution) reduces the number of degrees of freedom on two units. As a result there is no thermalization in such systems [14] and the corresponding mean multiplicity n
(s) should be equal to zero, see Appendix K. The existence of multiple production, n
(s)<1, testi"es to the statement that the thermalization phenomena exist in hadron processes, i.e. the system of Yang}Mills "elds is not completely integrable. But the most probable process with n&n
(s) did not lead to the "nal state with maximal entropy since n
(s);n , i.e. the de"nite restrictions on the dissipation dynamics should be taken

 into account. Such problems, being intermediate, are mostly complicated ones. The quantitative theory of these phenomena may lead to deep revision of the main notions of the existing quantum "eld theory [15,16], see Appendix K. So, the dynamical display of hidden conservation laws of the hadron system are probably unstable since we expect that the system is not completely integrable, solitary "eld con"gurations u (x, t) [17]. Then the quantum theory  should be able to describe quantum excitations of these "elds, i.e. to count the #uctuations of &curved' manifolds. The canonical perturbation theory methods, formulated in terms of creation and annihilation of particles in the external "eld u (x, t), are too complicated, see [18] and  references cited therein. For this reason, existing calculations usually do not exceed the semiclassical approximation. We hope that, as described in Appendix K, the quantization scheme would be able to solve this problem (see also the example described there). Another approach assumes that the special &t-channel' ladder-type Feynman diagrams are able to describe the n&n
(s) region [19]. This approach did not take into account the con"nement nonperturbative e!ects introducing the hardly controllable supposition that the free quarks and gluons may form a complete set of states. Formally this is right, but from all evidence, the decomposition on this Fock basis is realized in the nonAbelian gauge "eld theories on zero measure [16,20] (see Appendix K). Nevertheless, one may reject this argument assuming that the process is happening at a su$ciently small distance. Corresponding contribution came from the so-called &hard Pomeron' [21]. But the intrinsic problems of the accuracy of chosen logarithmic approximations [22], the understanding of the so-called nonlogarithmic corrections [23], of the fate of the infrared divergences remain unsolved till now.

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2.1.3. Multiple production as a process of dissipation So, the multiple production of soft hadron phenomena seems unsolvable on the day-to-day level of understanding and we may with a clear conscience move it away. This is why the hadron inelastic reactions lost some popularity, migrating the last two decades to the class of &noninteresting' problems. Yet, in a number of modern fundamental experiments, multiple production plays, at least, the role of background to the investigated phenomena and for this reason we should be ready for the quantitative estimation of it. Our hope to describe such a complicated problem as the multiple production phenomenon in the VHM conditions is based on the following idea. At the very beginning of this century, a couple P. and T. Ehrenfest, had o!ered a model to visualize Boltzmann's interpretation of the irreversibility phenomena in statistics. The model is extremely simple and fruitful [24]. It considers the two boxes with 2N numerated balls. Choosing the label of the balls randomly one must take the ball with the
corresponding label from one box and put it into another one. One may repeat this action an arbitrary number of times t. Starting from the highly &nonequilibrium' state with all balls in one box, N <1, it is seen to be
stationary with t tendency to equalization of the number of balls in the boxes (Fig. 1). The stationarity means that the number of balls in the other box rises &t at least on an early stage of

Fig. 1. Predictions of the Ehrenfest model. Four simulations are displayed.

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the process. This signi"es the presence of an irreversible #ow (of balls) toward the preferable (equilibrium) state. One can hope [24] that this model re#ects a physical reality of nonequilibrium processes with the initial state very far from equilibrium. A theory of such processes with (irreversible) #ow toward a state with maximal entropy should be su$ciently simple being close to the stationary Markovian. The VHM production process may be, at least in an early stage, stationary Markovian. If this is so then one may neglect long-range e!ects, nonperturbative as well, since they are not Markovian as follows from the experience described in Section 2.1.2. This is possible if the VHM process is happening so fast (being the short-range phenomenon) that the con"nement forces became &frozen'. It can be shown that the quantitative reason for these phenomena is a fast (exponential) reduction with n of the soft channel contribution into the hadron production process. So, we expect a change of the multiple production dynamics in the VHM region. Thus, the main input idea consists of two general propositions. The "rst of them is the following: (I). The hadron VHM production processes should be close to the stationary Markovian. &Freezing' the con"nement constraints, the entropy S may exceed for given energy (s its available maximum in the VHM domain. Then one can assume that the VHM "nal state is in &equilibrium', or is close to it. So, (II). The VHM xnal state should be close to equilibrium is our second basic proposition. We would select and appreciate particle physics models in accordance with these propositions. 2.2. S-matrix interpretation of thermodynamics The "eld-theoretical description of statistical systems at a "nite temperature is based usually on the formal analogy between imaginary time and inverse temperature "1/¹ [26]. This analogy is formulated by Schwinger [11] as the &euclidean postulate' and it assumes that (i) the system is in equilibrium, i.e. it should allow the arbitrary rearrangement of states of temporal sequence in the described process, and (ii) there are no special space-time long-range correlations among states of the process, i.e., for instance, the symmetry constraints should not play a crucial role. We do not know ad hoc whether or not to apply the &euclidean postulate' for given n and s, even if (1.1) is satis"ed. For this reason we are forced to formulate the theory in natural real-time terms. The "rst important quantitative attempt to build the real-time "nite-temperature "eld theory [27] discovered the formal problem of the so-called &pinch-singularities'. Further investigation of the theory has allowed to demonstrate the cancellation mechanism of these unphysical singularities [28]. This is attained by doubling the degrees of freedom [29}31]: the Green functions of the theory represent 2;2 matrix [32]. It surely makes the theory more complicated, but the operator formalism of the thermo-"eld dynamics [32] shows the unavoidable character of this complication.

 One can say that the opposite #ow is never seen. &What never? No never! What never? Well, hardly ever'. This dialog was taken from [25].  In other terms, one may have the possibility to apply the ergodic hypothesis.

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Notice that the canonical real-time "nite temperature "eld-theoretical description [29,30] of the statistical systems based on the Kubo}Martin}Schwinger (KMS) [29,33,34] boundary condition for a "eld is (t)"(t!i) .

(2.2)

It, without fail, leads to the equilibrium #uctuation}dissipation conditions [35] (see also [36]). Due to this it cannot be applied in our case, where the dissipation problem is solved. The origin of this boundary condition is shown in Appendix A. We will use a more natural microcanonical formalism for particle physics. The thermodynamical &rough' variables are introduced in this approach as the Lagrange multipliers of corresponding conservation laws. The physical meaning of these &rough' variables is de"ned by the corresponding equations of state. We shall use the S-matrix approach which is natural for the description of the time evolution. (The S-matrix description is used also in [38,39].) For this purpose the amplitudes p , p ,2, p q , q ,2, q "a (p, q) (2.3)   L   K LK of the m- into n-particles transition will be introduced. The in- and out-states must be composed from mass-shell particles [40]. Using these amplitudes we will calculate R (p, q)"a (p, q)"p , p ,2, p q , q ,2, q q , q ,2, q p , p ,2, p . (2.4) LK LK   L   K   K   L This will lead to the doubling of the degrees of freedom. The temperature description will be introduced (see also [41]) noting that, for instance, d "a (p, q)d (p) , L LK L L dp G , (p)"(p#m) d (p)"

(2.5) L (2
)2(p ) G  is the di!erential measure of the "nal state. It is a "rst example where the usefulness of the probability-like quantity &a  is seen. LK Measure (2.5) is de"ned on the energy}momentum shell L  p "P . (2.6) G G It should be underlined that a (p, q) are the translationally invariant amplitudes and four LK equalities: L K  p " q G G G G

 The statistical methods for particles physics are discussed also in [37].

(2.7)

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are obeyed identically. So, Eqs. (2.6) are the constraints and to take them into account one may multiply d on L L

e ?NI , I where  is the time-like 4-vector. It is evident that integration over  with factor e\ ?. gives the constraints (2.6). One may simplify the calculation assuming that all calculations are performed, for example, in the CM frame P"(E, 0). Then one may ignore the space components considering "( , 0). This  is the equivalent of the assumption that only the energy conservation law is important. The last step is the substitution  "i, where  is our Lagrange multiplier. To de"ne its  physical meaning one should solve the equation of state





L R R (2.8) E"! ln d e\@CNI ,! ln d () . L L R R I Such a de"nition of temperature as the Lagrange multiplier of the energy conservation law is obvious for microcanonical description [33]. The initial-state temperature will be introduced in the same way, taking into account (2.7). So, we will construct the two-temperature theory. It is impossible to use the KMS boundary condition in such a two-temperature description (the equation of state can be applied at the very end of the calculations). It should be noticed that the &density matrix' R (p, q), de"ned in (2.4), describes the &closed-path LK motion' in the functional space. So, if p , p ,2, p q , q ,2, q "n, oute 1> m, in   L   K

(2.9)

and p , p ,2, p q , q ,2, q H"q , q ,2, q p , p ,2, p   L   K   K   L  "m, ine\ 1 \ n, out

(2.10)

then, by de"nition,  ( )" ( )"( ) (2.11) >  \   with some &turning-point' "elds ( ), where  is the remote hypersurface. The value of ( )    speci"es the environment of the system. We will show that (2.11) coincides with the KMS boundary condition in some special cases. Here consequences of the vacuum boundary condition: ( )"0 (2.12)  are analyzed. One should admit also that the boundary conditions given below are not unique: one can consider arbitrary organization of the environment of the considered system. The S-matrix interpretation is able to show the way as an arbitrary boundary condition may be adopted. This should extend the potentialities of the real-time "nite-temperature "eld-theoretical methods.

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2.2.1. Example It seems useful to illustrate the above microcanonical approach by the simplest example, see also [41]. By de"nition, the n particles production cross section








L (2.13)  (s)" d (p) q #q !  p a (p, q) , L   G L L G where a (p, q),a (p, q) is the ordinary n particle production amplitude in accelerator L L experiments. Considering the Fourier transform of energy}momentum conservation -function one can introduce the generating function , see [41] and references cited therein. We may "nd in the L result that  is de"ned by the equality L >  d e@# (), E"(q )#(q ) , (2.14)  (E)" L   L 2
\  where









L dp e\@CNG  G a " d () . (2.15)

L L (2
)2(p ) G G The most probable value of  in (2.14) is de"ned by the equation of state (2.8). Inserting (2.15) into (2.14) we "nd expression (2.13) if the momentum conservation shell is neglected. The last one is possible since the cross sections are always measured in the de"nite frame. Let us consider the simplest example of noninteracting particles [41]:

()" L

()&2
mK (m)/L , L  where K is the Bessel function. Inserting this expression into (2.8) we can "nd that in the  nonrelativistic case (n&n )

 3 (n!1) ,  "  2 ((s!nm) i.e., we "nd the well-known equality E "¹ ,   

(2.16)

where E "((s!nm)/(n!1) is the mean kinetic energy and ¹"1/ is the temperature    (the Boltzmann constant was taken to be equal to one). It is important to note that Eq. (2.8) has a unique real solution  (s, n) rising with n and  decreasing with s [33]. The expansion of the integral (2.14) near  (s, n) unavoidably gives an asymptotic series with zero  convergence radii since () is the essentially nonlinear function of , see also Section 2.2.2. This L

 The generating functionals method was developed in [42].

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means that, generally speaking, #uctuations in the vicinity of  (s, n) may be arbitrarily high and in  this case  (s, n) has no physical sense. But if the #uctuations are Gaussian, then () coincides  L with the partition function of the n particle state and  (s, n) may be interpreted as the inverse  temperature. We will put the observation of this important fact in the basis of our thermodynamical description of the VHM region. 2.2.2. Relaxation of correlations The notion of &equilibrium' over some parameter X in our understanding is a requirement that the #uctuations in the vicinity of its mean value, XM , have a Gaussian character. Notice, in this case, that one can use this variable for a &rough' description of the system. We would like to show now that the corresponding equilibrium condition would have the meaning of the correlations relaxation condition of Bogolyubov [42], see also [43]. Let us de"ne the conditions when the #uctuations in the vicinity of  are Gaussian [44]. Firstly, to estimate integral (2.14) in the vicinity  of the extremum,  , we should expand ln (# ) over :  L  1 R 1 R ln ( )!  ln ( )#2 ln (# )"ln ( )!(s#  L  L  L  L  2! R 3! R  

(2.17)

and, secondly, expand the exponent in the integral (2.14) over, for instance, Rln ( )/R,2 , L   etc. In the result, if higher terms in (2.17) are neglected, the kth term of the perturbation series








Rln ( )/R I 3k#1 L   . 

& LI (Rln ( )/R) 2 L  

(2.18)

Therefore, because of Euler's ((3k#1)/2) function, the perturbation theory near  leads to the  asymptotic series. The supposition to de"ne this series formally, for instance, in the Borel sense is not interesting from the physical point of view. Indeed, such a formal solution assumes that the #uctuations near  may be arbitrarily high. Then, for this reason, the value of  loses its   signi"cance: arbitrary values of (! ) are important in this case.  Nevertheless, it is important to know that our asymptotic series exists in some de"nite sense, i.e. we can calculate the integral over  by expanding it over (! ). Therefore, if the considered series  is asymptotic, we may estimate it by "rst term if Rln ( )/R;(Rln ( )/R) . L   L  

(2.19)

One of the possible solutions of this condition is R ln ( )/R+0 . L  

 The term &vanishing of correlations' was used by N.N. Bogolyubov for this phenomenon.

(2.20)

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15

If this condition is satis"ed, then the #uctuations are Gaussian with dispersion &Rln ( )/R , L   see (2.17). Let us consider now (2.20) carefully. We will "nd by computing derivatives that this condition means the following approximate equality



  ( ) L !3 L L #2 L +0 , (2.21)



 L L L where I means the kth derivative. For identical particles, L I (2.22)

I( )"nI(!1)I d ( ) (q ) . L  G L  G Therefore, the left-hand side of (2.21) is the 3-point correlator K since d ( ) is a density of states  L  for given 
















  

(q ) !3 (q ) (q )  #2 (q )  , (2.23) G G  @ G @   @ @ G G G where the index  means that averaging is performed with the Boltzmann factor exp! (q).   Notice, in distinction with Bogolyubov, K is the energy correlation function. So, in our  interpretation, one can introduce the notion of temperature 1/ if and only if the macroscopic  energy #ows, measured by the corresponding correlation functions, are to die out. As a result, to have all the #uctuations in the vicinity of  Gaussian, we should have  K +0, m53. Notice, as follows from (2.19), that the set of minimal conditions actually appears K as follows: K , d (q)  

K ;K K, m53 . (2.24) K  If the experiment con"rms these conditions then, independent of the number of produced particles, the "nal state may be described with high enough accuracy by one parameter  and the energy  spectrum of particles is Gaussian. In these conditions one may return to the statistical [1] and the hydrodynamical models [45]. Considering  as a physical (measurable) quantity, we are forced to assume that both the total  energy of the system, (s"E, and the conjugate to it, variable  , may be measured simultaneously  with high accuracy. 2.2.3. Connection with Matsubara theory We would like to show now that the ordinary big partition function of the statistical system coincides with



R (p, q)" (, z) ,  LK     LK @ X _@ X 

(2.25)

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where R (p, q) is de"ned by (2.4). The summation and integration are performed with constraints LK that the mean energy of particles in the initial("nal) state is 1/ (1/ ). One may interpret 1/ in the   "rst approximation as the temperature and z (z ) as the activity for initial("nal) state.   Direct calculation, see Appendix B, gives the following expression for generating functional:

(, z)"e\ N(HG (H R () ,  where the particle number operator (K (x)" / (x))

(2.26)



N(H )"! dx dx(K (x)D (x!x,  , z )K (x)!K (x)D (x!x,  , z )K (x)) G H > >\   \ \ \>   > (2.27) and R ()"Z( )ZH(! ) ,  > \ where Z() is de"ned in (B.10):

(2.28)



Z()" De 1\ 4>( and, for the vacuum boundary condition ( )"0, 



(2.29)

D (x!x, , z)"i d (q)e\ OV\VYe\@COz(q) , \> 

(2.30)

D (x!x, , z)"!i d (q)e OV\VYe\@COz(q) , >\ 



are, respectively, the positive and negative frequency correlation functions at z"1. It is evident that



 

 

L K R (p, q)" e@ CNI 

e@ COI 

(, z) . (2.31) LK z (p ) z (q ) G X   I  I I I Notice, de"ning R (p, q) through the generating functional we extract the Boltzmann factors LK e\@C since the energy}momentum conservation -functions were extracted from amplitudes a (p, q). LK We suppose that Z() may be computed perturbatively. As a result, ( jK " / j is the variational derivative) R(, z)"e\ 4\ HK > > 4\ HK \ e V VYHG V"GI V\VY_ @ XHI VY ,

(2.32)

where D (x!x) is the matrix Green function. These Green functions are de"ned on the Mills [46] I time contours C in the complex time plane (C "CH ), see Fig. 2. This de"nition of the time ! \ > contours coincides with the Keldysh' time contour [30]. The generating functional (2.32) has the same structure as the generating functional of Niemi and Semeno! [28]. The di!erence is only in the de"nition of Green functions D . This choice is I

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17

Fig. 2. Keldysh time contour.

a consequence of the boundary condition (B.6). So, if (B.32) is used, then the Green function is de"ned by Eq. (B.45). Notice also that if  " " then a new Green function obeys the KMS   boundary condition, see (B.48). Following Niemi and Semeno! [28] one can write (2.32) in the form



()" D e 1,1  , ,1

(2.33)

where the functional measure D  and the action S () are de"ned on the closed complex time ,1 ,1 contour C , see Fig. 3. The choice of initial time t and t is arbitrary. Then one can perform shifts: ,1  t P!R and t P#R. In result, (i) if  " ", (ii) if contributions from imaginary parts    C and C of the contour C have disappeared in this limit, (iii) if the integral (2.33) may be >\ \@ ,1 calculated perturbatively then this integral is a compact form of the representation (2.32). Notice that the requirements (i)}(iii) are the equivalent of the Euclidean postulate of Schwinger. In this frame one can consider another limit t Pt . Then the C contour reduces to the  ,1 Matsubara imaginary time contour, Fig. 4. Later on we will use this S-matrix interpretation of thermodynamics. But one should keep in mind that corresponding results will hide assumptions (i)}(iii). We would like to mention the ambivalent role of external particles in our S-matrix interpretation of thermodynamics. In the ordinary Matsubara formalism the temperature is measured assuming that the system under consideration is in equilibrium with the thermostat, i.e. temperature is the energy characteristics of interacting particles. In our de"nition the temperature is the mean energy of produced, i.e. non-interacting, particles. It can be shown that both de"nitions lead to the same result. Explanation of this coincidence is the following. Let us consider the point of particle production as the coordinate of "ctitious &particle'. This &particle' interacts since the connected contributions

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Fig. 3. Niemi}Semeno! time contour.

Fig. 4. Matsubara time contour.

into the amplitudes a only are considered, and has a momentum equal to the produced particles LK momentum and so on. The set of these &particles' forms a system. Interaction among these &particles' may be described by the corresponding correlation functions, see Section 2.3.3. Let us consider now the limit t Pt . In this limit (2.33) reduces to 



()" D e\1+  , +

(2.34)

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19

where the imaginary time measure D  and action S () are de"ned on the Matsubara time + + contour. The periodic boundary condition (2.2) should be used for calculating integral (2.34). The rules and corresponding problems in the integral (2.34) can be calculated as described in many textbooks, see also [47]. In the limit considered the time was eliminated in the formalism and the integral in (2.34) performed over all the states of the &particles' system with the weight e\1+ . Notice that the doubling of degrees of freedom has disappeared and our "ctitious &particles' became real ones. On the other hand, the produced particles may be considered as the probes through which we measure the interacting "elds. As was mentioned above, their mean energy de"nes the temperature, if the energy correlations are relaxed. If even one of the conditions (i)}(iii) is not satis"ed then one cannot reduce our S-matrix formalism to the imaginary time Matsubara theory. Then one can ask: is there any possibility, staying in the frame of S-matrix formalism, to conserve the statistics formalism. This question is discussed in Appendix C, where the Wigner functions approach is applied. It may be shown that the formalism may be generalized to describe the kinetic phase of the nonequilibrium process, where the temperature should have the local meaning [49]. The comparison with the &local equilibrium hypothesis' is discussed at the end of Appendix C. 2.3. Classixcation of asymptotics over multiplicity Our further consideration will be based on the model-independent (formal) classi"cation of asymptotics [50]. 2.3.1. &Thermodynamical' limit We will consider the generating function L  (2.35) ¹(s, z)"  zL (s), s"(p #p )<m, n "(s/m . L  

 L This step is natural since the number of particles is not conserved in our problem. So, the total cross section and the averaged multiplicity will be



d  (s)"¹(s, 1)"  (s), n
(s)" n( (s)/ )" ln ¹(s, z) .  L L  dz X L L At the same time, the inverse Mellin transform gives



1 RL ¹(s, z)  " L n! RzL

1 " 2
i



dz 1 ¹(s, z)" zL> 2
i



dz e\L  X>  2QX . z

(2.36)

(2.37)

X The essential values of z in this integral are de"ned by the equation (of state) n"z

R ln ¹(z, s) . Rz

(2.38)

Taking into account the de"nition of the mean multiplicity n
(s), given in (2.36), we can conclude that the solution of (2.38) z is equal to one at n"n
(s). Therefore, z'1 is essential in the VHM  domain.

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The asymptotics over n (n;n is assumed) are governed by the smallest solution z of (2.38)

  because of the asymptotic estimation of the integral (2.37) (2.39)  (s)Je\L  X LQ . L Let us assume that in the VHM region and at high energies, (sPR, there exists such a value of z (n, s) that we can neglect in (2.35) the dependence on the upper boundary n . This formal trick 

 with the thermodynamical limit allows to consider ¹(z, s) as the nontrivial function of z for "nite s. Then, it follows from (2.38) that z (n, s)Pz at n3VHM , (2.40)   where z is the leftmost singularity of ¹(z, s) in the right-half plane of complex z. One can say that  the singularity of ¹(z, s) attracts z (n, s) if n3VHM. We will put this observation in the basis of  VHM processes phenomenology. We would like to underline once more that actually ¹(z, s) is regular for arbitrary "nite z if s is "nite. But z (n, s) behaves in the VHM domain as if it is attracted by the (imaginary) singularity z .   And just this z (n, s) de"nes  in the VHM domain. We want to note that actually the energy  L (s should be high enough to use such an estimation. 2.3.2. Classes and their physical content One can notice from estimation (2.39) that  weakly depends on the character of the singularity. L Therefore it is enough to classify only the possible positions of z . We may distinguish the following  possibilities: (A) z "R:  (O(e\L) ,  L (B) z "1:  'O(e\L) ,  L (C) 1(z (R:  "O(e\L) , (2.41)  L i.e., following this classi"cation, the cross section may decrease faster (A), slower (B), or as (C) an arbitrary power of e\L. It is evident, if all these possibilities may be realized in nature, then we should expect the asymptotics (B). As was explained in Section 2.2.1,  has the meaning of the n particle partition function in the L energy representation. Then ¹(z, s) should be the &big partition function'. Taking this interpretation into account, as follows from the Lee}Yang theorem [15], ¹(z, s) cannot be singular at z(1. At the same time, the direct calculations based on the physically acceptable interaction potentials give the following restriction from above: (D)  (O(1/n) . (2.42) L This means that  should decrease faster than any power of 1/n. L It should be noted that our classi"cation predicts rough (asymptotic) behavior only and did not exclude local increase of the cross section  . L One may notice that 1  (s) ! ln L "ln z (n, s)#O(1/n) .  n  (s) 

(2.43)

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21

Using thermodynamical terminology, the asymptotics of  is governed by the physical value of the L activity z (n, s). One can introduce also the chemical potential  (n, s). It de"nes the work needed for   one particle creation, ln z (n, s)" (n, s) (n, s), where 
(n, s)"1/ (n, s) is the produced particles     mean energy. So, one may introduce the chemical potential if and only if  (n, s) and z (n, s) may be   used as the &rough' variables. Then the above formulated classi"cation has a natural explanation. So, (A) means that the system is stable with reference to particle production and the activity z (n, s) is the increasing  function of n, the asymptotics (B) may be realized if and only if the system is unstable. In this case z (n, s) is the decreasing function. The asymptotics (C) is not realized in equilibrium thermo dynamics [52]. We will show that the asymptotics (A) re#ects the multiperipheral processes kinematics: created particles form jets moving in the CM frame with di!erent velocities along the incoming particles directions, i.e. with restricted transverse momentum, see Section 3.1.1. The asymptotics (B) assumes condensation-like phenomena, see Section 3.3. The third-type asymptotics (C) is predicted by stationary Markovian processes with the pQCD jets kinematics, see Section 3.2.2. The DIS kinematics may be considered as the intermediate, see Section 3.2.1. This interpretation of classes (2.41) allows to conclude that we should expect reorganization of production dynamics in the VHM region: the soft channel (A) of particle production should yield a place to the hard dynamics (C), if the ground state of the investigated system is stable with reference to the particle production. Otherwise we will have asymptotics (B). 2.3.3. Group decomposition Let us consider the system with several correlation scales. For example, in statistics one should distinguish correlation length among particles (molecules) and correlation length among droplets if the two-phase region is considered. In particle physics, one should distinguish in this sense correlation among particles produced as a result of resonance decay and correlations among resonances. In pQCD one may distinguish correlations of particles in jet and correlation among jets. There exist many model descriptions of this physical picture. In statistics Mayer's group decomposition [25] is well known. In particle physics one should note also the many-component formalism [2]. We will consider the generating functions (functionals) formalism [42] considering mostly jet correlations. In many respects it overlaps the above-mentioned approaches. The generating function ¹(z, s) may be written in the form   (z!1)I C (s)"  zJb , ln ¹(z, s)"  I J k! J I

(2.44)

where the coe$cients C are the moments of the multiplicity distribution I P (s)" (s)/ (s) . L L 

 The example considered in [48] illustrates this approach.

(2.45)

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So, C (s)" nP (s)"n
(s), C (s)" n(n!1)P (s)!n
(s) (2.46)  L  L L L and so on. Using the connection with the inclusive distribution functions f (q , q ,2, q ) I   I  (z!1)I ¹(z, s)"  d (q)f (q , q ,2, q ; s) , (2.47) I I   I k! I it is easy to "nd that



 

C (s)" d (q)f (q; s)"fM (s) ,     C (s)" d (q)f (q , q ; s)!f (q ; s)f (q ; s)"fM (s)!fM (s) ,           

(2.48)

etc. Generally,




  

1  (!1)J  J J fM (s) C (s)"  ,   k !k IG G k! I l  k! G J G I J  G where k "k , k ,2, k and J   J

(2.49)



fM (s)" d (q)f (q , q ,2, q ; s) . I I I   I One may invert formulae (2.49):









1   1 C (s) LI  I fM (s)"   kn !l

. (2.50) I l! I n ! k!   I I LI  I The Mayer group coe$cients b in (2.44) have the following connection with C : J I  (!1)I b (s)"  C (s) . (2.51) J l!k! I>J I It seems useful to illustrate the e!ectiveness of the generating function method by the following example. We will consider the transformation (multiplicity nP activity z) to show the origin of the Koba}Nielsen}Olesen scaling (KNO-scaling). If C "0, m'1, then  is described by the Poisson formulae: K L n
(s)L  (s)" (s)e\L
. (2.52) L  n! It corresponds to the case of absence of correlations.  In a private discussion with one of the authors (A.S.) in the summer of 1973, Koba noted that the main reason for investigation leading to the KNO-scaling was just the generating functional method of Bogolyubov [49].

J. Manjavidze, A. Sissakian / Physics Reports 346 (2001) 1}88

23

Let us consider a more weak assumption: C (s)" (C (s))K , K K 

(2.53)

where  is the energy-independent constant, see also [53], where a generalization of KNO scaling K on the semi-inclusive processes was o!ered. Then  ln ¹(z, s)"  K (z!1)n
(s)K . (2.54) m! K To "nd the consequences of this assumption, let us "nd the most probable values of z. The equation of state n"z

R ln ¹(z, s) Rz

has solution z
(n, s) increasing with n since ¹(z, s) is an increasing function of z, if and only if, ¹(z, s) is nonsingular at "nite z. As was mentioned above, the last condition has deep physical meaning and practically assumes the absence of the "rst-order phase transition [51]. Let us introduce a new variable: "(z!1)n
(s) .

(2.55)

The corresponding equation of state is as follows:






n R  " 1# ln¹() . n
(s) n
(s) R

(2.56)

So, with O(/n
(s)) accuracy, one can assume that K (n/n
(s)) . 

(2.57)

is essential. It follows from this estimation that such a scaling dependence is rightful at least in the neighborhood of z"1, i.e. in the vicinity of main contributions into  . This gives  n
(s) (s)" (s)(n/n
(s)) , L 

(2.58)

where (n/n
(s))K¹( (n/n
(s))) expn/n
(s) (n/n
(s))4O(e\L)  

(2.59)

is the unknown function. The asymptotic estimation follows from the fact that  " (n/n
(s))   should be a nondecreasing function of n, as follows from the nonsingularity of ¹(z, s). Estimation (2.57) is right at least at sPR. The range validity of n, where solution of (2.57) is acceptable, depends on the exact form of ¹(z, s). Indeed, if ln ¹(z)&exp(z), "const'0, then (2.57) is right at all values of n and it is enough to have the condition sPR. But if ln ¹(z, s)&(1#a(z))A, "const'0, then (2.57) is acceptable if and only if n;n
(s).

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Representation (2.58) shows that just n
(s) is the natural scale of multiplicity n [54]. This representation was o!ered "rst as a reaction on the so-called Feynman scaling for inclusive cross section I 1 . (2.60) f (q , q ,2, q )&

I   I (q ) G G As follows from estimation (2.59), the limiting KNO prediction assumes that  "O(e\L). In this L regime ¹(z, s) should be singular at z"z (s)'1. The normalization condition  R¹(z, s) "n
(s) Rz X gives: z (s)"1#/n
(s), where '0 is the constant. Notice, such behavior of the big partition  function ¹(z, s) is natural for stationary Markovian processes described by logistic equations [55]. In the "eld theory such an equation describes the QCD jets [56]. We wish to generalize expansion (2.44) to take into account the possibility of many-component structure of the multiple production processes [2]. Let us consider particle production through the generation, for instance, of jets. In this case decay of a particle of high virtuality q<m forms a jet of lower virtuality particles. It is evident that one should distinguish correlation among particles in the jet, and correlation among jets. Let  G (m ) be the probability that the ith jet of mass m includes n particles, 14n 4n, where G G G L G ,H  n "n . (2.61) G G The jets are the result of particles decay. Then let us assume that NM (m , p ) de"nes the mean  G G number of jets of mass m and momentum p : G G 1 ,H ,H dm dp G G e\@CNG NM (m , p ) (m ) ,   n !n

(2.62)

()" G  G G LG G L N H 2m (2
) G G G ,H H L , where n H "(n , n ,2, n H ). Notice that the Boltzmann factor e\@C, where (p)"m#p/2m is   , , the jets energy, plays the same role as the corresponding factor in (2.15) and is introduced to take into account the energy conservation law. We consider the VHM domain and for this reason (p/2m);1 is assumed. It is useful to avoid the particles number conservation law (2.61). For this purpose we will introduce






 



(, z)" ()"exp L L where t(z, m)" zL (m) . L L





dm dp e\@CNNM (m, p)(t(z, m)!1) ,  2m (2
)

(2.63)

(2.64)

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25

Comparing (2.63) with (2.44) we may conclude that t(z, m) plays the role of activity of jets. Then the generalization is evident:

(, z)" I(, z) I I "exp  dm d (p )e\@CNG (t(z, m )!1)NM (m , p ,2, m , p ) , G  G G I   I I I G where NM has the same meaning as C , i.e. NM is the correlation function of k jets. I I I

 



(2.65)

2.3.4. Energy-multiplicity asymptotics equivalence Let us consider the following &bootstrap' regime when (, z) is de"ned by the equation



(, z)J

dm dp e\@CNNM (m, p)t(z, m) .  2m (2
)

(2.66)

Inserting here the strict expression (2.65) we "nd a nonlinear equation for t(z, m). The solution of (2.66) assumes that





dm dp e\@CNNM t<  2m (2
)



dm dp  e\@CNNM t  2m (2
)

(2.67)

and









dm dp e\@CNM t<  2m (2
)



dm dp dm dp   e\@Ct   e\@Ct NM . (2.68)  2m (2
) 2m (2
)   To solve Eq. (2.66) in the VHM region, where the leftmost singularity over z is important, let us consider the anzats (z, m) t(z, m)" , (1!(z!1)a(m))G

 '0 , 

(2.69)

where (z, m) is the polynomial function of z, (z"1, m)"1. Using the normalization condition



R n
" t(z, m) H Rz

,

(2.70)

X

we can "nd



R a(m) "n
!(1, m), (1, m), (z, m)  H Rz

(2.71)

1 z (m)"1# .  a(m)

(2.72)

. X The partition function of the jet t(z, m) de"ned by anzats (2.69) is singular at

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This singularity would be signi"cant in the VHM region if z (m) is a decreasing function of m. This  means an assumption that (1, m) P0 at mPR . n
(m) H So, in "rst approximation we will choose a(m)"n
(m) . (2.73) H This choice may be con"rmed by concrete model calculations. Taking into account the energy conservation law, conditions (2.67) and (2.68) are satis"ed if





n
(s)!n
(s/4) H exp !n H ;1 (2.74) n
(s)n
(s/4) H H at n3VHM. Therefore, (2.69) obey Eq. (2.66) with exponential accuracy in the VHM region, i.e. if  n
(s)n
(s/4)  H " . (2.75) n< H n
(s)!n
(s/4) z (s/4)!z (s)   H H We assume here that one can "nd so large n and s that with exponential accuracy the factors &NM I do not play an important role. But at low energies condition (1.2) is important and the factors &NM should be taken into account. I Notice now an important consequence of our &bootstrap' solution: it means that we can leave production of the heavy jets only, if n3VHM. On the other hand, let us choose n"z n
(s), where  H z '1 is the function of s, and consider sPR. Then condition (2.75) de"nes z : if   n
(s/4) H , (2.76) z <  n
(s)!n
(s/4) H H then we are able to obey inequalities (2.67) and (2.68). The jet mean multiplicity, see Section 3.2.2, is ln n
(s)&(ln s . H

(2.77)

Then n
(s/4) H "eA(  Q!1\&(ln s;n
(s) (2.78) H n
(s)!n
(s/4) H H at sPR. Therefore, (2.76) may be satis"ed outside the VHM domain. Let us compare now the solutions of the equation of state. Inserting (2.69) into (2.38) we can "nd for a jet of mass (s that 1   !  . z"z!  "1#   n
(s) n n H

(2.79)

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27

The two-jet contribution of the masses &(s/2 gives  1  z"z!  "1# !  .   n n
(s/4) n H

(2.80)

At arbitrary "nite energies (z!z)'0 and, as follows from (2.78), they decrease &(1/n
(s)(ln s)   H with energy. Noting the normalization condition, ¹(z"1, s)" (s), and assuming that the vacuum is  stable, i.e.  4O(e\L), we can conclude that L } if n3VHM then z attracts z , i.e. z Pz , and if z !1;1 then these contributions should be      signi"cant in  ;  } if sPR, then z !1;1, and if n satis"es inequality (2.75), or if z satis"es inequality (2.76),   then the considered contributions are signi"cant in  .  It is the (energy-multiplicity) asymptotics equivalence principle. One of the simplest consequences of this principle is the prediction that the mean transverse momentum of created particles should increase with multiplicity at su$ciently high energies. This principle is the consequence of independence of contributions in the VHM domain on the type of singularity in the complex z plane and of the energy conservation law. Just the last one shifts the two-jet singularity to the right and z (s)(z (s/4).   We would like to mention also that this e!ect, when the mostly &energetic population' survives has been described mathematically by Volterra [55]. It is intuitively evident that one may "nd the &energetic population' when searching for the VHM one, or, it is the same, giving it a rich supply, i.e. to give the population enough energy. This is our (energy-multiplicity) equivalence ((!n)equivalence) principle. Notice, if the amount of supply is too high then few populations may grow. This is the case when the di!erence (z!z)'0 tends to zero at high energies.   We would like to note that singular at "nite z partition functions was predicted in the () -theory [57], in QCD jets [56], in the generalized Bose}Einstein distribution model [58]. In  all of these models decay of the essentially nonequilibrium initial state (highly virtual parton, heavy resonance, etc.) was described. One may distinguish the phases of the media by a characteristic correlation length. Then the phase transition may be considered as the process of changing correlation length. Our &bootstrap' solution predicts just such phenomena: at low multiplicities the long-range correlations among light jets are dominant. The &bootstrap' solution predicts that for VHM processes just the short-range correlations among particles of the heavy jet become dominant. The (!n)-equivalence means that this transition is a pure dynamical e!ect.

3. Model predictions Multiple production phenomena were "rst observed more than seventy years ago [59]. During this time vast experimental information was accumulated concerning hadron inelastic interactions, see the review papers [10].

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Now we know that at high energy (s: (i) The total cross section  (s) of hadron interactions is enhanced almost completely by the  inelastic channels; (ii) The mean multiplicity of produced hadrons n
(s) slowly (logarithmically) grows with (s; (iii) The interaction radii of hadrons bM slowly (logarithmically) increase with energy; (iv) The multiplicity distribution  (s) is wider than the Poisson distribution; L (v) The mean value of transverse momentum k of produced hadrons is restricted and is independent of the incident energy (s and produced particle multiplicity n; (vi) The one-particle energy spectrum d&d/. First of all, (i) means that the high-energy hadron interaction may be considered as an ordinary dissipation process. In this process the kinetic energy of incident particles is spent in produced particle mass formation. The VHM process takes place in vacuum and then it is assumed in the early stages that the multiple production phenomena re#ect a natural tendency of the excited hadron system to get to equilibrium with the environment [1]. In this way one can introduce as a "rst approximation the model that the excited hadron system evolves without any restrictions. In this model we should have n
(s)&(s. The dissipation is maximal in this case and the entropy S exceeds its maximum. This simple model has de"nite popularity up to 70-th. But the experimental data (ii) and (iii) prohibit this model and it was forgotten. Choosing the model we would like to hope that the considered model E takes into account experimental conditions (i)}(vi) in the n&n
(s) domain; E has natural asymptotics over multiplicity to the VHM region. It is necessary to remember also that E New channels of hadron production may arise in the VHM region. It is impossible to understand all possibilities without those o!ered in Section 2.3.2 in the classi"cation of asymptotics. Thus, we will observe predictions of E Multiperipheral models, distinguishing the soft Pomeron models, see [60]. E The dual-resonance model predictions for the VHM region are described also. It can be shown that this models predict asymptotics (A) if n
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29

VHM leads to the necessity to include low-x sub-processes. As a result we get out of the range of pQCD validity. E Multiple reduction of jets. We will see, that at very high energies in the VHM region the heavy jets creation should be a dominant process if the vacuum is stable with reference to the particle production. We will consider also decay of the &false vacuum' to describe the consequence of E Phase transition in the VHM domain. This channel is hardly seen for n&n
(s) since the con"nement constraints may prevent cooling of the system up to phase transitions condition. 3.1. Peripheral interaction 3.1.1. Multiperipheral phenomenology Later on multiple production physics was developed on the basis of experimental observation (iv). The Regge pole model naturally explains these experimental data and, at the same time, absorbs all experimental information, (i)}(vi). At the very beginning, adopting the Regge poles notion without its microscopical explanation, this description was self-consistent. The e!orts to extend the Regge pole model to the relativistic hadron reactions was ended by the Reggeon diagram technique, see [61] and references cited therein, and it was used later to construct the perturbation theory for  [62]. It was shown that the multiplicity distribution is wider than the L Poissonian one because of the multi-Pomeron exchanges. The leading energy asymptotics Pomeron contribution re#ects the created particle kinematics described in Appendix D, where the available kinematical scenario in the frame of pQCD is described. So, the longitudinal momentum of produced particles is large and is strictly ordered. At the same time, particles transverse momentum is restricted. Let us consider the inelasticity coe$cient introduced in (1.3) "1! /E(1, where  is



 the energy of the fastest particle in the laboratory frame. Then the strict ordering of particles in the Pomeron kinematics means that  is independent of the index of the particle. So, if the fastest particle has the energy  K(1!)(s, then the following particle should have the energy

  K(1!) K(1!)(s, and so on. Following this law, the (n!1)th particle would have 

 the energy  K(1!)L(s. In the laboratory frame the energy should degrade to  Km. L L Inserting here the above-formulated estimation of  we can "nd that if the number of produced L particles is n
(s)Kn , n "!ln(1!)'0, "ln(s/m) , (3.1)   then we may expect the total degradation of energy. This degradation is the necessary condition noting that the total cross section of slowly moving particles may depend only slightly on energy and it seems necessary for natural explanation of the weak dependence of the hadron cross sections on the energy. This consideration would be Lorentz covariant if one can "nd the slowly moving particle in an arbitrary frame. The resulting estimation of mean multiplicity has good qualitative experimental con"rmation.

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Notice that it was assumed when deriving (3.1) that the energy degrades step by step. In other words, if we introduce a time of degradation, then the time & is needed for complete degradation of energy. Assuming the random walk in the normal to incident particle plane, we can conclude that the points of particle production are located on a disk (in the moving frame) of radii bM &. This means that the interaction radii should grow with the energy of the colliding particles. If f (a#bPc#2) is the cross section to observe particle c inclusively in the a and b particles collision, then it was found experimentally that the ratio f (
>pP
\#2) f (K>pP
\#2) f (ppP
\#2) " " (3.2)  (K>p)  (pp)  (
>p)    is universal. This may be interpreted as the direct evidence of the fact that the hadron interactions have a large-distance character, i.e. that the interaction radii should be large. This picture assumes that the probability to have total degradation of energy is &e\@?YK ,

(3.3)

where  is some dimensional constant (the slope of Regge trajectory) and b is the two-dimensional impact parameter. This formula has also the explanation connected to the vacuum instability with reference to the real particle production in the strong color electric "eld. The above picture has natural restrictions. We can assume that each of the produced particles may be the source of above-described t-channel cascade of the energy degradation. This means that in the frame of the Pomeron phenomenology, we are able to describe the production of n(n
(s)

(3.4)

particles only. If n'n
(s) then the density of particles in the di!raction disk becomes large and (a) one should introduce short-distance interactions, or (b) rise interaction radii. It will be shown that just (a) is preferable. We will build the perturbation theory in the phenomenological frames (i)}(vi). Considering the system with variable number of particles the generating function ¹(z, s)" zL (s) , L L would be useful. One can use also the decomposition:

(3.5)

¹(z, s)" (s) exp(z!1)C (s)#(z!1)C (s)#2 ,     where, by de"nition,

(3.6)



R C (s)"n
(s)" ln ¹(z, s)  Rz

(3.7) X

is the mean multiplicity,



R C (s)" ln ¹(z, s)  Rz

X is the second binomial momentum, and so on.

(3.8)

J. Manjavidze, A. Sissakian / Physics Reports 346 (2001) 1}88

Our idea is to assume that all C , m'1 may be calculated perturbatively choosing K
P(0, s)"eX\LQ

31

(3.9)

as the Born approximation &superpropagator'. It is evident that (3.9) leads to Poisson distribution. Then, having in mind (ii), (iii) and (v) we will use the following anzats: P(q, s)"e?M\?YO  Q eX\K ,

(3.10)

where the transverse momentum q is conjugate to the impact parameter b. So, the Born term (3.10) is a Fourier transform of the simple product of (3.9) and (3.3). It contains only one free parameter, the Pomeron intercept (0). On the phenomenological level it is not important to know the dynamical (microscopical) origin of (3.10). For our purpose the Laplace transform of P would be useful. If n
(s)"n ln s, then   1 . (3.11) de\SKP(q, s)" P(, q)" #q# (z)   It is the propagator of two-dimensional "eld theory with mass squared



 (z)"(1!)#(1!z)n , n '0 .    Knowing Gribov's Reggeon calculus completed by the Abramovski}Gribov}Kancheli (AGK) cutting rules [63] one can investigate the consequences of this approach. The LLA approximation of the pQCD [19] gives 12 ln 2  +0.55,  "0.2 ,
"(0)!1"  


(3.12)

but radiative corrections give
+0.2 [64]. We will call this solution as the BFKL model. The quantitative origin of the restriction (3.4) is the following. The contribution of the diagram with  Pomeron exchange gives, since the di!raction radii increase with s, see (3.7), mean value of the impact parameter decreasing with : n
(s) bM K4 ln (s/m)/"a ,  where a"4/n . On the other hand, the number of necessary Pomeron exchanges &n/n
(s) since  one Pomeron gives maximal contribution (with factorial accuracy) at nKn
(s). As a result, bM &a

n
(s) . n

(3.13)

Therefore, if the transverse momentum of created particles is a restricted quantity, i.e.  bM &1,  where  is a constant, then the mechanism of particle production is valid up to  n&n
(s) . (3.14) Following our general idea, it will be enough for us to "nd the position of singularity over z. Analysis shows that (3.12) predict the singularity at in"nity.

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In Appendix E Gribov's Reggeon diagram technique with cut Pomerons is described and (3.14) is derived. It can be shown using this technique that the model with the critical Pomeron,
"0, is inconsistent from the physical point of view [62]. As was mentioned above, the model with
"(0)!1'0 is natural for the pQCD. The concrete value of
will not be important for us. We will assume only that 0(
;1 .

(3.15)

It is evident that the Born approximation (3.10) with
'0 violates the Froissart boundary condition. But it can be shown that the sum of &eikonal' diagrams solves this problem, see [66] and references cited therein. The interaction radii may increase with increasing number of produced particles if
'0 and then the restriction (3.14) is not important. In the used eikonal approximation, see Appendix F, n 1 zKz "1# ln  n
(s) n
(s)

(3.16)

is essential. Then the interaction radii bM &BK4(#ln(n/n
(s))) for these values of z. Note that B&< ,

(3.17)

even for n&n &(s.

 Nevertheless, using (3.16) one can "nd that the cross section decreases faster than any power of e\L:




n n  (s) L " ln (1#O(n
(s)/n)) . (3.18) n
(s) n
(s)  (s)  Generally speaking, although this estimation is right in the VHM region, there may be large corrections because of Pomeron self-interactions. But careful analyses show [62] that these contributions cannot drastically change estimation (3.18). !ln

3.1.2. Dual resonance model The search of dynamical source of the Regge description shows the di!erent dynamical nature of the Regge and Pomeron poles. The established resonance-Regge pole duality, e.g. [67] led to the Veneziano representation of the Regge amplitudes [68]. The Reggeon pole gives the decreasing &s\ contribution, but careful investigation shows that the mass spectrum of dual to Regge pole resonances increases exponentially. This prediction was con"rmed by experiment, see the discussion of this topic in [69]. The "eld theory development is marked by considerable e!orts to avoid the problem of color charge con"nement. Notice that the classical string has the same excitation spectrum. A remarkable attempt in this direction is based on the string model, in its various realizations, see, e.g. [70].

 The eikonal approximation in a quantum "eld theory has been developed in [65].

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33

But, in spite of remarkable success (in formalism especially) there is no experimentally measurable prediction of this approach till now, e.g. [71]. We would like to describe in this section production of &stable' hadrons through decay of resonances [72,73]. Our consideration will use the following assumptions. A. The string interpretation of the dual-resonance model indicates that the mass spectrum of resonances, i.e. the total number (m) of mass m resonance excitations, grows exponentially (3.19)

(m)"(m/m )Ae@ K,  "const, m'm .    Note also that the same hadron mass spectrum (3.19) was predicted in the &bootstrap' approach [69,74]. Moreover, it predicts that 5 "! . 2

(3.20)

B. The mass m resonance creation cross section 0(m) has the Regge pole asymptotics m 0(m)"g0  , g0"const . m

(3.21)

It has been assumed here that the intercept of the Regge pole trajectory 0". So, only the meson  resonances would be taken into account. C. If 0(m) describes the decay of a mass m resonance into the n hadrons, then the mean L multiplicity of hadrons  n0(m) . n
0(m)" L L 0(m)

(3.22)

Following the Regge model, m n
0(m)"n
0 ln . (3.23)  m  D. We will assume that there is a de"nite vicinity of n
0(m) where 0(m) is de"ned by n
0(m) only. L So, in this vicinity 0(m)"0(m)e\L
0K(n
0(m))L/n! . (3.24) L This is the direct consequence of the Regge pole model, if m/m is high enough.  Following our idea, we will distinguish the &short-range' correlations among hadrons and the &long-range' correlations among resonances. The &connected groups' would be described by resonances and the interactions among them should be described, introducing for this purpose the correlation functions among strings. So, we will consider the &two-level' model of hadron creation: the "rst level describes the short-range correlation among hadrons and the second level is connected to the correlations among resonances. The exact calculations are given in Appendix G. Comparing A and B solutions we can see the change of attraction points with rising n: at nKn
(s)"n
0 ln((s/m ) the transition from (A) asymptotics to (C) in (2.41) should be seen.  

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At the same time one should see the strong KNO scaling violation at the tail of the multiplicity distribution. We have neglected the resonance interactions when deriving these results. This assumption seems natural since at n
(s);n(n
(s) inequality (G.17) should be satis"ed, see the discussion of inequalities (2.67) and (2.68) in Section 2.3.4. 3.2. Hard processes 3.2.1. Deep inelastic processes The role of soft color partons in the high-energy hadron interactions is the most intriguing modern problem of particle physics. So, the collective phenomena and symmetry breaking in the nonAbelian gauge theories, con"nement of colored charges and the infrared divergences of the pQCD are the phenomena just of the soft color particles domain. It seems natural that the very high multiplicity (VHM) hadron interaction, where the energy of the created particles is small, should be sensitive to the soft color particle densities. Indeed, the aim of this section is to show that even in the hard-by-de"nition deep inelastic scattering (DIS), see also [21], the soft color particles role becomes important in the VHM region [75]. To describe the hadron production in pQCD terms the parton}hadron duality is assumed. This is natural just for the VHM process kinematics: because of the energy}momentum conservation law, produced ("nal-state) partons cannot have high relative momentum and, if they were created at small distances, production of qq
pairs from the vacuum will be negligible (or did not play an important role). Therefore, if the &vacuum' channel is negligible, only the pQCD contributions should be considered [50,76]. All this means that the multiplicity, momentum, etc. distributions of hadron and colored partons are the same. (This reduces the problem practically to the level of QED.) Let us consider now n particles (gluons) creation in the DIS [77]. We would like to calculate D (x, q; n), where ?@  D (x, q; n)"D (x, q) . (3.1) ?@ ?@ L As usual, let D (x, q) be the probability to "nd parton b with virtuality q(0 in the parton a of ?@ & virtuality, < and  ();1. We may always choose q and x so that the leading logarithm  approximation (LLA) will be acceptable. One should assume also that (1/x)<1 to have the phase space, into which the particles are produced, su$ciently large. Then D (x, q) is described by ladder diagrams. From a qualitative point of view this means the ?@ approximation of random walk over coordinate ln(1/x) and the time is ln lnq. LLA means that the &mobility' &ln(1/x)/ln lnq should be large ln(1/x)
(3.2)

But, on the other hand [78], ln(1/x);lnq/ . See also Appendix D.

(3.3)

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35

The leading contributions, able to compensate the smallness of  ();1 ,  give integration over a wide range ;k;!q, where k'0 is the &mass' of a real, i.e. G G time-like, gluon. If the time needed to capture the parton into the hadron is &(1/) then the gluon should decay if k<. This leads to the creation of (mini)jets. The mean multiplicity n
in the G H QCD jets is high if the gluon &mass' k is high: ln n
K(ln(k/). H Raising the multiplicity may (i) raise the number of (mini)jets  and/or (ii) raise the mean value mass of (mini)jets kM . We will see that the mechanism (ii) would be favorable. G But increasing the mean value of gluon masses, k , the range of integrability over k decreases, G G i.e. violates the condition (3.2) for "xed x. One can retain the LLA taking xP0. But this may contradict (3.3), i.e. in any case the LLA becomes invalid in the VHM domain and the next to leading order corrections should be taken into account. Note that the LLA gives the main contribution, that the rising multiplicity leads to the infrared domain, where the soft gluon creation becomes dominant. First of all, neglecting the vacuum e!ects, we introduce de"nite uncertainty to the formalism. It is reasonable to de"ne the level of strictness of our computations. Let us introduce for this purpose the generating function ¹ (x, q; z): ?@ dz 1 ¹ (x, q; z) . (3.4) D (x, q; n)" ?@ 2
i zL> ?@



At large n, the integral may be calculated by the saddle point method. The smallest solution z of  the equation R n"z ln ¹ (x, q; z) ?@ Rz

(3.5)

de"nes the asymptotic over n behavior D (x, q; n)Jexp!n ln z (x, q; n) . (3.6) ?@  Using the statistical interpretation of z as the fugacity it is natural to write  C (x, q; n) ln z (x, q; n)" ?@ . (3.7)  n
(x, q) ?@ Notice that the solution of Eq. (3.5) z (x, q; n) should be an increasing function of n. At "rst glance  this follows from the positivity of all D (x, q; n). But actually this assumes that ¹ (x, q; z) ?@ ?@ is a regular function of z at z"1. This is a natural assumption considering just the pQCD predictions. Therefore,





n D (x, q; n)Jexp ! C (x, q; n) . (3.8) ?@ n
(x, q) ?@ ?@ This form of D (x, q; n) is useful since usually C (x, q; n) is a slowly varying function of n. So, for ?@ ?@ a Poisson distribution C (x, q; n)&ln n. For KNO scaling we have C (x, q; n)"const. over n. ?@ ?@

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We would like to note that, neglecting e!ects of vacuum polarization, we introduce into the exponent such high uncertainty assuming nKn that it is reasonable to perform the calculations N with exponential accuracy. So, we would calculate n D (x, q; n) " C (x, q; n)(1#O(1/n)) . (3.9) !
(x, q; n)"ln ?@ ?@ n
(x, q) ?@ D (x, q) ?@ ?@ The n dependence of C (x, q; n) de"nes the asymptotic behavior of 
(x, q; n) and calculation ?@ ?@ of its explicit form would be our aim. We can conclude, see Appendix H, that our LLA is applicable in the VHM domain till (, z);ln(1/x);"ln(!q/) ,

(3.10)

where



(, z)"

O d wE(, z) .  O

(3.11)

and wE(, z)" zLwE () (3.12) L L is the generating function of the multiplicity distribution in a gluon jet. In the frame of constraints (3.10), F?@(q, x; w)Jexp4(N(, z)ln (1/x) .

(3.13)

The mean multiplicity of gluons created in the DIS kinematics R n
(, x)" ln F?@(q, x; w) " ()(4N ln(1/x)/ln < () , E X   Rz

(3.14)

where



O d  n
() H  O  and the mean gluon multiplicity in the jet n
() has the following estimation [79]: H  ()" 

ln n
()K( . H Inserting (3.16) into (3.15),

(3.15)

(3.16)

 ()"n
()/( .  H Therefore, noting (3.3), n
(, x)Kn
()(4N ln(1/x)/ ln ;n
() . H E H

(3.17)

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37

This means that the considered &t-channel' ladder is important in the narrow domain of multiplicities n&n
;n
. (3.18) E H So, in the VHM domain n


&exp !c



n
(M)!n
(M/2) H H n , H n
(M)n
(M/2) H H

(3.20)

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(3.19) would dominate in the VHM domain since the mean multiplicity n
(M) increases H with M. The experimental observation of these phenomena crucially depends on the value of aH, aH,2 but if (3.19) is satis"ed then one can expect that the events in the VHM domain would be enhanced by QCD jets and the mass of jets would have a tendency to be high with growing multiplicity. The singular at "nite z solutions arise in the "eld theory, when the s-channel cascades (jets) are described [56]. By de"nition, ¹(z, s) coincides with the total cross section at z"1. Therefore, the nearness of z to one de"nes the signi"cance of the corresponding processes. It is evident that both  s and n should be high enough to expect the jets creation. Summarizing the above estimations, we may conclude that O(e\L)4 (O(1/n) , (3.21) L i.e. the soft Regge-like channel of hadron creation is suppressed in the VHM region in the high-energy events with exponential accuracy. 3.3. Phase transition } condensation The aim of this section is to "nd the experimentally observable consequences of collective phenomena in the high-energy hadron inelastic collision [86]. We will pay attention mainly to the phase transitions, leaving out other possible interesting collective phenomena. The statistics experience dictates that we should prepare the system for the phase transition. The temperature in a critical domain and the equilibrium media are just these conditions. It is evident that they are not a trivial requirement considering the hadrons inelastic collision at high energies. The collective phenomena by de"nition suppose that the kinetic energy of particles of media are comparable, or even smaller, than the potential energy of their interaction. It is a quite natural condition noting that, for instance, the kinetic motion may decay even completely at a given temperature ¹, necessary for the phase transition long-range order. This gives, more or less de"nitely, the critical domain. The same idea as in statistics seems natural in the multiple production physics. We will assume (A) that the collective phenomena should be seen just in the very high multiplicity (VHM) events, where, because of the energy}momentum conservation laws, the kinetic energy of the created particles cannot be high. We will lean at this point on the S-matrix interpretation of statistics [87], see Section 2.2. It is based on the S-matrix generalization of the Wigner function formalism of Carruthers and Zachariazen [39] and the real-time "nite-temperature "eld theory of Schwinger and Keldysh [29,30], see Appendixes A}C. It was mentioned that the n-particle partition function in this approach coincides with the n-particle production cross section  (s) (in the appropriate normalization condition). Then, the L cross section  (s) can be calculated applying the n-point Wigner function = (X , X ,2, X ). In L L   L the relativistic case X "(u, q) are the 4-vectors. So, the external particles are considered as the I I &probes' to measure the state of the interacting "elds, i.e. the low mean energy of probes means that the system is &cold'. The multiple production phenomena may be considered also as the thermalization process of incident particle kinetic energy dissipation into the created particle mass. From this point of view

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39

the VHM processes are highly nonequilibrium since the "nal state of this case is very far from the initial one. It is known in statistics [24] that such processes aspire to be the stationary Markovian with a high level of entropy production. In the case of complete thermalization, the "nal state is in equilibrium. The equilibrium we will classify as the condition in the frame of which the #uctuations of corresponding parameter are Gaussian. So, in the case of complete thermalization, the probes should have the Gaussian energy spectra. In other terms, the necessary and su$cient condition of the equilibrium is the smallness of the mean value of energy correlators [42,87]. From the physical point of view, the absence of these correlators means depression of the macroscopic energy #ows in the system. The multiple production experiment shows that the created particle energy spectrum is far from a Gaussian law, i.e. the "nal states are far from equilibrium. The natural explanation of these phenomena consists in the presence of (hidden) conservation laws in the interacting Yang}Mills "elds: it is known that the presence of su$cient number of "rst integrals in involution prevents thermalization completely. Nevertheless, the VHM "nal state may be equilibrium (B) in the above formulated sense. This means that the forces created by the nonAbelian symmetry conservation laws may be frozen during the thermalization process (remembering its stationary Markovian character in the VHM domain). We would like to take into account that the entropy S of a system is proportional to the number of created particles and, therefore, S should tend to its maximum in the VHM region [1]. One may consider following the small parameter (n
(s)/n);1, where n
(s) is the mean value of the multiplicity n at a given CM energy (s. Another small parameter is the energy of the fastest hadron  . One should assume that in the VHM region ( /(s)P0. So, the conditions



 n
(s)  ;1,  P0 (3.1) n (s would be considered as the mark of the processes under consideration. We can hope to organize the perturbation theory over them having there small parameters. In this sense VHM processes may be &simple', i.e. one can use for their description semiclassical methods. So, considering VHM events one may assume that conditions (A) and (B) are satis"ed and one may expect the phase transition phenomena. The S-matrix interpretation of statistics is based on the following de"nitions. First of all, let us introduce the generating function [49] ¹(z, s)" zL (s) . (3.2) L L Summation is performed over all n up to n "(s/m and, at "nite CM energy (s, ¹(z, s) is

 a polynomial function of z. Following our idea, see Section 2.3, let us assume now that z'1 is su$ciently small and for this reason ¹(z, s) depends on the upper boundary n only weakly. In

 this case one may formally extend summation up to in"nity and in this case ¹(z, s) may be considered as a whole function. This possibility is important since it is the equivalent of the thermodynamical limit and it allows to classify the asymptotics over n in accordance with the position of singularities over z.

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Let us consider ¹(z) as the big partition function, where z is &activity'. It is known [51] that ¹(z) should be regular inside the circle of unit radius. The leftist singularity lies at z"1. This singularity is the manifestation of the "rst order phase transition [51,24,52]. The origin of this singularity was investigated carefully in the paper [24]. It was shown that the position of singularities over z depends on the number of particles n in the system: the two complex conjugated singularities move to the real z-axis with rising n and in the thermodynamical limit, n"R, they pinch the point z"1 in the "rst order phase transition case. More general analysis [52] shows that if the system is in equilibrium, then ¹(z) may be singular only at z"1 and z"R. The position of the singularity over z and the asymptotic behavior of  are closely related. L Indeed, for instance, inserting into (3.2)  Jexp!cnA we "nd that ¹(z) is singular at z"1 if L (1. Generally, using the Mellin transformation (2.37) one can "nd an asymptotic estimation (2.39)  Je\L  X L, z '1 ,  L where z is the smallest solution of the equation of state  R n"z ln¹(z) . Rz

(3.3)

(3.4)

Therefore, to have the singularity at z"1, we should consider z (n) as a decreasing function of n.  On the other hand, at constant temperature, ln z (n)& (n) is the chemical potential, i.e. is the   work necessary for creation of one particle. So, the singularity at z"1 means that the system is unstable: the less work is necessary for creation of one more particle if (n) is a decreasing function of n. The physical explanation of these phenomena is the following, see also [88]. The generating function ¹(z) has the following expansion:

 

(3.5) ¹(z)"exp  zJb , J J where b are known as Mayer's group coe$cients [25]. They can be expressed through the J inclusive correlation functions and may be used to describe the formation of droplets of correlated particles, see Section 2.3.3. So, if droplets consist of l particles, then b &e\@KJB\B (3.6) J is the mean number of such droplets. Here lB\B is the surface energy of d-dimensional droplet. Inserting this estimation into (3.5), ln ¹(z)& e@JI\KJB\B, "ln z . (3.7) J The "rst term in the exponent l is the volume energy of the droplet and being positive it tries to enlarge the droplet. The second surface term !lB\B tries to shrink it. Therefore, the singularity at z"1 is the consequence of instability: at z'1 the volume energy abundance leads to unlimited growth of the droplet.

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In conclusion we wish to formulate once more the main assumptions. (I) It was assumed "rst of all that the system under consideration is in equilibrium. This condition may be naturally reached in the statistics, where one can wait for an arbitrary time till the system reaches equilibrium. Note, in the critical domain, the time of relaxation t &(¹ /(¹!¹ ))JPR, (¹!¹ )P#0, '0, ¹ is the critical temperature. P     We cannot give any guarantee that in the high-energy hadron collisions the "nal-state system is in equilibrium. The reason for this uncertainty is the "nite time the inelastic processes and the presence of hidden (con"nement) constraints on the dynamics. But if the con"nement forces are frozen in the VHM domain, i.e. the production process is &fast', then the equilibrium may be reached. We may formulate the quantitative conditions, when the equilibrium is satis"ed [42]. One should have the Gaussian energy spectra of created particles. If this condition is hardly investigated in the experiment, then one should consider the relaxation of &long-range' correlations. This excludes the usage of relaxation condition for the &short-range' (i.e. resonance) correlation. (II) The second condition consists in the requirement that the system should be in the critical domain, where the (equilibrium) #uctuations of the system become high. Having no theory of hadron interaction at high energies we cannot de"ne where the &critical domain' lies and even whether it exists or not. But, having the VHM &cold' "nal state, we can hope that the critical domain is achieved. The quantitative realization of this picture is given in Appendix J. It is important to note that the semiclassical approximation used there is rightfully in the VHM domain.

4. Conclusion 4.1. Discussion of physical problems It seems useful to start the discussion of models by outlining the main problems, from the authors point of view. A. Soft color parton problems. The infrared region of soft color parton interactions is a very important problem of high-energy hadron dynamics. Such fundamental questions as the infrared divergences of pQCD, collective phenomena in the colored particles system and symmetry breaking are the phenomena of the infrared domain. The standard (most popular) hadron theory considers pQCD at small distances (in the scale of K0.2 Gev) as the exact theory. This statement is con"rmed by a number of experiments, namely deep-inelastic scattering data, hard jets observation. But the pQCD predictions have a "nite range of validity since the nonperturbative e!ects should be taken into account at distances larger than 1/. It is natural to assume, building the complete theory, that at large distances the nonperturbative e!ects lay on the perturbative ones. As a result, pQCD loses its predictability screened by the nonperturbative e!ects.  The corresponding formalism has been described, e.g. in [18].

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Notice, the pQCD running coupling constant  (q)"1/b ln(q/) becomes in"nite at q"  and we do not know what happens with pQCD if q(. There are few possibilities. For instance, there is a suspicion [13] that at q& the properties of theory changed so drastically (being de"ned on a new vacuum) that even the notions of pQCD disappeared. This means that pQCD should be truncated from below on the &fundamental' scale . It seems natural that this infrared cut-o! would in#uence the soft hadrons emission. The new possibility is described in Appendix K. This strict formalism allows to conclude that pure pQCD contributions are realized on zero measure, i.e. it is the phenomenological theory only. The successive approach shows that the Yang}Mills theory should be described in terms of (action,angle)-like variables. The last one means that the self-consistent description excludes such notions as the &gluon'. As a result of this substitution new perturbation theory would be free from infrared divergences, i.e. there is no necessity to introduce the infrared cuto! parameter . (Moreover, in the sector of vector "elds (without quarks) the theory is ultraviolet stable.) It seems important for this reason to investigate experimentally just VHM events, where the soft color partons production is dominant. It is important to try to raise the role of pQCD in the &forbidden' area of large distances. The VHM processes are at highly unusual condition, where the nonperturbative e!ects must be negligible. B. Dissipation problems. The highly nonequilibrium states decay (thermalization) which means in pQCD terms that the process of VHM formation should be enhanced, at least in asymptotics over multiplicity and energy, by jets. It is the general conclusion of nonequilibrium thermodynamics and it means that the very nonequilibrium initial state tends to equilibrium (thermalized) as fast as possible. The entropy S of a system is proportional to the number of created particles and, therefore, S should tend to its maximum in the VHM region [1]. But the maximum of entropy testi"es also to the equilibrium of the system. C. Collective phenomena. We should underline that the collective phenomena may take place if and only if the particles interaction energy is comparable to the kinetic one. The VHM system considered may be &cold' and &equilibrium'. For this reason the VHM state is mostly adopted for investigation of collective phenomena. One of the possible states in which the collective e!ects, see, e.g. [89], may be important is the &could colored plasma' [90]. The fundamental interest presents the problem of vacuum structure of Yang}Mills theory. For instance, if the process of cooling is &fast', since the dissipation process of VHM "nal-state formation should be as fast as possible, then one may consider formation of vacuum domains with various properties. Then decay of these domains may lead to large #uctuations, for instance, of the isotopic spin. Another important question is the collective phenomena in the VHM "nal state. The last one may be created &perturbatively', for instance, by the formation of heavy jets. Then the color charges should be con"ned. There are various predictions about this process. One of them predicts that there should be a "rst-order phase transition. 4.2. Model predictions Now we can ask: what can models say concerning the above problems?

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A. Soft colored partons production. The multiperipheral models predict fast decreasing of topological cross section in the VHM domain n
(s);n;n ,  (O(e\L). At the same time the

 L mean transverse momentum should decrease in this domain since the interaction radii should &increase' with n. The BFKL Pomeron predicts the same asymptotics,  (O(e\L), but the pQCD jet predicts L  "O(e\L). The naive attempt to insert into the BFKL Pomeron the production of particles via L (mini)jets seems impossible. This &insertion' can be done into the DIS ladder but investigation of the LLA kinematics in the VHM domain allows the conclusion that the &low-x' contributions should be taken into account. All this experience allows the assumption that in the VHM domain no &t-channel ladder' diagrams play su$cient role. This implies the existence of a transition to the processes with jet dominance. The pQCD is unable to predict the transition mechanism. B. Transition into &equilibrium'. If the t-channel ladders are &destroyed' in the VHM region, then jets, despite the small factor O(1/s) in the cross sections, are the only mechanism of particle production in the VHM domain. Dominance of heavy jets in the VHM domain may naturally explain the tendency towards equilibrium. But the description of thermalization in terms of jets of massless gluons production destroys this hope: the jet contribution  "O(e\L) assumes &bremsstrahlung' of soft gluons [91]. This prevents L the equilibrium since ordering without fail introduces the nonrelaxing correlations. C. Collective phenomena. Considering the collective phenomena, we propose to distinguish (a) the collective phenomena connected with the vacuum and (b) the collective phenomena produced in the VHM system. Following the experience of Section 3.3 we can conclude that the signal of vacuum instability is inequality:  'O(e\L). L Case (b) will not e!ect the cross section  . But if the system reached equilibrium in the VHM L domain then the collective phenomena may be investigated using ordinary thermodynamical methods. For instance, noting that !ln(¹( , z)/¹( , z))/ "F( , z) is the free energy one can     measure the thermal capacity R F( , z)"C( , z) . (4.8)   R  Then, comparing capacities of hadron and -quanta systems we can say whether or not the phase transition happened. The connection of the equilibrium and relaxation of correlations is well known [42]. Continuing this idea, if the VHM system is in equilibrium one may assume that the color charges in the pre-con"nement phase of VHM event form the plasma. One should note here that the expected plasma is &cold' and &dense'. For this reason no long-range con"nement forces would act among color charges. Then, being &cold', in such a system various, collective phenomena may be important. 4.3. Experimental perspectives The experimental possibilities in the VHM domain are not clear till now. Nevertheless, "rst step toward formulation of trigger system was done, see [92]. For this reason we would like to restrict

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ourselves by following two general questions. It seems that these questions are most important, being in the very beginning of VHM theory. I. For what values of multiplicity at a given energy the VHM processes become hard? The answer to this question depends on the value of the incident energy. If we know the answer then it will appear possible to estimate E the role of multiperipheral contributions, E the jet production rate, E the role of vacuum instabilities. It seems that the experimental answer to this question is absent since the produced particles are soft, their mean transverse and longitudinal momenta have the same value. In our understanding this question means: the total transverse energy may be extremely high. It is interesting also to search the heavy jets, i.e. to observe the #uctuations of particle density in the event-by-event experiment, but this program seems vague since, for all evidence, fractal dimensions tend to zero with increasing multiplicities. II. For what values of multiplicity does the VHM xnal state reach equilibrium? We hope that having the answer for this question we would be able E to investigate the status of pQCD, E to observe the phase transition phenomena directly, E to estimate the role of con"nement constraints. The equilibrium means that the energy correlation function mean values are small. Another point of interest, for example, is the charge equilibrium, when the mean value of charge correlation functions is small. Notice, the e!ect of the phase space boundary may lead to &equilibrium'. Indeed, if (p)Km#p/2m and p;m then one may neglect the momentum dependence of the amplitudes a . In this case the momentum dependence is de"ned by the Boltzmann exponent e\@NK, PR L only and we get naturally to the Gaussian law for momentum distribution. Correlators should be small in this case since there are no interactions among particles (a are constants). L But our question assumes that we investigate the possibility of equilibrium when n;n and

 p<m.

Acknowledgements The authors would like to take the opportunity to thank J. Allaby, A.M. Baldin, J.A. Budagov, R. Cashmore, G.A. Chelkov, M. Della Negra, D. Denegri, A.T. Filippov, D. Froidevaux, F. Gianotti, I.A. Golutvin, P. Jenni, V.G. Kadyshevski, O.V. Kancheli, Z. Krumstein, E.A. Kuraev, E.M. Levin, L.N. Lipatov, L. Maiani, V.A. Matveev, A. Nikitenko, V.A. Nikitin, A.G. Olchevski, G.S. Pogosyan, N.A. Russakovich, M.V. Saveliev, V.I. Savrin, Ju. Schukraft, N.B. Skachkov, S. Tapprogge, A.N. Tavkhelidze, V.M. Ter-Antonyan, H.T. Torosyan, N.E. Tyurin, E. Sarkisian, D.V. Shirkov, O.I. Zavialov, G. Zinovyev for valuable discussions and important comments. The

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45

scienti"c communications with them on the various stages were important. They are gratefull also to members of the seminar &Symmetries and integrable systems' of the N.N. Bogolyubov Laboratory of theoretical physics (JINR), to the scienti"c seminars of the V.P. Dzelepov Laboratory of nuclear problems, D.V. Scobeltsyn INP of the Moscow state University, of the ATLAS experiment community for a number of important discussions. They would also like to thank M. Gostkin and N. Shubitidze for the help in the formulation of the Monte Carlo simulation programs and for the discussions. One of the authors (J.M.) was supported in part by Georgian Academy of Sciences.

Appendix A. Matsubara formalism and the KMS boundary condition There are various approaches to build the real-time "nite-temperature "eld theories of Schwinger}Keldysh type (e.g. [47]). All of them use various tricks for analytical continuation of imaginary-time Matsubara formalism to real time [93]. The basis of the approaches is the introduction of the Matsubara "eld operator  (x, )"e@& (x)e\@& , + 1

(A.1)

where  (x) is the interaction-picture operator introduced instead of the habitual Heisenberg 1 operator (x, t)"e R& (x)e\ R& . 1 Eq. (A.1) introduces the averaging over the Gibbs ensemble instead of averaging over zerotemperature vacuum states. If the interaction switched on at the moment t adiabatically and switched o! at t then there is  the unitary transformation (x, t)";(t , t );(t , t) (x);(t, t ) . G  G 1 G

(A.2)

Introducing the complex Mills time contours [46] to connect t to t, t to t and t to t we form G   G a &closed-time' contour C (the end points of the contours joined together). This allows to write the last equality (A.2) in the compact form (x)"¹ (x)e !VY*  VY , 1 ! where ¹ is the time-ordering on the contour C operator. ! The generating functional Z( j) of correlation (Green) functions has the form Z( j)"R(0)¹ e !V*  V>HVV1 , ! where  means averaging over the initial state. If the initial correlations have a little e!ect, we can perform averaging over the Gibbs ensemble. This is the main assumption of the formalism: the generating functional of the Green functions Z( j)

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has the form in this case



Z( j)" D; t e\@&¹ e !VHVV; t G ! G with "(x). In accordance with (A.1) we have ; t e\@&"; t !i G G and, as a result,



Z( j)" De !@V*V>HVV ,

(A.3)

where the path integration is performed with KMS periodic boundary condition (t )"(t !i) . G G In (A.3) the contour C connects t to t , t to t and t to t !i. Therefore it contains an @ G   G G G imaginary-time Matsubara part t to t !i. A more symmetrical formulation uses the following G G realization: t to t , t to t !i/2, t !i/2 to t !i/2 and t !i/2 to t !i (e.g. [28]). This case G     G G G also contains the imaginary-time parts of the time contour. Therefore, Eq. (A.3) presents the analytical continuation of the Matsubara generating functional to real times. One can note that if this analytical continuation is possible for Z( j) then representation (A.3) gives good recipe of regularization of frequency integrals in the Matsubara perturbation theory, e.g. [47]. But it gives nothing new for our problem since the Matsubara formalism is a formalism for equilibrium states only.

Appendix B. Constant temperature formalism The starting point of our calculations is the n- into m-particles transition amplitude a , the LK derivation of which is the well-known procedure in the Lehmann}Symanzik}Zimmermann (LSZ) reduction formalism [94] framework, see also [95]. The (n#m)-point Green function G are LK introduced for this purpose through the generating functional Z [96] H L K G (x, y)"(!i)L>K jK (x ) jK (y )Z , (B.1) LK I I H I I where jK (x)" j(x)

(B.2)

and



Z " De 1H  . H

(B.3)

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The action



S ()"S ()!<()# dxj(x)(x) , H 

(B.4)

where S () is the free part and <() describes the interactions. At the end of the calculations one  can put j"0. To provide the convergence of the integral (B.3) over the scalar "eld  the action S () must H contain a positive imaginary part. Usually for this purpose Feynman's i-prescription is used. But it is better for us to use the integral on the Mills complex time contour C [46,47]. For example, > C : tPt#i, P#0, !R4t4#R (B.5) ! and after all the calculations return the time contour on the real axis putting "0. In Eq. (B.3) the integration is performed over all "eld con"gurations with standard vacuum boundary condition





d RI"0 , I N which leads to zero contribution from the surface term. Let us introduce now "eld  through the equation dxR (RI)" I

S () !  "j(x) (x)

(B.6)

(B.7)

and perform the shift P# in integral (B.3), conserving boundary condition (B.6). Considering  as the probe "eld created by the source



(x)" dyG (x!y)j(y) ,  (R#m) G (x!y)" (x!y) , V  only the connected Green function G will be of interesting to us. Therefore, LK L K G (x, y)"(!i)L>K jK (x ) jK (y )Z() , LK I I I I where



Z()" De 1\ 4>(

(B.8)

(B.9)

(B.10)

is the new generating functional. To calculate the nontrivial elements of the S-matrix we must put the external particles on the mass shell. Formally, this procedure means amputation of the external legs of G and further LK multiplication on the free particle wave functions. As a result, the amplitude of n- into m-particles

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transition a in the momentum representation has the form LK L K a (q, p)"(!i)L>K K (q ) K H(p )Z() . LK I I I I Here we introduce the particle distraction operator

(B.11)



. K (q)" dxe\ OVK (x), K (x)" (x)

(B.12)

Supposing that the momentum of particles is insu$cient for us the probability of n- into m-particles transition is de"ned by the integral








L K 1 d (q) d (p)   q !  p a  , r " I I LK L K LK n!m! I I where

(B.13)

dq L L I , (B.14) d (q)" d(q )"

L I (2
)2(q ) I I I is the Lorentz-invariant phase space element. We assume that the energy}momentum conservation -function was extracted from the amplitude. Note that r is the divergent quantity. To avoid this problem with trivial divergence, connected LK integration over reference frame, let us divide the energy}momentum "xing -function into two parts:



(q !p )" dP (P!q ) (P!p ) I I I I

(B.15)

and consider a new quantity











L L 1 d (q) d (p)  P!  q  P!  p a  (B.16) R(P)" L K I I LK n!m! I I LK de"ned on the energy}momentum shell (2.6). Here we suppose that the number of particles is not "xed. It is not too hard to see that, up to a phase space volume



R" dPR(P)

(B.17)

is the imaginary part of the amplitude vacvac . Therefore, computing r(P) the standard renormalization procedure can be applied and the new divergences will not arise in our formalism. The Fourier transformation of -functions in (B.16) allows one to write R(P) in the form



R(P)"

d d   e .? >?  () , (2
) (2
)

(B.18)

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where



1 K L

d(q )e\ ? OI  d(p )e\ ? NI a  . (B.19)

()" I I LK n!m! I I LK Introducing the Fourier-transformed probability () we assume that the phase-space volume is not "xed exactly, i.e. it is proposed that the 4-vector P is "xed with some accuracy if  are "xed. G The energy and momentum in our approach are still locally conserved quantities since an amplitude a is translationally invariant. So, we can perform the transformation LK  q "( ! )q # q P( ! )q # P , (B.20)  I   I  I   I  since 4-momenta are conserved. The choice of  "xes the reference frame. This degree of freedom  of the theory was considered in [11]. Inserting (B.11) into (B.19) we "nd that

 

()"exp !i dx dx(K (x)D (x!x,  )K (x) > >\  \ !K (x)D (x!x,  )K (x))Z( )ZH(! ) , (B.21) \ \>  > > \ where D and D are the positive and negative frequency correlation functions, respectively: >\ \>



D (x!x, )"!i d (q)e OV\VY\? >\ 

(B.22)

describes the process of particles creation at the time x and its absorption at x , x 'x , and  is     the CM 4-coordinate. Function



D (x!x, )"i d (q)e\ OV\VY>?  \>

(B.23)

describes the opposite process, x (x . These functions obey the homogeneous equations   (R#m) G "(R#m) G "0 , (B.24) V >\ V \> since the propagation of mass-shell particles is described. We suppose that Z() may be computed perturbatively. For this purpose the following transformations will be used (XK , / X at X"0): e\ 4("e\  VHK V(K YVe  VHV(Ve\ 4(Y "e V(V(K YVe\ 4(Y"e\ 4\ HK e  VHV(V ,

(B.25)

where jK was de"ned in (B.2) and K in (B.12). At the end of the calculations, the auxiliary variables j,  can be taken equal to zero. Using the "rst equality in (B.25) we "nd that Z()"e\  VHK VK Ve\ 4>(e\  V VYHV">> V\VYHVY , where D is the causal Green function: >> (R#m) G (x!y)" (x!y) . V >>

(B.26)

(B.27)

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Inserting (B.26) into (B.21) after simple manipulations with di!erential operators, see (B.25), we may "nd the expression

()"e\ 4\ HK > > 4\ HK \ exp



i dx dx 2

( j (x)D (x!x,  )j (x)!j (x)D (x!x,  )j (x) > >\  \ \ \>  >



!j (x)D (x!x)j (x)#j (x)D (x!x)j (x)) , > >> > \ \\ \

(B.28)

where D "(D )H (B.29) \\ >> is the anticausal Green function. Considering the system with a large number of particles, we can simplify the calculations choosing the CM frame P"(P "E, 0). It is useful also [41,33] to rotate the contours of  integration over 

:  "!i , Im  "0, k"1, 2 . I I I I For the result, omitting the unnecessary constant, we will consider " (). External particles play a double role in the S-matrix approach: their interactions create and annihilate the system under consideration and, on the other hand, they are probes through which the measurement of a system is performed. Since  are the conjugate to the particles energies I quantities we will interpret them as the inverse temperatures in the initial ( ) and "nal ( ) states   of interacting "elds. They are &good' parameters if and only if the energy correlations are relaxed. B.1. Kubo}Martin}Schwinger boundary condition The simplest (minimal) choice of ( )O0 assumes that the system under consideration is  surrounded by black-body radiation. This interpretation restores Niemi}Semeno! 's formulation of the real-time "nite temperature "eld theory [28]. Indeed, as follows from (B.21), the generating functional () is de"ned by corresponding generating functional



( )"Z( )ZH(! )" D D e 1 > \ 1 \ e\ 4> >(> > 4\ \(\  , > \  ! > \

(B.30)

see (B.21). The "elds ( ,  ) and ( ,  ) were de"ned on the time contours C and C . > > \ \ > \ As was mentioned above, see (2.11), the path integral (B.30) describes the closed-path motion in the space of "elds . We want to use this fact and introduce a more general boundary condition which also guarantees the cancelation of the surface terms in the perturbation framework. We will introduce the equality



N





d  RI " I > >

N



d  RI . I \ \

(B.31)

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The solution of Eq. (B.31) requires that the "elds  and  (and their "rst derivatives R  ) > \ I ! coincide on the boundary hypersurface    ( )"( )O0 , (B.32) !   where, by de"nition, ( ) is the arbitrary `turning-pointa "eld.  In the absence of the surface terms, the existence of a nontrivial "eld ( ) has the in#uence only  on the structure of the Green functions G "¹  , G "  , >> > > >\ > \ G "  , G "¹I   , (B.33) \> \ > \\ \ \ where ¹I is the antitemporal time ordering operator. These Green functions must obey the equations: (R#m) G (x!y)"(R#m) G (x!y)"0 , V >\ V \> (R#m) G (x!y)"(R#m)HG (x!y)" (x!y) V >> V \\ and the general solutions of these equations

(B.34)

G "D #g , GG GG GG G "g , iOj , (B.35) GH GH contain the arbitrary terms g which are the solutions of homogenous equations GH (R#m) g (x!y)"0, i, j"#,! . (B.36) V GH The general solutions of these equations (they are distinguished by the choice of the time contours C ) !



g (x!x)" d (q)e OV\VYn (q)  GH GH

(B.37)

are de"ned through the functions n which are the functionals of the &turning-point' "eld ( ): GH  if ( )"0 we must have n "0.  GH Our aim is to de"ne n . We can suppose that GH n &( )2( ) . GH   The simplest supposition gives n &  &( ) . (B.38) GH G H  We will "nd the exact de"nition of n starting from the S-matrix interpretation of the theory. GH It was noted previously that the turning-point "eld ( ) may be arbitrary. We will suppose that  on the remote  there are only free, on the mass-shell, particles. Formally, it follows from  (B.35)}(B.37). This assumption is natural also in the S-matrix framework [40]. In other respects the choice of boundary condition is arbitrary.

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Therefore, we wish to describe the evolution of the system in a background "eld of mass-shell particles. The restrictions connected with energy}momentum conservation laws will be taken into account and in other respects background particles are free. Then our derivation is the same as in [11]. Here we restrict ourselves to mentioning only the main quantitative points. Calculating the product a aH we describe time-ordered processes of particle creation and LK LK absorption described by D and D . In the presence of the background particles, this >\ \> time-ordered picture is slurred over because of the possibility to absorb particles before their creation occurs. The processes of creation and absorption are described in vacuum by the product of operators K K and K K . We can derive (see also [11]) the generalizations of (B.21). The presence of the > \ \ > background particles will lead to the following generating functional: R "e\ N(HG (H R ( ) ,  ! AN

(B.39)

where R ( ) is the generating functional for the vacuum case, see (B.30). The operator  ! N(H ) G H describes the external particles environment. The operator K H(q) can be considered as the creation and K (q) as the annihilation operator and G G the product K H(q)K (q) acts as the activity operator. So, in the expansion of N(K HK ) we can leave G H G H only the "rst nontrivial term



N(H )" d(q)K H(q)n K (q) , G GH H G H

(B.40)

since no special correlation among background particles should be expected. If the external (nondynamical) correlations are present then the higher powers of K HK will appear in expansion G H (B.40). Following the interpretation of K HK , we conclude that n is the mean multiplicity of G H GH background particles. Computing we must conserve the translation invariance of amplitudes in the background AN "eld. Then, to take into account the energy}momentum conservation laws one should adjust to each vertex of in-going a particles the factor e\ ? O and for each out-going particle we have LK correspondingly e\ ? O. So, the product e\ ?I Oe\ ?H O can be interpreted as the probability factor of the one-particle (creation#annihilation) process. The n-particles (creation#annihilation) process probability is the simple product of these factors if there are no special correlations among background particles. This interpretation is evident in the CM frame  "(!i , 0). I I After these preliminaries, it is not too hard to "nd that in the CM frame we have n

 ne\@ >@ O L (q )"n (q )" L >>  \\   e\@ >@ O L L




1  #  " "n q    e@ >@ O !1 2



.

(B.41)

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Computing n for iOj we must take into account the presence of one more particle GH  ne@ >@  O L  ne\@ >@  O L #(!q ) L n (q )"(q ) L   e@ >@  O L >\    e\@ >@  O L L L "(q )(1#n (q  ))#(!q )n (!q  )       and (q )"(q )n (q  )#(!q )(1#n (!q  )) . \>        Using (B.41)}(B.43), and the de"nition (B.35) we "nd the Green functions: n



dq e OV\VYGI (q, ()) G (x!x, ())" GH GH (2
)

53

(B.42)

(B.43)

(B.44)

where




iGI j(q, ())" G

i q!m#i

0

0

i ! q!m!i




n

#2
(q!m)




 #   q   2



n ( q )a ( )   \ 

and

 n ( q )a ( )   >   #  q  n   2








a ()"!e@(q $q ) . !   The corresponding generating functional has the standard form

(B.45)

(B.46)

(B.47) R ( j )"e\ 4\GHK > > 4\ HK \ eGV VYHG V%GH V\VY@HH VY , N ! where the summation over repeated indexes is assumed. Inserting (B.47) in the equation of state (2.8) we can "nd that  " "(E). If (E) is a &good'   parameter then G (x!x; ) coincides with the Green functions of the real-time "nite-temperature GH "eld theory and the KMS boundary condition: G (t!t)"G (t!t!i), G (t!t)"G (t!t#i) , >\ \> \> >\ is restored. Eq. (B.48) can be deduced from (B.45) by direct calculations.

(B.48)

Appendix C. Local temperatures We start this consideration from the assumption that the temperature #uctuations are large scale. We can assume that the temperature is a &good' parameter in a cell whose dimension is much

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J. Manjavidze, A. Sissakian / Physics Reports 346 (2001) 1}88

smaller than the #uctuation scale of temperature. (The &good' parameter means that the corresponding #uctuations are Gaussian.) Let us divide the remote hypersurface  on an N and let us propose that we can measure the   energy and momentum of groups of in- and out-going particles in each cell. The 4-dimension of cells cannot be arbitrarily small because of the quantum uncertainty principle. To describe this situation we decompose the -function of the initial state constraint (2.6) on the product of (N #1) -functions:  , K KJ ,  P!  Q ,  P!  q " dQ Q !  q I J J IJ J I J I J where q is the momentum of the kth in-going particle in the th cell and Q is the total IJ J 4-momenta of n in-going particles in this cell, "1, 2,2, N . Therefore, J 




  







, KJ   q "P . IJ J I The same decomposition will be used for the second -function of outgoing particle constraints. We must take into account the multinomial character of particle decomposition on N groups. This will give the coe$cient











n! m! , , n!  n m!  m , ) J ) J n !2n ! m !2m !  ,  , J J where is the Kronecker symbol. The summation over ) n , n ,2, n  "n  , m , m ,2, m  "m    , ,   , , is assumed. As a result, the quantity













d J (p ) , d J (q ) KJ LJ K I  Q !  q L I  P !  p R  (P, Q)"  a 

, LK J IJ J IJ m ! n !  J J LK, J I I

 (C.1)

de"nes the probability to "nd in the th cell the #uxes of in-going particles with total 4-momentum Q and of out-going particles with the total 4-momentum P . The sequence of these two measureJ J ments is not "xed. The Fourier transformation of -functions in (C.1) gives



, d d J J e  ,J /J ?J >.J ?J  () , R  (P, Q)"

, , (2
) (2
) I where

 ()"  ( ,  2,   ;  ,  ,2,  ) ,   ,   , ,

J. Manjavidze, A. Sissakian / Physics Reports 346 (2001) 1}88

55

has the form

 



, KJ d J (q) LJ d J (p) K e\ ?J OIJ

L e\ ?J NIJ a  .

 ()"

, LK m ! n ! J J J I I Inserting

(C.2)

K L a (p, q)"(!i)L>K K (q ) K H(p )Z(!) LK IJ IJ I I into (C.2) we "nd

 

,

 ()"exp i  dx dx[K (x)D (x!x;  )K (x) , > >\ J \ J



!K (x)D (x!x;  )K (x)] () ,  \ \> J >

(C.3)

where D (x!x; ), and D (x!x; ) are the positive and negative frequency correlation >\ \> functions. We must integrate over sets Q  and P  if the distribution of momenta over cells is not , , "xed. As a result,



R(P)" D (P)D (P)  () ,   ,

(C.4)

where the di!erential measure , d J K(P,  ) D(P)"

, (2
) J takes into account the energy}momentum conservation laws








, , , K(P,   )" dQ e  J ?J /J  P!  Q . J , J J J Explicit integration gives that , K(P,   )& (! ) , , J J where
is 3-vector of the CM frame. Choosing CM frame, "(!i, 0),



K(E,   )" , In this frame








 , , ,

dE e J @J #J E!  E . J J  J J

 (P)" D (E)D (E)  () , ,   ,

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J. Manjavidze, A. Sissakian / Physics Reports 346 (2001) 1}88

where , d J K(E,  ) D(E)"

, 2
i J and  () was de"ned in (C.3) with  "(!i , 0), Re  '0, k"1, 2. , IJ IJ IJ We will calculate integrals over  using the stationary phase method. The equations for the I most probable values of  : I R R ! ln K(E,   )" ln  (), k"1, 2 , (C.5) , , R R IJ IJ always have unique positive solutions  (E). We propose that the #uctuations of  near  are IJ I IJ small, i.e. are Gaussian. This is the basis of the local-equilibrium hypothesis [97]. In this case 1/ J is the temperature in the initial state in the measurement cell  and 1/ is the temperature of the J "nal state in the th measurement cell. The last formulation (C.4) implies that the 4-momenta Q  and P  cannot be measured. It is , , possible to consider another formulation also. For instance, we can suppose that the initial set Q  is "xed (measured) but P  is not. In this case we will have a mixed experiment:  is , J , de"ned by the equation R E "! ln  J , R J and  is de"ned by the second equation in (C.5). J Considering the continuum limit, N PR, the dimension of the cells tends to zero. In this case  we are forced by quantum uncertainty principle to assume that the 4-momenta sets Q and P are not "xed. This formulation becomes pure thermodynamical: we must assume that just   and    are measurable quantities. For instance, we can "x   and try to "nd   as a function of    the total energy E and the functional of  . In this case, Eqs. (C.5) become the functional  equations. In the considered microcanonical description, the "niteness of temperature does not touch the quantization mechanism. Indeed, one can see from (C.3) that all thermodynamical information is con"ned in the operator exponent eN(HG (H " e (K G "GH (K H , J G$H the expansion of which describes the environment, and the &mechanical' perturbations are described by the functional (). This factorization was achieved by the introduction of the auxiliary  "eld  and is independent of the choice of boundary conditions, i.e. una!ected by the choice of the systems environment. C.1. Wigner functions We will adopt the Wigner functions formalism in the Carruthers}Zachariazen formulation [39]. For the sake of generality, the m into n particles transition will be considered. This will allow the inclusion of the heavy ion}ion collisions.

J. Manjavidze, A. Sissakian / Physics Reports 346 (2001) 1}88

57

In the previous section, the generating functional  () was calculated by means of dividing the , &measuring device' on the remote hypersurface  into N cells   (C.6)

 ()"e\ N(_@X () ,  , where

 

, N(; , z)"  dx dx(K (x)D (x!x;  , z )K (x) > >\ J  \ J



!K (x)D (x!x;  , z )K (x)) \ \> J  >

(C.7)

is the particle number operator. The frequency correlation functions D and D are de"ned by >\ >\ equalities



D (x!x;  , z )"!i d (q)e OJ V\VYe\@J COJ z (q ) , >\ J    J



D (x!x,  , z )"i d (q)e\ OJ V\VYe\@J COJ z (q ) . \> J    J

(C.8) (C.9)

It was assumed that the dimension of the device cells tends to zero (N PR). Now we wish to  specify the cells coordinates. As a result we will get to the Wigner function formalism. Let us introduce Wigner variables [98] x!x"r, x#x"2y : x"y#r/2, x"y!r/2 .

(C.10)

Then



, N(; , z)"!i  d(q) dr(K (y#r/2)K (y!r/2)z (q )e OJ Pe\@J COJ  > \  J J #K (y#r/2)K (y!r/2)z (q )e\ OJ Pe\@J COJ ) dy . (C.11) \ >  J The Boltzmann factor, e\@GJ COGJ , can be interpreted as the probability to "nd a particle with the energy (q ) in the "nal (i"2) or initial (i"1) state. The total probability, i.e. the process of GJ creation and further absorption of n particles, is de"ned by multiplication of these factors. Besides, e OJ P is the out-going particle momentum measured in the th cell. Generally, it is impossible to adjust the 4-index of cell  with coordinate y. For this reason the summation over  and the integration over r are performed in (C.11) independently. But let us assume that the 4-dimension of the cell ¸ is higher than the scale of the characteristic quantum #uctuations ¸ , O ¸<¸ . (C.12) O One can divide the four-dimensional y space into the ¸-dimensional cells. Then, because of (C.12), the quantum #uctuations cannot take away particles from this cell. Then we can adjust the index of the measurements cell with the index of the y space cell.

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J. Manjavidze, A. Sissakian / Physics Reports 346 (2001) 1}88

As a result,

 

N(; , z)"!i dy d (q) dr(K (y#r/2)K (y!r/2)z (q , y) e O Pe\@ WCO  G > \   #K (y#r/2)K (y!r/2)z (q , y)) e\ O Pe\@ WCO  , \ >  

(C.13)

where





dy (C.14) dy" J !J and C() is the dimension ¸ of the y space cell with index . Notice that the momentum q did not carry the index  (or the index y of the space cell). Our formalism allows the introduction of more general &closed-path' boundary conditions. The presence of external black-body radiation will only reorganize the di!erential operator expNK (H ) and a new generating functional has the same form G H AN N

(, z)"e\ (_@X () .  AN The calculation of operator NK (H ) is strictly the same as in Appendix B. Introducing the cells G H we will "nd that



NK (H )" dr dyK (r#y/2)n (y)K (r!y/2) , G GH H G H where the occupation number n carries the cell index y: GH



n (r, y)" d (q)e OPn (y, q) GH  GH and (q "(q))  n

1 (y, q )"n (y, q )"n (y, ( # )q /2)" , >>  \\     e@ >@ WO !1

(y, q )"(q )(1#n (y,  q ))#(!q )n (y,! q ) ,        n (y, q )"n (y,!q ) . \>  >\  For simplicity the CM system was used. Other calculations are the same as the constant temperature case. n

>\

Appendix D. Multiperipheral kinematics First of all [21], two light-like 4-momenta p



"P !P m/s  

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59

are introduced. Here P are momenta of colliding particles. The "nal-state particles momenta  have the following representation: p " p # p #p ,      ,

p " p # p #p ,      ,

k " p # p #k . G G  G  G,

(D.1)

Sudakov's parameters, , , are not independent. The mass-shell conditions and the energy} momentum conservation laws give s  "m#(p )"E , s  "E , s  "E ,   , ,   , G G G,  # # "1,   G

 # # "1 ,   G

(D.2)

where E is the transverse energy. G, We have for the multiperipheral kinematics m , 1+ < <2< < &   L  s m ; ; ;2; ; &1   L  s

(D.3)

and the transverse momenta are restricted: p &k &m . G, G,

(D.4)

It corresponds to small production angles in the considered CM frame k   " G, ,  <  , G G G (s G

(D.5)

if the particle moves along P , and a similar expression exists for particles moving in the opposite  direction, where  ; . In the ¢ral region' of the CM frame  & &(E /E);1 the G G G G G, angles of produced particles are large and energies are small. It should be underlined that all this excludes the (mini)jets formation. The "nal-state particles phase space volume element is dq 1 dq dq (2 )>L   L> CL 2 d " Q >L )# q # 16L 4 q #m (q !q )# (q !q   L>  L L>




1 d d L> s ?OG  G " ;  ( ! )2 L

dZ ,   s q # L   L>  L G 

(D.6)

where C "3 and the 4-momentum of produced particle 4 k "( ! )p #( ! )p #(q !q ) "! p # p #(q !q ) . G G G>  G G>  G G> , G>  G  G G> ,

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J. Manjavidze, A. Sissakian / Physics Reports 346 (2001) 1}88

The square of pairs invariant mass E  G> . s "(p #k )"s , s "(k #p )" L, , s "E     L> L  G G\,   L G\ The energy conservation law takes the form s s 2s "sE 2E .   L , L, The trajectory of reggeized gluon is



q dk (q)" Q , 2
 (k#)((q!k)#) where  is the gluon &mass'. If this virtuality is large, <m, then the gluon decays creating a pQCD jet, but the constraint on the multiperipheral kinematics prevents this possibility. D.1. Deep inelastic reactions For the pure deep inelastic case, when one of the initial hadrons which is scattered at the angle  has the energy E in the cms of beams whereas another which is scattered at small angle and the large transfer momentum Q"4EE sin(/2)<m, is distributed to the same number of the emitted particles due to evolution mechanism we have [91] ( is small) 4E dD dEd cos  , d"'1" L L QM


   






 L /dk IL dk I dk   L L\ 2  d  d  2 dD " Q L\ L\ L L L 4
k k k K L K L\ K  V @L   1#z L 2P( ), P(z)"2 , ; d P     1!z @ L\ where G"( ! ) and the emission angle  "k /(E max( . )). G> G G G G G

(D.7)

D.2. Large angle production For the large-angle particle production process the di!erential cross section (as well as the total one) decreases with CM energy (s. Let us consider for de"niteness the process of annihilation of electron}positron pairs to n photons [99]: 2
 dF , d"*" L L s




 L L

dx dy (x !y )(y !y )(y )(x ) dF " G G G G G G\ G G L 2
G 1 s ;( !x )( !y ), y "ln , "ln , L L G  m G

(D.8)

J. Manjavidze, A. Sissakian / Physics Reports 346 (2001) 1}88

61

Similar formulae can be written for subprocess of quark-antiquark annihilation into n large-angle moving gluons. At the end, one can consider the following possibilities: (a) (b) (c) (d) (e)

Pomeron regime (P); Evolution regime (DIS); Double logarithmic regime (DL); DIS#P regime; P#DL#P regime.

The description of every regime may be performed in terms of e!ective ladder-type Feynman diagrams. This can be done using the blocks dZ , dD and dF . L L L Appendix E. Reggeon diagram technique for generating function We will consider, see (3.11)





1 , "ln(s/m) . de\SKP(q, ; z)" # q# (z)    as the &propagator of the cut Pomeron'. It will be assumed also that P(q, ; z)"

(E.1)

 (z)"!
#(1!z)n , n '0 .    So, the resonance short-range correlations will be ignored in this de"nition or propagator. It was assumed also that the &bare' slope  is z independent. It should be underlined that the &propagator' (E.1) is written phenomenologically. It absorbs the assumptions that (i) the di!raction cone shrinks with energy and (ii) the inclusive cross sections are universal, see (3.2). The set of principal rules concerning multiperipheral kinematics of Feynman diagrams is given in Appendix D. The reggeon calculus supposes that the virtuality of each line of the Feynman diagram is restricted. This ignores &hard jets', later known as the pQCD jets. Then the  Pomeron exchange eikonal diagram has only (#1) ways of being cut. If the cut line goes through  Pomerons, then the corresponding contributions are:



I J I(, q)" d (MI (q ,2, q ))YI P(q ,  ; z) P(q ,  ; z"1) , J J  J J G G ' G J J GI> where MI(q ,2, q ) is the &vertex function', the combinatorial coe$cient is J  J

(E.2)

(!1)J\I2J! YI" J !(!)! and the phase space element is









J d dq J J J J !    q!  q . d "

J J J (2
)i G J G

(E.3)

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J. Manjavidze, A. Sissakian / Physics Reports 346 (2001) 1}88

As usual, contribution (E.2) leads to the following mean multiplicity of produced particles:





R n
(s)I" ln d (s/m)SI(, q"0) J J J Rz

X

&n
(s) .

(E.4)

Therefore, &n
(s)

(E.5)

is essential in the VHM region, where n&n
(s) is assumed. The impact parameter representation:




s S dq qb I(s, q)" d e I(, q) J J m (2
)

(E.6)

would be useful also. The contribution (E.2) describes interactions with impact parameter b K4 ln(s/m)/ .

(E.7)

Notice that b is the number of cut pomerons independent of . But, remembering that 5 and that the Regge model is only able to describe large-distance interactions, mb 51, one can conclude that the Regge pole description is valid only for n4n
(s) .

(E.8)

This is why the VHM region is de"ned by n
(s). Appendix F. Pomeron with  > 0 Then the cut Pomeron propagator in the impact parameter representation g (b, ; z)"g(b, )eX\L
Q ,

(F.1)

where 1
b g(b, )" e Ke\ ?YK 2

(F.2)

is the uncut Pomeron pro"le function. Using this de"nition one can "nd that the contribution of the eikonal diagrams gives a contribution 1 F (b, ; z)"(1!e\HEbK)! (1!eHEbKX\L
Q\) ,  2

(F.3)

where  is a constant. The "rst bracket is essential for b44
. So, with exponential accuracy, the "rst term is equal to (4
!b) .

J. Manjavidze, A. Sissakian / Physics Reports 346 (2001) 1}88

63

Let us now consider the second bracket. For z(1





1 b44
 1# ln(1!eX\L
Q) "4
(, z) 

(F.4)

are essential. It is not hard to see that (, z) decreases if zP1 and (, z) is equal to zero for z"1. In this case, by the de"nition of the generating function, the integral over b of F (b, ; z"1) de"nes  the contribution to the total cross section. So, the model predicts the production of particles in the ring 4(, z)4b44
.

(F.5)

if z(1, i.e. if n(n
(s). Notice also that (, z"0)"
#O(e\K). Then, (F.6) F (b, ; z"0)"F (b, ; z"1) .    So, the elastic part of the total cross section is half of the total cross section. This means, using the optical analogy, that the scattering on the absolutely black disk is well described. The last conclusion means that the interaction radii should increase with n in the VHM region. Indeed, as follows from (F.3), 1 F (b, ; z)K (eHEbKX\L
Q\!1) ,  2

(F.7)

at z'1 and, therefore, 04b4B"4(
#(z!1)n
(s))

(F.8)

are essential.

Appendix G. Dual resonance model of VHM events Our purpose is to investigate the role of the exponential spectrum (3.19) in the asymptotic region over multiplicity n. In this case one can validate heavy resonance creation and such a formulation of the problem has de"nite advantages. (i) If creation of heavy resonances at nPR is expected, then one can neglect the dependence on the resonance momentum q . So, the &low-temperature' expansion is valid in the VHM region. G (ii) Having the big parameter n, one can construct the perturbations expanding over 1/n. (iii) We will be able to show at the end the range of applicability of these assumptions. For this purpose, the following formal phenomena will be used. The grand partition function ¹(z, s)" zL (s), ¹(1, s)" (s), n4(s/m ,n (s) 

 L  L will be introduced, see (2.35). Then the inverse Mellin transformation, see (2.37)



1 dz ¹(z, s) .  (s)" L 2
i zL>

(G.1)

(G.2)

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J. Manjavidze, A. Sissakian / Physics Reports 346 (2001) 1}88

will be performed expanding it in the vicinity of the solution z '0 of the equation of state,  see (2.38): R n"z ln¹(z, s) . Rz

(G.3)

It is assumed, and this should be con"rmed at the end, that the #uctuations in the vicinity of z are  Gaussian. It is natural at "rst glance to consider z "z (n, s) as an increasing function of n. Indeed, this   immediately follows from the positivity of  (s) and the "niteness of n (s) at "nite s. But one can L

 consider the &thermodynamical limit', see Section 2.3.1, or the limit m P0. Theoretically, the last  one is right because of the PCAC hypotheses and nothing should happen if the pion mass m P0.  In this sense, ¹(z, s) may be considered as the whole function of z. Then, z "z (n, s) would be an   increasing function of n if and only if ¹(z, s) is a regular function at z"1. The proof of this statement is as follows. We should conclude, as follows from Eq. (G.3), that z (n, s)Pz at nPR and at s"const , (G.4)   i.e. the singularity point z attracts z in asymptotics over n. If z "1, then (z !z )P#0, when      n tends to in"nity [50]. The concrete realization of this possibility is shown in Section 3.3. But if z '1, then (z !z )P!0 in VHM region, see Sections 3.1, 3.5.    One may use the estimation, see also (2.39): 1  (s) (G.5) ! ln L "ln z (n, s)#O(1/n) ,  n  (s)  where z is the smallest solution of (G.3). It should be underlined that this estimation is independent  of the character of singularity, i.e. the position z is only important with O(1/n) accuracy.  G.1. Partition function Introducing the &grand partition function' (G.1) the &two-level' description means that



1 I ¹(z, )

d (q) dm (q , z)e\@CG N (q , q ,2, q ; ),!F(z, s) , " (G.6) I G G I   I k!  (s)  G I where (q )"(q#m). This is our group decomposition. The quantity (q, z) may be considered G G G as the local activity. So, ln

¹ (q, z)



& B (q) . (G.7)   K If the resonance decay forms a group of particles with total 4-momentum q, then B (q) is the mean  number of such groups. The second derivative gives



¹ & B (q , q )!B (q )B (q ), K (q , q ) ,             (q , z) (q , z)   K

(G.8)

J. Manjavidze, A. Sissakian / Physics Reports 346 (2001) 1}88

65

where K (q , q ) is the two groups correlation function, and so on. One can consider B as Mayer's    I group coe$cients, see Section 2.3.3. The Lagrange multiplier  was introduced in (G.6) to each resonance: the Boltzmann exponent exp! takes into account the energy conservation law   "E, where E is the total energy of G G colliding particles, 2E"(s in the CM frame. This conservation law means that  is de"ned by the equation R (s" ln ¹(z, ) . R

(G.9)

So, to de"ne the state one should solve two equations of state (G.3) and (G.9). The solution  of Eq. (G.9) has the meaning of inverse temperature of the gas of resonances if  and only if the #uctuations in the vicinity of  are Gaussian, see Section 2.2.2.  On the second level, we should describe the resonance decay into hadrons. Using (3.24) we can write in the vicinity of z"1:




m (q, z)" zL0 (q)"g0  eX\L
K . , m L The assumptions B and D, see (3.21), were used here So,

(G.10)



I !F(z, s)" dm (m , z)BI (m; ) , G G I I G where m"(m , m ,2, m ) was de"ned in (G.10) and   I I BI (m; )" d (q)e\@CG OG B (m; q) . I I I G Assuming now that q ;m are essential, G 2m I G e\@KG . BI (m; )KB (m)

I I  G Following the duality assumption, one may write

(G.11)





(G.12)



(G.13)

I B (m)"BM (m) mAe@ KG  I I G G and BM (m) is a slowly varying function of m"(m , m ,2, m ): I   I NM (m)Kb . I I As a result, the low-temperature expansion is as follows: !F(z, s)" I

2ImI (g0)Ib  I I





K



(G.14)



dmmA>eX\L
0K\@\@ K

I

.

(G.15)

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We should assume that (! )50. In this sense one may consider 1/ as the limiting   temperature and the above-mentioned constraint means that the resonance energies should be high enough. G.2. Thermodynamical parameters Remembering that the position of the singularity over z is essential, let us assume that the resonance interactions cannot renormalize it, i.e. that the sum (G.15) is convergent. Then, leaving the "rst term in the sum (G.15),



m g0C   dm(m/m )A> eX\L
0K\@\@ K . !F(z, s)"    K

(G.16)

We expect that this assumption is satis"ed if





2m (g0)b   dmmA> eX\L
0K\@\@ K< b  K 





K

dmmA> eX\L
0K\@\@ K





(G.17)

for nPR, sPR,

nm n , ;1 . n (s



(G.18)

So, we would solve our equations of state with the following free energy:





m   d !F(z, s)"   m K 

m AY\
e\ KK  , m 

(G.19)

where, using (3.20), "#2(z!1)n
0 #5/2"2(z!1)n
0 ,
"m (! )50, "const .    

(G.20)

We have in terms of these new variables the following equation for z: 2n
0 R (,
) n"z  .  R
AY

(G.21)

The equation for  takes the form n



m (#1,
) "  , 
AY>

where n "((s/m ) and (
, ) is the incomplete -function 





(,
)"






dxxAY\e\V .

(G.22)

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67

G.3. Asymptotic solutions Following physical intuition, one should expect the cooling of the system when nPR, for "xed (s, and heating when n PR, for "xed n. But, as was mentioned above, since the solution of

 Eq. (G.22)  is de"ned by the value of the total energy, one should expect that  decreases in both   cases. So, the solution R
R
 (0 at sPR  (0 at nPR,
50,  Rs Rn

(G.23)

is natural for our consideration. The physical meaning of z is activity. It de"nes at "const the work needed for the creation of one particle. Then, if the system is stable and ¹(z, s) may be singular at z'1 only, Rz Rz  '0 at nPR,  (0 at sPR . Rs Rn

(G.24)

One should assume solving Eqs. (G.21) and (G.22) that R ( ,
)   ;( #1,
) . (G.25) z
A >   
A R   This condition contains the physical requirement that n;n . In the opposite case, the "niteness

 of the phase space for m O0 should be taken into account.  As was mentioned above, the singularity z attracts z at nPR. For this reason one may   consider the following solutions: A. z "R: z <
,
;1.   In this case
\AY(,
)& eAY AY
 .

(G.26)

This estimation gives the following equations:




 n n"C  ln(/
) eAY AY
, "C  ln ;1 , (G.27)  
n

 where C "O(1) are the unimportant constants. The inequality is a consequence of (G.25). G These equations have the following solutions: n ;1,  &ln n<1 .
K   n ln n

 Using them one can see from (G.5) that it gives  (O(e\L) . L B. z "#1: z P1,
;1.   

(G.28)

(G.29)

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One should estimate (,
) near the singularity at z"1 and in the vicinity of
"0 to consider the consequence of this solution. Expanding (,
) over
at P0, 1 (,
)"()!
AY e\
#O(
AY>)K #O(1) . 

(G.30)

This gives the following equations for : n"C 

 ln(1/
)!1 eAY 
 . 

(G.31)

The equation for
has the form "C eAY> 
 ,

  where C "O(1) are unimportant constants. G At n

(G.32)

0( ln(1/
)!1;1, i.e. at ln(1/
);n; ln(1/
) ,

(G.33)

we "nd 1  & .  ln(1/
)  Inserting this solution into (G.32)

(G.34)

1
& . (G.35)  n

 It is remarkable that
in the leading approximation is n independent. By this reason  becomes   n independent also 1 1 : z "1# .  &  ln(n )  n
0 ln(n ) 



 This means that  "O(e\L) L and obeys the KNO scaling with mean multiplicity n
"n
0 ln(n ). 



(G.36)

(G.37)

Appendix H. Correlation functions in DIS kinematics Considering particle creation in the DIS processes, one should distinguish correlation of particles in the (mini)jets and the correlations between (mini)jets. We will start from the description of the jet correlations. One should introduce the inclusive cross section for the  jets creation PJ (k , k ,2, k ; q, x) , J   J

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69

where k , i"1, 2,2, n are the jets 4-momentum in the DIS kinetics, !q<. Having  we can G J "nd the correlation functions NP(k , k ,2, k ; q, x) , J   J where (r)"r ,2, r and r "(q, g) de"nes the sort of created color particle. It is useful to introduce  J G the generating functional



L (H.1) F?@(q, x; w)" d (k) wPG (k )a?@(k , k ,2, k ; q, x) , L G L   L G L where a?@ is the amplitude, d (k) is the phase space volume and wPG (k ) are the arbitrary functions. G L L It is evident that F?@(q, x; w) "D?@(q, x) . U The inclusive cross sections J F?@(q, x; w) . P(k , k ,2, k ; q, x)"

U J   J wPG (k ) G G The correlation function

(H.2)

(H.3)

J ln F?@(q, x; w) . (H.4) NP(k; q, x)"

U J wPG (k ) G G We can "nd the partial structure functions D?@(q, x; n), where n is the number of produced (time-like) gluons, using their de"nitions. It will be useful to introduce the Laplace transform over the variable ln(1/x):



F?@(q, x; w)"




dj 1 H f ?@(q, j; w) . 2
i x

(H.5)

0 H The expansion parameter of our problem  ln(!q/)&1. For this reason one should take  into account all possible cuts of the ladder diagrams. So, calculating D?@(q, x) in the LLA all possible cuts of the skeleton ladder diagrams are de"ned by the factor [78]: 1 ?@G ?@ ,
P P P

(H.6)

i.e. the cut line may not only get through the exact Green function G (k) but through the exact P G vertex functions ?@(q , q , k ) also (q, q are negative). We have in the LLA (see Appendix D) P G G> G G G> ;!q;!q ;!q G G> and x4x 4x 41 . G> G

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Following our approximation, see the previous section, we could not distinguish the way in which the cut line goes through the Born amplitude a?@"(?@)G  . P P P We will simply associate wP Im a?@ to each rung of the ladder. P Considering the asymptotics over n, the time-like partons virtuality k K!q/y should be G G G maximal. Here y is the fraction of the longitudinal momentum of the jet. Then, slightly limiting the G jets phase space, ln k"ln q (1#O(ln(1/x)/q )) . (H.7) G G> G> As a result, introducing  "ln(q/), where  (q)"1/, "(11N/3)!(2n /3) in the LLA G G  D variable, we can "nd the following set of equations: R  f (q, j; w)" P ( j)wP()f (q, x; w) , ?A ?@ R ?@ AP where

(H.8)



 dx xHP (x) (H.9) ?A x  and P (x) is the regular kernel of the Bethe}Salpeter equation [78]. At w"1 this equation is the ?A ordinary one for D?@(q, x). We will search the correlation functions from Eq. (H.8) in terms of the Laplace transform P ( j)" ( j)" ?A ?A

nPJ (k , k ,2, k ; q, j)"nPJ (k; q, j) . ?@   J ?@ Let us write

 






1 J d G (wPG ( )!1) nPJ (k; q, j) . f (q, j; w)"d (q, j) exp  (H.10)

?@ ?@ ?@ G !  G J G Inserting (H.10) into (H.8) and expanding over (w!1) we "nd the sequence of coupled equations. Omitting the cumbersome calculations, we write in the LLA that ). PJ ( ,  ,2,  ; q, j)"d  ( j,  )P  ( j)d   ( j,  )2PJJ J> ( j)d J> ( j,  ?@   J ?A AA J>  AA AA  A @ One should take into account the conservation laws:

(H.11)

 )  2 ",  ( (2( ( .   J>   J> Computing the Laplace transform of this expression we "nd

(H.12)

PJ ( ,  ,2,  ; q, x) . ?@   J The kernel d ( j, ) was introduced in (H.11). Let us write it in the form ?@ d ( j) JN H , d ( j, )"   ?@ ?@  ! \ N! >

(H.13)

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where dN " ! , dN " ! , dN " , dN " OO  EE OE  OO OE EO EO OE

(H.14)

and  " # #[( ! )!4n   ] . EE OO EE D OE EO N  OO If x;1, then (j!1);1 are essential. In this case [78],

(H.15)

 & < & "O(1) . EE EO OE OO This means that the gluon jets dominate and

(H.16)

nE " #O(1) . EE EE One can "nd the following estimation of the two-jet correlation function:

(H.17)

(H.18) nP P ( ,  ; , )"O(max( /)PEE , ( /)PEE , ( / )PEE ) . ?@       This correlation function is small since in the LLA  ( (. This means that the jet correlation   becomes high if and only if the masses of the correlated jets are comparable. But this condition shrinks the range of integration over  and for this reason one may neglect the &short-range' correlations among jets. Therefore, as follows from (H.10),

 



O d wE() .  O We will use this expression to "nd the multiplicity distribution in the DIS domain. f (q, j; w)"d (, j) exp  EE ?@ EE

(H.19)

H.1. Generating function To describe particle production, one should replace: wP Im a?@PwP Im a?@ , P L P where wP is the probability of n particle production, L  wP "1 . (H.20) L L Having  jets, one should take into account the conservation condition n #n #2#n "n.   J For this reason, the generating functions formalism is useful. As a result, one can "nd that if we take (H.19) wE"wE(, z), wE(, z)

"1 , (H.21) X then f (q, j; w) de"ned by (H.19) is the generating functional of the multiplicity distribution in the ?@ &j representation'. In this expression wE(, z) is the generating function of the multiplicity distribution in the jet of mass k"eO.

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As a result, see (H.5),



F?@(q, x; w)J

0 H

dj (1/x)HePEE SOX , 2
i

(H.22)

where



O d wE(, z) .  O Noting the normalization condition (H.21), (, z)"

(H.23)

(, z"1)"ln  .

(H.24)

The integral (H.22) may be calculated by the steepest descent method. It is not hard to see that jKj "1#4N(, z)/ln(1/x) (H.25)  is essential. Notice that j!1;1 should be essential but we "nd, instead of the constraint (3.2), that (, z);ln(1/x) .

(H.26)

In the frame of this constraint, F?@(q, x; w)Jexp4(N(, z)ln1/x) .

(H.27)

Generally speaking, there exist such values of z that j !1&1. This is possible if (, z) is  a regular function of z at z"1. Then z should be an increasing function of n and consequently  (, z ) would be an increasing function of n. Therefore, one may expect that in the VHM domain  j !1&1.  Then jK1#(, z)/ln(1/x) would be essential in the integral (H.22). This leads to the following estimation: F?@(q, x; w)J e\SOX . But this is impossible since F?@(q, x; w) should be an increasing function of z. This shows that the estimation (H.27) has a "nite range of validity. Solution of this problem with unitarity is evident. One should take into account correlations among jets considering the expansion (H.10). Indeed, smallness of nPJ may be compensated by ?@ large values of JwPG ( , z) in the VHM domain. G G Appendix I. Solution of the jets evolution equation One may neglect quark jets in the VHM region since the gluons mean multiplicity n
'n
E O quarks multiplicity [81,85] and in the VHM region the leftmost singularities are important. Then we can write [84]



R O 12 ¹ (, z)" ¹ (, z) d(¹ (, z)!1) , H R H 11 H O

(I.1)

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where "ln(q/) and ¹ (, z) is the generating function of the distribution over the number of H gluons w (tau): L ¹ (, z)" zLw (), ¹ (, z"1)"1 . H L H L We search a solution in the VHM region, where

(I.2)

n
(I.3)




n A w " e\?LL
H . L n
H It is useful to introduce 1   ()"  nI\w () L I k! L for this solution. Inserting (I.4) into this expression,  ()"n
I() , I I where  (i) should be positive and (ii)  independent. I These conditions are satis"ed for the following values of k. Indeed, at k<1:

(I.4)

(I.5)

(I.6)




1  1 n I>A (k#)  "  e\?LL
H K\I>A . (I.7) I k! n n
(k#1) H L The generating function ¹ has the following form in terms of  : H I  ¹ (, z)"  (ln z)I () . (I.8) H I L Inserting (I.8) into (I.1) and assuming that  is a -independent quantity, we "nd the following I recurrent equation for  : I 4 I 1 (a  "    !2  . I k k I I\I k I I   Therefore, if k<(a ,

(I.9)

(I.10)

then we can neglect the last term on the right-hand side of (I.9) and in this case  are positive and I  independent. Noting that in (I.7) n&kn
are essential inequality (I.10) means that the solution H (I.4) is correct if n
(I.11)

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i.e., only for this value of n w has the form (I.4) and the corresponding generating functional has L singularity at  z "1# .  n
H

(I.12)

Appendix J. Condensation and type of asymptotics over multiplicity It is important for the VHM experiment to have an upper restriction on the asymptotics. We wish to show that  decreases faster than any power of 1/n: L  (O(1/n) . (J.1) L To prove this estimation, one should know the type of singularity at z"1. One can imagine that the points, where the external particles are created, form the system. Here we assume that this system is in equilibrium, i.e. in this system, there are no macroscopical #ows of energy, particles, charges and so on. The lattice gas approximation is used to describe such a system. This description is quite general and does not depend on details. Motion of the gas particles leads to the necessity to sum over all distributions of the particles on cells. For simplicity, we will assume that only one particle can occupy the cell. So, we will introduce the occupation number  "$1 in the ith cell:  "#1 means that we G G have no particle in the cell and  "!1 means that a particle exists in a cell. Assuming that the G system is in equilibrium, we may use the ergodic hypothesis and sum over all &spin' con"gurations of  , with the restriction: "1. It is evident that this restriction introduces the interactions [100]. G G The corresponding partition function in temperature representation [52]



(, H)" De\1HN ,

(J.2)

where integration is performed over (x)4R and, considering the continuum limit, D" d(x). The action V 1 S ()" dx ()!#g! , (J.3) H 2

 



where









    & 1!  , g&  , &  H .   

(J.4)

and 1/ is the critical temperature.  J.1. Unstable vacuum We start this consideration from the case '0, i.e. assuming that ' . In this case the  ground state is degenerate if H"0. The extra term &H in (J.3) can be interpreted as the

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75

interaction with external magnetic "eld H. This term regulates the number of &down' spins with "!1 and is related to the activity z"e@& ,

(J.5)

i.e. H coincides with the chemical potential. The potential v()"!#g, '0 ,

(J.6)

has two minima at  "$(/2g . ! If the dimension d'1, no tunnelling phenomena exist. But choosing H(0 the system in the correct minimum (it corresponds to the state without particles) becomes unstable. The system tunnels into the state with an absolute minimum of energy. The partition function (, z) becomes singular at H"0 because of this instability. The square root branch point gives a () Im (b, z)"  e\? @&, H

a '0 . G

(J.7)

Note, Im (b, z)"0 at H"0. Deforming the contour in the Mellin integral over z on the branch line,



1  dz 8a   e\? @  X .

()" L
zL> ln z  In this integral



z Jexp 



8a    n

(J.8)

(J.9)

is essential. This leads to the following estimation:

Je\? @L(O(1/n) . L It is useful to note at the end of this section that

(J.10)

(i) The value of is de"ned by Im (b, z) and the metastable states, the decay of which gives L a contribution to Re (b, z), are not important. (ii) It follows from (J.9) that in the VHM domain H&H &ln z &(1/n)P0 . (J.11)   So, the calculations are performed for the &weak' external "eld case, when the degeneracy is weakly broken. It is evident that the lifetime of the unstable (without particles) state is large in this case and for this reason the semiclassical approximation is correct. This is an important consequence of (3.1).

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J.2. Stable vacuum Let us consider now (0, i.e. ( . The potential (J.6) has only one minimum at "0 in this  case. The inclusion of an external "eld shifts the minimum to the point  " (H). In this case the   expansion in the vicinity of  should be useful. As a result, 



(, z)"exp



dx !=( ) ,  

(J.12)

where =( ) can be expanded over  :   1 =( )" dx  (x ; H)bI (x ,2, x ) . (J.13)  I  I J  J l J I In this expression, bI (x ,2, x ) is the one-particle irreducible Green function, i.e. bI is the virial J  J J coe$cient. Then  can be considered as the e!ective activity of the correlated l-particle group.  The sum in (J.13) should be convergent and, therefore, s PR if HPR. But in this case the  virial decomposition is equivalent to the expansion over the inverse density of particles [25]. In the VHM region it is high and the mean "eld approximation becomes correct. As a result,






  : s PR if PR  K!   4g

(J.14)

and


 

(, z)JeHE 12g

  \ . 4g

We can use this expression to calculate . In this case L  z JeEL PR at nPR ,  is essential and in the VHM domain

(J.15)

(J.16)

Je\EL(O(e\L) . (J.17) L This result is an evident consequence of vacuum stability. It should be noted once more that the conditions (3.1) considerably simplify calculations.

Appendix K. New multiple production formalism and integrable systems K.1. S-matrix unitarity constraints To explain our idea, let us consider the spectral representation of the one-particle amplitude: H(x ) (x ) A (x , x ; E)" L  L  , P#0 .    E!E !i L

(K.1)

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77

It describes the transition of a particle with energy E from point x to x . According to our general   idea, see introduction to Section 2.1, we will calculate



R (E)" dx dx A (x , x ; E)AH(x , x ; E) .         

(K.2)

The integration over the end points x and x is performed only for the sake of simplicity.   Inserting (K.1) into (K.2) and using ortho-normalizability of the wave functions (x) we L "nd that







1  1 1 1 "  ! R (E)"  E!E !i 2i E!E !i E!E #i L L L L L



1 "Im "
 (E!E ) . L E!E !i L L L

(K.3)

On the other hand, the closed-path amplitude, o!ered for calculation in [101],

 

1 H(x) (x) L " C (E)" dx L  E!E !i E!E !i L L L L



1 1 " P #i
(E!E ) " P #iR (E) . L  E!E E!E L L L L

(K.4)

So, we wish to calculate only the imaginary part of the closed-path contribution R(E)"Im C (E) .  Notice the extra factor  on the left-hand side. The reason for this choice is evident: the real part of C (E) is equal to zero at E"E , i.e. did not  L contribute to the measurable. To calculate the bound states energy spectrum, it is enough to know only the imaginary part of the closed-path amplitude. This property is not accidental. It is known as the optical theorem and is the consequence of the total probability conservation principles. The formal realization of this is the unitarity condition for the S-matrix: SS>"I. In terms of the amplitudes A, S"I#iA, the unitarity condition presents an in"nite set of nonlinear operator equalities: iAAH"A!AH .

(K.5)

Notice that expressing the amplitude by the path integral one can see that the left-hand side of this equality o!ers the double integral and, at the same time, the right-hand side is the linear combination of integrals. Thus, the continuum contributions into the amplitudes should be canceled to provide the conservation of total probability. In this sense it is a necessary condition.

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Indeed, to see the integral form of our approach, let us use the proper-time representation:



 A (x , x ; E)" (x ) H(x )i d¹e #\#L > C2    L  L   L and insert it into (K.2):

(K.6)



 R (E)" (K.7) d¹ d¹ e\2> >2\ Ce #\#L 2> \2\  .  > \ L  We will introduce new time variables instead of ¹ : ! ¹ "¹$ , (K.8) ! where, as follows from the Jacobian of transformation, 4¹, 04¹4R. But we can put 4R since ¹&1/PR is essential in the integral over ¹. As a result,





 > d d¹e\C2 e #\#L O . (K.9)

(E)"2
 
\ L  In the last integral, the continuum of contributions with EOE are canceled. Note that the L product of amplitudes AAH was &linearized' after the introduction of &virtual' time [103] "(¹ !¹ )/2. > \ We wish to calculate the density matrix (, z) including the consequence of the unitarity condition cancelation of unnecessary contributions. Here we demonstrate the result and the intermediate steps we will formulate, without proof, as the statements o!ered in [16,15], where the formalities are described. K.2. Dirac measure The statement, see [15] and references cited therein, S1. The unitarity condition unambiguously determines contributions in the path integrals for looks like a tautology since e 1V, where S(x) is the action, is the unitary operator which shifts a system along the trajectory. So, it seems evident that the unitarity condition is already included in the path integrals. The rule as the path integrals should be calculated is well known, see e.g. [102]. Nevertheless, the general path-integral solution contains unnecessary degrees of freedom (unobservable states with EOE in the above example). We would de"ne the path integrals in such a way that the condition L of absence of unnecessary contributions in the "nal (measurable) result be loaded from the very beginning. Just in this sense, the unitarity looks like the necessary and su$cient condition unambiguously determining the complete set of contributions.

 It is well known that this unitary transformation is the analogy of the tangent transformations of classical mechanics [103].

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S2. The m- into n-particles transition (unnormalized) probability R would have on the Dirac LK measure the following symmetrical form





K L R (p ,2, p , q ,2, q )" (q ; u) (p ; u) I I LK  L  K S I I K L K L "e\ KHC DM(u)e 1- S\ 3SC (q ; u) (p ; u),OK (u) (q ; u) (p ; u) . I I I I I I I I (K.10)



Here p(q) are the in(out)-going particle momenta. It should be underlined that this representation is strict and is valid for arbitrary Lagrange theory of arbitrary dimensions. The eikonal approximation for inelastic amplitudes was considered in [104]. The operator OK contains three elements, the Dirac measure DM, the functional ;(x, e) and the operator K( j, e). The expansion over the operator



1 K( j, e)" Re 2



1 dx dt , Re j(x, t) e(x, t) 2 >

dx dtjK (x, t)e( (x, t) (K.11) ! !> generates the perturbation series. We will assume that this series exists (at least in Borel sense). The functionals ;(u, e) and S (u) are de"ned by the equalities -



S (u)"(S (u#e)!S (u!e))#2 Re  



;(u, e)"<(u#e)!<(u!e)!2 Re

dx dte(x, t)(R#m)u(x, t) ,

(K.12)

!> dx dte(x, t)v(u) ,

(K.13)

!> where S (u) is the free part of the Lagrangian and <(u) describes interactions. The quantity S (u) is  not equal to zero if u have nontrivial topological charge (see also [105]). According to S1, considering motion in the phase space (u, p) the measure DM(u, p) has the Dirac form







H (u, p) H (u, p) p # H DM(u, p)" du(x, t) dp(x, t) u ! H p u VR with the total Hamiltonian

 



1 1 H (u, p)" dx p# (u)#v(u)!ju . H 2 2



(K.14)

(K.15)

This last one includes the energy ju of quantum #uctuations. The measure (K.14) contains the following information: a. Only strict solutions of the equations H (u, p) H (u, p) u ! H "0, p # H "0 p u

(K.16)

80

b. c. d. e. f. g.

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with j"0 should be taken into account. This &rigidness' of the formalism means the absence of pseudo-solutions (similar to multi-instanton, or multi-kink) contribution.

is described by the sum of all solutions of Eq. (K.16), independent of their &nearness' in the LK functional space;

did not contain the interference terms from various topologically nonequivalent contribuLK tions. This displays the orthogonality of the corresponding Hilbert spaces; The measure (K.14) includes j(x) as the external adiabatic source. Its #uctuation disturbs the solutions of Eq. (K.16) and vice versa since the measure (K.14) is strict; In the frame of the adiabaticity condition, the "eld disturbed by j(x) belongs to the same manifold (topology class) as the classical "eld de"ned by (K.16) [105]. The Dirac measure is derived for real-time processes only, i.e. (K.14) is not valid for tunneling ones. For this reason, the above conclusions should be taken carefully. It can be shown that theory on the measure (K.14) restores ordinary (canonical) perturbation theory.

The parameter (q; u) is connected directly with external particle energy, momentum, spin, polarization, charge, etc., and is sensitive to the symmetry properties of the interacting "elds system. For the sake of simplicity, u(x) is the real scalar "eld. The generalization would be evident. As a consequence of (K.14), (q; u) is the function of the external particle momentum q and is a linear functional of u(x):



(q; u)"! dxe OV



S (u)  " dxe OV(R#m)u(x), q"m , u(x)

(K.17)

for the mass m "eld. This parameter presents the momentum distribution of the interacting "eld u(x) on the remote hypersurface  if u(x) is the regular function. Notice, the operator (R#m)  cancels the mass-shell states of u(x). The construction (K.17) means, because of the Klein}Gordon operator and the external states being mass-shell by de"nition [40], that the solution "0 is possible for a particular topology LK (compactness and analytic properties) of quantum "eld u(x). So, (q; u) carries the following remarkable properties: 䢇 䢇 䢇

it directly de"nes the observables, it is de"ned by the topology of u(x), it is the linear functional of the actions symmetry group element u(x).

Notice, the space-time topology of u(x, t) becomes important in calculating integral (3.2) by parts. This procedure is available if and only if u(x, t) is the regular function. But the quantum "elds are always singular. Therefore, the solution (q; u)"0 is valid if and only if the quasiclassical approximation is exact. Just this situation is realized in the soliton sector of sin-Gordon model.

 The following trivial analogy with ferromagnetic may be useful. So, the external magnetic "eld H&
, if 
is the magnetics order parameter, and the phase transition means that 
O0. (q, u) has just the same meaning as H.

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81

Despite evident ambiguity (q; u) carries the de"nite properties of the order parameter since the opposite solution "0 can only be the dynamical display of an unbroken symmetry, i.e. of the LK nontrivial topology of interacting "elds, as the consequence of unbroken symmetry. If (K.16) have nontrivial solution u (x, t), then this &extended objects' quantization problem [107]  arises. We solve it by introducing convenient dynamical variables [15]. The main formal di$culty, see e.g. [108], of this program consists of transformation of the path-integral measure which was solved in [105]. Then S3. The measure (K.14) admits the transformation u : (u, p)P(, )3="G/G .   and the transformed measure has the form




(K.18)






h (, ) h (, ) # H , (K.19) DM(u, p)" d(t) d(t) Q ! H   VR! where h (, )"H (u , p ) is the transformed Hamiltonian. H H   It is evident that (, ) are parameters of integration of Eqs. (K.16) and they form the factor space ="G/G . For instance, if one particle dynamics is considered, then one may choose "x(0) and  "p(0). One may consider also the following possibility:



"

V

du (2(!v(u))

and "p/2#v(x) . In these terms h "!j(t)u (, ) and new Hamilton equations have the form H  Ru (, ) Ru (, ) Q "1!j  , "j  . R R

(K.20)

So, we have at j"0: "t#t and " . For this reason   S4. The (action, angle)-type variables are most useful.

 The S-matrix was introduced &phenomenologically', see also the example considered in [106,95], postulating the LSZ reduction formulae, see Eq. (B.1). So, the formal constraints, e.g. the Haag theorem, would not be taken into account on the chosen level of accuracy.  A number of problems of quantum mechanics were solved using also the &time sliced' method [109]. This approach presents the path integral as the "nite product of well-de"ned ordinary integrals and, therefore, allows to perform arbitrary space and space-time transformations. But the transformed &e!ective' Lagrangian gains an additional term & . The latter crucially depends on the way in which the &slicing' was performed. This phenomenon considerably complicates calculations and the general solution of this problem is unknown to us. It is evident that this method is especially e!ective if the quantum corrections & play no role. Such models are well known. For instance, the Coulomb model in quantum mechanics, the sine-Gordon model in "eld theory, where the bound-state energies are exactly quasiclassical.

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According to (K.18) there exists transformation of the perturbation generating operator: S5. The operator K has the following transformed form



2K" dtjK ) e( #jK ) e(  , K K E E

(K.21)

in the factor space, where j , e , X", , are new auxiliary variables. 6 6 As a result of mapping of the perturbation generating operator K on the manifold = the equations of motion became linearized:






h() DM" Q ! !j (!j ) . K E  R

(K.22)

Then S6. If Feynman's i-prescription is adopted, then the Green function of Eq. (K.22) g(t!t)"(t!t) .

(K.23)

Later on we will consider the soliton sector of the sin-Gordon model. In this case  is the G coordinate and  is the momentum of the ith soliton and N is the number of solitons. G Expansion of expK( je) gives the &strong coupling' perturbation series. Its analysis shows that S7. Action of the integro-diwerential operator OK leads to the following representation:



R (p, q)" LK

d(0) )



R R RK (p, q)#d(0) ) RE (p, q) . R(0) LK R(0) LK

(K.24) 5 This means that the contributions into R (p, q) are accumulated strictly on the boundary LK &bifurcation manifold' R= [110], i.e. depends directly on the topology of =. K.3. Multiple production in sin-Gordon model Let us consider now the completely integrable sin-Gordon model. For the sake of simplicity the integral



R (q)"e\ )K HC DM(u, p)(q; u)e Q S\ 3SC , 

(K.25)

where (q; u) was de"ned in (3.2), will be calculated. The e!ective potential of the sin-Gordon model



2m ;(u ; e )"! dx dt sin u (sin e!e) ,  , 

(K.26)

Ru Ru e "e )  !e )  . E R  K R

(K.27)

with

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83

Performing the shifts in (K.22):

 

 (t)P (t)# dtg(t!t)j (t), (t)# (t) , G G KG G G  (t)P (t)# dtg(t!t)j (t), (t)# (t) , G G EG G G

(K.28)

we can get the Green function g(t!t) into the operator exponent:



1 K(ej)" dt dt(t!t)K (t) ) e( (t)#( (t) ) e( (t) . K E 2

(K.29)

Note the Lorentz noncovariantness of our perturbation theory with Green function (K.23). As a result , Rh D,M(, )" d (t) d (t) (Q !(#)) ( ), ()" G G G G R G R

(K.30)

with u "u (x; #, #) . , , Using the de"nition





(K.31)



Dx (x )" dx(0)" dx , 

the functional integrals on the measure (K.30) are reduced to the ordinary integrals over initial data (, ) . These integrals de"ne zero modes volume. Notice that the zero-modes measure was de"ned  without the Faddeev}Popov anzats. We divide the calculations into two parts. First of all, we consider the quasiclassical approximation and then we will show that this approximation is exact. This strategy is necessary since it seems to be important to show the role of quantum corrections noting that for all physically acceptable "eld theories R "0 in the quasiclassical approximation. LK The N-soliton solution u depends upon 2N parameters. Half of them, N, can be considered as , the position of solitons and the other N as the solitons momentum. Generally, at tPR the u solution decomposed on the single solitons u and on the double-soliton bound states u [111]: , 
  L L u (x, t)"  u (x, t)#  u (x, t)#O(e\R) . H
I , H I Note that this asymptotic is achieved if  PR or/and  PR. This latter de"nes the bifurcation G G line of our model. So, the one soliton u and two-soliton bound state u would be the main elements 
of our formalism. Its (, ) parametrizations have the form 4  u (x; , )"! arctanexp(mx cosh !), "   8

(K.32)

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and



4  mx sinh  /2 cos  /2!    u (x; , )"! arctan tan 
 2 mx cosh  /2 sin  /2!    Performing the last integration we "nd



, R (q)" d d  e\ )K e 1- S, e\ 3S, _CK CE (q; u )  ,   , G where u "u ( #,  #(t)#) , ,  



.

(K.33)

(K.34)

(K.35)

and



(t)" dt(t!t)( #)(t) . 

(K.36)

In the quasiclassical approximation ""0 we have u "u (x;  ,  #( )t) . , ,    Notice that the surface term

(K.37)



dxIR (e OVu )"0 . I ,

(K.38)

Then





dxe OV(R#m)u (x, t)"!(q!m) dxe OVu (x, t)"0 , , ,

(K.39)

since q belongs to the mass shell by de"nition. The condition (K.38) is satis"ed for all q O0 since I u belongs to Schwarz space (the periodic boundary condition for u(x, t) does not alter this , conclusion). Therefore, in the quasiclassical approximation R "0.  Expanding the operator exponent in (K.34) we "nd that action of operators K , (  creates terms



& dxe OV(t!t)(R#m)u (x, t)O0 . ,

(K.40)

So, generally, if the quantum corrections are included, R O0.  Now we will show that the quasiclassical approximation is exact in the soliton sector of the sin-Gordon model. The structure of the perturbation theory is readily seen in the &normal-product' form



, R (q)" d d  : e\ 3S, _HK Ge Q S, (q; u ) ,   G ,  , G

(K.41)

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85

where Ru Ru Ru , jK "jK ) , !jK ) , "jK K R E R 6 RX

(K.42)

and



jK " dt(t!t)XK (t) 6

(K.43)

with the 2N-dimensional vector X"(, ). In Eq. (K.42)  is the ordinary symplectic matrix. The colons in (K.41) mean that the operator jK should stay to the left of all functions in the perturbation theory expansion over it. The structure (K.42) shows that each order over jK G is 6 proportional at least to the "rst order derivative of u over the variable conjugate to X . , G The expansion of (K.41) over jK can be written [105] in the form (omitting the quasiclassical 6 approximation):







, , R P (u ) , (K.44) R (q)" d d     G  RX 6G , G G , G where P G (u ) is the in"nite sum of &time-ordered' polynomials (see [105]) over u and its 6 , , derivatives. The explicit form of P G (u ) is complicated since the interaction potential is non6 , polynomial. But it is enough to know, see (K.42), that Ru , . P G (u )& GH RX 6 , H Therefore,

(K.45)

R (q)"0 (K.46)  since (i) each term in (K.44) is the total derivative, (ii) we have (K.45) and (iii) u belongs to , Schwarz space.

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89

NEW SUPERCONFORMAL FIELD THEORIES IN FOUR DIMENSIONS AND N"1 DUALITY

M. CHAICHIANa,b, W.F. CHENc, C. MONTONENb a

High Energy Physics Division, Department of Physics, FIN-00014 University of Helsinki, Finland b Helsinki Institute of Physics, P.O. Box 9 (Siltavuorenpenger 20 C), FIN-00014 University of Helsinki, Finland c Winnipeg Institute of Theoretical Physics and Department of Physics, University of Winnipeg, Winnipeg, Manitoba, Canada R3B 2E9

AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO

Physics Reports 346 (2001) 89}341

New superconformal "eld theories in four dimensions and N"1 duality M. Chaichian 
*, W.F. Chen, C. Montonen
 High Energy Physics Division, Department of Physics, FIN-00014 University of Helsinki, Finland
Helsinki Institute of Physics, P.O. Box 9, FIN-00014 University of Helsinki, Finland Winnipeg Institute of Theoretical Physics and Department of Physics, University of Winnipeg, Winnipeg, Manitoba, Canada R3B 2E9 Received June 2000; editor: A. Schwimmer Contents 1. Introduction 2. Some background knowledge 2.1. Four-dimensional conformal algebra and its representation 2.2. Scale symmetry breaking and renormalization group equation 2.3. Chiral symmetry in massless QCD 2.4. Various phases of gauge theories 2.5. N"1 superconformal algebra and its representation 3. Non-perturbative dynamics in N"1 supersymmetric QCD 3.1. Introducing N"1 supersymmetric QCD 3.2. Holomorphy of supersymmetric QCD 3.3. Classical moduli space of supersymmetric QCD 3.4. Quantum moduli space and low-energy non-perturbative dynamics 4. N"1 supersymmetric dual QCD and non-Abelian electric}magnetic duality 4.1. Dual supersymmetric QCD

92 98 98 103 107 113 126 141 142 145 154 159 183 183

4.2. Non-Abelian electric}magnetic duality 4.3. Duality in Kutasov}Schwimmer model 5. Non-perturbative phenomena in N"1 supersymmetric gauge theories with gauge group SO(N ) and Sp(N "2n ) A A A 5.1. N"1 supersymmetric SO(N ) gauge A theory with N #avours D 5.2. Dynamically generated superpotential and decoupling relation 5.3. Non-perturbative dynamical phenomena when N 54, N 4N !2 A D A 5.4. N 54, N 5N !1: dual magnetic A D A SO(N !N #4) description D A 5.5. Electric}magnetic}dyonic triality in supersymmetric SO(3) gauge theory 5.6. Dyonic dual of SO(N ) theory with A N "N !1 #avours D A 5.7. A brief introduction to Sp(N "2n ) gauge A A theory with N "2n quarks D D 6. New features of N"1 four-dimensional superconformal "eld theory and some of its relevant aspects

187 195

207 207 212 215 247 264 281 284

294

* Corresponding author. Tel.: #3580191-8441; fax: #358-191-8366. E-mail address: masud.chaichian@helsinki." (M. Chaichian).  Present address: Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1.  On leave from the Theoretical Physics Division, Department of Physics, University of Helsinki. 0370-1573/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 0 1 - 0

M. Chaichian et al. / Physics Reports 346 (2001) 89}341 6.1. Critical behaviour of anomalous currents and anomaly matching 6.2. Universality of operator product expansion 6.3. Possible existence of a four-dimensional c-theorem 6.4. Non-perturbative central functions and their renormalization group #ows

294 309 314 318

7. Concluding remarks 7.1. A brief history of electric}magnetic duality 7.2. Possible application of non-perturbative results and duality of N"1 supersymmetric QCD to nonsupersymmetric case Acknowledgements References

91 331 331

333 338 338

Abstract Recently, developments in the understanding of low-energy N"1 supersymmetric gauge theory have revealed two important phenomena: the appearance of new four-dimensional superconformal "eld theories and a non-Abelian generalization of electric-magnetic duality at the IR "xed point. This report is a pedagogical introduction to these phenomena. After presenting some necessary background material, a detailed introduction to the low-energy non-perturbative dynamics of N"1 supersymmetric S;(NA ) QCD is given. The emergence of new four-dimensional superconformal "eld theories and non-Abelian duality is explained. New non-perturbative phenomena in supersymmetric SO(NA ) and Sp(NA "2nA ) gauge theories, such as the two inequivalent branches, oblique con"nement and electric-magnetic-dyonic triality, are presented. Finally, some new features of these four-dimensional superconformal "eld theories are exhibited: the universal operator product expansion, evidence for a possible c-theorem in four dimensions and the critical behaviour of anomalous currents. The concluding remarks contain a brief history of electric-magnetic duality and a discussion on the possible applications of duality to ordinary QCD.  2001 Elsevier Science B.V. All rights reserved. PACS: 11.15.!q

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1. Introduction During the past several years there has been a great progress in understanding the nonperturbative dynamics of supersymmetric gauge theories. The breakthrough has been made in two aspects. One is in N"2 supersymmetric gauge theories pioneered by Seiberg and Witten [1,2]. Based on the low energy e!ective action obtained by Seiberg earlier from the non-perturbative
function analysis [3], which has now been veri"ed up to at most two derivatives and not more than four-fermion coupling by various calculation methods [4}6], Seiberg and Witten found that in the Coulomb phase N"2 supersymmetric Yang}Mills theory can admit a self-dual electric}magnetic duality. This kind of duality was originally proposed by Montonen and Olive [7] and previously it was believed that it could only exist in an N"4 supersymmetric Yang}Mills theory [8]. Furthermore, with this duality Seiberg and Witten explicitly showed that the con"nement mechanism is indeed provided by the condensation of magnetic monopoles and justi"ed in supersymmetric gauge theory the original conjecture by 't Hooft and Mandelstam [9,10]. In the presence of N"2 matter multiplets, they also showed that the chiral symmetry breaking is driven by the condensation of magnetic monopoles. Another direction in which progress has been made is N"1 supersymmetric gauge theory, mainly based on ideas by Seiberg. From an analysis of the quantum moduli space, a series of exact results has been obtained in N"1 supersymmetric gauge theory, and an almost complete phase diagram of N"1 supersymmetric gauge theory including the dynamical features and the particle spectrum in each phase has been worked out [11}16]. In particular, Seiberg found that the electric}magnetic duality can also exist in the IR "x point of N"1 supersymmetric gauge theory, where the theory is described by an interacting four-dimensional superconformal "eld theory. This report concentrates on these new four-dimensional superconformal "eld theories and N"1 duality. It is not accidental that so many non-perturbative results can be obtained in supersymmetric gauge theories. Supersymmetric quantum "eld theory is much more tractable than usual quantum "eld theory. One of the most remarkable characteristics of supersymmetric gauge theory is the non-renormalization theorem [17}19], which claims that an interaction vertex of a supersymmetric gauge theory is not spoiled by quantum corrections. If the theory is expressed in super"eld form, this immediately implies that the classical superpotential withstands quantum corrections and thus remains holomorphic. This property imposes a very strict constraint on the form of the interaction vertex at the quantum level. In addition, there is another direct consequence of the non-renormalization theorem. Since the vertex receives no quantum corrections, one can always de"ne the vertex renormalization constant to be one. Thus the renormalization of the coupling constants is only related to the wave function renormalization constants. This means that the beta function of the theory depends only on the anomalous dimensions of the "elds. In supersymmetric gauge theory with gauge group G and N species of matter "eld in the representation ¹ (i"1, 2,2, N ), this fact D G D is quantitatively manifested in the NSVZ (Novikov}Shifman}Vainshtein}Zakharov) beta function [20,21]: g 3¹(G) dim G!,D ¹(R )[1! ] G G G ,
,148(g)"! 1!¹(G)g/(8
) 16


(1.1)

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where  are the anomalous dimensions of the matter "elds, and the group invariants ¹(G) and G ¹(R) are de"ned as follows: Tr (¹?¹@)"¹(R)?@, ¹(G)"¹(R"adjoint representation) . (1.2) 0 The explicit form of this beta function provides a possibility to determine the non-trivial IR "xed points. It should be emphasized that the NSVZ beta function (1.1) is a non-perturbative result and is valid to all orders. Furthermore, recalling that there exists a direct connection between the trace anomaly and the beta function, for example in a pure Yang}Mills theory [22],
(g) F FIJ , I " I 2g IJ

(1.3)

one can see that at "xed points, the trace anomaly vanishes and hence conformal symmetry emerges since I is the measure of quantum conformal symmetry [23]. Therefore, an interacting I superconformal "eld theory can arise at the IR "xed point. The NSVZ beta function says that for an asymptotically free supersymmetric gauge theory, such an IR "xed point must exist. The origin of the non-renormalization theorem lies in the supersymmetry, which ensures the cancellation of quantum #uctuations from the bosonic and fermionic modes, since in a supersymmetric "eld theory the same number of bosonic and fermionic degrees of freedom occur in a supermultiplet, and they contribute to a virtual process with the same amplitude but with opposite sign. The basic building blocks of a quantum "eld theory are the Green's functions. As manifested in the Green functions supersymmetry, just like a usual local internal symmetry, leads to relations between various Green functions, which take the form of Ward}Takahashi (WT) identities. With the assumption that the supersymmetry is not spontaneously broken, these WT identities impose strong constraints on the form of the Green's functions. For example, the chiral supersymmetric WT identities not only determine that the Green's function of the lowest component "eld in a chiral multiplet is space}time independent, but also, together with the internal symmetries, renormalization group invariance and other physical requirements, specify the explicit dependence of the Green functions on the parameters of the theory such as masses and coupling constants [24,25]. This is another important reason why supersymmetric gauge theories are under better control. The algebraic foundation of a supersymmetric "eld theory, superalgebra, is an extension of the usual PoincareH symmetry. It uni"es some of the most fundamental conserved currents such as the energy-momentum tensor, the supercurrent and the axial type R current into a supermultiplet. Consequently, the quantum anomalies of these currents should also "t into a supermultiplet [26,27]. Furthermore, there exists a new type of anomaly, which was originally found in Ref. [28] and independently re-derived by Konishi [29]. This anomaly gives a connection between the squark condensation and the gluino condensation. All these new features provide possible ways to explore the non-perturbative aspects of a supersymmetric gauge theory. The vacuum structure of a supersymmetric gauge theory and the relevant non-perturbative dynamics have a rich physical content. Unbroken supersymmetry requires that there exists at least one ground state with zero energy [30,31]. In a four-dimensional supersymmetric gauge theory there usually exists a continuous degeneracy of inequivalent ground states [32,33]. Classically, these ground states correspond to the #at directions of the scalar potential and thus form a classical moduli space. Along these #at directions, some of the squarks can acquire expectation values which

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break the gauge symmetry. As a consequence, some "elds will acquire masses due to the superHiggs mechanism and the moduli space will be characterized by the light degrees of freedom. At the origin of the classical moduli space the gauge symmetry will be restored fully. Quantum mechanically, a dynamically generated superpotential can arise and dynamical supersymmetry breaking occurs. Consequently, the vacuum degeneracy will be lifted [32,33]. Note that the non-renormalization theorem only refers to the perturbative quantum correction and it does not pose any restriction on the non-perturbative quantum e!ects. The vacuum degeneracy may still persist after inclusion of non-perturbative e!ects, and then the theory has a quantum moduli space of vacua. The holomorphy of the quantum superpotential makes it possible to determine the light degrees of freedom and hence the quantum moduli space [34]. By analyzing the structure of the quantum moduli space, one can get a handle on some of the important non-perturbative features such as con"nement and chiral symmetry breaking since many non-perturbative physical phenomena are related closely to the vacua such as mass generation and fermion condensates, etc. If the dynamically generated superpotential has erased all vacua, then dynamical supersymmetry breaking occurs. Supersymmetry breaking induced by non-perturbative quantum e!ects was proposed by Witten nearly two decades ago [30,31]. Only in recent years have its possible physical applications been considered. This kind of supersymmetry breaking mechanism introduces very few physical parameters and thus has a great advantage over the soft supersymmetry breaking mechanism. At present dynamical supersymmetric breaking is a popular topic in supersymmetry phenomenology [35]. The non-perturbative aspects of an N"1 supersymmetric gauge theory exhibit a rich phase structure, depending heavily on the choice of gauge group and the matter contents [14}16]. The most remarkable non-perturbative dynamical phenomenon is that in a special range of colour number and #avour number, i.e. in the so-called conformal window [14], the theory may have a non-trivial infrared "xed point implied from by NSVZ beta function (1.1), at which the theory becomes a superconformal "eld theory. In particular, it now admits a physically equivalent dual description but with the strong and weak coupling exchanged [14]. The emergence of N"1 superconformal symmetry in the IR region has a great signi"cance. In addition to N"4 supersymmetric Yang}Mills theory and N"2 supersymmetric gauge theory with vanishing beta function, this is a new non-trivial conformal "eld theory in four-dimensional space}time. Especially, the N"1 supersymmetric gauge theory has in general the property of asymptotic freedom, at high energy it can be regarded as a free "eld theory. Thus the existence of the IR "xed point means that an o!-critical quantum "eld theory can be regarded as a radiative interpolation between a pair of four-dimensional conformal "eld theories [36]. A conventional method to observe the physical phenomena at di!erent energy scales is to investigate the #ow of some physical quantity along the renormalization group trajectory from the UV region to the IR region. In two-dimensional quantum "eld theory, there is a famous `c-theorema on the renormalization group #ow proposed by Zamolodchikov [37]. It states that there exists a c-function C(g) of the coupling g associated with the two-point function of the energy-momentum tensor, which decreases monotonically along the renormalization group #ow from the UV region to the IR region and becomes stationary at the "xed point, where the theory is a two-dimensional conformal invariant quantum "eld theory and the c-function coincides exactly with the central charge (the coe$cient of the conformal anomaly). Since the central charge actually counts the number of dynamical degrees of freedom of the theory [38,39], this theorem shows precisely how the

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information about the dynamical degrees of freedom at short distance is lost in the renormalization group #ow to long distance. Therefore, it would be helpful for the study of some non-perturbative dynamical phenomena in four-dimensional quantum "eld theory if a four-dimensional analogue of the c-theorem could be found. For example, one could get insight in a number of non-perturbative phenomena such as con"nement, chiral symmetry breaking, supersymmetry breaking and even the Higgs mechanism. However, unexpectedly the search and veri"cation of a four-dimensional c-theorem turned out to be extremely di$cult. Soon after the proposal of the two-dimensional c-theorem, much e!ort was put into looking for a candidate for a c-function and checking its evolution along the renormalization group #ow was made [40}42], but a general proof of the existence of a monotonically decreasing c-function is still lacking. Recently, some progress has been made in this direction [43]. The availability of a series of exact results in N"1 supersymmetric gauge theory and especially the discovery of superconformal symmetry in the IR "xed point has provided useful tools for testing c-theorems and "nding the right c-function. As in the two-dimensional case, a candidate for a c-function should be related to the central function [44], which is the coe$cient function of the most singular term in the operator product expansion of the energy-momentum tensor and should coincide with the conformal anomaly coe$cients at the "xed points. A qualitative analysis using the exact results give the "rst indications of the possible existence of a four-dimensional c-theorem in N"1 supersymmetric gauge theory [45]. Furthermore, asymptotic freedom implies that various quantities at the UV region can be determined in the framework of a free "eld theory, while the superconformal invariance at the IR "xed point allows an exact calculation on the anomaly coe$cients at the IR region [36]: the trace anomaly of the energy-momentum tensor and the chiral anomaly of R-current lie in the same supermultiplet and the latter can be exactly calculated through the 't Hooft anomaly matching. An explicit calculation suggests that the central function coinciding with the coe$cient of the Euler term of in the trace anomaly at the IR "xed point is the most appropriate candidate for the c-function [36,46]. The N"1 four-dimensional superconformal "eld theory arising at the IR "xed point has also some remarkable features in comparison with the two-dimensional case [47,48]. The operator product expansion of two energy-momentum tensor operators or of an energy-momentum tensor and a conserved current does not close, another current called the Konishi current must be introduced to ensure closure. Consequently, a four-dimensional superconformal "eld theory is characterized by two central charges and a conformal dimension. The central charges are fourdimensional superconformal invariants in the sense that they receive no higher-order (more than two-loop) quantum correction and remain at their one-loop values (i.e. always proportional to the number of dynamical degrees of freedom), while the conformal dimension does not, it can receive quantum corrections from every order [48]. The dual theory arising at the IR "xed point is of signi"cance since it gives an equivalent physical description to the low-energy dynamics of the original theory but in terms of a fundamental Lagrangian of a supersymmetric gauge theory with one (or several) additional superpotential(s). The dynamics of the mesons and baryons in the original theory can be equivalently replaced by the dynamics of dual fundamental dynamical degrees of freedom and one (or several) gauge singlet particle(s). Especially, the gauge couplings of the original and the dual theories are inversely related. This provides a possibility to study the low-energy non-perturbative dynamics of the original theory through the perturbation theory of the dual theory.

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There have already appeared several excellent reviews on N"1 duality with di!erent emphasis such as those by Intriligator and Seiberg [16], Shifman [49], Shifman and Vainshtein [50] and Peskin [51]. The present review mainly emphasizes superconformal "eld theory. It is written a pedagogical manner, giving detailed mathematical derivations and explanations of these new developments and including much of the preliminary knowledge needed to understand the new progress such as the representation of conformal algebra, the renormalization group equation, anomaly matching, phases of gauge theory and the Wilson e!ective action, etc. This report is based on a series of discussions and seminars in the theoretical physics group of the University of Helsinki and Helsinki Institute of Physics. We hope that it may prove helpful for beginners interested in this topic. The organization of this report is as follows: Section 2 contains some background material. We "rst review the de"nition of conformal transformations, the derivation of the conformal algebra and the "eld representation of conformal algebra in terms of Wigner's little group method. In a renormalizable relativistic quantum "eld theory, scale symmetry implies conformal symmetry, the anomalous breaking of scale symmetry means the violation of conformal symmetry at quantum level. We thus introduce scale symmetry breaking in the context of a massless scalar "eld theory and the renormalization group equation, which plays the role of anomalous Ward identity of scale symmetry. Since new developments occur in supersymmetric QCD, the supersymmetric extension of the non-supersymmetric theory, we introduce the chiral symmetry of ordinary QCD and its breaking. We discuss the possible (external) chiral anomaly in QCD and the 't Hooft anomaly matching. We shall see that the 't Hooft anomaly matching is an important tool to investigate electric}magnetic duality. The non-perturbative dynamical structure of a quantum "eld theory is quite rich and the theory can present several phases. The discovery of new non-perturbative phenomena in supersymmetric gauge theory has veri"ed this. Hence it is necessary to introduce the various phase structures and their dynamical behaviour. The N"1 superconformal algebra is the algebraic foundation of constructing a superconformal "eld theory, and thus based on the current supermultiplet and the Jacobi identities, we give the full superconformal algebra. Further, we review the representation of the superconformal algebra in "eld operator space. It is much more complicated than the ordinary conformal algebra: the ordinary conformal algebra can be realized on one type of "elds, whereas the superconformal algebra must be realized on a supermultiplet. In particular, the representations realized on di!erent supermultiplets, such as the chiral and the vector ones, have their own speci"c features. In particular, the representation of the superconformal algebra on the chiral multiplet yields a simple relation between the conformal dimension of the chiral super"eld and its R-charge, which has played a key role in determining Seiberg's conformal window. From Section 3, we begin to introduce the low-energy dynamics of supersymmetric QCD. Supersymmetric gauge theory exhibits some new characteristics compared with the non-supersymmetric case, the most important of which are R-symmetry and holomorphicity. Thus we "rst give a detailed introduction to these two aspects. We explain in detail how to combine the anomalous R-symmetry and the axial vector ; (1) symmetry to get an anomaly-free R-symmetry, and then  introduce the low-energy dynamics of supersymmetric QCD. Since supersymmetric QCD is a theory sensitive to #avour number N and colour number N , we analyze the theory with respect D A to di!erent ranges of N and N . Using the Georgi}Glashow model to illustrate the general D A de"nition of a classical moduli space, we explain how to describe the classical moduli space of

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supersymmetric QCD in the cases of N (N and N 5N , respectively. Especially for N "N D A D A D A and N "N #1 the constraint equations characterizing the classical moduli space are explicitly D A given. A large part of this section is devoted to describing the quantum moduli space and low-energy dynamics of supersymmetric QCD. In the case N (N , we give a detailed derivation of A D the ADS (A%eck}Dine}Seiberg) superpotential and argue the reasonableness of this non-perturbative superpotential by considering its various limits and the physical consequences obtained from this superpotential. When N "N , we show how the quantum corrections have modi"ed the D A classical moduli space and what physical e!ects are produced, and check the reasonableness of these physical pictures using the anomaly matching condition. For N "N #1, we introduce the D A e!ective superpotential to determine the quantum moduli space and the physical e!ects resulting from the quantum moduli space, 't Hooft anomaly matching providing a strong support. For the case N 'N , we show that the NSVZ beta function implies a nontrivial IR "x point in the D A conformal window 3N /2(N (3N . Then we brie#y explain the situation in the range A D A N '3N , where the asymptotic freedom of the theory is lost. D A In Section 4, we concentrate on non-Abelian electric}magnetic duality in the conformal window by "rst introducing the dual description of supersymmetric QCD and showing the reasonableness of this duality conjecture using 't Hooft's anomaly matching. Then we explain what non-Abelian electric}magnetic duality is and how the duality arises in the conformal window and show how the duality remains valid in various limits. The duality in the Kutasov}Schwimmer model is brie#y reviewed since it provides a possibility to study the non-perturbative aspects of supersymmetric extensions of the standard model. This model also gives a deep understanding to the duality of N"1 supersymmetric QCD. Section 5 mainly summarizes the non-perturbative phenomena of N"1 supersymmetric SO(N ) gauge theory. In contrast to the S;(N ) case, the gauge group SO(N ) ! A A is not simply connected and has only real representations. Depending thus heavily on the number of colours and #avours, the theory has richer and more novel non-perturbative dynamical phenomena. When N 4N !5, a dynamical superpotential is generated by gaugino condensaD A tion. In the cases N "N !3 or N !4, the theory has two inequivalent ground states, and some D A A exotic particle states appear. When N "N !2, the theory is in a Coulomb phase and the particle D A spectrum contains magnetic monopoles and dyons, and the oblique con"nement conjectured by 't Hooft occurs naturally. As in the S;(N ) case, in the range N 54, N 5N !1, the theory has A D D A a dual magnetic description, a supersymmetric SO(N !N #4) gauge theory. The most novel D A dynamical phenomenon is that the SO(3) gauge theory exhibits electric}magnetic}dyonic triality. In the one-#avour and two-#avour cases, a quantum symmetry with a non-local action on the matter "eld arises. If the theory has three #avours, it was surprisingly found that the N"1 duality in the SO(3) theory can actually be identi"ed as the electric}magnetic duality of N"4 supersymmetric Yang}Mills theory. Further, the concrete form of the dyonic dual of the SO(N ) gauge theory A with N "N !1 is introduced since its dual magnetic theory is an SO(3) gauge theory, while the D A SO(3) gauge theory has a dual dyonic theory. This means that the SO(N "N #1) theory should A D also admit a dual dyonic theory. Another typical supersymmetric model showing duality, the Sp(N ) gauge theory, is brie#y reviewed. In Section 6 we introduce some of new progress in A exploring four-dimensional superconformal "eld theory and non-Abelian electric}magnetic duality including 't Hooft anomaly matching in the presence of higher quantum corrections, the universality of the operator product expansion in four-dimensional superconformal "eld theory and the explicit evidence supporting a possible four-dimensional c-theorem in supersymmetric

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gauge theory. In the concluding remarks, Section 7, we brie#y recall the history of searching for electric}magnetic duality symmetry in relativistic quantum "eld theory and explore the possible applications of the non-perturbative results of N"1 supersymmetric gauge theory to ordinary QCD by softly breaking the supersymmetry.

2. Some background knowledge 2.1. Four-dimensional conformal algebra and its representation Conformal transformations preserve angles but change the scale. In a #at space}time, they are de"ned by the following transformation of the line element [52], ds" dxI dxJ"f (x) ds"f (x) dx? dy@ , IJ ?@

(2.1.1)

where f (x) is a scalar function and  is the space}time metric. The concrete form of the conformal IJ transformation can be found by considering in"nitesimal transformations xI"xI!I(x) .

(2.1.2)

Eqs. (2.1.1) and (2.1.2) give R  (x)#R  (x)"!( f (x)!1) ,!h(x) I J J I IJ IJ

(2.1.3)

and further, after contracting IJ with (2.1.3), 2 h(x)"! R I(x) , n I

(2.1.4)

2 R  (x)#R  (x)" R ?(x) , IJ I J J I n ?

(2.1.5)

where n is the dimension of space}time. Thus 2 (n!2)R R h(x)"! (n!2)R R RM "0 . I J M I J n

(2.1.6)

For n'2, Eq. (2.1.6) implies that h(x) is at most linear in x and from (2.1.4) that  (x) is at most I quadratic in xI. The general form of  (x) is I I(x)"aI#xI#IJx #2b ) xxI!bIx , J

(2.1.7)

where a and b are constant n-dimensional vectors,  is an in"nitesimal constant and  "! . IJ JI The "nite form of the above in"nitesimal conformal transformations is listed in Table 2.1.1. Counting the number of the generators, the dimension of the conformal group is (n#1)(n#2)/2. It

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Table 2.1.1 Conformal transformations xI"xI!aI xI"e\CxI xI"
I xJ J xI#bIx xI" 1#2b ) x#bx

Translations Scale transformation Lorentz transformation Special conformal transformation

h(x)"0 h(x)"!2 h(x)"0 h(x)"!4b ) x

is isomorphic to the orthogonal group O(n, 2) in Minkowski space. For n"4, it is a 15-dimensional space}time symmetry group. The in"nitesimal version of the conformal transformation clearly shows that the conformal group is a generalization of the PoincareH group. Thus, in addition to P and M , the set of I IJ generators consists of n#1 new ones, D and K , which correspond to scale and special conformal I transformations, respectively. According to the relation between symmetry and conservation law (Noether's theorem), corresponding to each continuous global invariance, there exists a conserved current jI . The space I integral of the time component gives a conserved charge, QI" dx jI , and the conserved charges  yield a representation of the generators of the symmetry group. It is well known that the conserved current for P and M are, respectively, the energy-momentum tensor  and the moment I IJ IJ M of  , IJM IJ M " x ! x . IJM IJ M IM J

(2.1.8)

The generators of the PoincareH group are hence





P " dx  , I I

M " dx M (x, t) . IJ  IJ

(2.1.9)

In a quantum "eld theory with conformal symmetry, the energy-momentum tensor  is IJ traceless, I "0 . I

(2.1.10)

Using this condition of tracelessness, one can construct the conserved currents for the scale and special conformal transformations, d "xJ , I JI

k "2x xM !x , IJ J MI JI

(2.1.11)

and the corresponding generators



D" dx d (x, t), 



K " dx k (x, t) . I I

(2.1.12)

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The full conformal algebra can be worked out in a model independent way, [M , M ]"i( M # M ! M ! M ) , IJ ?@ I@ J? J? I@ I? J@ J@ I? [M , P ]"!i P #i P , IJ M IM J JM I [M , K ]"!i K #i K , IJ M IM J JM I [D, P ]"!iP , I I

[D, K ]"iK , I I

[K , P ]"!2i( D#M ) , I J IJ IJ [K , K ]"[M , D]"[P , P ]"0 . I J IJ I J

(2.1.13)

The four-dimensional conformal group is isomorphic to the pseudo-orthogonal group O(4, 2), whose covering group is S;(2, 2). The conformal algebra (2.1.13) can be brought into a form which exhibits the O(4, 2) (or S;(2, 2)) structure by the identi"cation M "(M , M , M , M ) , ?@ IJ I I  M "(P #K ), I  I I

M "(P !K ), I  I I

M "D . 

(2.1.14)

Then M satisfy the algebraic relation of the group O(4, 2) (or S;(2, 2)) ?@ [M , M ]"!i M #i M #i M !i M , ?@ AB ?A @B ?B @A @A ?B @B ?A

(2.1.15)

where  "diag(#1,!1,!1,!1,#1,!1), a, b, c, d"0,23, 5, 6. Therefore, the representa?@ tions of the conformal algebra can be obtained through those of O(2, 4) (or S;(2, 2)). The conformal group in a quantum "eld theory is realized through unitary operators ¹(g), which are exponential functions of the generators and which transform the "eld operators as ¹(g) (x)¹(g)\"S (g, x) (g\x) . P PQ Q

(2.1.16)

Here r is a generic index labelling the "elds and the matrices S form a representation of the PQ conformal group. The in"nitesimal form of Eq. (2.1.16) involves the commutators of the generators with the "elds, and thus the action of the conformal group on the "elds is determined by these commutation relations. We can deduce the form of these commutators using the method of induced representations. The starting point are the translations, generated by the momentum operators P , I [P ,  (x)]"!iR  (x) , I P I P

(2.1.17)

or, equivalently, e .V (0)e\ .V" (x) . P P

(2.1.18)

From (2.1.16) we see that if g is a transformation belonging to the stability group or little group of x"0, i.e. leaves x"0 invariant, then the commutator of a generator of the little group with a "eld

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operator at x"0 will only involve the "eld at that point. Let S? be the generators of the little group obeying the algebra [S?, S@]"i f ?@ASA .

(2.1.19)

If we now posit the commutation relations [S?,  (0)]"!( ?)  (0) , P PQ Q the matrices ? will form a representation of the little group algebra (2.1.19): [ ?, @]"if ?@A A ,

(2.1.20)

(2.1.21)

as can be easily checked from the Jacobi identities involving two generators S?, S@ and the "eld  (0). Translating relations (2.1.20) to a general point x, P [e .VS?e\ .V,  (x)]"!( ?)  (x) , (2.1.22) P PQ Q and evaluating  iL exp[ixIP ]S? exp[!ixIP ]"  xI 2xIL [P  , [2, [P L , S?]2]]] , (2.1.23) I I I I n! L the equations for the commutators [S?,  (x)] are obtained, i.e. a representation of the conformal P algebra on the "eld operators is induced. From Table 2.1.1 we see that for the conformal group, the generators of the little group are M , IJ D and K . Thus we can write I [M ,  (0)]"!( )  (0) , IJ P IJ PQ Q [D,  (0)]"!i  (0) , P PQ Q [K ,  (0)]"!( )  (0) , (2.1.24) I P I PQ Q and the matrices  ,  and obey the same algebra as the corresponding generators, viz. IJ I [ , ]"[,  ]"0 , (2.1.25) I J IJ [, ]"i , (2.1.26) I I [ ,  ]"i( ! ) , (2.1.27) M IJ MI J MJ I [ ,  ]"i(  !  !  #  ) . (2.1.28) MN IJ NI MJ MI NJ NJ MI MJ NI Note that if the  , which according to (2.1.28) form a representation of the Lorentz algebra, IJ generate an irreducible representation, Schur's lemma implies, by virtue of (2.1.25), that the matrix  is proportional to the unit matrix:  "id . (2.1.29) PQ PQ Here d is called the scale or conformal dimension of the "eld. In this case it follows from (2.1.26) that "0. I

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Relations (2.1.24) can now be translated to a general x. We need to evaluate e .VM e\ .V"M !x P #x P , IJ IJ I J J I e .VDe\ .V"D!x ) P , e .VK e\ .V"K !2x D#2xJM #2x (x ) P)!xP . I I I JI I I

(2.1.30)

Inserting (2.1.30) into (2.1.22) and using (2.1.17), we "nally get [M ,  (x)]"!i(x R !x R ) (x)!()  (x) , IJ P I J J I P PQ Q [D,  (x)]"!i(x ) R) (x)!  (x) , P P PQ Q [K ,  (x)]"i(xR !2x (x ) R)) (x)!2x   (x) I P I I P I PQ Q # 2xJ( )  (x)!( )  (x) . IJ PQ Q I PQ Q

(2.1.31)

Together with (2.1.17), these give the action of the conformal group on the "elds. Finally, let us brie#y mention the "nite dimensional representations of the conformal algebra on the state space of a quantum "eld theory. Usually, to "nd a representation of a Lie algebra, one should "rst "nd the lowest (or highest) weight state, then use the raising (or lowering) operators to construct the whole irreducible representation. Each irreducible representation is labelled by the quantum numbers associated with the eigenvalues of the Casimir operators. For the conformal group, the "nite dimensional irreducible representations of its subgroup, the Lorentz group, are labelled by the angular momentum quantum numbers ( j , j ), since the Lorentz group is locally   isomorphic to S;(2);S;(2). The lowest weight states are (!j , !j ). Since P and M consti  I IJ tute the PoincareH algebra, the quantum numbers classifying its representations, the particle mass m and spin s for m'0 or the helicity  for m"0, also play a role in characterizing the representations. In particular, the commutation relations [D, K ]"iK and [D, P ]"!iP I I I I imply that the conformal dimension d, the eigenvalue associated with the scale transformation generator D, is another (magnetic) quantum number labelling the representations of the conformal algebra and P , K are the raising and lowering operators, respectively. This can be easily seen as I I follows. Assume that (0) is an eigenstate of !iD, !iD(0)"d(0) .

(2.1.32)

Then DK (0)"([D, K ]#K D)(0)"i(d#1)K (0) , I I I I DP (0)"([D, P ]#P D)(0)"i(d!1)P (0) . I I I I

(2.1.33)

Thus, given a conformal dimension d, the lowest weight is "(d,!j ,!j ) .  

(2.1.34)

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Table 2.1.2 Unitary representation of conformal algebra Weight (magnetic) quantum numbers (d, j , j )  

PoincareH quantum numbers (m, s) or (m, )

d"j "j "0   j O0, j O0, d'j #j #2     j j "0, d'j #j #1    j O0, j O0, d"j #j #2     j j "0, d"j #j #1   

Trivial 1-dimensional representation m'0, s" j !j  ,2, j #j (integer steps)     m'0, s"j #j   m'0, s"j #j ;   m"0, "j !j .  

It was found that there are only "ve classes of unitary irreducible representations of the fourdimensional conformal algebra. They are listed in Table 2.1.2. They di!er in their PoincareH content (m, s) or (m, ) [54]. A state in the Hilbert space is generated by the action of an operator on the vacuum, O"O0 .

(2.1.35)

Without spontaneous (conformal) symmetry breaking, the quantum states and the quantum operators are in one-to-one correspondence. An operator generating a quantum state with the lowest weight (d,!j ,!j ) is called a primary "eld. An operator with conformal dimension d#n   is called an nth-stage descendant "eld. Although the conformal symmetry must be broken in physics, we see that these unitary representations of the conformal algebra have imposed highly non-trivial constraints on the conformal dimensions of the "elds. 2.2. Scale symmetry breaking and renormalization group equation The scale symmetry is at the heart of conformal symmetry. In fact, in a renormalizable relativistic quantum "eld theory, scale invariance implies conformal symmetry. However, scale symmetry cannot be an exact symmetry in the nature, since in a "eld theory with exact scale symmetry the mass spectrum must be continuous or massless. To break scale invariance, three roads are open to us. The "rst would be to break the symmetry explicitly by introducing terms containing dimensional parameters into the Lagrangian. We shall not consider this case. A second way is spontaneous breaking, D0O0. In this case, Goldstone's theorem implies that there will be a corresponding massless Goldstone boson, which is awkward from a phenomenological point of view. There remains the third alternative, anomalous symmetry breaking. This means that although the classical theory is symmetric, there is no quantization scheme that would respect the symmetry. In a quantum "eld theory, renormalization introduces a scale into the theory and scale symmetry is broken. To discuss the anomalous breaking of scale symmetry, we shall "rst derive the naive Ward identity corresponding to scale transformation and then compare it with the renormalization

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group equation. The renormalization group equation can in fact be thought of as a kind of anomalous scaling Ward identity since it re#ects the scale dependence of a physical amplitude. We take the massless scalar "eld theory 1 1 L" R RI!  , 4! 2 I

(2.2.1)

as an example. Consider a general Green's function with the dilatation current, 0¹[d (y)(x )2(x )]0 . I  L

(2.2.2)

Taking the derivative with respect to yI gives R 0¹[dI(y)(x )2(x )]0"0¹[R dI(y)(x )2(x )]0  L I  L RyI L #  (x!y)0¹[(x )2[d(y), (x )]2(x )]0 . G  G L G

(2.2.3)

Integrating over y, using the scale transformations i[D, (x)]"(x)"(d #xIR ) , ( I

(2.2.4)

the classical relation I "RId "0 and discarding the surface term, we get the Ward identity I I



L dy 0¹[I (y)(x )2(x )]0"i  0¹[(x )2(x )2(x )]0 I  L  G L G

 






 

L R (x )2(x ) 0 "i  0 ¹ (x )2 d #xI  ( G RxI G L G G






R R "i nd #xI #2#xI 0¹[(x )2(x )]0"0 . (  RxI L RxI  L  L

(2.2.5)

In momentum space, this gives






L\ R ! p ) #D GL(p ,2, p )"0 , G Rp  L\ G G

(2.2.6)

where D,nd !4n#4 is just the canonical dimension of the Fourier transform of the Green ( function 0¹[(x )2(x )]0. Parameterizing the momenta p "eRp with p being certain  L G G G "xed momenta, and considering the corresponding 1PI Green's function L(eRp), we have G






R ! #D L(eRp)"0 , G Rt

(2.2.7)

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and hence L(eRp)"e"RL(p) . G G

(2.2.8)

This means that the Green function has canonical scaling dimension. However, this is not correct, since the naive scaling Ward identity ignores quantum e!ects. As a consequence of renormalization, anomalous dimensions have to be added to the canonical ones. Renormalization is a necessary procedure to deal with UV divergences. Its basic idea is to absorb the divergences into a rede"nition of the parameters and a rescaling of the "elds. To extract the UV divergences, one should impose renormalization conditions on the renormalized Green functions. Under di!erent renormalization prescriptions, the renormalized Green functions can di!er by a "nite quantity. Physical results should, however, be independent of the renormalization prescription. Since the renormalized parameters depend on the renormalization prescription, a change in the prescription is compensated by simultaneous changes of the renormalized parameters of the theory. Hence the physical amplitude can remain invariant and this is described by renormalization group equations (RGE). For massive  theory, the renormalization group equation for the 1PI part of the renormalized Green function GL(p ,2, p ) is  L\







R R R #
( ) # ( )m !n( ) L(p ,  , m , )"0 , 0 R K 0 0 Rm 0 0 G 0 0 R 0 0

(2.2.9)

where
( ), ( ) and  ( ) are the
-function of the scalar self-coupling, the anomalous 0 0 K 0 dimensions of wave function and mass renormalization. However, this equation is of little practical use since it only describes the dependence on . It allows us, however, to derive an equation describing the behaviour of Green functions under a variation of the external momenta. We rescale the momenta, p ,p, i"1, 2,2, n , G G

(2.2.10)

with p being certain "xed momenta, and get G








R R R # #m !D L(p, m , ,  )"0 . 0 0 G 0 0 R R Rm 0

(2.2.11)

The combination of (2.2.11) and (2.2.9) gives the RGE we prefer,







R R R !
( ) #(1! ( ))m #n( )!D L(p, m , ,  )"0 . 0 K 0 0 0 0 G 0 0 R R Rm 0 0

(2.2.12)

The solution to the RGE (2.2.12) can be worked out by de"ning the running functions () and m() with the boundary condition, (1)" , 0

m(1)"m , 0

(2.2.13)

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and the running
-functions and the anomalous dimension  for the mass renormalization, K
(())"

d () , d

d [ (())!1]m()" m() . K d

(2.2.14)

With the replacement of the renormalized parameters m and  by the corresponding running 0 0 coupling, the RGE (2.2.12) is converted into an integrable di!erential equation,





d #n[()]!D L(p, (), m(), )"0 . 0 G d

(2.2.15)

The solution can be easily written out L(p ,  , m , )"L(p,  , m , ) 0 G 0 0 0 G 0 0

   

 

"" exp !n

HM () d L(p, (), m(), ) 0 G
() 0 H

"" exp !n

M d L(p, (), m(), ) . (()) 0 G  

(2.2.16)

Solution (2.2.16) shows why  is called the anomalous dimension. Comparing with the naive scale Ward identity (2.2.8), one can see that the scale symmetry is broken by renormalization e!ects. Only when in a massless theory the
-function and the anomalous dimensions vanish, i.e. the theory is "nite, can the Green's function have canonical scaling behaviour
""0,








R !D L(p, ,  )"0 , 0 G 0 R

L(p, ,  )"" L(p, ,  ) . 0 G 0 0 G 0

(2.2.17)

A main application of the RGE is to discuss the large or small momentum behaviour of quantum "eld theory, giving information about the physics at di!erent energy scales. According to the de"nition p ,p, the case PR is called the UV limit and P0 is called the IR limit. I I We assume Eq. (2.2.14)



ln "

HM d
() H0

(2.2.18)

is valid in the whole range 0((R. Otherwise, the renormalization scale  could not vary arbitrarily and the theory would need a cut-o!. Eq. (2.2.18) is divergent when PR or P0.

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If an integral over a "nite interval is divergent, the integrand must be singular at either endpoint (or both). Then one must have
()P0 when PR or P0. Thus we have lim ()" ,
( )"0 , (2.2.19) D D M   that is, () must approach a zero of the
function. The zeroes of
are called "xed points. If
( )(0 with
()"d
/d, and lim ()" ,  is called an UV stable "xed point. If D M D D
( )'0 and lim ()" ,  is called an IR stable "xed point. D M D D For theories with only one coupling constant,  "0 must be one of the "xed points. If  "0 is D D a UV stable "xed point, the theory is called asymptotically free. If  "0 is a IR stable "xed point, D the theory is said to be IR stable. For example, QCD is asymptotically free, while  theory and QED are IR stable. One can now see that some theories have asymptotic scale invariance at high energy. From the solution to the RGE of  theory,

 

L(eRp, , m, )"e" RL[p, (t), m(t), ] exp !n G 0 G

R



(2.2.20) ((t)) dt .  Near the UV "xed point  , with the de"nition ,eR, we can approximately write D R 1 R exp !n ((t)) dt "\LAHD >CR, (t)"! dt[((t))!( )] . (2.2.21) D t   If JO(1/t) as tPR, the theory is called asymptotically scale invariant. The asymptotic scale behaviour of L(eRp, , ) can be obtained by expanding (t) around  . The leading term is 0 G D  L(eRp, , m, )&" \LAHD L(p, , m, ) . (2.2.22) 0 G 0 G Rigorously speaking, only ( ) is called the anomalous dimension of the "eld. D

 





2.3. Chiral symmetry in massless QCD 2.3.1. Global symmetries of massless QCD The Lagrangian of massless QCD with N #avours and N colours (colour gauge group D A G"S;(N )) reads as follows: A 1 D "R  !igA? ¹? , (2.3.1) L"   M iI D   ! F? FIJ?, IPQ I PQ I PQ ?PG ?@ IPQ GH @QH 4 IJ ?@ PQ GH where for clarity we explicitly write the various indices; ,
"1,2, 4 are the spinor indices, a"1,2, dim G"N!1 are group indices, i, j"1,2, N are the #avour indices and A D r, s"1,2, N are the colour indices. The Lagrangian (2.3.1) has an explicit S; (N );; (1) A 4 D #avour symmetry, Pe ?R, Pe ?,

M PM e\ ?R , M Pe\ ?M ,

(2.3.2)

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where A"1,2, N !1 are the S; (N ) #avour group indices and t are N ;N matrices. The D 4 D D D Noether theorem gives the conserved vector current, baryon number current and the corresponding charges: j"M  t, I I

j "M   , I I







Q " dx j" dx Rt,  4



Q " dx j " dx R . 

(2.3.3)

Since  , "0 and there is no quark mass term, Lagrangian (2.3.1) possesses yet other global  I #avour symmetry S; (N );; (1):  D  Pe ?RA , Pe ?A ,

M PM e ?RA , M PM e ?A .

(2.3.4)

The corresponding conserved axial vector current, ;(1) axial current and charges are as follows: j"M   t, I I 



Q" dx R t,  

j"M    , I I 



Q " dx R  .  

(2.3.5)

The vector and axial vector conserved charges form representations of the Lie algebra of the S;(N ) group D [Q, Q ]"[Q, Q ]"i f  !Q!,  

[Q, Q ]"i f  !Q! ,  

(2.3.6)

where f ! are the structure constants of the Lie algebra of S;(N ). The Lagrangian (2.3.1) D explicitly exhibits the chiral symmetry S; (N );S; (N );; (1);; (1) * D 0 D * 0

(2.3.7)

if it is rewritten by means of chiral spinors, 1 L"M iID  #M iID  ! F? FIJ? , * I * 0 I 0 4 IJ

(2.3.8)

where the left- and right-handed chiral spinors are associated with the Dirac spinor as follows:  ,(1! ),  ,(1# ) . *   0  

(2.3.9)

The chiral transformation under (2.3.7) are:  "e ?R , *0 *0  "e ? , *0 *0

M  "M e\ ?R , *0 *0 M  "M e\ ? . *0 *0

(2.3.10)

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and the corresponding Noether currents and charges are, respectively, j "M  t , *0I *0 I *0

 

j

*0I

"M

  , *0 I *0

 

Q " dx j " dx R t , *0 *0 *0 *0 " dx R  . Q " dx j *0 *0 *0 *0

(2.3.11)

These conserved charges give the representations of the generators of symmetry groups (2.3.7), for example, [Q , Q ]"i f  !Q! , *0 *0 *0

[Q, Q ]"0 , * 0

(2.3.12)

and their relations with the vector and axial vector conserved charge are Q"(Q#Q), 0   Q "(Q#Q ), 0  

Q"(Q!Q) ; *   Q "(Q!Q ) . *  

(2.3.13)

Note that there are dynamical vector conserved currents, which correspond to global gauge transformation in colour space, J? "M  ¹?, I I



QI ?" dx J? , 

[QI ?, QI @]"iC?@AQI A ,

(2.3.14)

where ¹? is the N ;N matrix representation of the generators of the S;(N ) colour gauge group A A A and C?@A are the structure constants of S;(N ). A 2.3.2. Chiral symmetry breaking Although Lagrangian (2.3.1) possesses a large global #avour symmetry S; (N ); * D S; (N );; (1);; (1), the observed symmetry is only S; (N );; (1). This means that a chiral 0 D  4 D symmetry breakdown must occur: S; (N );S; (N )PS; (N ), and that the ; (1) symmetry * D 0 D 4 D  also breaks. Unfortunately, up to now the mechanisms for both breakdowns have not been understood completely. In fact, the observed hadron spectrum tells us that chiral symmetry should be broken. Otherwise there would naturally exist parity degenerate states, whereas this is not a feature of the hadron spectrum. The argument runs brie#y as follows. First, according to Coleman's theorem, if there is no spontaneous breakdown of symmetry, the symmetry of the vacuum should be that of the world, i.e. if Q0"0, then [H, Q]"0, H being the Hamiltonian of the theory. Then consider a state , which is an eigenstate of both the Hamiltonian H and the parity operator P, H"E,

P" .

(2.3.15)

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If chiral symmetry were not spontaneously broken, we would have HQ "EQ , * *

HQ "EQ  , 0 0

PQ "PQ PRP"Q  . *0 *0 0*

(2.3.16)

De"ning the state 1 (Q !Q ) , " * (2 0

(2.3.17)

we obtain H"E, P"! .

(2.3.18)

Therefore, we come to the conclusion that  and  describe parity degenerate particles, hence the assumption Q 0"Q 0"0 is not correct. The breaking pattern should be 0 * Q? 0O0, 

Q?0"0 ,

(2.3.19)

i.e. the axial symmetry is spontaneously broken. Although the mechanism of the spontaneous breaking of chiral symmetry is not yet understood, the consequences of the breaking pattern (2.3.19) agrees with experimental observations. The "rst consequence of the spontaneous chiral symmetry breaking which agrees with experimental data is the Goldberg}Trieman relation [55,57]. The second piece of evidence is that some pseudoscalar mesons can be explained as the Goldstone bosons of the spontaneous breakdown of the chiral symmetry. For example, if we consider the spontaneous breaking of S; (2);S; (2) in QCD, the triplet of pions has the right * 0 quantum numbers and are candidates for Goldstone bosons. A concrete phenomenological model which manifests chiral symmetry breaking is the model describing the interactions between nucleons and mesons [55,57]. Usually, a spontaneous symmetry breaking is described by a non-vanishing vacuum expectation value. However, unlike in electroweak theory, there is no scalar "eld in QCD. The characteristic of chiral symmetry breaking is the appearance of a non-vanishing quark condensate 0M 0O0 ,

(2.3.20)

or equivalently, 0M  0O0, * 0

0M  0O0 . 0 *

(2.3.21)

Roughly speaking, the dynamical reason for this condensation could be that the coupling constant of QCD becomes strong at low energy. Thus it is possible that in the ground state of QCD there is an inde"nite number of massless fermion pairs which can be created and annihilated due to the strong coupling. These condensed fermion pairs have zero total momentum and angular momentum and hence make the ground state Lorentz invariant. Therefore, the QCD vacuum has the property that the operators which annihilate or create such fermion pairs can have non-zero

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vacuum expectation values (2.3.20) or (2.3.21). This can be regarded as a qualitative explanation of the chiral symmetry breaking in QCD. 2.3.3. Anomaly in QCD QCD is a vector gauge theory. The couplings of left-handed and right-handed fermions with gauge "eld are parity-symmetric, thus no dynamical chiral anomaly arises. This is unlike a chiral gauge theory such as the electroweak model, where the left-handed and right-handed fermions can be either in di!erent representations of the gauge group or coupled to di!erent gauge groups, and the axial vector currents or the chiral currents are dynamical currents. At the quantum level, if we require that the vector current is conserved, the conservation of the axial vector currents is violated and this is re#ected in the violation of a Ward identity. One typical amplitude is given by the triangle diagram consisting of one axial vector current and two vector currents or three axial vector currents. An anomaly will make the quantum chiral gauge theory non-renormalizable. Thus one must choose appropriate fermion species to make the anomaly cancel, otherwise we have no way to quantize a chiral gauge theory. In QCD, there are two kinds of axial vector #avour currents, the non-singlet one, j, and the I singlet one, j . The singlet axial vector current usually becomes anomalous, while the non-singlet I one remains conserved. However, if the quarks participate in other interactions, the non-singlet axial vector current may become anomalous. Like the dynamical chiral anomaly, these nondynamical anomalies still re#ect the violation of Ward identities. We consider the triangle diagrams  jJ? J@  and  jJ@ JA . At the classical level all the currents are conserved I J M I J M RIj"RIj"RIJ? "0 . I I I In terms of the Fourier transforms of the triangle diagrams:

 

?@ (p, q, r)(2
)(p#q#r)" dx dy dz e PV>NW>OX j(x)J? (y)J@ (z) , I J M IJM @A (p, q, r)(2
)(p#q#r)" dx dy dz e PV>NW>OX j(x)J@ (y)JA (z) , I J M IJM

(2.3.22)

the namK ve Ward identities read as follows: (p#q)I?@ (p, q, r)"pJ?@ (p, q, r)"qM?@ (p, q, r)"0 , IJM IJM IJM (p#q)I@A (p, q, r)"pJ@A (p, q, r)"qM@A (p, q, r)"0 . IJM IJM IJM

(2.3.23)

Usually the gauge symmetry is required to be preserved, pJ?@ (p, q, r)"qM?@ (p, q, r)"0 , IJM IJM pJ@A (p, q, r)"qM@A (p, q, r)"0 . IJM IJM

(2.3.24)

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Eqs. (2.3.24) are actually the renormalization conditions for evaluating the triangle diagrams. With these (physical) renormalization conditions, explicit calculations yield i  p?q@ Tr(¹?¹@) (p#q)I?@ (p, q, r)" IJM 2
 JM?@ i  p?q@ Tr(t ¹@, ¹A)"0 , (p#q)I@A (p, q, r)" IJM 2
 JM?@

(2.3.25)

where the trace is taken over both colour and #avour indices. The corresponding operator equations in coordinate space are g IJ?@F? F@ Tr(¹?¹@), RIj"! IJ ?@ I 16


RIj"0 . I

(2.3.26)

Therefore, only the singlet axial current j is anomalous. However, if the quarks interact electroI magnetically, an anomaly for the non-singlet current may exist. As an example, consider the case of two #avours (N "2), D u . (2.3.27) " d




Correspondingly, t?" ?/2 and the ¹'s are replaced by the electric charge matrix Q




Q"

2/3



e , RIj"! IJ?@F F Tr(Q ) . I IJ ?@ 16
 !1/3

(2.3.28)

If we choose A"3, we obtain the operator anomaly equation, e IJ?@F F . RIj"! IJ ?@ I 32


(2.3.29)

It is well known that this anomaly contributes to the decay
P2. 2.3.4. Anomaly matching Anomaly matching [58] is a basic tool in testing non-Abelian duality conjectures of N"1 supersymmetric QCD. Roughly speaking, the matching condition means that in a con"ning theory like QCD, the anomaly equations should survive con"nement [58}60]. Concretely, let us consider S;(N ) QCD with N #avours. The fundamental building blocks are quarks represented by A D  with i and r being #avour and colour indices, respectively. The general form of the conserved GP #avour singlet current is j "M  [A (1! )#B (1# )]  , (2.3.30) I GP I GH  GH  PQ HQ where A and B are Hermitian matrices in #avour space. This singlet #avour current will su!er from a chiral anomaly. Since the observed particles in QCD are colourless bound states of

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quarks } mesons and baryons } we should consider the matrix elements of the above current between these particle states. Let u, p,  denote a massless baryon state, where u is a solution of the massless Dirac equation, p is the four-momentum of the particle and  labels the other quantum numbers of the baryon. The matrix elements of the current operator between these hadron states are u, p,  j u, p,
"u  [C (1! )#D (1# )]u . (2.3.31) I I ?@  ?@  If the symmetry associated with j does not su!er spontaneous breakdown, then the following I relation should hold: Tr(C!D)"N Tr(A!B) . (2.3.32) A This is the matching condition suggested by 't Hooft. Obviously, the matching condition is connected with the current triangle anomaly. Recalling the Fourier transformation of the triangle Green function,  (p, q, r)(2
)(p#q#r) IJM



, dx dy dz e NV>OW>PX0¹[J (x)J (y)j (z)]0 , I J M

(2.3.33)

J being the gauge symmetry current, we obtain the anomalous Ward identity, N 1 rM (p, q, r)" A Tr(A!B) p?q@" Tr(C!D) p?q@ . IJM IJ?@ IJ?@ 2
 2


(2.3.34)

It is well known that this anomaly equation is true to all orders of perturbation theory and it even survives non-perturbative e!ects such as instanton correction [56,60]. Although  can receive IJM contributions from every order of perturbation theory, the anomaly equation remains the same as given by the zeroth-order triangle diagram. Therefore, the anomaly matching can be formulated in a stricter way: If one treats the massless baryons as if they were fundamental spin 1/2 particles, i.e. quarks, and ignores all other particles, one still gets the correct anomaly. This is the reason why (2.3.32) is satis"ed. In fact, this matching condition reveals the deep origin of the anomaly [61]: the anomaly is connected with the IR singularity of the amplitude, which gets contributions only from the massless spin 1/2 particles. 2.4. Various phases of gauge theories The possible phases of a "eld theory model are associated with symmetries of the theory. Phase transitions are usually associated with changes of the symmetry. In di!erent phases, the particle spectrum and the dynamics of the theory can be greatly di!erent. The quantity characterizing the phase is the order parameter. In a gauge theory the phases are classi"ed by the symmetry that is realized in the phase. The order parameter should be gauge and Lorentz invariant. There are several ways to de"ne an order parameter. The most common choice is the Wilson loop, which was proposed by Wilson in 1974

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[62]. The de"nition of the Wilson loop can be illustrated by a simple example of an Abelian gauge "eld: Assume that two charges $e are created at some point in the Euclidean plane (x, ), then separated to a distance R and kept static. Finally they come together and annihilate at another point after some Euclidean time ¹. The world lines of these two charges will form a contour. The interaction of these two external charges with the gauge "eld is





S "e dx jIA "e A dxI .  I I

(2.4.1)

The current density for two point charges moving on the perimeter of a loop is jI"e

dxI (x!x(s)), 0(s42
, ds

(2.4.2)

with s being the parameter describing the contour. In the path integral formalism, this source will introduce into the vacuum functional integral an extra factor, the Wilson loop





= ,exp[iS ]"exp ie A dxI .   I

(2.4.3)

The generating functional with this point electric charge interaction is just the `quantum Wilson loopa, Z"=  . 

(2.4.4)

In the non-relativistic limit, the quantum Wilson loop is associated with the static potential of two particles with opposite charges. To see this, we calculate the quantum Wilson loop over a rectangle with width R and length ¹. Choosing the gauge condition A (x, t)"0, we then have [63]  = ,= (R, ¹)"(0)R(¹) ,  

(2.4.5)

where



(0)"P exp ie

0



ds



dx ) A(x, 0) , ds



(¹)"P exp ie



0 dx ds ) A(x, ¹) , ds 

(2.4.6)

P denoting the path ordering. Performing the sum over the intermediate states in Eq. (2.4.5) and using the translation invariance (in Euclidean space) (¹)"e\&2(0)e&2 ,

(2.4.7)

we get = (R, ¹)" (0)nnR(¹)" (0)ne\#L 2 ,  L L

(2.4.8)

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Table 2.3.1 Various phases in gauge theory ( is the string tension and
is the renormalization scale) Phase

Static potential <(R) at large R

Coulomb Free electric Free magnetic Higgs Con"ning

1/R 1/[R ln(R
)] ln(R
)/R Constant

R

where H is the Hamiltonian and E  is the energy spectrum of the system. In the limit ¹PR, L only the ground state with the lowest energy E contributes to = (R, ¹),   2 (2.4.9) = (R, ¹) P e\# 02 .  Wilson's prescription for obtaining the static potential between the pair of charges as a function of their distance is the following:





1 (2.4.10) <(R), lim ! ln=(R, ¹) . ¹ 2 Thus, the phase can be characterized by the static potential de"ned in this way. Since the vacuum expectation value of the Wilson loop operator is determined by the full quantum theory, the general form of the static potential at large R should be <(R)&(R)/R .

(2.4.11)

The generalization of the above discussion to the non-Abelian case is straightforward. The two test charges should be in conjugate representations r and r of the gauge group and the Wilson loop operator is the trace of the holonomy operator in the representation r, = "Tr P exp  P





A dxI . I

(2.4.12)

In Eq. (2.4.11),  is classically a constant, "g/(4
), with g being the gauge coupling constant, but the quantum corrections make  be a function of R, since  runs due to renormalization e!ects. Depending on the functional form of (R) at large R (up to a non-universal additive constant), the phases are classi"ed as listed in Table 2.3.1. In the following we shall give a detailed explanation of each phase and mention the known "eld theories possessing such a phase. 2.4.1. Coulomb phase This dynamical regime has long-distance interaction behaviour. One typical theory presenting this phase is massive QED, where the running of the coupling constant is given by the Landau

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formula, which in momentum space reads:   , (2.4.13)  (q)"  1! /(3
)ln(q/m)  where m is the mass of the charged particle. Thus  decreases logarithmically at large distances. However, it stops running at R&m\, the corresponding limiting value of  being H " (R&m\)   and the static potential at large R being H 1 <(R)&  J . R R

(2.4.14)

(2.4.15)

The Coulomb phase also appears in a non-Abelian theory with massless interacting quarks and gluons, and is then called the non-Abelian Coulomb phase. The Coulomb potential can emerge at a non-trivial infrared "xed point of the renormalization group. Thus, such a theory is a non-trivial interacting four-dimensional conformal "eld theory. One known "eld theory having this feature is supersymmetric QCD with an appropriate choice of #avours and colours. We shall give a detailed discussion of this theory later. 2.4.2. Free electric phase This dynamical feature is also familiar. It occurs in massless QED. From the running of the coupling constant in momentum space,   , (2.4.16)  (q)"  1! /(3
)ln(q/
)  one can see that the electric charge is renormalized to zero at large distance and thus there will be a factor ln(R
) in the static potential. An intuitive physical reason is that a strong screening occurs due to quantum e!ects, and the photon propagator is dressed by virtual pairs of electrons. This dressing makes the running coupling constant behave as follows at large R: 1 (R)& . ln(R
)

(2.4.17)

Note that this is greatly di!erent from the massive case, where the running of the e!ective charge is frozen at the distance R"m\. In the massless case, the logarithmic fallo! continues inde"nitely. Thus the asymptotic limit of massless QED is a free photon plus massless electrons whose charge is completely screened. This is why this phase is called a free phase. Strictly speaking, the model with this phase is ill-de"ned at short distances, since the e!ective coupling grows continuously and "nally hits the Landau pole. Usually, this kind of theory must be embedded into an asymptotically free theory. A free electric phase also occurs in the IR region of a non-Abelian gauge theory which is not asymptotically free, and it is then called a non-Abelian free electric phase. Ordinary QCD with N '16 can have this phase. D

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2.4.3. Free magnetic phase The free magnetic phase owes its existence to the occurrence of magnetic monopole states. Assume that a magnetic monopole behaves like a point particle and participates in interactions like an electron. Analogous to the free electric phase, the free magnetic phase occurs when there are massless magnetic monopoles, which renormalize the magnetic coupling to zero at large distance, g(R) 1  (R)" & .  4
ln(R
)

(2.4.18)

Due to the Dirac condition e(R)g(R)&1, the electric coupling constant is correspondingly renormalized to in"nity at large distance,  (R)&ln(R
) . 

(2.4.19)

There also exists a non-Abelian free magnetic phase. The known examples are N"2 S;(2) Supersymmetric Yang}Mills theory at the massless monopole points [1,2] and the dual magnetic theory of N"1 supersymmetric S;(N ) gauge theory with N #avours when N #24 A D A N 43/2N [16]. D A 2.4.4. Higgs phase In a Higgs phase, the gauge group G is spontaneously broken to a subgroup H by a scalar "eld or by the condensation of a fermionic "eld. The gauge bosons corresponding to the broken generators will become massive due to the Higgs mechanism. One typical model is the Georgi}Glashow model, which describes the interaction of an S;(2) gauge "eld with the scalar "eld in the adjoint representation of the gauge group, 1  1 L"! G? GIJ?# (D )?(DI)?! (??!v) . 2 I 4 4 IJ

(2.4.20)

The non-vanishing expectation value "v leads to the spontaneous breaking of the gauge symmetry, S;(2)P;(1). Corresponding to the two broken generators, two gauge bosons become massive with mass M "gv due to the Higgs mechanism, and one remains massless, correspond5 ing to the unbroken generator. The dynamics is a little complicated: at a distance less than M\, 5 the static potential is the Coulomb potential &1/R; at a distance larger than M\, the interaction 5 force is short-range and the potential behaves as a Yukawa potential, <(R)&exp(!M R)/R, i.e. 5 the electric charge is exponentially screened. The gauge coupling constant runs according to the Landau formula at distances shorter than M\, since the remaining theory is an Abelian gauge 5 theory, and the running is frozen at the distance M\. Thus, in the Higgs phase the static potential 5 between two test charges should tend to a constant value. This can also be seen from an explicit computation of the Wilson loop in lattice gauge theory, where the quantum Wilson loop obeys a `perimeter lawa, = &exp[!
;(perimeter)] . 

(2.4.21)

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2.4.5. Conxning phase Con"nement means that the particles corresponding to the "elds appearing in the fundamental Lagrangian are absent in the observed particle spectrum. One well-known example of a con"ning phase is low-energy QCD. In QCD, the microscopic dynamical variables are quarks and gluons, they carry colour charges responsible for the strong interactions. However, all the observed particles are colourless hadrons, i.e. bound states of quark}antiquark pairs (mesons) or three quarks (baryons) or gluons (glueballs). The microscopic degrees of freedom are always con"ned. This kind of dynamical feature is called a con"ning phase. Strictly speaking, the dynamical mechanism for colour con"nement is not quite clari"ed yet. However, a phenomenological picture is believed to be as follows. If we place a static colour charge and a conjugate charge at a large distance from each other, they will create a chromoelectric "eld formed like a thin #ux tube between the two charges, in contrast to the electric}magnetic "eld in the Abelian case, which is dispersed. The #ux tube is a string-like object with constant string tension . Both the cross section and the string tension are determined by the energy scale at which con"nement occurs. Thus, the potential between quarks grows linearly with the distance since the string tension is constant: <(R)" R .

(2.4.22)

The quarks cannot leave the hadron since this would need an in"nite energy. This dynamical feature manifests itself in the Wilson loop as the `area lawa. This physical picture is reminiscent of the Meissner e!ect in type II superconductivity. From the viewpoint of "eld theory, superconductivity can be thought of as the spontaneous breakdown of electromagnetic gauge symmetry [64] created by the Bose condensation of the Cooper electron pairs in the vacuum state. One of the most prominent features in the superconductor is the exclusion of magnetic "elds, the Meissner e!ect. However, if we put two static magnetic charges inside the superconductor, since the magnetic #ux is conserved, the magnetic "eld cannot vanish everywhere. The magnetic #ux will be pressed into narrow tubes connecting these two magnetic charges. A phenomenological model describing this dynamics is the Landau}Ginzburg theory,

 



1 1 1 S" dx ! (R  !2iet A  )# m  ! g(  ) , NO G O N N 4 N N 2 G N 2

(2.4.23)

where i"1, 2, 3; p, q"1, 2; g, m'0 and t is the Hermitian ;(1) generator,




t"

0 !i i

0



.

(2.4.24)

This model is the non-relativistic analogue of a ;(1) gauge "eld interacting with a scalar "eld. De"ning  #i " exp(2ie) ,   we can rewrite the above action as

 

(2.4.25)



1 1 1 S" dx ! (R )!2e(R #A )# m! g . G G 2 G 2 4

(2.4.26)

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The Landau}Ginzburg action shows spontaneous symmetry breakdown due to the condensation of electron pairs. The #ux tubes are just the vortex solutions of the classical equations of motion derived from the above action. One can calculate the energy carried by the vortex per unit length at a distance far away from the magnetic charge source with the result being a constant [64]. Thus, the energy between two magnetic charges in a superconductor grows linearly with the separation between them. We expect a similar dynamical mechanism to exist in QCD resulting in colour con"nement. However, there are two di$culties to overcome. First, in QCD, it is the chromoelectric "eld that forms #ux tubes. The vacuum medium should expel the chromoelectric "eld, and this can only be achieved by the condensation of particles carrying magnetic charge. Thus, if the colour con"nement is produced in this way, it should be a dual Meissner e!ect. It is well known that monopole solutions had been found in the Georgi}Glashow model independently by 't Hooft [65] and Polyakov [66]. In this model there exists a scalar "eld and the Higgs mechanism can break the original S;(2) gauge group to ;(1), leading to the emergence of monopoles as soliton solutions to the classical equations of motion. But QCD, whose gauge group S;(3) remains unbroken, does not have classical magnetic monopole solutions. A possible scheme to solve this problem was proposed by 't Hooft. His idea is quite simple [67]: choose an appropriate non-propagating gauge condition to reduce the original S;(N ) QCD to A a multiple Abelian theory with gauge group ;(1),A \, i.e. the maximal Cartan subgroup of S;(N ). A This procedure is called Abelian projection, which can arti"cially create monopole con"gurations. Note that the QCD monopoles obtained in this way are not physical objects, since they are gauge dependent. Nevertheless, they may play an important role in implementing the dual Meissner e!ect in QCD. A non-propagating gauge means a gauge that is "xed in such a way that no unphysical degrees of freedoms such as Faddeev}Popov ghosts emerge. A familiar example is the unitary gauge. Usually, as indicated by 't Hooft, this kind of gauge choice can render the theory non-manifestly renormalizable and therefore, in concrete calculations one should go over to a nicer `approximately non-propagatinga gauge condition. Another more important feature of the non-propagating gauge is that it can induce singularities in space}time. 't Hooft interpreted these singularities as the `monopolesa, additional physical dynamical variables. A concrete method to implement this gauge is as follows. Let us consider some operator that transforms covariantly under the local gauge transformation ;(x), XP;X;\ .

(2.4.27)

For instance, X"G GIJ, X"G DGIJ, or X"M  and so on, r, s"1,2, N being colour IJ IJ P Q A indices. Then by choosing ;(x) appropriately X can be diagonalized,




  X"  0

2 0 \



2 



,

(2.4.28)

,A where the eigenvalues are ordered,  5 525 A .   ,

(2.4.29)

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The gauge is not yet "xed completely, as any diagonal gauge rotation




e F

;(x)"  0

2 0 \



2 e F,A



(2.4.30)

leaves X invariant. These gauge transformations form the group ;(1),A \ ,

(2.4.31)

i.e. the maximal Abelian subgroup of the gauge group. Thus, with this gauge choice S;(N ) A gluodynamics is reduced to a dynamics of N !1 `photonsa (the gluon components corresponding A to diagonal generators) and N (N !1) `matter "eldsa (the gluon components corresponding to all A A o!-diagonal generators) charged with respect to the `photona "elds. Let us see how the QCD monopole con"gurations emerge. As mentioned above, the monopoles correspond to singularities in space induced by the gauge choice. Singularities arise when two consecutive eigenvalues of X coincide. Without loss of generality, we consider the case in which  and  coincide at some space point x , while the other eigenvalues remain distinct. Close to x ,     a non-singular gauge transformation brings X into a form comprising a 2;2 Hermitian matrix 1#?(x) ? ,

(2.4.32)

where 1 is the 2;2 unit matrix and ? are the Pauli matrices, in the upper left-hand corner; otherwise, X is diagonal. With respect to the S;(2) subgroup rotating the 1, 2-components into each other, the "elds ?(x) behave as the components of an isovector. The coincidence  " at   x means that  ?(x )"0, a"1, 2, 3 . (2.4.33)  The vanishing of these three real functions can only happen in isolated points in three-dimensional space. Thus, ?(x) carries the characteristic features of the Higgs "eld of the 't Hooft}Polyakov monopole. Indeed, performing the "nal step of the gauge "xing, making ""0 and '0, generates Dirac strings in the photon "elds A and A. I I If the ;(1) charges of the `mattera "eld AGH are I q "!q "g; q "0, kOi, j , (2.4.34) G H I then the magnetic charges of the singularity are






2
2
,! , 0,2, 0 . (h , h , h ,2, h A )"    , g g

(2.4.35)

Note that the `elementary monopolesa only have adjacent magnetic charges di!erent from zero. Monopoles with non-adjacent magnetic charges can only arise as bound states of these `elementarya poles. Although we have been able to identify states that could give rise to a dual Meissner e!ect through condensation, there remains the formidable problem of showing that this actually is what

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happens in QCD. Lattice simulations have given some quite encouraging results, but they are still far from complete and many aspects still remain unclear [68]. It was thus very encouraging that a demonstration of this con"nement mechanism could be given for N"2 supersymmetric Yang}Mills theory [1] and QCD [2]. Seiberg and Witten were able to deduce an exact low-energy e!ective action using the electric}magnetic duality conjecture, and showed that the dual Meissner e!ect indeed happens. 2.4.6. Oblique conxnement In addition to the various phases introduced above, there still exists another peculiar dynamical scenario in which states with fractional baryon charges can emerge. This phase is called the oblique con"nement phase. Since this phase indeed emerges in supersymmetric SO(N ) gauge theory, we A shall in the following give a detailed explanation. Roughly speaking, oblique con"nement is produced by dyon condensation. A dyon is a state carrying both electric and magnetic charges. As a soliton solution in the Georgi}Glashow model it was "rst found by Julia and Zee [69] in the gauge choice of non-zero A? , the time component of  the gauge "eld. The magnetic monopole is the static solution of the Georgi}Glashow model in the gauge condition A? "0 [65]. The non-zero A should give the static solution both electric and   magnetic charges since the electric "eld does not vanish. It was further shown by Witten that a non-vanishing vacuum angle  [70], g G? GI ?IJ L" 32
 IJ

(2.4.36)

a!ects the electric charge of magnetically charged states. Although this term is a surface term and does not a!ect the classical equations of motion, it makes a particle with the magnetic charge h necessarily acquire an electric charge, g q" h. 4


(2.4.37)

Before discussing the e!ects of the -vacuum, we shall show that for QCD monopole con"gurations the `electrica charges of o!-diagonal gluons and the magnetic charges of singularities form a (2N!2)-dimensional lattice. Consider two particles, (1) and (2), with magnetic charges h and h and electric charges G G q and q, i"1,2, N labelling the ;(1) group. The charges have to obey the Schwinger} G G A Zwanziger quantization condition, ,A  (hq!qh)"2
n G G G G G

(2.4.38)

with ,A ,A  h "  q "0 . (2.4.39) G G G G In 't Hooft's terms `the particle (1) has a Dirac quantum n with respect to particle (2)a.

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It is easy to show that a particle, with electric and magnetic charges being integer coe$cient linear combinations of the charges of the particles (1) and (2), still satis"es the Schwinger} Zwanziger quantization condition with respect to both others. Therefore, the whole particle spectrum satisfying (2.4.38) and (2.4.39) can be constructed by "nding a basis of 2(N !1) particles A with charges h and q, A"1,2, 2N !2, and then all allowed sets of charges are G G A ,A \ h "  k h,  G G 

,A \ q "  k q . G  G 

(2.4.40)

So they form (2N !2)-dimensional lattice. A Since in each of the N!1 ;(1) groups, the dynamics is described by Maxwell's equations, which are invariant under rotations of the electric and magnetic charges, h Ph cos  #q sin  , G G G G G q P!h sin  #q cos  , G G G G G

(2.4.41)

we can de"ne a `standarda basis by rotating away the magnetic part of the (N !1) basic charges: A h"0, G

A"1,2, N !1 . A

(2.4.42)

The electrically charged gluons provide us with a set of N!1 basic charges obeying (2.4.42): q"g!g>, A"1,2, N !1 . G G G A

(2.4.43)

The standard basis is completed by the magnetic monopoles with magnetic charges as in (2.4.35): 2
2
h" >\,A ! >\,A , A"N ,2, 2N !2 . G A A g G g G

(2.4.44)

Quarks, if they occur, would have only electric charges, g qG "g  ! , i"1,2, N , G GG A N A

(2.4.45)

where i labels a "xed ;(1) group and the last term is necessary in order to satisfy the constraint  (2.4.39). The lattice for the case N "2 is sketched in Fig. 1a. The electrically charged particles lie A on the horizontal axis. From (2.4.43) the gluons have charges 0 (corresponding to diagonal generators), $g (corresponding to o!-diagonal generators). Other particles composed of gluons, according to Eq. (2.4.40), have the charge $2g and so on. The quarks, according to Eq. (2.4.45), have charges $g/2 in the case N "2. The monopoles lie on the vertical axis with magnetic charge A h"4
/g and the electric charge q"0. Other states on the vertical axis are the anti-monopole, a pair of monopoles and so on. All the points which do not belong to the horizontal and vertical axis are bound states of the electric and magnetic quanta.

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Fig. 1. Electric-magnetic charge lattice for the S;(2) case, q"electric charge, h"magnetic charge.

Now we switch on the  vacuum term. According to Eq. (2.4.37), the QCD monopoles acquire fractional ;(1) charges, g h, A"N ,2, 2N !2 . q" A A G 4
 G

(2.4.46)

Consequently, if O0 or 2
, the square lattice shown in Fig. 1a will become oblique (this is why the name oblique con"nement is applied). Fig. 1b shows the lattice corresponding to "
#, 0(;
. From the above discussion, we infer that there are three physical dynamical scenarios. If one of the purely electrically charged objects is a Lorentz scalar, it can develop a non-vanishing vacuum expectation value and then the theory is in the Higgs phase. If the "eld representing a monopole develops a non-vanishing vacuum expectation value, the quarks will be con"ned, and the theory is in the con"ning phase. If no condensation occurs, the particle spectrum is given by the lattice, and the theory is in the so-called Coulomb phase. Therefore, all the phases, if they emerge in S;(N) gauge theory, can be characterized by designating those points on the charge lattice that develop vacuum expectation values. The quantum in the Schwinger}Zwanziger quantization condition for

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every pair of these points must vanish, hq!qh"0 . G G G G

(2.4.47)

For the case of S;(2) sketched in Fig. 1, Eq. (2.4.47) implies that these points should lie on a straight line passing through the origin. In the general S;(N ) case, they can span a linear subspace with A the dimension not larger than N !1. All particles whose charges lie in this subspace show only A short-range interactions, and their gauge "elds are screened by the Higgs mechanism. All the particles that have a non-vanishing Schwinger}Zwanziger quantum with respect to one of the points on this subspace are con"ned. Overall, all the possible physical phases of the theory can be characterized by the linear spaces spanned by at most N points on the electric}magnetic charges lattice. (In particular, the pure Coulomb phase corresponds to choosing only the origin.) Phase transitions correspond to moving from one linear subspace to another. With this electric}magnetic lattice and its phase description, we can explain what the oblique con"nement looks like. Increasing , we can deform the q, h lattice in a continuous way. Suppose we have a con"nement mode corresponding to the line I sketched in Fig. 1b, i.e. monopoles in this direction develop expectation values. If  runs from 0 to
, this line will become continuously more tilted. At "
, the other line II, corresponding to the parity image of I with respect to the vertical axis, is also a con"nement mode. When  is very near
but not equal to
, vacuum expectation values of the monopoles cannot be developed in the direction represented by the line I or II. This follows from the Schwinger}Zwanziger quantization condition. It seems that the monopole particles have to move collectively and carry large electric charges. However, the monopoles corresponding to I and II carry opposite electric charges. There is a possibility that they form a tight bound state, which in turn condenses and hence develops a vacuum expectation value. The possible direction in which this condensation happens is the line III as shown in Fig. 1b. This alternative is referred to as oblique con"nement by 't Hooft [67,71]. However, one can see that this con"nement mode is very peculiar. Some of the particles with the external quantum numbers of fundamental quarks exist in the observable spectrum. Of course, the fundamental quarks are con"ned, since they do not lie on the line III. But the bound state of a quark and a dyon can lie on this line and hence is not con"ned. Since the dyon has no baryon charge, or any other external quantum numbers, this kind of composite particle has exactly the same baryon charge as the fundamental quarks, i.e. fractional. In usual QCD, the vacuum angle is empirically very close to zero, and oblique con"nement cannot occur. However, in supersymmetric SO(N ) gauge theory, A this dynamical phenomenon indeed exists. 2.4.7. More order parameters In the following, we shall introduce two other order parameters, which are very convenient for describing monopole condensation (quark con"nement) and dyon condensation (oblique con"nement). The "rst order parameter is the 't Hooft loop = , which is de"ned by cutting a contour out of the  space}time and considering non-trivial boundary conditions around it. This de"nition itself has geometric meaning. In a fashion similar to the Wilson loop, it can be interpreted as the creation and annihilation of a monopole and anti-monopole pair. With the choice of a rectangular contour, one

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Table 2.3.2 Static potential obtained from the 't Hooft loop and corresponding phases in gauge theory ( is the string tension and
is the renormalization scale) Phase

Static potential <(R) at large R

Coulomb Free electric Free magnetic Higgs Con"ning

1/R ln(R
)/R 1/[R ln(R
)] R Constant

can de"ne the static potential between the monopole and anti-monopole at large distance R, lim = "e\24R 0 . (2.4.48)  2 It can be argued from some concrete calculations [16] that the potential between the monopoles can be classi"ed as that listed in Table 2.3.2. Comparing the static potential between electrically charged particles obtained from the Wilson loop with that of magnetic monopoles from the 't Hooft loop, one can see that the dynamical behaviour of free electric and free magnetic phases are exchanged. The Higgs phase and the con"ning phase are also exchanged. This is in fact a re#ection of electric}magnetic duality: the Wilson loop and the 't Hooft loops are exchanged with the exchange of electrically charged particles with magnetic monopoles. In fact, the exchange of the Higgs phase and con"nement phase is a conjecture by Mandelstam and 't Hooft [9,10] based on the exchange of the free electric phase and the free magnetic phase. As mentioned above, this conjecture provides a conceivable mechanism for colour con"nement. It can be understood as a dual Meissner e!ect produced by the condensation of monopoles, since the Higgs phase is associated with the condensation of electrically charged particles, and the magnetically charged particles are con"ned. The Coulomb phase goes into itself under the electric}magnetic duality transformation. This is the unique self-dual phase. For an Abelian Coulomb phase with free photons, this can be easily seen from a standard duality transformation. The search for duality in non-Abelian Coulomb phase has become very popular in recent years. The original conjecture for the existence of this duality was made by Montonen and Olive [7]. Osborn [8] found that this duality can exist in N"4 supersymmetric Yang}Mills theory. Some years ago a major progress was made by Seiberg and Witten [1,2]. They found that the electric}magnetic duality in the low-energy N"2 supersymmetric Yang}Mills theory play a crucial role in understanding the non-perturbative dynamics. The electric}magnetic duality transformation turned out to be the monodromy, i.e. the transformation of a complex function relevant to the coupling around the singularity in the Riemann surface of the moduli space of the theory, while the singularities represent the various particles implied from the duality such as monopole and dyon, etc. In this way an exact coupling of low-energy N"2 supersymmetric Yang}Mills theory is thus determined and consequently, the full low-energy quantum e!ective action (including the non-perturbative contribution) is given. This e!ective action is con"rmed by some explicit instanton calculations [89]. Furthermore, Seiberg has shown

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Table 2.3.3 Dynamical law of the phases described by the order parameters Phase

Dynamical behaviour

Coulomb Con"nement Oblique con"nement

=

= 

= 

Perimeter law Area law Area law

Area law Perimeter law Area law

Area law Area law Perimeter law

that the non-Abelian electric}magnetic duality can emerge in the infrared "xed point of N"1 supersymmetric QCD [11}13]. This is the main topic we shall discuss in the following sections. With the Wilson and the 't Hooft loops, we formally de"ne another gauge invariant order parameter as the product of them, = "= = , (2.4.49)    which is called a dyon loop since it can describe the dynamics of the particles with both electric and magnetic charges. This order parameter is particularly suitable for describing the oblique con"nement phase. To compare the dynamical behaviour of each phase re#ected in these parameters, we collect them in Table 2.3.3. 2.5. N"1 superconformal algebra and its representation 2.5.1. Current supermultiplet The algebraic foundation of a superconformal invariant quantum "eld theory is superconformal algebra. To introduce superconformal symmetry, a natural way is to start from the supercurrent supermultiplet. Like in the derivation of ordinary conformal algebra in Section 2.1, we exhibit the structure of the supercurrent multiplet without resorting to a particular model and only by demanding that the currents and their supersymmetric transformations yield the supersymmetry algebra. According to the Noether theorem, corresponding to supersymmetry invariance of a relativistic quantum "eld theory we have the conserved supersymmetry current j (x), R jI(x)"0, and the I? I? supercharge



(2.5.1)




(2.5.2)

Q" dx j . 

The supercharge generates the supersymmetric transformation, (x)"i[M Q, (x)]. Here we use four-component notation, Q"

Q ? , QM "(Q?, QM  ), M Q"QM  . ? QM ?

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In two-component notation, M Q"?Q #QM  M ? . The supersymmetric variation of the supersym? ? metric current gives j (x)"!i[ j (x), M Q] . (2.5.3) I I Taking the space integral of the time component of the above supersymmetry variation and requiring that the N"1 supersymmetry algebra Q, QM "2IP is reproduced, we have I





dx j (x)"!i[Q, M Q]"!i[Q, QM ]"!2iIP "!2iI dx  .  I I

(2.5.4)

In view of Lorentz covariance, the supersymmetric transformation of supersymmetry current j must have the following general form: I j "2J #RMR , (2.5.5) I JI IM where R "!R . The second term of (2.5.5) vanishes when taking "0 and integrating over MI IM dx. Furthermore, the chiral ; (1) rotation of the supercharge yields 0

 

[Q, R]" Q, R" dx j ,  

  



M Q, dx j "!i dx j "M  Q"M  dx j .     

(2.5.6)

Hence j"i j #RJr , (2.5.7) I I JI with r "!r . Eqs. (2.5.5) and (2.5.7) imply that the R-current j, the supersymmetry current JI IJ I j and the energy-momentum tensor  belong to a supermultiplet I? IJ ( , j , j) . (2.5.8) IJ I I At quantum level, each member of the above supermultiplet will become anomalous. It was shown that the trace anomaly I , the -trace anomaly of the supersymmetry current, Ij , and the chiral I I anomaly of the R-current, RIj, lie in a chiral supermultiplet, ,(Ij , I , RIj) [27,97]. I I I I In a concrete classical superconformal invariant "eld theory } the massless Wess}Zumino model } the explicit but model independent supersymmetry transformations among the members of the current supermultiplet (2.5.8) are as follows: JRMjN , j "!2iJ #iJ (R j! RMj)#i I IJ  JI IJ M  IJMN i j"iM  j ! M   Jj , I I 3  I J 1  " iM ( RMj # RMj ) . IJ 4 IM J JM I

(2.5.9)

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Consequently, the supersymmetry transformation of the members of the anomaly multiplet is










2 1  !i Ij "! I !i RIj  , I  I 3 I 3 

 

2 1 I "iM JR !i Ij , J 3 I 3 I

1 (RIj)"M  JR !i Ij . I  J 3 I

(2.5.10)

Like in the non-supersymmetric case discussed in Section 2.1, if we require that the conformal symmetry is preserved at the quantum level, "0, i.e. I "0, I

Ij "0, I

RIj"0 , I

(2.5.11)

then the supersymmetric transformation (2.5.9) reduces to the form of the naive supersymmetry transformations (2.5.5) and (2.5.7), i JRMjN , j "!2iJ #J R j#  I IJ  J I 2 IJMN 1  " iM ( RMj # RMj ), j"iM  j . IM J JM I I I IJ 4

(2.5.12)

In particular, with the vanishing of the quantum anomalies (2.5.11), we have not only the conserved currents d and k and their charges D and K shown in (2.1.12), but also a new conserved I IJ I fermionic current s ,ixJ j , I JI

RIs "iIj #ixJ RIj "0 , I I J I

(2.5.13)

and the corresponding supercharge



S, dx s . 

(2.5.14)

S is called the generator of special supersymmetry transformations. Like the supersymmetric charge, S is a Majorana spinor,




S"

S ? , QM "(S?, SM  ) . ? SM ?

(2.5.15)

2.5.2. N"1 superconformal algebra In this section, we "rst derive the whole superconformal algebra and then work out in detail the representation of the superconformal algebra following Ref. [27]. In particular, we shall introduce

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the relation between the conformal dimension and the R-charge of a chiral super"eld, which plays a crucial role in determining the conformal window in N"1 supersymmetric gauge theory. Like the ordinary conformal algebra, the superconformal algebra can be derived directly from the transformation property of the current given in (2.5.12). We "rst get



[Q, QM ]"

 

dx j , QM  "i dx j  









1 "2I dx  #iG  dx RGj !i  dx RGj! GHI  dx R j I    G G HI 2



"2I dx  "2IP , I I



[P , Q]" I

(2.5.16)

 



1 dx  , M Q " dx  " iM dx( RMj # RMj ) I I 4 M I IM 



1 " iM dx( RGj # RGj ! RGj )"0 , G I IG  I G 4

(2.5.17)



[M , M Q]"i dx(x  !x  ) IJ I J J I



1 1 "! M dx[( j ! j ! j !(  )]"! M Q , J I JI  I J 2 JI 4

(2.5.18)



(2.5.19)

[R, M Q]"

 



dx j , M Q "i dx j "!M  dx j "!M  Q ,     

where we have used



dx RG(anything)"0, i"1, 2, 3;

"! , RIj "RIj"0, IJ JI I I

x RMj "x ( Rj # RGj )"x RG( j ! j ) . I JM  I J  JG  I JG  J G

(2.5.20)

Eqs. (2.5.16)}(2.5.19) give the fermionic part of the super-PoincareH algebra: [Q, M ]" Q, [Q, P ]"0, IJ  IJ I

Q, QM "2IP , [Q, R]" Q . I 

(2.5.21)

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To obtain the whole superconformal algebra. we only need to calculate two new commutation relations, [Q, K ] and [Q, D]. The others can be determined from the Jacobi identity. Thus, we I have

           

[M Q, K ]" M Q, dx k "!i dx k I I I "!i dx(2x xJ  !x  )"M  S , I J I I

(2.5.22)



[M Q, D]" M Q, dx d "!i dx d "!i dx xJ    J 1 "! M dx[GJ( j # j ! j )] G J JG  J G 4



1 i 1 1 "! M dx jJ"! M dx i  jG" M dx j " iM Q , J  G  2 2 2 2

(2.5.23)

where the condition Ij "0 was used. Eqs. (2.5.22) and (2.5.23) yield new commutation relations, I i [Q, R]" Q, [Q, K ]" S, [Q, D]" Q . (2.5.24)  I I 2 In addition, since the R-symmetry is only related to supercoordinate rotations, it actually belongs to the internal symmetries. Thus the generator of R-symmetry must commute with the generators of the ordinary space}time transformation, [R, P ]"[R, M ]"[R, D]"[R, K ]"0 . (2.5.25) I IJ I With the algebraic relations listed in (2.5.21), (2.5.24) and (2.5.25), the Jacobi identities (S, M, K), (Q, P, K), (Q, D, K), (Q, K, K) and (Q, K, R) yield the following commutation relations, respectively: 1 i [S, M ]" S, [S, P ]" Q, [S, D]"! S, [S, K ]"0, [S, R]"! S . IJ IJ I I I  2 2 (2.5.26) As an illustrative example, consider the Jacobi identity (S, M, K). We have [[Q, M ], K ]#[[K , Q], M ]#[[M , K ], Q]"0 , IJ M M IJ IJ M  [Q, K ]! [S, M ]![i( K ! K ), Q]"0, M M IJ MI J MJ I  IJ   S! [S, M ]#i  S!i  S"0,  IJ M M IJ MI J MJ I i  [ ,  ]S! [S, M ]"0, M IJ 4 M I J [S, M ]" S , IJ  IJ where the -algebra operation    "2  !2  #   was used. I J M I JM IM J M I J

(2.5.27)

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The anticommutators S, QM  (or equivalently Q, SM ) and S, SM  need some special considerations. Since both S and Q are fermionic generators, S, QM  must be proportional to bosonic generators, so it must be of the following general form:

S, QM "aD#bIP #c IJM #dIK #eR . I IJ I

(2.5.28)

The Jacobi identities (QM , S, D); (QM , S, P) and (QM , S, QM ) "x the inde"nite coe$cients as b"d"0, a"2i, d"1 and e"3 . Hence we "nally obtain 

QM , S"2iD# IJM #3 R . IJ 

(2.5.29)

Similarly one shows that

S, SM "2IK . I

(2.5.30)

The fermionic part of the whole N"1 superconformal algebra consists of the collection of the above commutation relations, [Q, M ]" Q, IJ  IJ [Q, D]"iQ, 

[S, M ]" S , IJ  IJ

[S, D]"!iS , 

[Q, P ]"0, I

[S, P ]" Q , I I

[Q, K ]" S, I I

[S, K ]"0 , I

[Q, R]" Q, 

[S, R]"! S , 

[R, M ]"[R, P ]"[R, D]"[R, K ]"0 , IJ I I

Q, QM "2IP , I

S, SM "2IK , I

S, QM "2iD# IJM #3 R . IJ 

(2.5.31)

The N"1 superconformal algebra is isomorphic to S;(2, 21). For N-extended supersymmetry, the supersymmetric algebra is isomorphic to S;(2, 2N). Furthermore, de"ning a conformal spinor,




,

Q ? , SM ?

M "(S?, QM  ) , ?

(2.5.32)

and ,( , , , ), a, b"0,2, 3, 5, 6, ?@ IJ I I  "  , I I 

" , I I

" .  

(2.5.33)

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one can write (2.5.31) in the form of the fermionic part of an S;(2, 21) algebra: [M , M ]"!M ; [, M ]" ; ?@  ?@ ?@  ?@ [, R]"; [M , R]"!M ; [M , R]"0 ; ?@

, " M , M "0;

, M " ?@M !3R . ?@

(2.5.34)

2.5.3. Representations of N"1 superconformal symmetry The method of "nding a representation of the N"1 superconformal algebra on "eld operators is the same as for the ordinary conformal algebra, i.e. using Wigner's little group method. First, the commutation relation [Q, K ]" S shows that the conformal spinor charge S comes from the I I commutator of the supercharge Q with the special conformal transformation generators K . Thus I it is possible to construct a superconformal multiplet from a multiplet of PoincareH supersymmetry by working out the transformation of the "elds under the special conformal transformations generated by K . According to (2.1.31), the transformation of the "eld  under a special conformal I transformation is [K , (x)]"i(2x x RJ!xR )(x)#(k #2ix d#2xJ )(x) . (2.5.35) I I J I I I IJ As shown in Section 2, in the "eld operator representation of the ordinary conformal algebra, there is no restriction on the little group representation , d and  except that d must be a number due I IJ to [D, M ]"0. However, in the superconformal algebra, the situation is di!erent: there is an IJ important constraint for coming from the relation [Q, K ]" S, thus a  should be `separated I I I I outa from the representation of the commutator [Q, K ]. This constraint will restrict the possible I little group representations on the components of a superconformal multiplet. The little group of the superconformal algebra can still be found by requiring that x"0 stays invariant. Then we see that the little group is composed of not only the Lorentz group, scale transformations and the special conformal transformations, but also of a ; (1) group due to the 0 algebraic relations [R, M ]"[R, D]"[R, K ]"0. Another di!erence between the representaIJ I tions of the superconformal algebra and the ordinary conformal algebra is that owing to the supersymmetry, the representation of the superconformal algebra must be realized on a supermultiplet. This is unlike the ordinary conformal algebra, where only one type of "eld is enough. For the superconformal algebra, several kinds of "elds such as (pseudo-)scalar "elds, spinor "elds and vector "elds are required due to supersymmetry. Thus we consider the most general supermultiplet, which can be constructed by starting from a complex "eld C(x) and performing the famous `seven stepsa introduced in Ref. [27]. Since the concrete construction of this general supermultiplet is very lengthy, we shall not repeat it explicitly. The basic method of "nding the chiral multiplet is illustrated detaily in Ref. [27]. Here, we only display this general multiplet and the supersymmetry transformations among its component "elds [27], G(x)"(C(x), (x), M(x), N(x), A (x), (x), D(x)) , I G(x)"!i[G(x), M Q]"!i[G(x), QM ] , C(x)"!iM  (x) , 

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(x)"(M(x)!i N(x))!iI(A (x)!i R C(x)) ,  I  I M(x)"M ((x)!iIR (x)) , I N(x)"!iM  ((x)!iIR (x)) ,  I A (x)"iM  (x)#M R (x) , I I I (x)"!i IJR A (x)#i D(x) , I J  D(x)"!M IR  (x) . (2.5.36) I  Consequently, the supersymmetry transformations for the "elds at the origin (x"0) should be the following: G(0)"(C(0), (0), M(0), N(0), A (0), (0), D(0)) , I G(0)"!i[G(0), M Q]"!i[G(0), QM ] , C(0)"!iM  (0) ,  (0)"(M(0)!i N(0))!iIA (0) ,  I M(0)"M (0) , N(0)"!iM  (0) ,  A (0)"iM  (0) , I I (0)"i D(0) ,  D(0)"0 .

(2.5.37)

We "rst "nd the representation of the little group on this supermultiplet. Since this supermultiplet is generated from C(x) through a series of successive supersymmetry transformations, as the "rst step, we de"ne the action of the generators of the little group on C(0), [C(0), R]"nC(0) , [C(0), D]"idC(0) , [C(0), M ]" C(0) . (2.5.38) IJ IJ Then, translating C(0) from the origin according to C(x)"exp(!ixIP )C(0) exp(ixIP ), we obtain I I [C(x), R]"nC(x) , [C(x), D]"ixIR C(x)#idC(x) , I [C(x), M ]"i(x R !x R )C(x)# C(x) . IJ I J J I IJ

(2.5.39)

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The algebraic relations [R, M ]"[D, M ]"0 require IJ IJ [n,  ]"[d,  ]"0 . IJ IJ

(2.5.40)

Since  is, by hypothesis, an irreducible representation, according to Schur's lemma, n and d must IJ be numbers. They are the conformal dimension and the R-charge of C(x). The conformal dimensions and R-charges for other "elds can be obtained from (2.5.37), (2.5.38), the superconformal algebra (2.5.31) and Jacobi identities. For example, using the Jacobi identities (C, R, Q) and (, R, QM ), we have [[C(0), R], Q]#[[Q, C(0)], R]#[[R, Q], C(0)]"0 , [nC(0), Q]#[! (0), R]#[! Q, C(0)]"0 ,    [(0), R]"(n# )[C(0), Q]"(n# ) (0) ,     [(0), R]"(n# )(0) ; 

(2.5.41)

[ (0), QM , R]# [R, (0)], QM ! [QM , R], (0)"0 , [iM(0)# N(0)#IA (0), R]# !(n# )(0), QM # ! QM , (0)"0 ,  I   i[M(0), R]# [N(0), R]#I[A (0), R]"(n#2 )[iM(0)# N(0)#IA (0), R] ,  I   I [M(0), R]"(n#2 )M(0) ;  [N(0), R]"(n#2 )N(0) ;  [A (0), R]"(n#2 )A (0) . I  I

(2.5.42)

Eqs. (2.5.41) and (2.5.42) show that the R-charges of  and M, N, A are (n# ) and (n#2 ), I   respectively. Similarly, the Jacobi identities (M, R, Q), (N, R, Q), (A , R, Q), (, R, QM ) and (D, R, Q) I yield the R-charges of (0), M(0), etc. If we write G(0) as a column vector,




C(0) (0) M(0) G(0)" N(0) , A (0) I (0) D(0)

(2.5.43)

the R-charges of the component "eld operators can be written as a diagonal matrix, n "n1# diag(0, 1, 2, 2, 2, 3, 4) 

(2.5.44)

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where 1 denotes the 7;7 unit matrix. The conformal dimensions of the component "eld operators can be worked out in the same way. For example, from the Jacobi identities (C, D, Q) and (C, D, QM ), we have [[C(0), D], Q]#[[Q, C(0)], D]#[[D, Q], C(0)]"0 ,





i [idC(0), Q]#[ (0), D]# ! Q, C(0) "0 ,  2  [(0), D]"i(d#)[C(0), Q]"i(d#) (0) ,     [(0), D]"i(d#)(0) ; 

(2.5.45)

[ (0), QM , D]# [D, (0)], QM ! [QM , D], "0 , [iM(0)# N(0)#IA (0), D]# i(d#)(0), QM # iQM , (0)"0 ,  I   i[M(0), D]# [N(0), D]#I[A (0), D]"i(d#1)[iM(0)# N(0)#IA (0), R] ,  I  I [M(0), R]"i(d#1)M(0), [N(0), R]"i(d#1)N(0) , [A (0), D]"i(d#1)A (0) . I I

(2.5.46)

Thus, the conformal dimensions of  and M, N, A are respectively (d#1/2) and (d#1). FurtherI more, the Jacobi identities (M, D, Q), (N, D, Q), (A , D, Q), (, D, QM ), (D, D, Q) give the conformal I dimensions of other component "elds such as (0), M(0), etc. Therefore, the conformal dimension of the supermultiplet G(0) is dI "d1#diag(0, , 1, 1, 1, , 2) .  

(2.5.47)

Working out the matrix representation of K on the supermultiplet is more di$cult. The I I relation [K , D]"!iK requires that should satisfy I I I [ , dI ]"! . I I

(2.5.48)

In terms of matrix elements, since dI is a diagonal matrix, the above equation becomes ( ) dI  !dI  ( ) "!( ) , I GI I IH G GI I IH I GH (dI !dI #1)( ) "0 . H G I GH

(2.5.49)

This means that the matrix elements ( ) of will vanish unless dI "dI #1. According I GH I G H to (2.5.47), the matrix representation of in the basis (2.5.43) will be of following form I

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(with * representing a possible non-zero value),




 0 0 0 0 0 0 0 0 0 0 0 0 0 0

* 0 0 0 0 0 0 ( )" * 0 0 0 0 0 0 , I * 0 0 0 0 0 0 0 * 0 0 0 0 0 0 0 * * * 0 0

C(0)

0

(0)

0

M(0)

*C(0) *C(0)

( ) N(0) " I A (0) I (0) D(0)

.

(2.5.50)

*C(0) *(0)

*M(0)#*N(0)#*AI (0)

Eq. (2.5.50) explicitly leads to

C(0)" (0)"0, i.e. [C(0), K ]"[(0), K ]"0 . I I I I

(2.5.51)

It seems that M(0)O0 and N(0)O0, but a careful analysis shows that this not the case. The I I special conformal transformations and supersymmetry transformations of M, N and C imply that

M and N have the same dimension as C, however, M and N should be four vectors due to I I I I Lorentz covariance. Since the transformation is linear, it is not allowed to have a non-local operator such as 䊐\ in the action of K on M and N, hence there is no way of constructing I a vector with the same dimension as C. Thus, we must have

M(0)"0, I

N(0)"0, [M(0), K ]"[N(0), K ]"0 . I I I

(2.5.52)

The remaining components A,  and D will not vanish under the action of K . Since the calculation I is quite lengthy, we only list the main steps: First, we must know the matrix representation  of M on the supermultiplet G(0). This can IJ IJ be found by means of the Jacobi identities from the known representation (2.5.38) on C(0). For example, using the Jacobi identity (Q, C, M ), we get IJ [[Q, C(0)], M ]#[[M , Q], C(0)]#[[C(0), M ], Q]"0 , IJ IJ IJ [ (0), M ]#[! Q, C(0)]#[ C(0), Q]"0 , IJ  IJ  IJ [(0), M ]"( ! )(0) . IJ IJ  IJ

(2.5.53)

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Furthermore, the Jacobi identities such as (QM , , M ), (Q, M, M ), (Q, N, M ), (Q, A, M ), IJ IJ IJ IJ (QM , , M ) and (QM , D, M ) give IJ IJ [M(0), M ]"( ! )M(0) , IJ IJ IJ [N(0), M ]"( ! N(0)) , IJ IJ IJ [A (0), M ]"( ! )A (0) , M IJ IJ IJ M [(0), M ]"( ! )(0) , IJ IJ  IJ [D(0), M ]"(2 ! )D(0) . IJ IJ IJ

(2.5.54)

Secondly, we must know the matrix representation of the special supersymmetry charge S on the supermultiplet. The Jacobi identities still play a role. For example, from the Jacobi identity (Q, K, C) we have [[Q, K ], C(0)]#[[C(0), Q], K ]#[[K , C(0)], Q]"0 , I I I [ S, C(0)]#[ (0), K ]"0 , I  I [ S, C(0)]"0, I

[S, C(0)]"0 ,

(2.5.55)

where (2.5.51) was used. Further, using the Jacobi identity (QM , S, C), we determine (0), S, [ QM , S, C(0)]# [C(0), QM ], S! [S, C(0)], QM "0 , [2iD# IJM #3 R, C(0)]#  (0), S"0 , IJ  

(0), S" (!2d# IJ #3n )C(0) .  IJ 

(2.5.56)

The third step is to use the Jacobi identities (QM , , K ), (Q, M, K ), (Q, N, K ) and (QM , , K ) for I I I I "nding the matrix representation of K on A,  and D. For example, the Jacobi identity I I (QM , , K ) gives I [ QM , (0), K ]# [K , QM ], (0)! [(0), K ], QM "0 , I I I J[A (0), K ]" SM , (0)" 3nC(0)#J NMC(0) , J I I I JINM [A (0), K ]"3nC(0) ! NMC(0) . J I IJ IJNM

(2.5.57)

Thus, we can work out

A (0)"! NMC(0)#3nC(0) , I J IJNM IJ

(0)"! JM  (0)!d (0)!n  (0) , I JM I  I  I 

D(0)"!3nA (0)# NMAJ(0) . I I IJNM

(2.5.58)

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After translating (2.5.58) from the origin, we "nally obtain

A (x)"! NMC(x)#3nC(x) , I J IJNM IJ

(x)"! JM  (x)!d (x)!n  (x) , I  I JM I  I 

D(x)"!3nA (x)!2idR C(x)!2 RJC(x)# NMAJ(x) . I I I IJ IJNM

(2.5.59)

Translating [G(0), S, combining this special supersymmetry transformation with the PoincareH supersymmetry transformations in the following way: G(x)"!i[<(x), M Q# S] ,

(2.5.60)

and de"ning the convenient combination ,!ixI , I

X!,d!n $ IJ ,    IJ

(2.5.61)

we "nally get the superconformal transformation on the supermultiplet expressed in the following compact form: C"!i  ,  "(M!i N)!iI(A !i R C)!2iX> C ,  I  I  M"(!iIR )# X\!2  , I N"!i (#iIR )!i  X\!2i   ,  I   A "i #R ()#i X\  , I I I I "!i IJR A #i D!X>(M#i N)#iIX>(A !i  C) , I J   I  I D"!IR  !2i  X\(!iIR ) . I    I

(2.5.62)

Eq. (2.5.62) is the representation of the superconformal algebra on a general supermultiplet. This supermultiplet is in general reducible. One can impose reality and chirality conditions on this general multiplet to reduce it to the representations on vector and chiral supermultiplets, respectively. First we consider the reduction to a vector supermultiplet. If the lowest component "eld C is real, then the whole supermultiplet will be real, G"GR, i.e. G is a vector supermultiplet. Since C is a real "eld, its R-charge n must vanish n(C)"0 ,

(2.5.63)

as ; (1) acts non-trivially only on complex "elds. In particular, taking the Hermitian conjugate of 0 the last relation of (2.5.39), since C"CR and M "MR , we obtain IJ IJ [C, M ]"i(x R !x R )C!R C . IJ I J J I IJ

(2.5.64)

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This shows that 

must be a real representation of the generators of the Lorentz group, IJ  "!R . (2.5.65) IJ IJ Next we turn to the reduction to a chiral supermultiplet through imposing a chirality condition on the general supermultiplet G. A chirality condition means choosing only the chiral (left- or right-handed) part of Dirac spinor. This can be done by imposing the constraint, (1! )"0, ((1# )"0) . (2.5.66)   This is equivalent to the de"nition of a chiral (anti-chiral) super"eld : D "0 (DM  "0), with ? ? C being the lowest component. The chiral condition (2.5.66) imposes a constraint on the superconformal transformation of  and hence selects the chiral supermultiplet. According to the second equation in (2.5.62), we should have (1! )"0 ,  (1! )[(M!i N)!iI(A !i R C)!2iX> C]"0 ,   I  I  (1! )(M#iN)!(1! )I(A !iR C)!2i(1! )X> C"0 .   I I   Thus we obtain M#iN"A !iR C"0 , I I

(2.5.67)

(2.5.68)

and




3n 1 0"(1! )X>"(1! ) d!  # IJ  IJ   2 2






3n 1 "(1# ) d# # (1! ) IJ .   IJ 2 2

 (2.5.69)

Eq. (2.5.69) gives n"!2d/3 ,

(2.5.70)

and (1! ) IJ "0 . (2.5.71)  IJ Using the self-dual property of -matrices [55],  IJ"i/2IJNM , we can write (2.5.71) as  NM (IJ!i/2IJNM )"0, and hence we get IJ NM i NM . (2.5.72)  "  IJ 2 IJNM This means that a chiral supermultiplet must be in a self-dual representation of the Lorentz group. Eq. (2.5.70) shows that the R-charge of the chiral multiplet must be !2/3 times its conformal

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dimension. These are two important properties of the chiral conformal supermultiplet. Furthermore, the superconformal transformations (M#iN)"0,

A "iR C , I I

(2.5.73)

lead to (x)"D(x)"0 .

(2.5.74)

Thus we are left with a chiral supermultiplet "(C, (1# ), M).  Finally, we reproduce the important results (2.5.70) and (2.5.72) concerning the chiral supermultiplet from the viewpoint of coset space [27]. In PoincareH supersymmetry, the little group is the Lorentz group, and Minkowski space is the coset space of the super-PoincareH group modulo the Lorentz group. There exist chiral multiplets with arbitrary additional Lorentz indices, the chirality condition DM "0 is covariant for arbitrary representations of the Lorentz group. For example, there exists not only the scalar chiral supermultiplet "(, , F), but the supersymmetric Yang}Mills "eld strength ="(F , , D) is also a chiral supermultiplet. However, this is not the IJ case in conformal supersymmetry. Since a chiral super"eld can be written as follows: (x, , M )"(x#i M , )"(y, ) ,

(2.5.75)

a chiral super"eld is independent of the (anti-chiral) super-coordinate M and hence one could from the beginning use a superspace G/H with the little group H generated by both M and QM  , G being IJ ? the super-PoincareH group. Obviously, this special superspace is parameterized only by xI and  and hence this superspace is called a chiral superspace. For a general super"eld de"ned in chiral superspace, the action of QM  would be as follows: ? [(x), QM  "2( I)  R (x)#q  (x) , ? ? I ? [(0), QM  "q  (0) . ? ?

(2.5.76)

For a chiral super"eld, the matrix representation q  of QM  should vanish, ? ? q  "0 . ?

(2.5.77)

This is the case of PoincareH supersymmetry. For conformal supersymmetry, as discussed in Section 2.1, the ordinary conformal group can be realized on Minkowski space, and Minkowski space is the coset space of the conformal group modulo the Lorentz group, dilation and special conformal transformations. Thus the conformal supersymmetry can be realized on Minkowski superspace with a complicated x,  and M dependence of the elements of the little group. It is also possible to realize conformal supersymmetry on chiral superspace. The relevant little group will now be generated by M , K , D, S?, SM ? and QM  since the transformations generated by these ? IJ I generators keep x""0 invariant. In this case the question whether there exists a chiral

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conformal supermultiplet is equivalent to whether the constraint (2.5.77) is consistent. This is not naturally satis"ed since there is a non-vanishing anti-commutator from the superconformal algebra,






3

SM ? , QM Q "( IJ)? Q M !2? Q R!iD . @ IJ @ 2 @

(2.5.78)

Requiring (2.5.76) to be satis"ed when SM ? , QM Q  acts on a chiral supermultiplet , we have @ 0"[(0), SM ? , QM Q ] @








3 " (0), ( IJ)? Q M !2? Q R!iD @ IJ @ 2






3 "( IJ)? Q  (0)!2? Q n#d (0) . @ IJ @ 2

(2.5.79)

Furthermore, considering the anti-self-dual property of  [27], IJ"!i/2 NM, the relations IJ IJNM (2.5.70) and (2.5.72) are reproduced.

3. Non-perturbative dynamics in N ⴝ 1 supersymmetric QCD We shall next introduce the non-perturbative dynamical phenomena in supersymmetric gauge theory. This section is concentrating on supersymmetric QCD with gauge group S;(N ) and A N #avours. The methods of analysing the non-perturbative dynamics include the gauge and global D symmetries; the holomorphic dependence of the non-perturbative dynamical superpotential not only on the chiral super"elds but also on various parameters such as the coupling and mass; instanton computations and the exact NSVZ beta function given in (1.1) as well as the decoupling relation of the heavy modes at low energy. Furthermore, the 't Hooft anomaly matching between high and low energy can provide a test of the non-perturbative result. In addition to the global symmetries S; (N );S; (N );; (1);; (1) (some of them are broken or anomalous at the * D 0 D  quantum level) as in ordinary QCD, the N"1 supersymmetric QCD has another ;(1) axial vector symmetry called the R-symmetry, which becomes anomalous like the ; (1) at the quantum level.  However, this R-symmetry can be combined with the ; (1) symmetry to an anomaly-free ;(1)  symmetry. The global and gauge symmetries together with the holomorphy and instanton calculations uniquely "x the superpotential and hence the moduli space in the range N (N . By D A considering the decoupling relation, the moduli spaces in the cases N "N or N #1 can also be D A A exactly determined. Consequently, a series of non-perturbative physical phenomena such as dynamical supersymmetry breaking, chiral symmetry breaking and con"nement are exhibited. The 't Hooft anomaly matching con"rms the correctness of the physical pictures. The NSVZ beta function implies that in the range 3N /2(N (3N , the theory has a non-trivial IR "xed point, at A D A which the theory is a superconformal "eld theory and the dual theory gives a completely equivalent description of the low-energy physics of the original theory but with a weak coupling. This phenomenon is similar to the electric}magnetic duality and is thus called a non-Abelian electric} magnetic duality. It will be discussed in detail in the next section.

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3.1. Introducing N"1 supersymmetric QCD 3.1.1. Classical action of N"1 supersymmetric QCD Supersymmetric QCD (SQCD) is the generalization of ordinary QCD. As required by supersymmetry, corresponding to each particle, there exists a superpartner. Corresponding to the gluon A? , I the left-handed quark  and the right-handed quark I , we have their superpartners: the gluino ?, the left-handed squark  and the right-handed squark I . In super"eld form, the Lagrangian of (massive) SQCD is 1 L" Im( Tr =?=   )#[QReE4,A Q#QI ReE4,M A QI ]  M  ?F FF 8
# m(QI Q)  #mH(QRQI R) M  . F F Here, = is the super"eld strength which in the Wess}Zumino gauge takes the form ? i = "¹? !i? # D?! ( I J) F? # I  (D M ? )? . ? ? ? ? IJ ?? I 2






(3.1.1)

(3.1.2)

The quantity  4
" #i 2
g

(3.1.3)

combines the gauge coupling constant g and the CP-violating parameter  into what can be e!ectively regarded as a constant chiral super"eld. Q and QI are chiral left- and right-handed quark super"elds, respectively. < is the vector super"eld of the gluon and the gluino. In Wess}Zumino gauge, they take the following forms: Q (y)" (y)#(2? (y)#F (y), QI (y)"I (y)#(2?I (y)#FI (y) , ?P P P P ?P P P P
(3.1.4)

(3.1.5) yI,xI#i? I  M ? , <(N ),
(3.1.6)

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where D, F and FI are auxiliary "elds, a, b"1,2, N!1, r, s"1,2, N and we suppress the A A #avour index. The various covariant derivatives are de"ned as follows: (D ) "(R Q #igA? (¹?)Q ) , (D )HP"R HP!igHQ(¹?)P A? , I P I P I P Q I I Q I (DI I )P"R I P!igI Q(¹?)P A? , (DI I )H"(R Q #igA? (¹?)Q )I H , I I Q I I P I P I P Q D M ?? "R M ?? #gf ?@AA@ M A? . I I I

(3.1.7)

Eliminating the auxiliary "elds F, FH, FI , FI H and D through their equations of motion, F "!mHI H, FI P"!mHHP, D?"!g[HP(¹?)Q  !I P(¹?)Q I H] , P P P Q P Q

(3.1.8)

we can obtain the Lagrangian given in Ref. [25]. 3.1.2. Global symmetries of massless supersymmetric QCD Massless SQCD possesses the global symmetries of ordinary QCD, i.e., S; (N ); * D S; (N );; (1);; (1). In addition, it has a new ;(1) symmetry called R -symmetry. In terms of 0 D   super"elds [32,33]: = ()Pe\ ?= (e ?), Q()PQ(e ?), QI ()PQI (eG?) . @ @

(3.1.9)

For the component "elds, this means  Pe\ ? , P P

I PPe\ ?I P, ?Pe ?? .

(3.1.10)

The total global symmetry of massless SQCD at the classical level is then S; (N );S; (N );; (1);; (1);;  (1) . * D 0 D  0

(3.1.11)

However, like ; (1), the R -symmetry su!ers from an anomaly at the quantum level. The current   corresponding to the R -symmetry (3.1.10) is  (3.1.12) j0"!M  (  )? ? #I ?( )  MI ? #??( )  M ?? , ? I ?? I ?? I ? I or in four-component form j0 "j #jI "M   #M ?  ? , I I I I   I 

(3.1.13)

 being the Dirac spinor of the quarks and
the four component Majorana spinor of the gluino. The operator anomaly equation for the R-current is g IJNMF? F? . RIj0 "(N !N ) I A D 32
 IJ NM

(3.1.14)

This anomaly equation arises as follows. Eq. (3.1.13) shows that j0 is composed of two parts. The I "rst part is the ordinary chiral current j. The triangle diagram  j(x)J? (y)J@ (z) gives the familiar I I J M

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Fig. 2. Triangle diagrams I ?@ (r, p, q). IJM

contribution !gN /(16
)F? FI IJ?. The second part jI  is formed by the gluino . The anomalous D IJ I triangle diagram is (see Fig. 2) I ?@ (z, x, y)" jI  (z)J? (x)J@ (y) , IJM I J M



I ?@ (r, p, q)(2
)(r#p#q)" dx dy dz e NV>OW>PXI ?@ (z, x, y) . IJM IJM

(3.1.15)

Since  is in the adjoint representation of S;(N ), J? is the current corresponding to a global A J gauge transformation in the adjoint representation, J? "if ?@AM @ A , I I

(3.1.16)

which couples with the gluon "eld A? . Requiring I RIJ? "0, pJI ?@ (r, p, q)"qMI ?@ (r, p, q)"0 , I IJM IJM

(3.1.17)

we obtain the anomalous Ward identity 1 1 p?q@ "2N p?q@ ?@ . (p#q)MI ?@ (p, q, r)"2f ?BAf @BA IJ?@ A IJ?@ MIJ 2
 2


(3.1.18)

Combining (3.1.18) with the anomaly of j , we obtain (3.1.10). I Since there are two anomalous ;(1) transformations, ;(1) and ;(1)  , it is possible to combine  0 them to get an anomaly-free ;(1) R-symmetry. Requiring that the linear combination jI "mjI  #njI 0 0 

(3.1.19)

has vanishing anomaly, using Eq. (3.1.14) and the corresponding equation for jI gives  N !N Am . n" D N D

(3.1.20)

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Table 3.1.1 Anomaly-free R-charges of "elds

  I I 

; (1)

; (1) 

;  (1) 0

; (1) 0

#1 #1 !1 !1 0

#1 #1 #1 #1 0

0 !1 0 !1 #1

(N !N )/N D A D !N /N A D (N !N )/N D A D !N /N A D #1

The simplest choice in (3.1.20) is m"1. The charges of the "elds under this anomaly-free ; (1) 0 symmetry are thus given in terms of the ; (1) and ; (1) charges by 0  N !N AA . R"R # D  N D

(3.1.21)

The quantum numbers of every "eld are listed in Table 3.1.1. We thus have an anomaly-free global symmetry at the quantum level S; (N );S; (N );; (1);; (1) . * D 0 D 0

(3.1.22)

A special consideration should be paid to the N "2 case. Since the fundamental and antiA fundamental representations of S;(2) are equivalent, there is no di!erence between the left- and right-handed quarks. Thus, the theory has 2N quark chiral super"elds QG, i"1,2, 2N , the D D anomaly-free global symmetry is S;(2N );; (1) D 0

(3.1.23)

and the ; (1) charge of Q is (N !2)/N . 0 D D 3.2. Holomorphy of supersymmetric QCD Supersymmetric gauge theory possesses a powerful property: the superpotential is a holomorphic (or anti-holomorphic) function of chiral super"elds Q (or anti-chiral super"elds QI ); Furthermore, the supersymmetric Ward identities determine that some Green functions have a holomorphic dependence on the mass parameter and coupling constant [25,3], and so does the low-energy e!ective Lagrangian. This property plays a key role in looking for the nonperturbative superpotential [32,33]. 3.2.1. Supersymmetric Ward identity In supersymmetric theories, the matter "elds are described by the chiral super"elds. Chiral super"elds have the important property that the product of two chiral super"elds is still a chiral

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super"eld. Thus, a chiral super"eld (y)"(y)#(2? (y)#F(y) , ?

(3.2.1)

where y"x#i M , can be thought of as either a fundamental chiral super"eld or a composite one, and the same goes for the component "elds. Under a supersymmetry transformation the component "elds of a chiral super"eld transform as follows: (a) [QM ? , ]"0, (b) QM ? , ?(x)"!i(2( I)? ?R (x), (c) [QM ? , F(x)]"i(2RI (x) ?? , I I ? (d) [Q , F]"0, (e) Q?, @(x)"(2?@F(x), (f) [Q , (x)]"(2 . ? ? ?

(3.2.2)

Assuming that there is no spontaneous supersymmetry breaking, Q?0"0,

QM ? 0"0 ,

(3.2.3)

we can derive strong constraints on Green functions from the Ward identities corresponding to the above supersymmetry transformations. First, we consider Green functions of the lowest components  (x ) of some chiral super"elds  (x ), here i"1,2, n, G G G G G(x ,2x ),0¹[ (x )2 (x )]0 .  L   L L

(3.2.4)

From item (b) in (3.2.2) and from (3.2.3), we get i 0¹[ (x )2 QM ? , ?(x )2 (x )]0 0"   G G L L (2





R R "0¹  (x )2( I)? ?  (x )2 (x ) 0"( I)? ? G(x ,2, x ) .   L L  L RxI G G RxI G G

(3.2.5)

Note that when we take the derivative outside the ¹-product in (3.2.5), the equal time commutator terms arise but all vanish. Eq. (3.2.5) means that the Green function of the lowest component of chiral super"eld operators is space}time independent. Now we apply this result to supersymmetric QCD. 3.2.2. Holomorphic dependence of supersymmetric QCD In supersymmetric QCD, the quark super"elds Q(x), QI (x) and the gauge "eld strength super"eld = are chiral super"elds. Since the product of two chiral super"elds is still a chiral super"eld, ? = =? and Q QI PG are chiral super"elds, where i and j are #avour indices. Their lowest component ? PH are, respectively, g g ??(x)? (x), (x), ? 32
 32


I PG(x) (x),I G (x) . PH H

(3.2.6)

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Note that here and the following i, j"1,2, N denote #avour indices. Considering the Green D function of these operators, 2  GN (x ,2, x , x GNO,GNOG , ,x )  N N> 2 N>O H 2HN





g g ,0¹ I G   (x )2I GN  N (x) (x )2 (x ) 0 , H  H N> N>O 32
 32


(3.2.7)

we know from (3.2.5) that GNO is space}time independent. Furthermore, we shall show that GNO depends holomorphically on the mass parameters, that is, it depends only on m , but not on G mH, i"1,2, N . The path integral representation of GNO is G D



N O g (x )e\ L , GNO" DX  I GI  I (x )  H I N>J 32
 I J

(3.2.8)

where L is the gauge-"xed e!ective Lagrangian of supersymmetric QCD, X is a shorthand for  all "elds integrated over, including the ghost "elds and their superpartners. From Eq. (3.1.6) we see that the coe$cient FH of mH in the SQCD Lagrangian is the auxiliary "eld of the composite H H anti-chiral super"eld QRHQI R. Hence H



 



R O g N mH GNO" DX  I GI  I (x ) mH dy FH  (x )e\ L H RmH H H H I N>J 32
 H J I

 

 

N O g "0¹  I GI  I (x )  (x ) mH dy FH0 H I N>J H H 32
 I J





mH N O g " H dy0¹  I GI  I (x )  (x ) QM  ,M ? (y)0"0 , H I N>J ? H 32
 2(2 I J

(3.2.9)

where M is the spinor term of QRHQI R and we used item (a) and (the conjugate of ) item (e) in (3.2.2) H H and also (3.2.3). Thus GNO is holomorphic with respect to the parameters m . Moreover, we can H compute its explicit dependence on them. Write the complex parameter m as m "m e ?. Then H H H H






R R R R m GNO" m !mH GNO"!i GNO . H RmH H Rm H Rm R H H H H

(3.2.10)

The m -dependence of the Green function GNO is thus given by the dependence on the phase angle H of the quark mass of the jth #avour. This dependence can be determined by de"ning a ;H(1)  transformation, which is non-anomalous and is explicitly broken by the jth quark mass m : H Pe\ ?,A ,

(I J,  )Pe ?BJH (I J,  ), J J

(I J,  )Pe ?BJH \,A ( J,  ) . J J

(3.2.11)

From the process of combining the ; (1) and ; (1) symmetries to get the anomaly free ; (1) in  0 0 Section 3.1.2, we know that this is possible. It can easily be checked that this ;H(1) symmetry is 

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indeed anomaly free. The only terms in the Lagrangian which are not invariant under the transformation (3.2.11) are the quark mass term of the jth #avour. Performing the transformation of variables (3.2.11) in the path integral (3.2.8) is the equivalent to changing the phase of m : m Pm e\ ?. Thus H H H RGNO "qHGNO , (3.2.12) mH RmH where p#q 1 N qH" !  ( J # J ) . H H G H N 2 A J Integrating Eqs. (3.2.12) we "nd

(3.2.13)

dG ,D dm "  qH H , GNO m H H ,D N 2 2 GNO,GNOG 2  GN "CG 2GN  (m )N>O,A \ J BGJH >BHJH  .  N  N H H H H H H The last equation in (3.2.14) can be expressed as follows:

(3.2.14)

,D N 2 2 N "CG 2GN (, g)  (m )N>O,A , G (3.2.15)  (m J m J )GNOG 2 H HN H HN H G H H J where after taking into account renormalization e!ects we have written the integration constant C as depending on the coupling constant g and renormalization scale  explicitly. We can use dimensional analysis to determine the explicit dependence on . Since ,  and  (I ) have dimensions 1, 3/2 and 1, respectively, so GNO has dimension 2p#3q. Comparing both sides of (3.2.15), we get GN (, g)"CG2 GN (g)N>O\,D ,A  . (3.2.16) CG2 H 2HN H 2HN Furthermore, since the left-hand side of (3.2.15) is the vacuum expectation value of renormalization group invariant operators, its right-hand side should also be expressed in terms of the renormalization group invariant quantities:

  


" exp !

 



E dg E K dg . , m "m exp ! 
(g)
(g)

(3.2.17)

Hence (3.2.15) can be rewritten as






N ,D N>O,A 2 2   GN ,  (m J m J )GNOG GN "(
A D )N>O\,D ,A   m tNOG (3.2.18) G H H 2HN H 2HN , , J  J J where t is a dimensionless constant tensor in #avour space, depending only on p, q, the #avour number N and colour number N . D A

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So far t is undetermined. Since GNO is space}time independent, one can evaluate it in the the limit x !x PR for all iOj. In this limit, if the vacuum is unique, the clustering condition G H implies that the Green function factors into a product of the vacuum expectation values of each composite operator. Applying the clustering property to the Green function GNO, we obtain






O N  I GJ  J  . H J Taking p"0, q"1 in (3.2.18), we have 2  GN " GNOG H 2HO



g  32






(3.2.19)



g ,D ,A  "c (
A D )\,D ,A  m , H , , J  32
 J while taking q"0, p"1 in (3.2.18), we get



(3.2.20)



,D ,A tGJ J . (3.2.21) [(m J m J ) ]I GJ  J "(
A D )\,D ,A  m H H , , J  G H  J Inserting (3.2.19) and (3.2.21) into the left-hand side of (3.2.18) and comparing with the right-hand  2GN factorizes into a product of tensors tGJ , side of (3.2.18), we see that the tensor tNOG H 2HN HJ N 2 . (3.2.22) tNOG 2  GN "  tGJ  N H H HJ J Usually the vacuum is ;(1),D invariant with ;(1) being the rotation group in each #avour space, Q 0"0, i"1,2, N . G D

(3.2.23)

The operator I G  is also ;(1),D invariant and so is tG . Hence tGJ J should be proportional to H H H the identity matrix in #avour space, tG "c G . H ( H

(3.2.24)

In (3.2.20) and (3.3.24), we have introduced two undetermined coe$cients c and c . From the H ( Konishi anomaly, which we introduce next, we can see they are in fact identical. First we have to explain this anomaly and then discuss its consequences. 3.2.3. Konishi anomaly The Konishi anomaly is another important characteristic of supersymmetric gauge theory. In operator form, the anomaly equation is [29], g

QM  , M ? G(x) (x)"!m I G (x)# (x)G . ? H G H H 32
 2(2 1

(3.2.25)

A naive supersymmetric gauge transformation gives only the "rst term (see (3.2.2)). The second term on the right-hand side of the above equation is the anomalous term. This anomaly equation

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can generate a series of anomalous Ward identities when it is inserted into the operators of various Green functions. To prove this anomaly equation, we "rst look at the composite operator mI G(x) (x). Similarly H to regularizing operator products in a gauge invariant way by point splitting [72], one can write it in a gauge-invariant non-local operator form. To keep supersymmetry manifest, it is better to work with super"elds. De"ning OG (x, u, , M ),mQI GP(x, , M );Q (x, u, , M )Q (u, , M ), H P HQ






i V Q ;Q (x, u, , M ),P exp dzI @Q ?DM Q e\4D e4 , P I @ ? 4 S P

(3.2.26)

where P denotes path ordering. In a supersymmetric gauge choice [27], the superspace component A  of the super-gauge potential vanishes, and !1/4 @Q ?DM Q e\4D e4 is the usual Yang}Mills "eld I @ ? ? A . One can easily show that O(x, u, , M ) is indeed gauge invariant under the super-local gauge I transformation, QPe\
Q, QI 2PQI 2e
,

e4"e\
Re4e
,

(3.2.27)

where
(x, , M ) is an arbitrary chiral super"eld. Correspondingly, @Q ?DM Q e\4D e4 transforms as I @ ? follows: @Q ?DM Q e\4D e4Pe\
@Q ?(DM Q e\4D e4#DM Q D )e
. I @ ? I @ ? @ ?

(3.2.28)

Using the fact that @Q ?(DM Q D )
" @Q ? DM Q , D 
"! @Q ?(2i J Q R )
"!4iR
, I @ ? I @ ? I ?@ J I

(3.2.29)

where the de"nition of a chiral super"eld, DM
"0, has been used, one can discard the second term of (3.2.28) in the integration, and thus O(x, u, , M ) is gauge invariant. Now we concentrate on the lowest component of the super"eld operator O(x, u, , M ). In the Wess}Zumino gauge, @Q ?DM Q e\4D e4 M "! @Q ? J Q A "!2A (x) , I @ ? FF I ?@ J I

(3.2.30)

so we have






i V OG (x, u, "M "0) "I GP(x) P exp ! dzI A (z) H 58   I 2 S ,I GP(x);Q (x, u) (u) . P HQ

Q  (u) HQ P (3.2.31)

This is just the ordinary path-ordered integral for gauge invariant non-local operators. We can now de"ne the local product I G(x) (x) as H I G(x) (x),lim OG (x#, x!, "M "0) . H H 58   C

(3.2.32)

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Using the classical equation of motion FI GP(x)"!mI GP(x), one "nds mI G(x) (x)"!lim FHGP(x#);Q (x#, x!) (x!) H P HQ C 1 lim QM , M GP(x#);Q (x#, x!) (x!) "! P HQ 2(2 C 1 "! lim [ QM , M GP(x#);Q (x#, x!) (x!) P HQ 2(2 C !IM ? GP(x#)  Q @Q ?Q (x) (x!)] , ?@ I ?P HQ

(3.2.33)

where we have used the notation Q "? (¹?)Q and the supersymmetry transformations ?P ? P [QM  , A? ]"!i  Q @Q ?? (x), [QM  , ]"0 . ? I ?@ I ? Q?

(3.2.34)

It is possible that the second term does not vanish in the limit P0 since there is a Yukawa interaction vertex iR/(2 in the Lagrangian (3.1.6), so that M (x#)(x)(x!) contains a linear singularity &(J/). One can see this from simple dimensional analysis:



(x) dy ¹[(y)M (x#)](y)¹[(x!)R(y)]&(x)\\&(x)\ .

(3.2.35)

The exact form of the second term of (3.2.33) can be obtained from a simple Feynman diagram calculation in momentum space (see Fig. 3),







dk R Jk ? 1 g J ?(x)?(x) . !i g  Q @Q ?Q (x)@P(x) " Q (2
) RkI (k!m) 4 ?@ I ?P 32
 @

(3.2.36)

Combining (3.2.36) with (3.2.33), one can see that this gives (3.2.25). It is worth mentioning that the super"eld form of (3.2.25) is g 1 DM (QRGeE4Q )"!m QI GQ # =?= G , H G H 32
 ? H 4

(3.2.37)

whose lowest component ("M "0) is just the Konishi anomaly equation. "QReE4Q is called the Konishi supercurrent super"eld, which plays an important role in the operator product expansion in 4-dimensional superconformal "eld theory [47,48]. If one expands the above super"eld equation, one can see that the  component is just the usual ; (1) anomalous equation.  Hence the Konishi current (the lowest component of the Konishi supercurrent) is the superpartner of the ; (1) current. In this sense, the existence of the Konishi anomaly is not strange since the  anomalies also form a supermultiplet. Later when we discuss the superconformal current multiplet, we shall return to this equation.

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Fig. 3. Feynman diagram for Konishi anomaly.

Now we see the consequence of the Konishi anomaly equation. Since the supersymmetry is not spontaneously broken for mO0 [30,31], the operator anomaly equation implies





g  , m I G " G G 32


I G "0, iOj . H

(3.2.38)

Therefore from (3.2.18), (3.2.20)}(3.2.22) and (3.2.24), we have c "c , H (

(3.2.39)

and hence






,D ,A c (m J m J )I GJ  J " H (
A D )\,D ,A  (m ) GJ J . G H  H H J  32
 , , J

(3.2.40)

Thus the Green function GNO can be speci"ed by a single numerical constant c . Once GNO is H determined, most of the other Green functions can also be determined through supersymmetric Ward identities. 3.2.4. Decoupling relation Finally we brie#y introduce the decoupling theorem in supersymmetric QCD since it will be widely used in discussing the reasonableness of the non-Abelian electric}magnetic duality [14]. The decoupling theorem is a general result in "eld theory, it describes the e!ects of the heavy

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particles in the low-energy theory [73]. In general this theorem states that if, after integrating out the heavy particles, the remaining low-energy theory is renormalizable, the ewects of the heavy particles appear either as a renormalization of the coupling constants in the theory or are suppressed by powers of the heavy particle masses. In electroweak model, we have some obvious examples such as the decoupling of the heavy =! and Z, their e!ects at low-energy either renormalize the electric charge or are suppressed. Now we apply this decoupling theorem to supersymmetric QCD. We assume that one #avour, say the N th #avour, becomes heavy, i.e. m D <
. In this large-m D limit, , D ,





g  32






,D ,A "c (N , N )(
A D )\,D ,A  (m ) H A D , , J  ,A ,D J





K,D <
,A ,D ,D \ ,A P c (N , N !1)(
A D )\,D \,A  (m ) , J  H A D , , \ J

(3.2.41)

where the explicit #avour number N and colour number N dependence of c is indicated in order D A H is the renormalization group invariant to show the decoupling of the N th heavy #avour.
A D D , , \ scale of supersymmetric S;(N ) QCD with N !1 #avours. We shall see that the e!ect of the N th A D D heavy #avour is re#ected in
A D , since the running coupling constant depends on it. At the , , \ scale that the decoupling takes place, these two theories should coincide. This means that the coupling constants g D (q) and g D (q) should be identical at the scale q"m D , , , \ , g D (m D )"g D (m D ) . , , , \ ,

(3.2.42)

From the one-loop
-function of N"1 supersymmetric QCD, g g
A D "! (3N !N )"!
, , , A D 16
 16
 

(3.2.43)

and the running coupling constant 4

q "  ln , g(q) 4



(3.2.44)

we obtain at q"m D , m m D , (3N !N ) ln ,D "[3N !(N !1)] ln . A D A D
A D
A D , , , , \

(3.2.45)

Thus






m D ,A \,D > , .
A D "
A D , ,
, , \ A D , ,

(3.2.46)

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Later we shall see that this relation imposes a restrictive constraint on the form of the nonperturbative superpotential. This relation in fact gives a link between the energy scales of theories with di!erent number of #avours. 3.3. Classical moduli space of supersymmetric QCD 3.3.1. Classical moduli space A "eld theory, be it classical, quantum or a low-energy e!ective theory, is in general one of a whole family of theories, parametrized by a number of parameters. Especially important for us are the vacuum expectation values of scalar "elds, called moduli, which can range over a moduli space. Let us illustrate this concept by considering the classical moduli space of a simple theory, the Georgi}Glashow model with gauge group SO(3). The classical action is

 



1 1  S" dx ! G?IJG? # DI?D ?! (!a) , IJ I 4 2 4

(3.3.1)

where G? "R =? !R =? #g?@A=@ =A , the Higgs "eld  is in the adjoint (vector) representaIJ I I J I I I tion of gauge group of SO(3), D ?"R ?#g?@AA@ A and "??. The classical ground sates I I I are given by =?"0 (up to a gauge transformation) and "a.

(3.3.2)

The classical moduli space is thus the 2-sphere (3.3.2). Each point on this space de"nes a semiclassical quantum "eld theory with the chosen point being the expectation value of the Higgs "eld in the vacuum state of the theory. The (semi)classical dynamics of all these theories are equivalent. Quantum e!ects will, in general, change the dependence of the (e!ective) theory on the moduli, and might even alter the topology of the moduli space. For the Georgi}Glashow model, the structure of the quantum moduli space is not known, but in supersymmetric theories we can often make de"nite statement about the quantum moduli space. 3.3.2. Classical moduli space of supersymmetric QCD We now turn to supersymmetric QCD. We "rst consider the classical moduli space and then turn to the quantum case. Recall the Lagrangian (3.1.6) of supersymmetric QCD. The scalar potential is g <" D?D? , 2 D?"HP(¹?)Q  !I PI (¹?)Q I HI . / P /Q / P /Q  To emphasize that a "eld is a component of the chiral super"eld Q (QI ), we add an index Q (QI ).

(3.3.3)

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Since the chiral super"elds Q and QI have both colour and #avour indices, one can arrange them in the form of an N ;N matrix according to #avour and colour indices: A D



 Q  

QI   

Q,D  

2 \

2 Q,DD , , \ 

(Q)" Q D ,  Q A ,

\

QI ,D  

2 QI ,DD . , \ 

(QI )" QI  D ,  QI  A ,

Q,DA ,

2

2

2

(3.3.4)

QI ,DA ,

Both (Q) and (QI ) can be globally rotated in the colour and #avour spaces to make them diagonal since the Lagrangian is globally S;(N ) and S;(N ) invariant. The D-#atness condition D?"0 A D does not lead to  " I "0. This is because, unlike ordinary QCD, supersymmetric QCD is very / / sensitive to the relative number of #avours N and colours N . Depending on the numbers of D A #avour and colour, the restrictions on Q and QI posed by the D-#atness conditions are di!erent. In the following we give a detailed analysis of the classical moduli spaces for di!erent ranges of N D and N : A 3.3.2.1. N (N . In this case, since the chiral super"eld matrix Q and QI can be rotated in colour D A space and #avour space, we can always reduce the matrices Q and QI to the following diagonal forms:



 0

2

0

a

2

0



 

a  0

\



(Q)" 0

0

0

0

2 a D , , 2 0





\

0

0

2

a  0

a ,2, a D O0 ,  ,

0

2

0

2

0



a  

\



(QI )" 0

0

0

0

2 a D , , 2 0







\



0

0

0

2

0

a ,2, a D O0 .  ,

(3.3.5)

The D-#atness condition in this case requires  "I RI , / /

(3.3.6)

whose super"eld form is Q"QI .

(3.3.7)

If a "a O0, the gauge symmetry (S;(N ) symmetry) will be spontaneously broken and the G G A super-Higgs mechanism will occur: some of the chiral super"elds will be eaten up and the same number of gauge "elds and their partners will become massive.

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Now the problem is what kind of quantity describes this D-#at moduli space. From the viewpoint of dynamics, this means how the low-energy dynamics is described. According to the idea of the e!ective "eld theory [75], a general method to get the low-energy e!ective action is to integrate out the heavy modes (massive "elds), the light modes (massless "elds) being the quantities describing the low-energy dynamics. Therefore, for the case at hand, a S;(N ) global gauge A invariant quantity would be ,A MG ,  QI QPG,QI QG, i, j"1,2, N . HP H D H P

(3.3.8)

In the following we give further arguments why this assumption is reasonable. (3.3.5) and (3.3.7) give MG " a  a  "a a  "a . H G PG H PH G H GH G GH P

(3.3.9)

(3.3.9) can be written in the explicit matrix form




a  0

0

2

0

a  

2

(MG )" H 

0

\



0

0

2 a D ,



.

(3.3.10)

Furthermore, (3.3.5) and (3.3.6) imply that the gauge symmetry S;(N ) is broken to S;(N !N ). A A D The original gauge group has N!1 generators and the remaining gauge group has A (N !N )!1 generators, the number of eaten chiral super"elds is thus [N!1]! A D A [(N !N )!1]"2N N !N . This number is also the number of particles becoming massive. A D A D D After these massive particles have been integrated out, only the massless particles are left. The number of the original matter "elds is 2N N (Q , QI , i"1,2, N , r"1,2, N ), so there are D A GP GP D A 2N N !(2N N !N )"N massless particles left. This is exactly the number of degrees of D A A D D D freedom of MG , thus we can use MG to describe the moduli space. Later we shall return to the H H dynamics. 3.3.2.2. N 5N . In this case, after an appropriate rotation in #avour space and colour space, the D A (Q) and (QI ) matrix takes the following diagonal form:




(Q)"

a  0




0

2

0

0 2 0

a

2

0

0 2 0



 





0

0

2 a A ,

a ,2, a A O0 ,  ,

\

\



0 2 0

a ,2, a A O0 .  ,

,

(QI )"

a  0



0

2

0

0 2 0

2

0

0 2 0



a  

\





0

0

2 a A ,

\



,

0 2 0

(3.3.11)

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157

Let us analyze the D-#atness condition, ,D ,A D?"   [HP (¹?)Q  !I PI (¹?)Q HI ] P /GQ /G P /GQ /G G PQ ,A ,A "   [H P(¹Y?)Q   !I  P (¹Y?)Q I H  ] ?G G P ?G GQ ?G G P ?G GQ G PQ ,A ,A "  [ (¹Y?)G !I  (¹Y?) G]"  ( !I /)(¹Y?) G , G ?G G ?G ?G ?G G G G

(3.3.12)

where  (I ) is the scalar component of chiral super"eld a , ¹Y? is the rotated ¹? in colour space, ?G ?G G ¹Y?";R¹?; with ; being certain unitary matrix. So the the condition de"ning a D-#at con"guration is  !I "C (constant) , ?G ?G D?"C Tr ¹Y?"C Tr ¹?"0 ,

(3.3.13)

where we have used the fact that ¹? is the generator of gauge group S;(N ). Since N 5N , the A D A possible gauge invariant chiral super"eld operators parameterizing the moduli space are not only the meson-type chiral super"eld operators, but also the baryon-type chiral "eld operators: MG "QI ) QG , H H 1 P 2P,A QG 2QG,,AA "Q G 2QG,A , BG 2G,A " P P N! A 1 BI H 2H,A " Q 2Q,A QI H 2QI H,,AA "QI H 2QI H,A , Q Q N! A

(3.3.14)

where i , j "1,2, N are #avour indices and r , s "1,2, N , are colour indices. Since the I I D N N A #avour indices in the baryons are antisymmetric, the number of BG 2G,A (or BI G 2G,A ) "elds is N! D C,AD " . , N !(N !N )! A D A

(3.3.15)

We shall see that not all of M, B and BI can be independently used to label the moduli space, there are additional constraints imposed on them. We "rst consider two simple cases. a. N "N . Since now the S;(N ) gauge symmetry is completely broken, the number of the D A A eaten particles (also the number of particles becoming massive) is N!1"N !1. The original A D number of chiral super"elds is 2N N "2N , so the dimension of moduli space is 2N !(N !1) D A D D D "N #1"N#1. However, from (3.3.14), the number of mesons and baryons is N #2, so D A D they are not independent in parameterizing the moduli space. From the de"nitions of these mesons

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and baryons, one can easily see that they satisfy the constraint det M"BBI .

(3.3.16)

This equation is obvious from the diagonal form (3.3.11), since N "N , B (BI ) has only one D A non-vanishing component, M"q 2q,

MG " a  a  "a a  , H G GP H PH G H GH P

,D det M"  a a "a a 2a A a A , G G   , , G

B"a 2a A ,  ,

BI "a 2a A .  ,

(3.3.17)

We shall see that at the quantum level, the constraint (3.3.16) will be modi"ed owing to the non-perturbative quantum correction coming from instantons. b. N "N #1. In this case, like in N "N , the S;(N ) gauge symmetry is also completely D A D A A broken. From (3.3.11), the number of the eaten super"elds is N!1"(N !1)!1, the number A D of the original chiral super"elds is still 2N N "2N (N !1), so the number of massless particles is D A D D 2N (N !1)![(N !1)!1]"N . However, the number of parameters describing the moduli D D D D space is N #2C,DD \"N #2N , so 2N chiral variables in M, B and BI are redundant. However, , D D D D one can exactly "nd 2N constraints to remove them. First we write the baryon operators in D (3.3.14) in their Hodge dual form 1  2 BG, 2G,A , BM ,A> ,A> 2 ,D , 2 G G G (N !N )! G G,A G,A> G,D D A 1 G 2G,A G,A> 2G,D B , 2 ,A . BMI G,A> G,A> 2G,D , G G (N !N )! D A

(3.3.18)

For the case at hand N "N #1, the baryon operators are B (BI ). With de"nition (3.3.18), one D A G G can easily "nd that they satisfy the constraints BM MG "0, G H

MG BIM H"0, H

M I G "B BI G , H H

(3.3.19)

where M I is the matrix whose elements are de"ned as (!1)G>H; the determinant of the matrix obtained from M by deleting the ith row and the jth column. Eq. (3.3.19) can be written in the following explicit form: 1 G 2G,A G  2 ,A MH 2MH,,AA "B BI G. H H H G H G N! A

(3.3.20)

The constraint equations can be checked easily, 1 BM MG " ( P 2P,A QG 2QG,,AA QG )QI P "0 . P P P H G H N ! GG 2G,A A

(3.3.21)

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159

Using the fact that MG M I I "det MG , M I G "det M(M\)G , I H H H H we can formally write (3.3.19) in the following form:

(3.3.22)

det M(M\)G "BGBI . (3.3.23) H H Note that since in this case det M"0, (3.3.23) is only a formal expression and the true meaning is given by (3.3.19). Finally, we consider the special case, N "2. In the classical direction, due to the general relation A (3.3.13), the matrix from of the quark super"eld is




(Q)"



a 0 2 0 2 0 0 a 2 0 2 0

.

(3.3.24)

The classical moduli space is parameterized by the gauge invariant

(3.3.25)

  2 ,D
(3.3.27)

3.4. Quantum moduli space and low-energy non-perturbative dynamics In the previous section we have discussed various classical moduli spaces parametrized by classical composite "elds } mesons and baryons. In some cases these composite "eld should satisfy some constraints. At the quantum level, the moduli space will be parametrized by the vacuum expectation values of the corresponding composite operators. The constraints may be changed due to possible non-perturbative quantum corrections. The quantum moduli space may di!er from the classical one and hence the corresponding physical consequences may also change. To see the quantum e!ects on the moduli space, it is necessary to investigate the non-perturbative low-energy dynamics. Since the moduli spaces vary with the relative numbers of colours and #avours, we shall discuss them according to di!erent ranges of the numbers of #avour and colour. 3.4.1. N (N : erasing of vacua D A Consider the non-perturbative dynamics for this case. Recall that the global symmetry of supersymmetric QCD at the quantum level is S; (N );S; (N );; (1);; (1). The dynamically * D 0 D 0

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generated superpotential should respect this symmetry. Under the chiral symmetry S; (N );S; (N ), QI &(0, NM ) and Q&(N , 0), MG &(N , NM ), and the simplest invariant under * D 0 D D D H D D S; (N );S; (N ) is the determinant det M. Let us determine the quantum numbers of det M * D 0 D under the ;(1) transformations. Since the ;(1) quantum numbers are additive, Table 3.1.1 gives B(MG )"B(Q)#B(QI )"0, H B(det M)"0,

N !N A , R(MG )"R(Q)#R(QI )" D H N D

N !N A "2(N !N ) . R(det M)"2N D D A D N D

(3.4.1)

To construct the superpotential, we need another quantity with mass dimension. The natural choice is the non-perturbative dynamical scale
. It is S; (N );S; (N ) invariant, and its * D 0 D transformation under the ;(1) symmetries is related to the vacuum angle . We can argue this from the perturbative viewpoint despite the fact that
is a non-perturbative energy scale. The running gauge coupling in perturbative QCD is g(q )  g(q)" , 1#g/(16
)
ln(q/q )   4
4
g(q ) q 4

q
q 
ln  " " 1# #  ln  ,  ln , g(q) g(q ) 16
  q g(q ) 2
q 2

  

(3.4.2)

where
is the coe$cient of one-loop
-function and
is the perturbative energy scale.   As a result,




8
 q @ "ln ,
@ "q@ e\
 EO .  g(q)


(3.4.3)

For the non-perturbative scale
, the  parameter will arise since it is associated with the complex coupling constant (3.1.3)
@ "q@ e\
 EO > F"q@ e

E>F
 ,q@ e
O .

(3.4.4)

Thus
@ should transform as e F under the ;(1) transformations. Note that the non-perturbative scale
is a complex parameter. For clarity the quantum numbers of det M and
@ are listed in Table 3.4.1. Since the low-energy superpotential must be a holomorphic function of Q and QI (i.e. det M, not QM or QIM ) and the scale parameter
, it should also be S; (N );S; (N );; (1) invariant. Its * D 0 D R charge is 2 since the superpotential is the F-term of the e!ective action

 

= M & dx d = .  

(3.4.5)

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161

Table 3.4.1 Quantum numbers of det M and
@

det M
@

; (1)

; (1) 

;  (1) 0

; (1) 0

0 0

2N D 2N D

0 2(N !N ) A D

2(N !N ) D A 0

Thus this superpotential must be composed only of det M and
. In addition, from (3.4.5) the mass dimension of = should be 3 owing to the mass dimensions } [\]"[d]"1/2. Since the  dimension of
is [
]"1, the coe$cient of one-loop beta function in N"1 supersymmetric QCD is
"3N !N , [MG ]"2, [det M]"2N , the R-charge-R()"R(d)"!1, the only possible  A D H D combination with R"2 and mass dimension 3 should be







@ ,A \,D  , = "C(N , N )  A D det M

(3.4.6)

where C(N , N ) is a dimensionless constant depending only on N and N , and the factor A D D A 1/(N !N ) comes purely from dimensional analysis. The superpotential (3.4.6) is called the A D A%eck}Dine}Seiberg (ADS) superpotential [32,33]. Owing to the famous non-renormalization theorem in supersymmetric theory, this superpotential cannot be generated from perturbative quantum corrections. However, the non-renormalization theorem only concerns at perturbation theory and imposes no restrictions on non-perturbative quantum corrections. Thus this superpotential can only possibly come from instanton contributions and the constant C can be determined from a one-instanton calculation. The superpotential can be further determined by considering various limiting cases. First, we consider the case that the expectation value of the N th #avour Q D "QI D "a becomes very , D D , large. Using the diagonal form (3.3.5) and the fact that det M is a rotational invariant, as well as the D-#atness condition (3.3.7), we have ,A ,A ,D QI QH"  QI QPH"  a  a PH"  a  a PH"a H , G GP G GP H G GP H G G P P P det M"det QI ) Q"a 2a D a D "det QI  ) Qa D ,  , \ , ,

(3.4.7)

where Q or (QI ) only contains N !1 #avours. At an energy scale less than a D , the N !1 #avours D D , can be thought as the light #avours since a D is very big. Compared with a D , a , i"1,2, N !1, , G D , can be regarded as approximately zero on this energy scale. Due to the super-Higgs mechanism, the S;(N ) gauge theory with N #avours is broken to S;(N !1) supersymmetric QCD with N !1 A D A D #avours. According to (3.4.2), at the energy q'a D the running coupling is , 4
3N !N q D ln , " A g(q) 2



(3.4.8)

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while at the energy scale q(a D , the running coupling is , 4
3(N !1)!(N !1) q A D " ln , g(q) 2

*

(3.4.9)

since now the theory becomes supersymmetric QCD with gauge group S;(N !1) and N !1 A D #avours, the N th heavy #avour having been integrated out. At the energy q"a D , the running D , coupling constants should match, 4
3N !N a 3(N !1)!(N !1) a D D ln ,D " A D " A ln , .

g(a D ) 2
2
* ,

(3.4.10)

Thus one can obtain the relation between the energy scales,
,A \,D .
,A \\,D \" * a D ,

(3.4.11)

Requiring that the ADS potentials should coincide at q"a , we have from (3.4.6), (3.4.7) and D (3.4.11),












,A \,D ,A \,D 
,A \\,D \ ,A \,D  "C(N !1, N !1) * . C(N , N ) A D A D det QI  ) Qa det QI  ) Q ,D

(3.4.12)

This implies C(N , N )"C(N !1, N !1)"C(N !N ) . A D A D A D

(3.4.13)

i.e. C(N , N ) should only be a function of N !N . A D A D Further, the explicit form of C(N !N ) can be determined from another limit: giving Q D and A D , QI D a large mass by adding a mass term (only the holomorphic part) to the superpotential at tree , level, = "mM,DD "mQ,D ) QI D . , , 

(3.4.14)

Similarly to the previous case, consider the energy scale m. When the energy q'm, the theory is a S;(N ) supersymmetric QCD with N #avours and the running coupling constant is (3.4.8). A D When the energy q(m, the theory is S;(N !1) supersymmetric QCD with N #avours. The A D running coupling constant is now 3(N !1)!N q 4
A D ln " . 2

g(q) *

(3.4.15)

Matching the coupling constants at q"m gives 4
3N !N m 3(N !1)!N m D ln " A D ln " A . g(m) 2

2

I *

(3.4.16)

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163

Hence we obtain
I ,A \,D \"m
,A \,D . *

(3.4.17)

Now with the mass term mM,DD for the N th #avour, at the energy q'm, the superpotential is , D







@ ,A \,D  = "C(N !N ) #mM,DD .  A D det M ,

(3.4.18)

As we know, the F-term associated with the superpotential is given by R= F" , R -

(3.4.19)

where  is the lowest component of an elementary or composite chiral super"eld O. Since the form of the superpotential in terms of chiral super"eld is the same as that of its lowest component, in the following we shall discuss the F-#atness condition of the corresponding chiral super"eld to manifest supersymmetry. Unbroken supersymmetry requires that the F-term must vanish (i.e. F-#atness). The F-#atness conditions for M D and M D , R= /RM D "0 and R= /RM D "0,  ,G G,  G, ,G lead to M

,D G

M D "0 , G,

"0,

(3.4.20)

where we have used that for a matrix M  1 Tr M\M" ln det M"  det M, det M

R det M "det MM H\ . G RMG H

(3.4.21)

The meson operator hence takes the following form:




M I

0



, det M"det M I M,DD . , M,DD , As a consequence, the superpotential (3.4.18) becomes M"

0





 

(3.4.22)


@ ,A \,D  #mM,DD = "C(N !N ) ,  A D det M
@ ,A \,D  (M,DD )\,A \,D #mM,DD . "C(N !N ) , , A D det M I  This formula can be derived as follows: det (M#M)  ln det M"ln det (M#M)!ln det M"ln det M "ln det (1#M\M)&ln(1#Tr M\M)&Tr M\M .

(3.4.23)

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The F-#atness condition for M,DD , ,






R= C(N !N )
@ ,A \,D  A D (M,DD )\ >,A \,D  #m"0 , " , RM,DD N !N , A D det MI

(3.4.24)

gives








m(N !N ) \,A \,D >,A \,D  A D M,DD " , C(N !N ) A D






@ >,A \,D  . det M I

(3.4.25)

Inserting this expectation value into the superpotential (3.4.23), we get





  


m(N !N ) >,A \,D 
@ >,A \,D  A D = "C(N !N )  A D C(N !N ) det M I A D



#m



"

 

m(N !N ) \,A \,D >,A \,D  A D C(N !N ) A D






@ >,A \,D  det M I



m
@ >,A \,D  C(N !N ) ,A \,D >,A \,D  A D (1#N !N ) . A D N !N det M I A D

(3.4.26)

Using (3.4.17), we can write the superpotential as follows:











@I   ,A \,D \ C(N !N ) ,A \,D  ,A \,D \

* A D [N !(N !1)] , = " A D  N !N det M I A D

(3.4.27)

where
I "3N !(N !1) is the one-loop beta function coe$cient of S;(N !1) supersymmetric  A D A QCD with N #avours. Recalling that when m becomes big, the theory will become an S;(N ) D A theory with N !1 #avours. Requiring that (3.4.27) leads to the correct superpotential in the D low-energy theory (q(m), we must have C(N !N )"(N !N )C,A \,D  , A D A D

(3.4.28)

with C being a universal constant. Hence we get a more transparent form of the superpotential







,A \,D ,A \,D  = "(N !N )C,A \,D  .  A D det QI ) Q

(3.4.29)

The universal constant C can be determined by a concrete instanton calculation. Here we only cite the result of Ref. [80]. For N "N !1, the superpotential (3.4.29) is proportional to the D A one-instanton action and thus the constant C can be exactly computed in a one-instanton background. In particular, in this case the gauge group S;(N ) is completely broken since A det MO0. There is no infrared divergence and the instanton calculation is reliable. Ref. [80] has presented a detailed calculation in the dimensional regularization method and in the modi"ed

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165

minimal subtraction scheme, and the result shows that C"1. Thus, for the case of N (N we D A "nally obtain the exact ADS superpotential,







,A \,D ,A \,D  . = "(N !N )  A D det QI ) Q

(3.4.30)

Note that this superpotential is the Wilsonian e!ective potential due to the scale
[81]. For the case N (N !1, this superpotential is associated with the gaugino condensate of the unbroken D A S;(N !N ) gauge group. A D For N "2, the meson matrix < is a 2N ;2N antisymmetric matrix, det < is not the simplest A D D gauge and global invariant to constitute the superpotential, since it can be written as the square of a simpler invariant, the Pfa$an of <, 1   <   <   2< ,D\ ,D , Pf <"(det <" . GG GG G G 2,D N ! D .

(3.4.31)

where P denotes the permutation i ,2, i D  and  the signature of P. A similar procedure to the  , . derivation of (3.4.30) yields the dynamical superpotential of the S;(2) case,







\,D \,D  . = "(2!N )  D Pf <

(3.4.32)

Let us see what are the physical consequences of the dynamical superpotential (3.4.30). As we know, the relation between the usual potential and the superpotential is

 

R=   , <"F " /GP R /GP









R= R
,A \,D ,A \,D \ F "  "(N !N ) QI ) Q det QI ) Q Tr (QI ) Q)\ GP A D det QI ) Q RQ RQ GP GP





1 ,A \,D  Q\ "(N !N )(
,A \,D ),A \,D \ GP A D det QI ) Q






1 1 ,A \,D  & , Q det QI ) Q

(3.4.33)

where (3.4.21) was employed again. Eq. (3.4.33) shows that the dynamically generated superpotential leads to a squark potential, which tends to zero only when det MPR. Therefore, the quantum theory does not have a stable ground state. In classical theory we have a vacuum con"guration, but at the quantum level no vacuum state exists! Finally, we consider the massive case. The mass term is one part of the tree-level superpotential, = "Tr m ) M"mH MG .  G H

(3.4.34)

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In the weak coupling and small mass limit, the full superpotential is







,A \,D ,A \,D  #mH MG . = "= #= "(N !N ) G H    A D det QI ) Q

(3.4.35)

The vacua are still labeled by M,M, which is determined by the F-#atness condition






R=
,A \,D ,A \,D \ R 1  "
,A \,D #mH G det M RMG RMG det M H + H




"!




,A \,D ,A \,D  (M\) H#m H"0 . G G det M

(3.4.36)

This gives




mG " H




,A \,D ,A \,D  (M\)G , H det M




det m"









,A \,D ,D ,A \,D  1 1 ,A ,A \,D  , "(
,A \,D ),D ,A \,D  det M det M det M

1 "(det m),A \,D ,A (
,A \,D )\,D ,A . det M

(3.4.37)

Taking 1/det M back into (3.4.36), we get mG "(
,A \,D det M),A (M\)G , H H MG "[(det m)
,A \,D ],A (m\)G . H H

(3.4.38)

When N "2, the quark mass term is w "m
(3.4.39)

Now we consider the case q(mG . This means that the matter "elds get very big masses and H hence will decouple. The theory will become an S;(N ) Yang}Mills theory. (3.4.35) and (3.4.37) give A the full superpotential of this case, =(m) "(N !N )[
,A \,D det m],A #N [(det m)
,A \,D ],A D  A D "N [(det m)
,A \,D ],A "N
 , A A *

(3.4.40)

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where
 "(det m
,A \,D ),A is the energy scale of the low energy S;(N ) Yang}Mills theory, * A a many-#avour generalization of (3.4.17). Comparing with (3.2.41), one can see that the superpotential is generated by gaugino condensation in the low energy S;(N ) Yang}Mills theory. Thus in the A case that N (N !1, the superpotential is associated with gaugino condensation, while when D A N "N !1, the superpotential arises from instanton contributions. D A Furthermore, we can show that in the massive case, the Wilsonian e!ective superpotential (3.4.30) is the same as the 1PI e!ective superpotential. Let us "rst explain the de"nition of 1PI e!ective superpotential in a general supersymmetric theory. Consider a supersymmetric theory with the tree-level superpotential = " J XG .  G G

(3.4.41)

XG can be fundamental or composite super"elds or their gauge invariant polynomials. (3.4.41) is similar to the source terms in the usual functional integration with J being the background G external sources. The generating functional is




 



Z[J ]"e % (G " D[ f (X )] exp iS#i dx d J XG , G G G G

 

M [J] . G[J]"2# dx d =(J),2#=

(3.4.42)

G[J] is the generating functional of connected Green functions. Here we only write out its part related with the quantum superpotential. Correspondingly, =(J) is the connected superpotential. Using the expectation value calculated from = M (J) = M (J) "XG,XI . G J G

(3.4.43)

If the omitted part (2) is independent of J , XI is the usual vacuum expectation value in the G G presence of the external sources J , G




 



G(J) " D[ f (X )]X exp iS#i dx d J XG . XI " G G G G J G G

(3.4.44)

Performing a Legendre transformation, we can get the 1PI e!ective action for XI : G

 

   




 (XI G)" G(J)! dx d J XI G  G (G G " 2# dx d =(J)! J XI G G G



(

, G

(3.4.45)

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where J are solutions to (3.4.43). Correspondingly, the dynamical superpotential part is G

 




dx d =(J)! J XI G , (3.4.46) G (G G and obviously, =(J ) can be obtained from = (XG) by the inverse Legendre transformation, G  = M (XI )" 

 

= M (J )"= M (XI G)# dx d J XI G , G G  G where XI G is the expectation value of the operator XG satisfying the following equation: R= M  #J "0 . G RXI G The 1PI e!ective potential is de"ned as

(3.4.47)

(3.4.48)

 

(3.4.49) = M (XI , J)"= M (XI G)# dx d J XI G . G   G This procedure is similar to the calculation of e!ective potential [55,57]. For the case at hand, X"MG , J"mH . From (3.4.38) and (3.4.40), we obtain H G R(det m) R= (m)  "[(det m)
,A \,D ],A \
,A \,D RmG RmG H H "[(det m)
,A \,D ],A (m\)H "MH . (3.4.50) G G With the de"nition (3.4.46), using (3.4.50), (3.4.40) and the second equation in (3.4.37), we have = (M)"N
 !MG mH "N
 !(det m
,A \,D ),A (m\)G mH  A * H G A * H G A D A "(N !N )[(det m)
, \, ], A D
,A \,D ,D ,A \,D 
,A \,D ,A "(N !N ) A D det M det M "(N !N ) A D





 


,A \,D ,A \,D  . det M



(3.4.51)

Comparing (3.4.51) with the ADS superpotential (3.4.30), which is a Wilsonian e!ective superpotential, we can see they are identical. Therefore, in the massive case, the 1PI e!ective superpotential is the same as the Wilsonian e!ective superpotential. 3.4.2. N "N : Conxnement with chiral symmetry breaking or baryon number violation D A We have seen that in the case N (N , the non-perturbative superpotential lifts the vacuum D A degeneracy. All the classical vacua disappear. What the situation for N 5N ? We shall see that in D A

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this case no non-perturbative superpotential can be generated dynamically and hence the vacuum degeneracy remains. The reasons are as follows: For the case N "N , Table 3.1.1 shows that the R-charges of the chiral super"eld Q(QI ) and of D A
@ "
,D both vanish. However, the superpotential should have R-charge 2, since it is an F-term. Thus in this case it is not possible to construct a superpotential. For the case N 'N , considering only the R-charge and dimensionality, we could have D A a dynamically generated superpotential




=J




,A \,D ,A \,D  . det M

However, since N !N (0,
,A \,D will be in the denominator of the superpotential, and this A D cannot match the expression generated by instantons. It is known that the contribution from instantons is proportional to
,A \,D [80]; it is not possible to have a superpotential of the form
\,A \,D . In particular, when N 'N , from the previous diagonal form, det M"0, no nonD A perturbative superpotential can be generated and the D-#atness still remains. However, for N 5N , some more interesting phenomena will arise. First, we see that in the case D A N "N , although the vacuum degeneracy cannot be lifted, owing to non-perturbative quantum D A e!ects, the quantum moduli space will be di!erent from the classical one. This is re#ected in the change of the constraint, det M!BI B"
,A ; Pf <"
 for N "2 . A

(3.4.52)

The above constraints must be manifested in the low-energy e!ective Lagrangian. One natural way is to introduce a Lagrange multiplier "eld X to add the following superpotentials to the e!ective Lagrangian: = "X(det M!BI B!
,A ) ;  = "X(Pf
(3.4.53)

The reasonableness of the modi"ed constraint (3.4.52) can be argued as follows. We "rst consider a superpotential at tree level by adding a large mass term for the N th #avour, D = "mM,DD . , 

(3.4.54)

Since for N "N , no dynamical superpotential is generated, this tree level superpotential should D A be the full superpotential. At the energy q(m, after integrating out the N th #avour, the theory D is an S;(N ) gauge theory with N !1 #avours. A dynamical e!ective superpotential (3.4.30) is D D generated by instanton contributions,
,A \,D \ m
,A " , = " *  det M I det M I

(3.4.55)

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where we have used (3.4.17) and M I gets contributions from the N !1 light #avours. The F-#atness D conditions R= /RM D "R= /RMI D "0 (i(N ) lead to  G, D  G, M I 0 . (3.4.56) M,D "MG D "0, M" G , 0 M,DD , This gives




det M"det M I M,DD , ,



det M M,DD " . , det M I

(3.4.57)

Inserting (3.4.57) back into (3.4.54), we obtain m det M = " .  det M I

(3.4.58)

Comparing (3.4.58) with (3.4.55), one can see that only by choosing det M"
,A , can one get the low-energy e!ective superpotential. This is the case for BI B"0. In the case that BI BO0, it can also be proved that (3.4.52) is satis"ed [84]. Since the right-hand side of (3.4.52) is proportional to the one-instanton action [32,33], the quantum modi"cation of the classical constraint must arise from the one-instanton contribution. The quantum constraint (3.4.52) has important physical consequences. The singular point B"BI "M"0 (M"0 means that the eigenvalues of M vanish) has been eliminated by quantum e!ects through the generation of a mass gap since the point B"BI "M"0 does not satisfy the constraint. A vivid explanation was given by Intriligator and Seiberg by considering a twodimensional surface de"ned by X>" in three-dimensional space [16]. If X>"0, then either X"0 or >"0, and the surface is X-plane or >-plane. If O0, these two cones are smoothed out to a hyperboloid, and the origin is expelled from the surface. So the only massless particles are the moduli, the quantum #uctuations of M, B, BI satisfying the constraint. In geometric language, they are the tangent vectors of the surface determined by the constraint (3.4.52) in (composite) chiral super"eld space. In the region of large M, BI and B, the gauge symmetry is spontaneously broken and the theory is in the Higgs phase. In the region of small M, BI and B (near the origin), the theory is in the con"nement phase due to the quantum constraint (3.4.52). In particular, the anomaly-free global symmetry S; (N );S; (N );; (1);; (1) is broken since now the origin * D 0 D 0 B"BI "M"0 is not on the quantum moduli space. Di!erent points on the quantum moduli space exhibit di!erent dynamical pictures. In the following we shall consider two typical points in the quantum moduli space: 1. MG "
G , B"BI "0: Conxnement and chiral symmetry breaking. Obviously, this point lies H H in the quantum moduli space. In this case a quark condensation occurs since MG "
G O0, so H H the chiral symmetry S; (N );S; (N ) is spontaneously broken to the diagonal S; (N ). How* D 0 D 4 D ever, the ; (1);; (1) symmetry still remains. Thus the breaking pattern is 0 S; (N );S; (N );; (1);; (1)PS; (N );; (1);; (1) . (3.4.59) * D 0 D 0 4 D 0  Strictly speaking, one should write MG "
G , B"BI "0. H H

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Table 3.4.2 S; (N );; (1);; (1) quantum numbers of elementary and composite "elds 4 D 0

Q QI  / I /  M B BI  +  I

S; (N ) 4 D

; (1)

; (1) 0

N D NM D N D NM D 1 N !1 D 1 1 N !1 D 1 1

1 !1 1 !1 0 0 N D !N D 0 N D !N D

0 0 !1 !1 #1 0 0 0 !1 !1 !1

Let us analyze the transformation behaviours of M, B and BI under S;(N ) ;; (1);; (1). From D4 0 MG "QI G ) Q one may naively think that the number of the mesons is N . However, since we are H H D considering quantum #uctuations of the moduli "elds around the expectation values MG "
G , H H

B"BI "0 ,

(3.4.60)

the #uctuation matrix MG !
G should be traceless, H H Tr(MG !
G )"0 . H H

(3.4.61)

Hence there are actually N !1 (super-)mesons; the #uctuations MG span the adjoint representaD H tion of the vector group S; (N ). The ; (1) quantum numbers of these #uctuations are 0 since 4 D they are meson operators and their ; (1) quantum numbers should also be 0 from Table 3.4.1. 0 Consequently, the fermionic component  is also in the adjoint representation of S;(N ) and its + D baryon number is 0. In particular, the R quantum number of  is !1, since MG "QI GQ is + H H a chiral super"eld MG "G #G #FG , H + +H +H

(3.4.62)

and R()"1 ( is the super-coordinate). The quantum #uctuations of B and BI do not carry any #avour index and hence are in the trivial representation of S; (N ). Their baryon numbers are respectively N and !N due to the 4 D D D additivity of the ;(1) quantum number. For clarity, we list the quantum numbers of the various "elds in Table 3.4.2. A strong support to the dynamical pattern comes from 't Hooft anomaly matching. As introduced in Section 2.3.3, in a theory with con"nement, the anomalies contributed by massless composite fermions at the macroscopic level and those from the elementary massless con"ned fermions should match. Now we give a detail check whether the anomalies match. Corresponding

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Table 3.4.3 Currents corresponding to global symmetry S; (N );; (1);; (1) 4 D 0

 / I /   +  I

S; (N ) 4 D

; (1)

; (1) 0

j"M t I / I / jI "M I tM  I I / I / 0 j(M)"f !M ! I + I + 0 0

j "M  I / I / jI  "!M I  I I / I / 0 0 j "N M  I D I jI  "!N M I  I I D I

j0"!M  I / I / jI 0"!M I  I I / I / j0()"M ? ? I I j0"!M   I + I + j0"!M  I I jI 0"!M I  I I I

Table 3.4.4 Energy-momentum tensor composed of the fermionic components of chiral super"elds; L[]"i/2(M I! !! M I), I I ! "R ! )*/2, )*"1/4[ ), *], )"e) I I I )*I I ¹ IJ  / I /   +  I

i/4[(M !  !! M  )#(  )]!g L[ ] / I J / J / I / IJ /  P I / / i/4[(M ? ! ?!! M ? ?)#(  )]!g L[] I J J I IJ i/4[(M  !  !! M   )#(  )]!g L[ ] + I J + J + I + IJ + i/4[(M !  !! M  )#(  )]!g L[ ] I J J I IJ  P I

to the global symmetry S; (N );; (1);; (1) and the quantum numbers listed in Table 3.4.2, 4 D 0 the currents for elementary fermions and massless composite fermions are collected in Table 3.4.3. In addition, the fermionic part of the energy-momentum tensor is in Table 3.4.4 to allow for a discussion of a possible axial gravitational anomaly. The above tables give the currents corresponding to the global symmetry S; (N ); 4 D ; (1);; (1) at both fundamental and composite levels: 0 E For the elementary fermions: S; (N ) current: 4 D J,j(Q)#jI (QI )"M t  #M I tM   I , I I I /GP I GH /HP /GP I GH /HP A"1,2, N !1, i, j"1,2, N , r"1,2, N . (3.4.63) D D A ; (1) current: J ,j (Q)#jI  (QI )"M  !M I  I . I I I /GP I /GP /GP I /GP ; (1) current: 0 J0,j0(Q)#jI 0(QI )#j0() I I I I "!M  !M I  I #M ? ?; a"1,2, N!1 . I A /GP I /GP /GP I /GP

(3.4.64)

(3.4.65)

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Table 3.4.5 S; (N );; (1) 't Hooft anomaly coe$cients 4 D 0 Triangle diagram

Elementary anomaly coe$cient

Composite anomaly coe$cient

(; (1)) 0 (; (1)); (1) 0 0 (S; (N )); (1) 4 D 0 ; (1) 0

!N !1 D !2N D !N Tr(tt ) D !N !1 D

!N !1 D !2N D !N Tr(tt ) D !N !1 D

E For the composite fermions: S; (N ) current: 4 D J,j(M)"f !M ! . I I + I + ; (1) current:

(3.4.66)

(3.4.67) J ,j  (B)#jI  (BI )"N M  !N M I  I . I I I I D I D ; (1) current: 0 (3.4.68) J0,j0(M)#j0#jI 0"!M   !M  !M I  I . I I I I I + I + I One can directly calculate the anomaly coe$cients of various triangle diagrams composed of the above currents. In order to calculate the anomaly coe$cient, one must introduce the gauge "elds associated with S; (N ), ; (1) and ; (1), that is, turn these global groups into local ones. In 4 D 0 general, these new gauge "elds are not physical ones, except in some special cases such as electroweak theory, where the gauge symmetry is the local #avour symmetry. Therefore, 't Hooft called these assumed gauge "elds `spectator gauge "eldsa [58]. Also the gauge coupling constants associated to these `spectator gauge "eldsa should be very small so that the dynamics of the real physical strong coupling gauge interaction cannot be a!ected. As 't Hooft pointed out, one may either think of these `spectator gauge "eldsa as completely quantized "elds or simply as arti"cial background "elds with non-trivial topology. The calculation of the possible anomalous triangle diagrams shows that the anomaly coe$cients really are identical. They are listed in Table 3.4.5. Note that in Table 3.4.5 (and in what follows) we use the groups to represent the triangle diagrams composed of the corresponding currents, for example, S; (N ); (1) represents J0JJ  and ; (1) means the axial gravitational anomal4 D 0 I J M 0 ous triangle diagram J0¹ ¹ , etc. In principle, there are many possible triangle combinations I JM ?@ of the currents, but only the anomaly coe$cients listed in Table 3.4.5 do not vanish. The calculation of the anomaly coe$cients listed in Table 3.4.5 is straightforward. For example, considering the ; (1) triangle diagram. For elementary fermions, the amplitude is 0  J0J0J0" j0(Q)j0(Q)j0(Q)# j0(QI )j0(QI )j0(QI ) I J M I J M I J M # j0()j0()j0() , (3.4.69) I J M

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and the anomaly coe$cient is R(Q)R(Q)R(Q) Tr(1)# R(QI )R(QI )R(QI ) Tr(1)#R()R()R() Tr(1)



" 2      (!1)#?@@AA?" !2N #N !1"!N !1 . GH HI IG PQ QR RP D D D

(3.4.70)

For composite fermions, the corresponding triangle diagram amplitude is J0J0J0" j (B)j (B)j0(B)# j0(BI )j0(BI )j0(BI ) I J M I J M I J M #  j0(M)j0(M)j0(M) . I J M

(3.4.71)

The anomaly coe$cient is R(B)R(B)R(B)#R(BI )R(BI )R(BI )#R(M)R(M)R(M) Tr(1) " 2(!1)#(!1) Tr(1)





" !2#N #1"!N !1 . D D

(3.4.72)

Thus the anomaly coe$cients at both elementary and composite levels are equal. As another illustrative example, take the axial gravitational anomaly ; (1). At fundamental 0 level, the amplitude is J0¹ ¹ , and the anomaly coe$cient is I JM NB !Tr(1) Tr(1) !Tr(1) Tr(1) #Tr(1) "!2N #N !1"!N !1 , (3.4.73)         

D D D where the subscripts `c.f.a and `f.f.a denote the fundamental representations of colour gauge group S;(N ) and #avour group S;(N ), respectively. At composite level, the anomaly coe$cient is A D !1!1!!!  "!2!(N !1)"!(N #1) . D D

(3.4.74)

The anomaly coe$cients again match exactly. 2. MG "0, B"!BI "
,D : Conxnement and baryon number violation. In this case, the symmetry H breaking pattern is S; (N );S; (N );; (1);; (1)PS; (N );S; (N );; (1) . * D 0 D 0 * D 0 D 0

(3.4.75)

The chiral symmetry does not break since MG "0. The R charges R(M)"R(B)"R(BI )"R(
)"0 H due to N "N , so R symmetry still remains. Obviously the baryon number symmetry is broken, D A usually B(B)"N , B(BI )"!N , B(
)"0, and the equation B"!BI "
,D does not satisfy the D D baryon number conservation law. Note that a breaking of baryon number conservation is not possible in ordinary QCD, as Vafa and Witten proposed a strict theorem stating that the spontaneous breaking of vector symmetries is forbidden [76]. However, their proof of this theorem is based on the vector nature of the quark}quark}gluon vertex, while in supersymmetric QCD, there exist scalar quarks, and the quark}squark}gluino interaction vertex, which is an axial vector vertex, so this spoils the initial assumption of the theorem and hence the baryon number symmetry can be spontaneously broken.

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Table 3.4.6 S; (N );S; (N );; (1) quantum numbers for elementary "elds * D 0 D 0

Q QI  / I / 

S; (N ) * D

S; (N ) 0 D

; (1) 0

N D 1 N D 1 1

1 NM D 1 NM D 1

0 0 !1 !1 1

Table 3.4.7 Currents composed of the fermionic components of elementary chiral super"elds

 / I / 

S; (N ) * D

S; (N ) 0 D

; (1) 0

j (Q)"M t *I / I / 0 0

0 jI ? (QI )"M I tM  I 0I / I / 0

j0(Q)"!M  I / I / jI 0(QI )"!M I  I I / I / j0()"M ? ? I I

Table 3.4.8 Representation quantum numbers for composite chiral super"elds

M B BI  +  I

S; (N ) * D

S; (N ) 0 D

; (1) 0

N D 1 1 N D 1 1

NM D 1 1 NM D 1 1

0 0 0 !1 !1 !1

Table 3.4.9 Currents composed of the fermionic components of composite chiral super"elds

 +  I

S; (N ) * D

S; (N ) 0 D

; (1) 0

j (M)"M t *I + I + 0 0

j (M)"M tM  0I + I + 0 0

j0(M)"!M  I + I + j0(B)"!M  I I jI 0(BI )"!M I  I I I

Let us check the 't Hooft anomaly matching conditions. In this case the quantum number for the elementary and composite "elds are obvious since all the quantum numbers remain intact. We list the quantum numbers and the currents for the elementary and composite "elds in Tables 3.4.6}3.4.9.

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Table 3.4.10 S; (N );; (1) 't Hooft anomaly coe$cients, d !, Tr(t t , t!) *0 D 0 Triangle diagram

Elementary anomaly coe$cient

Composite anomaly coe$cient

S; (N ) *0 D S; (N ); (1) *0 D 0 ; (1) 0 ; (1) 0

#(!)d !N D !N Tr(tt ) D !(N #1) D !N !1 D

#(!)d !N D !N Tr(tt ) D !(N #1) D !N !1 D

E At elementary level: The currents corresponding to the global symmetries S; (N );S; (N );; (1) are as * D 0 D 0 follows: J ,j (Q)"M t  ; *I *I /GP I GH /HP J ,jI  (QI )"M I tM   I ; 0I 0I /GP I GH /HP J0,j0(Q)#jI 0(QI )#j0() I I I I " !M  !M I  I #M ? ? . /GP I /GP /GP I /GP I

(3.4.76) (3.4.77)

E At composite level: The currents corresponding to S; (N );S; (N );; (1) are as follows: * D 0 D 0 J ,j (M)"M G t H ; *I *I + I GH + J ,j? (M)"M tM   ; 0I 0I +G I GH +H J0,j0(M)#j0(B)#jI 0(BI ) I I I I "!M G t H !M  !M I  I . I +H I GH +G I

(3.4.78)

One can easily check that for both elementary and composite fermions only the triangle diagrams S; (N ), S; (N ); (1), ; (1) and the axial gravitational anomaly ; (1) do not vanish. *0 D *0 D 0 0 0 The corresponding anomaly coe$cients can be calculated in the same way and the results are listed in Table 3.4.10. As expected, the anomaly coe$cients match exactly. Note that we have used the fact that the #uctuations of B and BI are not independent due to the constraint B!,D "BI #,D .

(3.4.79)

Thus in calculating the anomaly coe$cient of ; (1), only the contribution from  is considered. 0 It should be emphasized that in the above section (and in what follows) we have assumed (will assume) that the global symmetries are realized linearly on M, B and BI .

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Table 3.4.11 Sp(4);; (1) 't Hooft anomaly coe$cients 0 Triangle diagram

Elementary anomaly coe$cient

Composite anomaly coe$cient

Sp(4); (1) 0 ; (1) 0 ; (1) 0

!2 Tr (tt )  2;4;(!1)#3 2;4;(!1)#3

! Tr(tt )  5;(!1) 5;(!1)

For N "2, the point A




!i 

<"


0



"


0

!i 






0 !1 0

0

1

0

0

0

0

0

0 !1

0

0

1

0

(3.4.80)

is obviously in the moduli space. Consequently, the global #avour symmetry S;(4) breaks to Sp(4), while R-symmetry is preserved. The 't Hooft anomaly matching associated with the global symmetry Sp(4);; (1) can be checked. The fundamental massless fermions are quarks and 0 gauginos, their quantum numbers with respect to Sp(4);; (1) are 4 and 1 , respectively. The 0 \  low-energy "elds are the quantum #uctuations of < around the expectation value (3.4.80) subject to the constraint (3.4.52). Their fermionic component transform as 5 under Sp(4);; (1). With \ 0 these quantum numbers, the various conserved currents can be easily constructed. Considering the relation between the quadratic Casimir operators of the 4- and 5-dimensional representations of Sp(4), 2 Tr(tt ) " Tr(tt ) , we see that the anomaly coe$cients at low energy and high energy   levels are indeed equal, as shown in Table 3.4.11.

3.4.3. N "N #1: Conxnement without chiral symmetry breaking D A At the classical level, the gauge invariant chiral super"elds describing the moduli space in this case are MG and the baryon super"elds H B "  2 ,A P 2P,A QH 2QH,,AA , P P G GH H BI "  2 ,A P 2P,A QI H 2QI H,,AA . P P G GH H

(3.4.81)

Under the S; (N );S; (N ), the composite super"elds M, B and BI transform as follows: * D 0 D M: (N , NM ), B: (NM , 1), BI : (1, N ) . D D D D

(3.4.82)

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Let us "rst analyse what the quantum moduli space is in this case. The S;(N ) gauge symmetry and A the global symmetry S; (N );S; (N );; (1);; (1) as well as the mass dimension 3 deter* D 0 D 0 mine that the e!ective superpotential = must be of the following form:  1 (a det M#bBGM HBI ) , = , G H 
@

(3.4.83)

where a and b are non-vanishing constants which need to be determined. Note that this superpotential is not the dynamically generated superpotential, it is rather an arti"cial one for describing the moduli space. Owing to the holomorphic decoupling, if we give one #avour, say, the N th D #avour, a large mass m, then at the energy scale q(m, the low-energy theory should be an S;(N ) A theory with N #avours. Thus, after integrating out this heavy #avour, we should get the non-trivial D constraints (3.4.52) of the N "N case. After giving the last #avour a mass, the superpotential D A becomes 1 (a det M#bBGMHBI )!mM,DD . = (m)" , G H 
@

(3.4.84)

The F-#atness conditions (due to the unbroken supersymmetry) for MG D and M,D with i(N G D , reduce M to the following form:




M"

M I

0

0

M,DD ,



.

(3.4.85)

With MG D "0 and M ,D "0, the F-#atness conditions for B and BI yield , G G G




B"

0

B D ,




, BI "

0

BI

.

(3.4.86)

,D

So the e!ective superpotential takes the following form: 1 = " (a det M I M,DD #bB,D M,DD BI D )!mM,DD 
@ , , , , 1 , (a det M I M,DD #bM,DD BBI )!mM,DD , , , ,
@

(3.4.87)

where we denoted B D ,B and BI D ,BI . The F-#atness condition for M,DD gives , , , a det M I #bBBI "m
@ .

(3.4.88)

At the energy q"m, the running coupling constant of the S;(N ) theory with N "N #1 A D A #avours should match with that of the S;(N ) theory with N ("N ) #avours: A D A 4
3(N #1)!N
3N !N
A A ln " A A ln * . " g(m) 2
m 2
m

(3.4.89)

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Table 3.4.12 Representation quantum numbers for elementary "elds

Q QI  / I / 

S; (N ) * D

S; (N ) 0 D

; (1)

; (1) 0

N D 1 N D 1 1

1 NM D 1 NM D 1

0 0 0 0 0

1/N D 1/N D 1/N !1 D 1/N !1 D 1

This gives
,A "m
,A \ , *

(3.4.90)


@I  "m
@ . *

(3.4.91)

i.e.

If we choose the undetermined parameters a"1, b"!1, (3.4.88) will lead to the constraint (3.4.52) in the case N "N . Therefore, the e!ective superpotential in the case N "N #1 should D A D A be of the form 1 (det M!BGM HBI ) . = " G H 
@

(3.4.92)

The moduli space of vacuum states is described by the F-#atness conditions, R= /RBG"  R= /RBI " R= /RM H"0,  G  G M ) B"BI ) M"0, det M(M\)GH"BGBI H .

(3.4.93)

Obviously, the origin M"B"BI "0 is on the moduli space since it satis"es the above conditions. Thus the whole global symmetry S; (N );S; (N );; (1);; (1) is preserved in the origin of * D 0 D 0 the moduli space. If the above dynamical picture is correct, 't Hooft's anomaly matching with respect to this global symmetry must be satis"ed. According to Table 3.1.1, we list the quantum numbers for the elementary fermions and composite fermions in Tables 3.4.12 and 3.4.13, respectively. The corresponding conserved currents corresponding to the global symmetry S; (N );S; (N );; (1);; (1) are collected in * D 0 D 0 Table 3.4.14. One can easily "nd that the non-vanishing anomaly coe$cients at both the elementary and composite fermion levels are identical. They are listed in Table 3.4.15. In the case N "2, the superpotential in the low-energy e!ective Lagrangian is A 1 = "! Pf < . 


(3.4.94)

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Table 3.4.13 Representation quantum numbers for composite "elds

M B BI  +  I

S; (N ) * D

S; (N ) 0 D

; (1)

; (1) 0

N D NM D 1 N D NM D 1

NM D 1 N D NM D 1 N D

0 N ("N !1) A D N ("N !1) A D 0 N ("N !1) A D 0

2/N D N /N "(N !1)/N A D D D N /N "(N !1)/N A D D D 2/N !1 D !1/N D !1/N D

Table 3.4.14 Currents corresponding to the global symmetry S; (N );S; (N );; (1);; (1) * D 0 D 0

S; (N ) * D S; (N ) 0 D ; (1) ; (1) 0

Elementary fermions

Composite fermions

J "M t *I / I / J "M I tM  I 0I / I / J "M  !M I  I I / I / / I / J0"!N /N M  ! N /N M I  I I A D / I / A D / I / #M ? ? I

J "M t #M tM  *I + I + I J "M tM  #M I t I 0I + I + I J "N M  #N M I  I I A I A I J0"(!1#2/N )M  ! 1/N M  I D + I + D I ! 1/N M I  I D I

Table 3.4.15 't Hooft anomaly coe$cients

(S; (N )) *0 D (S; (N )); (1) *0 D (S; (N )); (1) *0 D 0 (; (1)); (1) 0 (; (1)) 0 (; (1)); (1) 0 ; (1) 0

Elementary level

Composite level

#(!) Tr(t t , t!)N A N Tr(tt ) A !N/N Tr(tt ) A D !2N A !N #6N !12#8/N !2/N D D D D 0 !N #2N !2 D D

#(!) Tr(t t , t!)N A N Tr(tt ) A !N/N Tr(tt ) A D !2N A !N #6N !12#8/N !2/N D D D D 0 !N #2N !2 D D

The quarks and the gaugino transform as 6 and 1 , respectively, and the quantum #uctuation of   the composite "eld < as 15 under the global symmetry group S;(6);; (1). The 't Hooft  0 anomaly conditions are satis"ed as can be seen from Table 3.4.16. 3.4.4. Supersymmetric QCD for N 'N #1 D A Now we continue to add the #avours and the more interesting phenomena will arise. In the following we shall concentrate on several typical ranges of the #avour and colours.

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Table 3.4.16 't Hooft anomaly coe$cients

(S;(6)) (S;(6)); (1) 0 ; (1) 0 ; (1) 0

Elementary level

Composite level

2 Tr(t t , t!)  2;(!2/3) Tr (tt )  12;(!2/3)#3 12;(!2/3)#3

Tr(t t , t!)  !1/3 Tr(tt )  15;(!1/3) 15;(!1/3)

3.4.4.1. 3N /2(N (3N : Non-Abelian Coulomb phase, conformal window and electric}magnetic A D A duality. We now specialize to N"1 S;(N ) supersymmetric QCD with N #avours. The NSVZ A D beta function (1.1) and anomalous dimension of the quark and squark "elds are g 3N !N #N (g) A D D ,
(g)"! 16
 1!N g/(8
) A g N!1 A #O(g) . (g)"! 8
 N A

(3.4.95)

Thus we see that
(g)(0, and the theory in this range is asymptotically free. However, the  anomalous dimensions of the quark and squark "elds imply
(g)'0, i.e. the one-loop beta  function is negative and the two-loop beta function is positive. This fact will make the beta function have non-trivial zero points, i.e. non-trivial "xed points. One explicit non-trivial "xed point can be observed in the following way: taking the limit N PR and N PR but keeping N g and D A A N /N "3! with ;1 "xed, we have D A N g (g)K! A , 8
 g 3N !N #N (g) g 3!N /N #N /N (g) A D D D A D A
(g)"! K! !g/(8
) 16
 1!N g/(8
) 16
 A





g N g " #(3!) A . 2 8


(3.4.96)

Accordingly, the beta function has a second zero at 8
 N g"! #O() . A 3

(3.4.97)

So, at least for large N , N and "3!N /N ;1, the theory has a non-trivial IR "xed point. At A D D A this non-trivial IR "xed point a four-dimensional superconformal "eld theory will arise due to the

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relation between the trace of the energy-momentum tensor and the
-function. Therefore, in the range 3N /2(N (3N , the infrared region of N"1 supersymmetric QCD is described by A D A a four-dimensional superconformal "eld theory and this range is called Seiberg's conformal window. The quarks and gluons are not con"ned but appear as interacting (e!ective) massless particles. The e!ective non-relativistic interaction potential between two static electric charged quarks takes the form of the Coulomb potential: <(r)&1/r. Consequently, the theory is now in the non-Abelian Coulomb phase. 3.4.4.2. N '3N : Non-Abelian free electric phase. In this range, since the one-loop beta function D A
'0, the theory is not asymptotically free. At large distance (low energy) the coupling constant  becomes smaller, and the particle spectrum of the theory consists of elementary quarks and gluons. Hence in this range of N , the theory is in a so-called `non-Abelian free electric phasea. As D mentioned in Section 2.4, a theory in a free electric phase is not well de"ned due to the Landau singularity. This can be easily seen from the de"nition of the beta function (in the modi"ed minimal subtraction scheme) Rg (N !3N )g A " .
" D  R 16


(3.4.98)

Integrating this equation, we obtain g  g" . 1!g (N !3N )/(16
)ln(/ )  D A 

(3.4.99)

The coupling constant increases with , and theory breaks down at the Landau pole [55] 8
 " exp .  g (N !3N )  D A

(3.4.100)

However, it can be regarded as the low-energy limit of another theory. To conclude this section, we explain why one can discuss non-trivial four-dimensional superconformal theory in the range 3/2N (N (3N . N (3N ensures that the theory is asymptotically A D A D A free. As for the lower bound 3/2N (N , this requires some knowledge about the representations A D of the four-dimensional superconformal algebra on the chiral supermultiplet introduced in Section 2.5. As we know, for the representation of the superconformal algebra on the chiral supermultiplet, there is a simple relation between the R-charge and the conformal dimension of a gauge invariant chiral super"eld operator O (see (2.5.70)), R(O)"!d(O), or 

d(O)"R(O) . 

(3.4.101)

However, from the unitary representations listed in Table 2.1.2, we see that except for the trivial representation, which is not interesting, the conformal dimension d of a physical operator should

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183

be larger than 1. Otherwise the highest weight representation of the conformal algebra will have a negative norm state, and this is not allowed in a unitary representation. The simplest gauge invariant chiral super"eld is the meson M. The R-charges listed in Table 3.1.1 give N !N 3 A '1 , d(M)" R(M)"3 D N 2 D

(3.4.102)

and hence one gets N '3/2N . This is the reason why a superconformal "eld theory arises in the D A range 3/2N (N (3N . A D A 4. N"1 supersymmetric dual QCD and non-Abelian electric}magnetic duality The global symmetry and the 't Hooft anomaly matching require that for N 'N #1 the D A low-energy phenomena of supersymmetric S;(N ) QCD with N #avours need a distinct descripA D tion through the introduction of a dual theory, which takes the same form as the fundamental supersymmetric gauge theory except for a gauge singlet "eld and an additional superpotential. In this section, we shall concentrate on Seiberg's conformal window 3N /2(N (3N and show A D A how the non-Abelian electric}magnetic duality arises at the IR "xed point. We shall discuss why the two dual theories have inverse couplings and how the theory behaves when some of the heavy modes decouple. A generalization of supersymmetric QCD with a matter "eld in the adjoint representation of the gauge group and a non-trival superpotential and its duality will also be discussed. 4.1. Dual supersymmetric QCD In general, when the #avour N 'N #1, the gauge invariant chiral super"elds parameterizing D A moduli space are the meson super"elds MG and the baryon super"elds B 2 and BI I I 2 I . Naively GH I H GH I one may think that the e!ective superpotential should have the following form: = &(det M!B 2 MGGI MHHI 2MIII BI I I 2 I ) .  GH I GH I

(4.1.1)

Although this superpotential is S; (N );S; (N ) and gauge invariant, it cannot be regarded as * D 0 D an e!ective superpotential since its R-charge is not equal to 2. In addition, one can easily "nd that the 't Hooft anomaly matching condition for the fermionic components of (Q, QI , ) and (M, B, BI ) cannot be satis"ed if we adopt the e!ective superpotential (4.1.1). A clever way out has been found by Seiberg and this has led to the invention of dual supersymmetric QCD. From the Hodge dual form of the baryon super"elds (3.3.18), 1  2 BG, 2G,A , BM ,A> ,A> 2 ,D , 2 G G G (N !N )! G G,A G,A> G,D D A BMI G,A> G,A> 2G,D ,

1 G 2G,A G,A> 2G,D B , 2 ,A , G G (N !N )! D A

(4.1.2)

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one can see that BM and BMI have NI "N !N A D A

(4.1.3)

indices. Thus one can assume that these baryon super"elds are bound states of NI chiral super"elds A q and q , 1 BM ,A> ,A> 2 ,D ,B  2 ,I A ,    2  ,I A qP  qP  2qP ,,I I AA , G G G G G G NI P P G G A 1 BMI G,A> G,A> 2G,D ,BI  2 ,I A ,    2  ,I A qQ  qQ  2qQ ,,II AA , H H H NI Q Q H H A i, j"1,2, N , r , s "1,2, NI "N !N . D A D A

(4.1.4)

Obviously qG and q G belongs to the fundamental representation of S;(N );S;(NI ). To bind these P D A P elementary constituents into gauge invariant baryon super"elds, we must construct a dynamical Yang}Mills theory with gauge group S;(NI ), which provides the dynamics. A Seiberg proposed that the low-energy supersymmetric S;(N ) QCD with N 'N #1 can be A D A described by a supersymmetric Yang}Mills theory with gauge group S;(NI ) coupled to the chiral A super"elds qP and q P as well as a new colour singlet chiral super"eld MH together with an additional G H G gauge invariant e!ective superpotential = "q ) M ) q "qP MGH   q Q . G PQ H 

(4.1.5)

Note that this new colour singlet super"eld M cannot be directly constructed from q and q like that the meson super"eld M is constructed from Q and QI . Its quantum numbers can be determined from the superpotential (4.1.5) once we have determined the quantum numbers for q and q . From (4.1.4) and the quantum numbers of the original quarks of the meson and baryon super"elds listed in Table 4.1.1, we can determine the quantum numbers of the elementary dual super"elds and their fermionic components listed in Table 4.1.2. Obviously, the superpotential is S;(NI ) gauge invariant and globally S; (N );S; (N ) invariant. Further, the superpotential A * D 0 D should be ; (1);; (1) invariant, i.e. the superpotential should have baryon number 0 and 0 R-charge 2. This requires that the quantum numbers of the new meson super"eld M under S; (N );S; (N );; (1);; (1) should be * D 0 D 0




N M: N , NM , 0, 2!2 A D D N D



.

(4.1.6)

To summarize, the low-energy supersymmetric S;(N ) QCD with N 'N #1 #avours A D A can be described by a dual theory. The dynamical variables in the dual theory are the gauge singlet operator M, the S;(NI ) gluons and the dual quarks q and q . In addition to the standard A

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Table 4.1.1 Representation quantum numbers of the chiral super"elds in the original supersymmetric QCD

Q QI  / I /  MGH  + B  2 ,I A G G BI  2 ,I A H H

S; (N ) * D

S; (N ) 0 D

; (1)

; (1) 0

N D 1 N D 1 1 N D N D NM NI D A 1

1 NM D 1 NM D 1 NM D NM D 1 N NI D A

1 !1 1 1 0 0 0 N A N A

1!N /N A D 1!N /N A D !N /N A D !N /N A D 1 2!2N /N A D 1!2N /N A D N (1!N /N ) A A D N (1!N /N ) A A D

Table 4.1.2 Representation quantum numbers of the chiral super"elds and their fermionic components in dual supersymmetric QCD

q q  O  O I

S; (N ) * D

S; (N ) 0 D

; (1)

; (1) 0

NM D 1 NM D 1 1

1 N D 1 N D 1

N /NI A A !N /NI A A 1 !N /NI A A 0

N /N A D N /N A D N /N !1 A D N /N !1 A D 1

form of the supersymmetric QCD Lagrangian, the dual Lagrangian includes a kinetic energy term of the colour singlet M and the e!ective superpotential (4.1.5), which is S; (N );S; (N );; (1);; (1) invariant as the original supersymmetric QCD. Seiberg further * D 0 D 0 found that these two theories are dual in the sense of electric}magnetic duality in the conformal window, i.e. as the #avour number N decreases, the original supersymmetric S;(N ) QCD D A becomes more strongly coupled, but the dual supersymmetric S;(NI ) QCD becomes more weakly A coupled. We shall give a detailed explanation in the next subsection. This relation between the S;(NI ) gauge theory and the original S;(N ) is called non-Abelian electric}magnetic duality. The A A original theory is usually called the electric theory and the dual theory is called the magnetic theory. In addition, the ;(1) quantum numbers (i.e. the baryon numbers and R-charges) listed in Tables 4.1.1 and 4.1.2 show that there is a very complicated relation between the baryon numbers and R-charges of the quarks in the electric and magnetic theories. Since the ;(1) quantum numbers such as baryon number and R-charge are additive quantum numbers, this relations implies that the quarks q and q cannot be simply expressed as polynomials of the quarks in the electric theory. This connection between `electrica and `magnetica quarks is highly non-local and complicated. Only in some special cases can the explicit connection between them be worked out [83]. The `magnetica quarks q and q and the S;(N !N ) gluons can be interpreted as solitons of the electric theory, i.e. D A as non-Abelian magnetic monopoles.

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Now one question naturally arises: does this S;(NI ) theory describe the real physical dynamics? A From the above statements, it seems as if this S;(NI ) gauge theory is only a formal device to A introduce dual quark super"elds q and q supposedly bound together to form the baryon super"eld. However, in two-dimensional space time, there exist examples where the gauge "eld parametrizing a constraints becomes dynamical. The most famous example is the two-dimensional CP, model [82]. Thus we can assume that this dual S;(NI ) supersymmetric QCD is a dynamical theory in A which there are gauge bosons and their superpartners } gauginos. A strong support for this picture is still the 't Hooft anomaly matching: the anomaly coe$cients of supersymmetric QCD and of the dual supersymmetric QCD are identical, so the dual supersymmetric QCD has indeed provided a dynamical description of the composite super"elds of the electric theory. In the following we shall explicitly show this. The quantum numbers for dual quarks super"elds listed in Table 4.1.2 mean that the currents composed of the dual quarks, the mesons and the dual gauginos I corresponding to the global symmetry S; (N );S; (N );; (1);; (1) are the following: * D 0 D 0 E S; (N ) * D JI  ,M  tM    #M M t M ; G I GH H *I OGP I GH OHP

(4.1.7)

E S; (N ) 0 D JI  ,M   t    #M M tM  M ; 0I OGP I GH OHP I GH H

(4.1.8)

N N A M    ! A M      ; JI  , I N !N OGP I OGP N !N OGP I OGP D A D A

(4.1.9)

E ; (1)

E ; (1) 0












N N JI 0, !1# A M    # !1# A M      I OGP I OGP OGP I OGP N N D D



2N # IM ? I ? # 1! A M M M . G I G I N D

(4.1.10)

Note that in the above currents the range for the #avour indices is i, j"1,2, N , for the colour D indices r "1,2, N !N , for the magnetic gauge group indices a "1,2, (N !N )!1 and for D A D A the #avour group indices A"1,2, N !1. D The S; (N );S; (N );; (1);; (1) currents composed of the electric quarks and gauginos * D 0 D 0 are listed below: E S; (N ) * D J "M t ; *I / I /

(4.1.11)

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Table 4.1.3 Anomaly coe$cients of the original and dual supersymmetric QCD, d !" Tr(t t , ¹!)

(S; (N )) *0 D (S; (N )); (1) *0 D (S; (N )); (1) *0 D 0 (; (1) (1)); (1) 0 (; (1)) 0

S;(N ) (or elementary fermions) A

S;(NI ) (or composite fermions) A

#(!)d !N A N Tr(tt ) A !N/N Tr(tt ) A D !2N A N!1!2N/N A A D

d !N A N Tr(tt ) A !N/N Tr(tt ) A D !2N A N!1!2N/N A A D

E S; (N ) 0 D J "M I tM  I ; / 0I / I

(4.1.12)

J "M  #M I  I ; I / I / / I /

(4.1.13)

N N J0"! A M  ! A M I  I #M ? ? . I I N / I / N / I / D D

(4.1.14)

E ; (1)

E ; (1) 0

It can be easily checked that the non-vanishing anomaly coe$cients are those collected in Table 4.1.3. 4.2. Non-Abelian electric}magnetic duality Section 4.1 gives a dual description of low-energy supersymmetric QCD. However, only at the IR "xed point of the range 3N /2(N (3N , can the `electrica theory and `magnetica theory A D A describe the same physics. In the following we give a detailed analysis of this non-Abelian electric}magnetic duality. First, the range 3N /2(N (3N implies the inequality A D A NI (N (3NI , D A  A

(4.2.1)

where NI "N !N is the number of colours in the dual theory. Hence the range A D A 3N /2(N (3N for the `electrica theory, for which a non-trivial IR "xed point exists, implies the A D A range 3/2NI (N (3NI for the magnetic theory. Thus the "xed point of the magnetic theory is A D A identical to that of the original electric theory, i.e. they have the same non-Abelian Coulomb phase! Note that these two theories have di!erent gauge groups and di!erent numbers of interacting particles, but they have the same "xed point, and at the "xed point they describe the same physics. In the non-relativistic case, both theories have a 1/r potential in these two ranges, and there is no

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experimental way to tell whether the electric or magnetic gauge bosons are mediating the interaction. Secondly, we look at the dynamics. Eq. (4.1.5) gives an additional superpotential of the magnetic theory. It can be argued that at the IR "xed point there exists a simple relation between an `electrica meson and a colour singlet of the `magnetica theory M G "MG . H H

(4.2.2)

Since both S;(N ) and S;(NI ) QCD are asymptotically free in the range 3N /2(N (3N , both A A A D A of them have an UV "xed point g"0. (3.4.95) shows that the anomalous dimensions of M in the electric theory and of M in the magnetic theory vanish at the UV "xed point. Thus, their dimensions at the UV "xed point are the canonical dimensions. From its de"nition (3.3.14) the canonical dimension of M is 2 at the IR "xed point, while from the beta function (3.4.95), the vanishing of the beta function gives the anomalous dimension N "!3 A #1 . N D

(4.2.3)

Thus, the full dimension of M at the IR "xed point is N !N A . D(M) "2#"3 D '0     N D

(4.2.4)

This can also be obtained from the relation between the conformal dimension and R-charge of a chiral meson super"eld operator in a superconformal "eld theory (see (2.5.70)), 3 N !N 3 A . D(QI Q)" R(QI Q)" [R(QI )#R(Q)]"3 D 2 N 2 D

(4.2.5)

In the magnetic theory, the discussion in Section 4.1 shows that M cannot be constructed from the elementary magnetic quarks q and q . It should be regarded as an elementary chiral super"eld, thus its canonical dimension is 1 and so its dimension at the UV "xed point is 1. At the IR "xed point, M and M should become identical under the renormalization group #ow and M should have the same dimension as M, i.e. 3(N !N )/N . This can be seen from its R-charge in (4.1.6). In order to D A D relate M to M at the UV "xed point, we must introduce a scale  and hence (4.2.2) arises. In the following, we shall replace M by M/, and then the additional superpotential (4.1.5) in the magnetic theory can be written as follows: 1 =" q M G q H .  G H

(4.2.6)

We shall see that the introduction of this parameter with mass dimension will make the nonAbelian electric}magnetic duality explicit. From dimensional analysis, the scale
I of the magnetic theory and
of the electric theory should satisfy
,A \,D
I ,D \,A \,D J,D .

(4.2.7)

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The explicit relation involves a phase factor (!1),D \,A ,
,A \,D
I ,D \,A \,D "(!1),D \,A ,D .

(4.2.8)

The necessity of introducing this phase factor will be explained in the following. Let us "rst see the consequences of the relation (4.2.8). First, (4.2.8) implies that the gauge coupling of the electric theory becomes stronger while the coupling of the magnetic theory will become weaker and vice versa. This can be seen from the running coupling constant (3.4.5),
,A \,D "q,A \,D e\
 EO ,


I ,D \,A \,D "q,D \,A \,D e\
 E O .

(4.2.9)

The relation (4.2.8) leads to (!1),D \,A ,D "q,D e\
 EO e\
 E O .

(4.2.10)

At a certain "xed scale q"
, we have (!1),D \,A




 ,D    e
 E O  "e\
 E O .


(4.2.11)

This can be thought of as the analogue of the usual electric}magnetic duality gP1/g in an asymptotically free theories. Secondly, (4.2.8) gives the connection between gluino condensations of the electric theory and the magnetic theory. (4.2.8) gives ln
,A \,D #ln
I ,D \,A \,D "ln(!1),D \,A ,D , d ln
,A \,D "!d ln
I ,D \,A \,D .

(4.2.12)

As it is well known, the quantum one-loop e!ective action of supersymmetric QCD can be expressed in the following form:

 
 

1 1 dx d Tr(=?= )#h.c. " ? 4 g  1 q @ " ln dx d Tr(=?= )#h.c. , ? 8



(4.2.13)

where
"3N !N for the electric theory and
"3(N !N )!N for the magnetic theory.  A D  D A D Di!erentiating the e!ective action with respect to ln
@ , we obtain ="!= I  , ? ?

(4.2.14)

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whose lowest component shows that the gluino condensates of the electric and magnetic theories are related by "!I I  .

(4.2.15)

Eq. (4.2.14) is very similar to the ordinary electric}magnetic duality if we consider its  component, which contains the term F FIJ and is in electric}magnetic components IJ E!B"!(EI !BI ) . (4.2.16) Now let us see the e!ect of the phase factor (!1),D \,A . A duality transformation of the relation (4.2.8) (i.e. N PN !N ) gives A D A
,D \,A \,D
I ,A \,D "(!1),A ,D "(!1),A \,D ,D . (4.2.17) Hence "! .

(4.2.18)

Eqs. (4.2.17) and (4.2.18) imply that the dual of the magnetic theory (i.e. the dual of the dual) is an S;(N ) theory with scale
. We denote its quark "elds as dGP and dI . Now there are two A GP independent colour singlets, one is the original MG and another is constructed from the magnetic H quarks NH "q ) q H . (4.2.19) G G The mass dimension 3 and R-charge 2 of the superpotential imply that the possible gauge invariant and S; (N );S; (N );; (1);; (1) invariant superpotential for the dual description of the * D 0 D 0 magnetic theory must be of the following form: 1 1 1 =" N H dGdI # M G NH " N H (!dGdI #MG ) . H H  G H  H G  G

(4.2.20)

M and N are massive chiral super"elds and they can be integrated out by their equations of motion: M G "dGdI , N"0 . (4.2.21) H H This shows that the duals of the magnetic quarks can be identi"ed with the original electric quarks Q and QI . Therefore, the dual of the magnetic theory coincides with the original electric theory. The phase factor (!1),D \,A plays an important role in revealing this. Finally, we stress the correspondence between the operators in the electric and magnetic theories. As mentioned above, the explicit connection between the electric quarks and magnetic quarks is very di$cult to "nd. However, the relation between gauge invariant composite operators in both theories is obvious. The meson operator MG "QGQI is identical to the colour singlet H H operator MG at the infrared "xed point. As for the baryon operators, (4.1.2) shows that at the IR H "xed point the baryon operators are related as follows: BG 2G,A "QG 2QG,A "CG 2G,A H 2H,I A GB  2 ,I A "CG 2G,A H 2H,I A q  2q ,I A , H H H H BI G 2G,A "QI G 2QI G,A "CG 2G,A H 2H,I A B  2 ,I A "CG 2G,A H 2H,I A q  2q ,I A , H H H H

(4.2.22)

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where the normalization constant C is C"(!(!),A \,D
,A \,D ,

(4.2.23)

which is required by the various limits to be discussed in following subsection. Note that the dual relations (4.2.8) and (4.2.22) respect the idenpotency of the duality transformation. 4.2.1. Various deformations of non-Abelian electric}magnetic duality By deformation we mean the various limiting cases mentioned in Section 3.4.1. We shall show how in these limits the electric}magnetic duality exchanges strong with weak coupling and the Higgs phase with the con"nement phase in the dual theories. First, we consider the limit of a large mass of the N th #avour in the electric theory (S;(N ) D A supersymmetric QCD with N #avours) by introducing a superpotential = "mM,DD . After , D  integrating out this heavy mode, the low-energy theory will be an S;(N ) supersymmetric QCD A with N !1 light #avours. As stated in Section 3.4.1, the matching of coupling constants at the D energy scale q"m gives a connection between the scale
of the high energy theory and the scale
of the low-energy theory (see (3.4.17)),
,A \,D \"m
,A \,D . Since supersymmetric QCD in * * the range 3N /2(N (3N is an asymptotically free theory, the low-energy electric theory will A D A have the stronger coupling. Let us see how the dual magnetic theory (S;(NI ) supersymmetric QCD A with N #avours) behaves. With the added tree-level superpotential, the full superpotential is D 1 =" q MG q H#mM,I D , i, j"1,2, N . D  G H ,D

(4.2.24)

The F-#atness conditions for M,I DD , MGI D and M,I D yield , G , q D q ,D "q D q ,D "!m, q q ,D "q D q G"0 . G ,P P , ,

(4.2.25)

So the lowest components of q D and q D will break the magnetic gauge group S;(N !N ) to , , D A S;(N !N !1) with N !1 light magnetic quarks. The equations of motion of the massive D A D quarks q D , q ,D and the composite quantity q D q ,D lead to , , M,I DD "MGI D "M,I D "0 . , G ,

(4.2.26)

Thus the low-energy superpotential is composed of the light "elds M K , q( and q( , 1 1 =" q( M K G q( H" q( M K G q( H , G H   GQ H Q

i, j"1,2, N !1, s"1,2, N !N !1 . D D A

(4.2.27)

Similarly to the electric theory, the coupling constants of the high-energy theory (S;(NI ) with A N #avours) and the low-energy magnetic theory (S;(NI !1) with N !1 #avours) should match D A D

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at the energy scale q"q D q ,D , , 3NI !N (q D q ,D  4
D ln , " A 2

I g(q D q ,D ) , 3(NI !1)!(N !1) (q D q ,D  A D , " ln ,
I 4
*
I ,I A \,D
I ,I A \\,D \" . * q D q ,D  ,

(4.2.28)

Using the above relation, one can easily "nd
I ,I A \,D
,A \,D \
I ,I A \\,D \" m
,A \,D * * q D q ,D  , "

m(!1),D \,A ,D "(!1),D \,A \,D \ . !m

(4.2.29)

Thus relation (4.2.8) is preserved in the large mass limit. Further it can be easily veri"ed that (4.2.22) is also preserved. Therefore, under the mass deformation duality is preserved. From the running gauge coupling, one can see that the duality makes a more strongly coupled electric theory equivalent to a more weakly coupled magnetic theory at the IR "xed point. However, for the case N "N #2, the magnetic theory is an S;(2) gauge theory and has D A only two colours. In this case the above discussion of the mass deformation is incomplete. If we introduce a large mass term for the N ("N #2)th #avour, the S;(2) magnetic gauge D A symmetry will be completely broken. The low-energy electric theory contains the mesons M K G , i, j"1,2, N #1, while in the low-energy magnetic theory, after integrating out the massive H A quarks, there are only massless, colourless magnetic quarks q( and q( left. The low-energy G G superpotential is still of the form (4.2.6) but with no summation over colour indices. From the relation (4.2.22), which gives the connection between the baryons B and BI G of the low-energy G electric theory and the colour singlet magnetic quarks of low-energy magnetic theory, we have BG 2G,A "QG 2QG,A "CK G 2G,A Gq( , G 2 A 2 BI G G, "QI G 2QI G,A "CK G G,A Gq( , G

(4.2.30)

where CK "(
,A \/. If we adopt the dual form (4.1.2) of the baryon, one can see

 

BM " G

 


,A \ 1
,A \ * *   2 ,A BG 2G,A " q( , G  N! G G G  D


,A \ 1
,A \ ( * *   2 ,A BI G 2G,A " q . BIM " G G  N! G G G  D

(4.2.31)

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This means that the gauge singlet magnetic quarks are in fact the baryons of the low-energy electric theory, i.e. the baryons are magnetic monopoles of elementary quarks and gluons. This idea was actually proposed many years ago by Skyrme. Later Witten further showed that at least in the large N case, the baryons can be regarded as solitons of the low-energy e!ective Lagrangian of ordinary A QCD [77]. In supersymmetric gauge theory, this idea comes out naturally in the context of electric}magnetic duality. Furthermore, with N "N #2, we can obtain the superpotential (3.4.92) of the case of D A N "N #1 light #avours from the duality between the low-energy electric and magnetic theory. D A From (4.2.29) and (4.2.31), the superpotential in low-energy magnetic theory is 1 1 K G q( H" BM M K G BM H , =" q( M
,A \ G H  G H *

(4.2.32)

where
is the scale of the low-energy electric theory with N ("N #1) light #avours. However, * D A since the gauge symmetry in the magnetic theory is completely broken, the low-energy e!ective action should include the contribution from instantons for the broken magnetic group, since in this case the magnetic theory is completely Higgsed and the instanton calculation is reliable. According to (3.4.29), the instanton contribution in the magnetic theory is K ) det M K
I \,A > det (\M "! , = " *  q,D >q ,D >
,A \,A > *

(4.2.33)

where (4.2.29) and (3.4.57) were used. Putting (4.2.32) and (4.2.33) together, we obtain the superpotential of the low-energy theory with N "N #1 light #avours (the index L is omitted) D A 1 (B MG BI H!det M) . ="
,A \ G H

(4.2.34)

This is exactly the superpotential (3.4.92), where we got it from the strongly coupled electric theory, while here we rederived it from the weakly coupled magnetic theory. In a similar way one can consider the general mass deformation by introducing a mass term mH MG for all the (electric) quarks. The full superpotential ="q MG q H#mH MG implies that the G H G H G H dual magnetic quarks get the mass (matrix) MG mG " H ,

H 

(4.2.35)

so the low-energy magnetic theory is a pure S;(N !N ) Yang}Mills theory. From (3.4.40) we D A have
I  "[det (\M)
,I A \,D ],I A , *
I ,A \,D "\,D
I ,D \,A \,D det M . *

(4.2.36)

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Correspondingly, there exists an e!ective superpotential produced from gluino condensation, = "NI
I  "(N !N )[\,D
I ,D \,A \,D det M],D \,A   A * D A "(N !N ) A D







,A \,D ,A \,D  , det M

(4.2.37)

where we have used (4.2.8). One can see that (4.2.37) is the same as (3.4.29), but (4.2.37) is obtained from the dual magnetic theory. This implies that the superpotential (3.4.29) and the expectation values (3.4.28) are reproduced correctly when mass terms are added to the magnetic theory. It should be emphasized that the factor (!1),D \,A plays a crucial role in getting (4.2.37), otherwise the sign will be opposite. There is another possible deformation, which is realized by making the quark super"elds have big expectation values along the D-#at direction (3.3.13) in the electric theory. For simplicity, we only choose one #avour, say, the N th #avour, to have a large expectation value, i.e. D Q,D "QI ,D  is large. The lowest components of Q,D and QI ,D will break the electric S;(N ) A theory with N #avours to S;(N !1) with N !1 #avours. From (3.4.11), the matching of D A D running coupling constant at Q,D "QI ,D  leads to
,A \,D
,A \\,D \" * . * Q,D QI ,D 

(4.2.38)

In the magnetic theory, similarly to (4.2.35), a large M,DD  yields a large mass \M,DD  for the , , N th magnetic quarks, q,D and q ,D . The low-energy magnetic theory is S;(N !N ) with N !1 D A D D light #avours and the low-energy scale is
I ,D \,A \,D \"\M,DD 
I ,A \,D \,D . * ,

(4.2.39)

One can easily check that relation (4.2.8) is satis"ed and hence that duality is preserved. Similar discussions for BO0 show that the duality is also preserved in this case [78]. 4.2.2. Non-Abelian free magnetic phase: N #24N 43/2N A D A Now we consider a special range of colour and #avour numbers, N #24N 43/2N . From A D A the NSVZ beta function of the dual theory g 3(N !N )!N #N (g) D A D D
(g)"! , 16
 1!(N !N )g/(8
) D A g (N !N )!1 D A (g)"! #O(g) , 8
 (N !N ) D A

(4.2.40)

one can see that when N 43/2N , 3(N !N )4N , the beta function is positive, so the magnetic D A D A D S;(N !N ) theory is not asymptotically free and is weakly coupled at low-energy. Thus at D A low-energy the magnetic quarks are not con"ned and the particle spectrum consists of the singlet M, and the magnetic quarks q and q . Relations (4.2.22) show that the massless magnetic particles

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are composites of the elementary electric degrees of freedom. Comparing with the case N 53N in D A the electric theory, which is in the non-Abelian free electric phase, we say that the theory is in a non-Abelian free magnetic phase since there are massless `magnetica charged "elds. 4.3. Duality in Kutasov}Schwimmer model 4.3.1. Kutasov's observation The Kutasov}Schwimmer model is the usual N"1 supersymmetric S;(N ) QCD but with an A additional matter "eld X in the adjoint representation of the gauge group and an associated superpotential of the general form [85,86,83] I s G Tr XI>\G . ="  k#1!i G

(4.3.1)

It seems that the presence of these non-renormalizable interactions described by this superpotential is irrelevant for the short-distance behaviour of the theory, but they may have strong e!ects on the infrared dynamics. Thus this model has a rich electric}magnetic duality structure. In the following we give a brief introduction to the duality present in this model. We start from the original observation made by Kutasov [85]. According to the NS
(4.3.2)

where i labels the species of "elds, "g/(4
) is the "ne structure constant, and  () are the G anomalous dimensions of  , G   ()"!C (r ) #O() G  G


(4.3.3)

Tr (¹?¹@)"¹(R)?@, ¹?¹?"C (R)1, ¹(G),¹(R"adjoint) . 

(4.3.4)

with

Non-trivial "xed points arise when 3¹(G)! ¹(r )[1! ()]"0 . G G G

(4.3.5)

Note that now these "xed points are not necessarily the infrared "xed points. At a "xed point the physics is described by a superconformal "eld theory, and the scaling dimension of  should be the G sum of the canonical and anomalous dimensions d "1# /2 . G G

(4.3.6)

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On the other hand, for a superconformal theory there exists a simple relation between the scaling dimension and the anomaly-free R-charge R "d . G  G With (4.3.6) and (4.3.7), (4.3.5) can be written as

(4.3.7)

¹(G)# ¹(r )[R !1]"0 . (4.3.8) G G G (4.3.8) actually provides a condition for an R-symmetry to be anomaly free, under which the supercoordinates  have the R-charge 1 and the "elds  have R-charges R . Thus (4.3.8) is also ? G G called the anomaly cancellation condition. In general, there may be many ; (1) symmetries whose 0 R-charges satisfy (4.3.8) and thus they are anomaly free. However, since we are only interested in infra-red (IR) "xed points, it is necessary to know which of the R-symmetries is the right one, i.e. which R-symmetry becomes a part of the IR superconformal algebra. There are some cases in which the answer is known, for instance, if all the matter "elds  belong to the same representation G r "r, i"1,2, M, then we have G ¹(G) . (4.3.9) R "R"1! G M¹(R) The S;(N ) supersymmetric QCD with N quark and anti-quark super"elds, Q and QI , is just this A D G G case, where we have ¹(S;(N ))"N , ¹(N )"1/2, M"2N . (4.3.10) A A A D Thus anomaly-free R charges for quark and anti-quark super"elds are R"1!N /N , which A D naturally agrees with the R-charges listed in Table 3.1.1. However, there is no general method of determining R in (4.3.8) to pick the right R-charge so that it becomes a generator of the IR G superconformal algebra. It is not even clear in general whether the theory ends up in the infrared region of a non-Abelian Coulomb phase. Among the phases introduced in Section 2.4, we know that only the Coulomb phase allows a self-dual description. Based on the above observation, Kutasov considered a straightforward generalization of supersymmetric QCD [85], i.e. supersymmetric S;(N ) Yang}Mills theory with a matter superA "eld X in the adjoint representation of the gauge group, N multiplets QG in the N and D A N supermultiplets QI in the NM representations; i"1,2, N . Due to the presence of the matter D G A D "eld in the adjoint representation, the one-loop
function coe$cients for this model becomes
"2N !N . (4.3.11)  A D Thus the theory is asymptotically free only for N (2N . It is natural to assign the same R charge D A R to all the fundamental multiplets QG, QI and a di!erent R charge R to the adjoint one, X. From D G ? (4.3.10) and ¹(adjoint)"N , the anomaly cancellation condition (4.3.8) takes the form A N #2N (R !1)#N (R !1)"0 , A ? A D D N R #N R "N . (4.3.12) D D A ? D

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There are many possible assignments for R and R satisfying (4.3.12), the problem is which D ? R-symmetry becomes part of the IR superconformal algebra. Kutasov used the following technique to solve this problem [85]: "rst formally "nding the dual description, then making use of the consistency of the theory to work out the correct dynamics so that the right anomaly-free R-symmetry can be distinguished. The method of searching for the dual magnetic description is the same as in the supersymmetric QCD case discussed in Section 4.1. Since the baryons can reveal the form of the duality transformation, one "rst de"nes the baryon-like operators in the above model, 2QG,,AA , B G 2GI GI> 2G,A "? 2?,A X@ X@ 2X@II QG 2QGII QGI> ? ? ? @ @ ?I> ?

(4.3.13)

where  ,
"1,2, N are colour indices. For a given k, there are G H A



N D k



N D N !k A

baryons B G 2GI GI> 2G,A , so the total number of the baryon operators is





,A ND  k I




N 2N D D . " N !k N A A

(4.3.14)

Eqs. (4.3.13) and (4.3.14) show that k4N , N !k4N , and hence the baryon operators (4.3.13) D A D exist only for N 5N /2. D A Eq. (4.3.14) reveals that the spectrum of baryons has a symmetry under N  2N !N (with the A D A #avour number N "xed) since D







2N 2N D " D . N 2N !N A D A

Thus the corresponding dual (`magnetica) theory should be an S;(NI "2N !N ) gauge theory A D A with the fundamental supermultiplets q , q G and adjoint supermultiplet > as well as some other G colour singlets. Similarly to (4.2.22), an identi"cation of the baryon operators in electric and magnetic theories should be possible: B G 2GI GI> 2G,A "B H 2HN HN> 2H,D\,A , 



(4.3.15)

with p"N !N #k. Assume that the magnetic quark super"elds q and q G have the R-charges D A G RI and that > has the same R-charges R as X. Then the anomaly cancellation condition (4.3.12) in D ? both electric and magnetic theories and the identi"cation (4.3.15) yield N R #N R "N , D D A ? D N RI #(2N !N )R "N , D D D A ? D kR #N R "(N !N #k)R #(2N !N )RI . ? A D D A ? D A D

(4.3.16)

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Table 4.3.1 Representation quantum numbers of the matter "elds of the electric theory under S; (N );S; (N );; (1);; (1) * D 0 D 0

Q QI X

S; (N ) * D

S; (N ) 0 D

; (1)

; (1) 0

N D 1 1

1 N 1D

1 !1 0

1!2N /(3N ) A D 1!2N /(3N ) A D 2/3

Hence we get 2N 2 2N !N 2 A , RI "1! D A . R " , R "1! D D ? 3 N 3N 3 D D

(4.3.17)

The R-charge of the adjoint supermultiplets in (4.3.17) seems to present a puzzle to us: the theory we are considering is asymptotically free, so g"0 is the UV "xed point, at which the R-charge of X is 2/3, whereas (4.3.17) shows that at the IR "xed point the R-charge of X is also 2/3. Usually this is not possible since (4.3.2) and (4.3.3) show that even for a perturbative "xed point the conformal dimension and hence the R-charge of X can receive a contribution at least at "rst order in . It was realized by Kutasov that the reason for this lies in the ambiguous action of ; (1) on the 0 chiral supermutiplets in the adjoint and fundamental representations, and only an interaction superpotential relevant to the chiral supermutiplet in the adjoint representation can "x this ambiguity. Thus Kutasov's resolution to this puzzle was to assume that the model considered should be subject to a Wess}Zumino superpotential composed of the adjoint matter [85] = (X)JTr X . 

(4.3.18)

A perturbative calculation in the Wess}Zumino model shows that the interaction provided by this superpotential contributes to the anomalous dimensions of X and hence to the
function of the gauge coupling. Thus with the superpotential (4.3.18) there is a possible additional "xed point at which X has the canonical conformal dimension 1 and hence the R-charge 2/3. In particular, superpotential (4.3.18) respects the R-symmetry given by (4.3.17), since as a superpotential it should have R-charge 2. Therefore, the introduction of this superpotential also chooses the right R-symmetry so that it becomes a part of the IR superconformal symmetry. Of course, the R-charge (4.3.17) at the IR "xed point is actually found by requiring the existence of a duality, which is independent of the discussion of superpotential (4.3.18). Next we brie#y review the duality mentioned above. First, the electric theory has an anomalyfree global symmetry S; (N );S; (N );; (1);; (1) at the IR "xed point under which the * D 0 D 0 quantum numbers of the matter "elds are listed in Table 4.3.1. The dual description is the S;(2N !N ) gauge theory. The matter "elds include not only the D A dual quarks q , q G and the adjoint "eld >, but also two gauge singlet chiral super"elds MG and NG . G H H At the IR "xed point, these gauge singlets are the meson and meson-like operators of the original

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Table 4.3.2 Representation quantum numbers of the matter "elds of the magnetic theory under S; (N );S; (N );; (1);; (1) * D 0 D 0

q q > M N

S; (N ) * D

S; (N ) 0 D

; (1)

; (1) 0

NM D 1 1 N D N D

1 N D 1 NM D NM D

N /(2N !N ) A D A !N /(2N !N ) A D A 0 0 0

1!2(N !N )/(3N ) D A D 1!2(N !N )/(3N ) D A D 2/3 2!4N /(3N ) A D 8/3!4N /(3N ) A D

Table 4.3.3 't Hooft anomaly coe$cients Triangle diagrams

Anomaly coe$cients

S; (N ) *0 D S; (N ); (1) *0 D 0 S; (N ); (1) *0 D ; (1) 0 ; (1) 0 ; (1); (1) 0

#(!)N Tr(t t , t!) A !2N/(3N ) Tr(tt ) A D N Tr(tt ) A !2(N#1)/3 A 26(N!1)/27!16N/(27N ) A A D !4N/3 A

electric theory MG "QG QI P "QG ) QI , NG "QG XP QI Q "QG ) X ) QI . H P H H H P Q H H

(4.3.19)

The quantum numbers of all the matter "elds under the global symmetry S; (N ); * D S; (N );; (1);; (1) are listed in Table 4.3.2. 0 D 0 Strong support to this duality pattern comes from the 't Hooft anomaly matching for the above global symmetry group. Using the quantum numbers listed in Tables 4.3.1, 4.3.2 and the quantum numbers of the electric and magnetic gauginos, : (1, 1, 0, 1); I : (1, 1, 0, 1) ,

(4.3.20)

one can easily write down the conserved currents and the energy-momentum tensor composed of the massless fermionic components and calculate the anomaly coe$cient. The explicit calculation shows that the coe$cients are indeed equal in both theories and they are listed in Table 4.3.3. Note that now there are more non-vanishing triangle diagrams in comparison with the usual N"1 supersymmetric QCD. As in the usual supersymmetric QCD case, the global symmetry and holomorphy determine that the superpotential of the dual theory should be of the form = "MG q ) > ) q H#NG q ) q H#Tr > .

 H G H G

(4.3.21)

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Note that the "rst term of the above superpotential is an operator with UV dimensions 4. However, as argued in the following, MG q ) > ) q H is actually not always irrelevant, i.e. its dimension can H G become less than 4. The dynamical structure of the theory can be qualitatively analysed through the scaling dimension of the meson "elds MG at the IR "xed point, H 3 2N d(M)" R"3! A . (4.3.22) 2 N D The one-loop beta function coe$cient (4.3.11) shows that when N '2N , the electric theory is not D A asymptotically free and thus in a free electric phase. At N &2N , d(M)&2, (4.3.21) shows that the D A electric theory should be weak coupled. As N decreases, the running coupling in the IR electric D theory increases and d(M) decreases. At N "N , d(M) becomes 1 and M behaves as a free scalar D A "eld. It is then natural to expect that the dimension of M in the IR region remains 1 for N (N D A as well. In the dual magnetic description, the dynamical behaviour of M is also remarkable. From the one-loop beta function coe$cient of the magnetic theory,
I "2(2N !N )!N "3N !2N , D A D D A one can see that for

(4.3.23)

(4.3.24) N KN D  A the coupling is weak, and at low-energy M will become a free "eld coupled to the gauge sector through the "rst term of the non-renormalizable superpotential (4.3.21). If N '2N /3, the D A coupling of Mq>q will decrease at large distances. For N not much larger than 2N /3, perturbaD A tion theory works. One can see that M becomes a free "eld with dimension 1. However, as N increases, the coupling in the magnetic theory becomes stronger, the anomalous dimension of D Mq>q may become more and more negative until, if the coupling is strong enough, this irrelevant operator actually becomes relevant, i.e. the full (canonical # anomalous) dimension may become less than 4. Hence it will have e!ects on the IR dynamics of M and the strongly interacting magnetic gauge degrees of freedom. Obviously, if this phenomenon occurs, it is completely due to non-perturbative e!ects in the magnetic theory, since usually the IR magnetic gauge coupling should be larger than the critical coupling so that Mq>q becomes relevant. Note that the interaction of M with the gauge sector is a #avour interaction and the gauge interaction occurs through the colour degrees of freedom. Usually at low energy the colour interaction is much stronger than the #avour one. One can perform a similar discussion for the operators NG and the related Nqq interaction. H From Table 4.3.2 the operators NG have dimension H 2N 3 (4.3.25) d(N)" R(N)"4! A , N 2 D which will go to 1 at N "2N /3. For N (2N /3, (4.3.23) shows that the magnetic theory is not D A D A asymptotically free, and hence the full theory is free in the infrared region. Consequently,

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NG become free "eld operators when N (2N /3. Otherwise they will participate in the interaction H D A non-trivially. In summary, the general dynamical structure of this set of theories can be stated as follows: the theory is free in the ranges N '2N and N (2N /3. The former corresponds to a free electric D A D A phase, with the "elds M and N corresponding to quark composites with dimensions 2 and 3, respectively, due to (4.3.22) and (4.3.25); the latter corresponds to a free magnetic phase with M and N being free "elds with dimension 1. The theory is in an interacting non-Abelian Coulomb phase for the range 2N /3(N (2N , in which there is a dual magnetic description, i.e. supersymmetric A D A S;(2N !N ) QCD with two colour singlets M and N. Obviously, the model with N "N is A D D A self-dual under the duality N 2N !N , which exchanges the range N (N (2N with A D A A D A 2N /3(N (N . (4.3.22) shows that as N (N the "eld operators MG are free "eld operators A D A D A H since they have dimension 1, but the full theory is not. A discussion similar to the S;(N ) QCD case can show that the duality is preserved under A a mass deformation and in the #at directions. Note that the #at directions of the magnetic theory are composed of the D-#at directions of the gauge sector and the F-#at directions obtained through the superpotential (4.3.21) [85]. 4.3.2. Duality in the Kutasov}Schwimmer model 4.3.2.1. Kutasov}Schwimmer model. The Kutasov}Schwimmer model is a generalization of the model introduced in last section. The "eld content is the same, only superpotential (4.3.18) is replaced by a general one, ="g Tr XI> . (4.3.26) I For simplicity, only the case k(N will be considered. (4.3.26) shows that for k"1 this superA potential is the mass term for X, so one can integrate out the adjoint super"eld in the IR region and return back to the usual supersymmetric QCD. The k"2 case is just the one discussed in the last section. We "rst have a look at the moduli space of this model. As in the usual supersymmetric QCD, the moduli space is labeled by two kinds of gauge invariant operators: the meson operators (M )G "QG (XJ\)? QI @ , l"1,2, k , J H ? @ H

(4.3.27)

with (XJ\)? ,X?  X?  2X?J\ , ? @ @ ? and the baryon-like operators I  n "N , J A J are the `dresseda quarks

BL L 2LI "QL 2QLI ;  I

where Q J QG ,(XJ\QG) "(XJ\) @QG , l"1,2, k , J? ? ? @

(4.3.28)

(4.3.29)

(4.3.30)

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Table 4.3.4 Representation quantum numbers of the matter "elds of the electric theory under S; (N );S; (N );; (1);; (1) * D 0 D 0

Q QI X

S; (N ) * D

S; (N ) 0 D

; (1)

; (1) 

; (1) 0

N D 1 1

1 N 1D

1 !1 0

1 1 0

1!2N /(3N ) A D 1!2N /(3N ) A D 2/(k#1)

and consequently 1 QLJ " ? 2?J QG  2QGLJ J . J l! J? J?

(4.3.31)

Eqs. (4.3.29) and (4.3.31) show that the total number of the baryon operators is






N N kN D 2 D " D .  (4.3.32) n N

 n J L  I A Before discussing the duality, we "nd the anomaly-free global symmetry. Classically, the action of N"1 the theory has the global symmetry S; (N );S; (N );; (1);; (1);;  (1) . (4.3.33) * D 0 D  0 The standard form of the classical action of the N"1 supersymmetric gauge theory with quark super"elds in the fundamental representation and a matter "eld in the adjoint representation and the superpotential (4.3.26) imply that the quantum numbers of matter "elds under (4.3.33) should be as listed in Table 4.3.4. At the quantum level, the ; (1) and ;  (1) will become anomalous. Like in the S;(N ) QCD A  0 case, there exists an anomaly-free ;  (1) symmetry coming from a combination of ;  (1) and 0 0 ; (1) transformations. The method of searching for such an anomaly-free ; (1) is the same as the  0 one discussed in Section 3.1.2. The classical conserved currents corresponding to the classical ;  (1) and ; (1) transformations are, respectively,  0 k!1 j  "M    ! M    !M ?  ? ; 0 I / I  / k#1 6 I  6 I  j "M    . I / I  / Their operator anomaly equations read




(4.3.34)



2N g A !N F? FI IJ? ; R jI "2 D 32
 IJ I 0  k#1 g F? FI IJ? . R jI "2N I  D 32
 IJ

(4.3.35)

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Table 4.3.5 Anomaly-free ; (1) combination of quantum numbers of the matter "elds in the electric theory 0

Q QI X

; (1)

; (1) 

; (1) 0

1 !1 0

1 1 0

1!2N /(3N ) A D 1!2N /(3N ) A D 2/(k#1)

Table 4.3.6 Representation quantum numbers of the matter "elds in the magnetic theory under the global symmetry S; (N );S; (N );; (1);; (1) * D 0 D 0 S; (N ) * D

S; (N ) 0 D

q

NM D

1

q

1

N D

>

1

1

N D

NM D

M

H

; (1) N A kN !N D A N A ! (kN !N ) D A 0

0

; (1) 0 2 kN !N D A 1! N k#1 D 2 (kN !N ) D A 1! N k#1 D 2 N#1 4 N 2 A# 2! ( j!1) N#1 N k#1 D

According to (3.4.4) the anomaly-free ; (1) charge should be the following combination: 0






2 N A A. R"R # 1!  k#1 N D

(4.3.36)

Therefore, the full quantum symmetry is S; (N );S; (N );; (1);; (1), with the anomaly* D 0 D 0 free R-charges listed in Table 4.3.5. 4.3.2.2. Duality. The discussion in the last section and (4.3.32) show that the dual magnetic description should be an S;(kN !N ) gauge theory with the following matter content: D A N #avours of dual quarks q , q G, an adjoint "eld > and gauge singlets M , which are identical to D G H (4.3.27) at the IR "xed point. The quantum numbers of these matter "elds under the global symmetry are listed in Table 4.3.6. They are determined by (4.3.27) and the identi"cation of the baryon-like operators in both the electric and magnetic theories, BL L 2LI &BK K 2KI   



(4.3.37)

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Table 4.3.7 't Hooft anomaly coe$cients Triangle diagrams

Anomaly coe$cients

S;(N ) D S;(N ); (1) D 0

N Tr(t t , t!) A 2 N A Tr(tt ) ! k#1 N D

S;(N ); (1) D ; (1) 0

N Tr(tt ) A 2  16 N A !1 #1 (N!1)! A k#1 (k#1) N D 4 ! N k#1 A

; (1); (1) 0 ; (1) 0




 

2 ! (N#1) k#1 A

with BK K 2KI "qK 2qKI ;  m "kN !N , l"1, 2,2, k .  I J D A

 J

(4.3.38)

Eqs. (4.3.37) and (4.3.38) suggest that m "N !n , l"1, 2,2, k . J D I>\J

(4.3.39)

As in the usual supersymmetric QCD, this dual picture is supported by the 't Hooft anomaly matching. Considering the various currents corresponding to the global symmetry S; (N );S; (N );; (1);; (1) and the energy-momenta composed of the fermionic compo* D 0 D 0 nents of the above matter "elds and gauginos in both the electric and magnetic theories and calculating the non-vanishing triangle diagrams, one can easily "nd that the anomaly coe$cient are indeed identical. They are listed in Table 4.3.7. The quantum numbers given in Table 4.3.6 and the holomorphy determine that the superpotential of the dual, magnetic theory should be of the following form: I = "Tr >I>#  M q >I\Hq ,

 H H

(4.3.40)

where the normalization coe$cients are chosen equal to 1. In principle these coe$cients can be calculated and they are relevant for a more clear understanding of duality. Especially, the case k"1 corresponds to the Seiberg duality discussed in Section 4.2, since now the superpotential (4.3.26) and (4.3.40) show that X, > are massive and can be integrated out. The case k"2 is just the Kutasov model discussed in the previous section.

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4.3.2.3. Deformation. There are many interesting deformations, which can provide a non-trivial check on the above duality. Two cases will be discussed [86]. The "rst one is the mass deformation, which is implemented by giving a mass to the N th electric quarks; thus, the whole superpotential D of the electric theory at tree level becomes = "g Tr XI>#mQI D Q D .  I , ,

(4.3.41)

Consequently, the low-energy theory becomes an S;(N ) gauge theory with N !1 #avours since A D the heavy #avour should be integrated out. On the other hand, in the magnetic theory the introduction of the mass term for the electric quarks gives a new term to the magnetic superpotential (4.3.40) due to the duality: I = "g Tr >I>#  M q >I\Hq#m(M ),DD .

 I H , H The equations of motion for (M ),DD show that the vacuum should satisfy , q D >J\q ,D "! m, l"1,2, k , , JI

(4.3.42)

(4.3.43)

which together with the following conditions determine the expectation values: q ,D " ; q? D "?I ; ? ? ,



? ,
"1,2, k!1 , >? " @> @ 0 otherwise .

(4.3.44)

Eqs. (4.3.43) and (4.3.44) imply that the Higgs mechanism occurs. Consequently, the low-energy magnetic theory is the S;(kN !N !k) theory with N !1 #avours. It corresponds exactly to D A D the low-energy electric theory. Thus the duality is preserved under this mass deformation. Another deformation is achieved by perturbating the electric superpotential (4.3.26) by the following term:






m =(X)" Tr X# X#X , 2

(4.3.45)

where the last term  Tr X is a Lagrange multiplier term since the S;(N ) adjoint "eld X is A traceless. Due to the relation between the scalar potential and the superpotential, <()"R=/RX 6 , the vacuum solutions are given by diagonal matrices X"(x P ) with P Q 6( eigenvalues x satisfying the quadratic equations: P 3x#mx#"0 .

(4.3.46)

Thus there are two solutions, !m$(m!12 x " , ! 6

(4.3.47)

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which correspond to the two minima of the scalar potential. Since the supersymmetry is not broken, the Witten index implies that there should exist N #1 possible vacua labelled by A k"0, 1,2, N . Without loss of generality, we assume that there are k eigenvalues x equal to A G x and N !k ones equal to x . Consequently, the gauge group is broken, > A \ S;(N )PS;(k);S;(N !k);;(1) . A A

(4.3.48)

In each vacuum the theory is reduced to the usual supersymmetric QCD since X becomes massive and can be integrated out. In the dual magnetic theory, the gauge group is S;(2N !N ) and hence there seems to exist D A 2N !N #1 supersymmetric vacua. A similar analysis shows that the introduction of the D A superpotential (4.3.45) breaks the symmetry of the magnetic theory as follows: S;(2N !N )PS;(l);S;(2N !N !l);;(1) . D A D A

(4.3.49)

However, as indicated by ADS superpotential in Refs. [32,33], a supersymmetric vacuum is not stable if the number of #avours is smaller than the number of colours in supersymmetric QCD. Therefore, the lth vacuum is stable if and only if l4N and 2N !N !l4N . Thus the true D D A D vacua are labelled by l"N !N ,2, N and there are only N #1 vacua. This coincides exactly D A D A with the requirement of duality. One can naturally think that the explicit correspondence between the vacua given in (4.3.48) and (4.3.49) is l"N !k. D Special consideration should be paid to two particular cases, k"0 or N . Now the S;(N ) gauge A A group remains unbroken. Consequently, the trivial vacuum X"0 in the electric theory corresponds to a magnetic vacuum with >O0, in which the breaking pattern of the magnetic gauge group is S;(2N !N )PS;(N );S;(N !N );;(1) . D A D D A

(4.3.50)

However, we can still get a consistent duality mapping. Eq. (4.1.5) shows that in the magnetic theory there exists a superpotential ="MGHq P q . The equations of motion for M will set G HP q ) q"0. Furthermore, according to Eq. (3.4.52), the quantum moduli space is given by det(q ) q)!BBI "
,D , which relates q and q to the baryons BI , B. Thus in the quantum moduli space of vacua, q ) q"0 means BBI "!
,D . This non-vanishing expectation value of B will break the ;(1) symmetry of (4.3.50). Therefore, Seiberg's duality pattern arises: S;(N );S;(N )  S;(N );S;(N !N ) . D A D D A

(4.3.51)

In the case k'2, there are many possible choices for the superpotential = leading to deformations. The situation becomes more complicated, but the analysis is conceptually similar to the case k"2. By Eq. (4.3.26), the vacua are determined by the equation =(x)"0. Since =(x) is a polynomial of degree k'2, there exist k solutions, and in general these solutions are di!erent. The ground states are labelled by the number i (l"1,2, k) of eigenvalues x residing in the lth J minimum of the scalar potential (i 4i 424i and  I i "N ). Correspondingly, the electric J J A   I

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gauge group S;(N ) is broken by the non-vanishing expectation values as follows: A S;(N )PS;(i );S;(i );2;S;(i );;(1)I\ . A   I

(4.3.52)

Note that each of the S;(i ) factors describes a supersymmetric QCD model since X becomes J massive and can be removed. In the dual magnetic theory, a similar breaking pattern occurs. Assume that there are j eigenvalues in the lth minimum ( j "kN !N ) and the breaking of the J J D A J magnetic gauge group is S;(kN !N )PS;( j );S;( j );2;S;( j );;(1)I\ . D A   I

(4.3.53)

The duality mapping is just given by j "N !i . J D J 5. Non-perturbative phenomena in N ⴝ 1 supersymmetric gauge theories with gauge group SO(Nc ) and Sp(Nc ⴝ2nc ) In this section we shall review the non-perturbative dynamics of N"1 supersymmetric gauge theories with gauge groups SO(N ) and Sp(N "2n ). We shall see that in these supersymmetric A A A gauge theories, some new dynamical phenomena arise such as inequivalent branches, oblique con"nement and electric}magnetic}dyonic triality [15]. In addition, there emerges an explicit phase transition from the Higgs phase to the con"ning phase [15] in supersymmetric SO(N ) gauge A theory with quarks in the fundamental representation (i.e. the vector representation). This is di!erent from the supersymmetric S;(N ) gauge theory, where the transition from the Higgs phase A to the con"ning phase is smooth. The reason behind this is that in the SO(N ) supersymmetric A gauge theory, the dynamical quarks cannot screen and therefore, at large distances con"nement occurs suddenly. In the following, the non-perturbative phenomena in the SO(N ) supersymmetric gauge theory A will be discussed in detail. The new dynamical phenomena di!erent from the S;(N ) case will be A emphasized. The non-perturbative dynamics of the Sp(2n ) supersymmetric gauge theory is similar A to the S;(N ) and SO(N ) cases. In fact, it is proposed that by using the trick of extrapolating the A A colour parameter N of SO(N ) to a negative value, the dynamics of the Sp(N "2n ) supersymmetA A A A ric gauge theory can be read out from that of the SO(N ) theory. Thus the Sp(2n ) case will be only A A brie#y introduced. 5.1. N"1 supersymmetric SO(N ) gauge theory with N yavours A D 5.1.1. Global symmetries of N"1 supersymmetric SO(N ) QCD A The fundamental representation of the SO(N ) group is its vector representation and hence it is A always real. Thus in SO(N ) gauge theory there is no di!erence between left- and right-handed A quarks. This is di!erent from the S;(N ) case, where the right-handed quark is the left-handed A anti-quark. The classical Lagrangian of N"1 supersymmetric SO(N ) QCD is A M ?= M  M  )#QReE4,A Q  M  . L" Tr(=?=   #= ?F ?F FF 

(5.1.1)

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In the Wess}Zumino gauge, the (four-) component "eld form of above Lagrangian reads 1 1 L"! F? F?IJ# iM ?ID?@@#iM GPI(D ) #(DI)HGP(D ) IJ I I GP I GP 4 2 g i ! i(2g[ (¹?)P M GQ?!HGP(¹?)Q M ? ]! [R,]# IJHMF? F? , GP Q P GQ IJ HM 2 32


(5.1.2)

where the group indices a"1,2, N (N !1)/2; the colour indices r, s"1,2, N and the #avour A A A indices i"1,2, N . D is the covariant derivative in the adjoint representation and D in the D I I vector representation, D?@"R ?@!gf ?@AAA ; DI "RI !igAI?¹? . The classical Lagrangian I I I PQ PQ PQ (5.1.2) possesses the global symmetry S;(N );; (1);;  (1) corresponding to the following D  0 transformations: E S;(N ): D  P(e ?R)H  , M GPPM HP(e\ ?R)G ; G HP H GP  P(e ?R)H  , HGPPHHP(e\ ?R)G . G HP H GP E ; (1):  Pe A ?, M PM e A ? .

(5.1.4)

E ;  (1): 0 Pe A ?, M PM e A ?; ?Pe\ A ??, M ?PM ?e\ A ? .

(5.1.5)

(5.1.3)

Like in the S;(N ) case, the ; (1) and ;  (1) symmetries su!er from an Adler}Bell}Jackiw A  0 (ABJ) anomaly, and they can be combined into an anomaly-free ; (1) symmetry. To do this, let us 0 "rst calculate the fermionic triangle anomaly diagrams of the relevant currents and get the coe$cients in the anomaly operator equations for jI and jI  , and then try to construct an  0  anomaly-free ; (1) symmetry. The Noether currents corresponding to the above ; (1) and ;  (1) 0  0 transformations are j "M GP   ,j , I I  GP I j  "M GP   !M ?  ?,j!J . (5.1.6) I  GP  I  0 I I I Considering the triangle diagrams  j J? J@ , JJ? J@  with J? and J? being the dynamical I I J I I J I I currents corresponding to the global gauge transformations, which are composed of the quarks and the gaugino, respectively, J? "M GP (¹?) Q ; J? "f ?@AM @ A , I I P GQ I I we "nd the operator equation for the anomalies of j

(5.1.7) I

and j

0 I

:

1 RIj "2N IJHMF? F? ; I D 32
 IJ HM

(5.1.8)

1 IJHMF? F? . RIj  "[2N !2(N !2)] D A IJ HM 0 I 32


(5.1.9)

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Table 5.1.1 Combination of anomaly-free R-charges for fundamental "elds;  and  denote the lowest component and the (two-) / / component spinor of the quark chiral super"eld

 /  / Q 

; (1) 

;  (1) 0

; (1) 0

#1 #1 #1 0

0 !1 0 #1

(N #2!N )/N D A D !(N !2)/N A D (N #2!N )/N D A D #1

Table 5.1.2 Representation quantum numbers of fundamental "elds under global symmetries S;(N );; (1);; (1) D 0

 /  / Q 

S;(N ) D

R

N D N D N D 0

(N #2!N )/N D A D !(N !2)/N A D (N #2!N )/N D A D #1

Using the same arguments as in the S;(N ) case, namely "nd the above anomalies lead to a shift of A the vacuum angle  in the Lagrangian (5.1.1) and hence the absence of anomalies is re#ected in the vanishing of the shift, one can easily obtain the non-anomalous ; (1) symmetry with the charge 0 being the linear combination N !N #2 A A. R"R # D  N D

(5.1.10)

The R -charges and the ; (1) charges of all the "elds (and even of some parameters) should be   combined in such ways so that an anomaly-free ; (1) symmetry is achieved. Therefore, at the 0 quantum level the anomaly-free global symmetry is S;(N );; (1). The anomaly-free R-charges of D 0 some fundamental "elds are listed in Table 5.1.1 and the transformation properties of every fundamental "eld corresponding to the global symmetry are given in Table 5.1.2. In addition, the theory has an explicit Z charge conjugation symmetry C [15]. In particular, the  ; (1) transformation (5.1.4) and the operator anomaly equation (5.1.8), show that at the quantum  level the theory is also invariant under a discrete Z D symmetry, , QPe L
,D Q, n"1,2, 2N . D

(5.1.11)

(For N "3, the symmetry is enhanced to Z D , which will be discussed in detail in Section 5.5.1.) A , In fact, the anomaly equation (5.1.8) means that under the ;(1) transformation (5.1.4) the vacuum 

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angle  in the Lagrangian (5.1.2) is shifted to P#2N  , D

(5.1.12)

and hence we have the discrete symmetry (5.1.11). However, since the even elements of Z D , i.e. , those with n"2, 4,2, 2N , also belong to S;(N ), this global #avour symmetry should be D D S;(N );Z D /Z D . D , , 5.1.2. Classical moduli space If written in two-component "eld form, the scalar potential of the supersymmetric SO(N ) QCD A has the same form as in the S;(N ) case, (3.3.3). The classical moduli space is still determined by the A D-#atness condition, D?"0. As in the S;(N ) theory, the D-#atness condition does not require A that the expectation values of the squarks vanish, only that they equal some constant values. To make the supersymmetry manifest we work in the super"eld form. Like in the S;(N ) case, we write A the quark super"eld in an N ;N matrix form in terms of the #avour and colour indices. Since the D A scalar potential is SO(N ) gauge and S;(N ) global transformation invariant, in the D-#at A D directions we can use these gauge and global rotations to make the quark super"eld matrix diagonal. In the following we shall discuss the classical moduli space for di!erent relative colour and #avour numbers. 5.1.2.1. N (N . The quark super"eld matrix can in this case be diagonalized as follows: D A




Q"

a  0



0

2

0

2 0

a

2

0

2 0



 

\



\ 0

0

0

2 a D ,

.

(5.1.13)

2 0

If all the a O0, their scalar components will break the gauge group SO(N ) to SO(N !N ) for G A A D N (N !2, and completely break S;(N ) for N 'N !2. The moduli space will still be D A A D A described by the expectation value of the `mesona super"elds MGH"QG QPH,QG ) QH , P

(5.1.14)

which is explicitly gauge invariant. We shall show that these N (N #1)/2 mesons are the D D appropriate low-energy dynamical degrees of freedom since they correspond precisely to the matter super"elds left massless by the Higgs mechanism. Due to the spontaneous symmetry breaking SO(N )PSO(N !N ), among the N N quark super"elds Q P there are A A D D A G N (N !1)!(N !N )(N !N !1)"N N !N (N #1)  A D A D A D  D D  A A

(5.1.15)

"elds which become massive through the Higgs mechanism. N N ![N N !N (N #1)/2]" D A D A D D N (N #1)/2 "elds remain massless. D D

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5.1.2.2. N 5N . In this case the D-#at directions are given by D A


 a  0

0

2

0

a

2

0



 

\



Q" 0

0

0

0

2 a A . , 2 0





\



0

0

2

0

(5.1.16)

The low-energy gauge invariant degrees of freedom that can be constructed to describe the moduli space are not only the meson super"elds MGH"QG ) QH, but also the baryon super"elds 1 P 2P,A QG  QG  2QG,A ,A . BG 2G,A " P P P N! A

(5.1.17)

Along the #at directions given by (5.1.16), we have


 a  0

0

2

0

2

0



a  

\



M" 0

0

0

0

2 a A ; , 2 0





\



0

0

2

0

B2,A "a a 2a A   ,

(5.1.18)

with all other components of M and B vanishing. Thus one can see that the rank of M is at most N . A If the rank of M is less than N , i.e. one a or several a 's vanish, then the baryon "elds B"0. If the A G G rank of M is equal to N , then B has rank 1 and its non-zero component is the square root of the A product of non-zero diagonal values of M, up to a sign. So the baryon "eld should not be regarded as an independent variable to parameterize the moduli space. Therefore, for N 5N , the classical D A moduli space is described by a set of M of rank at most N with an additional sign coming from A taking the square root B"$(detM

(5.1.19)

for M with rank N , where the prime denotes that only the non-zero diagonal values of M are A considered.

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5.2. Dynamically generated superpotential and decoupling relation The quantum moduli space and the corresponding non-perturbative dynamics will become more complicated than in the S;(N ) case due to the peculiarities of the SO(N ) group. Some new A A dynamical phenomena will arise. The quantum theory is also more sensitive to the relative numbers of colours and #avours than the S;(N ) gauge theory. In each case (N (N or N 5N ) A D A D A we must give a detailed classi"cation of the relative numbers of colours and #avours and discuss the corresponding non-perturbative phenomena. Before going into details we "rst present the general form of the dynamically generated superpotential in the case N (N and the connections between D A di!erent energy scales related by the decoupling limits in various ways. To construct the dynamically generated superpotential, we need a dynamical scale associated with the running coupling. For N '4, the coe$cient of one-loop beta function of the SO(N ) A A gauge theory with N quarks in the vector representation of the gauge group is D
"3(N !2)!N . (5.2.1)  A D With Eq. (3.4.4), the running of the gauge coupling is   A \\,D "q,A \\,D e\
E O > F . (5.2.2)
, ,A ,D (5.2.2) implies that the dynamically generated scale
becomes a complex number due to the presence of the vacuum angle, and further it can be formally thought of as the scalar component of a (space}time independent) chiral super"eld. The ; (1) charge and the R -charge of this super"eld   should be the same as those of , i.e. 2N and 2(N !N #2), respectively, since the anomalies of D D A ; (1) and ;  (1) give corresponding shifts in .  0 In the cases 2(N 44, there are some peculiarities. When N "4, since A A SO(4)KS;(2) ;S;(2) , (5.2.3) 6 7 with the subscripts X and > labelling two S;(2) branches, the theory must be decomposed into two independent gauge theories with gauge group S;(2). The quark super"elds, which before were in the vector representation of SO(4) before, should now constitute the fundamental (spinorial) representation (2, 2) of S;(2) ;S;(2) , 6 7 QG "QG , r"1,2, 4; ,
"1, 2 , (5.2.4) P ?@

i.e. under a gauge transformation, QG PA6AA7BQG , A6A3S;(2) ; A7B3S;(2) . (5.2.5) ?@ ? @ AB ? 6 @ 7 There are two independent gauge couplings, one for each S;(2) , s"X, >. Since the one-loop beta Q function coe$cient in an S;(2) gauge theory with N quarks in the fundamental representation is D 6!N , we have D
Q \,D . (5.2.6) e\
EQ O> F" q




Eq. (5.2.6) shows that the running of the gauge coupling in each S;(2) branch accidentally coincides with the general form of the SO(N ) theory with N "4. A A

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213

Table 5.2.1 ;(1) quantum numbers of the quantities composed of the dynamical superpotential (N '3) A


@ det M

; (1) 

;  (1) 0

; (1) 0

2N D 2N D

!2(N #2!N ) D A 0

0 2(N #2!N ) D A

When N "3, the vector representation of SO(3) coincides with its adjoint representation. In this A case the one-loop
-function coe$cient is
"6!2N , and the running of the gauge coupling is  D D
Q \, e\
EQ O> F" . (5.2.7) q




One can see that the running coupling in this case does not coincide with the general form for SO(N ), and thus it needs a special consideration. A A gauge invariant quantity composed of the dynamically generated superpotential should be a function of the low-energy dynamical degree of freedom M. The global symmetry S;(N );; (1) D 0 restricts it to be a function of det M. We list the various quantum numbers of
@ and det M in Table 5.2.1 for N O3. With the requirement that the superpotential should have R-charge 2 and A dimension 3, and should be holomorphic, the only possible form is ="C






D ,A \\,D 
,A A \\, , ,D . , , det M A

D

(5.2.8)

The coe$cient C A D can be determined, like in the S;(N ) case, through an explicit calculation. , , A Since the scale
is a complex quantity, the superpotential can pick up a Z A D phase factor due , \, \ to the power 1/(N !N !2), A D
,A \\,D ,A \\,D  ="C A D e L
,A \\,D  ,A ,D , , det M ,C

 ,A ,D ,A \\,D 











D ,A \\,D 
,A A \\, , ,D , n"1, 2,2, N !2!N . A D det M

(5.2.9)

The phase factor in (5.2.9) labels di!erent but physically equivalent vacua of the theory coming from the spontaneous breaking of a discrete symmetry induced by gaugino condensation in the low energy SO(N !N ) Yang}Mills theory. A D In the following sections, we shall see that a superpotential of the form (5.2.8) can indeed be generated by gaugino condensation, like in the S;(N ) case with N (N !1. For N "N !2, A D A D A the above superpotential does not makes sense. For N !2(N 4N , a superpotential (5.2.8) A D A cannot be generated, since it would lead to non-physical dynamical behaviour. For N 'N , det M"0, and superpotential (5.2.8) does not exist. Overall, there will be no dynamically D A generated superpotential for N 5N !2. For the special case N "3, we shall see that there is D A A also no dynamically generated superpotential for any N . Consequently, these theories, with no D

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dynamically generated superpotential, will have a quantum moduli space of exactly degenerate but physically inequivalent vacua, and they will present interesting non-perturbative dynamical phenomena di!erent from the S;(N ) case. A For later use, we give the relations between di!erent energy scales connected by the decoupling of heavy modes. Due to the peculiarity of the SO(N ) group when 2(N 44, we must give special A A consideration to the decoupling in SO(3) and SO(4) theories. Giving the N th quarks a large mass = "1/2mM,D ,D "1/2mQ,D QP,D , and making this heavy P D  mode decouple, the theory with N quarks will yield a low-energy theory with N !1 quarks. The D D running couplings should match at the scale m. When N '4, we have A 3(N !2)!N m 3(N !2)!(N !1) m 4
A D ln A D " " ln ;
A D 2

A D 2
g(m) , , ,  , \ A \\,D m"
,A \\,D \ .
, ,A  ,D ,A  ,D \

(5.2.10)

When N "4, due to Eq. (5.2.3), in each S;(2) branch the coupling constants should match, A m 6!(N !1) m 6!N D D ln " ln ;
D 2

D 2
Q, Q , \ D m"
\,D \,
\, s"X, > . Q ,D \ Q,D

(5.2.11)

In the case N "3, the quarks are in the adjoint representation of the gauge group and the A one-loop beta function coe$cient changes to
"6!2N , so we have  D m 6!2(N !1) m 6!2N D D ln " ln ;
D 2

D 2
 ,  , \ D \ .
\,DD m"
\,  ,  ,D \

(5.2.12)

Another way of decoupling is through the Higgs mechanism with a large expectation value a D in , (5.1.13). The SO(N ) theory with N quarks will decouple into an SO(N !1) theory with N !1 A D A D quarks. Under the requirement that the running couplings should match at the energy a D , we get , a relation between the high-energy scale
A D and the low-energy scale
A . In the case , , , \ ,D \ N '5, we have A 4
3(N !2)!N a 3[(N !1)!2]!(N !1) a D A D ln ,D " A D , " ln ;
A D
A g(a D ) 2
2
D , , , \ , \ , A \\,D a\"
,A \\,D \ .
, ,A ,D ,D ,A \,D \

(5.2.13)

Since in the moduli space, QG "a G and hence M,D ,D "a D , (5.2.13) can be written as , P G P A \\,D (M,D ,D )\"
,A \\,D \ .
, ,A ,D ,A \,D \

(5.2.14)

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When N "5, a decoupling SO(5)PS;(2) ;S;(2) occurs, and the high energy running A 6 7 coupling should match the low energy ones in each S;(2) branch, 9!N a 6!(N !1) a D D ln ,D " D , ln ;
D
D 2
2
, Q, \ D \ . "
\,D D (M,D ,D )\"
\,
\,D D a\ Q,D \ , ,D ,

(5.2.15)

For the case N "4, the decoupling pattern is S;(2) ;S;(2) PSO(3). This needs a special A 6 7 consideration. From (5.2.4) and M,D ,D "Q,D ) Q,D "Q,6D 7 Q,6D 7 ?6 @6 ?7 @7 , ? ? @ @

(5.2.16)

it follows that the sum of the two running couplings of each S;(2) branch should match the coupling of the low-energy SO(3) theory,






6!N a a 6!2(N !1) a D D ln ,D #ln ,D " D , ln ;
D
D
D 2
2
6, 7, , \ D \ .
\,D D
\,D D (a D )\"4
\,D D
\,D D (M,D ,D )\"
\, 6, 7, 6, 7, ,D \ ,

(5.2.17)

The numerical factor 4 is due to the fact that M,D ,D "Q,6D 7 Q,6D 7 ?6 @6 ?7 @7 "2(Q,D Q,D !Q,D Q,D ) , ? ? @ @    

(5.2.18)

and hence in the moduli space, M,D ,D "2a D . ,

(5.2.19)

In general,
O
since the dynamics of each S;(2) branch is independent, but for convenience, 6 7 we shall limit the discussions on SO(4) to the case
"
. 6 7 5.3. Non-perturbative dynamical phenomena when N 54, N 4N !2 A D A 5.3.1. N 4N !5: dynamically generated superpotential by gaugino condensation D A This range is similar to the S;(N ) case when N (N !1 [32,33]. A superpotential arises A D A generated by gaugino condensation in the SO(N !N ) supersymmetric Yang}Mills theory, which A D is the remainder of the SO(N ) QCD broken by the scalar component of Q. According to (5.2.8), ! we have [15]






D \ ,A \,D \ 16
,A A \, 1 , ,D , ="(N !N !2)" (N !N !2) A D \, \ A D , A D det M 2

(5.3.1)

where the phase factor  ,exp(i2n
/N). Like in the S;(N ) case when N (N !1, this , A D A quantum e!ective superpotential will lift the classical vacuum degeneracy and make the theory

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have no vacuum. This can be explicitly seen from the F-term relevant to this dynamical superpotential:






D \ ,A \,D \ R= 1 16
,A A \, ,  ,D F " "!  A D Q\ GP RQGP GP 2 , \, \ det M






1 ,A \,D \ 1 O0 . & Q det M

(5.3.2)

Thus all the supersymmetry vacua disappear and this is another typical example of dynamical supersymmetry breaking [30,31]. If we consider a mass term for the matter "elds, w "Tr(mM)/2, this situation will change  greatly. The full superpotential with this mass term is






D \ ,A \,D \ 16
,A A \, 1 1 , ,D # mH MG . = " (N !N !2) A D D , \, \  2 A det M 2 G H

(5.3.3)

The moduli space is still given by the following F-#at direction labelled by M,M,






D \ ,A \,D \ R= 16
,A A \, 1 1 , ,D FH "  "!  A D (M\)H # mH "0 , G G , \, \ RMG 2 G det M 2 H

(5.3.4)

which gives mH " A D G , \, \ det m"[






D \ ,A \,D \ 16
,A A \, , ,D (M\)H , G det M

,A \,D \

],D

D \),D ,A \,D \ (16
,A A \, , ,D (det M),A \,A \,D \

(5.3.5)

and (det m),A \,D \,A \ 1 "( A )\,D . , \ D \),D ,A \ (16
,A A \, det M D , ,

(5.3.6)

Inserting Eq. (5.3.6) into (5.3.4), we have D \],A \(m\)G . MG " A [16(det m)
,A A \, , ,D H H , \

(5.3.7)

Therefore, with this mass term we obtain a theory with N !2 supersymmetric vacua labelled by A M. This result can also be derived by calculating the Witten index [31]. Further, we can easily check that the superpotential is indeed generated by gaugino condensation in the SO(N !N ) Yang}Mills theory. Integrating all massive modes out by inserting (5.3.5) A D

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and (5.3.6) into (5.3.3), we can see that 1 D \],A \ = " (N !2) A [16(det m)
,A A \,  2 A , \ , ,D 1 " (N !2) A
 A D , , \ , \,  2 A

(5.3.8)

D \],A \ is the low energy scale for the SO(N !N ) where
 A D ,[16(det m)
,A A \, , ,D A D , \,  Yang}Mills theory, the many-#avour generalization of (5.2.10). If not all of the matter "elds are massive, we can integrate out the massive quarks and get the e!ective superpotential at low energy for the massless ones. It has the same form as (5.3.1) but with the scale replaced by the low energy one. For instance, if we only add a mass term for the N th D quark, = "m,DD M,DD /2, decoupling this heavy quark, we obtain the low-energy e!ective super, ,  potential:






D \\  ,A \,D \\

16
,A A \, 1 , ,D \ , = " [N !(N !1)!2] A D D , \, \\ * 2 A det M

(5.3.9)

where
A D is given by Eqs. (5.2.10), (5.2.11) and (5.2.12), respectively, depending on the , , \ concrete case. 5.3.2. N "N !4: Two inequivalent branches and novel dynamics D A In this range, Eq. (5.1.13) indicates that SO(N ) is broken to SO(4)KS;(2) ;S;(2) by the A 6 7 scalar component of Q. The phase factor for each S;(2) branch is  "e L
"e L
"$1, Q

s"X, >; n"1, 2 .

(5.3.10)

The scale for the low energy S;(2) Yang}Mills theory (i.e. all the matter "elds integrated out) of Q each branch is A \
,A \\,A \
,
 "
 " ,A ,A \ " ,A ,A \ , 6 7 det M det M

(5.3.11)

which is the many-#avour generalization of Eq. (5.2.16). The number 6 is the one-loop
-function coe$cient of S;(2) Yang}Mills theory. Note that we have used
"
as discussed above. The 6 7 dynamical superpotential is generated by gaugino condensation in the unbroken S;(2) ;S;(2) 6 7 Yang}Mills theory. According to Eqs. (5.3.1) and (5.3.11) the dynamical superpotential is ="= #= "(4!2)
 #(4!2)
 6 7  6 6  7 7




"2 #2 "( # ) 7 6 7  6



A \ 16
,  ,A ,A \ . det M

(5.3.12)

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Since the phase factor  labels di!erent vacua, there are four ground states, which are labelled by the four possible combinations of ( ,  ), i.e. 6 7 1. (1, 1); 2. (!1,!1); 3. (1,!1); 4. (!1, 1) .

(5.3.13)

The "rst two ground states, characterized by  " , are physically equivalent, since they are 6 7 related by a discrete R-symmetry given in Eq. (5.1.11). Similarly, the last two ground states with  "! are also physically equivalent. Therefore, the sign of   labels two physically 6 7 6 7 inequivalent branches of the low-energy e!ective theory. The non-perturbative dynamics in these two branches is greatly di!erent, as shown in the following. The dynamics in the branch with   "1 is the same as in the case N "N !4. The dynamical 6 7 D A A \ /det M) lifts all the vacuum degeneracy and the quantum theory superpotential ="(16
, ,A ,A \ has no vacuum. The two ground states with   "1 present a completely di!erent physical pattern [15]. 6 7 Superpotential (5.3.12) is zero, and hence the vacua in the quantum moduli space are still degenerate but physically inequivalent. These vacua are parameterized by M. The two di!erent values $1 of  ("! ) in this branch mean that for every M there are two ground states. 6 7 However, in the origin of the moduli space, M"0, there is only one vacuum. Classically, in this vacuum the gauge symmetry SO(4) is enhanced to SO(N ), i.e. at the origin of the moduli space, the A original SO(N ) symmetry does not break at all and the low-energy e!ective theory will have A a singularity, corresponding to the SO(N )/SO(4) vector bosons which become massless. This A singularity can show up in the kinetic term K (M, MR), the classical KaK hler potential. In   quantum theory, the situation will change: such a singularity is either smoothed out or it is associated with some "elds which become massless. In the theory we are considering, Intriligator and Seiberg conjectured that the classical singularity at the origin is simply smoothed out [15]. This means that the massless particle spectrum at the origin is the same as it is at other generic points, consisting only of the "elds M. Similar phenomena have happened in low energy N"2 supersymmetric Yang}Mills theory [1,2] and in a toy model proposed in Ref. [87]. This conjecture can be subjected to several independent and non-trivial tests. The "rst is the 't Hooft anomaly matching. Since at both microscopic and macroscopic levels the theory has a global S;(N );; (1) symmetry which is unbroken at the origin, one can check the massless D 0 (fundamental and composite) particle spectrum by looking whether the 't Hooft anomaly at the fundamental level matches with that at the composite level. The quantum numbers of the fundamental massless fermions (quark and gluino) under the global symmetry SO(N );S;(N );; (1) and the currents corresponding to S;(N );; (1) as well as the energyA D 0 D 0 momentum tensor are listed in Tables 5.3.1}5.3.3, respectively. The 't Hooft anomaly coe$cients contributed by the massless elementary fermions can be easily calculated as in the S;(N ) case and A they are collected in Table 5.3.4. At the macroscopic level, the only massless fermion is the fermionic component  of MGH, which + belongs to the N (N #1)/2-dimensional representation of the S;(N ) group. Its R-charge, from D D D Table 5.1.1, is N !2N #4 N #2!N A A !1" D . 2 D N N D D

(5.3.14)

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219

Table 5.3.1 Representation quantum numbers of fundamental "elds under the global symmetry SO(N );S;(N );; (1) A D 0

 / Q 

SO(N ) A

S;(N ) D

; (1) 0

N A N A N (N !1)/2 A A

N D N D 1

!(N !2)/N A D (N #2!N )/N D A D #1

Table 5.3.2 Currents composed of fundamental fermionic "elds corresponding to the global symmetry S;(N );; (1) D 0

 / 

S;(N ) D

; (1) 0

j(Q)"M t  I / I / 0

j (Q)"(2!N )/N M  I A D / I / j ()"M ? ? I I

Table 5.3.3 Contribution of the fundamental fermionic "elds to the energy-momentum tensor; L[]"i/2(M I! !! M I), I I ! "R ! )*/2, )*"i/4[ ), *] and )"e) I I I )*I I ¹ IJ  

i/4[(M !  !! M  )#(  )]!g L[ ] / I J / J / J / IJ / i/4[(M ? ! ?!! M ? ?)#(  )]!g L[] I J J I IJ

Table 5.3.4 't Hooft anomaly coe$cients from elementary massless fermions Triangle diagrams and gravitational anomaly

't Hooft anomaly coe$cients

; (1) 0 S;(N ) D S;(N ); (1) D 0 ; (1) 0

N (N !1)/2#N /N (2!N ) A A A D A N Tr(t t , t!) A (2!N )N /N Tr(tt ) A A D !N (N !3)/2 A A

Thus the S;(N );; (1) Noether currents are, respectively: D 0 N !2N #4 A j(M)"M N  t O , j0(M)" D M N  N , I + I NO + I + I + N D p, q"1, 2,2, N (N #1)/2 . D D

(5.3.15)

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Table 5.3.5 't Hooft anomaly coe$cients for the composite fermions Triangle diagrams and gravitational anomaly

't Hooft anomaly coe$cients

; (1) 0 S;(N ) D S;(N ); (1) D 0 ; (1) 0

(N #1)(N !2N #4)/(2N ) D D A D (N #4)Tr(t t , t!) D (N #2)(N !2N #4)/N Tr(tt ) D D A D (N #1)(N !2N #4)/2 D D A

The corresponding 't Hooft anomaly coe$cients are collected in Table 5.3.5, and one can easily see that when N "N !4, the anomalies of the macroscopic theory exactly match those of the D A microscopic theory. Note that in calculating the anomaly coe$cients of Table 5.3.5 we have used the relation between the quadratic and cubic S;(N ) Casimirs in the N (N #1)/2 dimensional D D D representation and in the fundamental (N -dimensional) representation: D "(N #4)Tr(t t , t!) D , Tr(t t , t!) D D D , , , > (5.3.16) "(N #2)Tr(tt ) D . Tr(tt ) D D D , , , > Thus the 't Hooft anomaly matching supports the conjecture: the classical singularity of the KaK hler potential near the origin is smoothed out by quantum e!ects and hence [1,87] Tr MRM K(MR, M)+ & , 


(5.3.17)

where the dynamical scale
is introduced from dimensional considerations. Another test of the above conjecture is the decoupling of a heavy mode. Giving Q D a large mass , and integrating it out, the resulting low-energy e!ective theory should agree with the N "N !4 D A case discussed in the last section. We "rst look at the branch with   "1. Adding the mass term 6 7 = "mM,DD /2 to the dynamical superpotential (5.3.12), we have ,  A \ 16
,  1 ,A ,A \ = " (5.3.18) # mM,D D . ,  2 det M






The F-#atness condition for M,D

,D A \ 16
,  1 ,A ,A \ M,D D " . , m det M




gives



(5.3.19)

Thus, the low-energy superpotential is






A \\ 3 16
, 3  ,A \,A \ =" mM,DD " , , 2 2 det M

(5.3.20)

where we have used the decoupling relation (5.2.10) and det M"det MM,DD . Eq. (5.3.20) coincides , exactly with the superpotential (5.3.1) with N "N !5. D A

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In the branch with   "!1 the dynamical generated superpotential is zero. Adding a mass 6 7 term, the full superpotential is = "= "mM,DD /2. It is not possible to return back to the ,   theory with N "N !5 using this superpotential and thus this branch should be eliminated D A from the low-energy e!ective theory with N 4N !5. This supports the conjecture, since if one D A requires that the KaK hler potential is smooth everywhere in M, this branch will have no supersymmetric ground state due to the fact that the scalar potential will be proportional to 1/K(MR, M). This means that the supersymmetry is dynamically broken. Further, if we suppose that in the branch with   "!1 there are new massless states somewhere, then the addition of this 6 7 Q,D mass term will result in additional ground states. Since all the ground states for N (N !4 D A are already exhausted by (5.3.7), there will be no such extra ground states. Therefore, one can conclude that the manifold of quantum vacua (i.e. the quantum moduli space) must be smooth everywhere and without any new massless "elds except M. The physical phenomena at the origin in M are very interesting. The above discussion shows that at the classical level there are massless quarks and gluons and their superpartners, while in the quantum theory, only the M quanta are massless. Since M are colour singlets, this clearly shows that the elementary degrees of freedom are con"ned. However, the global chiral symmetry S;(N );; (1) is not broken at the origin. Therefore, we have a novel physical phenomenon that D 0 there is con"nement but without chiral symmetry breaking. The same phenomenon has been observed in the S;(N ) case with N "N #1. A D A 5.3.3. N "N !3: Two dynamical branches and massless composite particles D A (glueball and exotic states) For this value it will be shown that the ground state of the theory still has two branches, but with di!erent dynamics. From (5.1.13), we know that the expectation value of the scalar component of Q breaks SO(N ) to A SO(3), but the dynamically generated superpotential at low-energy is not a simple continuation of (5.2.8), since in breaking SO(N ) to SO(3) by the Higgs mechanism, there is a special case A SO(4)KS;(2) ;S;(2) between SO(N ) and SO(3). The dynamically generated superpotential is 6 7 A not only contributed by the gaugino condensation of supersymmetric SO(3) Yang}Mills theory, but it also receives contributions from the instanton in each S;(2) branch. A concrete method to Q "nd the dynamically generated superpotential is as follows. First by choosing N !1 eigenvalues D of Q to be large we break the SO(N ) theory to S;(2) ;S;(2) with one quark super"eld A 6 7 Q,D left massless. Matching the running gauge couplings at the scales of the Higgs mechanism gives us the relation between the low-energy and high-energy scales:
,A \
 "
 " ,A ,A \ , 6 7 det M

det M det M" . M,D D ,

(5.3.21)

(This relation is actually the many-#avour generalization of (5.2.15).) Then the expectation value Q,D  breaks the S;(2) ;S;(2) gauge group of this intermediate theory to a diagonally 6 7 embedded SO(3) . According to (5.2.17), the low-energy scale is B
 "4

 (M,D ,D )\ . B 6 7

(5.3.22)

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The gaugino condensation in the unbroken SO(3) will generate a superpotential

,A \ = "2"2(
 )"4 67 "4 , "$1 . (5.3.23) B B M,DD det M , In addition instantons in the broken S;(2) generate another superpotential 6

,A A \ = "2 6 "2 , ,A \ , (5.3.24) 6 det M M,DD , and instantons in S;(2) give 7

,A A \ = "2 7 "2 , ,A \ . (5.3.25) 7 det M M,DD , Adding these three contributions together, we obtain the superpotential for SO(N ) with A N "N !3, D A
,A \ ="= #= #= "4(1#) ,A ,A \ . (5.3.26) B 6 7 det M Here the generation of the superpotential from the broken S;(2) needs some delicate explanaQ tion [15]. The instanton contributions in S;(2) contain those from the instantons in the broken Q part. Usually when a gauge group G is broken to a non-Abelian subgroup H along a #at direction, there is no need to consider instantons in the broken G/H part. This is because an instanton in the broken G/H is not well-de"ned, and can be rotated to become an instanton of the H gauge theory. But the dynamics described by the H gauge theory will be stronger than these instanton e!ects. However, when the instantons in the broken part like (S;(2) ;S;(2) )/SO(3) are well de"ned, 6 7 their e!ect must be taken into account when one integrates out the massive gauge "elds. This situation occurs when G (or one of its factors if G is fully reducible) is completely broken or broken to an Abelian subgroup, or when the index of the embedding of H in G is large than 1. In our case, the second index of SO(3) in the adjoint representation is 2 and thus one should consider the contribution from these instantons. Now let us see what physics the superpotential (5.3.26) describes. First, the low energy theory again has two physically inequivalent branches classi"ed by . The branch with "1 is the continuation of (5.3.1) to N "N !3. Thus at the quantum level the classical degeneracy will be D A lifted and there is no vacuum. The branch with "!1 has vanishing superpotential, and thus there exists a quantum moduli space of degenerate vacua. However, the dynamics in this case is greatly di!erent from the   "!1 branch of the N "N !4 case. This can be observed from 6 7 D A the decoupling of the heavy mode. The decoupling is done in the standard way. Adding a mass term = "mM,DD /2 and then ,  integrating out Q,D in the same way as above, the branch with "1 will give the two ground states of the   "1 branch of the case N "N !4. This is exactly what we expect. If we add the mass 6 7 D A  The index of a group is de"ned as the expansion coe$cient of the leading term when expanding the trace of the product of any number of its generators in some representation in terms of the fundamental symmetric invariant tensors of the group [88]. For the trace of n generators, the index is called nth index of this representation.

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term = "mM,DD /2 to the mass term in the "!1 branch, we should get the two ground states ,  of the   "1 branch of the N "N !4 case. However, since the dynamical superpotential 6 7 D A vanishes, a similar argument as in the N "N !4 case shows that this branch has no decoupling D A limit and must be eliminated upon adding = . In oder to avoid this disease, Intriligator and  Seiberg conjecture that there must be additional massless particles at the origin of the moduli space [15]. Since these massless "elds should not appear at generic points of the moduli space, there must be a superpotential responsible for their masses away from the origin M"0. The simplest way (perhaps the only possible way) to implement this conjecture is to introduce some (chiral super-) "elds k with i being the #avour indices, and make them couple to MGH. We will see that these "elds G indeed have a natural physical interpretation. Near the origin, the corresponding superpotential for the mass term of these new particles should have the asymptotic form: 1 =& MGHk k G H 2

when M&0 ,

(5.3.27)

where  is a scale with mass dimension. It is necessary to introduce this scale since the "eld k, as a scalar super"eld, should have dimension 1, M has dimension 2 and the superpotential has dimension 2. Requiring this superpotential to respect the global S;(N );; (1) symmetry, we see D 0 that the k should belong to the N -dimensional conjugate representation of S;(N ). Since the G D D superpotential should have R charge 2 and M has R-charge 2(N #2!N )/N , the "elds k should D A D G have R-charge





N #2!N N !2 1 1 A " A 2!2 D "1# . (5.3.28) N N N 2 D D D Now making the heavy mode M,DD decouple near the origin by adding the mass term = " ,  mM,DD /2 to (5.3.27), , 1 1 = " mM,DD # MGHk k , (5.3.29) ,  2 G H 2 and then integrating out the M,DD , we immediately obtain , k D "$i(m . (5.3.30) , The two sign choices in k D  can be interpreted as two physically equivalent ground states with , ="0, which exactly correspond to the two choices in the   "!1 branch of the low energy 6 7 N "N !4 theory. D A Eq. (5.3.27) is the approximate form of the superpotential near M"0. The most general superpotential respecting the S;(N );; (1) symmetry and having the right mass dimension is D 0 then





(det M)(MGHk k ) 1 G H MGHk k . (5.3.31) =" f t" G H
,A A \ 2 , ,A \ In order for this general superpotential to yield the ground states (5.3.30), f (t) must be a holomorphic function in the neighborhood of t"0. f (0) can be set to 1 by rescaling q Pq /f (0). G G

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Table 5.3.6 't Hooft anomaly coe$cients for composite fermions Triangle diagrams and gravitational anomaly

't Hooft anomaly coe$cients

; (1) 0 S;(N ) D S;(N ); (1) D 0 ; (1) 0

1/N D !Tr(t t , t!) 1/N Tr(tt ) D 1

How can we check the reasonableness of the conjecture that in addition to the massless "elds M there still exist other massless particles? We again resort to the 't Hooft anomaly matching. It will be a highly non-trivial veri"cation of the above conjecture if the anomalies contributed by the massless spectrum consisting of MGH and k match those of the fundamental massless particle G spectrum with N "N !3. The conserved S;(N );; (1) currents composed of the fermionic D A D 0 component of k are: G






1 M   . j( )"M t  ; j ( )" 1# I I I I I I I I I I N D

(5.3.32)

The relevant fermionic part of the energy-momentum tensor for the 't Hooft axial gravitational anomaly formally reads i ¹ " (M  !  !! M   )!g L[ ] . IJ 4 I I J I J I I I IJ I

(5.3.33)

The 't Hooft anomaly coe$cients corresponding to the triangle diagrams composed of these currents are listed in Table 5.3.6. Adding the anomaly coe$cients to those contributed by the "eld M listed in Table 5.3.5 and comparing them with the anomaly coe$cients from the massless elementary particles listed in Table 5.3.4, one can see that they are precisely equal for N "N !3. D A Finally let us see what physical objects these "elds k can be interpreted as k should be G G constructed from the fundamental chiral super"elds since the product of any chiral super"elds is still a chiral super"eld. We know that the fundamental chiral super"elds in the theory are the matter "elds QG and the gauge super"eld strength =? . From the mass dimensions and the P ? quantum numbers under S;(N );; (1), one can immediately identify k as D 0 G A b , k "
\, ,A ,A \ G G

(5.3.34)

where b "(Q),A \ Tr(= =?) G ? G 1 ,  2 P P 2P,A\ QG  QG  2QG,A\ ,A\ (=? =??) . P P P ? (N !4)! GG G G,A\ A

(5.3.35)

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Obviously, b has mass dimension N !1. Superpotential (5.3.31) can now be rewritten in G A terms of b , G





(det M)(MGHb b ) 1 G H MGHb b , f t" =" G H
,\A\ 2
,\A\ ,A ,A \

(5.3.36)

where the scale  appearing in (5.3.31) has been absorbed into the de"nition of f. Eq. (5.3.34) shows that the k "elds describe exotic particles. One can intuitively think of such G exotics as being some heavy bound states. They become massless at the origin of the quantum moduli space. These exotic particles are similar to the massless mesons and baryons in S;(N ) A supersymmetric QCD with N "N #1, as discussed in Section 3.4.3. Since they are colour D A singlets and respect the chiral symmetry S;(N );; (1), this is again a phase in which there exists D 0 con"nement but without chiral symmetry breaking. 5.3.4. N "N !2: Coulomb phase with massless monopoles and dyons and conxnement D A and oblique conxnement For this number of #avours, the dynamics is more complicated than in the cases discussed above. From Table 5.1.2, the R-charge of MGH vanishes, thus no superpotential of the form (5.2.9) can be dynamically generated from gaugino condensation. Consequently, the quantum theory has a moduli space of physically inequivalent vacua, which should still be parametrized by the expectation values MGH. Classically, according to (5.1.13) in the moduli space determined by the D-#at directions, the SO(N ) gauge group breaks to SO(2) ;(1). Thus the low-energy theory is in A the Coulomb phase with the gauge vector super"eld being a massless photon supermultiplet. Like in the various cases discussed above, there exists a singularity at the origin M"0 (or equivalently, det M"0) of the classical moduli space, which is associated with an unbroken gauge symmetry. In the quantum theory, we will see that a distinct kind of singularity arises at M"0, which is related to massless monopoles rather than massless vector bosons. In fact, from the discussion on the Coulomb phase in Section 2.4, we can imagine this situation arising since the Coulomb phase has a natural electric}magnetic duality, and hence monopoles should emerge. How can we explore the non-perturbative dynamics in the Coulomb phase? The Coulomb phase looks simple, but actually its non-perturbative dynamics, as a consequence of electric}magnetic duality, is very complicated. In the cases considered above, the dynamical superpotential and the decoupling limit give almost all of the non-perturbative phenomena, at least qualitatively. However, the situation in the Coulomb phase is completely di!erent. There is no dynamical superpotential to depend on. Fortunately, the investigation of N"2 supersymmetric gauge theories by Seiberg and Witten provided some clues [1,2]. The Coulomb phase of the theory can be investigated by determining the e!ective gauge coupling "/
#i8
/g of the massless photon supermultiplet on the moduli space of vacua. This is because the general form of the low-energy e!ective action in the Coulomb phase is



L&Im d = =? . ?

(5.3.37)

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The non-perturbative phenomena for the value N "N !2 have much in common with the D A Coulomb phase of the N"2 supersymmetric gauge theory. As will be discussed in the following, there are massless monopoles and dyons at some points of the moduli space, and their condensation will cause con"nement and oblique con"nement. It is then not so strange that both cases exhibit similar non-perturbative phenomena since their low-energy theories are both in the Coulomb phase. How can we determine ? In general, in the Coulomb phase  receives two contributions. When MGH is very large, the microscopic theory is weakly coupled and perturbation theory works. Thus, at the quantum level, one part of  in the Coulomb phase comes from the gauge running coupling evaluated at the energy scale characterized by MGH. The explicit form of this perturbative contribution is given by the one-loop beta function of the microscopic theory, i.e.
"3(N !2)!N "2N !4"2N .  A D A D

(5.3.38)

From the requirement of holomorphicity, the e!ective coupling constant  in the Coulomb phase should be a holomorphic function of the parameters MGH. Because of the global #avour symmetry S;(N ), it should depend on S;(N ) invariant combinations of MGH. The natural choice is D D ;,det MGH .

(5.3.39)

As shown in Eq. (3.4.4), the perturbative one-loop exact  at the energy scale characterized by MGH is e 
O det M"
@ ;





@ i "! ln ; 2


.

(5.3.40)

The general form of the one-loop running gauge coupling and dimensional analysis imply that the perturbative contribution to  must be of this form. However, for small MGH (or equivalently small ;), another non-perturbative contribution arises from the instantons, and hence the determination of  will become complicated. There are two ways to determine the explicit functional form of . The "rst one is using a straightforward instanton calculation [1,3,49], which gives an F-term of the form:

 




d (=?= ) ?


,A \ ;



,

(5.3.41)

where =? is the low-energy ;(1) photon "eld strength supermultiplet. The n-instanton contribution to the low-energy e!ective action in the dilute gas approximation is

 

d =?= ?







,A \ L . ;

(5.3.42)

Since the e!ective Lagrangian can always be written in the following general form:



1 Im d  = =?#2 , L "  ?  16


(5.3.43)

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the F-term gives the n-instanton correction to  of the form:







,A \ L . ;

(5.3.44)

To get the full non-perturbative , one should sum all the instanton contributions (5.3.42):







,A \ L  =?= ,  a ? L ; L

(5.3.45)

where the a are some constant coe$cients, which are determined by explicit instanton calcuL lations. Despite the fact that this method is very physical and can be carried out up to three instantons, it is actually not possible to perform the above summation. Thus this method can only be used as a check of the non-perturbative result [89]. A beautiful and powerful method to determine  was worked out by Seiberg and Witten [1] based on E E E E E

electric}magnetic duality conjecture in the Coulomb phase; global geometric structure of the quantum moduli space; holomorphicity; gauge invariance and various global symmetries; correct decoupling limit.

A crucial observation of Seiberg and Witten is that in the S;(2) case, the quantum moduli space of N"2 supersymmetric Yang}Mills theory can be determined exactly: the Riemann surface of the quantum moduli space is a torus and  is equivalent to the modular parameter of the torus, i.e. the ratio of the periods of the torus. Then one can use the global geometric property of the torus and the electric}magnetic duality to determine . Before going into the details of determining , we "rst see how the SO(N ) symmetry spontaneously breaks to SO(2) ;(1). A The spontaneous breaking from SO(N ) to SO(2) is not straightforward, since there are the two A special groups SO(4) and SO(3) between them. The breaking pattern should be as follows: "rst the breaking SO(N )PSO(4) S;(2) ;S;(2) occurs; then S;(2) ;S;(2) PS;(2) SO(3); "nally A 6 7 6 7 S;(2) breaks to ;(1). The "rst step can be taken by considering the region of the moduli space where N !4 A eigenvalues of MGH become large, breaking the SO(N ) theory to a low-energy A SO(4) S;(2) ;S;(2) theory with two light #avours (N "2) in the fundamental representation 6 7 D (2, 2) of the low-energy gauge group. Matching the running gauge couplings at the scale MGH gives the relation between the high-energy dynamical scale and the low-energy scale,
,A \
,A \ .
 "
 " ,A ,A \ , 6 7 det M ; ,A \ &

(5.3.46)

means Eq. (5.3.46) is the N !4 #avour generalization of (5.2.15) and (5.2.17). ; ,det M A D & , \ the determinant for N !4 heavy #avours, or equivalently, the product of the N !4 large D A

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eigenvalues of MGH. Explicitly, the S;(N "2) invariant combination of two light #avours in the D low-energy S;(2) ;S;(2) theory is 6 7 ; det M " "det M , (5.3.47) ;K " DE ; det M A & , \ where M are the S;(2) ;S;(2) singlets, DE 6 7 M "Q ) Q ,Q   Q   ? ? @ @ , f, g"1, 2;  "
"1, 2 . (5.3.48) DE D E  D? ? E@ @ G G The second step S;(2) ;S;(2) PS;(2) can be performed by considering the limit of large 6 7 ;K and taking one eigenvalue, say M , large. Then the gauge symmetry is broken to S;(2) , with   the light #avour Q decomposing into an S;(2) singlet S (i.e. in the trivial representation) and ?@  a triplet  (i.e. in the adjoint representation) [13],  Q " S . (5.3.49)   Eq. (5.3.49) and the de"nition M ,Q ) Q mean that    ; ,;I . (5.3.50) "  M  The subscript d indicates that this S;(2) is a diagonally embedded subgroup of S;(2) ;S;(2) . 6 7 This index will be omitted later. Integrating out the heavy #avour, we obtain the low-energy S;(2)  theory. According to (5.2.17), the relation between the low-energy dynamical scale
and the  intermediate dynamical scales
, s"X, > is Q 16

 6 7 .
" (5.3.51) M  Now we have an S;(2) theory with a triplet . It is very similar to the N"2 Seiberg}Witten model except for two extra gauge singlets (M and another one from Q ). The last step   S;(2)P;(1) is induced by the scalar potential of ?, a"1, 2, 3. This is a standard Higgs mechanism. Therefore, we can use the Seiberg}Witten method to determine the low-energy e!ective coupling in the Coulomb phase. There already exist several excellent reviews on the Seiberg}Witten solution [90], here we only repeat the main points. 5.3.4.1. Seiberg}Witten algebraic curve solution. The scalar potential of N"2 supersymmetric S;(2) Yang}Mills theory is g <()"! (D?), 2

D?"[R, ]? .

(5.3.52)

Classically, like in supersymmetric QCD, there is a ; (1) symmetry, under which the gaugino , the 0 triplet  and its superpartner  transform as Pe ?;

Pe ?;

Pe ? .

(5.3.53)

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At the quantum level, this symmetry is broken by a gauge anomaly, which is equivalent to a shift of the vacuum angle , P!2N !2N "!8 . (5.3.54) A A The shift of  by 2
is still a symmetry of the theory since the generating functional is invariant. Thus at the quantum level the ; (1) symmetry reduces to a discrete symmetry 0 P e
; P e
; P e
 . (5.3.55) In the moduli space of the vacua, the scalar potential vanishes, but ? may be non-vanishing. This will break S;(2) to ;(1), and as a consequence, the theory will be in the Coulomb phase. One can choose @"a@

(5.3.56)

by a gauge rotation with a being a complex number. The parameter labelling the moduli space will be the gauge invariant chiral super"eld u"a"Tr rather than a. The discrete symmetry (5.3.55) is realized on u as a Z symmetry,  uP e
u"!u .

(5.3.57)

(5.3.58)

In the BPS limit all the particles including the gauge bosons, monopoles and dyons have a universal mass formula: Z"an #a n , (5.3.59)  " where n and n are the electric and magnetic charges carried by the particle. n O0, n "0 



corresponds to the usual electrically charged particles; n O0, n "0 corresponds to the magnetic

 monopoles and n O0, n O0 to the dyons. In the weak-coupling region,

 a "a , (5.3.60) " the subscript D indicating the dual variable. Since there exists an electric}magnetic duality in the Coulomb phase, the duality transformation maps m"(2Z,

a  a , (a) !1/(a), , (5.3.61) " " which is a typical feature of electric}magnetic (S-)duality, i.e. the strong and the weak couplings are exchanged, while the mass spectrum of the particles remains the same. Notice that there exists another invariance of , P#1 ,

(5.3.62)

since this means, from Eq. (3.1.3), the vacuum angle is shifted by 2
and hence the generating functional remains unchanged.

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The transformations (5.3.61) and (5.3.62) of  can be implemented on (a, a )2 by the matrices " 0 1 1 1 S" , ¹" . (5.3.63) !1 0 0 1









These matrices generate the group S¸(2, Z), under which  is transformed as




P

a b c

a#b " , a, b, c, d3Z; ad!bcO0 . c#d d

(5.3.64)

One immediately recognizes that the invariance of  is the same as the modular invariance of one-loop amplitudes of strings [91]. This fact gives a hint that  can be interpreted as the modular parameter of a certain torus. Seiberg and Witten proposed the following formula [1,90] "da /da . (5.3.65) " We will see that a and a can indeed be regarded as the two periods of a torus. " Eq. (5.3.40) implies that when a is large, the S;(2) theory is weakly coupled. Hence  can be determined perturbatively. Since the one-loop beta function coe$cient
"4 for N"1S;(2)  gauge theory with a triplet matter "eld, one can according to (5.3.40) write down the holomorphic relation between  and u as i u (u)" ln .



(5.3.66)

In the region of the moduli space corresponding to strong coupling, there is not only the one-loop perturbative contribution, but also the non-perturbative corrections from instantons. This makes the u-dependence of  quite complicated. Fortunately, as Seiberg and Witten did [1,2], one can use the geometric structure of the quantum moduli space to determine (u). At large u (u) is a multi-valued function, i u (u)" ln #i2k
, k"0,$1,$2,2 ,



(5.3.67)

i.e. for each point u in the moduli space, there exists an in"nite number of (u). From complex analysis we know that to make the correspondence between (u) and u one-to-one, we must cut the u-plane along a line connecting two branch points. For example, both u"0 and u"R are branch points of ln(u/
), we cut the u-plane along the positive real axis. The moduli space divides into branches and (u) will be a single-valued function on each branch. These branches can be glued together by identifying the lower lip of the previous branch with the upper lip of the next branch and the lower lip of the last branch with the upper lip of the "rst branch. The complex surface obtained in this way is called the Riemann surface of the moduli space, and (u) will be a single valued function on this Riemann surface. Seiberg and Witten found the exact solution of (u) by determining the singularity structure of the above Riemann surface, and we shall follow their reasoning. Later Bonelli et al. and Flume et al. [92] showed that their solution is unique assuming only that supersymmetry is unbroken and that

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the number of singularities in u is "nite. They started from the solution (5.3.66) for (u) at large u. Eqs. (5.3.65) and (5.3.66) imply that in the weak coupling limit, a 2i 2i a " a ln ! a . "




(5.3.68)

Since u"R is a singular point of (u), moving along a closed path around u"R shifts ln u by ln uPln u#2i


(5.3.69)

and hence ln aPln a#i


(5.3.70)

due to u"a. Such a transformation of a complex function around a singularity is called a monodromy. From (5.3.68) and (5.3.70), we have a P!a #2a, aP!a . " " Therefore, there exists a non-trivial monodromy at in"nity on the Riemann surface,




M " 



!1

2

0

!1

"P¹\ ,

(5.3.71)

(5.3.72)

where ¹ is the generator of the S¸(2, Z) group given in (5.3.63) and P is the element !1 of S¸(2, Z). This non-trivial monodromy at u"R implies that there must exist other monodromies on the u-plane. Equivalently speaking, (u) has a branch point at large u, requiring further branch point or points in the interior of the u-plane, corresponding to strong coupling of the theory, since now u is "nite and non-perturbative e!ects have already become noticeable. Now let us analyse these extra singularities. Due to the discrete symmetry uP!u, singularities of the moduli space must emerge in pairs, i.e. for each singularity at u"u , there must exist another  one at u"!u . If there are only two singularities, they must be u"R and u"0 since they are  the only "xed points of the Z symmetry. We have already seen that u"R is a singularity and  that it corresponds to the weak coupling limit of theory. Since the monodromy around 0 is the same as the monodromy around R, M "M , u"a is not a!ected by any monodromy and   hence u would be a global coordinate of the moduli space. Consequently, (u) would be an analytic function on the moduli space, and its imaginary part Im (u)&1/g would be a harmonic function  due to the Cauchy}Riemann equation (5.3.73) R R  Im (u)"0 . X X Since the Laplacian R R  is a positive-de"nite operator, (5.3.73) means that Im (u) cannot be X X positive de"nite everywhere on the moduli space. Therefore, there must exist regions in the moduli space where the low-energy e!ective gauge coupling g &1/(Im  becomes imaginary. To avoid   It is a large loop around u"0, but we consider the region outside the loop.

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this unphysical conclusion, there have to exist at least two singularities in the interior of the moduli space [93]. So we can consider three singularities, i.e. R, u and !u for some u O0. u"0 will    not be a singular point, although its existence respects the Z symmetry.  What is the physical interpretation of the singularities at u"$u ? Classically, u"0  is a singular point since classically this means a"0, or that the full gauge symmetry S;(2) is restored and no Higgs mechanism occurs. The massive gauge bosons and their superpartners become massless. However, there is no singularity at u"0 in the quantum moduli space. The singularities at u"$u do not imply that the gauge bosons become massless. This is because  the theory at u"$u is in the strong coupling region. The existence of massless gauge bosons  would imply an asymptotically conformal invariant theory in the infrared limit, but conformal invariance implies that the dimensional parameter u"Tr "0. This is obviously inconsistent. Therefore, the singularities at u"$u do not correspond to massless gauge bosons. Seiberg  and Witten found that the singularities at u"$u in fact correspond to massless magnetic  monopoles and dyons. According to the BPS mass formula (5.3.59) [1], the mass of the magnetic monopole is m"2a  . "

(5.3.74)

Thus a massless magnetic monopole will give a "0. If we choose a to vanish at u , i.e. " "  that the magnetic monopoles become massless there, we will see that at the singularity u"!u  the dyon becomes massless. From the Montonen}Olive duality conjecture, the magnetic monopole hypermultiplet couples to the dual "elds of the original N"2 photon supermultiplet, and the dynamics is exactly the same as that of the N"2 supersymmetric QED with massless electrons. The magnetic coupling is weak due to the electric}magnetic duality and hence the perturbative method can be adopted. From the one-loop beta function of the N"2 supersymmetric Abelian gauge theory with a massless hypermultiplet, the magnetic coupling should run according to g d  g " " . " 8
d

(5.3.75)

Since the scale  is proportional to a and 4
i/[g (a )] is  when  "0, then near u"u " " " " "  (or near a "0) we have " a

d i i  "! ;  "! ln a . " da " " "

"

(5.3.76)

Eqs. (5.3.61) and (5.3.65) lead to da  "! , " da "

(5.3.77)

 Note that supersymmetric QED, unlike supersymmetric QCD, does not allow a non-vanishing vacuum angle, except if it is embedded into a larger gauge group.

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which can be integrated to give a near u"u ,  i i a"a # a ln a ! a . (5.3.78) 
" "
" Since a (u ) vanishes, in the vicinity of u , a should be a good coordinate like u and depend "   " linearly on u, so we obtain a "c (u!u ); "   i i a"a # c (u!u ) ln (u!u )! c ln (u!u ) .    



(5.3.79)

From these expressions, we immediately read o! the monodromy matrix as u turns around u counterclockwise, u!u Pe 
(u!u ),   






 




a a a " P " "M  " , S a a a!2a "

1 0 "S¹S\ . M " S !2 1

(5.3.80)

The monodromy matrix at u"!u is easy to "nd. Since there are only three singularities, the  contour around u"R can be deformed into a contour encircling u and a contour encircling  !u , both being counterclockwise and having same base point. The factorization of the mono dromy matrices gives M "M  M  .  S \S

(5.3.81)

and hence M






!1 2

\S

"

!2 3

"(¹S)¹(¹S)\ .

(5.3.82)

The physical interpretation of the singularity at u"!u can be found from the way the BPS  mass formula (5.3.59) transforms under the action of the monodromy matrix. We write the central charge Z as




a Z"(n , n ) " .

 a

(5.3.83)

The monodromy transformation



 a a " PM " a a

(5.3.84)

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can be equivalently thought of as changing the magnetic and electric quantum numbers as (n , n )P(n , n )M .





(5.3.85)

The state with vanishing mass should be invariant under the monodromy and hence should be a left eigenvector of M with unit eigenvalue. For the singularity u , we have assumed that it  corresponds to massless monopoles, so the monopole (1,0) should be a left eigenvector of M  with S unit eigenvalue. It is easy to check that the left eigenvector of M  with unit eigenvalue is \S (n , n )"(1,!1). This is a dyon since both the electric and magnetic charges do not vanish. Thus

 the singularity at !u can be interpreted as being due to a (1,!1) dyon becoming massless. The  general monodromy matrix corresponding to a massless dyon (n , n ) is






M(n , n )"



1#2n n

 !2n



2n  . 1!2n n



(5.3.86)

One special point should be emphasized. Since M  M  OM  M  the monodromy relation \S S S \S (5.3.81) seems not to be invariant under u P!u . This does not contradict the Z symmetry,    since we have not indicated the base point in de"ning the composition of two monodromies. Assume the composition M  M  happens in the base point u"P, then another choice of base S \S point u"!P will lead to M "M  M  .  \S S

(5.3.87)

Then from (5.3.72), (5.3.80) and (5.3.87) we get the monodromy M






3 2 " , \S !2 !1

(5.3.88)

one of whose left eigenvectors with unit eigenvalue is the dyon (1, 1). The Z symmetry on the  quantum moduli space not only exchanges the singularity, but also exchanges the base point P, hence M  M  M  M  . S \S \S S

(5.3.89)

At the same time the (1,!1) dyon is exchanged with the (1, 1) dyon. Using the monodromy corresponding to the fundamental monopole or dyon, one can construct the monodromy corresponding to the composite dyon [1,2]. For example, turning "rst n times  around R, then around !u , and then n times around R in the opposite direction, we obtain   the monodromy M\L M  ML "¹\L (¹S)¹(¹S)\¹L  \S 




"

!1!4n  !2



2#8n #8n   "M(1,!1!2n ) ,  3#4n 

(5.3.90)

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which corresponds to a massless dyon with n "1 and any n 3Z. Similarly, we have

 M\L M  ML "¹\L S¹S\¹L  S 




"

1!4n  !2



8n  "M(1,!1!2n ) .  1#4n 

(5.3.91)

We know that a (u) and a(u) are multiple-valued functions on the moduli space, and they are " single-valued function on each branch with the branch points being at R, u and !u , or   equivalently, single valued on the Riemann surface of the moduli space. Since the value of u depends on the choice of renormalization scheme, one can always choose u "1. There are two   approaches to determining a(u) and a (u) and hence  exactly by (5.3.65). The "rst one is to look for " a di!erential equation satis"ed by a(u) and a (u), since meromorphic functions with such singular" ities should satisfy the characteristic equations of the hypergeometric function, which is called the Picard}Fuchs equations [94]. This method is straightforward and one can easily write the explicit forms of a(u) and a (u) in terms of the integral representation of the hypergeometric function [1], "



(2 S (x!u dx ; a (u)" "
 (x!1



(x!u (2  . dx a(u)"
(x!1 \

(5.3.92)

Consequently, the low energy e!ective coupling is given by da (u)/du (u)" " . da(u)/du

(5.3.93)

However, Seiberg and Witten gave an indirect but theoretically more beautiful expression for above solutions } the algebraic curve of the Riemann surface of the moduli space [1,2]. They made use of the geometric structure of the Riemann surface of the moduli space in constructing the solutions. Here we shall only use the explicit expression (5.3.92) for the solution to understand the Seiberg}Witten construction. From (5.3.92), we see that the integrand has square-root branch cuts with branch points at #1, !1, u and R. The two branch cuts can be chosen from !1 to #1 and from u to R. The Riemann surface of the integrand is composed of two sheets glued along the cuts. If one adds the point at in"nity to each of the two sheets, the topology of the Riemann surface is that of two spheres connected by two tubes, this is just a torus. How can we describe a torus, or more generally, an arbitrary Riemann surface quantitatively? We know that the torus is formed by the compacti"cation of a complex plane, identifying zPz# , 

zPz# , 

(5.3.94)

with  / " and Im '0.  and  are called the periods of the torus.  is the modular     parameter of the torus, which determines the geometric structure of the torus. Two tori with

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modular parameters  and  related by a S¸(2, Z) transformation a#b , ad!bc"1, a, b, c, d3Z " c#d

(5.3.95)

have the same geometric shape (i.e. are conformally equivalent) [94]. Eq. (5.3.94) shows that the torus is in fact the coset space C/G of the complex plane C, with the group G consisting of the action on C zPz#n#m, 3C, Im '0 .

(5.3.96)

A natural choice to describe a Riemann surface is to use the meromorphic functions de"ned on it. For a torus, a meromorphic function should be elliptic (doubly periodic) functions with periods 1 and  due to (5.3.96). A typical example is the Weierstrass elliptic function with periods 1 and  [95,96]:





1 1 1 ! , n, m3Z, n, mO0 . "(z)" #  z (z!n!m) (n#m) LK

(5.3.97)

The function " satis"es the following di!erential equation: "(z)"4("!e )("!e )("!e ) ,   

(5.3.98)

with




1 , e ""  2




 e "" ,  2




e "" 

1# . 2

(5.3.99)

"(z) is again an elliptic function and hence is another meromorphic function de"ned on the torus. If we de"ne "(z),y, x,"(z), the di!erential equation (5.3.98) can be written as y"4(x!e )(x!e )(x!e ) .   

(5.3.100)

This is a plane cubic curve with the plane coordinates (x, y) being meromorphic functions [96]. In fact, every Riemann surface can be equivalently represented by such an algebraic curve. The domain of de"nition of an algebraic curve is a Riemann surface. Conversely, given an algebraic curve, one can immediately know what the Riemann surface is. The zero points of y are just the singular points of the Riemann surface. Therefore, a Riemann surface and its algebraic curve are in one-to-one correspondence. For example, the plane cubic curve (5.3.100) shows that y has three zero points x"e , i"1, 2, 3, in the x-space. From (5.3.100) G y"$(4(x!e )(x!e )(x!e ) .   

(5.3.101)

This shows that y is a double-valued function on x-space. The branch points are obviously x"e ,R. We choose one branch cut from e to e and another one from e to R. y is then G    a single-valued function on the Riemann surface de"ned by joining the two sheets of the x-plane

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along the cuts. It is now easy to see that the Riemann surface is a torus. A torus has two independent non-trivial closed paths called cycles. The loop on the two-sheeted covering of the x-plane that goes around one of the two cuts corresponds to one of the cycles and the loop which intersects both cuts, i.e. the loop goes around e and e pairing into the second sheet for half of the   way, corresponds to the other cycle. For the theory we are discussing, the singularities are at 1, !1 and R in the u-plane, and the corresponding Riemann surface is a torus, which, according to (5.3.100), should be parametrized by a family of curves with the parameter u, y"(x!1)(x#1)(x!u) .

(5.3.102)

The periods a (u) and a(u) of the torus represented by this algebraic curve and hence (u), as argued " by Seiberg and Witten, can be found as follows [1,90]. First, a torus is represented by two cycles, which we denote as  and  . These cycles form a local canonical basis for the "rst homology   group H (¹, C) of the torus, or equivalently, the "rst homology group of the curve, ¹ denoting  the torus or the algebraic curve family. According to Stokes' theorem, one can construct a homotopy invariant by pairing an element in the "rst homology group with an element in its dual, the "rst cohomology group. Since a (u) and a(u) should be homotopic invariant, we de"ne "



a " "

A

,



a"

A

,

(5.3.103)

where  is an element of the "rst cohomology group H(¹, C). As H(¹, C) is two dimensional, its basis must be provided by any two linearly independent elements. One typical choice is dx  " ,  y

x dx  " .  y

(5.3.104)

In general,  should be a linear combination of the above basis elements with u-dependent coe$cients, "a (u) #a (u) .    

(5.3.105)

In addition,  must be a form with vanishing residue so that, on encircling a singularity, a and " a transform in the way that  and  transform under a subgroup of S¸(2, Z). Especially, the   physical requirement Im '0 must be satis"ed. Eqs. (5.3.93) and (5.3.103) give da /du , " " da/du



da "" du

d , du A



da " du

d . du A

(5.3.106)

On the other hand, on a torus de"ned by the above curve there exists a natural de"nition for the periods,



b" G

A

G

 , i"1, 2 , 

(5.3.107)

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and the modulus parameter b  "  S b  should possess the fundamental property Im  '0. Therefore, one can choose S d dx "f (u) "f (u)  du y

(5.3.108)

(5.3.109)

with f (u) a function of u only. Then we get Z dx/y da /du b "  " " A , (5.3.110) " " S Z dx/y b da/du  A which naturally satis"es Im '0. The reverse is also true, i.e. if Im '0 everywhere, then " S and  satisfy (5.3.109) and (5.3.110). The function f (u) can be determined by demanding the asymptotic behaviours (5.3.68), (5.3.79) of a and a near u"1, !1, R, " f (u)"!(2/4
.

(5.3.111)

Therefore, with (5.3.102) and (5.3.111), one can integrate (5.3.109) over u and get (2 y (2 (x!u dx" dx . " 2
(x!1 2
x!1

(5.3.112)

We "nally have from (5.3.106)



a " "

A

(2 (x!u dx, 2
(x!1 



a"

(2 (x!u dx . 2
(x!1 A

(5.3.113)

Deforming the cycles  and  continuously into the branch cuts, we immediately get (5.3.92), the   same result as obtained from the di!erential equation approach. Since the Riemann surface and the algebraic curve are exactly in one-to-one correspondence, and given a Riemann surface, its modulus  is "xed up to an S¸(2, Z) transformation, one can conveniently use the algebraic curve to represent the solution of the low-energy e!ective theory in the Coulomb phase. The Seiberg}Witten method can be naturally generalized to the case with matter "elds, i.e. N"2 supersymmetric QCD [2]. The matter "elds belong to N"2 hypermultiplets, which are pairs of N"1 chiral supermultiplets (Q , QI ) in the fundamental representation of the gauge group and its G G conjugate (i"1,2, N is the #avour index) [27,97]. In N"1 language, the classical Lagrangian D consists of the standard coupling of the N"2 gauge supermultiplet to Q , QI , plus the following G G superpotential

 



,D ,D =" d (2  QI G?¹?Q #  m QI GQ , G G G G G

(5.3.114)

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239

 is the N"1 chiral supermultiplet part of the N"2 gauge supermultiplet, since an N"2 gauge supermultiplets is composed of an N"1 gauge supermultiplet (gauge "elds and gaugino) and N"1 chiral supermultiplet in the adjoint representation. The classical moduli space of the vacua will be determined by both D-#atness and F-#atness conditions. There are two types of possible solutions to these conditions [2]. The "rst type leads to a Coulomb branch of the moduli space with the expectation values of the scalar components of , Q and QI satisfying O0,

Q "QI R"0 . G G

(5.3.115)

The second one gives a Higgs branch in the moduli space in the massless case (m "0) with G "0,

Q "QI RO0, i"1,2, k4N , N 52 . G G D D

(5.3.116)

We are only interested in the Coulomb branch. To ensure that (n , n ) are both integers even in the

 presence of matter "elds, one should rescale the electric charge n by a factor 2 and simultaneously  divide a by 2 so that the BPS mass spectrum m"2a n #an  remains the same. This rescaling "  can be thought of as a transformation










a a 1 0 " P " " 0 1/2 a a/2

a a " ,S " . a a

(5.3.117)

Consequently, the monodromy matrix will transform as



 m n

m n

PS

p

q

p

q




S\"

1

0




m n

1 0

0 1/2

p

q

0 2



m

2n

p/2

q

"

.

(5.3.118)

Therefore, in the new convention the monodromy matrices (5.3.72), (5.3.80) and (5.3.82) with N "0 are D











1 0 !1 4 ; M " ; M " \S S !1 1 !1 3




M " 



!1

4

0

!1

.

(5.3.119)

With these new monodromy matrices the algebraic curve corresponding to (5.3.102) becomes [2,13] y"x!ux#
x , 

(5.3.120)

where u "
 was chosen. The curve (5.3.120) describes the same physics as the curve (5.3.102).  However this curve is more general since it can be applied to the case of matter "elds. Thus, one usually adopts the curve solution of the form (5.3.120) even in the case N "0. The branch points of D  For N "0, 1, the moduli space has no Higgs branch. When N "2, there are two Higgs branches in the moduli D D space, which coincide with the Coulomb branch at the origin of the moduli space. These two branches are exchanged by a parity-like symmetry. For N 53, there is only one Higgs branch, which meets the Coulomb branch at the origin of the D moduli space [2].

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the moduli space are the zeros of the plane cubic curve (5.3.120) x"0, x"x "(u!(u!
) , ! 

(5.3.121)

together with the point at in"nity. The natural choice for the two branch cuts is (x , x ) and (0,R), \ > and the Riemann surface of course remains a torus. When two branch points coincide, one cycle of the torus will vanish and hence the torus will become singular. This can be related to the singular points in the u-plane, where there will exist massless particles. Two solutions of a cubic equation can coincide only when the discriminant  vanishes. For a general cubic equation x#Bx#Cx#D"0 ,

(5.3.122)

the discriminant is 1 (BC!4C!4BD#18BCD!27D) . "! 108

(5.3.123)

The discriminant for the algebraic curve (5.3.120) is hence 1 (u)" (u!
)
 . 16

(5.3.124)

Therefore, the torus will become singular at u"$
, the zeros of (u). Massless monopoles and dyons will exist in these two singular points. In the Coulomb phase Seiberg and Witten worked out the explicit algebraic curve solutions for N "1, 2, 3 using a similar method as in the N "0 case [2]. D D In summary, in the Coulomb phase, we can get insight into the dynamics by determining the e!ective gauge coupling (u) with u"Tr  being the coordinate of the moduli space, while (u) is indirectly given by a family of elliptic curves parametrized by u. From the algebraic curve, one can determine the Riemann surface of the moduli space and its periods a (u) and a(u), and hence the " (u) through (5.3.93). In particular, the singular points in the moduli space, which will lead to a singular Riemann surface, correspond to the zeros of the discriminant of the vanishing elliptic curve. There exist massless particles in these singular points of the moduli space. Now we go back to the theory at hand. In the Coulomb phase the e!ective gauge coupling at large ;K is given by the curve (5.3.120), y"x!x;I #
x ,  B

(5.3.125)

in which ;I is the light "eld ;I "Tr "2;K /M , 

(5.3.126)

where we have used (5.3.50), and the factor 2 comes from taking the trace. Considering the relation (5.3.126) and rescaling yP(M /2)y, xP(M /2)x, we write the curve at large ; as   y"x!x;K #x

 . 6 7

(5.3.127)

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241

Using this asymptotic solution, we can "nd the exact curve solution. We "rst assume that the exact solution has the general form of a plane cubic curve, y"x#x#
x# .

(5.3.128)

The coe$cients ,
and  are functions of ;, the gauge coupling and the scales
,
. There exist 6 7 some important constraints on them [2,13]: 1. In the weak coupling limit
"0, the curve should give a singular Riemann surface for every ;, i.e. the curve should vanish. So generally one can assume that the curve should be of the form y "x(x!;) when
"0 . 

(5.3.129)

2. ,
and  should be holomorphic functions in ; and the various coupling constants, since this guarantees that  is also holomorphic in them. 3. The solution expressed by the curve (5.3.127) should be compatible with all the global symmetries of the theory including the discrete symmetry and those explicitly broken by the anomaly. 4. In various limits we should get the curves of the models obtained in the corresponding limit. 5. The curve should have the correct monodromies around the singular points. We use these constraints to determine the coe$cients. The global #avour symmetry S;(2) D and the discrete symmetry Z as well as the large ; limit (5.3.127) show that the coe$cient   must be [13] "!;K #(
 #
 ) 6 7

(5.3.130)

with  being some constant. In addition, the asymptotic form at large ; gives
"

 , 6 7

"0 .

(5.3.131)

This choice also ensures that the curve is singular when either
or
vanishes. So the exact 6 7 curve solution should be y"x#[!;#(
 #
 )]x#

 x . 6 7 6 7

(5.3.132)

To determine the parameter , we consider the limit
<
. In this limit the theory is 7 6 approximately an S;(2) gauge theory with three singlets M , f, g"1, 2, and a triplet "eld I . 6 DE This model is just the N "N "2 case of the low energy supersymmetric S;(N ) QCD with D A A N #avours. From the discussion in Section 3.4.2, the quantum moduli space is parametrized by D < with the constraint (3.4.52). So here we have det M # Tr (I )";K #;I "
 , DE  7

(5.3.133)

where  is a dimensional normalization necessary for making the dimensions right. The low-energy Coulomb phase of S;(2) with a triplet I is just the Seiberg}Witten model, whose exact solution is 6 given by the curve (5.3.125). Its branch points (5.3.121) and the discriminant (5.3.124) show that the solution is singular at ;I "$
 since two branch points coincide. Therefore, for S;(2) with the 6 6

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matter "elds containing not only I , but also M , and considering (5.3.125), the  of this S;(2) DE 6 theory should be singular at ;K +
 $
 7 6

(5.3.134)

in the
<
limit. On the other hand, the discriminant of the curve (5.3.132) is 7 6 1 (

) [!;K #(
 #
 )]!4

  , "! 6 7 6 7 108 6 7

(5.3.135)

and hence the curve is singular at ;K "(
 #
 )$2

 . 6 7 6 7

(5.3.136)

Comparing (5.3.134) with (5.3.135), we "nd that "1. So "nally we obtain the solution in the large ; limit, y"x#(!;K #
 #
 )x#

 x . 6 7 6 7

(5.3.137)

Usually, for convenience of discussion, we rescale
P(2
, s"X, >. Then the curve (5.3.137) is Q Q rewritten as y"x#(!;K #4
 #4
 )x#16

 x . 6 7 6 7

(5.3.138)

Expressing (5.3.138) in terms of the original high-energy scale (5.3.46), using (5.3.47) and rescaling yP;y, &

xP; x , &

(5.3.139)

we have y"x#(!;#8
,A A \ )x#16
,A A \ x, , ,A \ , ,A \

(5.3.140)

where relation (5.3.46) was used. It should be stressed that (5.3.140) is the curve where ; is large enough to compare with the scale
,A A \ . The exact curve solution should reproduce it in the , ,A \ large ; limit. As Seiberg and Witten did [1,2], assuming that the quantum corrections to (5.3.140) , as implied by (5.3.44), the holomorphy and the are polynomials in the instanton factor
,A A \ , ,A \ various global symmetries prohibit any corrections to (5.3.140). Therefore, the curve (5.3.140) is an exact solution [15]. We can now discuss the physical consequences. First the branch points given by the curve are x"0, R and $(;(;!16
,A A \ )] . x "[;!8
,A A \ !  , ,A \ , ,A \

(5.3.141)

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243

According to (5.3.109) and (5.3.110), the periods of the torus are

 

a (;)& "

V>

V\

"

V>

V\



a(;)&

V\



)#16
,A A \ x] dx[x#x(!;#8
,A A \ , ,A \ , ,A \ dx[x(x!x )(x!x )] , > \

dx[x(x!x )(x!x )] > \

(5.3.142)

and the e!ective coupling is given by the ratio V> dx/y(x) . (;)" V\\ V dx/y(x) 

(5.3.143)

The singularities of the e!ective gauge coupling (;) can be inferred from the vanishing of the discriminant 1 )!64
,A A \ ]"0 , (
,A \ )[(!;#8
,A A \ "! , ,A \ , ,A \ 108 ,A ,A \

(5.3.144)

which gives singularities at ;"0 and ;"16
,A A \ ,; . The monodromy matrices around , ,A \  ;"0 and ;"; can be immediately obtained by observing the change in the asymptotic  expansion form of a (;) and a(;) near ;"0 and ;"; when taking ;Pe
;. They are, "  respectively, [13]






1 0 , M "S\¹S"  !1 1






!1 4 M "(S¹\)\¹(S¹\)" ,  !1 3

(5.3.145)

up to an overall conjugation by ¹. According to (5.3.86) and (5.3.145) we can see that (1, 0) is the left eigenvector of M and (1,!1) the left eigenvector of M . This reveals that there must exist   massless monopoles or dyons in two subspaces MGH"MH of the moduli space of vacua determined by det MH"0 or det MH"; . Note that the spaces of these singular vacua MH are  non-compact. Ignoring the overall ¹ conjugation, which from (5.3.91) only shifts the electric charges of the monopole or dyon, we can consider that magnetic monopoles exist in the singular vacuum MH with det MH"0 and dyons in the singular vacuum MH with det MH"; .  The number of massless monopoles or dyons existing in the singular vacua MH with det MH"0 or det MH"; can be detected from the monodromy of  upon taking M around MH. We will see  that these two cases are di!erent. Let us "rst consider the vacua with ;"; . Since in this case  ;"f (M)"det M is a single-valued function in the M complex plane, moving M around MH, (M!MH)Pe
(M!MH) leads to (;!; )Pe
(;!; ) and gives the monodromy M .    From the above discussion, we know that this monodromy is associated with a single pair of dyons E! with magnetic charge $1. At ;"; the dyons are massless and away from ;"; the  

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dyons become massive. Thus the global symmetries, the holomorphy and mass dimension determine that the superpotential near ; should be 




;!;  ="(;!; ) 1#O  A \
, ,A ,A \



E>E\ .

(5.3.146)

The situation for the singular vacuum MH with det MH"0 is more interesting. det MH"0 means that r(N with r being the rank of MH, so MH has N !r zero eigenvalues. Thus when D D taking M around a vacuum MH, (M!MH)Pe
(M!MH), the complex function ;"det M will behave as ;Pe
,D \P;. Since the transformation ;Pe
; leads to the monodromy M , the  above transformation should yield the monodromy M,D \P. Therefore, there must exist N !r pairs  D of massless monopoles in the vacuum parameterized by M"MH with MH having rank r. This requires the superpotential for N pairs of monopoles q> and q\ with magnetic charge $1 to be of D G G the form






det M 1 MGHq>q\ , =" f t" G H A \
, 2 A A , , \

(5.3.147)

where f (t) should be holomorphic around t"0 and normalized as f (0)"1. The scale  is introduced to ensure the correct dimension 3 of the superpotential because M has dimension 2 and q! has dimension 1. The superpotential automatically makes N !r monopoles massless at G D MH since it has rank r. In addition, to make the above superpotential respect the global #avour symmetry S;(N ) and R-symmetry, the monopole q! must belong to the conjugate fundamental D G representation NM of S;(N ) and have R-charge 1. D D The magnetic monopole and dyon are the left eigenvectors of the monodromy matrices at the singularities obtained from the curve solution (5.3.140). Considering the various representation quantum numbers of the electric quark Q and the magnetic quark q under the global symmetry S;(N );; (1) and their electric and magnetic charges, one can regard the dyon E as the bound D 0 state of the electric quark Q and magnetic quark q [15], E!&q!QG . G

(5.3.148)

This construction can be checked from the left eigenvectors of M and M .   In addition, there exists a massless supermultiplet left by the breaking SO(N )PSO(2) ;(1) in A the Coulomb phase, i.e. an N"1 low-energy (e!ective) photon "eld, whose "eld strength is a chiral super"eld W and which can be given a gauge invariant construction on the moduli space of vacua ? in terms of the fundamental "elds, ,= (Q),A \ . W &   2 ,A\ PQP P 2P,A\ (= ) QG QG 2QG,,AA\ P \ ? PQ P P ? ? GG G

(5.3.149)

Later we will see that a similar relation also exists in the non-Abelian Coulomb phase. The reasonableness of the above massless particle spectrum is supported by two non-trivial consistency checks. The "rst one is still 't Hooft anomaly matching. At the origin M"0 of the moduli space of vacua, the global symmetry S;(N );; (1) is unbroken like in the SO(N ) theory. D 0 A The massless particle spectrum consists of the meson "eld MGH, the photon supermultiplet with the

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245

Table 5.3.7 Currents composed of the fermionic components of monopole and photino corresponding to the global symmetry S;(N );; (1) D 0

! OG W

S;(N ) D

; (1) 0

j"M >tM  \#M \tM  > I OG GH I OH OG GH I OH 1

0 j ()"M ?W ?W I I

Table 5.3.8 Energy-momentum tensor contributed by the fermionic components of q! and the photino; L[]" G i/2(M I! !! M I), ! "R ! )*/2, )*"i/4[ ), *] and )"e) I I I I I )*I I ¹ IJ ! OG W

i/4[(M > ! \!! M > \)#(  )]!g L[ ] OG I J OG J OG I OG IJ O i/4[(M W ! W !! M W W )#(  )]!g L[ ] I J J I I IJ 5

Table 5.3.9 't Hooft anomaly coe$cients Triangle diagrams and gravitational anomaly

't Hooft anomaly coe$cients

; (1) 0 S;(N ) D S;(N ); (1) D 0 ; (1) 0

1 !2 Tr(t t , t!) 0 1

"eld strength (5.3.149) and the monopole pair q! associated with the singularity ;"det M"0. G Let us check whether the 't Hooft anomalies contributed from this low-energy massless particle spectrum match those contributed by the massless fundamental quarks as listed in Table 5.3.4. The contributions from massless MGH have already been collected in Table 5.3.5, so we need to only consider the contributions from the fermionic components of the q! and the photino. Eq. (5.3.149) G shows that the R-charge of the photino W is 1#(N #2!N )(N !2)/N "1 for N "N !2. D A A D D A The relevant currents and energy-momentum tensors are listed in Tables 5.3.7, 5.3.8 and the corresponding anomalies in Table 5.3.9. Adding the contributions from the low-energy particles to the contributions from the "eld M given in Table 5.3.5, one can see the anomalies indeed match the high-energy anomalies given in Table 5.3.4 for N "N !2. D A Another check is to verify that the decoupling of a heavy #avour will yield the description of the N "N !3 case discussed in Section 5.3.3. Without losing generality, we choose the N th #avour D A D to be heavy by adding a large mass term = "mM,DD /2. There are two branches in the moduli , 

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space, a branch with det MH"; and a branch with det MH"0. We "rst discuss the branch with  det MH"; . According to (5.3.146), the full superpotential near ;"; with the above large   mass term is ="(;!; )E>E\#mM,DD . ,  

(5.3.150)

Integrating out M,DD by its equation of motion , R= det M 1 " E>E\# m"0 D D RM,D M,D 2 , ,

(5.3.151)

m , E>E\"! 2 det M K

(5.3.152)

gives

where M K denotes the mesons for the remaining N !1"N !3 #avours. Obviously, the nonD A vanishing expectation value (5.3.152) of E>E\ has made the electric charge con"ned. Since E! are a pair of dyons, according to the discussions in Section 2.4, this phenomenon is just the oblique con"nement proposed by 't Hooft. The low energy superpotential at ;"; is  m det MH m
,A A \ 1 , ,A \ . "8 =" mM,DD " , det M K 2 det MK 2

(5.3.153)

and the low-energy scale
A A , Using relation (5.2.10) between the high-energy scale
A A , , \ , , \ (5.3.153) is just the superpotential of the "1 branch of (5.3.26). As for the branch with det MH"0, we add a large mass term for the N th #avour to (5.3.147). D The classical equation of motion of M,DD , , R= 1 det M m 1 1 df " f (t)q>D q D # # "0 , , , RM,DD M,DD 2 2 dt
,A A \ 2 , ,A \ , ,

(5.3.154)

shows that q!D O0 and hence that the magnetic ;(1) group is Higgsed. From the discussion in , Section 2.4, the dual Meissner e!ects occurs and the original electric variables are con"ned. Due to the non-trivial function f (t) in (5.3.147) and the constraint from the magnetic ;(1) D-term, q>eE4" q\  M  , there is a di$culty in explicitly integrating out the massive modes. However, from the G G FF classical equations for M,DD , q>D and q\D , one can see that the low-energy massless modes are only , , , and q\ , iK , jK "1,2, N !1. Usually, for convenience of discussion, one de"nes the gauge M K GKK , q> H GK GK D invariant interpolating "elds 1 qK " (q>q\ !q\ q> ) G 2(m GK ,D GK ,D

(5.3.155)

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247

of q! to replace q! as massless modes. Considering the contribution from the magnetic ;(1) GK GK D-term, one can get the low energy e!ective superpotential





 

1 (det M K )(M K GHq K q K ) G H M =" fK tK " K GHq K q K G H A \ m
, 2 ,A ,A \ 1 (det M K )(M K GHq K q K ) G H M " fK tK " K GHq K q K . G H A \ 2
, ,A ,A \

(5.3.156)

This is just the superpotential of the "!1 branch of the low energy N "N !3 theory given D A by (5.3.36). Consequently, according to (5.3.35) we have A A q K "
\, b "
\, (Q),A \(=?= ) , G ,A ,A \ GK ,A ,A \ ?

(5.3.157)

i.e. the remaining massless monopoles can be identi"ed as massless exotics (glueballs). Note that in (5.3.156) fK (tK ) depends on f (t) in (5.3.147), and the condition from the ;(1) D-term is important in showing that a non-trivial f (t) leads to fK (tK ). In summary, the physical phenomena in the branch with det MH"0 are very interesting. Upon giving a heavy mass to Q,D , some of the massless magnetic monopoles q! condense and this G condensation leads to the con"nement of Q . Especially, the remaining massless magnetic monoG poles can be identi"ed as massless exotics (glueballs). According to the discussion in Section 4.2, this phenomenon was also observed in the S;(N ) theories where massless magnetic quarks A become massless baryons. So one can conclude that the following non-perturbative physical phenomenon is generic: some of gauge invariant composite operators such as baryons, glueballs and other exotics can be thought of as (Abelian or non-Abelian) magnetic objects. 5.4. N 54, N 5N !1: dual magnetic SO(N !N #4) description A D A D A 5.4.1. General introduction to magnetic description In this range, the infrared behaviour of these theories can be equivalently described by a dual magnetic theory. The reason we resort to a magnetic description is the same as in the S;(N ) case: A the `electrica mesons (5.1.14) and baryons (5.1.17) are not the appropriate variables to describe the moduli space of vacua and we cannot use them to construct a consistent dynamical superpotential. One can easily check that the 't Hooft S;(N );; (1) anomalies cannot match for the fermionic D 0 components of (Q, ) in the microscopic theory with those of (M, B). The matter "eld variables in the magnetic theory are the original `electrica variables MGH and magnetic quarks qP with the G superpotential 1 1 =" MGHq ) q " MGHq  qP . G H GP H 2 2

(5.4.1)

To ensure that = is S;(N );; (1) invariant, q must be in the conjugate fundamental representaD 0 G tion NM of S;(N ) and have R-charge (N !2)/N since the R-charge of M is 2(N !N #2)/N . D D A D D A D The subscript r is the magnetic colour index. Note that the magnetic quarks cannot be introduced

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from the baryons (5.1.17) as in the S;(N ) case. Before we explain the role played by the scale , we A "rst explain what the dual gauge group SO(NI ) is, which is not as easily found as in the S;(N ) case. A A This gauge group is restricted by the requirement that is ; (1) anomaly free in the magnetic theory. 0 Since the ; (1) charge of the magnetic gaugino I should be 1, so from the above assignments of the 0 ; (1) charge to M and the magnetic quarks q  , the anomaly-free ; (1) current in terms of four 0 0 GP component is






N !2 2(N !N #1) A A !1 M G      #IM ?   I ? # D M    . (5.4.2) OP I  OGP I  + I  + N N D D The dynamical vector current for this magnetic SO(NI ) gauge theory, i.e. the Noether current A corresponding to global SO(NI ) gauge transformations, is A    I  M I (5.4.3) J? "M G   ¹?    #f ?@AI @ I A . I OP I PQ OGQ I The triangle diagram j0J? J@I  gives the ; (1) anomaly I I I 0 N !2 1 A !1 #2f ? @I A f ? @I A IJHMF? F? RIj0" 2N IJ HM D I N 32
 D 1 IJHMF? F? . "[2(N !2!N )#2(NI !2)] (5.4.4) A D A IJ HM 32
 j0" I








That ; (1) is anomaly free means that the above anomaly coe$cient should vanish and hence we 0 have NI "N !N #4. Thus the magnetic gauge group should be SO(N !N #4) [15]. This A D A D A theory is asymptotically free since the one-loop
function coe$cient is
I "3(NI !2)!N "3(N !N #4!2)!N "2N !3N #6'0 (5.4.5)  A D D A D D A because N 54 and N 5N !1. The scale  is introduced for the following reason. In the electric A D A description the meson "eld MGH"QG ) QH has dimension 2 at the UV "xed point g"0 and generally acquires some anomalous dimension at the IR "xed point. In the magnetic description, since now M is thought of as an elementary matter "eld, its dimension should be its canonical dimension 1 at the UV "xed point g "0. We denote M in the magnetic description by M . In order to relate K M to M of the electric description, a scale  must be introduced with the relation M"M . Later K K all the expressions will be written in terms of M and  rather than in terms of M . K Dimensional considerations show that for generic N and N the scale
I of the magnetic theory A D should be related to
, the scale of the electric theory by
@
I @I  "
,A \\,D
I ,D \,A >\,D "C(!1),D \,A ,D ,

(5.4.6)

where C is a dimensionless constant which will be determined below. Like in the S;(N ) gauge A theory, scale relation (5.4.6) has several consequences: 1. It is preserved under mass deformation and along the #at directions, in which spontaneous symmetry breaking occurs. The phase (!1),D \,A is necessary to ensure that this relation is preserved.

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249

2. It reveals that as the electric theory becomes stronger the magnetic theory becomes weaker and vice versa, q,D e\
 EO e\
 E O J(!1),D \,A .

(5.4.7)

3. It implies that the dual theory of the magnetic description is identical to the original electric theory; 4. Combined with the quantum e!ective action, it yields a relation between the gaugino condensations in both the electric and magnetic theories, "!I I . Now let us look at how the discrete symmetry behaves in the dual magnetic theory. The discussion in Section 5.1.1 shows that there exist discrete symmetries Z D and Z generated by the  , transformation QPe 
,D Q and the charge conjugation C, respectively. In the magnetic theory, due to the kinetic term of the gauge singlet M/ and the superpotential (5.4.1) in the classical action, the corresponding discrete symmetries is generated by qPe\ 
,D Cq and C, respectively. Note that the Z D symmetry commutes with the electric gauge group SO(N ) but does not commute , A with the magnetic gauge group SO(N !N #4) [15]. D A Finally, it is worth seeing what the correspondences are in the magnetic theory for the gauge invariant chiral operators such as the meson, baryon and exotics used in the electric theory. The gauge invariant chiral operators appearing in the previous subsections, which can also be formally de"ned for N 5N !1 and N 54, are collected in the following: D A A MGH"QG ) QH ,  1 P 2P,A QG 2QG,,AA ,B G 2G,A , BG 2G,A " P P N! A 1 (=?= )P 2P,A\ QG 2QG,,AA\ ,b G 2G,A\ , bG 2G,A\ " P P \ ? (N !4)! A 1 = P 2P,A\ QG 2QG,,AA\ \,W G 2G,A\ . WG 2G,A\ " P ? ? P (N !2)! ? A

(5.4.8)

Based on S;(N );; (1) symmetry, these operators should be mapped to the following gauge D 0 invariant operators of the magnetic theory MGHPMGH , B G 2G,A PG 2G,D bI ,A> 2 ,D , G G

b G 2G,A\ PG 2G,D BI ,A> 2 ,D , G

G W G 2G,A\ PG 2G,D (= I ) ,A\ 2 ,D , G

? ? G

(5.4.9)

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where bI

G,A> 2G,D

1 ,  2 qP  2qP ,,DD\,A G (N !N )! P P,D\,A G,A> D A

for N 'N , D A

1 BI ,A\ 2 ,D ,  qP  2qP ,,DD\,A> , G G (N !N #4)! P  2P ,D\,A> G,A\ G D A 1 = I  qP  2qP ,,DD\,A> . (= I ) ,A\ 2 ,D , G (N !N #2)! ? P  2P ,D\,A> G,A\ G ? G D A

(5.4.10)

The R-charge of each `electrica operator in (5.4.9) is equal to that of its `magnetica image, for example, the R-charge of the baryon operator B G 2G,A of the electric theory is N (N !N #2)/N , A D A D while the R-charge of G 2G,D bI ,A> 2 ,D is 2#(N !N )(N !2)/N "N (N !N #2)/N . G G

D A A D A D A D Mappings (5.4.9) show that the baryons of the electric theory are dual to the exotics of the magnetic description, and the electric photon supermultiplet is dual to the magnetic photon supermultiplet. The above is a general introduction to the dual magnetic description of supersymmetric SO(N ) A gauge theory in the range of colours and #avours, N 54 and N 5N !1. It is shown that the A D A magnetic theory is a supersymmetric SO(N !N #4) gauge theory with colour singlets MGH and D A N magnetic quarks in the conjugate fundamental representation of S;(N ). Since the gauge D D groups SO(3) and SO(4) are special, we shall in the following two sections give a special discussion of the magnetic SO(3) and SO(4) gauge theories, and then return to the general case. 5.4.2. N "N !1: Dual magnetic SO(3) gauge theory D A N "N !1 means that the dual magnetic description is an SO(3) gauge theory. The matter D A "elds, as discussed in the general introduction, are the SO(3) singlets MGH and the magnetic quarks  qP. Note that now the magnetic quarks belong to the adjoint representation of SO(3) due to the G coincidence of the adjoint representation and vector representation of SO(3). We "rst introduce the dynamics of this case. Since from Table 5.1.1 the R-charge of MGH is 2/N for N "N !1, the D D A gauge invariant and S;(N );; (1) invariant superpotential consists of not only (5.4.1), but also of D 0 a term proportional to det M. Taking into account the mass dimension, the full superpotential should be of the form 1 1 det M , =" MGHq ) q ! G H 2
,A \ 2 ,A ,A \

(5.4.11)

where
,A ,D ,A \ is the dynamically generated scale and the normalization factor 2 is chosen to make various deformations of theory consistent, as shown in detail later. The scale relation here is also di!erent from the general case, since the one-loop
function coe$cient of the magnetic SO(3) theory is
I "6!2N , while for N "N !1 the factor
I ,D \,A >\,D in the general scale  D D A relation (5.4.6) is just
I \,D ; thus here the scale relation should be something like the square of the

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Table 5.4.1 Currents composed of the fermionic components of the singlets M, magnetic quarks and the magnetic SO(3) gluino corresponding to the global symmetry S;(N );; (1) D 0

GH + G  OP I

S;(N ) D

; (1) 0

GH t IJ + GHIJ I + P Q M G  tM  H  OP I GH OQ 0

(N !2N #4)/N M GH GH D A D + I + (N !2)/N P Q M G  H  A D OP I OQ MI ? I ? I

Table 5.4.2 Energy-momentum tensor contributed from the fermionic component of MGH, qG and the magnetic SO(3) gluino; P L[]"i/2P Q (M  I!   !! M  I  ), ! "R ! )*/2, )*"i/4[ ), *] and )"e) I. P I Q I P Q I I )*I I ¹ IJ  +  O I

i/4[(M GH ! GH !! M GH GH )#(  )]!g L[ ] + I J + J + I + IJ + i/4[P Q (M G  ! G  !! M G  G  )#(  )]!g L[ ] OP I J OQ J OP I OQ IJ O i/4[((MI ? ! I ? !! MI ? I ? )#(  )]!g L[] I J J I IJ

general scale relation for N "N !1, D A A \",A \ , )
I \, 2(
,A A \ , ,A \ ,A \

(5.4.12)

where the normalization factor 2 is determined in the same way as the factor 2 and will be explained later. At the origin of the moduli space M"0, the matter "elds MGH and qP and the SO(3) vector G bosons of the magnetic theory are massless. Both the electric and the magnetic theories have the global symmetry S;(N );; (1). One can verify the reasonableness of this magnetic theory by D 0 checking whether its 't Hooft S;(N );; (1) anomalies match those of the electric theory given in D 0 Table 5.3.4. The corresponding currents composed of the fermionic components and the energymomentum tensors for the gravitational anomaly are listed in Tables 5.4.1 and 5.4.2, respectively, and the 't Hooft anomaly coe$cients are collected in Table 5.4.3. It is shown that by explicit calculations that for N "N !1 the 't Hooft anomalies in the electric and magnetic theories D A indeed match. A special point for the magnetic SO(3) theory should be mentioned. The one-loop
-function coe$cient,
I "6!2N implies that when N 53 this magnetic SO(3) gauge theory is not  D D asymptotically free. Consequently, the value g "0 of the magnetic coupling is the infrared "xed point and thus the theory is free in the infrared. The "elds q and M behave as free scalar "elds and G should have dimension 1. In this case the superpotential (5.4.11) does not exist and hence the infrared theory has a large accidental symmetry. When N (3, the global symmetries (including D the discrete symmetry Z D ) and the holomorphy around M"q"0 determine that superpotential ,

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Table 5.4.3 't Hooft anomaly coe$cients Triangle diagrams and gravitational anomaly

't Hooft anomaly coe$cients

; (1) 0 S;(N ) D S;(N ); (1) D 0 ; (1) 0

(N #1)(N !2N #4)/(2N )#N (N !2)/N #(N !N #4)(N !N #3)/2 D D A D A A D D A D A [(N #4)!(N !N #4)]Tr(t t , t!) D D A 2[(N #2)(N !2N #4)/N #(N !2)/N ]Tr(tt ) D D A D A D (N !2N #4)(N #1)#N (N !2) D A D A A

(5.4.11) of the magnetic theory is unique. Unlike (5.3.31) and (5.3.147), (5.4.11) cannot be modi"ed by a non-trivial function of the global and gauge invariants. In the following we shall see that the det M term of (5.4.11) is very important in properly describing the theory when it is perturbed by a large mass term or along the #at directions. In particular, without the det M term, the term MGHq ) q would possess a Z D symmetry in contrast to the Z D symmetry (5.1.11) of the electric , G H , theory, while as a dual theory the magnetic theory should have a Z D symmetry. , In the following we shall show that the moduli space of vacua of the magnetic description agrees with that of the original electric theory. We shall see that some interesting phenomena will arise. 5.4.2.1. Flat directions. The moduli space of the magnetic theory is given by the F-#at directions of the superpotential (5.4.11) and the D-#at directions of the gauge theory part. Here we only consider the F-#at directions. Eq. (5.4.11) shows that for MO0 the magnetic quarks will have a mass matrix \M. At low energy, the heavy quarks will decouple and there are only k"N !rank(M) massless dual quarks q left. We "rst consider the case rank(M)"N . All the D D dual quarks become massive and hence decouple, so the low-energy theory is a pure SO(3) Yang}Mills theory with massless singlets M. According to (5.2.12) this low energy magnetic theory has a scale A \ det(\M) .
I  "
I \, ,D  Gluino condensation generates a dynamical superpotential

(5.4.13)

A \,A \ det M , = "2I I "2
I  "2
I \, (5.4.14)   ,D where "$1 means that the gaugino condensation leads to two vacua. Considering the term proportional to det M in (5.4.11) and adding it to (5.4.14), we have the full low energy superpotential

1 1 A \,A \ det M! det M"(!1) det M . (5.4.15) = "2
I \, ,D  A \ 2
,A A \ 2
, , ,A \ ,A ,A \ Therefore, the "1 branch reproduces the moduli space of supersymmetric ground sates represented by generic M. The "!1 branch will be discussed later. Eq. (5.4.15) also shows why the normalization factors 2 and 2 were chosen in (5.4.11) and (5.4.12), respectively.

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When rank(M)"N !1, the corresponding low-energy theory is a magnetic SO(3) with one D massless #avour, which can be taken to be the N th #avour, q D . This is just the magnetic version of D , the N"2 Seiberg}Witten model [1], whose exact solution is given by the algebraic curve (5.3.132). From the discussions in Section 5.3.4 we know that it has a massless photon and massless monopoles at the singularity u,u "q D "(16
I  "4
I  , "$1 ,    , where, according to (5.4.12) and (5.4.13), the low-energy scale
I

(5.4.16) 

is given by


I  "  2(
,A \ det MK )

(5.4.17)

with det M K "det M/M,D ,D , the product of the N !1 non-zero eigenvalues of M. Considering the D contribution of the massless magnetic monopoles to the low-energy superpotential, the full low-energy superpotential near the massless monopole points u+4
I  should be the sum of the  low-energy version of (5.4.11) and the monopole contribution (5.3.146), 1 1 1 =" M,DD (q D )! det M K ! (u!u )EI > EI \  C C 2 , , 64
,A A \ 2 , ,A \






1 1 1  det M K ! (u!4
I  )EI > EI \ , " M,DD u!  C C 32
,A A \ 2 2 , , ,A \

(5.4.18)

where the choice of the normalization factor f (u )"!1 of the EI > EI \ term is purely for  C C convenience. From this low energy superpotential, the equation for M,D ,D gives  det M K u" "4
I  .  2
,A A \ , ,A \

(5.4.19)

This has "xed the low-energy magnetic theory with "1 in which there are a massless photon supermultiplet and a pair of massless monopoles EI ! . The u equation of motion further yields > M,D ,D "EI > EI \ . > >

(5.4.20)

This relation can be understood from electric}magnetic duality: the monopoles EI ! are magnetic > relative to the magnetic variables; they will be electric in terms of the original electric variables. Now we consider the low-energy electric theory. rank(M)"N !1"N !2 means that the D A SO(N ) gauge group breaks to SO(2) ;(1) in the #at direction but with one of the elementary A quarks which is charged under ;(1) remaining massless. According to the duality relation, it should be a massless collective excitation in the magnetic description } the monopole. Eq. (5.4.20) is just a re#ection of this correspondence. For the case rank (M)4N !2, the low-energy magnetic theory is an SO(3) gauge theory D with k"N !rank (M)52 light #avours. When k"2,
"0, the low-energy theory is at a D  non-trivial "xed point of the beta function and hence it is described by a four-dimensional

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superconformal "eld theory. If k'2, the low-energy theory is in a free magnetic phase, it is not well de"ned due to the Landau pole. All the results in the magnetic theory of these cases are dual to those of the original electric description. 5.4.2.2. Mass deformations. Let us see whether the magnetic SO(3) theory leads to the correct description of the N "N !2 electric theory if one heavy #avour decouples. As usual, adding D A a large Q,D mass term = "mM,D ,D /2 to the superpotential (5.4.11) of the magnetic theory, we  have the full superpotential 1 1 1 = " MGHq ) q ! det M# mM,D ,D .  2 G H 2
,A \ 2 ,A ,A \

(5.4.21)

The M,D ,D equation of motion R= /RM,D ,D "0 gives  det M K q D " !m , 2
,A A \ , ,A \

(5.4.22)

with det M K "det M/M,D ,D . This non-vanishing expectation value breaks the magnetic SO(3) group to SO(2). After integrating out the massive "elds, the low-energy magnetic SO(2) gauge theory has matter "elds consisting of neutral "elds MGK HK and "elds q! of SO(2) charge $1 and the GK corresponding tree-level superpotential is = "MGK HK q K q K /2. However, according to the discussion G H  in Section 5.3.4, the contribution to the superpotential from the instantons in the broken magnetic SO(3) should also be included since there are well-de"ned instantons in the broken part of the SO(3) group. Therefore, the superpotential is modi"ed to





det M K 1 MGK HK q> q\ . =" f GK HK A \ 2
, ,A ,A \

(5.4.23)

can be identi"ed as the monopoles of the electric This is just the superpotential (5.3.147) and the q! GK theory with N "N !2. D A Now we consider another special point in the moduli space of vacua. After we choose a small mass m for the N th #avour but large det M K , the "rst N !1 #avours q will become very heavy and D D G must be integrated out. The low-energy theory is the magnetic SO(3) theory with the quarks q D in , the adjoint representation and the scale given by (5.4.17). This is just the Seiberg}Witten model discussed in Section 5.3.3 [1]. From its exact algebraic curve solution we know there exist massless monopoles or dyons EI ! at u "4
I  . Thus near one of these two vacua, the low-energy C   superpotential should include contributions from these solitons, 1 1 1 1 det M K # mM,D ,D ! ((q D )!4
I  )EK > EK \ =+ M,D ,D (q D )! ,  \ \ 2
,A A \ 2 2 , 2 , ,A \






1 1 1  det M K #m ! (u!4
I  )EK > EK \ , " M,D ,D u!  \ \ 2
,A A \ 2 2 , ,A \

(5.4.24)

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where we denoted u,q D . The M,D ,D and u equations of motion, respectively, give , 1  det M K !m; u" 2
,A A \ , ,A \

M,D ,D "EK > EK \ . \ \

(5.4.25)

Inserting (5.4.25) into (5.4.24) we get the low-energy superpotential as m becomes large





1 1 det M K . = " m!(1!) * 2 2
,A A \ , ,A \

(5.4.26)

This = shows that for "1 there is no supersymmetric vacuum since the F-term does not vanish. * The low-energy superpotential for the "!1 vacuum is






1 1 det M K EK > EK \ . = " m! \ \ * 2 A\ \ 2
, ,A ,A \

(5.4.27)

The theory can be identi"ed as the one described by (5.3.146) with EK ! as the dyons, which become \ A \\"16
,A \ . This conclusion is the same as that obtained massless at det M K "16m
, ,A ,A \ ,A ,A \ from low-energy electric theory discussed in Section 5.3.4. If we assign large masses to more #avours, the number of massless #avours in the low-energy theory is correspondingly reduced. The non-perturbative phenomena observed in the N "N !3 D A and N "N !4 electric theory will be produced: the monopoles or dyons condense and lead to D A con"nement or oblique con"nement. If there are N (N !4 massless #avours in the low-energy D A theory, the moduli space of vacua will not exist and the con"ning branch disappears [1]. However, when all the #avours are given large masses, there exist an oblique con"nement branch with the superpotential 1 det M . = "! 
32
,A A \ , ,A \

(5.4.28)

This superpotential can be formally obtained by setting m"0 in (5.4.27) and using the u equation of motion since all the #avours should be integrated out. Overall, the monopoles q! of the electric theory at the origin (u"0) of the low energy G N "N !2 electric theory have a weakly coupled magnetic description in terms of the magnetic D A quarks q! of dual theory. The massless dyons E! of the N "N !2 electric theory at G D A are described by strongly coupled dyons. u"u "det M K "16
,A A \ , ,A \  5.4.3. N "N : Dual magnetic SO(4) gauge theory D A The discussion in Section 5.1.2 shows that the classical moduli space of the electric SO(N ) theory A with N "N can be parametrized by the mesons MGH"QG ) QH and the baryon B"det Q but with D A the constraint B"$(det M. The quantum moduli space in this case should be described in the context of the dual magnetic description, which is an SO(4) S;(2) ;S;(2) gauge theory with 6 7 N #avours of quarks in the (2, 2) representation of the gauge group and the SO(4) singlets MGH. The D symmetries and the holomorphy around M"q"0 determine the superpotential as given by

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Table 5.4.4 Currents composed of the fermionic components of the singlets M, magnetic quarks and the magnetic SO(3) gluino corresponding to the global symmetry S;(N );; (1) D 0

GH + G  6  7 OP P I

S;(N ) D

; (1) 0

M GH t IJ + GHIJ I + P 6 Q 6 P 7 Q 7 M G  6  7 tM  H  6  7 OP P I GH OQ Q 0

M GH GH + I + !2/N P 6 Q 6 P 7 Q 7 M G  6  7 G  6  7 D OP P I OQ Q IM ? I ? I

Table 5.4.5 Energy-momentum tensor composed of the fermionic components of MGH, qG and the magnetic SO(3) gluino, P Q s"X, >; L[]"i/2P 6 Q 6 P 7 Q 7 (M  6  7 I!   6  7 !! M  6  7 I  6  7 ), ! "R ! )*/2, )*"i/4[ ), *] and )"e) I PP I QQ I PP QQ I I )*I I ¹ IJ  +  O I Q

i/4[(M GH ! GH !! M GH GH )#(  )]!g L[ ] + I J + J + I + IJ + i/4P 6 Q 6 P 7 Q 7 [(M G  6  7 ! G  6  7 !! M G  6  7 G  6  7 )#(  )]!g L[ ] P Q P Q OP I J OQ J OP I OQ IJ O i/4[(MI ? ! I ? !! MI ? I ? )#(  )]!g L[I ] Q I J Q J Q I Q IJ

(5.4.1) and it cannot be modi"ed by any non-trivial gauge invariant function. The one-loop beta function coe$cients of the S;(2) theory and the SO(N ) gauge theory with N "N #avours A D A together with dimensional analysis give the scale relation [1] D
,A \",A , s"X, > . 2
I @I  A
@A A "2
I \, ,A ,A Q, , , Q,A

(5.4.29)

The normalization factor 2 is chosen for consistency under deformation and symmetry breaking along the #at directions. The one-loop beta function coe$cient
I "6!N of S;(2) shows that  D the magnetic S;(2) ;S;(2) theory is not asymptotically free for N 56. Thus the magnetic 6 7 D theory in the case of N 56 is free in the infrared region. For N "N "4, 5, the theory is D A D asymptotically free and has a non-trivial infrared "xed point, at which the magnetic theory is in an interacting non-Abelian Coulomb phase. Thus the magnetic theory should be dual to the original SO(N ) electric theory with N "N quarks QG in a non-Abelian Coulomb phase. A D A The magnetic theory has an anomaly-free S;(N );; (1) global symmetry, under which the D 0 magnetic quarks q are in the conjugate fundamental representation NM and have the R-charge G D (N !2)/N , the singlets MGH are in the N (N #1)/2 dimensional representation and have RA A D D charge 4/N . The global S;(N );; (1) symmetry is unbroken at the origin of the moduli space of A D 0 both the electric and the magnetic theories and q and MGH are massless. We can verify this massless G particle spectrum by checking the 't Hooft anomaly matching between the electric and magnetic theories. The currents and the energy-momentum tensors are listed in Tables 5.4.4 and 5.4.5 and the 't Hooft anomalies from the massless fermions of magnetic theory are collected in Table 5.4.6. The anomaly coe$cients are indeed equal to those listed in Table 5.3.4 for N "N . D A

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Table 5.4.6 't Hooft anomaly coe$cients Triangle diagrams and gravitational anomaly

't Hooft anomaly coe$cients

; (1) 0 S;(N ) D S;(N ); (1) D 0 ; (1) 0

N (N !1)/2#1/N (2!N ) D D D D N Tr(t t , t!) D (2!N )Tr(tt ) D !N (N !3)/2 D D

5.4.3.1. Flat directions. The #at directions are given by the F-term of (5.4.1) and the D-term of the magnetic SO(4) gauge theory. The MGH equations of motion give q ) q "0. Furthermore, the G H vanishing of D-terms gives the solution of q "0. Thus the low energy theory around such G a point M should be a magnetic SO(4) gauge theory with k"N !rank(M) dual quarks q since D G the equations of motion for M and the vanishing of the D-term do not require them to vanish. In the following we consider several typical cases. When rank(M)"N , there are no massless dual quarks. The low-energy magnetic theory is D a pure S;(2) ;S;(2) Yang}Mills theory with the scale 6 7 det M ,A " , s"X, > , (5.4.30)
I  ,
I " Q A \ 2
,A A \ /det(\M) 2
, , ,A ,A ,A where scale relations (5.2.10) and (5.4.29) are used. The discussion in Section 5.3.2 shows that the low-energy supersymmetric SO(4) S;(2) ;S;(2) gauge theory have four vacua labelled by 6 7   "$1. The gaugino condensation in each S;(2) generates the superpotential 6 7 Q (det M) . (5.4.31) ="2(I I  #I I  )"2( # )
I "2( # ) 6 7 6 7 6 7 2
,A \ ,A ,A Eq. (5.4.31) shows that these two vacua with   "1 have the superpotential 6 7 1 (det M) (5.4.32) =+$ 4
,AA \A , , and hence are not supersymmetric vacua since F "R=/RM does not vanish unless M"0. + The other two vacua with   "!1 lead to ="0 and hence lead to two supersymmetric 6 7 ground states. Therefore, there exist two vacua for rank(M)"N corresponding to the sign $ of D $(= I ) != I ) , the super"eld form of the gaugino condensation I I  #I I  . By the ?6 ?7 6 7 identi"cation B&(= I ) !(= I ) , (5.4.33) ?6 ?7 one can see that these two vacua for M of rank N "N correspond to the sign of D A B"$(det M and hence are identical to the classical moduli space of vacua discussed in Section 5.3.2.

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For the case rank(M)"N !1, the low energy theory is a magnetic SO(4) theory with one D #avour, say, q D . This low-energy theory was discussed in Section 5.3.3, where the low-energy , theory has no massless gauge "elds but q &b . For the present case, the massless magnetic G G composite should be something like a glueball q &(= I ) !(= I ) . ?6 ?7

(5.4.34)

According to (5.4.1) one can construct a low-energy e!ective superpotential 1 1 1 =" M,D ,D q D ) q D ! q D ) q D q , N D D (M,D ,D !q ) . , , 2 , , 2 , , 2

(5.4.35)

Integrating out N D D "q D ) q D from the superpotential, we obtain ,, , , M,D ,D "q  ,

(5.4.36)

so the composite "eld q of the magnetic theory is a semi-classical construction in the electric theory. On the other hand, (5.1.18) implies that for rank(M)"N !1"N !1, the SO(N ) gauge D A A symmetry is broken to ;(1) in the moduli space and N !1 of the N quarks become massive. D D So the electric theory is completely Higgsed but one of the quarks, q,D , remains massless. Consequently, the baryon "eld is B"$(det M"$(M,D ,D ,

(5.4.37)

and thus from (5.4.34) and (5.4.36) we have B"$q &(= I ) !(= I ) . ?6 ?7

(5.4.38)

This means that the baryon "eld of the electric theory is indeed mapped to a massless glueball q &(= I ) !(= I ) of the magnetic theory under the duality, as expected. ?6 ?7 For rank(M)"N !2, the low-energy theory is a magnetic SO(4) S;(2) ;S;(2) with D 6 7 two #avours q , i"N !1, N in the representation (2, 2). This theory was also discussed in G D D Section 5.3.4. It is in the Coulomb phase and its exact solution is given by curve (5.3.137), y"x#(!;#
 #
 )x#

 x. Here for the magnetic theory ;"det N and 6 7 6 7 GH N "PQq q , i, j"N !1, N . The discriminant GH GP HQ D D "[;!(
 #
 )][;!(
 !
 )] , 6 7 6 7

(5.4.39)

shows that there exist massless monopoles or dyons at the points of the moduli space ;"0 and ;"; "(
 #
 ),4
. We only consider the massless monopoles near ;"; . According  6 7  to (5.3.146) and (5.4.1), the superpotential near ;"; should be  1 =" 2

,D 1  MGHN ! [det(N )!; ]EI >EI \ . GH  GH 2 GH,D \

(5.4.40)

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259

At ;"; , the NGH equations of motion give  MGH MGH EI >EI \" " &MGH . (5.4.41) ; 4
  This shows that the monopoles or dyons in the magnetic theory are identi"ed as some of the components of the elementary quarks of the electric theory. When rank(M)"N !3, the low energy theory is a magnetic SO(4) S;(2) ;S;(2) with D 6 7 three #avours q . This is just the case discussed in the last subsection since S;(2) SO(3). Due to G Q the vanishing of the one-loop beta function coe$cient:
I "6!2N "0, the low-energy theory is  D in a free non-Abelian magnetic phase with the gauge group SO(3) and three #avours of magnetic quarks. These magnetic quarks can be identi"ed as the quarks QG, i"N !2, N !1, N , of the D D D SO(N ) electric theory, since along the #at directions with rank(M)"N !3, the theory is Higgsed A D to an electric SO(3) gauge theory. It is very interesting that these elementary quarks and gluons emerge out of strong coupling dynamics in the dual magnetic theory. For the case rank(M)4N !4, the low-energy theory is a magnetic theory SO(4) S;(2) ; D 6 S;(2) with more than three #avours. Since the one-loop beta function coe$cient 7
I "6!2N (0, the theory is either not asymptotically free or a free theory in the infrared  D region. Overall, above discussions shows that for rank(M)"N there are two vacua, the same as in the D classical case, while for rank(M)(N there is a unique ground state which can be interpreted D either as the one of the electric or as the one of the magnetic theory. 5.4.3.2. Mass deformation. We add as usual a large mass term to the N th electric quark by D introducing a tree level superpotential = "mM,D ,D /2. With (5.4.1), the full superpotential  of the magnetic theory is thus = "(1/2)MGHq ) q #mM,D ,D . The M,D ,D equation of  G H  motion gives (5.4.42) q D "!m . , This non-vanishing expectation value breaks the gauge symmetry S;(2) ;S;(2) to the diagonal 6 7 subgroup S;(2) . Consequently, the N !1 quarks q K 6 7 will be decomposed into S;(2) triplets B D GP P B q( K and singlets S K as in (5.3.49). The q D equations of motion give MG,D q "0 for any q and hence G G , G G lead to MG,D "0 .

(5.4.43)

Further, the MGK ,D equations of motion give q K ) q D "0, iK "1,2, N !1 . (5.4.44) G , D With (5.4.42), (5.4.43) and (5.4.44), the remaining low-energy theory is the diagonal SO(3) S;(2) B gauge theory with N !1 SO(3) triplets q( K and the singlets MGK HK , iK , jK "1,2, N !1. The dynamics D D G of these "elds is described by the low-energy superpotential inherited from (5.4.1), 1 = K " MGK HK q( K q( K , a"1, 2, 3 . G? H? 2

(5.4.45)

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The instantons in the broken magnetic S;(2) ;S;(2) give an additional contribution to the 6 7 superpotential since they are well de"ned. The superpotential generated by the instantons can be analysed as follows. For det M K O0, superpotential (5.4.45) gives masses \M K to the "rst N !1 D dual quarks q K , hence the low-energy theory has one dual quark q A and the S;(2) scale, which, , Q G according to (5.2.10) and (5.4.29), is ,A  det M K
I  " " . Q 2
,A \/det(\M A K ) 2
,A \ ,A ,A , ,A

(5.4.46)

Eqs. (5.3.24) and (5.3.25) imply that the superpotential generated by the instantons in the broken magnetic S;(2) has the form Q
I  det M K
I  #
I  7 "4 "! , = "2 6  q D )q D 2m
,A A \ q D )q D , , , , , ,A

(5.4.47)

where we have used (5.4.46), (5.4.29) and the decoupling relation (5.2.10) for N "N , D A .
,A A \ m"
,A A \ , ,A , ,A \

(5.4.48)

Combining = with the superpotential (5.4.45), one can see that the low-energy dynamics  properly reproduces the magnetic SO(3) theory with N !1 #avours with the superpotential D (5.4.11). Moreover, the scale relation (5.4.29) reproduces the scale relation (5.4.12) for the lowenergy electric and magnetic theories. First, the square of scale relation (5.4.29) now becomes A )(
,A \)",A . 2(
I \, ,A ,A Q,A

(5.4.49)

Then in the magnetic theory, non-vanishing expectation value (5.4.42), according to (5.2.17), leads to A )(qN .qN )\"
A \, I \, 4(
I \, Q,A D D ,A \

m A )"
A \ (
I \, I \, . Q,A ,A \ 2

(5.4.50)

Inserting (5.4.48) and (5.4.48) into (5.4.49), we immeditely get (5.4.12). Having discussed these two speci"c cases, we shall review the general N 'N case for which the D A dual theory is an SO(N !N #4) gauge theory. D A 5.4.4. N 'N : General dual magnetic SO(N !N #4) gauge theory D A D A The superpotential in this range is given by (5.4.1), which, like before, is uniquely determined by the symmetries and holomorphy around M"q"0. Scale relation (5.4.6) between the electric and magnetic theories now becomes A \\,D
D \,A >\,D "(!1),D \,A ,D , I , 2
, ,A ,D ,D \,A >,D

(5.4.51)

where the normalization factor C"1/2 is chosen to get the consistent low-energy behaviour under large mass deformation and along #at directions.

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261

Table 5.4.7 Currents composed of the fermionic components of the singlet M, magnetic quarks and the magnetic SO(N !N #4) D A gluino corresponding to the global symmetries S;(N );; (1) D 0

GH + G  OP I ?

S;(N ) D

; (1) 0

M GH t IJ + GHIJ I + M G  tM  H  OP I GH OP 0

(3N !2N )/N M GH GH D A D + I + (N !N !2)/N M G  G  A  D D OP I OP  IM ? I ? I

Table 5.4.8 Energy-momentum tensor composed of fermionic components of MGH, qG and the magnetic SO(N !N #4) gluino; P D A L[]"i/2(M  I!   !! M  I  ),  "R ! )*/2, )*"i/4[ ), *] and )"e) I P I P I P P I I )*I I ¹ IJ  +  O I

i/4[(M GH ! GH !! M GH GH )#(  )]!g L[ ] + I J + J + I + IJ + i/4[(M G  ! G  !! M G  G  )#(  )]!g L[ ] I J OP J OP I OP IJ O OP i/4[(MI ? ! I ? !! MI ? I ? )#(  )]!g L[I ] I J J I IJ

We "rst have a look at the dynamical behaviour of the magnetic SO(N !N #4) gauge theory D A with N #avours. Its one-loop beta function coe$cient
I "3(N !N #2)!N reveals that for D  D A D N 43(N !2)/2,
I 40 and hence the theory is not asymptotically free. Consequently, in the D A  infrared region the theory will provide a weakly coupled magnetic description to the strongly coupled electric theory. When 3(N !2)/2(N (3(N !2) both the magnetic SO(N !N #4) A D A D A and the electric SO(N ) theories are asymptotically free and have the same interacting infrared "xed A point, at which both the electric and the magnetic descriptions are in a non-Abelian Coulomb phase and are physically equivalent. Due to scale relation (5.4.51), the magnetic description is at strong coupling as the number of #avours N increases while the electric description is at weak D coupling and vice versa. For N '3(N !2), the one-loop electric beta function coe$cient
I (0, D A  so the electric description is a free theory in the infrared region whereas the magnetic coupling is strongly coupled. Therefore, the high-energy magnetic theory has provided a dual low-energy description of the electric theory and vice versa. At the origin of the moduli space, the "elds MGH, the magnetic quarks q and the SO(N !N #4) G D A gauge "eld multiplets are all massless, and the global symmetry S;(N );; (1) remains unbroken. D 0 One can check that the 't Hooft anomalies of this massless spectrum of the magnetic theory indeed match the anomalies listed in Table 5.3.4 which receives contributions from the fundamental particles of the electric theory. The relevant particulars such as the currents, the energy-momentum tensor parts and the anomaly coe$cients are listed in Tables 5.4.7, 5.4.8 and 5.4.9, respectively. In the following, we further discuss the decoupling behaviour of the magnetic theory under large mass deformation and along the #at directions and show that these phenomena indeed coincide with the original electric description.

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Table 5.4.9 't Hooft anomaly coe$cients Triangle diagrams and gravitational anomaly

't Hooft anomaly coe$cients

; (1) 0 S;(N ) D S;(N ); (1) D 0 ; (1) 0

N (N !1)/2#N (2!N )N A A A A D N Tr(t t , t!) A N (2!N )/N Tr(tt ) A A D !N (N !3)/2 A A

5.4.4.1. Flat directions. In the #at directions parametrized by M, superpotential (5.4.1) shows that there are k"N !rank(M) massless magnetic quarks q . The F term, i.e. the M equation of D G motion from (5.4.44), and the vanishing of the D-term from the gauge coupling part of the magnetic theory require q "0. So along the #at directions the gauge symmetry SO(N !N #4) remains G D A unbroken but there are only k"N !rank(M) #avours in the low-energy magnetic theory. In the D following we discuss the dynamics of the low-energy magnetic theory along the #at directions according to the rank of M. For rank(M)'N , more than N quarks are massive, so the #avour number k in the low-energy A A magnetic theory satis"es k"N !rank(M)(N !N , and thus D D A k4(N !N )!1"(N !N #4)!5 . D A D A

(5.4.52)

Therefore, this low-energy magnetic SO(N !N #4) theory with k #avours is similar to the D A electric SO(N ) theory with N 4N !5 #avours discussed in Section 5.3.1. Thus a superpotential A D A like (5.3.1)






D \,A >\I ,D \,A \I> 16
, 1 ,D \,A >I =" (N !N !k#2) D A A , \, \I> det M 2 D

(5.4.53)

will be generated. Therefore, there exists no supersymmetric ground state at q "0 due to the G above dynamical superpotential. For rank(M)"N , the low-energy magnetic SO(N !N #4) gauge theory has A D A k"N !rank(M)"N !N "(N !N #4)!4 D D A D A

(5.4.54)

#avours. Thus it is analogous to the electric SO(N ) theory with N "N !4 #avours considered A D A in Section 5.3.2. Similarly, a superpotential






D \,A >  1 16
, ,D \,A >I =" ( # ) 6 7 det M 2

(5.4.55)

arises. Consequently, two supersymmetric ground states exist at the origin q "0, corresponding G to the two sign choices for  (or  ) in   "!1. There is no supersymmetric ground state for 6 7 6 7   "1. The same is also true in the underlying electric theory. 6 7

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263

For rank(M)"N !1, the low-energy magnetic theory is SO(N !N #4) with A D A k"N !N #1"(N !N #4)!3 D A D A

(5.4.56)

#avours. So it is analogous to the theory considered in Section 5.3.3. Thus the low-energy magnetic theory has no massless gauge "elds but has massless composites, and they can be interpreted as some of the components of the elementary electric quarks as in (5.3.35). For rank(M)"N !2, the low-energy magnetic theory is SO(N !N #4) with A D A k"(N !N #4)!2 #avours. It is analogous to the theory discussed in Section 5.3.4. A similar D A analysis shows that this magnetic theory has a massless photon which is con"ned because of the existence of magnetic monopoles at the origin q "0. This is just the dual description of the G corresponding low-energy electric theory when rank(M)"N !2, in which there is a massless A photon with massless elementary quarks. For 3N /2!N /2!34rank(M)(N !2, the number of massless #avours number k lies in A D A the range (N !N #4)!2(k"N !rank(M)4[(N !N #4)!2] . D A D  D A

(5.4.57)

The low-energy magnetic theory is still strongly coupled since the one-loop beta function coe$cient
I "3(N !N #2)!k'0. If we dualize this magnetic theory, the electric theory will be  D A a free SO(N !rank(M)) gauge theory with N !rank(M) massless quarks due to the fact that A D
"3(N !rank(M)!2)!(N !rank(M))(0. This result is obvious in the original electric  A D description. For rank(M)(3N /2!N /2!3, the
I (0 and hence the variables in the low-energy magA D  netic theory are the same as the free ones. In summary, when N 'N , the moduli space of supersymmetric vacua is described by M, D A whose rank is at most N along with an additional sign when rank(M)"N . Thus one has A A obtained a consistent description of the classical moduli space of the electric theory as discussed in Section 5.1.2 in terms of the strong coupling e!ects of the magnetic theory. 5.4.4.2. Mass deformation. Like in the above two special cases, we consider a tree-level superpotential = "mM,D ,D /2. In the electric theory this term gives a mass to the N th quark Q,D and D  the corresponding low-energy theory is an SO(N ) gauge theory with N !1 quarks. In the A D magnetic theory, combining this term with the superpotential (5.4.1), we have the full superpotential = "(1/2)MGHq ) q #mM,D ,D . The M,D ,D equations of motion lead to q D "!m.  G H  , This non-vanishing expectation value further breaks the magnetic SO(N !N #4) gauge theory D A with N quarks to an SO(N !N #3) gauge theory with N !1 quarks. The q D equation of D D A D , motion gives MGK ,D "0, iK "1,2, N !1, and the MGK ,D equations of motion yield q K ) q D "0. D G , Therefore, the remaining low-energy magnetic theory is a magnetic SO(N !N #3) gauge theory D A with N !1 #avours and the superpotential = "1/2MGK HK q K ) q K . This low-energy magnetic theory D * G H is dual to the low-energy SO(N ) gauge theory with N !1 massless quarks. This can be seen from A D following two aspects. First, scale relation (5.4.51) for the high-energy theories can be precisely reduced to the one that relates the low-energy electric and magnetic theories mentioned above.

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Eq. (5.2.10) gives the scale of the low-energy electric SO(N ) theory with N !1 massless quarks, A D A \\,D \"m
,A \\,D ,
, ,A ,D \ ,A ,D

(5.4.58)

while (5.2.14) gives the scale of the low-energy magnetic theory D \,A >\ \,D \"
 ,D \,A >\ \,D (!m)\ .
I  , ,D \,A >,D \ ,D \,A >,D

(5.4.59)

Inserting (5.4.58) and (5.4.59) into (5.4.51) we indeed get the scale relation that relates the SO(N ) A electric theory with N !1 #avours to the SO(N !N #3) magnetic theory with N !1 #avours, D D A D 2
,A \\,D \
I ,D \,A >\,D \"(!1),D \,A \,D \ .

(5.4.60)

Secondly, if we consider a concrete case, N "N #1, the corresponding magnetic description is D A an SO(5) theory with N #1 #avours. The mass term mM,D ,D /2 breaks the SO(N ) electric theory A D with N "N #1 #avours to the low-energy SO(N ) electric theory with N "N #avours, while at D A A D A the same time, the non-vanishing expectation value q D  breaks SO(5) with N "N #1 #avours D A , to the low-energy magnetic SO(4) S;(2) ;S;(2) with N "N , which was discussed in the last 6 7 D A subsection. This explicit low-energy pattern coincides with the general consideration [15]. 5.5. Electric}magnetic}dyonic triality in supersymmetric SO(3) gauge theory 5.5.1. Peculiarities of supersymmetric SO(3) gauge theory Supersymmetric SO(3) gauge theory is somehow exceptional compared to the general cases introduced above. New non-perturbative phenomena arise. The most remarkable of them is the occurrence of two theories dual to the original one. If the original SO(3) gauge theory is `electrica, one dual theory is `magnetica and the other one is called `dyonica. One refers to this dynamical pattern as electric}magnetic}dyonic triality. Moreover, a discrete symmetry (Z D ) which is explicit , in the original SO(3) theory can be realized in the dual magnetic and dyonic theories. However, such symmetries cannot be explicitly observed in the dual Lagrangians because they are implemented by non-local transformations on the "elds and hence are called `quantuma symmetries. In the following we "rst give a general introduction to the special points of supersymmetric SO(3) gauge theory and then, following the route of Ref. [1], introduce some concrete cases with the de"nite matter contents. Let us "rst have a look at the discrete symmetries of the SO(3) gauge theory with N quarks. It is D invariant under the Z charge conjugation transformation C and an enhanced Z D symmetry,  , QPe L
,D Q .

(5.5.1)

This is because the quarks QG are in the adjoint representation of SO(3). With the normalization of SO(3) generators Tr(¹? ¹@I )"? @I , a , bI "1, 2, 3, the ; (1) operator anomaly equation is  1 RIj"4N IJHMF? F? . IJ HM I D 32#

(5.5.2)

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This Z D symmetry will be realized non-locally in the dual theories. We can roughly understand , this as follows in the dual magnetic theory, which is an SO(N #1) theory with N dual quarks D D q and has the discrete symmetry Z D and charge conjugation Z generated by the operation  G , qPe\
,D Cq . (5.5.3) The full Z D symmetry in this dual magnetic theory should be generated by the `square roota of , the operation (5.5.3). There are many possible choices in the `square roota of the charge conjugation C, but Intriligator and Seiberg, using the concrete examples, found that it should be a special element of the S¸(2, Z) electric}magnetic duality modular transformation [15], A"¹S¹S .

(5.5.4)

The discrete Z D is thus a non-local `quantum symmetrya. , Another special point is that in the dual description of SO(3) gauge theory a new superpotential term proportional to det(q ) q ) (5.5.5) G H arises with q being the magnetic quarks. The necessity of adding this superpotential term stems G from ensuring that the dual of the dual of the SO(3) gauge theory should be SO(3) itself. The discussion in Section 5.4.2 shows that the dual description of SO(N #1) with N #avours is SO(3) D D with N #avours and an extra interaction term proportional to det M. By analogy, we can see that D only with the inclusion of (5.5.5) the dual of the dual of the SO(3) superpotential (5.4.11) is identical to that of the original theory. For N 53, the full superpotential of SO(N #1) should be of form D D (5.4.11) with the replacements MGH  qG ) qH ; D \\  !
D \\ , "
, I ,
,A A \ , ,A \ ,D >,D ,D >,D that is,

(5.5.6)

1 1 det(q ) q ) . (5.5.7) =" MGHq ) q # G H G H 2 D \\ 2
I , ,D >,D Scale relation (5.4.12) should remain the same with the exchange
I 
since now SO(3) is the original theory, and thus we get the scale relation D \\)
\,D ",D , 2(
I , ,D >,D ,D and its square root

(5.5.8)

D \\
\,D "(!1)\,D ,D , (5.5.9) 2
I , ,D >,D ,D where "$1 comes from taking the square root of the instanton factor
\,D D of the electric , SO(3) theory and the phase (!1)\,D preserves the relation (5.5.9) along the #at directions and under mass deformation.

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Term (5.5.5) in superpotential (5.5.7) brings some new phenomena into the dual SO(N #1) D theory. Despite the invariance of term (5.5.5) under the global symmetry S;(N );; (1), it breaks D 0 some of the discrete symmetries, since under the transformation q Pe L
,D  G det(q ) q )Pe L
det(q ) q ) . (5.5.10) G H G H This shows that only the Z D subgroup of the Z D symmetry and the charge conjugation , , C remain unbroken. Due to anomaly (5.1.9) in the SO(N #1) gauge theory, except for N "1, 2, D D the transformations q Pe 
,D q shift the vacuum angle : G G 2
"#
. (5.5.11) P#2N D 4N D Therefore, the symmetry transformation q Pe 
,D q in the original electric SO(3) theory, which G G should also exist in the dual SO(N #1) theory, changes the sign of (5.5.5) (i.e. the n"1 case of D (5.5.10)) and shifts the vacuum angle by
(see (5.5.11)) for N O1, 2. Intriligator and Seiberg D interpreted this phenomenon as follows [15]. The original electric SO(3) theory has, in fact, two dual descriptions corresponding to the two signs of this term for N O1, 2. One of them is D `magnetica, which is the electric SO(N #1) theory of discussed in Section 5.4.2; the other dual D theory is `dyonica, which will be reviewed later. These two dual theories are related to another by a duality transformation. We will see that these triality transformations are the extension of the Z group of N"1 duality transformations to the modular transformation group S¸(2, Z).  Precisely speaking, it is the subgroup of S¸(2, Z), S S¸(2, Z)/(2) , (5.5.12)  which permutes these three theories, where (2) is the subgroup of S¸(2, Z) generated by the monodromies (5.3.119). In one word, the full Z D includes the moduli transformation which , exchanges the magnetic and dyonic theories and which appears as a quantum symmetry in the dual descriptions. Since the theories with N "1, 2 are exceptional cases, we "rst discuss how the quantum D symmetry is realized in these two cases and then turn to the cases with more #avours. 5.5.2. One-yavour case: Abelian Coulomb phase and quantum symmetries Since the quark now is in the adjoint representation of the SO(3) group, the model is just the N"2 theory discussed by Seiberg and Witten [1]. The theory has a quantum moduli space labelled by the expectation value u"M/2"Q/2. The SO(3) gauge symmetry is spontaneously broken to SO(2) ;(1) on this moduli space. Consequently, the low-energy theory has a Coulomb phase with a massless photon. As discussed in Section 5.3.4, the low-energy gauge coupling is given by the algebraic curve solution (5.3.120), (5.5.13) y"x!Mx#
 x .    One usually chooses for convenience the normalization
P2
and MP2M. With this   convention the curve solution changes to y"x!Mx#4
 x . 

(5.5.14)

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As discussed in Section 5.3.4, the singularities of the quantum moduli space is given by the zeroes M ,$4
 of the discriminant (M)"M!16
 . There exists a pair of massless magnetic !   monopoles q! at M "4
 and a pair of massless dyons q! at M "!4
 . Correspond> >  \ \  ingly, the e!ective superpotentials, for the monopoles and dyons, according to (5.3.147), are, respectively







M M q> q\ ; = "f q> q\ . = "f > > \ \
 \ \ > >
  

(5.5.15)

Near the singularities M"M , from (5.3.146), the superpotentials are, respectively, ! 1 1 = + (M!4
 )q> q\ ; = + (M#4
 )q> q\ . > 2  > > \ 2  \ \

(5.5.16)

Comparing (5.5.16) with (5.5.7), one can see that the "rst term is the Mq>q\ term and the second one is (5.5.5). Let us see what the quantum symmetry in this model is. Classically, the N"2 S;(2) gauge theory has a global R-symmetry S; (2);; (1) [2,27]. At the quantum level, the ; (1) symmetry is 0 0 0 broken by an anomaly. Since all the fermionic "elds, which include the gaugino and quarks, are in the adjoint representation and their transformations under ; (1) have the same form, the operator 0 anomaly equation for this R-current in the adjoint representation is 1 1 IJHMF? F? "8 IJHMF? F? , a"1, 2, 3 . RIJ0"4N IJ HM IJ HM I A 32
 32


(5.5.17)

This anomaly shifts the vacuum angle:   #8, and hence the ; (1) is broken to Z0 under 0  which the "eld Q and its fermionic component transform as QPe L
Q,  PeGLL , n"1, 2,2, 8 . / /

(5.5.18)

The index R in Z0 indicates that it is the remanent from the ; (1). Since the center elements of  0 S; (2) are contained in Z0 , the full global symmetry including the charge conjugation C is 0  ((S; (2);Z0 )/Z );C . 0  

(5.5.19)

Now let us observe how the discrete symmetries of (5.5.19) are realized in a given vacuum. From (5.5.18), the Z0 generator e
 acts on the scalar component of M as the operation R,  R: MPe
M"!M .

(5.5.20)

Hence the Z0 symmetry is spontaneously broken to Z0 for MO0 since the elements e L
 of   Z0 with n even still leave M invariant. At the origin M"0 of the moduli space the full  Z0 symmetry is restored. It should be emphasized that for a given vacuum, i.e. a given point on the  moduli space, only a Z0 symmetry is left of the ; (1) symmetry, while on the whole moduli space  0

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the full Z0 symmetry is still preserved. R acts as charge conjugation on the scalar component of Q,  so does on Q, R: QPe
Q"!Q .

(5.5.21)

Thus for MO0 this remaining Z0 should be generated by RC. Intriligator and Seiberg further  observed that the generators of Z0 includes an S¸(2, Z) modular transformation [15]  w"RA

(5.5.22)

with




A"(¹S)\S(¹S)"

!1 1 !2 1



,

(5.5.23)

where ¹ and S are the S¸(2, Z) generators given in (5.3.63). Since that A"!1"C, C being the charge conjugation generators. Thus, w is the square root of the Z0 generator, RC, i.e. a generator  of Z0 .  The following consideration shows why the moduli transformation A necessarily appears in w. Consider the central charge Z"an #a n of the N"2 superalgebra at M"0. The action of  " ; (1) on the supercharge (two-component form), Q, is QPe ?Q. After ; (1) is broken to Z0 , the 0 0  above transformation is carried out by the elements of Z0 and is generated by the transformation  R: QPe
Q. Due to the N"2 superalgebra, Z& Q, Q, and Z must transform under w as ZPe
Z"iZ. On the other hand, from integral expressions (5.3.113) for a(M) and a (M), "



(2 +
 dx(x!M/(4
 )  , a " "
(x!1 



(2 > dx(x!M/(4
 )  , a"
(x!1 \

(5.5.24)

it is easily seen that near M"0 [15] an #a n "i(an #a n ) ,  "   "

(5.5.25)

if n and n are related to n and n by the modular transformation A"(¹S)\S(¹S). Since the 



modular transformation A can be rewritten as C¹(S\¹S) whereas according to (5.3.80) S\¹S is the monodromy around M "4
 , thus A is actually congruent to C¹ modulo the multiplica>  tion by the monodromy [15]. If MO0, the Z0 symmetry is broken to Z0 . The broken Z0 generator w maps the massless    monopole at the singular point M"4
 to the massless dyon at M"!4
 . At the origin the   Z0 symmetry is restored and these massless soliton states are degenerate and are mapped from one  to another by the Z0 symmetry. Since the monopoles and dyons are collective excitations, the "elds  representing them are not local, so it is not possible to give w a local realization. This fact further

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269

con"rms that A should be a modular transformation. In particular, A cannot be diagonalized by an S¸(2, Z) transformation, i.e. we cannot "nd a 2;2 matrix X with integer elements to make A diagonal by the operation X\AX. This means that there exists no photon "eld multiplet which is invariant under the action of A, since the photon supermultiplet is the only local object in the low-energy theory. Therefore, even A cannot be realized locally in the low-energy e!ective theory [15]. Now we consider how the above non-local symmetry is re#ected in the dynamics (5.5.15) of monopoles and dyons. The interpretation is as follows. The electric SO(3) theory has two dual theories, one of them, which is called the magnetic dual, describes the physics around M"4
  with the superpotential = in (5.5.15). The other dual theory, which can be called the dyonic dual, > gives the physics near M"!4
 with the superpotential = in (5.5.15). The magnetic dual is  \ related to the electric theory by the modular transformation S of S¸(2, Z) modulo (2), while the dyonic dual is related to the electric description by the S¸(2, Z) transformation S¹ modulo (2). 5.5.3. Two-yavour case: non-Abelian Coulomb phase and quantum symmetries In this case the classical moduli space is parametrized by the expectation value of the colour singlets MGH"QG ) QH, i, j"1, 2, which transform as the adjoint representation of the global S;(2) D #avour symmetry. For MGHO0 the S;(2) gauge symmetry is completely broken and the theory is in the Higgs phase. On the submanifold of the moduli space where det M"0 , the rank of M should be 1, and the S;(2) gauge symmetry will break to ;(1). There now only exists a photon supermultiplet in the low-energy theory, and the theory is in a Coulomb phase [15]. There are not only these two phases in the low-energy theory, a con"ning phase can also arise. To see this, we add a tree level superpotential = "MGHm /2. For det mO0, both Q and Q get  GH   a mass, and after integrating out the massive matter "elds, the low-energy theory is an N"1 pure S;(2) gauge theory, which is known to be in the con"ning phase. For det m"0 but mO0, only one of the matter "elds gets a mass, the low-energy theory is the N"2 supersymmetric Yang}Mills theory discussed by Seiberg and Witten. Since now the matter "eld is in the adjoint representation, the theory is in the Coulomb phase as shown in Ref. [1]. Thus we see that the theory can be in three di!erent phases, the Coulomb, con"ning and the Higgs phases. Note that since the theory has no "eld in the fundamental representation of gauge group, as discussed in Section 2.4, the con"ning phase and the Higgs phases are distinct. The order parameter, the Wilson loop, obeys an area law in the con"ning phase, but a perimeter law in the Higgs phase. In the following we discuss the dynamics of each phase. Let us "rst consider the con"ning phase. The low-energy e!ective Lagrangian for N"1 supersymmetric S;(N ) gauge theory was conA structed in the 1980s from the ; (1) anomaly [25,98]. Since locally SO(3) S;(2), the low-energy 0 e!ective action for the present N"1 pure SO(3) Yang}Mills gauge theory is the same as in the pure S;(2) case [12,84]:


  


 



 (det m) #2 , = "S ln  #2 "S ln   S S

(5.5.26)

where S"=?= is the composite (glueball) "eld and we have used decoupling relation (5.2.12) for ? the SO(3) theory,
 (det m)"
 . According to Intriligator's `integrating ina technique [84],  

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the low-energy superpotential for the SO(3) gauge theory with matter "elds can be obtained from




 

1
 (det m) 1 ="= # MGHm "S ln  #2 ! MGHm  2 GH GH S 2

(5.5.27)

by integrating out m . R=/Rm "0 gives GH GH m "4SM\ GH GH

(5.5.28)

and hence 16S det m" . det M

(5.5.29)

Inserting (5.5.28) and (5.5.29) into (5.5.27) we obtain the low-energy superpotential for the SO(3) gauge theory with two #avours,




="S ln

 

16
 S  !2 . (det M)

(5.5.30)

With the inclusion of the mass term for the two #avours, the full low-energy superpotential is




 

16
 S 1  !2 # MGHm . = "S ln GH  (det M) 2

(5.5.31)

Because S, as a glueball "eld, is always massive, it should be integrated out. The equation of motion for S, R= /RS"0, gives  det M . S"$ 16


(5.5.32)

Thus after integrating out S the full superpotential for the con"ning phase is det M 1 # Tr mM . = "G  2 8


(5.5.33)

The dynamics of the Coulomb and the Higgs phase can be discussed as follows. Integrating out M from (5.5.33) we obtain M "$4(
det m)m\, GH  GH

S"$
det m . 

(5.5.34)

Taking




m"

0

0

0 m 



,

(5.5.35)

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271

i.e. choosing only the second #avour Q to be massive. After integrating out Q , the low-energy   theory is the Seiberg}Witten model and hence is in the Coulomb phase [1]. Note that now det M"0 since from (5.5.34) det m"0. Nevertheless, (5.5.34) and (5.5.35) give Tr(Q )"Tr M "$4
m "$2
 ,     

(5.5.36)

where we have used scale relation (5.2.12) between
and
. The exact solution is given by the   curve (5.5.14), y"x!M x#4
m x. There exist massless monopoles and dyons at the    points M "$4m
. The Higgs phase is represented by the generic points in the moduli    space with mass matrix m"0 (i.e. m "0). GH In summary, the above discussion has shown that all the three phases can be described by the superpotential ="e

det M 1 # Tr mM , 2 8


(5.5.37)

where e"0,$1 labels the three branches. The branch with e"0 describes the Higgs or Coulomb phases of the theory. Both of these phases are obtained for det m"0. For m"0, the generic point in the moduli space is in the Higgs phase. When only m O0, the low-energy theory is in the  Coulomb phase, and it has a massless monopole at the point M "4m
and a massless dyon    at the point M "!4m
. When det mO0, the monopole (the dyon) will condense and lead    to con"nement (oblique con"nement). e"!1 represents the con"ning branch and e"1 the oblique con"nement branch [15]. Now let us turn to the dual description of the electric SO(3) theory with N "2. It has two dual D theories and they are SO(3) gauge theories with two #avours. The holomorphy, dimensional analysis, the global and gauge symmetries determine the superpotentials of the dual theories, which should take following form:






8
I 1 2  det M# det(q ) q) , =" Tr(Mq ) q)# 3 24
I 3 

(5.5.38)

where "$1 labels the two dual theories and the scales of the theories are related by the square root of (5.4.12), 64

I " .  

(5.5.39)

The sign ambiguity introduced by taking the square root is re#ected in (5.5.38) by the sign . The "rst term in (5.5.38) is the standard one (5.4.1) of the dual theory. The det M term, from (5.4.11), should appear for N "N !1 and the det(q ) q) term is as in (5.5.5). The coe$cients in (5.5.38) and D A (5.5.39) are chosen to guarantee the duality; later we shall explain in detail why they are chosen as those given in (5.5.38). The dual magnetic theory should also have three branches. In comparison with (5.5.37), the superpotential is not only of the form (5.5.38), there should also emerge another term characterizing

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the branches. Thus the full superpotential of the dual magnetic theory should be






8
I 2 1 det M  det M# det N #e , = " Tr(MN)# C 3 3 24
I 8
I  

(5.5.40)

where N "q ) q and e "0,$1 labels the three branches. Integrating out N by the equations of GH G H GH motion for N and adding a QG mass term, = "Tr(mM)/2, we immediately obtain GH 






e ! 8
I 1 # Tr(mM) . = "  C  1#3e  2

(5.5.41)

This superpotential is the same as (5.5.37) with the identi"cation e ! . e" 1#3e 

(5.5.42)

Eq. (5.5.41) shows that coe$cients in (5.5.38) are chosen to guarantee the identi"cation of this dual potential with (5.5.37). The e "0 branch describes the weakly coupled Higgs phase of the dual theory. On the other hand, when e "0 (5.5.42) gives e"!"G1. From the above discussion, we know that in the electric theory this corresponds to the two strongly coupled branches. In particular, when "1, the dual theory is in the Higgs phase, and e"!1 states that the corresponding electric theory is in the con"ning phase. The Higgs branch of the "!1 dual theory describes the oblique con"nement branch of the electric theory since now e"1. Therefore, the "1 branch should be regarded as the magnetic dual and the "!1 branch as the dyonic dual. These are exactly the duality patterns introduced in Section 5.5.1. The two other branches of the dual theories are strongly coupled. The branches with e ", which describe the oblique con"nement of the magnetic theory and con"nement of the dyonic theory, give the e"0 branch and hence correspond to the Higgs branch of the electric theory. The branches with e "!, which correspond to the con"nement phase of the magnetic theory and the oblique con"nement phase of the dyonic theory, also yield e", and hence give another description of the strongly coupled branches of the electric theory [15]. Overall, in this two-#avour case, there are three equivalent theories: electric, magnetic and dyonic. The moduli space of each of them has three branches: Higgs, con"nement and oblique con"nement. The map between the branches of the di!erent theories is the S permutation given  by (5.5.42). Let us argue this from the discrete symmetry, along the line of the discussion in Section 5.5.2. The electric theory has a Z0 symmetry generated by QPe
Q and charge  conjugation C. In the magnetic theory the Z0 symmetry takes the form: MPe
M and  qPe\
AQ, where A is the non-local moduli transformation and satis"es that A"C. Thus the Z0 symmetry is a `quantum symmetrya.  In the following we explicitly verify the above duality patterns by working out one example, the Coulomb phase of the electric theory. This phase is obtained by adding = "mM/2 to (5.5.40)  with e " and integrating out the massive "elds. The equations of motion for M, N , M and 

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273

N from the superpotential 






2 1 1 8
I  det M# det N # mM = #= " Tr(MN)#  3 CC 16
I 2 3 

(5.5.43)

give, respectively,  q )q   , q ) q "0, M"0 . M"!    16
I  (5.5.44)

2 8
I 1  M# m"0, (q ) q )#   3 3 2

Eq. (5.5.44) shows that for M#3m/(16
I )O0, the expectation value of q will break the   gauge group to SO(2). Eqs. (5.5.43) and (5.5.44) lead to the low-energy superpotential for the remaining massless "elds q , denoted as q> and q\ according to their SO(2) charges,   






 1 1 M!m q>q\" (M!4m
)q>q\ , ="      * 2 16
I 2 

(5.5.45)

where relation (5.5.39) was used. This low-energy superpotential will be modi"ed by quantum corrections from instantons in the broken magnetic SO(3) part. For large m, the instanton contribution is small and can be ignored. The superpotential shows that the low-energy theory has massless "elds q! at M"4m
"4m
 . These massless "elds can be interpreted as the    monopoles for "1 and the dyons for "!1 of the N "1 case. This is precisely the dual D interpretation according to which the "1 branch is the magnetic description and the "!1 the dyonic one. In addition to the above two monopoles in the Coulomb phase, one can still "nd other monopoles in the N "1 theory arising from the strong coupling dynamics of the dual theories. D Superpotential (5.5.43) shows that q gets an e!ective mass   4M # u m " 12
I 3 

(5.5.46)

for u"q O0. Thus q should be integrated out "rst and an N "1 theory is obtained.   D Eqs. (5.5.46) and (5.2.12) give the scale of the low-energy magnetic theory






4  
I  "
I # u .  3  12

(5.5.47)

There should exist massless monopoles at






4  16
I M  u"$4
I  "G4
I # u " .  3  12 ($3!)

(5.5.48)

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The M equation of motion in (5.5.44) gives 4
I 3  3  M! m"! \u"! M! m .  4 16
4 

(5.5.49)

From (5.5.48) and (5.5.49), we obtain G3# M"4m
.  1$

(5.5.50)

For mO0, since "$1, the above equation gives only one solution: M"!4m
"!4
 .  

(5.5.51)

Therefore, another monopole of the N "1 theory has been found, whose existence is a conseD quence of the strong coupling dynamics of the dual theory. In the m"0 case, a similar analysis shows that there exists a strongly coupled state in the dual theories along the #at directions with det M"0. This state can be interpreted as the massless quark of the electric theory. Finally, we consider the dual of dual description (5.5.38) and see how the triality behaves [15]. From scale relation (5.5.39), the dual of the dual theories (5.5.38) should be an SO(3) theory with N "2 quarks, say dG, gauge singlet "elds M and N and the scale D
I "
.  

(5.5.52)

By analogy with (5.5.38), the superpotential should be






det N 2 8
I det M 2  # ! Tr[N(d ) d)] =" Tr(MN)# 24
I 3 3 3 




#

det N det(d ) d) 8
I  # 3 24
I 




 

det M 2 det N 2 " Tr(MN)# # ! Tr[N(d ) d)] 24
3 24
I 3  




#



det N det(d ) d) # , 24
I 24
 

(5.5.53)

where "$1 and "$1 label di!erent duals. There are two possible choices for  and . The "rst one is "!. Then the superpotential shows that N is a Lagrangian multiplier implementing the constraint M"d ) d and consequently the superpotential is ="0. Thus with this choice the dual of dual theory is just the original electric theory with dG"QG. The other choice is ", and

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275

superpotential (5.5.53) becomes






2 det N det(d ) d) 2 det M # # . =" Tr(MN)! Tr[N(d ) d)]# 3 12
I 24
3 24
  

(5.5.54)

Now we integrate out N . The equations of motion for N give GH GH 8
I det NN\"  (d ) d !M ) , GH G H GH  N "
 [det(d ) d !M )](d ) d !M )\ . GH  G H GH G H GH

(5.5.55)

Inserting (5.5.55) into (5.5.54) we get the superpotential  =" 12


 MGH  (dI ) dJ)! [det M#(d ) d)] . GI HJ 24
 

(5.5.56)

Eq. (5.5.56) does not describe new dual theories. De"ning





 q , (  dH G GH
I 

(5.5.57)

and using scale relation (5.5.39), we rewrite (5.5.56) in terms of the new variables, 1 =" 12








 
 MGH(qG ) qH)! det M#  det(q ) q)
I 24

I    





2 24
I 8
I  (q ) q) .  det M# " Tr[M(q ) q)]# 3 det 3

 (5.5.58)

This is precisely the magnetic dual superpotential (5.5.58). Thus it does not give a new dual description. In summary, the supersymmetric SO(3) gauge theory with N "2 has three equivalent descripD tions: the original electric theory and the two magnetic dual ones. The above discussions show that taking the duals of the dual theories permutes these three descriptions. 5.5.4. N "N "3 case: Identixcation of N"1 duality with N"4 Montonen}Olive-Osborn duality D A This case has several special points [15]. First, the one-loop beta function vanishes, and hence the bare coupling constant  " /(2
)#4i
/(g ) is not renormalized at one-loop. The two-loop    beta function is negative, so the theory is not asymptotically free and the theory is free in the infrared region; Secondly, the (magnetic and dyonic) dual descriptions are SO(4) S;(2) ;S;(2) 6 7 gauge theories, thus we should discuss the duality for each S;(2) branch. In particular, since the Q electric quark super"elds Q can be written as a 3;3 square matrix in terms of their #avour and colour indices, the theory allows a cubic superpotential &det Q at tree level. With this cubic

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superpotential, the N"1 duality is in fact the duality proposed by Montonen and Olive [7] and found by Osborn [8] in N"4 supersymmetric Yang}Mills theory. Like in the discussions in previous sections, the various symmetries and holomorphy as well as the mass dimension determine the superpotential of the magnetic and dyonic theories to be 1 1 =" Tr[M(q ) q)]# 64
I 2

det(q ) q), s"X, > ,

(5.5.59)

Q

where q ) q"P 6 Q 6 P 7 Q 7 q  6  7 q  6  7 since the magnetic quarks q belong to the fundamental representaPP QQ tion of each S;(2) . The scales
I of the magnetic S;(2) are chosen to be equal and are given by Q Q Q the relation 2e
O
I  " , Q

(5.5.60)

where  is the bare gauge coupling mentioned above, and "$1 shows that the term e
O is the  square root of the SO(3) instanton factor. The sign of  determines whether the dual theory is magnetic or dyonic. Note that in the two-#avour case  only appears in superpotential (5.5.38) and not in scale relation (5.5.39). However, here  appears in the scale relation so that it can relate the instanton factor for the S;(2) to the square root of the instanton factor for the electric Q SO(3) theory. As usual, we analyse the duality by looking at the #at directions and mass deformation. The analysis in the #at directions is similar to the general N "N case and the det(q ) q) term in the D A superpotential does not modify the analysis essentially. Thus, in the following we only discuss the mass deformation. Introducing a large mass term = "mM/2 for the third #avour, according to (5.2.12) the  decoupling of this heavy #avour in the electric theory will lead to a low-energy theory with two #avours and the scale
 "me
O . 

(5.5.61)

In the dual theory, adding the above mass term to superpotential (5.5.59), we have 1 1 1 = " Tr[M(q ) q)]# det(q ) q)# mM .  2 64
I  2 Q

(5.5.62)

The equation of motion for M from (5.5.62) gives the expectation value, q "!m, which  breaks the dual SO(4) S;(2) ;S;(2) gauge group to a diagonal subgroup SO(3) . Eq. (5.2.17) 6 7 B gives the decoupling relation of the dual theory 4(
I  )(m)\"
I  . Q 

(5.5.63)

Eqs. (5.5.60), (5.5.61) and (5.5.63) immediately yield the relation (5.5.39) between the scale
I of  the low-energy magnetic theory and the scale
of low-energy electric theory. This is another  argument for the correctness of relation (5.5.60). Integrating out the massive "eld M using the

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equations of motion for N "q ) q , we have     det(q ) q)(q ) q)\ . M"! 32
I  Q

(5.5.64)

Inserting (5.5.64) into (5.5.62), we obtain 1 1  1 K (q( ) q( )]# mM" Tr[M K (q( ) q( )]# det(q( ) q( ) =" Tr[M 2 2 32
I 2 

(5.5.65)

where we have taken into account (5.5.60) and (5.5.63). q( denotes the two light #avours. In addition, the contribution generated by instantons in the broken part of the S;(2) ;S;(2) gauge group 6 7 should be included. To get the instanton contribution, we introduce the Lagrangian multiplier "eld ¸ and rewrite the second term as Tr[¸(q( ) q( )] !32
I det ¸. The original term can be recovered  upon integrating out ¸. Consequently, the superpotential (5.5.65) is rewritten as 1 K (q( ) q( )]#Tr[¸(q( ) q( )]!32
I det ¸ . =" Tr[M  2

(5.5.66)

K / and hence that they should be Eq. (5.5.66) implies that q( G 6 7 get the e!ective mass 2¸#M PP integrated out. The low-energy theory is a pure SO(3) gauge theory with the scale given by (5.2.12), B






M K
I  "
I  det #2¸   



"
I  

[det(M K #2¸)] . 

(5.5.67)

A superpotential contribution by instantons is 2
I  det(M K #2¸) . = "2
I  "   

(5.5.68)

The whole low-energy superpotential is the combination of this instanton generated one and (5.5.68), 1 2
I  det(MK #2¸) . =" Tr[(M K #2¸)(q( ) q( )]!32
I det ¸#  2 

(5.5.69)

Integrating out ¸, (5.5.69) gives





8
I 1 2  det M K (q( ) q( )]# K # det(q( ) q( ) , =" Tr[M 3 24
I 3 

(5.5.70)

which is just the superpotential of the dual SO(3) gauge theory of the N "2 case given by (5.5.38). D Next we add a perturbation to the electric theory in the form of a cubic superpotential [15] = "
det Q , 

(5.5.71)

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where
corresponds to the Yukawa coupling. This kind of superpotential is special for N "3 D since only in this case Q can be written as a square matrix in terms of its #avour and colour indices. If we choose
"(2, the gauge coupling will be identical to the Yukawa coupling, and the theory is very similar to the N"4 SO(3) supersymmetric Yang}Mills theory since all of these three #avours are in the adjoint representation and the one-loop beta function is zero. However, in general this theory is not identical to the N"4 supersymmetric SO(3) Yang}Mills theory since the two-loop beta function is negative, and thus only in the infrared region, the theory with the cubic superpotential agrees with the N"4 supersymmetric Yang}Mills theory. The choice
"(2 means that we rescale Q as




QP

(2  Q.


(5.5.72)

Due to the non-vanishing two-loop beta function, there is a conformal anomaly connected to the scale transformation (5.5.72) in the infrared region [22], which is proportional to 4 ln


 (2


d (=?= )#h.c. ?

(5.5.73)

This leads to a relation between  , which is the e!ective gauge coupling in the infrared region, and #  "/(2
)#4i
/g , the bare gauge coupling:   e
O# "e
O





 1 " e
O
 , 4 (2

(5.5.74)

since the classical Lagrangian takes the form 1/g d (=?= )#h.c.  ? What are the e!ects of the cubic superpotential in magnetic theory? Eq. (5.4.8) and the second relation in (5.4.9) mean that the electric operator det Q is mapped to (= I ) !(= I ) . Therefore, the ?6 ?7 addition of the above cubic superpotential to the electric theory makes the magnetic S;(2) ;S;(2) theory have
I  O
I  . Consequently, the various symmetries determine that 6 7 6 7 superpotential (5.5.59) should be modi"ed to 1 2e
O =" Tr[M(q ) q)]# f ( , ) det(q ) q) . # 2 

(5.5.75)

Similarly, scale relation (5.5.60) is modi"ed to 2e
O
I  g ( , )" , Q Q #

(5.5.76)

where f and g are functions of  and  whose explicit forms are not known. When
"0, the scales Q # should coincide:
I  "
I  , and thus from (5.5.74), (5.5.75) and (5.5.76) we should have 6 7  "iR, f ( , ) # "g ( , ) # "1 . Q # O  # # O 

(5.5.77)

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279

In the infrared region the magnetic SO(4) S;(2) ;S;(2) theory with
I O
I also #ows to 6 7 6 7 the N"4 supersymmetric gauge theory. This can be observed from the
I <
I limit. We denote 6 7  in this limit by  . Since supersymmetric S;(2) gauge theory with N "3 is free in the infrared # H D region, for
I "0 the S;(2) gauge symmetry has become a global symmetry and hence is not 7 7 a dynamical symmetry any more. Consequently, the magnetic quarks qG 6 7 can be written as q6 , P PP A"1,2, 6. Thus the magnetic theory is an S;(2) theory with six doublets coupled through the 6 superpotential (5.5.75). This superpotential breaks the global S;(6) to S;(3) ;S;(2) under which D 7 the S;(2) doublets q 6 are in the representation (3$ , 2). The strong S;(2) dynamics con"nes them 6 6 P to be the meson "elds N ,q ) q in the representation (6$ , 1), and G in the representation (3, 3) of GH G H S;(3) ;S;(2) . This can be seen from the decompositions of the fundamental representations of D 6 S;(3);S;(3) and S;(2);S;(2): 3;3"36, 2;2"13. The above global and S;(2) gauge 6 symmetries and the holomorphy determine that the interaction of these "elds should be given by the superpotential 1N 1 det N GH (G ) H)# = "! #2 det  ,  2
I 8
I  6 6

(5.5.78)

where the G have been rescaled to have mass dimension 1 instead of 2. Combining (5.5.78) with (5.5.75) and adding a mass term Tr(mM)/2, we have the full superpotential 2e
O 1 [f ( , )#2g ( , )]det N = " MGHN # GH H 6 H  2  1 det N 1 1 N (G ) H) GH #2 det # # Tr(mM) . ! 8
I  2 2
I 6 6

(5.5.79)

Now we gauge the group S;(2) , i.e. localize this group and introduce new degrees of freedom, 7 the S;(2) gauge "elds. If the energy is higher than
I , the coupling of the S;(2) gauge theory is 7 6 7 very weak and runs with the scale
I . If the energy is much lower than
I , the S;(2) gauge "elds 7 6 7 will couple to the three triplets G. The coupling  will become very strong and will not run. Thus it 7 should satisfy
I  e
O7 & 7 .
I  6

(5.5.80)

This means that for
I ;
I ,  KiR. 7 6 7 The "elds M and N in (5.5.79) are massive and should be integrated out. The M equations of GH motion set N "!m , GH GH

(5.5.81)

and the N equations of motion give GH G ) H 4e
O # [f ( , )#2g ( , )]m\ . M " H * H GH GH 
I *

(5.5.82)

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After integrating out M and N superpotential (5.79) becomes 1  m (G ) H)!2e
O [f ( , )#2g ( , )]det m . ="2 det # GH H 6 H 2
I 6

(5.5.83)

For m"0, this superpotential is proportional to the Yukawa potential det. Thus in the infrared region the magnetic theory can be identi"ed with an N"4 S;(2) supersymmetric gauge theory with weak coupling  . 7 The above discussion on the infrared magnetic theory was based on the limit
I <
I . Away 6 7 from this limit, in the infrared magnetic theory also #ows to an N"4 supersymmetric gauge theory with the coupling  being a function of  . 7 # It should be emphasized that the N"4 supersymmetric theory with  , as the infrared limit of 7 the magnetic theory, is the not the same as the N"4 supersymmetric gauge theory with  given # by (5.5.74), i.e. the infrared limit of the original electric theory. The original electric N"4 supersymmetric gauge theory, with coupling  , is weakly coupled for
;1; this can be seen # from (5.5.74): 2  & ln
&R . # i


(5.5.84)

(Note that "/(2
)#4i
/g.) The magnetic N"4 theory, with coupling  , is strongly coupled 7 for
;1. This is because when
;1, the mapping from the electric operator to the magnetic one leads to
det QP
[(= I ) != I ) ]K0 . ?6 ?7

(5.5.85)

Consequently
I +
I , 6 7

(5.5.86)

and hence according to (5.5.80),  +0. Conversely, the magnetic N"4 gauge theory is weakly 7 coupled when
I <
I . This limit occurs for
&1, where the original electric N"4 theory is 6 7 weakly coupled. Based on this fact, Intriligator and Seiberg assumed that [15] 1  "! . #  0

(5.5.87)

In this sense the N"1 duality can be interpreted as a generalization of the N"4 duality proposed by Osborn based on the Montonen}Olive conjecture. In this duality, the meson operator MGH of the electric theory can be related to the corresponding operator of the magnetic theory in the  PiR limit by di!erentiating superpotential (5.5.83) with 7 respect to m , GH  M " (G ) H)!4e
O [f ( )#2g ( )]det (m)m\ . GH
I H 6 H GH 6

(5.5.88)

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One notices that MGH has not the simple form of (G ) H) but is shifted. A similar shift was observed by Seiberg and Witten in discussing the #ow from the N"4 to the N"2 theory when m has one vanishing eigenvalue and the two other eigenvalues are equal. This gives a strong support to the above assumption that N"1 duality is related to N"4 duality [15]. 5.6. Dyonic dual of SO(N ) theory with N "N !1 yavours A D A It was shown in the last section that there exists a dyonic dual description in the SO(3) gauge theory, which is a new duality phenomenon. In this section we shall explore some aspects of this dual theory in more detail such as its #at directions and mass deformation, etc. We start from the electric SO(N ) (N '4) theory with N "N !1 #avours discussed in A A D A Section 5.4.2. Its dual magnetic description is an SO(3) gauge theory with N quarks and D superpotential (5.4.11). Now we consider the dual of this magnetic theory [15]. The discussion in Section 5.4.2 shows that the SO(3) theory has both magnetic and dyonic dual descriptions. Both of them are SO(N ) gauge theories with N matter "elds dG, gauge singlet "elds MGH and N and the A D GH superpotential 1 1 [det M! det(d ) d)] =" Tr[N(M!d ) d)]! A \\ 2
, 2 ,A ,A \

(5.6.1)

according to (5.4.11), where "$1 and the scales satisfy A \\"
,A \\
II , ,A ,A \ ,A ,A \

(5.6.2)

due to the scale relations (5.4.12) and (5.5.9). The equation of motion from (5.6.1) yields MGH"dG ) dH. One can easily see that the theory (5.6.1) with "1 gives ="0 and (5.6.2) shows that A \\"
,A \\ .
II , ,A ,A \ ,A ,A \

(5.6.3)

Thus this is just the original electric theory with the matter "elds identi"ed with the electric quarks QG. On the other hand, theory (5.6.1) with "!1 has the superpotential 1 1 ="! det(d ) d)" det(d ) d) A \\ 32
, A \\ 32
II , ,A ,A \ ,A ,A \

(5.6.4)

and the scale A \\"!
,A \\"e

,A \\ .
II , ,A ,A \ ,A ,A \ ,A ,A \

(5.6.5)

According to (3.1.3), we have 4
1 1 1  A \\,D " A \\ , ln
, " # i& ln
@A D " ln
, , , ,A ,D ,A ,A \ 2i
2i
2i
2
g

(5.6.6)

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so scale di!erence (5.6.5) with the original electric theory means a shift of the vacuum angle  by
, P#
.

(5.6.7)

Therefore, the theory with the superpotential (5.6.4) is called the dyonic description of the original electric theory. Let us next consider the physics in the #at directions of the dyonic dual theory and the e!ects of mass deformation. 5.6.1. Flat directions The #at directions of this theory are very subtle. The natural variables parametrizing the classical moduli space are the singlet "elds MGH"dG ) dH. The M equation of motion from (5.6.4) gives det M"0. Since M is an N ;N matrix, one might conclude that the classical moduli space of D D vacua is given by all the values of MGH subject to the constraints det M"0, i.e. rank(M)4N !1, D and the gauge symmetry at most breaks to SO(2) ;(1). However, this conclusion is not correct since from (5.6.4) this theory is scale
dependent. This can be made more clear from the following arguments [15]. Consider the #at directions where M is diagonal and has N !1 non-zero equal D eigenvalues a. In the case of the vacua far away from the origin of moduli space, i.e. a<
 A A , the , , SO(N ) gauge symmetry will break to SO(2) ;(1) due to the non-vanishing M. From superA potential (5.6.4) and the Higgs mechanism, some quarks will acquire masses of order a,D \/
,D \ while the massive gauge bosons are much lighter, their masses are of the order (a. In the case that the energy of the theory lies between these two values,
 A D (q(a, the gauge symmetry is , , neither broken nor are the quarks in the vector representation of SO(N ). This occurs because the A interaction described by superpotential (5.6.4) is not renormalizable. Therefore, the above symmetry breaking pattern cannot be applied to the large a case. On the other hand, if the the expectation values of dG are very large, the gauge symmetry is broken at a high-energy scale and the SO(N ) gauge interaction is weak. This is because its one-loop beta function is positive. However, in A this case superpotential (5.6.4) leads to strong coupling for the massive "elds so that they cannot be easily integrated out and hence the classical analysis gives the wrong conclusion. What will happen if we consider the origin of the moduli space? Near the origin, the expectation value M ;
 A D , and thus one can analyse the #at direction by "rst putting superpotential GH , , \ (5.6.3) aside. Then the dyonic dual description is similar to the electric theory. From the discussion on the electric theory in Section 5.5.2, we know that this dyonic dual theory should have several branches as does the electric theory. We only consider its oblique con"ning branch, which should still be described by superpotential (5.4.28), only with the scale replaced by
II . This = will di!er 
from (5.4.28) by a sign due to the relation (5.6.2) with "!1. Now considering superpotential (5.6.4) on this branch and adding to it = will give the full superpotential = "0. Thus on this 
 branch with oblique con"nement of the dyonic description, one "nds that the #at directions given by the space of MGH are identical with the ones in the original electric theory, except that this theory is strongly coupled. 5.6.2. Mass deformation To discuss the mass deformation, we again add a large mass term for the N th #avour, D = "mM,D ,D /2 to superpotential (5.6.4),  1 1 det M# mM,D ,D . (5.6.8) = "!  2 32
,A A \ A , , \

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283

The above discussion shows that in #at directions, near the origin of moduli space, the dynamics is strongly coupled, and the theory has a con"nement branch and hence there exist monopoles. Away from the origin, in the case m;
, the massive "elds can be integrated out. From (5.6.8) the equations of motion of these massive "elds lead to d,D ) d,D "dGK ) d,D "0,

iK "1,2, N !1("N !2) . D D

(5.6.9)

Eq. (5.6.9) implies that d,D "0 while dGK may not vanish. If dGK O0, the SO(N ) gauge symmetry will A break to SO(2) ;(1), the massless "elds being M K GK HK "dGK ) dHK . However, rewriting the superpotential 1 1 det M K M D D # mM,D ,D = "! ,,  2 32
,A A \ , ,A \











1 det M K 1 det M K " m 1! d,D ) d,D " m 1! d> ) d\ , A \ 2 16m
,A A \ 2 16
, A A A , , \ , , \

(5.6.10)

, where we have used decoupling relation (5.2.10), we see that in the region det M K "16
,A A \ , ,A \ there are also light "elds coming from d,D , denoted as d! according to their ;(1) charges. The "elds d! can be interpreted as the dyons E! of the low-energy N "N !2 theory. Recall that these D A dyons were found in Section 5.3.4 by means of a strong coupling analysis of the electric theory and in Section 5.4.2 by a strong coupling analysis of the magnetic theory, while here we recognize them in a weak coupling analysis of the dyonic theory. This means that in the dyonic dual theory, the dyon is a fundamental particle. As a natural consequence, an oblique con"ning superpotential like (5.4.28) should be present in the tree level Lagrangian of the dyonic theory (5.6.4). Finally, to clearly show the triality between electric, magnetic and dyonic theories, let us see what the magnetic dual and dyonic dual descriptions of this dyonic theory look like [15]. First we consider the magnetic dual of the dyonic theory with superpotential (5.6.4). It is an SO(3) gauge theory with N quarks q and singlet "elds MGH. Its superpotential should be composed of D G superpotential (5.4.11) of the magnetic SO(3) gauge theory and the tree level superpotential (5.6.4) of the dyonic theory, 1 1 1 det M# det M . (5.6.11) =" MGH(q ) q )! G H 2 II ,A \ 64
II ,A A \ 32
, ,A \ ,A ,A \ "!
,A A \ , we see that (5.6.11) is the According to (5.6.2) and square root of (5.4.12),
II ,A A \ , ,A \ , ,A \ same as (5.4.11), so the magnetic dual of the dyonic dual theory is exactly the magnetic dual of the original electric theory. If we take the dyonic dual of the dyonic theory with superpotential (5.6.4), according to (5.6.6) and (5.6.7), the vacuum theta angle will be shifted by
again and this will lead to a superpotential which cancels (5.6.4). Thus the dyonic dual of the dyonic dual theory is the original electric theory. To summarize, the SO(N ) theory with N "N !1 #avours has three equivalent descriptions: A D A the original electric SO(3) theory discussed in Section 5.3.4, the magnetic SO(3) theory described in Section 5.4.2 and the dyonic SO(N ) theory considered here. Taking the dual of the dual theory A permutes these three descriptions.

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5.7. A brief introduction to Sp(N "2n ) gauge theory with N "2n quarks A A D D In this section we shall give a brief introduction to the non-perturbative phenomena, especially the electric}magnetic duality, of N"1 supersymmetric Sp(N ) gauge theory with N #avours of A D matter in the fundamental representation. These phenomena such as the generation of a dynamical superpotential, the erasing of the classical vacuum degeneracy by non-perturbative quantum e!ects, the appearance of a conformal window and the relevant duality are qualitatively similar to those found in the S;(N ) gauge theory with matter in the fundamental representation and in the A SO(N ) gauge theory with matter in the vector representation [99]. In fact, it was shown that the A dynamics behaviours of the Sp(N ) gauge theory are parallel to that of the SO(N ) theory since by A A formally extrapolating the parameter N in the SO(N ) to negative value, the result obtained in the A A SO(N ) theory can be easily adapted to the Sp(N ) theory [100]. Thus in the following we shall only A A state the main results. 5.7.1. Some aspects of Sp(N "2n ) gauge theory A A 5.7.1.1. Sp(N ) gauge theory. First we brie#y introduce the unitary symplectic group Sp(N ). It is A A composed of the transformations that preserve the antisymmetric inner product  J " [101],  with




(J )"

0



1

!1 0

"1 A i , , ",A  

(5.7.1)

where the element 1 denotes N /2;N /2 unit matrix. Thus the number of colours N should be A A A even, N "2n . The dimension of this group is N (N #1)/2"n (2n #1). In particular, the A A A A A A number N of #avours must be even since for an odd number of (chiral) fermions there exists D a discrete global anomaly [102], which will make the theory inconsistent at the quantum level. So, we write N "2n . It should be emphasized that the fundamental representation of Sp(N ) is D D A always pseudo-real. The classical Lagrangian of supersymmetric Sp(2n ) gauge theory has the same form as (5.1.1) A and (5.1.2). The classical global #avour symmetry is S;(2n );; (1);;  (1) and the explicit D  0 transformations of the "elds are similar to those listed in (5.1.3), (5.1.4) and (5.1.5). At the quantum level, the ; (1) and ;  (1) symmetries will su!er from the ABJ chiral anomaly. The corresponding  0 operator anomaly equations are 1 R jI "4n IJHMF? F? , I  D 32
 IJ HM





1 IJHMF? F? , R jI  " 4n !4(n #1) D A IJ HM I 0 32


a"1, 2,2, n (2n #1) , A A

(5.7.2)

respectively. In the same way as in the S;(N ) and SO(N ) cases, one can combine ; (1) and ;  (1) A A  0 to get an anomaly-free R-symmetry with the R-charge [51] n !n !1 A R"R # D A.  n D

(5.7.3)

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285

Table 5.7.1 Representation quantum numbers of fundamental "elds S;(2n ) D

; (1) 

;  (1) 0

2n D 0

#1 0

0 #1

QG 

; (1) 0 (n !1!n )/n D A D #1

Thus the quantum theory has anomaly-free global symmetries S;(2n );; (1). According to D 0 (5.7.3), the various ;(1) quantum numbers of the quark super"eld and gaugino and their representation dimension under S;(2n ) are listed in Table 5.7.1. The perturbative theory gives the D one-loop beta function coe$cient [51]:
"3(N #2)!2N "3(2n #2)!2n .  A D A D

(5.7.4)

5.7.1.2. Classical moduli space. The classical moduli space is described by the D-#atness directions. Q may have non-vanishing expectation values in a D-#at direction. Up to gauge and global rotations, these expectation values QG  can be written in the following matrix forms [99] P




Q"

a  0



0

2

0

0 2 0

a

2

0

0 2 0



 

\





0

0

2 aD L

\ 0

(5.7.5)

0 2 0

for n (n and D A


 a  0

1 "

0

2

0

a

2

0



 

\



Q" 0

0

0

0

2 a A 1 " L 2 0





\

0

0

0

2

0

(5.7.6)

for n 5n . Eq. (5.7.5) shows that for generic a these expectation values break Sp(2n ) to D A G A Sp(2n !2n ) by the Higgs mechanism for n (n and completely break Sp(n ) for n 5n . Thus A D D A A D A the moduli space of vacua is described by the expectation values of the `mesona super"elds M "Q ) Q "PQQ ) Q "!M . GH G H GP HQ HG

(5.7.7)

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Note that the fundamental representation of Sp(2n ) is always pseudo-real, so the metric in the A colour space is antisymmetric. The number of "elds M is n (2n !1). For n (n , this is just the GH D D D A number of the matter "elds left massless after the Higgs mechanism. For n 'n , since from (5.7.6) D A and (5.7.7) rank(M)42n , the M should be subjected to the classical constraints [99] A GH G 2GLD M   M   2M LA> LA> "0 . GG GG G G

(5.7.8)

In particular, for n "n #1, the above constraint can be written as D A Pf M"0 ,

(5.7.9)

where Pf M is the Pfa$an of the antisymmetric matrix M. Note that the moduli space of M subject to the constraint (5.7.8) is singular on submanifolds with rank(M)42(n !1) since in A this case some of the a are zero and Sp(2n ) is not broken to Sp(2n !2n ). Consequently, there G A A D exist additional massless bosons on these singular manifolds. The number of the meson super"elds MGH subject to the constraints (5.7.8) is 4n n !n (2n #1). D A A A It is precisely the number of the matter "elds QG remaining massless after the Higgs mechanism. P Note that there are no baryons in distinction to the S;(N ) and SO(N ) cases since the invariant A A tensor P 2PLA of Sp(2n ) is always reducible, i.e. it can always be broken up into sums of products of A the second rank anti-symmetric tensor JGH. Therefore, the baryons always decompose into mesons. 5.7.2. Quantum moduli space and non-perturbative dynamics The non-perturbative quantum e!ects will modify the classical moduli space. Like in the S;(N ) A and SO(N ) cases, the dynamics is very sensitive to the relative numbers of colours and #avours. A Di!erent ranges of the colour and #avour numbers will present distinct physical pictures [99]. 5.7.2.1. n 4n : dynamically generated superpotential and erasing of classical vacuum. This range D A is very similar to the S;(N ) case. In the low-energy theory there is a dynamically generated A superpotential and it lifts all the classical vacuum degeneracy. The explicit form of this dynamically generated superpotential is determined by the holomorphicity and the anomaly-free global S;(2n );; (1) symmetry as well as the mass dimension 3. The representation quantum numbers D 0 of the quantities entering the superpotential are listed in Table 5.7.2. Similarly to the dynamical superpotential (3.4.32) of the S;(2) case, and since M is an antisymmetric matrix, the dynamical Table 5.7.2 Representation quantum numbers of the quantities composing the dynamical superpotential,
being the dynamical scale

MGH det M
@

S;(2n ) D

; (1) 

;  (1) 0

; (1) 0

n (2n !1) D D 0 0

#2 4n D 4n D

0 0 !4(n !1!n ) D A

2(n !1!n )/n D A D 4(n !1!n ) D A 0

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superpotential should be










@  LA >\LD 
LA >\LD LA >\LD  ="A(n , n ) "A(n , n ) LA LD . A D Pf M A D Pf M

(5.7.10)

A(n , n ) can be determined from the low-energy limit and the explicit instanton calculation. A D First, the one-loop beta function coe$cient (5.7.4) and (3.4.4) give the running coupling




e
O"e\
E\O> F"


LA >\LD . q

(5.7.11)

Considering the large a D limit, (5.7.5) shows that the Sp(2n ) theory with 2n #avours becomes the A D , low-energy Sp(2n !2) theory with 2n !2 #avours. In the low-energy theory A D Pf M K "a\ Pf M , LD

(5.7.12)

where M K denotes the mesons corresponding to the light #avours. With (5.7.11) the identi"cation of the running coupling at the energy q"a D gives the relation between the high-energy and L low-energy scales 2
LA \LD
LA A \LDD \" LA LD . L \L \ aD L

(5.7.13)

As in the S;(N ) case discussed in Section 3.4.1, the requirement that superpotential (5.7.10) A properly reproduces the superpotential of the low-energy theory, restricts the coe$cients A(n , n ) A D to the form: A(n , n )"2LD LA >\LD A(n !n , 0)"2LD LA >\LD A(n !n ) . A D A D A D

(5.7.14)

Further, giving the (2n !1)th and 2n th #avours a large mass by introducing a tree level D D superpotential, = "mM D , the low-energy theory is the Sp(2n ) theory with 2n !2 A D  L \LD #avours with the scale given by the matching of the running couplings of the high- and low-energy theories at the energy q"m: A >\LD \"m
LA >\LD .
L LA LD LA LD \

(5.7.15)

After integrating out the two heavy #avours, the low-energy superpotential coincides with the general form of (5.7.10) only when the A(n , n ) satisfy A D











A(n , n ) LA >\LD A(n , 0) LA > A D A . " n #1!n n #1 A D A

(5.7.16)

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To ensure (5.7.14) and (5.7.16) A(n , n ) must be equal to A D A(n , n )"(n #1!n )e L
LA >\LD (2LA \A)LA >\LD , A D A D n"1, 2,2, n #1!n . (5.7.17) A D The constant A can be determined from the instanton contribution. An explicit calculation in the modi"ed dimensional regularization scheme shows that A"1 [103]. The dynamically generated superpotential is now "nally "xed: ="(n #1!n )e L
LA >\LD  A D






A >\LD LA >\LD  2LA \
L LA LD , Pf M

n"1, 2,2, n #1!n . (5.7.18) A D As mentioned in the S;(N ) case, the concrete dynamical mechanisms generating the superA potential (5.7.18) for n (n and n "n are di!erent. For n "n , the gauge group Sp(2n ) is D A D A D A A completely broken, and (5.7.18) is generated by an instanton in the broken Sp(2n ). A similar A calculation as in the S;(N ) case can explicitly verify this [32,33]. For n (n , (5.7.18) is associated A D A with gaugino condensation in the Sp(2n !2n ) supersymmetric Yang}Mills theory, A D 1 ="(n #1!n ) ! =?= . (5.7.19) A D ? 32
 1NLA \LD  Like in the N (N case of the S;(N ) theory, the dynamically generated superpotential (5.7.19) D A A has lifted all the classical vacua since it leads to non-vanishing F-term, F"R=/RQO0. As the S;(N ) and SO(N ) cases, this is another typical example of dynamical supersymmetry breaking. A A However, if we add a mass term = "m MGH/2 to (5.7.18) and integrate the massive "elds, the  GH low energy Sp(2n !2n ) Yang}Mills theory has n #1 supersymmetric vacua represented by the A D A expectation values of M , GH A A >\LD )LA >(m\) . (5.7.20) M "e L
L >(2LA \ Pf m
L LA LD GH GH This conclusion can also be obtained from calculating the Witten index [31]. Note that now the mass matrix of the Sp(2n ) quarks is antisymmetric. A






5.7.2.2. n "n #1: Smoothing of classical singular moduli space and chiral symmetry breakD A ing. Superpotential (5.7.18) does not make sense for n 5n #1, so the dynamically generated D A superpotential ="0. Consequently, the vacuum degeneracy is not lifted for n 5n #1 and there D A exists a continuous quantum moduli space parametrized by the expectation values of M . Giving GH masses to all of these 2n #avours, their expectation values should still be given by (5.7.20). This D leads to a constraint to the quantum moduli space expressed in terms of the Pfa$an . (5.7.21) Pf M"2LA \
LA A > L LA > This is a quantum deformation of the classical constraint (5.7.8) as for the N "N case of the D A S;(N ) theory. This quantum correction is due to the contribution from instantons. Because of the A

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Table 5.7.3 't Hooft anomaly coe$cients, tI  here denoting the generators of Sp(2n ) D Triangle diagrams and gravitational anomaly

Anomaly coe$cients

Sp(2n ); (1) D 0 ; (1) 0 ; (1) 0

!2n Tr(tI tI ) A !n (2n #3) A A !n (2n #3) A A

quantum deformation, there are no longer any classical singularities on the quantum moduli space. Thus the quantum moduli space of vacua is smooth and there are no other additional massless "elds. This conclusion can be veri"ed by the 't Hooft anomaly matching. It is easy to see from (5.7.6) and (5.7.7) that the non-vanishing M  satisfying (5.7.21) break the GH global S;(2n );; (1) chiral symmetry to Sp(2n );; (1) since M is antisymmetric. We can D 0 D 0 GH check the 't Hooft matching conditions for this unbroken global symmetry. The fundamental massless fermions are the gaugino ? and the quarks QG . Their representation dimensions and P anomaly-free R-charges under the gauge group Sp(2n ) and the global symmetry Sp(2n );; (1) A D 0 are (n (2n #1), 1) and (2n , 2n ) , respectively. The massless composite fermions are the A A \ A D \ fermionic components of the #uctuations of M around M satisfying (5.7.21) and their representation quantum numbers are (1, n (2n !1)!1) . We can write down the Sp(2n );; (1) Noether D D \ D 0 currents and the energy-momentum tensors composed of the fundamental massless fermions and the fermionic components of quantum moduli as in the S;(N ) and SO(N ) cases. It can be easily A A calculated that for n "n #1, the anomalies do match. The explicit non-vanishing anomaly D A coe$cients are listed in Table 5.7.3. . From the high-energy We make two #avours heavy by adding a mass term = "mM D  L \LD and low-energy scale relation (5.7.15) and constraint (5.7.21), after integrating out the two heavy #avours, we immediately get superpotential (5.7.18) for the low-energy n "n theory. D A 5.7.2.3. n "n #2: Conxnement without chiral symmetry breaking. Before discussing the quanD A tum moduli space let us "rst have a look at the classical moduli space given by constraints (5.7.8), which now become G 2GLA GLA> GLA> GLA> GLA> M   2M LA> LA> "0 . GG G G

(5.7.22)

Eq. (5.7.22) shows that the number of constraints is ()"6. Thus the classical low-energy theory  has n (2n !1)!6"(n #2)(2n #3)!6"n (2n #7) light "elds M. In the quantum theory, D D A A A A the light "elds in the low-energy theory are n (2n !1)"(n #2)(2n #3)"n (2n #7)#6 D D A A A A antisymmetric "elds M but subject to the dynamics given by the superpotential Pf M . ="! 2LA \
LA A > A L L >

(5.7.23)

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Table 5.7.4 't Hooft anomaly coe$cients for both high- and low-energy theories, t being the generators of S;(2n ) D Triangle diagrams and gravitational anomaly

Anomaly coe$cients

S;(2n ) D S;(2n ); (1) D 0 ; (1) 0 ; (1) 0

2n Tr(t t , t!) A !2n (n #1)/(n #2)Tr(tt ) A A A !n(2n #3)/(n #2) A A A !n (2n #3) A A

The classical constraints can arise as the equations of motion from this superpotential. This can be easily seen for rank(M)"2n . The superpotential gives masses to A




2n D 2



!2n A "6

components of M, hence only n (2n #7) "elds remain massless. This coincides with the classical A A case as it should be. A new phenomenon is that there will be singular submanifolds in the quantum moduli space when rank(M)42(n !1). In distinction from the classical physical interpretation, here the A singularity implies that some of the quantum components of M become massless on these submanifolds. At the origin of the moduli space, rank(M)"0, all the (n #2)(2n #3) compoA A nents of M are massless and the full global S;(2n );; (1) chiral symmetry is unbroken. Thus we D 0 can check whether the 't Hooft anomalies match for the massless fermions at both fundamental and composite levels. The massless fundamental fermions are the gaugino and quarks, and their quantum numbers under S;(2n );; (1) are given in Table 5.7.1. The massless composite fermions D 0 are the fermionic components of the quantum moduli around the origin M"0 and their S;(2n );; (1) quantum numbers are (n (2n !1)) . One can easily calculate the 't Hooft D 0 D D \ triangle anomaly diagrams and the ; (1) axial gravitational anomaly. The non-vanishing anomaly 0 coe$cients are collected in Table 5.7.4 and they indeed match. Note that at M "0 the chiral GH symmetry does not break but the colour degrees of freedom are con"ned. A similar phenomenon was also observed in the N "N #1 case of the S;(N ) theory. D A A to the superpotential (5.7.23) and Making two #avours heavy by adding = "mM D  L \LD integrating out these two massive "elds, we can get the constraint (5.7.21) in the low-energy n "n #1 theory as an equation of motion. D A 5.7.2.4. n 'n #2: Conformal window and duality. Increasing the number of #avours, we now D A reach the theories with n 'n #2. Eq. (5.7.4) shows that for n 53(n #1) the one-loop beta D A D A function coe$cient
40, the theory is not asymptotically free. It is free in the infrared region. So  this range is not interesting. In the range 3(n #1)/2(n (3(n #1), there is a non-trivial infrared A D A "xed point of the renormalization group #ow at which the theory is in an interacting non-Abelian Coulomb phase. From the discussions on the phases of gauge theory in Section 2.4, the theory in

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Table 5.7.5 't Hooft anomaly coe$cients contributed by the massless fermions of the electric and magnetic theories, t being the generators of S;(2n ) D Triangle diagrams and gravitational anomaly

Anomaly coe$cients

S;(2n ) D S;(2n ); (1) D 0 ; (1) 0 ; (1) 0

2n Tr(t t , t!) A !2n (n #1)/n Tr(tt ) A A D !n (2n #3) A A n (2n #1)!4n (n #1)/n A A A A D

this phase can have a self-dual description. It was found [99] that the dual description is an Sp(2n !2n !4) gauge theory with 2n matter "elds qG in the fundamental conjugate representaD A D tion of S;(2n ), gauge singlets M "!M and a superpotential D GH HG 1 =" M qG qH JPQ , 4 GH P Q

(5.7.24)

where  is a parameter with mass dimension. The one-loop beta function coe$cient of the gauge coupling of the dual theory is, according to (5.7.4),
I "3[2(n !n !2)#2]!2n "6(n !n !1)!2n .  D A D D A D

(5.7.25)

Eq. (5.7.25) shows the reason to require n '3(n #1)/2: this choice makes
I '0 and the dual D A  theory has an identical "xed point of the renormalization group #ow as the electric theory, at which the electric and magnetic theories give physically equivalent description. Similarly to the S;(N ) A and SO(N ) cases, the relation between the dynamical scale
of the electric theory and the scale A
I of magnetic theory is A >\LD
D \LA \\LD "C(!1)LD \LA \LD . I L
L LA LD LD \LA \LD

(5.7.26)

This scale relation will lead to typical electric}magnetic duality features as described for the SO(N ) A and S;(N ) theories. The low-energy electric theory is equivalent to high-energy magnetic theory A and vice versa. Taking the singlet M to transform as the meson "elds M "Q ) Q of the electric theory, the GH G H magnetic theory with superpotential (5.7.24) has a global anomaly-free S;(2n );; (1) #avour D 0 symmetry as the electric theory, under which M is in the representation (n (2n !1)) D D \LA >LD  and the magnetic quark super"elds q in (2n ) A . At M"0, the global S;(2n );; (1) is G D L >LD D 0 unbroken in both the electric and magnetic theories. One can easily check that the 't Hooft anomalies contributed by the massless fermions in the electric Sp(2n ) theory, gaugino and quarks, A match those contributed from the massless fermions in the magnetic theory: magnetic gaugino, magnetic quarks and the fermionic components of the singlet M. Both sets of massless fermions give identical 't Hooft anomaly coe$cients as listed in Table 5.7.5.

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Along the lines of discussion of the S;(N ) and SO(N ) theories, we shall discuss the dynamical A A behaviour in the #at directions and under the mass deformation of the magnetic theory. Let us "rst "nd the #at directions in the magnetic theory. The equations of motion for M from superpotential (5.7.24) and the D-terms of the Sp(2n !2n !4) gauge theory give the #at directions parametrized A D a large expectation value, the quark by qG"0 and arbitrary M . Now giving M D GH L \LD and Q D of the electric theory will get large expectation values and they will super"elds Q D L L \ break the electric Sp(2n ) theory with 2n #avours to a low-energy Sp(2n !2) theory with 2n !2 A D A D #avours and the scale 2
LA >\LD ,
LA A \LDD \" L \L \ M D  L \LD

(5.7.27)

which is obtained from matching the running couplings at the energy M D . On the other L \LD  will give a large mass (2)\M  hand, in the magnetic theory, a large M D LD \LD L \LD D D to qL \ and qL . The low-energy magnetic theory is an Sp(2n !2n !4) theory with 2n !2 D A D #avours, a superpotential of the form (5.7.24) and a scale D \LA \\LD \"(2)\M
I LA >\LD .
I L LD \LD LD \LA \LD \

(5.7.28)

Eqs. (5.7.24) and (5.7.28) show that the scale relation is preserved for the low-energy electric and magnetic theories and hence the duality relation remains. Another interesting example is giving large values to rank(M) eigenvalues of M. Then rank(M) magnetic quarks get heavy masses, and the low-energy magnetic theory is the Sp(2n !2n !4) gauge theory with 2n !rank(M) #avours. From the above discussions, we D A D know that the electric Sp(2n ) gauge theory with n 5n #2 has a vacuum at the origin, Q"0, A D A while the Sp(2n ) theory with n 4n #1 does not: when n 4n , the dynamical superpotential has A D A D A erased all the vacua and the classical vacuum at the origin cannot escape from this fate; when n "n #1, the instanton correction smoothes out the singularity at the origin. In the magnetic D A theory, for 2n !rank(M)42n !2n !2"2(n !n !2)#2, i.e. rank(M)5n #1 there D D A D A A should exist no vacuum at q"0 due to strong coupling e!ects. Since the equation of motion of M requires the vacuum to be at q"0, there is no supersymmetric vacuum for rank(M)5n #1. This is an obvious classical constraint in the electric theory, while in the A magnetic theory it is recovered from the strong coupling dynamics and the equations of motion of M. In the following we consider mass deformation. Giving a mass to the (2n !1)th and 2n th D D , the low-energy electric theory is an #avours by adding the superpotential = "mM D  L \LD Sp(2n ) gauge theory with 2n !2 #avours and the scale A D A >\LD \"m
LA >\LD .
L LA LD LA LD \

(5.7.29)

On the other hand, the equations of motion for M D obtained from = and superpotential L \LD  (5.7.24) yield qLD \ ) qLD "!2m .

(5.7.30)

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This expectation value breaks the dual Sp(2n !2n !4) theory to Sp(2n !2n !6). FurtherD A D A and M K D , iK "1,2, 2(n !1), yield more, the equations of motion of M K D GL D GL \ qGK ) qLD \"qGK ) qLD "0 .

(5.7.31)

This means that (5.7.32) M K D "M K D "0 . GL GL \ Eqs. (5.7.30) and (5.7.32) imply that the light "elds in the low-energy Sp(n !n !3) theory are the D A singlets M K K and the 2n !2 magnetic quarks qGK with a superpotential of the form (5.7.24). GH D The matching of the running couplings at the energy qLD \ ) qLD  gives the scale relation between the low-energy and high-energy magnetic theories, D \LA \\LD \"!(m)\
D \LA \\LD .
I L I L (5.7.33) LD \LA \LD LD \LA \LD \ Eqs. (5.7.29) and (5.7.33) show that the dual scale relation (5.7.26) is still satis"ed and hence the duality is preserved in the low-energy theory. Another kind of mass deformation consists of assigning masses to some of the #avours by adding a mass term = "mGHM /2 with rank(mGH)"2r. The low-energy electric theory is an  GH Sp(2n !2r!2n !4) gauge theory with 2n !r #avours. If one chooses the mass matrix m to D A D satisfy n !r!n !2"0, the magnetic gauge group will be completely broken and there will D A arise a contribution to the superpotential from the instanton in the broken magnetic gauge group. On the other hand, this also occurs in the low-energy electric theory when n "n #2. Turning on D A in the electric Sp(2n ) theory with n "n #3 and integrating the mass term = "mM D A D A  L \LD out the (2n !1)th and 2n th #avours, we get the low-energy Sp(2n ) gauge theory with D D A n "n #2. In the magnetic theory, with the addition of = , the equation of motion for D A  yields qLD \ ) qLD "!2m. The other magnetic quarks qGK , iK "1,2, 2n !2 M D D L \LD "2(n #2) get masses M K GK HK /(2). This is implied by the superpotential (5.7.24). Thus Sp(2n ) A A breaks to Sp(2) and the instanton in this magnetic Sp(2) gauge group yields a low-energy superpotential

K /(2)) K
I \LA > Pf M
I \LA > Pf(M "! LA > =" LA > qLD \qLD (2)LA >m C Pf M K C Pf M K "! "! , (5.7.34) 2LA >(m
LA A A ) 2LA >
LA A > L L > L LA > A >"CLA >, and the decoupling where we have used the scale relation (5.7.26)
LA A A
I \L L L > LA > A A "m
LA A . Eq. (5.7.34) shows that if the constant appearing in (5.7.26) is C"16, relation
LA > L L > L LA > superpotential (5.7.34) is precisely superpotential (5.7.23) for the n "n #2 case of the electric D A theory. Therefore, all the results for n 4n #2 in the electric theory can be obtained from the dual D A magnetic theory with n 4n #3 by #owing down through introducing mass terms. D A To summarize, supersymmetric Sp(2n ) gauge theory with matter "elds in the fundamental A representation is another typical example that exhibits duality and other interesting non-perturbative dynamical phenomena. Therefore, all the N"1 supersymmetric gauge theories have

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conformal windows and dual descriptions. This is in fact a generalization of Montonen} Olive}Osborn duality of the N"4 supersymmetric theory and Seiberg}Witten duality of the low-energy N"2 supersymmetric theory. However, there are some di!erences between N"1 duality and N"2, 4 duality. In the N"1 duality the original electric theory and the dual magnetic description have di!erent gauge groups and matter "eld contents. Especially, the N"4 and N"2 dualities are thought to be exact, while N"1 duality only arises in the infrared region. In spite of this limitation, the potential application of N"1 duality in exploring the non-perturbative dynamics should not be underestimated since N"1 supersymmetry can be easily broken to get a non-supersymmetric theory. In addition, the associated conformal windows also have great signi"cance since we get a large number of non-trivial interacting four-dimensional conformal quantum "eld theories [104].

6. New features of N ⴝ 1 four-dimensional superconformal 5eld theory and some of its relevant aspects The discussion in the previous sections shows that non-Abelian electro-magnetic duality emerges in the IR "xed point of N"1 supersymmetric gauge theory, where the theory is described by an interacting superconformal invariant "eld theory. This is a new conformal invariant quantum "eld theory in four dimensions [104]. The only known non-trivial superconformal "eld theories known before in four dimensions were the N"4 supersymmetric Yang}Mills theory [105] and the special N"2 theories with vanishing
-function [106]. In fact, conformal invariance and duality depend on each other: superconformal symmetry, manifested by the vanishing of the NSVZ
-function, is the necessary environment for the survival of electric}magnetic duality, otherwise the running of the coupling will ruin the electric}magnetic duality. On the other hand, duality is very useful in understanding the dynamical structure of superconformal "eld theory. The combination of electric}magnetic duality and superconformal symmetry has revealed a large number of non-perturbative dynamical phenomena. Furthermore, some non-perturbative information beyond the IR "xed point can be acquired by investigating the renormalization group #ow. Note that the realistic colour number N "3 and #avour number N "6 satisfy the conformal A D window condition, 3N /2(N (3N . In this section we shall introduce some of the new features A D A of these non-trivial four-dimensional superconformal "eld theories and some related aspects including the critical behaviour of the various anomalous currents, anomaly matching in the presence of higher-order quantum corrections, the universality of the operator product expansion and the evidences for the existence of a four-dimensional c-theorem provided by duality and superconformal symmetry. 6.1. Critical behaviour of anomalous currents and anomaly matching As shown in (2.5.8), in a superconformal "eld theory, the supersymmetry algebra determines that the energy-momentum tensor, the supersymmetry current and the chiral R-current all lie in a single supermultiplet. Consequently, the axial anomaly of the R-current, the -trace anomaly of the supersymmetry supercurrent and the trace anomaly of the energy-momentum tensor also

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belong to the same supermultiplet. However, the trace anomaly of the energy-momentum tensor is proportional to the
-function and thus cannot be saturated by the one-loop quantum correction, while it was for a long time believed that the axial anomaly only receives contributions from the one-loop quantum correction [107]. This used to be the famous anomaly puzzle in supersymmetric gauge theory. A series of investigations have concluded that this paradox is actually due to the di!erence between the operator form and the matrix element form of the chiral anomaly equation [108}114]: the operator form of the anomaly equation is one-loop only, while the matrix element form can receive multi-loop contributions. It was pointed out in Ref. [114] that this di!erence actually occurs in any gauge theory, since the gauge invariance of the regularization schemes adopted in calculating the anomaly can only unambiguously "x the form of the renormalized matrix elements, while the form of the operator equation is conditional. Only in supersymmetric gauge theory, has this anomaly paradox been exposed because of the anomaly supermultiplet. In previous sections we only considered the one-loop 't Hooft anomaly matching as a check of the duality and other non-perturbative dynamical phenomena. Since N"1 duality only exists in the IR "xed point of the theory, where the theory is in a strong coupling region, the e!ects of higher-order quantum corrections cannot be neglected. Therefore, we must now check 't Hooft's anomaly matching including higher-order quantum corrections. Among the various anomalous currents participating in 't Hooft anomaly matching, only the singlet chiral R-current is a!ected by higher-order quantum corrections [115]. Eq. (3.1.21) shows that in supersymmetric S;(N ) QCD with N #avours the anomaly-free R-current at the one-loop A D level is a combination of two classically conserved but quantum mechanically anomalous axial vector currents. One is the current R lying in the same supermultiplet with the energy-momentum  and the supersymmetry supercurrent. It was given by Eq. (3.1.13) and can be rewritten in the following two-component "eld form [115]: 2 ,D R  " Tr(R  )!  (GR G #I GR I G ) ? ? ? ? ? ? ?? g G

(6.1.1)

which is the fermionic part of the lowest ("M "0) component of the supercurrent super"eld [108,115], 2 Z J  "! Tr(= e4=R e\4)# [( [D (e\4Q)]e4DM  (e\4QR)# Qe\4D [e4DM  (e\4QR)] ?? ? ? ? ? ? ? g 4 # QDM  (e\4D QR)! QPQR,
(6.1.2)

where Z is the wave function renormalization constant of the quark chiral super"elds. Note that here and in what follows, for the convenience of discussion, we have rescaled the vector supermultiplet
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following super"eld [29]: Z KI  "! ?? 4




1 [D (e\4Q)]e4DM  (e\4QR)! Qe\4D [e4DM  (e\4QR)] ? ? ? ? 2





!QDM  (e\4D QR)!(QPQR,
(6.1.3)

which corresponds to the transformation invariance of the chiral super"elds: = P= , QPe @Q, QI Pe @QI . ? ?

(6.1.4)

The anomaly of this current comes only from the one-loop quantum correction, and can be read from the Konishi anomaly relation [29], N ,D DM KI "DM Z  (QRe4Q #QI e\4QI R)" D Tr =, G G G G 2# G here and in what follows, ="=?= . The super"eld ?

(6.1.5)

,D (6.1.6) K"  (QRe4Q #QI e\4QI R) G G G G G is usually called Konishi supercurrent due to the Konishi anomaly relation [29]. The higher-order e!ects of the R anomaly can be easily inferred from the operator anomaly  equation of the supercurrent J  found a long time ago [115], ??





3N !N 1 ,D A D Tr =#DM Z  (QRe4Q #QI e\4QI R) , DM ? J  "! D (6.1.7) ?? ? G G G G 2
 8 G with  being the anomalous dimension of the matter "elds, "!R ln Z/R"!d ln Z/d ln . The many loop e!ects are re#ected in the term proportional to  in Eq. (6.1.7). The combination of (6.1.5) and (6.1.7) gives the multi-loop anomaly equation for R ,  1 [3N !N (1!)]FIJ?FI ? . RIR " A D IJ I 16


(6.1.8)

The anomaly coe$cient is proportional to the NSVZ beta function of the gauge coupling, given by Eq. (3.4.96). The  component of the Konishi anomaly relation (6.1.5) gives the axial anomaly of the Konishi current, 1 N FIJ?FI ? . RIK " IJ I 16
 D

(6.1.9)

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Eqs. (6.1.8) and (6.1.9) suggest that the anomaly-free R-current with the inclusion of the higherorder e!ects should be the following combination:






3N (6.1.10) R "R# 1! A ! K . I I I N D In the dual magnetic theory, complications will arise in constructing the anomaly-free R-current due to the cubic superpotential (4.1.5) involving the magnetic quarks q, q and the colour singlet M. The magnetic Konishi current consists of a magnetic quark part and a singlet "eld part, K  ,KO  #KM , ?? ?? ?? ,D ,D \,A KO  "  (GP R G  #I GP R I G  ) , O? OP? O? OP? ?? G P  ,D ,D > KM "  IR M I M . ?? I The R -current in the magnetic theory has the same form as in the electric theory,  2 RI  " Tr (I  I )!KO  . ?? g  ? ? ??

(6.1.11)

(6.1.12)

The superpotential provides an additional classical source for current non-conservation. The anomaly equations of the supercurrent super"eld JI and the Konishi relation are thus modi"cations of Eq. (6.1.7) and (6.1.5), respectively [115]:




 



,D RW 1 ,D
I  Tr = 3W!  G ! I #   Z DM (GRe4I G) , (6.1.13) G G RG 8 16
 G G D 1 ,D 1 , RW C   Z DM (GRe4I G)"  G # G Tr= I  , (6.1.14) G G 8 2 RG 16
 G G where G is a certain general chiral super"eld contained in the superpotential W,  and Z are the G G anomalous dimension and the wave function renormalization for  ,
I "3N !,D C G G G  A is the coe$cient of the "rst-order
-function, CG is de"ned by the normalization Tr(¹I ? ¹I @I )"C ? @I G with ¹? being the matrix representation to which the Gs belong, and W is a general classical superpotential. According to (6.1.13) and (6.1.14), the conserved R-current of the magnetic theory with the inclusion of the higher-order quantum corrections should be the following combination [115]: DM ? JI  "D ?? ?










3(N !N ) A ! (KO !2K+)!(2 # )K+ RI "RI # 1! D I I O I I O + I N D ,RI #c KO #c K+, I O I + I 3N !2N 3N !2N D ! ,c! ; D ! "c ! , c " A c "!2 A O O O O + + + + N N D D

(6.1.15)

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where  and  are the anomalous dimensions of the "elds q, q and M, respectively. The meaning O + of the terms and their coe$cients in (6.1.15) is as follows: the special combination KO !2K+ is I I classically conserved and its coe$cient is the numerator of the NSVZ beta function of the magnetic gauge theory; the coe$cient of K+ is proportional to the beta function
"f (2 # ) of the I D O + Yukawa coupling f of the cubic superpotential W "fqG M q HP . D P GH Although the higher-order quantum corrections in constructing the anomaly-free R-current are considered, they actually do not a!ect 't Hooft's anomaly matching. Since the 't Hooft anomaly is an external gauge anomaly, the theory should be put in a background of some external gauge "elds. It was explicitly demonstrated that all the higher-order contributions to the external anomalies of the R-current cancel exactly [115]. We "rst consider the ; (1); (1) triangle diagram. Introducing 0 the external ; (1) gauge "eld G to couple to the baryon number current, one can easily calculate I the anomaly of the R -current of the electric theory [115],  1 N N (1!)GIJGI , (6.1.16) R RI "! IJ I  48
 D A where G "R G !R G is the external ; (1) gauge "eld strength. The anomaly for the Konishi IJ I J J I current K in the external "eld background, like the internal anomaly, comes only from one-loop I quantum corrections, 1 R KI" N N GIJGI . I IJ 48
 D A

(6.1.17)

Eqs. (6.1.10), (6.1.16) and (6.1.17) lead to 1 (!2N)GIJGI . R RI"! A IJ I 16


(6.1.18)

Thus the external anomaly of the R-current in the triangle diagram ; (1); (1) receives no 0 higher-order contributions. This cancellation is actually attributed to the de"nition (6.1.10) of R , I which ensures that the higher-order contribution of the triangle diagram containing R is  dismissed. Note that the anomalies we used to derive Eq. (6.1.10) are the internal ones. On the magnetic theory side, it can also easily be found from (6.17) that R RI"N/(8
)GIJGI [115]. I A IJ Thus the 't Hooft anomalies for the triangle diagram ; (1); (1) match exactly as in the one-loop 0 case. Similarly, the other two triangle diagrams containing the R-current, the axial gravitational anomaly ; (1) and the triangle diagram ; (1)S;(N ), also match exactly as in the one-loop case. 0 0 D There remains the ; (1) triangle diagram. This diagram only concerns the ; (1) current. It was 0 0 argued from the holomorphic dependence of the quantum e!ective action on the external "eld that the external anomaly for this triangle diagram gets only contributions from one-loop quantum corrections [115]. It is well known that the Wilson e!ective action of a pure supersymmetric Yang}Mills theory depend holomorphically on the gauge coupling and "eld strength [112,113]. In  The form of this beta function can be easily understood: due to the non-renormalization theorem of W , the Yukawa D vertex f(    #    #    ) is not renormalized, hence the renormalization of the coupling constant f is O + O O + O + + O determined by the wave function renormalization constants, f "Z\Z\f. Thus the beta function is
"2 # . 0 O + D O +

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the presence of matter "elds such as quarks and mesons, the Wilson e!ective action of a supersymmetric gauge theory with the inclusion of an external gauge "eld takes in Pauli}Villars regularization the following general form [113,115]:








8
 M M 1 ! 3C ln  ! ln G dx d Tr= S ()" % 5   16
 g  G M 1  C ln G dx d Tr = # G   16
 G Z 1 # G dx d dM GRe4G# dx d W()#h.c. , (6.1.19) 4 2 G where g is the bare gauge coupling,  is the renormalization scale, it also plays the role of infrared  cut-o! in the Wilson e!ective action [113]. M and M are the Pauli}Villars regulator masses for  G the ghost super"elds and matter "elds, respectively. The ghost "elds arise due to the (super-)gauge"xing. C is the C in the adjoint representation of gauge group; = is the super"eld strength % G  corresponding to the external gauge "eld; C are de"ned similar to C for the generators when G G they appear in the interaction term with the external gauge "eld. W(G) is the superpotential, and its presence or absence depends on the concrete model. The holomorphy of the vector part of the Wilson e!ective action determines that the coe$cients in front of = and = are saturated by only the one-loop quantum correction. Higher-order  quantum corrections only enter the wave function renormalization constant Z, and are absent for the coupling constant. Note that the Wilson e!ective action is actually an operator action. Its expectation value yields the usual quantum e!ective action, i.e. the generating functional of the 1PI Green functions,

 







e 15 I"e I .

(6.1.20)

The one-loop contribution to the R anomaly (including both the internal and external anomalies)  can be obtained through the action of the operator M R/RM # M R/RM on the vector "eld   G G G part of the one-loop quantum e!ective action:








M M 8
 1 ! 3C ln  ! ln G dx d Tr = U " %   16
 g  G M 1  C ln G dx d Tr (= ) . (6.1.21) # G   16
 G The anomaly for the Konishi current is also given by the di!erentiation M (R/RM )U . This G G way of calculating the anomaly con"rms that the R-currents constructed in Eqs. (6.1.10) and (6.1.15) indeed have no internal anomaly, since the =?= term of U  is invariant under the ? action of the operator



M

R R # M (1#c),  RM G RM G  G G



c"c, c . G O M

(6.1.22)

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The non-invariance of the = part in (6.1.21) under the action of (6.1.22) gives the external  anomaly of the R-current. The higher-order quantum e!ect is re#ected in the presence of the wave function renormalization factors Z , which can be included by the replacement [112] G M  . M PM "   (g /g) 

M M PM " G , G G Z G

(6.1.23)

Eq. (6.1.23) also implies that the role of Z for the ghost regulator mass M is played by the factor G  (g /g). Consequently, the multi-loop quantum e!ective action is 






M M 1 8
  ! ln G ! 3C ln    U " % (g /g) Z 16
 g   G G






M 1 G  C ln # G Z 16
 G G



dx d  Tr= . 

dx d Tr=

(6.1.24)

The = part of this multi-loop quantum e!ective action is invariant under the action of the modi"ed operator R R # M (1#c) . M G RM  RM G G  G

(6.1.25)

This means that the R-current remains internal anomaly-free in the presence of higher-order quantum correction, as it should be, while the action of (6.1.25) on the = part of    U  yields  the external anomaly of the R-current. With the relation (6.1.23) it can be easily found that





R R M # M (1#c)    U   RM G RM G  G G





R R # M (1#c) U  . " M  RM G RM G  G G

(6.1.26)

This implies that the external anomaly of the R-current is exhausted by the one-loop quantum correction, and thus the anomalies of the ; (1) triangle diagram match like in the one-loop case. 0 The above discussion is a detailed analysis of the infrared behaviour of the R-current with the inclusion of the higher-order quantum correction carried out in the operator form of the anomaly equation. One can also start from the matrix element form of the anomaly equation [116]. The matrix element form of the anomaly equation for the Konishi current has indeed revealed some more interesting features than its operator form. The operator anomaly equations for the Konishi current, Eqs. (6.1.9) and (6.1.17), imply that its internal and external anomalies have only one-loop character, but from the matrix element form one sees that the external anomaly of the Konishi current is proportional to the
-function and hence presents multi-loop character. Furthermore, the matrix element form shows that the Konishi current must be renormalized and thus it eceives an anomalous dimension. This implies that the Konishi current remains anomalous at the critical

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point, despite the fact that the matrix element of its operator anomaly equation is proportional to the
-function, which vanishes at the critical point. In the following we shall illustrate this by working out two examples, supersymmetric QED and QCD. For supersymmetric QED, its generating functional in the presence of an external vector super"eld < is 



Z" D
(6.1.27)

where S[<, < , , I ] is the classical action of supersymmetric QED, 

   1 4g 

S" dx

 



d =#h.c. # d dM [Re\4>4 #I e\4>4 I R]

# gauge "xing part ,

(6.1.28)

and  is the quantum e!ective action. We consider the case that the external momentum k<= (k), then only the terms quadratic in = survive in an expansion in powers of   = (k)/k. Thus the quantum e!ective action in this energy region is [47] 



1 [< ]" dx d = #h.c.  4g (k)  

(6.1.29)

Eqs. (6.1.5) and (6.1.7) show that the operator form of the Konishi anomaly DM K and the supercurrent anomaly DM ? J  are proportional to the vector super"eld strength operators = and ?? D =, respectively. Thus to determine the matrix element of the operator anomaly, one should ? "rst compute the expectation value =. From Eqs. (6.1.27) and (6.1.28), this expectation value can be expressed in terms of the external "eld < , integrated over dx d, and it is obtained by  taking the logarithmic derivative of Z with respect to 1/g , 



dx d=#h.c."!4i

R ln Z . R(1/g ) 

(6.1.30)

Omitting the integration over x and , (6.1.29) and (6.1.30) imply that the matrix element of the chiral super"eld operator = is directly connected to the external super"eld strengths = [114,115], 



="






R 1 1 Rg Rg R   = " =   R(1/g ) g (k) R(1/g ) Rg Rg g      

g
[g (k)]
(g )
(g )     = . "  = " 
(g )
(g )  g (k)
(g )     

(6.1.31)

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Here
"g/(2
) is the one-loop
-function of supersymmetric QED [114,116]. The factor 
(g )/
(g ) depends on the UV cut-o!
, and hence it can be identi"ed as a renormalization factor    of the operator =, i.e.
(g ) ="   = .
(g )  

(6.1.32)

Consequently, Eq. (6.1.31) becomes
(g )  = . = " 
(g )   

(6.1.33)

Eq. (6.1.33) and the Abelian analogue of the operator anomaly equations, (6.1.5) and (6.1.7) [113],
(g )
(g ) DM ? J  "!  D =, D DM K"   D = , ?? ? ? 6g ? g  

(6.1.34)

lead to the renormalized matrix elements of the operator anomaly equation in the external background "eld < , 
(g ) 1
(g ) 1  D ="! D =  DM ? J  "!  D ="! ?? ? 6g 6(2
)
(g ) ? 6(2
) ?   
(g ) 1
(g )  D = "!  D = , "! ?  6(2
)
(g ) ?  6g    DM K


(g ) "  = .   g 

(6.1.35)

Eqs. (6.1.31) and (6.1.35) imply that the supersymmetry supercurrent is not renormalized and it is conserved at the "xed point, whereas the Konishi current must be renormalized with the renormalization factor Z (g )"
(g )/
(g ), and the renormalized Konishi current is )     K "Z K .  )

(6.1.36)

Therefore, the Konishi current will stay anomalous at the IR "xed point despite the fact that the matrix element of its anomaly equation is proportional to the beta function. For supersymmetric QCD, the analysis is not so simple as in the Abelian case because the external background gauge "eld is charged with respect to the gauge group, otherwise it cannot participate in the interaction with the matter "elds. Furthermore, the matrix element of a current operator depends on gauge "xing. However, in the kinematic region where the currents carry zero momentum, their matrix elements can be reduced to a form similar to (6.1.35) [47]. With the

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one-loop form of the Wilson e!ective action (6.1.19)








1
S []" dx d #
ln 5  2g  

Tr(=#= ) 



,D Z # dx d dM  G (QGRe4QG#QI Ge\4QI GR) 4 G



1 " dx d Tr(=#= )  2g U 



,D Z # dx d dM  G (QGRe4QG#QI Ge\4QI GR) , 4 G

(6.1.37)

and the relation between S [] and [], (6.1.20), the di!erentiation of the 1PI e!ective action 5 with respect to the one-loop coupling 1/g gives U  1
()
( )   Tr =" Tr = , 
()
( ) 1! N /(2
)   A 

(6.1.38)

where
() is the NSVZ
function, and
is the "rst-order one. The matrix elements form of  anomaly equations (6.1.5) and (6.1.7) at the renormalization scale , can be derived in the same way as in the QED case,
() N
() DM ? J  " Tr(D = ), DM K " D Tr = , ?? ?    24
 2

() 

(6.1.39)

where the renormalized Konishi current is de"ned as [47]






 N K " 1!  A Z ( )K )   2


(6.1.40)

and the renormalization factor Z ( )"
( )/
( ). )     Furthermore, the renormalization group invariance of the two-point correlator of the Konishi current shows that its anomalous dimension is related to the slope of the beta function at the critical point. The scale dimension for a general local operator O(x) at the "xed point is the sum of its canonical dimension d and anomalous dimension (夹 ) at the critical value of the coupling  constant, which can be determined from the scaling behaviour of its two-point correlator O(x)O(y) at large distance (x!yPR). The Callan}Symanzik equation (in coordinate space)






R R x!y #2d #2()#
() O(x)O(y)"0  Rx!y R

(6.1.41)

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determines that the correlator should be of the form O(x)O(y)"[Z (x!y)][(x!y)] , -

(6.1.42)

where Z is the renormalization factor for the operator O, (x!y) is the running coupling constant de"ned at the scale &1/x!y, [(x!y)] is an unknown function up to some space}time or internal indices. At the critical point,
(夹 )"0, ()"夹 , correlator (6.1.42) behaves as 1 O(x)O(y)& . x!yB >A夹

(6.1.43)

For the Konishi current, as discussed above, its renormalization factor depends on the beta function. It is enough to look at the two-point correlator of its axial vector component, . a &[DM  , D ]J I ? ? F

(6.1.44)

Eqs. (6.1.36), (6.1.40), the Lorentz covariance and the renormalization group equation for a (x)a (y) lead to [116] I J

 



(x!y) (x!y)  ()
()   ()g I J   IJ # , a (x)a (y)" I J x!y
() x!y 

(6.1.45)

where the form factors  () cannot be explicitly determined from the Callan}Symanzik equa tion. At large distance, the running gauge coupling (x!y) #ows to  , i.e. the value at the IR H "xed point. Without losing generality, we assume that the
-function has only a simple zero. Hence near the critical point,
()"
( )(! ) . H H

(6.1.46)

The de"nition of the
function at the scale &1/x!y,
[()]"d()/d ln , gives  !"x!y\@Y?H near x!yPR . H In the case that 



(6.1.47)

have no pole at the critical point, substitution (6.1.47) into (6.1.45) yields [116],

(x!y) (x!y)  ()  ()g I J   IJ # , a (x)a (y)& I J x!y>@Y?H  x!y>@Y?H 

(6.1.48)

Thus the anomalous dimension of the Konishi current is related to the slope of the beta function. In the dual magnetic theory, similarly to the operator form, the matrix element form of the Konishi current anomaly equation will become quite complicated due to the #avour interaction superpotential among the magnetic quarks and the singlet "elds. The conservation of the Konishi current is spoiled by both the superpotential and the gauge anomaly. To extract the matrix element form of the Konishi current anomaly, a technique was invented [116] which consists of "rst

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constructing an anomaly-free Konishi current for the Kutasov}Schwimmer model introduced in Sections 4.3 [83,85,86], then making the adjoint matter "eld decouple by introducing a large mass term for it. In this way, one can obtain the Konishi current in the usual supersymmetric QCD, which is thus called minimal supersymmetric QCD as in Refs. [83,85,86]. Recall that the electric theory side of the Kutasov}Schwimmer model allows the superpotential (4.3.26), = "g Tr XI>, X being the matter "eld in the adjoint representation of the S;(N ) gauge group.  I A I q >I\Fq, g , The magnetic theory has a superpotential (4.3.40) = "g Tr >I>#I t M N N N I

 I g and t are the corresponding coupling constants, which are explicitly written out here. In the I N critical point of the model, the singlet "eld M I is identical to the meson "eld M in the electric N N theory de"ned by (4.3.27) due to the duality. When the Kutasov}Schwimmer model is at the critical point, the various couplings including both the electric and magnetic gauge coupling and those appearing in the superpotentials = and  = take their critical values, " , " , s"s , etc. The anomaly-free Konishi currents in

 both the electric and magnetic theories are given by the Noether currents corresponding to the non-anomalous ; (1) transformations on the matter "elds. According to Eqs. (4.3.12) and (4.3.16),  the absence of the ;(1) anomalies in both the electric and magnetic theories requires that N q #N q "0 , D / A 6 N q !(N !N )q "0 , D O A D 7

(6.1.49)

where q ("q I ), q , q ("q  ), q are the corresponding ;(1) charges of Q, QI , X, q , q  , respectively. O 7 O O / / 6 O At the same time, the ;(1) invariance of the superpotentials = and = in both the electric and 

 magnetic theories assign the coupling s, s , t with the charges N q "!(k#1)q , q  "!(k#1)q , 7 Q 6 Q









2N N !N A #1!p q # p!k#2 D A q , p"1, 2,2, k , q N" 6 7 R N N D D

(6.1.50)

where use was made of the "rst relation of (6.1.49) and the identi"cation of the singlet "elds M I in N the magnetic theory with the meson "elds M of the electric theory at the critical point. FurtherN more, matching the coupling constant t at the scale of decoupling one heavy #avour requires that N the charges q N remain identical under a deformation N PN !1. This leads to [116] D D R q "q , 6 7

(6.1.51)

and hence q "q  , Q Q

3!k q N "!q . R Q k#1

(6.1.52)

Relations (6.1.50), (6.1.51) and (6.1.52) determine that the non-anomalous Konishi supercurrent in the critical electric and magnetic Kutasov}Schwimmer model is a linear combination of the Konishi current in the minimal supersymmetric QCD and a current constructed from the adjoint

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matter "elds [116]: ,D N K "  (QGRe4QG#QI Ge\4QI GR)! D Tr(XRe4Xe\4) ,  N A G N ,D K "  (qGRe4qG#q Ge\4q GR)! D Tr(>Re4>e\4)

 N A G





N # 2#(1!k) D Tr (MRM) , N A

(6.1.53)

where the ;(1) charges of the electric quark supermultiplet are chosen as 1 as shown in (6.1.4). We decouple the adjoint "eld X in the electric theory by introducing its mass term m Tr X. After X is integrated out, the theory returns to the minimal supersymmetric QCD with only quark super"elds Q, QI . Furthermore, due to the duality map of the chiral operators at the critical point of the Kutasov}Schwimmer model [83,85,86], > also acquires a mass term m Tr >, thus in the magnetic theory the minimal supersymmetric QCD will be reproduced in the low-energy limit with the matter "elds q, q and M. There are two delicate points to make clear in realizing this: "rst, to precisely produce the minimal dual magnetic theory, one must choose a phase to ensure that the magnetic gauge group S;(kN !N !k) breaks down to S;(N !N ). This in principle should D A D A be under control. Second, despite starting from a critical Kutasov}Schwimmer model and decoupling the heavy "eld X (in the electric theory) or > (in the magnetic theory), usually one does not get the minimal supersymmetric QCD still at the critical point. This can be understood from another viewpoint: the heavy "elds X and > can be thought of as the regulator "elds of the minimal supersymmetric QCD away from its critical point and their masses m as the UV cut-o!. Consequently, the resulting Konishi operators in the minimal theory should be thought of as bare operators de"ned at the scale given by m. Equivalently speaking, the Konishi operators in the Kutasov}Schwimmer model may be thought of as operators of the minimal supersymmetric QCD regularized in a particular way. Therefore, the Konishi operators in the minimal (electric or magnetic) theories can be de"ned by directly dropping the "elds X and >. This immediately gives ,D K "  (QGRe4QG#QI Ge\4QI GR) ;  G ,D N !N AJ . K "2Tr(MRM)#  (qGRe4qG#q Ge\4q GR),2J # D (6.1.54)

 + O N D G Eq. (6.1.54) shows that the electric Konishi current obtained in this way has its usual form, and hence is expected to #ow in the infrared region to the corresponding current in the critical minimal theory. The magnetic Konishi current needs a delicate analysis. Since the K given in (6.1.54) is

 obtained by the reduction of the magnetic Konishi current in the critical Kutasov}Schwimmer model, it is de"ned at the scale m. At this scale the various couplings of the magnetic minimal supersymmetric QCD are equal to their critical values with the heavy "eld > present, i.e. the critical values of the coupling constants in Kutasov}Schwimmer model, " , " . Below the scale m,

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as stated above, the low-energy theory is the usual non-critical minimal supersymmetric QCD. Thus the Konishi operator in (6.1.54) is actually de"ned on a particular renormalization group trajectory, which is a path in the space parametrized by the couplings (, ) of the non-critical minimal supersymmetric QCD. After this point is clear, one can easily see how the Konishi current #ows to the IR "xed point of the magnetic theory of the minimal supersymmetric QCD. Note that the Konishi current operator is a composite operator, thus in general its renormalization will lead to mixing with operators of lower dimensions. However, before the decoupling of the heavy "elds X and >, the electric and magnetic Konishi current operators are identical in the critical Kutasov}Schwimmer model. Thus their dynamical behaviour along the renormalization group trajectory, i.e. at any distance (or energy scale), should be identical. This means that the magnetic Konishi current operator is renormalized multiplicatively, and hence it cannot mix with operators of lower dimension as soon as this is true for the electric Konishi current operator in the o!-critical electric theory. Therefore, the magnetic Konishi current operator can be rewritten as a linear combination of renormalization group invariant operators in an o!-critical minimal magnetic theory, K "AK #BK ,

  

(6.1.55)

where K and K are two fundamental renormalization group invariant operators constituting   a basis for any other renormalization group invariant operators. These two operators diagonalize the anomalous dimensions matrix, i.e. that they do not mix with each other along the trajectory of renormalization group #ow. A and B are two constants determined by the original critical Kutasov}Schwimmer model, thus they depend only on the critical values of the couplings, " , " . The decomposition (6.1.55) is explicitly suitable for analysing the IR behaviour of the magnetic Konishi current near the "xed point. K and K can be constructed by inspecting the various ;(1)   currents and their anomalies as well as their relations with the Konishi current operator. In the o!-critical minimal supersymmetric QCD, one well known renormalization group invariant operator is the non-anomalous R-current. However, it does not mix with the Konishi current since they belong to current supermultiplet with di!erent superspins. Thus this R-current should be excluded from the candidates for K and K . Two other possible candidates are combinations of   K+ and KO de"ned in (6.1.15). One is K ,KO!2K+, whose divergence is given by the gauge 5 anomaly, and the other current is K ,K+, whose divergence is only proportional to the  superpotential. The magnetic Konishi current (6.1.54) can be rewritten in the following form in terms of these two new currents, N N !N A K #2 D K . K " D 5

 N  N A A

(6.1.56)

Unfortunately, K and K mix under the renormalization group #ow, so they cannot play the 5  roles of K and K . It is found that K is actually the following combination [116],    2N !3N #N  A D O K #
I K "
I  K #
I K , K " D 5 H  ? 5 H   48


(6.1.57)

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where
I "( #2 )/2 is the beta function of , and  ,  are the anomalous dimensions of the H + O O + "elds q, q and M, respectively. This form of K explicitly remains invariant along the trajectory of  the renormalization group #ow since its D DM  divergence is proportional to the anomaly of the ? supercurrent J  , ?? 2N !3N #N  A D O D DM  Tr = #
I D DM W , D DM K "DM ? J  "! D ?  ?? ?

 H ? 48


(6.1.58)

where = is the gauge super"eld strength of the magnetic gauge theory and W is the interaction

 superpotential among the magnetic quarks and the gauge singlet, W"q MG q H. To "nd the G H renormalization group invariant combination K requires determining the matrix I of anomalous  dimensions. A similar procedure to the one used to derive (6.1.36) and (6.1.40) shows that the entries of I are proportional to linear combinations of the beta functions in the IR limit. Thus the magnetic Konishi current (6.1.56) in the minimal supersymmetric QCD can formally be written as K "A( ,  , m)K #B( ,  , m)K .

  

(6.1.59)

In particular, A( ,  , m) and B( ,  , m) have well-de"ned non-vanishing limits at mPR. The form of Eq. (6.1.59) makes it possible to compute the anomalous dimension of the magnetic Konishi current. Like what was done in the electric theory, we consider the two-point correlator a (x)a (y) of its axial vector component a . The large distance behaviour of this correlator is I J I determined by the matrix of anomalous dimensions [116] and hence by the renormalization factors ZI . These renormalization factors take into account the mixing of the operators Tr = and W.

 Since, as shown above, the entries of the ZI matrix are given by a combination of the beta functions, the correlator a (x)a (y) at large distance must depend on the beta functions of the coupling I J constants taken at the scale &1/x!y. Assuming that the beta functions only have simple zeros at the critical point, one has the following asymptotic expansion in the neighbourhood of the critical point, "夹 , I "I 夹 :
I "
I  (夹 , 夹 )(!夹 )#
I  (夹 , 夹 )(!夹 ) , ? ?? ?H
I "
I  (夹 , 夹 )(!夹 )#
I  (夹 , 夹 )(!夹 ) , H H? HH

(6.1.60)

where the constants
I  ,
I  ,
I  and
I  are the elements of the matrix
I  (夹 , 夹 )" ?? ?H H? HH GH R
I (i)/Rj i, j", . A similar calculation as in the electric theory gives 1 , a (x)a (y)& I J x!y@I   >

(6.1.61)

where
I  is the minimal eigenvalue of the matrix (
I  ). The identi"cation of the correlators

 GH a (x)a (y)&a (x)a (y) at the critical points yields I J I J ,
(夹 ) "
I (夹 , 夹 ) 



(6.1.62)

i.e. the anomalous dimensions of the electric and magnetic Konishi currents are identical at the critical points and are given by the slope of the beta functions evaluated at the critical points. This

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is one of the most important conclusions revealed by the matrix element form of the anomaly equation of the Konishi current. 6.2. Universality of operator product expansion The operator product expansion (OPE) is an axiomatic-algebraic way to study conformal invariant quantum "eld theory. It had achieved great success in two dimensions [117], where the quantum conformal algebras, the Virasoro and Kac-Mody algebras, are derived with the use of the OPE, and all of the two-dimensional conformal "eld theories are classi"ed according to the unitary representations of the conformal algebra. Thus a natural idea is to look at the OPE in the four-dimensional case. Some new features of the four-dimensional superconformal "eld theories have indeed been revealed by the OPE: First, in contrast to the two-dimensional case, the OPE of products of energy-momentum tensors ¹ does not form a closed algebra. A new operator  with lower dimension arises. An IJ explicit calculation in a free superconformal "eld theory shows that  is the Konishi current operator. In particular,  develops an anomalous dimension. Thus, the discussions in Section 6.1 suggests that even in an interacting four-dimensional superconformal "eld theory  may also be identi"ed with the Konishi current operator [47]. Secondly, the OPE of ¹ 's and the OPE of 's show that there are two central charges, c and c, IJ where c is related to the gravitational trace anomaly of the theory. In addition, the OPE of ¹ and IJ  implies that  has a non-vanishing conformal dimension h. These three numbers, c, c and h characterize a four-dimensional superconformal "eld theory. In the context of a free fourdimensional "eld theory, an explicit calculation gives [118] 1 (12N #6N #N ) , c"    120(4
)

(6.2.1)

where N , N and N are the numbers of real vector "elds, Majorana spinor "elds and real scalar    "elds. For a supersymmetric gauge "eld theory with gauge group G, there are N ,dim G T components of the vector super"eld and N ,dim ¹ components of the chiral super"eld in the Q representation ¹. Central charge (6.2.1) hence becomes 1 1 c" (3dim G#dim ¹)" (3N #N ) . T Q 24 24

(6.2.2)

It was found that in a free supersymmetric "eld theory [116], "R, h"0, and c"N "dim ¹ . Q

(6.2.3)

Eqs. (6.2.1), (6.2.2) and (6.2.3) mean that c depends on the total number of degrees of freedom of the theory, while c counts the degree of freedom of the chiral matter super"elds. Thirdly, higher-quantum corrections show that c and c are universal, i.e. they are constants independent of the couplings, while h is not.

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In the following we shall illustrate these features by working out several examples of four dimensional conformal "eld theories such as free massless scalar and spinor "eld theories and N"1 supersymmetric gauge theory at the IR "xed point. The simplest example of a four-dimensional conformal "eld theory is a massless free scalar "eld theory with the following Lagrangian and propagator in Euclidean space [116], 1 1 , L" RIR , (x)(y)" I (x!y) 2

(6.2.4)

and the energy-momentum tensor 1 1 2 ¹ (x)" R R !  (R )! R R  . I J IJ M IJ 6 3 I J 3

(6.2.5)

The correlator (x)(y) immediately leads to the OPE of ¹ 's, IJ 1 1 1 1 X ! (y)X #2 ¹ (x)¹ (y)"!c IJ MN IJMN (x!y)\F 360 IJMN (x!y) 36

(6.2.6)

with c"1 and h"2, where the omitted terms are less singular terms, and the tensor operator X is X (x)"2  䊐!3(  #  )䊐!2( R R # R R )䊐 IJMN IJ MN IM JN JM IN IJ M N MN I J ! 2( R R # R R # R R # R R )䊐!4R R R R . IM J N IN J M JN I M JM I N I J M N

(6.2.7)

The OPE (6.2.6) is "xed by the symmetry of ¹ , the conservation RI¹ "RJ¹ "0 and the IJ IJ IJ tracelessness ¹I "0. In particular, it shows that a new operator " arises. The OPE algebra I is closed by the expansions, 2c 2 (x)(y)" # (y)#2, (x!y)F (x!y)F 1 h #2 ¹ (x)(y)"! (y)R R I J (x!y) IJ 3

(6.2.8)

with c"1. Note that in this special case c"c"1, but they are in general not equal. The generalization to the case of n free massless scalar "elds G, i"1, 2,2, n, is straightforward, where now c"c"n, h"2 and " GG. G Another familiar example of a four-dimensional conformal "eld theory is the free massless fermionic "eld (in Euclidean space), x. !y. 1 , L" (M R. !R M  ), (x)M (y)" I I (x!y) 2 1 ¹ " (M  R #M  R !R M  !R M  )!g L . J I I J J I IJ IJ 2 I J

(6.2.9)

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The correlator, the symmetry and conservation of ¹ and the parity determine that the form of the IJ OPE of ¹ 's should be of the form [116], IJ 1 1 X ¹ (x)¹ (y)"!c IJMN IJ MN (x!y) 360 1 1 # J (y) [ R R R #(  )]#[  ] #2 IM?@ J N ? 4 @ (x!y)

(6.2.10)

with c"6. The -term involves the axial vector current operator J "M   . This is required by I  I the conservation of parity since the parity odd tensor  appears in the OPE. These two simple IJHM examples show that the non-closure of the OPE of ¹ is a general feature of four-dimensional IJ conformal "eld theory and that the OPE should be the form of (6.2.6) and (6.2.8) with c, c and h generic. Now we turn to the supersymmetric case. There is a long known four-dimensional superconformal "eld theory, N"4 supersymmetric Yang}Mills theory. There also exist some N"2 superconformal gauge theories with certain matter "eld contents. They all have identically vanishing beta functions, and their "rst central charges do not receive any higher-order quantum corrections. Thus to reveal the special features of superconformal "eld theory we need to consider an N"1 supersymmetric gauge theory in the IR "xed point. Here we choose a general classical (fourcomponent form) Lagrangian including both supersymmetric QCD (3.1.6) and a cubic superpotential ="> QPQQQR/6 [48], PQR 1 1 1 L" (F )# M D. #(DI)RD # M D.  IJ I 2 2 4 #i(2g[M ?夹P(¹?)Q (1! ) !M P(1# )(¹?) Q ?] P  Q  P Q 1 ! [M P(1! )> RQ#M P(1# )>夹 夹RQ]  PQR  PQR 2 1 1 # g[HP(¹?) Q ]# > >HPKLQRH H , P Q K L 4 PQR 2

(6.2.11)

where  and  are the four-component form of the quark super"eld QG and the #avour indices are P suppressed. Since the energy-momentum tensor ¹ , the R-current R and the supersymmetry IJ I current lie in the same supermultiplet, it is enough to work out the OPE of the lowest component of this supermultiplet, i.e. the R -current, R (x)"M ?  ?/2!M   /2. Then the OPE of the  I I  I  whole supermultiplet can be obtained through supersymmetry transformations. Note that at the IR "xed point the R -current is identical to the anomaly-free R-current, R (x). The dimension and  I conservation of the R-current require that the OPE of R (x) should be [48] I c 2 c 1 (R R !䊐 ) # (y)(R R !䊐 ) #2 , R (x)R (y) " IJ (x!y) 9
 I J IJ (x!y)\F I J VW 3
 I J (6.2.12)

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where the new operator (x) is the lowest component of the real super"eld (z) (z"(x, , M )), and is related to the renormalized Konishi operator by (z)"(g, >)K (z) . 

(6.2.13)

(g, >) is a function of the couplings that can be determined from an explicit perturbative calculation [48,38]. It is dimensionless since K is a renormalized operator and carries the power  F of the renormalization scale . Similarly, the OPE of (x)'s should be of the following form [47,38]: c 1 #2 , (x)(y) " VW 16
 (x!y)\F

(6.2.14)

where the omitted parts in (6.2.12) and (6.2.14) denote less singular terms. The second central charge c and anomalous dimension h can be obtained by an explicit perturbative calculation. There are two independent methods [48]. The "rst one is calculating the connected four-point correlation function R (x)R (y)R (z)R (w) in the asymptotic region where I J M N x!y, z!w;x!z, y!w. However, this method cannot give the proportionality function (g, >). The second method is to consider the two- and three-point correlators K (x)K (y)   and R (x)R (y)K (z), K (x) being the lowest component of K (z). Since for a conformal I J    invariant "eld theory, the two- and three-point functions can be "xed up to a constant, the general forms of K (x)K (y) and R (x)R (y)K (z) can be easily written out. Concretely, scale and   I J  translation invariance lead to A , K (x)K (y)"   16
(x!y)

(6.2.15)

where A"c/ due to (6.2.13). The tensor form of R (x)R (y)K (z) is "xed by the conservation I J  of the R-current and the correct transformation properties under inversion, xIPxI"xI/x, since any conformal transformation can be generated by combining inversions with rotations and translations [53]. Thus 1 B R (x)R (y)K (z)" I J  36
 (x!y)\F(x!z)>F(y!z)>F







h 1! I (x!y) 4 IJ



1 h ! 1# I (x!z)I (z!y) , MJ 2 2 IM

(6.2.16)

where I (x)"Rx /RxJ" !2x x /x. In the limit x&y, the most singular term is IJ I IJ I J 1 B 1 R (x)R (y)K (z)& (R R ! 䊐) . I J  IJ (x!y)\F 72
(h!2) (y!z)>F I J

(6.2.17)

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The comparison of (6.2.15) and (6.2.17) with the OPEs (6.2.12) and (6.2.14) leads to the relations B B c(g, >)" , " . (h!2)A (h!2)A

(6.2.18)

A and B can be obtained through an explicit perturbative calculation. We thus "nally get [48]: c"N #2P , Q P

3 h" > >HPQR .
N PQR Q

(6.2.19)

The quantum corrections to the "rst central charge was calculated from a general renormalizable theory containing vector, spinor and scalar "elds in curved space}time [38]. Specialized to the N"1 supersymmetric gauge theory (6.2.11), the result is [48]






1
(g) c" 3N #N #N !P . T Q T g P 24

(6.2.20)

In (6.2.19) and (6.2.20), N "dim G and N "dim ¹ as shown in (6.2.2) and (6.2.3). The gauge beta T Q function
(g) and the anomalous dimensions P of the chiral matter super"elds at the one-loop Q level of model (6.2.11) are the following:





Tr C(¹) 16

(g)"g !3C(g)# , dim G 1
" (> K#> K#> K) , KRP Q KQR P PQR 3! KPQ R 1 16
P " >PKL>H !2g[C(¹)]P , QKL Q Q 2 C(G)?@"f ?ABf @AB, [C(¹)]P "(¹?¹?)P . Q Q

(6.2.21)

Comparison of (6.2.19) and (6.2.20) with the free "eld case, (6.2.2) and (6.2.3), shows that the quantum corrections to c and c are proportional to a combination of anomalous dimension P and P the beta function
(g). As we know, if the conformal symmetry is preserved at quantum level, the beta function and anomalous dimensions must vanish. Hence from (6.2.19) and (6.2.20), the "rst and second central charges c and c are identical to their classical values. Conversely, if both c and c receive no quantum correction, then the theory remains conformally invariant, whereas second relation of (6.2.19) shows that under these conditions the anomalous dimension h remains non-vanishing if we consider the cubic superpotential composed of the chiral super"eld. This means that c and c are independent of the couplings and hence are universal, while the anomalous h is not. The above conclusion is the one-loop result. From the NSVZ beta function (1.1) and the relation between
and the anomalous dimension P given in (6.2.21), we conclude that to all PQR Q orders in the couplings g and > there exists a "xed surface of the renormalization group #ow provided that the matter representation is chosen so that 3dim GC(G)!Tr C(¹)"0 and g, > are

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such that P "0. Therefore, an important feature of four-dimensional superconformal "eld theory Q has been revealed: there exists a space of continuously connected conformal "eld theories and their central charges are universal, i.e. c and c are constants, independent of the couplings in this space. Such quantities are called invariants of a four-dimensional superconformal "eld theory. The third quantity contained in the OPE, the anomalous dimension h is, as stated above, not an invariant of a four-dimensional superconformal "eld theory, it actually manifests the inequivalence of the continuously connected four-dimensional superconformal "eld theories in this space. These may be the essential features of a superconformal "eld theory in four dimensions [48]. 6.3. Possible existence of a four-dimensional c-theorem The c-theorem was proposed by Zamolodchikov in the context of two-dimensional conformal quantum "eld theory [37]. Its main idea is to de"ne an appropriate function c(g) on the space of the couplings, which monotonically decreases along the trajectory of the renormalization group #ow from the UV region to the IR one. At the "xed point g"g of the beta function,
(g )"0, the D D c function c(g ) should be equal to the central charge of the Virasoro algebra of the resulting D two-dimensional conformal "eld theory. Since this theorem can show how the dynamical degrees of freedom are lost along the trajectory of the renormalization group #ow from the UV region to the IR region, a theorem, if it exists in four dimensions, will be quite helpful to understand the dynamical mechanisms of some non-perturbative phenomena such as con"nement and chiral symmetry breaking. The search for a four-dimensional c-theorem began immediately after the invention of the two-dimensional c-theorem. It was "rst suggested by Cardy that a c-function in the four-dimensional case should depend on the expectation value of the trace of the energymomentum tensor of a quantum "eld in a curved space}time, integrated over a 4-dimensional sphere S with constant radius [40]



C"N

1

¹I (g dx . I

(6.3.1)

Here N is a positive numerical factor and its value depends on the normalization of the c-function. In a curved background space}time, the trace ¹I  takes the following general form: I ¹I "!aG#bF#d䊐R#eB , I

(6.3.2)

where G"1/4(IJNM R?@ )"(RI ) is the Euler topological number density, R is the Riemann ?@AB IJ IJHM scalar curvature, F"CIJNMC with C the Weyl tensor, which in a general n-dimensional IJNM IJNM space}time can be expressed in terms of the Riemann curvature, 2 (g R !g R !g R #g R ) C "R ! IM JN JN MI JM NI IJNM IJNM n!2 IN JM 2 (g g !g g )R . # IM JN (n!1)(n!2) IN JM

(6.3.3)

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The last term in (6.3.2) represents the contribution from other external background "elds. For example, in a background gauge "eld A? , B"F? F?IJ [39]. The coe$cients a and b in (6.3.2) have I IJ an universal meaning and they have been calculated [41], 1 (N #11N #62N ) , a"   360(4
)  1 (N #6N #12N ) . b"   120(4
) 

(6.3.4)

One can see that the coe$cient b is just the central charge c given in (6.2.1). The coe$cient d has no universal meaning and its value can be de"ned at will by introducing a local counterterm proportional to the integral R, so it is renormalization scheme dependent coe$cient. A four-dimensional c-theorem was "rst attempted in the S;(N ) gauge theory coupled to A N massless Dirac fermions in the fundamental representation by choosing the coe$cient a of the D Euler number density to be the c-function [40]. If the #avour number N is su$ciently small, the D theory is asymptotically free and g"0 is the UV "xed point. At gP0, the particle spectrum contains N!1 gauge bosons and N N fermions. In the low energy case, the theory will be A A D strongly coupled, gPR is the IR "xed point. The chiral symmetry S; (N );S; (N ) breaks * D 0 D to S; (N ) due to quark condensation, yielding N !1 Goldstone bosons. Therefore, at the UV 4 D D "xed point 1 [62(N!1)#11N N ] , c "lim c(g)" A A D 34 360(4
) E

(6.3.5)

and at the IR "xed point 1 c " lim c(g)" (N !1) . '0 360(4
) D E

(6.3.6)

Eqs. (6.3.5) and (6.3.6) show that the requirement of a c-theorem, c !c '0, is not satis"ed 34 '0 identically. Later it was proposed by Osborn et al. [41] that one can directly choose a coe$cient a in the trace anomaly as a c-function even away from the "xed point and modify it to satisfy the renormalization group #ow equation. This scheme is an exact analogue of Zamolodchikov's procedure in two dimensions. This c-function can reduce to the coe$cient a of the trace anomaly when approaching the critical point [39]. However, the monotonically decrease of the c-function along the trajectory of renormalization #ow cannot be explicitly shown. Another attempt was to de"ne a c-function by using the spectral representation of the two-point correlator of the energy-momentum tensor and constructing a reduced spectral density for the spin-0 intermediate state [42]. The concrete steps are as follows. First work out the spectral representation of two-point correlator of energy-momentum tensors in n-dimensional (Euclidean)

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space}time, ¹ (x)¹ (0)"¹ (x)¹ (0) #¹ (x)¹ (0) IJ NM IJ NM Q IJ NM Q "A

L









d c()  (R)G(x, )# IJNM







d c()  (R)G(x, ) , IJNM

(6.3.7)

where 2
L A " , L 2L\(n/2)(n#1)(n!1)

(6.3.8)

and s"0, 2 denotes spin.



G(x, )"




1 dLx e NV  L\ " K (x) L\ (2
)L p# 2
2
x

(6.3.9)

is the two-point correlator of the "elds involved. Lorentz covariance and the conservation of the energy-momentum tensor determine that there are only two possible Lorentz structures for the intermediate states: s"0 and s"2. The tensors Q are, respectively, IJNM





n!1 1 (# # ## # ## # )!# # ,  (R)" IN JM IM NJ IJ MN IJ MN IJNM 2 (n!1) 1 # , # "R R !䊐 .  (R)" IJ I J IJ IJNM (n) IJ NM

(6.3.10)

Away from criticality, owing to the UV divergence, the spectral density must be scale dependent, cQ()"cI(,
). The analysis shows that the candidate for c-function should be the `reduceda spin-0 spectral density [42], c(,
)"c(,
)/L\, and this indeed satis"es (6.3.11) lim c(,
)"(c !c )(),%c() . 34 '0  This c-function candidate indeed monotonically decreases along the trajectory of the renormalization group #ow from short distance to short distance and becomes stationary at the "xed point, but its physical meaning at the "xed points is not clear. It has only been checked that in free scalar and spinor "eld theories, c(,
) coincides with the coe$cients a of the trace anomaly (6.3.2). The above attempts at establishing a four dimensional c-theorem imply that if the c-function exists it will reduce to the coe$cients a or b of the trace anomaly at the "xed points. The new non-perturbative results make it possible to test a c-theorem in four-dimensional supersymmetric gauge theory [45]. The introduction in Sections 3.4, 4.1 and 4.2 shows that the particle spectrum of N"1 supersymmetric QCD in the IR region can be deduced, hence a theorem can be explicitly checked for most of the ranges of N and N . The analysis supports the existence of D A a four-dimensional c-theorem [45]. At the UV "xed point the theory is a free theory with a particle spectrum composed of N!1 vector supermultiplets and 2N N scalar supermultiplets. Since for A D A


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a free theory the c-function reduces to the sum of the central charges carried by the free "elds, one has in the UV "xed point c "(N!1)c #2N N c , 34 A 4 D A 1

(6.3.12)

where c and c are central charges corresponding to the scalar and vector supermultiplets. At the 1 4 IR "xed point, the low-energy dynamics of the theory depends heavily on the relative number of N and N and one must perform the analysis according to di!erent values of N and N . First, D A D A considering the N "0 case, the low-energy pure supersymmetric Yang}Mills theory is strongly D coupled, the gluons and gluoninos condensate into massive colour singlets and hence they develop a mass gap [51], so the IR theory contains only the vacuum state. Consequently, c "0 . '0

(6.3.13)

For 0(N (N , the dynamically generated superpotential erases all the vacua and the particle D A spectrum is not clear. For N "N , there exists a smooth moduli space described by the D A expectation values of N meson supermultiplet operators MG , the baryon B and antibaryon D H super"eld operators BI with the constraint (3.4.52), det M!BBI "
,A , so that at the IR "xed point, there are N #2!1 massless scalar super"elds, since the quantum #uctuations must satisfy D the constraint. Thus we get c "(N #1)c . '0 D 1

(6.3.14)

For N "N #1, the quantum moduli space has singularities associated with additional massless D A particles. The maximum number of massless particles is at the origin of the moduli space, there are N free massless mesons and 2N free massless baryons. Their contribution to the central charge is D D c "(N #2N )c . '0 D D 1

(6.3.15)

In the range N #24N 43/2N , the low-energy theory is e!ectively described by a magnetic A D A S;(N !N ) gauge theory with N magnetic quarks, N magnetic antiquarks and N mesons. All D A D D D these particles are free at the IR "xed point and hence the central charge is c "[(N !N )!1]c #[2N (N !N )#N ]c . '0 D A 4 D D A D 1

(6.3.16)

In the range 3/2N 4N 43N , the IR region is described by a superconformal "eld theory, and A D A can be parametrized equivalently by the magnetic or electric variables. However, both versions of the theory are interacting theories, the one-loop results (6.3.2) and (6.3.4) for the trace anomaly are not enough to extract the central charge c . A technique of computing exactly the trace anomaly '0 and hence c for these interacting superconformal "eld theories will be introduced in next section. '0 In the range N 53N , the theory ceases to be asymptotically free, the particle spectrum at the UV D A "xed point is not clear and hence the central charge at the UV "xed point cannot be calculated. The above known central charges at the IR and UV "xed points all give c !c '0 . 34 '0 This hints at the possible existence of a c-theorem for a supersymmetric gauge theory.

(6.3.17)

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More explicit and concrete evidence for the existence of a four-dimensional c-theorem was provided by calculating exactly the renormalization group #ows of the central charges of N"1 supersymmetric QCD in the conformal window 3/2N 4N 43N [36]. The explicit formula for A D A the #ows of the trace anomaly coe$cients were given. It was shown that the coe$cient a of the Euler number density in the trace anomaly (6.3.2) is always monotonically decreasing a !a '0. This is a strong support for the existence of a four-dimensional c-theorem in N"1 34 '0 supersymmetric gauge theory. The following section will give a detailed introduction. 6.4. Non-perturbative central functions and their renormalization group yows A quantitative investigation of the existence of a four-dimensional c-theorem was made by working out the explicit non-perturbative formula for the renormalization group #ow of the central functions. The central functions are the central charges away from the conformal criticality [44]. If an asymptotic gauge theory has a non-trivial infrared "xed point, at which the theory becomes an interacting conformal "eld theory, then the quantum "eld theory away from criticality can be regarded as a radiative interpolation between a pair of four-dimensional conformal "eld theories. To extract certain non-perturbative information about this quantum "eld theory, one should "rst identify relevant physical quantities and observe their renormalization group #ow from the UV "xed points to the IR ones. As stated in previous sections, at the "xed points, the c-functions coincide with the central charges. These central charges are the coe$cients of the leading terms in the operator product expansion of the various conserved quantities in the "xed points. In the conformal window 3N /2(N (3N of N"1 supersymmetric QCD, the NSVZ
-function A D A shows that the theory has a non-trivial IR "xed point, where the theory admits an equivalent dual magnetic description but with strong/weak coupling exchanged. In the following, we shall review the derivation of the renormalization group #ow of the non-perturbative central functions associated with the various conserved quantities developed in Ref. [36]. The explicit non-perturbative formula for the central functions shows that the appropriate candidate for the c-function should depend on the coe$cient of the Euler term in the trace anomaly. In a supersymmetric gauge theory, several facts help to determine non-perturbatively the renormalization group #ow of the central functions. First, the energy-momentum tensor ¹ (x) and IJ the R -current R (x) lie in the same supermultiplet, so do the trace anomaly ¹I and the  I I anomalous divergence of the R -current, R RI . This fact makes the coe$cients of the trace  I  anomaly and R RI relevant. Secondly, if one introduces external source "elds for the #avour I  current j (x) and the energy-momentum tensor ¹ (x), the trace anomaly has a close relation with I IJ the two-point correlators  j (x) j (y) and ¹ (x)¹ (y) and their central charges. Finally, the I J IJ HM anomalous divergence of the R -current can be exactly calculated at an infrared "xed point  through 't Hooft anomaly matching. As shown in Eq. (6.1.10), the R -current can be combined with  the Konishi current to an (internal) anomaly-free R-current, and the combination coe$cient of the Konishi current is just the numerator of the NSVZ
-function. Thus at the IR "xed point the R -current will coincide with the (internal) anomaly-free R-current, whereas the 't Hooft anomalies  for this internal anomaly-free R-currents can be calculated in the whole energy range from only the one-loop triangle diagram. Therefore, the 't Hooft anomalies of the R -current can be exactly  determined at the IR "xed point. In the UV region, g"0 must be the UV "xed point due to asymptotic freedom, so all the anomalies can be computed in the context of a free "eld theory. We

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319

are going to show the non-perturbative derivation of the renormalization group #ow of the central functions. The N"1 supersymmetric QCD has the anomaly-free global symmetry S; (N ); * D S; (N );; (1);; (1). The fermionic parts of #avour currents corresponding to S; (N ); 0 D 0 * D S; (N );; (1) given in Section 3 are rewritten in four-component form [36], 0 D j"M  (1! )t, jI "IM  (1! )tM I ,  I  I  I  I 1 (M   !MI   I ) . j " I  I  I 2N A

(6.4.1)

If written in two-component form they are the M component of the current super"elds QRtQ, QI tM QI R and (QRQ#QI QI R), respectively; t and tM  being the generators of S;(N ) in the fundamental D and conjugate fundamental representation. The anomaly-free R-symmetry current is the combination (6.1.10) of the anomalous Konishi current and the R -current, and the fermionic parts of their  four-component forms are K "M   #MI   I , R "M ?  ?!(M   #MI   I ) .  I  I  I   I  I  I  I 

(6.4.2)

Note that in (6.4.1) and (6.4.2) the #avour and colour indices carried by the quarks are suppressed. Let j denote one of the #avour currents listed in (6.4.1). The conservation of the current, the I dimensional analysis and the renormalization e!ects restrain the two-point correlator of j to be of I the form b[g(1/x)] 1 (R R !R ) .  j (x) j (y)" IJ I J x (2
) I I

(6.4.3)

Since j is conserved at quantum level, it has no anomalous dimension. As a consequence, the I Callan}Symanzik equation





R R  #
(g)  j (x) j (y)"0 I J R Rg

(6.4.4)

and the
-function
[g()]" dg()/d determine that the function b depends only on the running coupling g(1/x). At the UV and IR "xed points of the renormalization group #ow, g and g , the 34 '0 following limits exist: b

34

,lim b[g(1/x)]"b[g ] , 34 V

b , lim b[g(1/x)]"b[g ] . '0 '0 V

(6.4.5)

To determine the renormalization #ow b !b , we "rst consider the correlation function 34 '0 (6.4.3) to any order in perturbation theory, and then make reasonable arguments to extract the all orders result. In perturbation theory, one unavoidable result is the emergence of UV divergences.

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To regulate all the sub-divergences contained in b[g(1/x)], we "rst expand the function b[g(1/x)] to any "nite order of g, b[g(1/x)]"  b [g()]tL, t,ln(x) , L LY where b [g()] is a polynomial in g(). Evaluating Eq. (6.4.6) at x"1/ gives L b[g()]"b [g()] . 

(6.4.6)

(6.4.7)

Eq. (6.4.3) and the Callan}Symanzik equation (6.4.4) yield 

Rb[g(1/x)] "0 , R


(g)

db (g) L #(n#1)b (g)"0 , L> dg

b


(g) db (g) L (g)"! . n#1 dg

L>

(6.4.8)

Eq. (6.4.8) shows that all b (g) with n51 can be expressed in terms of
(g), b (g) and its derivatives. L  However, when inserting (6.4.6) into (6.4.3), one can easily see that at short distance tL/x is too singular to have a Fourier transform in momentum space. This is actually the re#ection of the UV divergence in coordinate space. With a newly developed regularization method in coordinate space called di!erential regularization [119], the overall divergence near x"0 can be regulated as follows [36,44]: n! L 2ItI> [ln(x)]L "! R  !a (x) . L 2L> (k#1)!x x I The fully regulated form factor of correlator (6.4.3) thus becomes



(6.4.9)



b[g(1/x)] n! L 2ItI> "! b [g()] R  #a (x) . (6.4.10) L L x 2L> (k#1)!x L I Consequently, the scale derivative of b[g(1/x)]/x can be written as the sum of the local contribution of the k"0 term plus a non-local term proportional to
(g) due to Eq. (6.4.8),

 

F(x) R b[g(1/x)] "2
bI [g()](x)#
[g()]R ,  x x R

(6.4.11)

where n! bI [g()]" b [g()] , L 2L L

(6.4.12)

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321

and F(x) is the sum of all the terms with k51 in the scale derivative. In particular, with Eq. (6.4.8), the scale derivative of bI [g()] yields a di!erential equation for bI [g()] [36],
(g)

dbI (g) #2bI (g)"2b(g) . dg

(6.4.13)

Eq. (6.4.13) means that the functions b[g()] and bI [g()] coincide at the "xed points of the renormalization group #ow. This result is an important step towards the non-perturbative determination of the renormalization #ow of the b-function. It should be emphasized that Eqs. (6.4.8)}(6.4.13) were derived within perturbation theory, but they can be regarded as non-perturbative results [36]. The function bI [g()] turns out to be the coe$cient e of the external "eld part of the trace anomaly (6.3.2). This can be easily shown by writing down the generating functional for the current correlation function (6.4.3),



e\ I " [d]e\1 P > V HIV I V ,

(6.4.14)

where  denotes all the "elds appearing in the functional integral including the ghost "elds associated with gauge "xing. B is the external "eld coupled to the #avour current j listed in (6.4.1). I I Since the action of the scale derivative is equal to the insertion of the trace of the energymomentum tensor with zero momentum, i.e. the insertion of dx ¹I [101], I R R , (6.4.15) R d?"¹? "
(g) " ? ? R Rg we have 



 



R  3N !N (1!) 1 D e\ " [d]e\1 P > V HIV I V dz ! A (F? )# q(B ) , IJ IJ R 32
 4

(6.4.16)

where the general form of the trace anomaly of N"1 supersymmetric gauge theory plus an external anomaly was used: 1 3N !N (1!) D (F? )# q(B ) , ¹I "! A IJ IJ I 4 32


(6.4.17)

q being a coe$cient needing to be determined. On the other hand, the scale derivative of the #avour current correlator (6.4.3) gives

 









  R R e\   j (x) j (y)" j (x) j (y) dz I (z) " J I J I R iB (x) iB (y) R I I J  3N !N (1!) 1 D " [d]e\1 P > V HIV I V dz ! A (F? )# q(B ) IJ IJ B (x)B (y) 32
 4 I J 3N !N (1!) D "q(R R ! 䊐)(x)! A J (x) J (y) dz(F? ) . (6.14.18) I J IJ I J IJ 32


  







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Comparing the above equation with (6.4.11), one can immediately identify the coe$cients of their local terms up to contributions O(
[g()]) which will vanish at the "xed points, 1 bI [g()]#O(
[g()]) . q" 8


(6.4.19)

The renormalization group #ow of bI [g()], can be computed from the anomalous divergence R RI , which lies in the same supermultiplet with the trace anomaly and hence their coe$cients I  should be identical. In particular, as shown in Eq. (6.1.10), the RI -current can be combined with the  Konishi current to form an (internal) anomaly-free R-current. The (external) anomalous divergence R RI can be calculated exactly from the one-loop triangle diagram R (x) j (y) j (z). Note that I I J M 't Hooft anomaly matching holds only for the correlator R (x) j (y) j (z), not for I J M R (x) j (y) j (z) or K (x) j (y) j (z). This is because R (x) and K (x) have internal anomalies, I J M I J M I I so in the higher-order triangle diagrams, there will arise so-called `rescattering graphsa [114] containing an internal triangle diagram in which the axial vector current will communicate with a pair of gluons due to the internal anomaly coming from the sub-fermionic triangle diagram. This will produce higher-order non-local contributions to the chiral anomaly. In view of this, the expectation values of the anomalous divergence of R , R and K in the presence of the external I I I "eld B coupled to the #avour current j are as the following: I I 1 sBIJBI , RIR , IJ I 48
 1 bI [g()]BIJBI #2 , RIR "! I IJ 48
 1 RIK "! kI [g()]BIJBI #2 . I IJ 48


(6.4.20)

The omitted terms in RIR  denote the non-local contributions proportional to the internal I anomaly
[g()]F FI IJ. There is a similar non-local contribution to RIK , which will cancel the IJ I corresponding terms of RIR  in the linear combination (6.1.10) and leads to RIR . In fact, I I these non-local contributions are irrelevant for the following analysis since their local terms are of the order O(
[g()]); the quantity s is a constant independent of the renormalization scale. The combination of (6.1.10) and (6.4.20) implies that the external anomaly coe$cients satisfy the relation






3N bI [g()]# 1! A ![g()] kI [g()]"!s . N D

(6.4.21)

This relation holds along the whole trajectory of the renormalization group #ow since s is independent of the renormalization scale. For an asymptotically free supersymmetric gauge theory, g"0 is the UV "xed point and the anomalous dimension (g ) vanish at the UV "xed point. 34 Consequently, bI and kI can be calculated from the one-loop triangle diagrams R JJ and 34 34 

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323

KJJ in the context of a free "eld theory. The coincidence of Eq. (6.4.21) at scale  and in the UV limit gives






3N bI [g()]"b #[g()]kI ! 1! A ![g()] (bI [g()]!kI ) . 34 34 34 N D

(6.4.22)

As for the IR aspect, in the conformal window 3N /2(N (3N , the theory #ows to a non-trivial A D A "xed point g夹 . The vanishing of the NSVZ
-function gives the exact infrared limit value of the anomalous dimension, 3N  "1! A . '0 N D

(6.4.23)

Then, at the IR "xed point, Eq. (6.4.22) becomes: b !b " kI . '0 34 '0 34

(6.4.24)

Furthermore, based on the observation that the gaugino does not contribute to the #avour current correlator, and that the quark and antiquark contribution to the combination R #K cancels, I I we obtain in the UV limit, RIR #RIK "0 . I I

(6.4.25)

This equation together with (6.4.20) yields b

34

"!kI . 34

(6.4.26)

The calculation of the one-loop RIR (x) j(y) j (z) gives b "2N /N . Eqs. (6.4.23), (6.4.24) and I J M 34 D A (6.4.26) lead to b "6. The total renormalization group #ow of the central function b from the UV '0 limit to the IR "xed point is thus obtained, b

34






N !b "6 D !1 . '0 3N A

(6.4.27)

One can easily see that this #ow is always negative in the whole conformal window, 3N /2(N (3N . This contradicts the c-theorem and hence the b-function is excluded from being A D A a candidate for a c-function. In the magnetic theory, the four-component form of the #avour currents corresponding to the global symmetry S;(N ) ;S;(N )  ;; (1);; (1) is 0 DO DO jI  "M  (1! )tM  , jI  "MI   (1! )tI   O OI  O I  O OI  O I 1 jI " (M    !IM    I  ) . I N !N  O I  O  O I  O D A

(6.4.28)

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The anomaly-free magnetic R-current is given by the combination (6.1.15), where the fourcomponent form of the Konishi current and the original R-current are K "KO #K+; KO "M    #MI    I  , I I I I  O I  O  O I  O K+"M    , RI "M ?  ?!K . I  + I  + I  I  I

(6.4.29)

The #avour interaction superpotential W "fqPGM q H makes the analysis of the identi"cation of D GH P the external trace anomaly coe$cient e with the central function b quite complicated, since now b is a function of both g(1/x) and f (1/x). However, the same analysis as in the electric theory shows that they indeed coincide at the "xed points [36]. The derivation of the renormalization group #ow of the central function bI in the magnetic #avour current correlator is similar to the electric theory, and the di!erence is only in the Konishi current part. The combination (6.1.15) and the matrix elements of the external anomaly equations 1 s BIJBI , RIRI " IJ I 48
 1 RIRI "! bI [g(), f ()]BIJBI #2 , I IJ 48
 1 kI O[g(), f ()]BIJBI #2 , RIKO "! IJ I 48
 1 RIK+"! kI +[g(), f ()]BIJBI #2 I IJ 48


(6.4.30)

as well as the scale independence of s , lead to the relation bI "bI # kI O # kI +#(2 # ) (kI +!kI +) 34 O 34 + 34 O + 34





3(N !N ) A (kI O!2kI +!kI O #2kI +) , ! 1! D 34 34 N D

(6.4.31)

where, as in the case of the electric theory, a quantity with subscript UV means that it is evaluated from the lowest-order triangle diagrams at high energy, while the other quantities are de"ned at an arbitrary renormalization scale . At the IR "xed point, the vanishing of the
-functions for both the gauge magnetic coupling and the Yukawa coupling gives 3(N !N ) 1 A . '0"! '0 "1! D + O N 2 D

(6.4.32)

The above anomalous dimensions and the relation (6.4.31) lead to the renormalization group #ow bI !bI "'0kI O #'0 kI + . + 34 '0 34 O 34

(6.4.33)

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325

Eq. (6.4.33) is the non-perturbative formula for the #ow of the central function of the #avour currents in the magnetic theory. For the baryon number current listed in (6.4.28), the one-loop triangle diagram  jI j j  and RIRI #RIK "0 in the UV limit yield IJM I I bI

34

2N A , "!kI O " 34 N !N D A

kI +"0 . 34

(6.4.34)

Eqs. (6.4.33) and (6.4.34) determine that b "6. Therefore, the #ow of the central function of the '0 baryon number current is





N 2N !3N D A . "2 D bI !bI "6 1! '0 34 3(N !N ) N !N D A D A

(6.4.35)

This #ow is again positive throughout the whole conformal window of the magnetic theory, 3(N !N )/2(N (3(N !N ) (i.e. 3N /2(N (3N ). Moreover, the central functions in the D A D D A A D A electric and magnetic theories have the same IR values, b "bI "6. The above results are '0 '0 actually an inevitable outcome of the electric}magnetic duality in the conformal window. They can be regarded as support for Seiberg's electric}magnetic duality conjecture. Having veri"ed explicitly that the central functions of the #avour currents are not suitable candidates for the c-function, we turn to the gravitational central functions [44]. These central functions are contained in the operator product expansion of the energy-momentum tensor ¹ . IJ The conservation of ¹ implies that the two-point correlator of the energy-momentum tensor IJ should be of the form [44]: 1 c[g(1/x)] f [ln(x), g(1/x)] #

, ¹ (x)¹ (0)"! IJ MN IJ MN 48
 IJMN x x

(6.4.36)

where
"R R ! 䊐 and is the transverse traceless spin 2 projective tensor, the n"4 IJ I J IJ IJMN case given in Eq. (6.3.10) "2# # !3(# # ## # ) , IJMN IJ MN IM JN IN JM I " M"0, RI "RJ "RM "RN "0 . IMN IJM IJMN IJMN IJMN IJMN

(6.4.37)

With a non-local rede"nition [44] 1 1 T "¹ (x)#
¹M , IJ IJ 3 䊐 IJ M

(6.4.38)

one can get the two-point correlator of the energy-momentum tensor with an improved trace property and a simple correlator: c[g(1/x)] 1 . T (x)T (0)"! IJ MN 48
 IJMN x

(6.4.39)

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As for the #avour current, the central function c[g(1/x)] is connected to the coe$cient of the Weyl tensor part of the gravitational anomaly. For N"1 supersymmetric QCD, we introduce the background metric g (x) for the energy-momentum tensor ¹ (x) and the external ;(1) gauge "eld IJ IJ < for the R -current. According to the general form (6.3.2) of the trace anomaly in the presence of I I external "elds, the trace anomaly at the "xed point is a c c C CIJMN! RI RI IJMN# <
(6.4.40)

where < "R < !R < is the "eld strength of < . The coe$cients of (C ) and (< ) are IJ I J J I I IJMN IJ identical since RIR and the trace anomaly are in the same supermultiplet. The coe$cient of the I Euler number density is an independent constant [39]. The renormalization group #ow of the c-function and the a-function can be determined using a similar technique as in the #avour current case. First, the c- and a-functions in the UV limit can be calculated in a free "eld theory due to the asymptotic freedom. In a free supersymmetric gauge theory with N vector and N chiral supermultiplets, the central function c at the UV "xed point is T Q given by (6.2.2) and the a-function is [39,118], a

1 " (9N #N ) . T Q 34 48

(6.4.41)

For N"1 supersymmetric QCD, N "N!1 and N "2N N . O! criticality there will arise T A Q A D internal trace anomaly terms in ¹I , but they are proportional to the
function for gauge I coupling,
[g()], and hence give no contribution to the #ow. The central charges thus depend on the running gauge coupling, c"c[g()], a"a[g()]. The next step is still to resort to the R anomaly since the coe$cient of the RIR is connected to the trace anomaly coe$cient. In I I particular, RIR can be exactly evaluated in the IR region. In the presence of the background I gravitational "eld and the external gauge "eld coupled to the R -current, the external "eld part of  the RIR anomaly is of the following general form: I RIR "(uc#va)RIJMNRI #(rc#sa)
(6.4.42)

where u, v, r and t are universal (or model independent) coe$cients, and they can be calculated in the UV limit from free "eld theory due to asymptotic freedom. With the observation that gauginos and quarks have opposite R-charges, #1 and !1, respectively uc #va and rc #sa can 34 34 34 34 be evaluated from the triangle diagrams R ¹¹ and R R R , respectively:     3N !N Q , uc #va " T 34 34 24
 27N !N T Q . rc #sa " 34 34 24


(6.4.43)

M. Chaichian et al. / Physics Reports 346 (2001) 89}341

With the values of c

34

327

and a given in (6.4.41), it can be easily found that 34

c !a 5a !3c 34 RIJMNRI 34
(6.4.44)

The same procedure as for deriving Eq. (6.4.19) shows that o! criticality the coe$cient c [g()] of (= ) and (< ) in the trace anomaly (6.4.40) is related to the central function c[g()] as IJMN IJ c [g()]"c[g()]#O[
(g)] .

(6.4.45)

So they coincide at the "xed points of the renormalization group #ow. The anomaly coe$cients of the terms in RIR appear as the combination c!a and 5a!3c. The renormalization group #ow I of c [g()]!a[g()] and 5a[g()]!3c [g()] can be determined in the same way as deriving Eq. (6.4.27). First, the triangle axial gravitational anomalies R¹¹, R ¹¹ and K¹¹ give  1 s IJHMR RNB , R ((gRI)" IJNB HM I 12
  1 R ((gRI )" (c [g()]!a[g()])IJHMR RNB #2 , I  IJNB HM 12
 1 R ((gKI)" k[g()]IJHMR RNB #2 , I IJNB HM 12


(6.4.46)

where the omitted are the non-local O[
(g())] terms coming from the internal anomaly. In the UV limit, the quantities k[g()] and c [g()]!a[g()] can be calculated exactly from the one-loop triangle diagrams, R ¹¹ and K¹¹ [36],  34 34 k

34

1 1 "! N "! N N , Q 8 D A 16











1 1 1 2 c !a "! N ! N "! N!1! N N . 34 34 T 3 Q A 16 16 3 D A

(6.4.47)

The scale independence of s and the combination Eq. (6.1.10) lead to  c [g()]!a[g()]"c

1 !a # [g()]k 34 34 34 3






1 3N ! 1! A ![g()] (k[g()]!k ) . 34 3 N D

(6.4.48)

The non-perturbative formula for 5a!3c can be obtained from the triangle diagram R R R .    Eq. (6.4.2) shows that the fermionic part of the R -current is contributed by the gaugino, the left and right-handed quarks, all of which are Majorana spinors, and these three contributions have the same form M   /2. The amplitude of the triangle diagram formed by three identical axial I 

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currents composed of Majorana spinors has a Bose symmetry. A calculation in the UV limit yields [120] R 1 R R J (x)J (y)J (z) "!  (x!z)(y!z) I J M 34 Rz 12
 IJHM Rx Ry M H M 16 , C (x, y, z) , 9 IJ C (x, y, z)"C (y, x, z) . IJ JI

(6.4.49)

With Eq. (6.4.49), rewriting the combination (6.1.10) as R "R #(! )K /3 and considering I I '0 I the various Bose symmetric contributions of gaugino and quarks to the anomalous divergence of the triangle R R R , one can write down the following anomaly equations [36]:    R R (x)R (y)R (z)"(5a[g()]!3c[g()])C (x, y, z) , I J M IJ Rz M R R (x)R (y)R (z)"s C (x, y, z) , I J M  IJ Rz M R [R (x)R (y)K (z)#K (x)R (y)R (z)#R (x)K (y)R (z)]"3k [g()]C (x, y, z) , I J M I J M I J M  IJ Rz M R [R (x)K (y)K (z)#K (x)K (y)S (z)#K (x)R (y)K (z)]"3k [g()]C (x, y, z) , I J M I J M I J M  IJ Rz M R K (x)K (y)K (z)"k [g()]C (x, y, z) , I J M  IJ Rz M

(6.4.50)

where the coe$cient s stays constant along the trajectory of the renormalization group #ow since  the R -current is internal anomaly free. The above Bose symmetric anomaly equation and the I combination (6.1.10) yield 5a!3c"s #k (! )#(! )k #  (! )k . '0   '0    '0 

(6.4.51)

In the UV limit, the coe$cients of the triangle anomaly involved in (6.4.50) can be exactly calculated in a free theory,




N  9 N 9 A , k "! N A ,  "0, k " N 34 34 34 16 Q N 16 Q N D D k

9 " N , N "2N N . 34 16 Q Q D A

(6.4.52)

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329

Thus Eq. (6.4.52) becomes 5a

34

!3c

34

"s !k  # k !   k .  34 '0  '0 34  '0 34

(6.4.53)

Eqs. (6.4.51) and (6.5.53) give 5a[g()]!3c[g()]"5a !3c #h , 34 34 h, k !k  #  k  #k [g()](! ) '0 34  34 '0  34 '0  '0 #k [g()](! )#  k [g()](! ) .   '0   '0

(6.4.54)

With Eqs. (6.4.48) and (6.4.54), the non-perturbative formula for the central functions turns out to be c"c

34

#(! )k[g()]# k #h , '0  '0 34  

a"a #(! )k[g()]# k #h . 34  '0  '0 34 

(6.4.55)

Evaluating c[g()] and a[g()] at the IR "xed point, we get the renormalization group #ow of the central charges, 5 1 1 1 c !c "  k #  k !  k #  k '0 34 6 '0 34 2 '0 34 6 '0 34 54 '0 34






N  N NN " A D  9 A #3 A !4 , '0 N N 48 D D 1 1 1 1 !  k #  k a !a "  k #  k '0 34 2 '0 34 2 '0 34 6 '0 34 54 '0 34




NN N "! A D  2#3 A 48 '0 N D



.

(6.4.56)

In deriving (6.4.56) we have used the values listed in (6.4.47) and (6.4.52). In fact, the values of the central functions at both the UV and IR "xed points can be calculated directly. Since the NSVZ
-function vanishes at the IR "xed point, the combination (7.1.10) shows that R and R coincide at the IR "xed point. Because the anomaly-free R has one-loop exact I I I external anomalies and the coe$cients are scale independent, the IR values of the external R anomaly coe$cients must be equal to the UV ones. This fact means I R R R R (x)R (y)R (z) " R (x)R (y)R (z) " R (x)R (y)R (z) . I J M '0 Rz I J M '0 Rz I J M 34 Rz M M M

(6.4.57)

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Using R34"R #(1!3N /N )K with R and K given in (6.4.2) and considering the relative I I A D I I I contributions from the gaugino and quarks, one can "nd [36]




 




N  N  9 9 A A N !N " N!1!2N N . 5a !3c " Q N A A D '0 '0 16 T N 16 D D

(6.4.58)

Calculating R/Rz R (x)R (y)R (z) in the UV limit as in free "eld theory, we obtain: M I J M 5a

34

!3c

34






2 9 N!1! N N . " A 27 A D 16

(6.4.59)

Applying a similar procedure to the axial gravitational triangle diagram ¹¹R, we "nd 1 1 c !a " (N!1#2)" (N#1) ; '0 '0 16 A 16 A






1 2 c !a "! N!1! N N . 34 34 16 A 3 D A

(6.4.60)

The values of the central functions at the "xed points are thus "xed:

 


 

 


N  1 N 1 A #5 " 7(N!1)!9N N 7N!2!9 A c " A A D N A '0 16 16 N D D N  3 3 N A #1 " a " 2(N!1)!3N N 2N!1!3 A '0 16 A A D N A 16 N D D

 

1 1 c " [3(N!1)#2N N ], a " [9(N!1)#2N N ] . 34 24 A D A 34 48 A D A

,

;

(6.4.61)

It can be easily checked that #ow equation (6.4.56) is satis"ed with the above explicit values for the central charges. In a similar way, the #ow of central charges in the dual magnetic theory can be worked out:








3N 1 A 1! c !c " '0 34 24 2N D



N 9 A !6N#6N #N N , A D A D N D

1 3N  A (3N#4N N #3N ) . a !a "! 1! '0 34 A A D D 12 2N D

(6.4.62)

The result given in Eqs. (6.4.56) and (6.4.62) shows that the #ow of the c-function can be both positive and negative. For example, in the electric theory, c !c is negative near the lower edge '0 34 of the conformal window, N &3N /2, but positive near the upper edge, N &3N , while in the D A D A magnetic theory c !c is positive in the entire conformal window. Thus there is no c-theorem '0 34 for this c-function in supersymmetric gauge theory. However, the #ow of the central charge function a always satis"es a !a (0 for both the electric and magnetic theories in the whole '0 34

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range of the conformal window. This fact seems to suggest the existence of an `a-theorema, i.e. a suitable c-function would be the coe$cient of the Euler term of the trace anomaly. In Section 6.3 we have already introduced the partial quantitative result obtained by Bastianelli that both c !c and a !a are negative [45]. Now the explicit non-perturbative formula shows that '0 34 '0 34 the c-theorem is only applicable to the a-function. This coincides with the initial choice made by Cardy [40].

7. Concluding remarks 7.1. A brief history of electric}magnetic duality The status of QCD as a true theory describing strong interaction has been generally accepted for more than 20 yr. Due to the property of asymptotic freedom, its predictions at small distances such as deep inelastic scattering processes can be calculated and coincide with the results of experimental tests. However, its description for the low-energy dynamics of quarks is not clear yet. It is well known that the remarkable strong interaction phenomena at low-energy are colour con"nement and chiral symmetry breaking [121], but the mechanisms for both of them are still not fully understood. Although some phenomenological models have been proposed, the "nal truth should be determined by an exact solution of QCD. However, with the present methods, it is impossible to "nd an exact quantum action of QCD since it is a highly non-linear theory. Perturbative calculations become invalid at low energy due to the strong coupling. Therefore, understanding the non-perturbative dynamics of strong interaction theory is a big problem awaiting to be solved. Duality has provided a possibility to tackle this problem since it relates the strong coupling in one theory to the weak coupling in another theory. The other feature of duality is that it interchanges fundamental quanta in the theory with the solitons of its dual theory. It has long been known that such a duality relation really occurs in two-dimensional relativistic quantum "eld theories. Based on Skyrme's conjecture, Coleman and Mandelstam found that the bosonic sine-Gordon model is completely equivalent to the massive Thirring model [122,123]. The strong and weak coupling in these two theories are exchanged and the solitons of the sine-Gordon theory corresponds to the fundamental fermions of the massive Thirring model. The search for duality in four-dimensional gauge theory has a long history. It was noticed early one, that classical electrodynamics, formulated in terms of "eld strengths, possesses an electric} magnetic duality symmetry, if magnetic charges and currents are introduced as sources. The quantum theory of magnetic charges was found by Dirac [124]. He found the famous quantization condition ensuring the consistency of the quantum mechanics of a magnetically charged particle moving in an electromagnetic "eld. This quantization condition implies that the electric}magnetic duality, if it exists, exchanges strong and weak couplings. However a consistent quantum "eld theory with magnetic monopoles was not found until after more than 40 yr. In 1974, 't Hooft [65] and Polyakov [66] independently found "nite energy classical solutions in the Georgi}Glashow model and that they can be interpreted as monopoles. At large distance, this classical solution behaves as a Dirac monopole. Furthermore, another kind of soliton solutions carrying both electric charge and magnetic charge (dyon) was found by Julia and Zee [69]. In the Prasad}Sommer"eld limit, the explicit analytic solutions for the magnetic

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monopole and the dyon were worked out. In this limit the classical masses of the magnetic monopole and dyon saturate the Bogomol'nyi bound and hence such monopole and dyon solutions are called BPS solutions [74]. A comparison of the classical masses and charges of the BPS monopoles with those of the fundamental quantum particles such as the massive gauge bosons and Higgs particles produced by spontaneously symmetry breaking in the Georgi}Glashow model [74] shows that the whole particle spectrum including the BPS magnetic monopoles is invariant under electric}magnetic duality provided that the BPS monopoles are interchanged with the massive gauge vector bosons [7]. This observation motivated the Montonen}Olive duality conjecture that there should exist a dual description of the Georgi}Glashow model where the elementary gauge particles should be BPS magnetic monopoles and they should form a gauge triplet together with the photon, while the massive magnetic bosons should behave as `electric monopolesa. This conjecture was further reinforced by the fact that two very di!erent calculations for the long-range force between the massive gauge bosons (done by computing the lowest order Feynman diagrams contributed by photon and Higgs particle exchange [7]), and that between the BPS magnetic monopoles (a calculation due to Manton [125]) yielded identical results. Later Witten considered the strong CP violation e!ects (i.e. including a -term) in the Georgi}Glashow model and found that the -term shifts the allowed values of the electric charge in the monopole sector [126]. As a consequence, Montonen}Olive duality was extended from Z to S¸(2, Z) duality  since the -parameter and the coupling constant of the theory can be combined into one complex parameter. However, the Montonen}Olive conjecture su!ers from several serious drawbacks. First, owing to the Coleman}Weinberg mechanism [127], a non-zero scalar potential will be generated by quantum corrections even if the classical potential vanishes, consequently, the classical mass formula will be modi"ed. Thus there is no reason to believe that the electric}magnetic duality of the whole particle spectrum is not broken by radiative corrections through a renormalization of the Bogomol'nyi bound. Secondly, it is well known that the massive gauge bosons have spin one, while the magnetic monopoles are spherical symmetric solutions and should have spin zero. Thus although the mass spectrum is invariant under duality, the quantum states and quantum numbers in two dual theories do not match. In addition, there is the di$culty to test this conjecture since the duality relates weak to strong coupling and we know very little about solutions of four-dimensional strongly coupled theory. Fortunately supersymmetry provides a way to circumvent these problems. The main physical motivation to introduce supersymmetry is to solve the gauge hierarchy problem [128]. To some extent, it prevents the generation of quantum corrections to the mass terms and provides a naturalness to mass relations in quantum "eld theory. In (extended) supersymmetric gauge theories with central charges, the supersymmetry has another e!ect: the Bogomol'nyi bound is a property of some representations of the supersymmetry algebra. Due to the work of Witten and Olive [129], the physical meaning of the central charges was made clear: they are precisely the electric and magnetic charges. Thus supersymmetry protects the Bogomol'nyi bound against quantum corrections, guaranteeing that the bound is true both at the classical and the quantum levels. The main reason for this is that the BPS mass formula is a necessary condition for the existence of the short representations of the supersymmetry algebra. A typical example with a non-vanishing central charge is N"2 supersymmetric Yang}Mills theory, which is obtained by the dimensional reduction of N"1 supersymmetric Yang}Mills theory in six dimensions and

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whose bosonic part is just the Georgi}Glashow model [130]. So the supersymmetric generalization of the BPS magnetic monopole and dyon solutions can be naturally obtained. Observing the massive supermultiplets arising out of higgsing this model, one can see that they are only four-fold degenerate and not sixteen-fold degenerate as usually expected from the representation of supersymmetry algebra realized on massive particle states. This fact indicates that the central charge of N"2 supersymmetry saturates the bound. Nevertheless, this also shows that the states in the short massive supermultiplet have spin 1/2 and 0, but not spin 1, or 1/2 and 1, but not 0 [79]. Thus N"2 supersymmetric Yang}Mills theory is not an appropriate theory admitting Montonen}Olive duality. This obstacle is surmounted in N"4 supersymmetric Yang}Mills theory, which can be obtained through a dimensional reduction of ten-dimensional N"1 supersymmetric Yang}Mills theory [8]. This model also admits the Higgs mechanism and has BPS monopole solutions. The massive vector multiplets naturally saturate the Bogomol'nyi bound. However, in this model, thanks to the larger supersymmetry, the short supermultiplet containing the BPS states and the one containing the massive vector bosons are isomorphic, both of them have spin s"0, 1/2 and 1 states. Further, the N"4 theory possesses another remarkable feature: the
-function vanishes identically [105]. This means that the gauge coupling is a genuine constant and does not run. This property not only makes the relation between the couplings of the electric and magnetic theories generally valid, but also present a "rst non-trivial superconformal "eld theory in four dimensions, since the trace of the energy momentum tensor is measure of conformal symmetry and it is proportional to the
function [22]. This is far from the end of the duality story. After more than a decade, Seiberg and Witten, in two papers [1,2] which already have become classical, based on the general form of the low-energy Wilson e!ective action of N"2 supersymmetric Yang}Mills theory [3], made the conjecture that there exists an e!ective electric}magnetic duality in N"2 supersymmetric Yang}Mills theory. Further using this conjecture they determined the explicit instanton coe$cients appearing in the low-energy e!ective action and the whole structure of quantum vacua. The results of Seiberg and Witten were supported by explicit instanton calculations and hence this duality conjecture is convincing. As a consequence, Seiberg}Witten's work set o! a new upsurge in the search for dualities in quantum "eld theory [131]. In addition, in an exciting development Seiberg found that if the numbers of colours and #avours satisfy 3N /2(N (3N , the N"1 supersymmetric S;(N ) QCD has a de"nite infrared "xed A D A A point, where the theory becomes an interacting superconformal "eld theory. A duality also then arises in the IR "xed point. Moreover, a series of non-perturbative results including those in SO(N ) A and Sp(2n ) gauge theories have been obtained [11,14}16]. N"1 supersymmetric theories have the A possibility for practical physical applications after breaking the supersymmetry. Therefore, a con"rmation of N"1 duality would be remarkably signi"cant. 7.2. Possible application of non-perturbative results and duality of N"1 supersymmetric QCD to non-supersymmetric case To conclude this report, we mention some possible physical applications of Seiberg's N"1 duality. A natural expectation is that we can get some understanding on the non-perturbative aspects of ordinary QCD from the exact results of supersymmetric QCD deformed by breaking the supersymmetry. It was "rst investigated in Ref. [132] how to extend Seiberg's exact results on

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supersymmetric QCD to nonsupersymmetric models by considering the e!ects of soft supersymmetry breaking terms. In the electric theory a soft breaking term is composed of the mass terms of squark "elds and the gaugino: L "!m (HP #HI P  I )#m ??#m夹M ?M ? . E E 1 / /G /PG /G /PG

(7.1)

This term can be rewritten in a super"eld form:





L " d M (QReE4Q#QI e\E4QI R)! d M Tr(=?= ) 1 / E ?



M = ! dM MH Tr(= M ? ) , E ?

(7.2)

where M is a vector super"eld whose D-component equals !m and M (MH) is a chiral E / / E (anti-chiral) super"eld whose F-component equals m (mH). At the low energy, there are many E E possible gauge invariant soft supersymmetry breaking terms that can be built from the meson M and baryons, B and BI . For the case N 4N #1, the soft term (precisely speaking, the "rst order D A expansion of soft supersymmetry breaking terms near the origin of moduli space) was proposed to be [132],



L " d[C M Tr (MRM)#C M (BRB#BI RBI )#CM M(M, B, BI )#h.c.] 1 + / /



!



d M Tr (=?= )#h.c. , E ?

(7.3)

where C , C and CM are normalization constant coe$cients, M(M, B, BI ) is a function + of the composite chiral super"eld and is invariant under the global symmetry S; (N ) * D ;S; (N );; (1) ;; (1). The expectation value Tr(=?= ) in (7.3) should be understood as 0 D 0 ? a combination of the various chiral super"elds appearing in the low-energy e!ective Lagrangian which has the same quantum numbers as Tr(=?= ). The rationale in introducing this soft breaking ? term lies in that S,!Tr(=?= ) exists in the low-energy e!ective Lagrangian of supersym? metric gauge theory [98]. As pointed out in Ref. [132], the soft breaking terms in (7.3) are not the most general terms that can be written down. There are terms of higher order in the "elds, suppressed by powers of
, of the form Tr(MRM)L>/
L, Tr(BRB)L>/
L and Tr(BI RBI )L>/
L. However, (7.3) may be true near the origin of the moduli space M" B"BI "0, thus the vacuum under consideration with the above soft breaking terms is the one near the origin of the moduli space. For the range N 5N #2, as seen in Section 4.1, the D A theory has a dual description and the composite super"eld is e!ectively replaced by the dual magnetic quarks. Correspondingly, near the origin of the moduli space, M"q"q "0, the soft breaking term is LI "CI m Tr(R  )#C m(R #R   ) , 1 + + + + O O O O O O

(7.4)

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where CI and C are normalization coe$cients. With these soft breaking terms at the fundamental + O and composite "eld levels, it is found that in the case of N (N , the standard vacuum of ordinary D A QCD with both con"nement and chiral symmetry breaking can be obtained. However, when N 5N , some strange phenomena arise: there appear new vacua with spontaneously broken D A baryon number for N "N and a vacuum state with unbroken chiral symmetry for N 'N , D A D A These exotic vacua contain massless composite fermions and especially, in some cases dynamically generated gauge bosons. This indicates that it is not straightforward to obtain a complete understanding of non-perturbative QCD starting from supersymmetric QCD. However one encouraging result is that Seiberg's electric}magnetic duality seems to persist in the presence of small soft supersymmetry breaking. In fact, due to the existence of the superpotential W"q MG q H in the dual magnetic theory, the G H Lagrangian (7.4) does not exhaust all the possible soft breaking terms. Another trilinear interaction term (called A-term) is allowed by soft supersymmetry breaking [133] (7.5) ¸I  "h G H , 1 OG +H O where the trilinear coupling h is introduced as a free parameter. Note that this A-term, like the gaugino mass term, breaks the R-symmetry. The e!ects of (7.5) on the phases of the range N 5N #1, the vacuum structure and the fate of duality in the soft breaking N"1 supersymmetD A ric QCD was investigated in Refs. [134,135]. With the inclusion of (7.5), the whole scalar potential containing the soft breaking term is: 1 1 <( ,   ,  )" Tr( RR   )# Tr(  R R#R R    ) O O O O O + + O O + + O O O + k k 2 2 g  # (Tr R ¹I ? !Tr   ¹I ?R )#m Tr(R )#m Tr(R   ) O O O O O O O O O O 2 (7.6) #m Tr(R  )!(h Tr G H #h.c.) , + + + OG +H O where ¹I ? is the generator of the magnetic gauge group S;(N !N ) and g is the gauge coupling D A constant. The third term is the D-term of the magnetic gauge theory. k , k are the normalization O + parameters for q, q and M so that their kinetic terms (KaK hler potentials) take the canonical form [132,133] (7.7) K(q, q , M)"k Tr(qReE 4I q#q e\E 4I q R)#k (MRM) . + O The phase structure can be revealed by analyzing the minimum of the above scalar potential. With the assumption that h is real, the minimum of the potential can be obtained along the diagonal direction

 

 P P " OG G OG 0



i, r"1,2, N !N   P D A P " OG G OG 0 i"N !N #1,2, N , D A D

 G , i, j"1,2, N !N , D A G " +G H +H 0 i, j"N !N #1,2, N . D A D

i, r"1,2, N !N , D A i"N !N #1,2, N , D A D (7.8)

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It is found that the trilinear term plays an important role in the realization of the broken phase and leads to a rich vacuum structure [134,135]. E In the direction  "q and   "0 (or  "0 and   "q), the vacuum expectation value OG OG OG OG  "0. If m(0 (or m (0), the scalar potential will be unbounded from below [134] and +G O O the theory becomes unphysical. If m'0 (or m '0), the scalar potential has the minimum O O <"0 at q"0 and thus the theory is in the gauge and chiral symmetric phase. E In the #at direction of the D-term,  "  "X . A broken phase arises when the solution of G OG OG the expectation value equation






2 2 G[ ]"  !h f ( )! m  "0 +G +G +G + +G k k O +

(7.9)

satis"es the following inequalities: 2 f ( ),  !2h #m#m 40 . +G +G O O +G k O

(7.10)

Otherwise, the theory will be in a (gauge and chiral) symmetric phase. Eq. (7.9) requires that the soft breaking parameters should satisfy 2 h5 (m#m ) , O k O O

(7.11)

having then a non-trivial solution: h#(h!2(m#m )/k h!(h!2(mO#m )/kO O O O . O 4 4 +G 2/k 2/k O O

(7.12)

The su$cient condition for the existence of a broken phase is that the local maximum point of G[ ] should be within the region (7.12) and further at that point the value of G[ ] +G +G should not be negative. This requires







1 m  2m 2 m#m  O ! + h 50 . h# + ! O k 3 k 3k 3 O + +

(7.13)

Depending on the ratio ,2m /(m#m )k /k , the phase structure in these D-#at directions + O O O + can present various patterns [134]. For example, in the phase diagram labelled by (h, (m#m )/2), when "1, the theory has only a chiral symmetry breaking phase and an O O unbroken phase if all m, m and m are positive, whereas when "20, e.g., the theory presents + O O two unbroken phases and two kinds of broken chiral symmetry phases. In addition, in this direction, if all m, m and m are negative, the scalar potential is still unbounded from below O O + and hence the theory becomes unphysical. We still use the 't Hooft anomaly matching to check the survival of duality in the presence of the trilinear term. Since the trilinear term violates the R-symmetry explicitly, it can be checked that

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in the unbroken phase the 't Hooft anomalies S; (N ) and S; (N ); (1) still match as *0 D *0 D in the supersymmetric limit. This seems to imply the existence of Seiberg's duality in this phase after soft supersymmetry breaking with the inclusion of the trilinear term. In the broken phase, the situation will become complicated since a large symmetry breaking takes place. However, there is a simple case where conclusions can be drawn. It should be "rst emphasized that when adding the soft supersymmetry breaking terms we break the global #avour symmetry S; (N !N );S; (N !N ) into S; (N !N !1);; (1);S; (N !N );; (1) breaking * D A 0 D A * D A * 0 D A 0 terms [133]. Assume that in the dual magnetic theory the "rst #avour has soft scalar masses m and m  , di!erent from the other N !1 #avours, m and m  and that only m and m  satisfy O O D O O O O the conditions (7.12) and (7.13) for the broken phase. In this case only the vacuum expectation  do not vanish. Consequently, the following gauge values  "  "X and   + O O symmetry breaking occurs: S;(N !N )PS;(N !N !1) . (7.14) D A D A Furthermore, the other N !1 #avours of the dual magnetic quarks and (N !1) singlet fermion D D components  remain massless. Thus the resulting theory has the global symmetry + S; (N !1);S; (N !1);; (1), under which massless dual quarks,  ,   and the singlet * D 0 D Y O O fermion  transform as (NM , 0, N /(N !N !1)), (0, N ,!N /(N !N !1)) and (N , NM , 0), + D A D A D A D A D D respectively. Now let us consider the corresponding electric theory. The soft breaking Lagrangian consisting only of the mass terms of squarks and gaugino is (7.1). If at the supersymmetry breaking scale, the #avour symmetry is broken in the same way as in the magnetic theory, i.e. the remaining global #avour symmetry is S; (N !1);S; (N !1), then nothing can prevent the appearance of the * D 0 D superpotential = "M Q ) QI . Consequently, a soft breaking term corresponding to this super   potential can be added to (7.1). Especially, a B-term, !M Q ) QI can also arise as a soft breaking  term. Thus the (mass) matrix of the "rst #avour of squarks,   and  I  can be written as follows, / / m  #M !M  M " / . (7.15)  !M mI  #M  / If det(M )'0, the potential minimum corresponds to   " I  "0 and the gauge  / / symmetry S;(N ) remains unbroken. In this case it can be checked that the 't Hooft anomalies A S; (N !1) and S; (N !1); (1) match with those in the magnetic theory, *0 D *0 D S; (N !1) and S; (N !1); (1). This seems to suggest that the N"1 duality remains *0 D *0 D Y after supersymmetry breaking with a trilinear term even in the broken phase. In the case that det(M )(0 and m  #mI  #2M '2M , the squarks   and  I  will /  / /  / acquire the vacuum expectation values and the gauge symmetry is broken to S;(N !1). This case A seems to correspond to the dual magnetic theory with the scalar potential unbounded from below and hence the duality disappears [134]. It was further shown that the trilinear soft breaking term plays an important role in determining the vacuum structure in the cases N 4N #1 [135]. In particular, for the range N "N #1, the D A D A trilinear term is just the #avour interaction among the scalar components of the baryon and mesons,




LI "h G H . D1  G +H



(7.16)

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The chiral symmetry can be broken if h is su$ciently large. This is completely di!erent from the case that the soft breaking Lagrangian is only composed of mass terms of superpartners [132,133], where the chiral symmetry is always preserved for N "N #1. D A Overall, the duality should have physical applications in exploring non-perturbative dynamics, but ahead of us there is still a long way to go.

Acknowledgements The "nancial support of the Academy of Finland under the Project No. 37599 and 163394 is greatly acknowledged. W.F.C is partially supported by the Natural Sciences and Engineering Council of Canada. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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FORCE SPECTROSCOPY ON SINGLE PASSIVE BIOMOLECULES AND SINGLE BIOMOLECULAR BONDS

Rudolf MERKEL Lehrstuhl fuK r Biophysik E22, Physik Department, Technische UniversitaK t MuK nchen, James-Franck-Str. 1, D-85747 Garching, Germany

AMSTERDAM } LONDON } NEW YORK } OXFORD } PARIS } SHANNON } TOKYO

Physics Reports 346 (2001) 343}385

Force spectroscopy on single passive biomolecules and single biomolecular bonds Rudolf Merkel Lehrstuhl fu( r Biophysik E22, Physik Department, Technische Universita( t Mu( nchen, James-Franck-Str. 1, D-85747 Garching, Germany Received August 2000; editor: E. Sackmann

Contents 1. Introduction 1.1. Physiological examples 1.2. Physical considerations 2. Technical aspects 2.1. Scanning force microscopy 2.2. Optical traps 2.3. Hydrodynamic techniques 2.4. Magnetic forces 2.5. Micropipette aspiration-based techniques 2.6. Immobilization of biomolecules

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3. Case studies 3.1. Stretching of polymers 3.2. Force-induced dissociation of speci"c bonds 3.3. Unfolding of multidomain proteins 4. Discussion and outlook Acknowledgements References

361 361 368 375 377 378 378

Abstract This article describes the recent development of the rapidly expanding "eld of single molecule force spectroscopy. As the advancement of this "eld was dictated by technical progress, a large part of this review is focussed on methodological aspects. Moreover, several instructive experiments will be presented. However, it is already impossible to cover the whole "eld in one review article. Experiments on stretching of single polymers, on force induced dissociation of single bonds, and on unfolding of multidomain proteins are covered whereas active molecules (molecular motors) will be excluded. The physical interpretation of the selected experiments is addressed. The connection between the experimental results and the underlying microscopic properties of the molecules and their respective bonds will be discussed. Especially kinetic e!ects

E-mail address: [email protected] (R. Merkel). 0370-1573/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 0 3 - 4

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are of high importance for the interpretation of these experiments.  2001 Elsevier Science B.V. All rights reserved. PACS: 87.15.By; 87.15.La; 87.15.He; 82.20.Mj; 87.64.!t Keywords: Biophysics; Single molecule experiments; Highly sensitive force measurement devices

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1. Introduction Mechanical forces are of profound importance for the function of biological organisms. Cells and organisms have to cope with external forces (e.g. blood cells subjected to hydrodynamic shear "elds in the circulation). Moreover, many essential functions of living organisms demand active force production (e.g. chromosome separation in cell division or engul"ng food). For these reasons the study of the in#uence of mechanical forces on the diverse functions of di!erent parts of living organisms has a rich history. However, interpreting these experiments is a formidable task because many di!erent e!ects combine in a highly complicated way. For example, living cells will react to the application of mechanical forces. Changes in cell metabolism, morphology and even the protein composition will result. These changes are a necessary and highly important adaptation of living cells to their environment [30]. However, it is a hopeless enterprise to identify unequivocally a unique cause for speci"c e!ects. Simpli"ed model systems have been introduced to overcome this problem. For example, receptors and ligands were reconstituted into model membranes; cell surface receptors were bound to solid support and the peeling o! of bound cells was analyzed. Finally, the mechanical properties of in vitro networks of biological polymers were studied to infer the mechanical properties of the isolated polymer. In these experiments, macroscopic or mesoscopic material properties were measured. It is not obvious how to deduce the properties of the smallest entities from these experiments (e.g. the sti!ness of an isolated polymer or the strength of one speci"c bond). In some cases, it was shown that this procedure is not possible at all. For example, in experiments where model membranes are connected via mobile receptor and ligand pairs, it is impossible to calculate the a$nity or mechanical strength of the single bonds from the macroscopic measurement of the adhesion energy density. To overcome these problems many research groups focused on the mechanical study of single molecules or bonds. In these studies minuscule forces of some piconewtons must be exerted and measured. In many experiments tiny lengths (&nm) must be measured at the same time. Therefore, advances in the "eld of single molecule force spectroscopy were made only recently and coupled tightly to the evolution of instrumentation. However, the advent of highly sensitive force measurement techniques has spurred the development of this "eld and highly promising results were presented. In this review three di!erent types of experiments are discussed, namely stretching single polymers, breaking single molecular bonds, and unfolding single proteins. The vast "eld of experiments on the force production by single motor proteins was excluded because for the discussion of those experiments di!erent approaches are necessary. Reviews that cover the "eld of molecular motors can be found in references [81,111,112]. Another "eld that is only brie#y addressed is the theoretical basis of the experiments which will be treated only in a very basic way. Bond breakage is thoroughly discussed in the review articles of Bongrand et al. and Zhu who also discuss the question of bond formation which will be entirely neglected in this review [129,21,180]. A further review on force spectroscopy on single molecules was written by the group of Gaub [33]. Reviews covering technical aspects will be cited in the appropriate sections. 1.1. Physiological examples Whenever something is moved or deformed a mechanical force must act. This `somethinga may be a part of a molecule, a protein, a cell organelle, a cell or a part of a whole tissue. Thus many

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Fig. 1. A crude sketch of the structure of a sarcomere, the ultrastructural unit of skeletal muscle. Thin actin "laments (thin straight lines) radiate from the Z-disks (thick vertical lines) like teeth from a comb. At the midplane between adjacent Z-disks thick myosin "laments are intercalated between the thin "laments. In normal relaxed muscle (left) thin and thick "laments overlap. It is the interaction of myosin with actin that is responsible for force production in muscle. In far overstretched muscle (right) titin (curly lines connecting Z-disks and thick "laments) is stretched. The retracting force of the elastic titin molecules is necessary for the overstretched muscle to recover its correct ultrastructure.

physiological processes involve mechanical forces. In most cases, the molecular origins and results of force action are yet to be explored. However, some cases may serve as vivid illustration for the importance of mechanical forces in biology and biophysics. As "rst example, I will discuss the role of the giant protein titin (Fig. 1). In skeletal muscle it stabilizes the position of the thick myosin "lament in the middle between adjacent Z-discs. A resting muscle can be stretched so far that myosin and actin "laments no longer overlap. Following such an overstretching thick and thin "laments are sorted again and the regular ultrastructure of the skeletal muscle is reestablished. This sorting is produced by the elastic titin molecules that are stretched together with the muscle and subsequently pull the Z-disks together and guide the myosin "laments to their correct positions. Thus one function of this protein is to be stretched and to recover its normal conformation subsequently. For a closer description of titin see e.g. [58,93]. A second example is the rolling of leukocytes at the walls of blood vessels. Leukocytes are white blood cells that usually circulate with the blood stream. However, at the location of an in#ammation these cells invade the connective tissue to "ght the invading pathogenic organisms. To this end the leukocytes must "rst be stopped (i.e. they must adhere to the vessel wall) before they can penetrate the blood vessel and enter the tissue. The very "rst step in this process is the expression of P-selectin on the surface of the endothelium cells lining the vessel. P-selectin binds to speci"c molecular motifs on the surface of leukocytes. However, the leukocytes are exposed to the hydrodynamic force of the blood stream which results in rapid failure of these bonds to P-selectin. For each bond that breaks a new one is established. As a result the leukocytes roll along the vessel wall with much reduced velocity. During this process the leukocytes are activated by P-selectin binding and express di!erent cellular receptors which eventually mediate cell arrest followed by penetration of the vessel wall. For the purpose of this review the most interesting aspect is the mechanical loading and failure of speci"c bonds between cells. For a detailed account of the involved molecules see [157] (Fig. 2).

 This is no normal physiological situation but occurs in injuries.

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Fig. 2. A schematic view on the rolling motion of leukocytes (sphere) in contact to the vessel wall (bar). Speci"c bonds between leukocyte surface molecules (arrows radiating out of the cell) and P-selectin (V) on the surface of the blood vessel wall form. Due to the hydrodynamic force of the blood stream (symbolized by the arrows on the left-hand side) the cell experiences a drag force (thick arrow) which breaks the speci"c bonds. However, the torque of the force couple (bond versus hydrodynamic drag) rotates the cell so that new bonds can form and the cell exhibits a rolling motion along the vessel wall.

1.2. Physical considerations Force spectroscopy experiments on single biomolecules must be performed at temperatures where the respective molecules are in their functional state (usually between & 03C and & 403C). Thus the thermal energy, k ¹, amounts to 4.3;10\ J. On the other hand, biomolecules are stabilized by weak interactions only, i.e. the di!erence in Gibbs' free enthalpy between folded and unfolded forms of typical proteins ranges from 5 to 10 kcal/mol [35]. Free enthalpies for binding of typical receptor-ligand bonds are of comparable magnitude. Thus, it is obvious that thermal motion is important in force spectroscopy on single molecules. In fact, we will see from the examples given in this review that force spectroscopy experiments on single molecules and single biomolecular bonds probe the microscopic thermodynamics of the systems under study. The outcome of the experiments is dominated by the respective thermodynamic potentials for molecule or bond deformation, cf. Fig. 3. If one or more barriers in Gibbs' free enthalpy exist, the respective processes are kinetically hindered. Examples are force-induced dissociation of single bonds and force-induced unfolding of proteins. In these cases, the experiments are dominated by the time scale of the force application. In some cases seemingly con#icting results were obtained by `fasta and `slowa techniques. Thus, special care must be taken to identify and investigate kinetic e!ects in all force spectroscopy experiments. In this review, I will discuss three di!erent examples. The "rst is stretching of polymers. Here potential barriers are usually not present and therefore kinetic e!ects play little role. The second example is force induced dissociation of single receptor}ligand bonds. In these experiments kinetic e!ects are of paramount importance. In fact, if we apply a force on a time scale exceeding the statistical lifetime of an unloaded bond we will "nd a vanishing yield force. However, if the force is much faster applied, nonzero forces will be found. In the last example, I will discuss force induced unfolding of multidomain proteins. In these experiments several potential barriers for unfolding are found, at least one for each domain.

 This corresponds to 2.6 kJ/mol or 0.62 kcal/mol.

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Fig. 3. Schema of the possible shapes of Gibbs free enthalpy as function of bond or molecule deformation. These curves do not include the external force. Solid line: stretching of polymers, no barriers. Dotted line: dissociation of single biomolecular bonds, this process is in general dominated by one enthalpic barrier. Dashed line: unfolding of multidomain proteins, each domain contributes at least one enthalpic barrier, the unfolded polypeptide chain behaves like a stretched polymer. The latter two examples exhibit kinetic hindering, the "rst one not.

2. Technical aspects Force spectroscopy on single molecules is technically most demanding. Minute forces on the order of a few piconewton must be measured. As we will see below, the natural force unit for polymer stretching ranges from &0.1 to &20 pN, whereas the natural force unit for forceinduced bond dissociation falls in the range from &2 to &200 pN. The minute size of the forces to be measured is one of the major problems of single molecule force spectroscopy. To date, several methods have been developed that are capable of such superb force resolution. Out of these, I will discuss scanning force microscopy (SFM), hydrodynamic shear techniques, micropipette aspiration-based techniques, optical trapping and magnetic forces. In each case, force is not applied directly to single molecules but on micron-sized objects like microbeads or SFM tips. Thus, biomolecules must be immobilized onto these objects. This procedure poses another major challenge for the experimentalist because the bonding of biomolecules to the respective substrate must (i) preserve the native structure and function of the biomolecules, (ii) leave active sites (e.g. binding sites) accessible, and (iii) exhibit exceptional stability. Additionally, nonspeci"c interactions between these micron-sized `handlesa must be negligible compared to the forces applied to single molecules. Finally, the whole design of the experiment must enable the experimentalist to

 For comparison, the photons emitted by a 1 mW laser pointer exert a force of 3.3 pN on the pointer due to the conservation of momentum.  For this purpose, one must work under physiological conditions which imply for most proteins an aqueous bu!er exhibiting a neutral pH (&7.4), an ionic strength of &300 mM, and a temperature in the range from &03C to &403C.  Here we are dealing with seemingly tiny forces. Still, the demands on the toughness of the biomolecule anchoring are drastic. For example, a force of only 5 pN acting on each atom of an iron crystal will lead to a stress of about 0.2 GPa, well exceeding the elastic threshold of this familiar material.  Nonspeci"c interactions are forces like van der Waals attraction that do not depend on single molecule binding or stretching but on the presence of the micron-sized `handlesa.

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control the number of molecules involved and to uniquely identify a regime where single molecules are probed. This last point is of utmost importance because it is usually not possible to derive properties of single molecules from measurements on several molecules for the following reason. In real-world situations, the mechanical load will be distributed unequally between di!erent molecules that are loaded in parallel. Calculating single molecule properties from such `convoluteda data requires highly nontrivial assumptions on the underlying processes. In view of this impressive list of requirements, it is not surprising that there is no single technique suitable for all experiments in dynamic force spectroscopy. It is just the opposite, for most experiments dedicated techniques were developed or existing techniques were improved, adapted, and "ne-tuned. In the following, I will discuss several techniques that are presently used in dynamic force spectroscopy. It is very likely that new techniques will come up in the near future. 2.1. Scanning force microscopy 2.1.1. Classical scanning force microscopy Scanning force microscopy (SFM), alternatively termed atomic force microscopy (AFM), is a modern microscopy technique with tremendous impact on many "elds of science and technology. In this technique, a sharp tip is mounted to a soft spring and scanned across a sample. Interaction forces between tip and sample result in spring de#ections that are detected, see Fig. 4. This de#ection signal may be used either directly to construct images or as feedback signal. With the feedback mode surfaces of constant interaction force between tip and substrate may be measured. In many instances, these iso-force surfaces can be considered as a good approximation for the geometric contour of the substrate. In the original version of SFM a scanning tunneling microscope was used to measure the de#ection of the spring [17]. This is a very complicated way to

Fig. 4. Principle of the scanning force microscope. A piezoelectric tube is used to move the sample in three dimensions (x, y, z). A microfabricated tip connected to a soft cantilever is scanned across the specimen. Cantilever de#ections are detected by a quadrant photodiode onto which a laser beam is re#ected from the back of the cantilever. After appropriate calibrations the photodiode signal can be used to determine the acting force.

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perform this measurement and accordingly such instruments were troublesome to use. However, improvements were developed at a breath-taking speed. The invention of an optical scheme for spring de#ection monitoring resulted in much simpler and more reliable instruments [114,4]. The `#uid cella allows the user to image samples immersed in #uids, e.g. physiological bu!ers [41]. Finally, microfabricated integrated tips were invented that are nowadays commercially available in many di!erent shapes and sti!nesses [2]. This development resulted in reliable instruments which can be much easier handled than other instruments in force spectroscopy on single molecules. Since several very good reviews about SFM exist [73,67,133,25,134,68,42], I will not go into great detail here. For the purpose of single molecule force spectroscopy the superb height resolution of SFM combined with its speed are of paramount importance. In force spectroscopy experiments, the SFM tip is not scanned across the surface but is repeatedly moved perpendicular to the substrate [178,134,68]. At the present moment, SFM tips are commercially available exhibiting sti!nesses as low as 10 mN/m. Using such tips interaction forces of 10 pN may be detected under favorable circumstances. SFM is widely used in single molecule force spectroscopy and I will present many examples in the further course of this review. 2.1.2. Pullers and related techniques In dynamic force spectroscopy on single molecules the imaging power of SFM is not of particular importance. Accordingly, custom designed instruments were developed that use standard SFM technology (i.e. integrated tips and the optical detection scheme) but allow exclusively the recording of force distance curves [123,139,126]. Such instruments need to have piezoelectric translators for one dimension only. Thus, more compact and more rugged instruments can be designed. Thermal drift and mechanical vibrations can be minimized while maintaining the advantage of using standard parts that are commercially available. In other instruments, the authors went one step further and used home designed ultrasoft levers for force application. Here glass microneedles [84,44] and etched optical "bers [34] were utilized. Lever de#ection is measured by optical microscopy and image processing [84,44] or by shining light through the optical "ber and imaging its light emitting end onto a split photodiode [34]. Using such techniques extremely soft springs can be fabricated, sti!nesses as low as 1.7
N/m [44] and 1.5
N/m [84] were reported. Such ultrasoft levers allow measurements of tiny forces (&1 pN) while keeping problems with thermal drift at a manageable level. However, the hydrodynamic drag on these long soft needles results in fairly sluggish response to sudden force changes. Thus force resolution is bought by sacri"cing speed. 2.2. Optical traps All molecules and particles exhibit a polarizability,
, i.e. electric "elds induce an electric dipole moment p"
E. Therefore, a mechanic force F"(pE)"2
E acts on a particle in an electric "eld gradient. For the existence of a stable position of a particle a maximum of the electric "eld within

 On mica, an atomically #at substrate, spring de#ections may be determined with a standard deviation as low as 0.8 As using commercial microscopes.

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Fig. 5. The geometric ray optics approximation of optical trapping. A dielectric sphere is located o!-axis below the focal point of a lens. Following the paths of the optical rays (solid lines) and calculating the momentum change of the light rays at the refractive interfaces (dashed arrows), one "nds that the net force acts radially towards the more intense beam (symbolized by a thicker line) and vertically towards the focus.

the medium is necessary. Unfortunately, this cannot be achieved by static electric "elds where the maximum always occurs at some surface. Using temporal varying "elds, i.e. light waves, creating a maximum of the electric "eld amplitude in free space is as simple as collecting a beam of light by a lens. This e!ect was used to trap and manipulate particles in transparent media [11,13]. Several very instructive reviews of the technique exist [31,18,164,149], therefore I will focus on the basics only. In optical trapping two forces act on particles, namely the scattering force and the gradient force. These are most easily explained for very small particles (size;wavelength), cf. Ref. [164]. The scattering force arises because the Rayleigh scattering of photons implies randomizing their momentum. Thus, the average momentum of all photons is reduced by scattering. The `missinga momentum is taken up by the scattering particle which therefore experiences a mechanical force acting along the direction of propagation of the incoming light. The second force, the gradient force, is simply the force arising due to the interaction of the induced dipole moment of the particle with the spatially varying electric "eld. This force pulls particles of higher refractive index than the surrounding medium towards the maximum of the electric "eld amplitude. For very large particles the forces may be calculated using geometrical optics and summing up the momentums carried by all beams of light [12] (Fig. 5). In the intermediate size regime, which is very important for practical applications, force calculation is extremely di$cult because the full boundary problem of electrodynamics must be solved [164]. Another important aspect for particle trapping is the fact that scattering force and gradient force follow di!erent power laws regarding sphere size and refractive index. Thus, the balance between both forces depends strongly on the exact parameters of the experiment, i.e. bead size, refractive index and the intensity distribution within the trap. Optical traps were designed in two di!erent con"gurations: dual counter-propagating beam traps [11] and single-beam gradient traps [13] (Fig. 6). In traps using two counter propagating beams the scattering forces of both beams cancel at the midplane and stable trapping is achieved by the gradient force. In single-beam gradient traps the scattering force is fully active and must be

 In the small particle regime the scattering force scales approximately as rn  whereas the gradient force scales as rn . Here r denotes the particle radius and n the di!erence between the refractive index of particle and solution.

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Fig. 6. Principle of optical traps. Left: single beam gradient trap. Right: dual counter propagating beam trap. Crosses denote foci of lenses, "lled circles the center position of the trapping potential. For the single-beam gradient trap scattering force is symbolized by full arrows, gradient force by dotted arrows.

overcome by the gradient force. This is only possible if light is collected over a very large angular range, i.e. lenses of highest numerical aperture must be used. Thus, the trapping region of single-beam gradient traps (also called `optical tweezersa) is very small (below 1
m in radial direction), located close to the cover slip, and relatively large light intensities are necessary for trapping. Due to these high intensities radiation damage (via heat and photochemistry) of biomaterials is a serious concern in single-beam gradient traps. Direct light absorption by biomaterials is most important in the ultraviolet and blue spectral region. Water strongly absorbs in the middle and far infrared region. Thus, energy absorption is minimum in the near infrared region. For this reason, near-infrared lasers are now nearly exclusively used in biophysical optical tweezers experiments. The setup of laser tweezers is much simpler compared to a dual-beam trapping setup where the optical alignment of both beams is a constant nuisance. Therefore, dual-beam traps are only used in cases where their advantages (large working distance and large size of the trapping zone, far less light intensity at the same trapping force) are of crucial importance for the experiment. In order to measure forces using optical traps one has to calibrate the strength of the trapping potential and one has to measure the de#ection of the trapped bead from the trap center [164]. Calibration may be performed by observation of the Brownian motion of the trapped bead or subjecting trapped beads to #uid #ows of known velocity. In the "rst case, trap sti!ness at small bead excursion is measured. In the second case much larger deviations are tested. Thus, the linearity of the trapping force can be explicitly tested. Most often both methods are used to calibrate optical traps, Brownian motion as routine check to correct for particle to particle variations and hydrodynamic forces to identify the linear region of the trap. For dual counterpropagating beam traps direct calibration of forces from the de#ected light exciting the trap is possible [155]. Di!erent techniques are used to measure bead motions. The simplest method is projecting the image of the bead onto a camera [91] or a four quadrant photodiode [53,117,151] and measuring the bead de#ection directly. Alternatively an optical trapping interferometer may be

 For this purpose, the bead motion is modeled as harmonic oscillator subjected to Brownian forces. Both the mean square amplitude and the cuto! frequency of this Brownian oscillator can be measured and used for calibration of the trap sti!ness.

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used [39,166,165]. This system is related to di!erential interference contrast microscopy because it is a shear interferometer containing two Wollaston prisms [131]. In this technique, the laser light of the trap is used directly to detect bead displacements. This displacement sensor is unidirectional because only motions in the optical shear direction are sensed. Finally, one may use the condenser of the light microscope to collect the laser light leaving the trap. If the trapped bead is displaced relative to the trap center the angular distribution of the light leaving the specimen is changed. This angular distribution may be determined by a four quadrant photodiode positioned in the back focal plane of the microscope condenser [59,61]. The output of all particle detection systems (besides cameras) must be calibrated to yield absolute displacements. For this purpose calibrated piezoelectric translation stages are a popular choice. One of the major advantages of optical traps is the impressive speed of displacement detection. In systems where the trapping laser is used at the same time as displacement sensor, roll-o! frequencies as high as 100 kHz may be reached [61]. At the same time, spatial resolution better than 1 nm is achieved in many setups. These advantages allow the construction of elaborate instruments like isometric picotensiometers exploiting feedback strategies [53]. 2.3. Hydrodynamic techniques It is surprisingly simple to apply very small forces using hydrodynamic drag. For example, from Stokes' equation it follows that a sphere of radius 1
m subjected to a uniform #ow of water experiences a force of only &0.02 pN for 1
m/s #ow velocity. However, application of forces to single molecules requires two surfaces between which the molecules may be suspended. There are only two geometries for which the forces between two surfaces may be accurately determined: doublets of identical spheres in shear #ow and spheres bonded to the wall of a parallel wall laminar #ow chamber. In most other cases only rough estimates of forces are possible. 2.3.1. Bead doublets in shear yow In this technique, a dilute suspension of identical spheres is subjected to shear #ow [169]. Up to now this technique was exclusively used to study the force induced dissociation of receptor}ligand bonds [171,167,168,92]. In these experiments, a multivalent receptor molecule (e.g. an immunoglobulin) bridges two identical beads bearing ligand molecules. Due to the shear #ow this bead doublet undergoes rotation. During each orbit the interbead bond is twice subjected to tension and compression. Moreover, shear forces occur as well. Experimental studies were performed in two di!erent setups. In one case, the shear #ow within a tube of circular cross-section was used [171]. In this case, the center of gravity of both beads undergoes uniform translation. To compensate this a glass microcapillary is translated vertically during the observation with a stationary microscope. Hence the name of this technique `traveling microtube apparatusa. In the second approach, a shear #ow is created in a plate-and-cone-rheometer [167]. In this rheometer type, the #uid shear does not depend on radial position. If plate and cone are counter-rotated at equal angular velocities

 This is most easily understood for a very large particle which acts as lens. Shifting this lens amounts to steering the direction of the exiting beam.

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a stationary layer exists at the midplane between plate and cone. For observation of doublet orbits in this stagnant layer a stationary microscope is used and the rheometer is manufactured of some transparent material. In both geometries, smallest forces can be applied to bead doublets. However, doublet formation occurs by chance and cannot be externally induced. This results in major problems in collecting enough data for good statistics. Additionally, it is very di$cult to work on single molecules because to reach a single bond regime very low densities of receptors must be used which result in exceedingly small probabilities of doublet formation. Moreover, the complicated force history of the bead doublets makes data interpretation very di$cult [103]. Due to all these problems, the method never found widespread use but seminal results were achieved by it. 2.3.2. Parallel wall yow chamber In this alternative hydrodynamic method, a parallel wall laminar #ow chamber is used. This technique was mostly used to study force-induced dissociation of single bonds. One type of adhesion molecules, let us say the receptors, resides on microbeads or cells that move with the hydrodynamic #ow. The complementary adhesion molecules, the ligands, are immobilized on the bottom of the #ow chamber (cf. for example Refs. [79,10]). Immobilization may be achieved by either direct binding of molecules or by plating appropriate cells on the substrate. Bond formation is detected via light microscopic observation through the bottom of the chamber, cf. Fig. 7. In this technique, one measures the statistical distribution of the lifetime of bonds subjected to force. For bond formation, one has to rely on random particle}wall contacts. The single bond regime is reached by diluting one type of adhesion molecules until particle arrests occur infrequently. The form of the distribution of durations of arrests is often used as additional criterion. Usually, a single exponential distribution is taken as indication that single molecular bonds are probed. From these distributions the rates of bond dissociation are determined. The tensile force on a given bond is calculated by the balance of force and torque. An arrested spherical particle in contact with the chamber wall experiences the following force, F , and torque,  ¹ [62,172,79]:  F "32.05rG, ¹ "11.86rG . (1)  

Fig. 7. Principle of bond loading by use of a laminar #ow chamber. Left: An arrested spherical particle in contact with the wall experiences a force F and a torque ¹ . Right: The balance of force and torque requires that the bond (thick line) is   located o!-center. If the distance l is known, the tensile force acting on the bond, F , can be calculated from the force and
torque on the bead, see text.

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Here  is the #uid viscosity, r the sphere radius, and G the wall shear rate (i.e. the velocity gradient Rv/Rx of the #uid #ow at the wall). From the geometry of a sphere attached to a wall, see the right-hand side of Fig. 7, we "nd that the balance of force and torque is given by [9] F "F cos
, F r#¹ "F l sin
. (2) 
 
These equations may only be satis"ed if the distance, l, between the center of the projected area of the sphere and the anchor point of the bond is not zero. For the calculation of the force on the bond, F , the distance l has to be measured. This may be done by frequently reversing the #ow direction.
If such a #ow reversal occurs during a bead arrest the bead will be translated by 2l [9]. In summary, the parallel wall laminar #ow chamber is a powerful tool for single molecule force spectroscopy. Forces on bonds may be preselected via the wall shear rate. The statistical distributions of bond lifetimes at "xed force are measured directly. However, calibration of forces is not simple because the distance, l, between the center of the projected area of the sphere and the anchor point of the bond must be determined. The technique is well suited for the work on living cells. 2.4. Magnetic forces Most biomaterials react barely or not at all to magnetic "elds. This implies that magnetic "elds do not interfere with biomaterial structure and function. Thus, artifacts due to force application are minimized. All magnetic force techniques that are relevant for this review are based on superparamagnetic beads that are commercially available from di!erent sources. These beads consist of polystyrene with embedded microcrystals of iron oxide. The microcrystals are too small for the formation of a permanent magnetic dipole moment, yet so large that the coupling between the spins in the microcrystals results in extremely high paramagnetic susceptibilities. Force, F, and torque, T, on a particle subjected to a magnetic "eld H are given by F"(M ) )H,

T"MH ,

(3)

where M denotes the magnetic moment of the particle and  the vector product. For an ideal paramagnetic particle M"H holds where  is a scalar factor. Thus, the torque on an ideal paramagnetic particle vanishes. However, it was reported that superparamagnetic beads subjected to a magnetic "eld exhibit a body "xed axis that orients parallel to the magnetic "eld [156,158,5]. This implies that these beads exhibit a residual permanent magnetic moment. Forces on such superparamagnetic beads were used by two di!erent groups to investigate the stretching of DNA [156,158]. In the setup of the Paris group the magnetic "eld H is oriented in the plane of focus of the observation microscope, whereas the tensor of the magnetic "eld gradient RH /Rx exhibits G H only one eigenvector with nonzero associated eigenvalue, see Fig. 8. This eigenvector is oriented perpendicular to the plane of focus. Therefore, the superparamagnetic bead is pulled upwards from the substrate. The magnitude of the force can be tuned by changing the distance between the permanent magnets and the specimen. Moreover, the DNA may be twisted by rotating the pair of

 In many experiments whole cells were used instead of rigid beads. This results in an additional complication because cells exhibit viscoelastic behavior, i.e. l may depend on the duration of the arrest.

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Fig. 8. Principle of the setup used by the Paris group for DNA stretching [158]. Left: The magnetic "eld is oriented within the plane of focus. The superparamagnetic bead exhibits a small permanent moment M and a large induced  moment. Thus, the magnetic force, F, on the bead points in the direction of the magnetic `"eld gradienta, i.e. perpendicular to the plane of focus of the microscope. Right: Force calibration. The length, b, of the DNA at force F is determined by microscopy. For excursions within the plane of focus, the tethered bead experiences a backdriving force F "F tan +F r/b. By using the equipartition theorem one "nds for the thermally driven mean square excursion , r"k ¹b/F. Thus, the magnetic force can be calculated from the measured Brownian motion of the tethered bead.

magnets around the microscope. Di!erent "eld geometries are possible as well [156]. The setup of the Paris group is a prime example for an instrument designed speci"cally for one purpose, i.e. stretching and twisting of DNA. The latter deformation mode is of highest physiological importance as the tightly coiled DNA must be partially unwound during replication and transcription. Using this setup exceedingly small forces as low as 10 fN may be measured [158,5]. 2.5. Micropipette aspiration-based techniques The micropipette aspiration technique [115,52] may be used to measure piconewton forces under physiological conditions (Fig. 9). In this technique, an open cylindric glass microtube is micromanipulated in the "eld of view of a light microscope. Small suction pressures may be applied between the inner medium of the pipette and the measurement chamber by a hydrostatic manometer. Pipettes exhibiting an inner diameter ranging from &1 to &10
m are commonly used. Suction pressures ranging from &1 Pa to 50 kPa can be easily applied and accurately measured. This results in suction forces (P R ) in the range from &1 pN to &1
N. Such . aspiration forces may be directly applied to microscopic objects, e.g. by using a cell as hydraulic piston, driving it against a binding surface, and breaking the resulting bonds by subsequent aspiration [170,147,148]. However, the accuracy and ease of use of the force measurements can be tremendously improved by introducing a closed biomembrane capsule. In the so-called biointerface force probe technique a biomembrane capsule is aspirated by the pipette and a microbead carrying the molecules of interest is glued to the apex opposing the pipette opening (see Ref. [50] and Fig. 10). The aspiration

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Fig. 9. Principle of the micropipette aspiration technique. The object to be studied (3), e.g. a cell, is immersed in physiological bu!er (2) which forms a drop suspended between two cover slips. The sample is located in the "eld of view of a light microscope, i.e. between condenser (1) and objective (7). The micropipette (4) is an open cylindric tube of glass. It is maneuvered with the help of a three axis micromanipulator (5). Pressure di!erences between inside and outside medium are produced by adjusting the height of an hydraulically connected water reservoir (6). This pressure di!erence P results in forces on the object. Motion or deformation due to this mechanical force is analyzed by video microscopy and image processing.

Fig. 10. Principle of the biointerface force probe. A red blood cell is aspirated by micropipette suction and a microbead is attached to the cell apex. Left: Micrograph. Scale bar, 5
m. Middle: The suction pressure of the pipette, P, is transformed into membrane tension, , which enforces a spherical shape of the membrane capsule. Right: Application of a force to the microbead results in a deformation of the spherical shape. This deformation is controlled by membrane tension and applied force. Therefore the deformation may be utilized to measure the force.

pressure results in a mechanical surface tension of the membrane capsule. This endows the membrane capsule with a resistance against forces applied to the microbead (e.g. by loading a single biomolecular bond). Therefore, the membrane capsule acts as a spring. Biomembranes possess three characteristic deformation modes, area expansion, in plane shear, and bending. The force}extension relation of the membrane capsule may be calculated as long as surface tension dominates the deformation, i.e. as long as the contributions of in plane shear and membrane bending to the deformation energy are negligible. For technical reasons only phospholipid vesicles and red blood cells are used as force sensors [51,50,153]. Phospholipid vesicles in the #uid phase exhibit no shear rigidity, therefore they may be safely used as force transducer. For red blood cell membranes in plane shear relaxes by plastic #ow if the shear tension exceeds approximately 20
N/m. For mechanical surface tensions beyond this value red blood cells may be used as force transducers. Membrane bending does not play a role in either case because the bending module of

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phospholipid membranes and red blood cell membranes is very low [141]. However, this holds only if the applied forces are distributed over a large area by the microbead [46]. Therefore, the latter serves not only as protein carrier but acts also to prevent mechanical instabilities of the soft biomembrane due to point forces. While in plane shear and membrane bending are irrelevant for the biointerface force probe, area expansion and surface tension are of paramount importance. In fact, the area expansion module of biomembranes is so high that we may safely calculate the force}extension relation by assuming a "xed surface area of the membrane capsule. These considerations show that no material parameter of the membrane capsule is of relevance for the "nal force}extension relation. Therefore, uncertainties in the values of these parameters do not impart the accuracy of force measurements using the biointerface force probe. The details of the calculation of the force}extension relation may be found in [153]. In this work, it is shown that the force}extension relation of the biointerface force probe is linear for small extensions (4200 nm), and proportional to the suction pressure of the pipette. The last point is noteworthy because it implies that the sti!ness of this force transducer can be tuned at will during the experiment by changing the pipette's suction pressure. Only suction pressure and geometrical parameters enter the expressions for the force}extension relation. As these can be accurately determined by simple measurements of lengths in the microscopic image, the force}extension relation can be easily determined to an accuracy of approximately 15% [153]. For elongations exceeding &0.3
m the force}extension relation is nonlinear. This is caused by two constraints: preservation of membrane area and of enclosed volume. The resulting equations can be e$ciently integrated by numerical methods resulting in a full force}extension relation [153]. The lower limit of the force probe's sti!ness was found to be 45
N/m if a red blood cell was used as membrane capsule. The transverse sti!ness of the force transducer does barely depend on membrane tension and amounts to about 10
N/m. If phospholipid vesicles are used the transverse sti!ness vanishes and there is no lower limit for the transducer sti!ness besides the practical problems of working with lowest suction pressures and phospholipid vesicles. This force probe is fully compatible with the requirements of biomolecules. The molecules under investigation experience no additional stress besides being immobilized to a microbead. Thus, this technique is ideally suited to study force-induced dissociation of single adhesive bonds. 2.6. Immobilization of biomolecules As pointed out before all force measuring techniques presented here rely on the presence of a micron-sized `handlea for force application. Molecules under investigation must be bound to the surface of these `handlesa. In most experiments on single biomolecules the surface properties of these `handlesa are decisive for success or failure of the experiment. Thus biomolecule immobilization is an important, albeit di$cult, topic in force spectroscopy on single molecules. The methods for biomolecule immobilization fall in three di!erent classes: physisorption, speci"c binding to surface groups, and covalent binding to surfaces. Physisorption is the technique most  A mechanical force on a single molecule is a prime example for a point force.  For comparison: the weight of a tiny piece of standard laser copier paper, size 7.6 mm;7.6 mm, would extend such a soft spring by 1 m!

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widely used. However, direct adsorption of proteins to solid surfaces often results in substantial changes of protein structure and function. In general, high adsorption energies result in massive loss of protein function. For example, it was shown that physisorption of the enzyme -chymotrypsin to polystyrene resulted in nearly complete loss of activity [121]. Similar results were found for antibodies adsorbed to polystyrene where only 3% of the adsorbed antibodies were still able to bind their antigen [26]. Thus studies on physisorbed proteins may result in misleading and arti"cial results. Moreover, the mechanical strength of the `physisorption bonda is not de"ned. Yield forces close to 2 nN were found in SFM studies [138,109,99], albeit loosely adsorbed molecules were most likely not detected in these experiments. In cases where a linkage between force probe and substrate exists during the whole experiment (e.g. unfolding of proteins) relying on physisorption is perfectly adequate. If one tries however to investigate single-bond dissociation, it is impossible to distinguish between the desorption of biomolecules from the surface and `truea bond failure. For such experiments physisorption of biomolecules is clearly no good choice for the method of immobilization. Problems with functional loss and structure changes due to the proximity of a solid substrate may be overcome by introducing an intermediate layer of coupling molecules that bind the target molecules speci"cally [26,102]. Here antibodies to target molecules [26,148], the avidin}biotin system [15,142], and metal-chelating complexes involving oligo-histidin-tags on genetically engineered proteins [144,143] were successfully used. The "rst two approaches are very versatile because antibodies are available to many target molecules and close to all biomolecules may be biotinylated [142]. For the use of metal-chelating complexes histidin tags must be present at the surface of the respective proteins. This is not a severe problem because an increasing number of di!erent proteins are produced by genetic engineering where an a$nity handle like a histidin tag is routinely introduced to facilitate puri"cation. Additionally, this immobilization strategy has the additional bene"t that proteins are bound in an oriented fashion. It is very likely that alternative a$nity handles from genetic engineering, like target sequences for bacterial biotin transferring proteins, will be used for biomolecule immobilization in the near future. While using speci"c binding to an intermediate layer minimizes surface induced loss of activity, the mechanical strength of most of these bonds remains to be explored. For one example, biotin}avidin, surprisingly low yield forces were found at slow loading of bonds [113]. This implies that biomolecule immobilization via noncovalent bonds has its dangers for force spectroscopy. The problem of insu$cient mechanical strength may be overcome by covalent immobilization strategies. Many established procedures and commercially available reagents exist for this purpose. The respective reagent target speci"c groups at the surface of biomolecules. Most often primary amino groups (}NH ), free sulfhydryl groups (}SH), and carboxyl groups (}COOH) are used. The  substrate must exhibit suitable functional groups as well. Alternatively, these groups may be introduced via self-assembled monolayers of silanes on glass or thiols on gold [174,150,143,176]. These functionalized self-assembled monolayers are usually composed of a mixture of molecules bearing reactive and nonreactive endgroups (e.g. mixtures of amino-silanes and hydroxy-silanes). By this design, the density of anchor points for biomolecule immobilization may be directly controlled. In order to minimize the number of separate steps in the preparation small biomolecules like oligomeric DNA strands can be chemically attached to molecular moieties that form self-assembled monolayers prior to monolayer formation [122]. The problem of biomolecule immobilization is of utmost importance for biosensor applications as well. This has spurred many

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experimental studies [150,76,102,77]. It should be mentioned that even covalent bonds fail under mechanical load. In SFM studies yield forces as high as 2 nN for Si}C bonds and 1.4 nN for Au}S bonds were found [63]. For the latter bond a yield force of 1 nN was found in an alternative study [116]. However, bond yield forces depend on the rate of force application (cf. Section 3.2). Unfortunately, this aspect was not investigated in the aforementioned studies. Following the concepts for bond dissociation to be described below (Section 3.2) and known potentials for covalent bonds [128], we may estimate that a rate of bond dissociation of 1 s\ is reached at a load of 0.9 nN for Au}S bonds and at 3 nN for C}C bonds. These forces are much higher than the ones usually encountered in force spectroscopy on single molecules. Moreover, the dissociation rate depends strongly on force. Thus, covalent bonds are a `safea choice for most experiments in force spectroscopy. However, a protein in direct contact with a solid substrate may still experience drastic structural changes because the same surface forces are active as in physisorption of proteins. To circumvent this problem covalently bound layers of hydrated polymer may be introduced between biomolecule and solid surface [76,74,64,161,162]. This procedure indeed combines mechanic strength with minimized functional loss. Moreover, nonspeci"c interactions like van der Waals attraction between the respective molecule `handlesa are further reduced. Please note that in all studies a further agent to block nonspeci"c interactions was used. Popular choices are serum albumin and casein. In some cases, `irrelevanta biomolecule mixtures may be used as well (e.g. "sh sperm DNA in experiments on -phage DNA stretching). From the above considerations, it is obvious that selecting and establishing the `besta method of biomolecule immobilization may be very di$cult and time consuming. However, for some types of experiments this is of decisive importance. It is impossible to cover all possible methods and reagents in this review. Much more information and details can be found in Refs. [27,83,69].

3. Case studies 3.1. Stretching of polymers A wealth of experimental data exists for single polymers subjected to mechanical forces. Since polymers are composed of many identical subunits, experiments on polymers may be e$ciently modeled. This in turn helped enormously to guide and interpret experiments. Three di!erent phenomena were found, namely entropic elasticity, backbone elasticity and conformational changes in the molecules. Entropic elasticity is caused by the #exibility of polymers. Many conformations exhibit the same or nearly the same internal energy and thus entropy becomes the dominant e!ect at low forces. If the polymer is su$ciently #exible the entropic elasticity is adequately described by the freely jointed chain model [65,90]. This model is reasonable for most synthetic polymers and many

 However, one should not forget the toughness of the substrate. For example, evidence was presented indicating that piconewton forces are su$cient to extract strands of polymer from polystyrene latex beads [36].

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Table 1 Summary of data on single polymer stretching. Unless stated di!erently all studies were performed in water Polymer

Conformation transition

Model

l I

f 

Ref.

Carboxymethyl-dextran Dextran

250}350 pN 700}850 pN

EFJC EFJC

0.6 nm

4 nN , 10 nN
5 nN , 24 nN


[139]

Dextran Amylose Pullulan

700}900 pN 230}320 pN Very broad

EFJC EFJC EFJC

0.44 nm 0.45 nm 0.45 nm

6 nN 3 nN 5 nN

[110]

Carboxymethyl-amylose Carboxymethyl-cellulose Heparin

&300 pN None &700 pN

EFJC EFJC

0.5 nm 4 nm

6 nN , 15 nN
200 nN

[100]

Poly(methacrylic acid)

None

FJC WLC

0.3 nm 0.6 nm

Poly(acrylic acid)

None

EFJC

0.6 nm

Poly(dimethyl siloxane) in heptane

None

FJC WLC

0.2 nm 0.5 nm

Poly(ethylene glycol) in hexadecane in water

None Very broad

EFJC

0.7 nm

100 nN

Poly(vinyl alcohol)

&100 pN

EFJC

0.6 nm 


10 nN 


[128] 8 nN

[98] [146] [126]

[101]

Denotes below the conformational transition.
Denotes above the conformational transition. The following force}extension laws were used: Freely jointed chain model (FJC) x"¸ [coth( f l /k ¹)!k ¹/- ], worm-like chain model (WLC) f"2k ¹/l [x/¸ #0.25(1!x/  I I I  ¸ )\!0.25], and the variants of these models where backbone elasticity is taken into account by replacing  ¸ P¸ (1#f/f ). These latter versions of the extension laws are indicated by EFJC and EWLC, respectively. Please    note that the Kuhn length, l , of a worm-like chain is twice its persistence length. The other symbols are: x polymer I extension, f acting force and ¸ polymer contour length. 

natural polysaccharides. For the freely jointed chain model, the force}extension relation is given by [54] x"Nl [coth( f l /k ¹)!k ¹/f l ] , (4) I I I where x denotes polymer extension, N the degree of polymerization, l the length of the e!ectI ive repeat unit (sometimes also called `Kuhn lengtha), and f the acting force. The natural unit of force in this equation is k ¹/l and ranges from 1 to 20 pN for the polymers studied up to now I (cf. Table 1). However, there are certain polymers that exhibit chain sti!ness. This holds especially for `thicka biopolymers where the monomeric units are held together by several bonds in parallel. Examples are DNA, "lamentous actin, microtubules, and intermediate "laments. In this case, polymer bending costs energy. These polymers may be modeled by the worm-like chain model. The energy

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of a given polymer con"guration r(s) is in this model [88,108] E"k ¹



*




Rr(s)  ds!fx , Rs

(5)  where ¸ is the contour length of the polymer, s the coordinate measured along the contour, and  l the persistence length of the polymer. In this model, the force}extension relation may be  calculated if the polymer is very long (¸ /l <1) and free volume e!ects may be neglected   [54,24,108]. The result is 0.5l 

f"k ¹/l g(r/¸ ) ,   (6) g(x)+x#1/[4(1!x)]! .  The exact function g(x) may be calculated numerically [108,23]. The approximate equation given above deviates from the exact solution by as much as 7% for forces in the range from 0.2k ¹/l to  10k ¹/l . Eq. (6) has been widely used to interpret measurements on DNA and other sti!  polymers (Fig. 11). Again the characteristic force k ¹/l amounts to about 0.1 pN for DNA and  ranges from 1 to 20 pN for other polymers, (cf. Table 1). It was noted in many experiments on polymers that at forces substantially exceeding the characteristic forces for entropic elasticity the data may no longer be described using a constant contour length of the polymer. To account for this e!ect linear elasticity of the polymer backbone was introduced [124]. Most experimentalists simply replaced the contour length ¸ in Eqs. (4) and  (6) by ¸ "¸ (1#f/f ) , (7)    where f is the elastic module of the chain and ¸ its unstretched contour length. This procedure   is not well justi"ed from a theoretical point of view but yields a very good description of data in most experiments. 3.1.1. Experiments on DNA The whole "eld of single polymer stretching started with the seminal experiments of Smith et al. [156], who used a combination of magnetic and hydrodynamic forces to stretch single polymers of DNA that were suspended between a cover slip and a superparamagnetic bead. These experiments showed that DNA is an ideal model system because of its enormous length and the existence of the elaborate tools for DNA modi"cation from molecular biology. In these experiments, -phage DNA exhibiting a contour length of 16.4
m was used. Using the appropriate DNA oligomers and speci"c hybridization biotin was bound to one end of the DNA and digoxigenin, a small hapten, to the other end. Exploiting speci"c binding of biotin to streptavidin (immobilized on the bead) and of digoxigenin to a speci"c antibody on the coverslip DNA was suspended by its endgroups in a polar

 Without applied force the mean square radius of a polymer is given by R"2l ¸ for the worm-like chain model  and R"l ¸ for the freely jointed chain model. Thus the Kuhn length l of a worm-like chain is given by l "2l . I I I   Bouchiat et al. [23] give an improved interpolation equation g(x)+x#1/[4(1!x)]!1/4# a xG with G G a "!0.5164228, a "!2.737418, a "16.07497, a "!38.87607, a " 39.49944, and a "!14.17718 which       approximates the exact solution to an accuracy of 10\.

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Fig. 11. Left: Schematic view of the di!erent polymer models. In the freely jointed chain model (blobs and lines) the distance between successive blobs is "xed whereas the angles between successive bonds (lines) are entirely free. In the worm-like chain model (continuously bend curve) polymer bending costs energy. Shown are the coordinates of the polymer used in de"ning its energy, Eq. (5). Right: Calculated force}extension relations for the freely jointed chain model (Eq. (4), full line), the approximate solution of the worm-like chain model, Eq. (6) (dotted line), and the improved version due to Bouchiat et al. (dashed line). Please note that the Kuhn length of a worm-like polymer is twice its persistence length.

way. It is impossible to achieve such an excellent control over the experimental conditions with other polymers. These experiments provoked theoretical treatments of the stretching of worm-like polymers [24,108]. It was shown that the force}extension relation of DNA can be well described by a worm-like chain model for forces up to 20 pN. In subsequent experiments using higher forces it became apparent that enthalpic elasticity of the polymer backbone must be taken into account [155,14,177]. Eq. (7) was used and elastic modules f in the range 1.0}1.3 nN were found for ionic  strengths exceeding 20 mM [155,177,14]. Please note that DNA is a highly charged polymer. Therefore, the normal double stranded DNA structure is destabilized at low ionic strength which is indicated by a reduced elastic module [14]. Moreover, the persistence length, l , is in#uenced by  low salt concentrations and multivalent ions. In sodium chloride at concentrations exceeding 10 mM the persistence length of DNA is about 50 nm [24,155,177,14], at lower salt concentrations l is increased by charge repulsion of the polymer [14], whereas multivalent ions in general reduce  l yet exceptions of this rule exist [14,177]. At still higher forces of 65}70 pN a highly cooperative  conformation transition sets in. This transition was found by microcantilever bending [34], optical tweezers [155], and SFM [135,32]. During the transition DNA lengthens by a factor of 1.7 at nearly constant force. At low ionic strength (410 mM) the plateau force is reduced. The molecular origin of the transition is fairly well understood. DNA consists of two strands and can assume many di!erent conformations. At physiological conditions B-DNA is the prevalent form. It is a right-handed -helix with 10.4 base pairs per helix turn and a helical pitch of 25.4 As per turn. Other forms with di!erent helical pitch are possible but exhibit a higher free energy. However, under high tension such longer DNA forms are energetically favored. Model calculations show  DNA is a chiral and polar polymer, i.e. its two ends are not equivalent.  The mechanical energy stored in DNA during the conformational transition amounts to 3.4 As ;0.7;70 pN which corresponds to 2.4 kcal/mol and base pair. This is comparable to the free energy of base pairing which is 3 kcal/mol for guanine}cytosine and 1.5 kcal/mol for adenine}thymine.

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that DNA assumes a ladder-like structure with correspondingly much reduced twist under such high tension. This new conformation is called S-DNA. Here the base pairs exhibit a considerable tilt angle from the normal with respect to the helical axis. This is not the case for B-DNA which explains the high cooperativity of the transition } tilted and nontilted base pairs can only coexist at considerable energetic cost [34,94,86,1,87]. During the so-called BPS transition the DNA double helix is partially unwound. As both ends of the DNA are "xed, i.e. not free to rotate, this can only happen if one of the DNA strands has at least one `nicka. At the position of such a nick the DNA is only held together by a single covalent bond which is free to rotate. Such types of defects are fairly common in normal DNA. However, if all nicks are repaired using special enzymes (ligases) this is no longer the case and the linkage number of DNA is a topological constant. The linkage number counts how often both DNA strands are wound around each other. Using some techniques (magnetic forces or adapted microcantilever setups) DNA may be twisted. As DNA can be immobilized in a polar way, negative and positive twisting can be distinguished. The excess linkage number can be used either to change the twist of the DNA or to wind the DNA around itself in so-called `plectonemesa. These latter structures take up a lot of the DNA length and are familiar from telephone cords. Thus, twisting and DNA elongation under force are coupled which was found for the "rst time by Strick et al. [158}160]. At low tensions length consuming plectonemic structures prevail whereas at higher tensions (&10 pN) conformational changes of the DNA are dominant. It was shown that at negative supercoiling (i.e. unwinding the DNA) a part of the DNA denatures and forms a molten blob where the strands are not wound around each other [6,160]. This way, the topological constraint of "xed linkage number may be satis"ed while avoiding large elastic deformations which would increase the free energy considerably. For positive supercoiling a fraction of the DNA assumes a di!erent double helical conformation where the base pairs are pointing to the outside and much higher twist is achieved [6]. This conformation of DNA was "rst proposed by Pauling and is therefore termed P-DNA. To understand all conformational changes that occur in super- or under-coiled DNA at various forces one has to consider at least seven di!erent conformations of DNA: B-DNA, plectonemic B-DNA, S-DNA, P-DNA, plectonemic P-DNA, denatured DNA and Z-DNA [97]. In the latter conformation DNA assumes a left-handed double helix in contrast to all other conformations which are right handed. The theoretical description of DNA supercoiling under tension is very complicated and will not be described here. Instead the interested reader is referred to the original literature [175,104}107,118,78,22,127]. Surprisingly, hysteresis is absent in all these transitions, i.e. kinetic hindering is unimportant here. The two strands forming native DNA are kept together by hydrogen bonding and dispersion interaction between the bases on the di!erent strands. Piconewton forces are su$cient to separate the strands. Heslot and coworkers constructed a special DNA from two -phage DNAs and several DNA oligomers [44,19,20]. In this construct, one single DNA strand was continuous throughout the whole DNA whereas the other strand was broken in the middle and exhibited functionalized dangling ends at the break. This DNA construct was "xed to a coverslip through one end of the continuous strand. The broken strand was "xed in the middle of the DNA to an ultrasoft microneedle. By translating the cover slip relative to the needle interstrand bonds could be loaded

 Denaturation of DNA means that the base pairing is broken and both strands are no longer connected.  Only the reannealing of denatured DNA appears to take some time.

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and eventually broken. Forces were measured by detecting needle de#ections. Surprisingly, low forces of approximately 13 pN were su$cient for strand separation. The force exhibited #uctuations of about 1 pN on length scales of 0.5
m [44]. These force #uctuations were reproducible. Moreover, a di!erent DNA construct where -phage DNA was separated from the opposite end exhibited closely related features, albeit in reversed sequence. From these observations the authors concluded that the force #uctuations were speci"c for the DNA sequence. Indeed, if one assumes that 1000 base pairs are broken per
m translation, the force #uctuations correlate well with the content of guanine}cytosine base pairs in the sequence of -phage DNA. As the free energy of base pairing is 3 kcal/mol for guanine}cytosine pairs and only 1.5 kcal/mol for adenine}thymine pairs it seems natural that higher forces are required to separate GC base pairs. Despite the complexity of the interactions, unzipping of DNA is a reversible process [20] and may be modeled by an equilibrium statistical approach. The free energies of (i) both single strands, (ii) the base pairing and (iii) the microneedle must be considered. The free energy of the full system was minimized under the appropriate boundary conditions. This procedure yielded equilibrium needle de#ections that closely coincided with the data [19]. Please note that the measuring device (the needle) must be included in the statistical modeling of the process. Statistical #uctuations of the length of the separated strand occur at "xed elongation. Therefore, the GC content of the sequence is measured on a scale of 20}350 base pairs in these experiments [19]. Melting and unzipping of DNA was also studied by SFM [135,32]. In this work, short pieces of -phage DNA and engineered DNAs exhibiting either a -A-T-A-T- or a -G-C-G-C- sequence were used. The BPS transition was again found at 65 pN for -phage DNA at rates of force application ranging from 1.5 to 30 nN/s. For the pure GC sequence the BPS transition was not altered whereas the more weakly bound AT sequence exhibited the BPS transition at a much lower force of 35 pN. At forces beyond the BPS transition a separate melting transition of lower cooperativity was observed for -phage DNA and the pure GC sequence. The position of this transition depends on the rate of force application. For -phage DNA the transition force varied from 150 pN at a rate of 7 nN/s to 270 pN at a rate of 30 nN/s. The pure GC sequence melted at about two times higher forces, the pure AT sequence did not exhibit a separate melting transition. Pronounced hysteresis between stretch and release occurs if DNA is stretched into the melting transition, whereas remarkably little hysteresis was found in experiments where the melting transition was not reached. Thus the melting transition is kinetically hindered on the time scale of these experiments. Furthermore compelling evidence was presented that the AT sequence melts already during the BPS transition. Working on single strands of AT or GC sequence the authors could determine the equilibrium force necessary to separate GC base pairs (20 pN) and AT base pairs (9 pN). In earlier SFM work on DNA strand separation short single strands of DNA with complementary sequences were bound to SFM tip and substrate [122,95]. In the work of Lee et al. [95] broadly distributed separation forces ranging from 0.5 to 1.5 nN were found. The spread of the data was explained by a variable number of base pairs forming in the tip-substrate attachments. In later work, Noy et al. [122] used sequences that were designed to suppress such incomplete pairing. Here a yield force of &0.5 nN for a DNA with 14 base pairs was found. At lower forces DNA

 Single strands and microneedle are elastic and may be modeled as harmonic oscillators for this purpose.

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elasticity was found. In these data a broad transition at &120 pN is apparent. As it is not clear to what extend the DNA oligomers were rotationally restricted in these experiments a further interpretation of this transition is di$cult. In the latest experiments along these lines Strunz et al. [162] separated double stranded DNA of 10, 20 and 30 base pairs length using a SFM. In these experiments yield forces that depend on the rate of force application were found. In other words, kinetic hindering of detachment is the dominant feature here. This is entirely analogous as in the case of force-induced dissociation of single speci"c bonds and I will come back to these experiments in the respective section. 3.1.2. Experiments on other polymers Nearly, all polymers besides DNA exhibit at least one position per monomeric unit where the system is free to rotate. Thus polymer twisting is unimportant here. Following the outstanding experiments on DNA many more polymers were studied by force spectroscopy on the single molecule level. Because of the small size of most polymers these studies were performed with scanning force microscopy techniques. For the synthetic polymers poly(methacrylic acid) [128], poly(acrylic acid) [98], poly(dimethyl siloxane) [146] and poly(vinyl alcohol) [101] no conformational transitions were found. In most experiments the freely jointed chain model (Eq. (4)) including backbone elasticity (Eq. (7)) was used for data interpretation. However, in one case it was reported that the data could be equally well "tted by an extensible freely jointed chain or an extensible worm-like chain [128]. In another case, it was reported that a worm-like chain model gave better "ts that a freely jointed chain model, however backbone elasticity was not taken into account [146]. Data interpretation is very di$cult in these experiments because synthetic polymers are comparatively short (typical lengths are &100 nm) and thus a small data interval limited by resolution and polymer length hampers data "tting. In order to decide whether the worm-like chain model or the freely jointed chain model describes the data better one needs data spanning the force range from well below k ¹/l to well above this characteristic force (cf. Fig. 11). As k ¹/l I I amounts to only &10 pN for most synthetic polymers (cf. Table 1) we must await further developments in SFM technology to answer this question. The experimental situation is much more satisfying for biopolymers like polysaccharides because these exhibit a larger contour length (&1
m) compared to synthetic polymers. Due to the more complex architecture of these polymers conformational transitions occur frequently. In the "rst study in this "eld Gaub and coworkers examined the force}extension relation of dextran and carboxymethyl-dextran by SFM [139]. The linkage between tip and polymer was created either by streptavidin-biotin bonds or via nonspeci"c bonding. Using these methods no control over the length of the strand linking tip and substrate is possible. Thus, the contour length of the strand was an additional important "t parameter. The data were "tted to an extensible freely jointed chain model (Eq. (4) and (7)). At higher forces a region of reduced polymer sti!ness was found. This transition region extends from 200 to 300 pN for carboxymethyl-dextran and from 700 to 850 pN for dextran. No hysteresis between stretch and release was found. Therefore kinetic barriers are  Some other biopolmyers like "lamentous actin exhibit similar topological constraints like DNA, however here it is much more di$cult to immobilize single "laments by biochemical manipulation. Therefore, the corresponding twisting experiments are not feasible at the present time.

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unimportant on the time scale of the experiment (&0.1 s). Moreover, all curves collapsed onto a master curve after normalization of the polymer extension by the contour length. This shows that single polymers were probed. Parallel to the experimental work it was shown by molecular dynamics that this transition can be explained by a rotation of one C}C bond which extends the length of the monomeric glucose unit [139]. Later work followed these lines. Conformational transitions were found in carboxymethyl-amylose and heparin, but none in carboxymethylcellulose [100]. Again, the experiments were analyzed by molecular dynamics and the underlying conformational transitions were identi"ed as the chair}boat transition of the glucose unit [70]. The same conformational transition was observed in a parallel study for the polysaccharides dextran, amylose and pullulan [110]. In the latter study, it was shown that chemical cleaving of the glucose ring abolished the transition. Moreover, ab initio calculations were used to determine which conformation of the glucose ring is thermodynamically stable under the respective tensions. The only example of a synthetic polymer exhibiting a comparable conformational transition is poly(ethylene glycol). In water, hydrogen bonding induces helix formation of the ethylene glycol repeat units. These helices are `stretched outa by mechanical tension. The stretched conformer is about 0.8 As longer and its equilibrium free enthalpy is about 3 k ¹ higher compared to the helical segments. Because of this low enthalpy di!erence the transition is continuous and covers basically the full force spectrum. Taking into account, the change in enthalpic di!erence due to an external force the force spectrum could be calculated by assuming an equilibrium population of both states and a freely jointed chain behavior for the whole chain [126]. Essentially, the same picture was found in theoretical work [89], albeit additional conformational states were found. As in the case of polysaccharides no hysteresis was found. The absence of kinetic barriers is the reason for the remarkable success in modeling the conformational transitions of polymers (cf. [139,70,110,89]) because one can search for the absolute minimum of the free energy. For this purpose, highly e$cient tools of theoretical physics exist. Up to now native xanthan is the only example where substantial hysteresis is present in a conformational transition of a biopolymer other than DNA [99]. 3.2. Force-induced dissociation of specixc bonds One of the most stunning facts in biology is the precision with which biomolecules recognize their target molecules. For example, the receptors of the immune system distinguish perfectly between `owna and `foreigna cells, but also between this winter's in#uenza virus and last year's. The tremendous speci"city of biomolecular interactions allowed the evolution of highly e$cient and well di!erentiated cells and organisms [3,163]. The so-called `speci"c bonda is in fact the result of many weak local interactions summing up into an overall stable bond. The relevant interactions (e.g. electrostatic attraction, hydrogen bonding, dispersion interaction, hydrophobic force) are all of short range (&1 nm) under physiological conditions and result in binding enthalpies on the order of the thermal energy k ¹ [35]. Biomolecules (e.g. proteins, glycolipids, DNA) are macromolecules with a complex and well-de"ned three-dimensional structure [40]. Therefore, their  Physiological conditions imply an aqueous bu!er exhibiting a neutral pH (&7.4), an ionic strength of &300 mM, and a temperature in the range from about 03C to about 403C.

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surfaces are highly structured as well. Only in cases where two molecules possess topologically and chemically complementary surfaces it will be possible that the weak and localized interactions add up to an overall strong bond with a binding energy of some 10 k ¹ or more. Compared to covalent bonds, these interactions are weak, especially if one considers that large molecules consisting of several thousands of atoms are involved. However, the probability that two unrelated molecules exhibit complementary surfaces is very remote. Therefore, this type of bonds is the basis for the superb speci"city of biomolecular interactions [3]. From a thermodynamic perspective, speci"c bonds can be understood by the standard concepts of chemical equilibrium. However, as soon as a force is applied, the system is no longer in thermodynamic equilibrium and poses therefore a challenging experimental and conceptual problem. Experiments involving many bonds at one time are straightforward to perform because relatively large forces may be applied. Yet here many new phenomena appear due to collective e!ects (e.g. lateral interactions between di!erent cell}cell bonds by the elastic deformation of the cell surface). A recent theoretical study provides an example for the wealth of phenomena that occur due to multiple bonds. In that study the most simple limiting case was considered, that is all bonds were assumed to experience the same force. Despite this simple model the dependence of the net yield force on the number of bonds switched from logarithmic to direct proportionality, depending on the rigidity of the surface anchoring of the proteins, the sti!ness of the force probe, the reversibility of bonds, and the retraction speed [145]. On the one hand, these collective e!ects are known as the principle of multivalency in biological interactions and are of great importance here. On the other hand, the existence of collective e!ects makes it impossible to extract singlebond properties from experiments on the many-bond level [21]. Still, the properties of single bonds are the very basis for bioadhesion and thus we must understand them before we can try to model collective adhesion. Investigating force-induced dissociation of single speci"c bonds is most challenging. The basic experimental phenomenon that is observed is the disruption of a linkage between two surfaces, e.g. the loss of adhesion between SFM tip and substrate. On the level of single events there is no criterion to distinguish the speci"c bonds under investigation from the unwanted nonspeci"c events and to determine the locus of bond failure. Therefore, one must optimize the preparation to a level where nonspeci"c events and multiple bonds are truly negligible; moreover, the locus of bond failure must be de"ned in all events. The problem is enhanced by the relatively small size of the involved molecules which enforces working in close proximity to the solid substrate where nonspeci"c events are most prominent. We shall see below that force-induced dissociation of single speci"c bonds is entirely dominated by kinetic e!ects because all known speci"c bonds exhibit large kinetic barriers. Moreover, as soon as we apply a force to a bond we basically provide an in"nite source of energy, i.e. once the bond is broken the two binding partners separate to in"nite distances. Thus, the process of force-induced bond dissociation is for practical considerations irreversible. The bound state is no longer the state  Please note that the situation is much more favorable in the case of protein unfolding or polymer stretching because there a full force}extension relation is measured per single event and not just one yield force or time of bond dissociation. If one assumes that unfolding follows the same pathway in all events, the shape of each single curve and its reproducibility may be used as criterion for single molecule events and to discriminate nonspeci"c events. Moreover, the question of a locus of failure is not relevant.

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of stable thermal equilibrium as soon as a force is present. It is at best a metastable state. Thus, the study of force-induced bond dissociation amounts in essence to measuring the statistical lifetime of a metastable state. To understand the experiments, we must consider the in#uence of an external force on the rate of bond dissociation, k ( f ). This problem cannot be solved in a rigorous way, still, a meaningful  expression for k ( f ) may be motivated by the following considerations that were "rst put forward  by Bell in a seminal article on cell adhesion [16] and later discussed in a much more quantitative way by Evans and coworkers [48]. The phase space of a speci"c bond possesses 6, coordinates, where N is the number of atoms involved. N is on the order of 10 000. The bound state must be a local minimum of free energy separated from the free state by an energy barrier. Despite the complexity of this phase space, there will be in many cases a unique way connecting these two states along which the system has to cross the lowest possible energy barriers. Therefore, the vast majority of unbinding reactions occurs along this path in phase space. This path is called the reaction path. If the time scales of the internal modes within the protein and those of the motion along the reaction path (i.e. the time scale of the experiment) are well separated, the free energy along this reaction path is well de"ned. For rigid proteins this should be a good assumption, because here the internal modes are
sec processes and faster. However, for proteins exhibiting large conformational #uctuations this may not hold. In fact, it was shown that some very large proteins may lose their three-dimensional structure in response to an external force, cf. Section 3.3. In that case, the whole concept presented here is obviously meaningless. Our description of force-induced dissociation relies on the assumption that the sole action of the force is adding a term !fx to the thermodynamic potential of the system, other changes (e.g. unfolding) are neglected [16,48] (Fig. 12). If we further assume that a single barrier dominates the dissociation process, it is possible to give approximate expressions for the rate of bond dissociation in two limiting cases. The "rst case corresponds to a frictionless motion in the potential. For this case, the transition state theory, sometimes also called Eyring theory, applies (see, for example, Ref. [60]). Here it is assumed that all molecular complexes dissociate that reach the activation energy maximum. The state of the molecule at this barrier is termed the `transition statea, hence the name of the theory. The corresponding result is





!GH k ¹ exp , k "  k ¹ h

(8)

where GH is the di!erence in free enthalpy between transition state and bound state.  Here and in the following, I apply a classical description in the spirit of the Born}Oppenheimer model of molecules.  In general, dissociation may occur along several di!erent pathways exhibiting energy barriers of comparable magnitude. In this case, an external force might lead to a gradual change of the frequencies of occurrence of the di!erent paths. This would result in a much more complicated dependence of the dissociation rate on force compared to the simple picture to be described in the following.  Otherwise the potential depends on the history of the system, see e.g. Ref. [66].  Please note that the in#uence of torque is excluded as well. Torque may arise in experiments where the biomolecules under study are connected to the surfaces by bonds of limited #exibility. Modelling this e!ect is only possible if the microscopic details of the surface anchoring are known, which is generally not the case.

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Fig. 12. Schematic variation of the thermodynamic potential along the reaction coordinate in the most basic approach. Shown are the undisturbed potential (solid line) and the potential under in#uence of a force f (dotted line). The force contributes a term !fx to the potential (dashed line). For the undisturbed potential, the activation energy barrier GH is indicated. The location of the bound state of the undisturbed potential was used as origin of the x-axis.

The second limiting case that may be dealt with is overdamped motion of the system along the reaction path (see, for example, Ref. [66]). In this case, the theory of Kramers may be used which predicts







  

!GH k "   exp . 

k ¹ Here is the friction experienced by the system moving along the reaction coordinate, and the factors are the oscillation frequencies at the bound state and the transition state, respectively. If the potential barrier is steep, adding a term !fx to the free enthalpy will reduce the activation energy barrier by an amount of f l, where l is the distance between bound state and transition state along the reaction coordinate. Thus, we get the approximate result [48] !GH fl k +   exp exp "k eDD ,  

k ¹ k ¹

(9)

where k denotes the dissociation rate of the unstressed bond, and the characteristic force of the  bond, f , is de"ned by k ¹/l.  Please note that the same "nal expression, k +k eDD , can be derived from the transition state   theory (cf. Eq. (8)) albeit with a di!erent expression for the prefactor k . From the experiments, it is  impossible to conclude which theory applies because we can neither measure nor in#uence the friction in the interior of the protein complex. However, this question is of little interest here because both approaches result in equivalent expressions. While in this approach some basic assumptions about the free energy along the reaction path are made (e.g. a single steep barrier is assumed) a way of reconstructing more details of this free energy pro"le without a priori assumptions was proposed in a recent publication [71].

 This implies the further assumption that the force acts along the reaction path. In view of the elasticity of biomolecules this is most likely a good assumption.

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In experiments using hydrodynamic shear force bonds are exposed to a constant force. The basic quantity in these experiments is the time of bond dissociation. By repeating the experiment very often one determines the distribution of bond lifetimes, i.e. the rate of bond dissociation at given force, k ( f ). In all other techniques force is increased gradually and one determines the yield force  of a single bond as basic quantity. After repeating the experiment many times the distribution of yield forces is constructed from the data. This design of the experiment obscures the underlying stochastic nature of bond dissociation. As a consequence, the misleading concept of `a de"ned yield force for a given type of bonda came to be and is slow to fade despite compelling experimental evidence against it. In order to relate the rate of bond dissociation to the spectra of yield forces one has to consider the kinetics of the bound population [48]. A standard "rst-order kinetic scheme may be used. dN/dt"!k t#k ) (1!N), N(t"0)"1 , (10)   where N is the fraction of bonds that existed at time zero and remain intact up to time t, k is the  rate of bond formation. As soon as a "nite force is applied bond formation may be safely neglected because once the bond is broken both binding partners are separated with "nite velocity to macroscopic distances. We must further perform a coordinate transformation from time, t, to force, f. As force is linearly increased, f (t)"fQ t, this is a trivial task. The formal solution for N( f ) is given by

 



1 D k() d . (11) N( f )"exp ! fQ  The quantity N( f ) can be directly determined from the experiments as it is the cumulative distribution of yield forces. Most often, however, distributions of yield forces are displayed. Using Eqs. (9) and (11), it is straightforward to calculate the median of yield forces, f , and the most  probable yield force, f H, i.e. the maximum of the distributions.






fQ ln 2 , f "f ln 1#   k f  fQ for fQ 'k f , f H" f ln k f    else . 0



(12)

Both quantities exhibit a characteristic dependence on ln fQ . This is the direct result of the stochastic nature of bond dissociation. A statement like `a bond between x and y fails at z pNa is meaningless unless the rate of bond loading, fQ , is speci"ed. Given enough time any bond will fail, even at vanishing load. Please note that a nonlinearly increasing force results in equations that di!er from Eqs. (12). This occurs, for example, if the adhesion molecules are linked to the substrate by very long polymers [49]. For most early experiments on force-induced dissociation of single bonds just a single yield force was reported. However, in these "rst experiments the feasibility of force spectroscopy on single

 However, in the case of covalent bonds the average lifetime of an unstressed bond may be billions of years and more.

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Table 2 Summary of reported bond yield forces from experiments where the rate of force application was not varied Bond

Technique

fQ

Yield force

Ref.

Antibody}antigen

Hydrodynamic shear

Not given

&20 pN

[171]

Antibody}antigen

Micropipette aspiration

Not given

&50 pN

[170]

Actin}actin

Microcantilever bending

Not given

110 pN

[84]

Biotin}avidin Iminobiotin}avidin

SFM SFM

Not given Not given

160 pN 90 pN

[55]

Biotin}streptavidin Desthiobiotin}avidin Iminobiotin}streptavidin

SFM SFM SFM

Not given Not given Not given

260 pN 90 pN 140 pN

[119]

Biotin}streptavidin

SFM

Not given

340 pN

[96]

Antibody}antigen

SFM

40 nN/s

240 pN

[72]

Biotin}avidin

SFM

Not given

120 pN

[56]

Au}S or Au}Au Si}C

SFM SFM

10 nN/s 10 nN/s

1.2 nN 2.0 nN

[63]

bonds was demonstrated. Hydrodynamic shear [171], micropipette aspiration [170], SFM [55,119,72,38,37,56,7], and microcantilever bending [84] were used to this end (Table 2). In a second line of experimental development, the stochastic nature of force-induced bond dissociation was investigated. Evans and coworkers used an early version of the biointerface force probe (cf. Section 2.5) to investigate force-induced bond dissociation and found that a bond loaded with constant force survived a certain time. These bond lifetimes exhibited a statistical distribution [46]. Using hydrodynamic shear on bead doublets Goldsmith and coworkers were able to "nd the same e!ect for another type of bonds and could for the "rst time give estimates for the parameters k and f in Eq. (9) [167,168,92]. In these experiments bonds experienced a complicated force   history, additionally more than one bond was present. Therefore, the experiments were later reanalyzed using a Master equation approach which yielded re"ned results [103]. Using laminar #ow chambers the results may be interpreted much more directly. These experiments were mostly performed on whole cells subjected to shear #ow. Cells adhered to ligands immobilized to the bottom of the #ow chamber. The single-bond regime was reached by successively reducing the density of ligands until `quantizeda cell arrests were found. The distributions of the durations of arrests for several ligand densities below the threshold appeared single exponential and yielded coinciding dissociation rates. Thus, these experiments probed most likely single bonds using constant forces. In the pioneering work of Alon et al., the in#uence of force on bonds between neutrophils and di!erent selectins was studied [8}10]. The results could be well described with Eq. (9). In later work, many more di!erent bonds were studied using whole cells, cf. Table 3. In the context of this review the results of Puri et al. are very interesting [132]. In this work, the adhesion molecule CD34 was immobilized to the chamber wall and chemically modi"ed. Investigating the force dependence of bonds between these modi"ed forms of CD34 and neutrophils one example

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Table 3 Summary of measured kinetic parameters of bonds as given in the respective publications Bond

Technique

k 

l

Ref.

P-selectin}ligand on neutrophils L-selectin on neutrophils}ligand E-selectin}ligand on neutrophils

HSFC HSFC HSFC

1 s\ 7 s\ 0.7 s\

0.4 As 0.2 As 0.3 As

[10,9] [9] [9]

L-selectin on neutrophils-CD34 -periodate treated CD34 -periodate and borohydrite treated CD34

HSFC HSFC HSFC

8 s\ 6 s\ 6 s\

0.2 As 0.1 As 0.3 As

[132]

L-selectin}ligand on neutrophils

HSFC

7 s\

0.2 As

[8]

P-selectin}ligand on neutrophils E-selectin}ligand on neutrophils L-selectin}ligand on neutrophils L-selectin on neutrophils}ligand

HSFC HSFC HSFC HSFC

2 s\ 3 s\ 3 s\ 4 s\

0.4 As 0.2 As 1.1 As 0.6 As

[154]

Antibody-cell bound CD15

HSFC

2 s\

0.9 As

[29]

Extraction of blood group B antigen from erythrocytes Antibody-blood group B antigen Protein G-IgG

HSBD

0.06 s\

3 As

[167,103]

HSBD HSBD

0.08 s\ 0.01 s\

1.4 As 3 As

[168,103] [92,103]

Extraction of L-selectin from neutrophils Extraction of CD18 from neutrophils Extraction of CD45 from neutrophils

MA MA MA

0.06 s\ 0.05 s\ 0.005 s\

3 As 0.9 As 3 As

[148]

P-selectin}ligand

SFM

0.02 s\

2.5 As

[57]

Streptavidin-biotin

SFM

1.2 As

[179]

Nitrilotriacetate-His-tag (NTA-His )  L-selectin}ligand

SFM

2 As

[82]

12, 3, 0.6 As H

[45]

Protein A-IgG

BIP

7 As

[152]

Streptavidin-biotin Avidin-biotin

BIP BIP

5, 1.2 As H 30, 3, 1.2 As H

[113]

Extraction of SOPC from SOPC membrane Extraction of SOPC from SOPC/chol membrane

BIP BIP

7, 12 As H 17 As

[47]

0.07 s\

BIP 0.1 s\

0.9, 0.09 s\H 0.016 s\

The equation k "k exp( f/f ) was used. Here f denotes the acting force, f "k ¹/l, and l is the distance between     bound state and transition state. Multiple results (marked with H) refer to experiments where several di!erent barrier widths were observed. HSFC hydrodynamic shear using a #ow chamber, HSBD hydrodynamic shear on bead doublets, MA micropipette aspiration, SFM scanning force microscopy, BIP biointerface force probe, SOPC L--1-stearoyl-2oleoyl- phosphatidylcholine, and chol cholesterol.

was found were the dissociation rate, k , remained una!ected but the characteristic force, f , was   changed by almost a factor of 3. This proves that the absolute value of the rate of bond dissociation, k , is no indicator for mechanical toughness of the respective bond. In a di!erent work, the force  dependence of adhesion bonds was probed in a cell free assay using otherwise identical methods.

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375

Here again the data were well described by Eq. (9) [130]. In a recent study clear deviations from this equation were found at high forces (&250 pN) for neutrophils bonded to selectins [154]. These results may be explained by a change in the locus of bond failure. Shao and Hochmuth investigated the force dependence of receptor extraction from neutrophils [148], their results indicate that a crossover to receptor extraction should occur at high forces. Tests of Eq. (9) were also performed using techniques where force is linearly increased. The corresponding distributions of yield forces were compared to Eq. (12). Using surface force microscopy (SFM) P-selectin}ligand bonds were studied [57]. By the same technique double stranded DNA was `unzippeda [162]. In this case, the distance l between bound state and transition state and the spontaneous dissociation rate k depend on DNA length. Avidin}biotin  and streptavidin}biotin bonds were investigated using the biointerface force probe, cf. Section 2.5. For avidin}biotin bonds several linear regimes were found for the dependence of f H on ln fQ [113,45]. The corresponding characteristic forces agreed with the shape of the thermodynamic potential along the reaction coordinate which was calculated by molecular dynamics simulations [75]. Crossover is caused by an inner barrier in this potential. Similar crossovers were observed for the dynamic strength of L-selectin}ligand bonds [45]. Using the same technique force-induced dissociation of single protein A-IgG bonds was studied [152,161]. Here, it was possible to demonstrate that the rate of spontaneous bond dissociation, k , as determined form force-induced  bond dissociation and a "t to Eq. (9) coincided with the dissociation rate of unloaded bonds. Thus, it was shown that force-induced and spontaneous bond dissociation are indeed closely related processes as implied by our theoretical excursus at the beginning of this section. 3.3. Unfolding of multidomain proteins Proteins fold into a unique three dimensional structure, the so-called `native conformationa. This conformation is a minimum of the free enthalpy of the protein. It is stabilized by weak localized interactions like electrostatic interaction, van der Waals force, hydrogen bonds and hydrophobic e!ect } exactly the same forces that stabilize speci"c bonds (cf. Section 3.2). Therefore, it is not surprising that force-induced unfolding of proteins may be modeled in close analogy to force-induced dissociation of speci"c bonds. However, there is one important di!erence. After bond dissociation the force separates receptor and ligand to in"nite distances whereas both ends of an unfolded protein are still connected by the polypeptide chain. Therefore, force-induced unfolding of proteins is reversible and we cannot neglect refolding as we have neglected rebinding in deriving Eq. (11). Yet folded and unfolded conformation are separated by a pronounced barrier in the thermodynamic potential. Thus kinetic e!ects are very important in force-induced unfolding of proteins. Additionally, as the maximum separation between both ends of the polypeptide chain is limited the unfolded state is favored by a xnite energy upon loading the protein. Therefore, the folded conformation is still thermodynamically stable if "nite but small enough forces are applied  For DNA lengths of 10, 20 and 30 base pairs the following correlations were found: l+7 As #0.7 As ;n and k +10\L s\. Here n stands for the number of base pairs in the double stranded DNA.   This technique was also used to test the mechanical anchoring strength of single lipid molecules in arti"cial membranes. These are not single molecular bonds but self-assembled supramolecular systems, yet the same basic features were observed [47].

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[43,85] whereas a loaded speci"c bond can be at best a metastable state. The unfolded polypeptide chain acts as #exible linker exhibiting a force}extension relation like any other #exible polymer (cf. Section 3.1). Thus this section on unfolding of multidomain proteins is a synthesis of both preceding sections. The experiments on force-induced unfolding are best to perform on large proteins exhibiting many homologous domains. Using such proteins one may search for repetitive features in the force}elongation curves and there is ample de#ection before the polypeptide chain is pulled taut. Therefore, the "rst experiments in this "eld were performed on the giant muscle protein titin which consists of about 150 immunoglobulin domains and 132 "bronectin-III domains. This molecule was stretched using optical traps [173,80] and SFM [137]. The experiments were performed very di!erently and provide a vivid example of how di!erent experimental techniques may yield seemingly con#icting results that can still be explained by one and the same underlying mechanism. In the SFM experiments [137,138] titin was suspended between tip and substrate and stretched by retracting the tip with constant speed (0.06}3
m/s, cantilever sti!ness 80 mN/m). The resulting cantilever de#ections revealed a characteristic sawtooth pattern: every 25 nm a force peak of 150}300 pN amplitude occurred. The rising #ank of each peak was gradual and could be "tted by the force}extension relation of a worm-like chain, Eq. (6). The persistence length of 4 As coincided with the length of a peptide bond. Thus, it is the unfolded peptide chain that gives raise to the gradual increase of force at the rising #ank of each peak. Between successive peaks the contour length of the worm-like chain increased by about 28 nm. This is exactly the length increase that one expects from unfolding a single domain. Moreover, the authors showed that such curves could be repeatedly measured if the force was released for about 1 s between successive measurements. Thus refolding of titin domains occurred rapidly. The height of the force peaks depended on the rate of force application. Higher rates resulted in higher forces. This e!ect was explained by a kinetic barrier for unfolding. Eq. (9) may be used to model the in#uence of force on the rate of unfolding. This is due to the close analogy between unfolding of proteins and bond breaking. By modeling the process of unfolding the authors determined the distance between the folded state and the transition state for unfolding, l, and found a value of 3 As . The rate of spontaneous unfolding in absence of force, k , exhibited a value of 3;10\ s\.  The group of Bustamante suspended single titin molecules between two microbeads and stretched the protein continuously using an optical trap [80]. The optical trap used in their experiment had a 500 times lower sti!ness compared to the SFM cantilevers in the above described study. Therefore, the force was much slower applied than in the SFM study which resulted in unfolding of titin at much lower forces of approximately 25 pN. Moreover, unfolding of single domains could not be detected because the corresponding elongation of the polypeptide chain reduced the acting force by only a few piconewton, which is not su$cient to suppress force-induced dissociation of further domains. The data could be well described by worm-like chain elasticity coupled with continuous unfolding at high forces during stretching and refolding at low forces during release. In another optical trap study [173], the trap was rapidly displaced and the subsequent force relaxation was observed. Here, peak forces up to 200 pN were reached. Force relaxation occurred in discrete steps which were attributed to single domain unfolding.

 Single domain unfolding can be observed only if a stiw enough spring is used as force probe.

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377

All these observations on titin unfolding can be explained using a unique description that takes into account the force dependences of the unfolding rate and the refolding rate plus the elasticity of the polypeptide chain [136]. Subsequently more proteins were unfolded using SFM. The extracellular matrix protein tenascin contains "bronectin-III domains and therefore exhibits during unfolding phenomena very similar as in the experiments on titin [123]. Spectrin, however, exhibited unfolding in SFM studies at about "ve times lower forces compared to titin [140]. This indicates that the unfolding pathway is very di!erent for this protein. In studies of the proteins of a bacterial surface layer (hexagonally packed intermediate layer of Deinococcus radiodurans) [120] and bacteriorhodopsin [125] highresolution SFM imaging and unfolding of speci"c structures within the images were combined. Titin was reinvestigated and an unfolding intermediate was identi"ed as hump in the raising #ank of the sawtooth pattern. This intermediate was also found in steered molecular dynamics simulations [109]. The interpretation of the aforementioned studies on native proteins is hampered by the fact that the protein domains are highly homologous but not identical. Thus, domain-to-domain variations in unfolding experiments are expected. This limitation was overcome by genetic engineering a protein which consisted of one speci"c titin domain repeated several times [28]. This engineered multidomain protein was unfolded using SFM. In the force}distance traces similar sawtooth patterns as for titin were found. Again, worm-like chain parameters and the kinetic parameters for unfolding were determined (k "3.3;10\ s\, l "3.5 As ). Additionally, protein refolding was   studied in presence of a constant force. From these experiments the parameters of the refolding kinetics (k "1.2 s\, l "23 As ) were deduced. Using transition state theory, Eq. (8), the free   enthalpies of transition state and unfolded state were determined from these kinetic parameters. The values were compared to the results of equilibrium denaturation studies on the same protein using guadinium chloride. Good coincidence was found. This shows that force induced unfolding is closely related to equilibrium unfolding. The sole action of a mechanic force is shifting the equilibrium and changing the reaction rates.

4. Discussion and outlook One of the most important messages of this review is the revelation that our measurement tools are integral parts of the experiment. Thus, we do not study a single molecule but this single molecule attached to a spring and a surface of speci"c properties. Very lucid examples for this point are the studies on titin unfolding [173,80,137] and on DNA unzipping [44,19,20]. A second very important aspect is the discussion of kinetic e!ects which is one of the main foci of this review. This latter point requires that experiments must be performed on many very di!erent time scales. For these two reasons (force probe speci"c e!ects and kinetic e!ects) it is important to compare the results obtained by very di!erent techniques on one and the same physical object. Very often seemingly con#icting results will be found but a unique description of the phenomena will be possible in the long run. It is impossible to foretell the development of such a thriving "eld as single molecule force spectroscopy. However, some lines of development seem predictable. First of all, surface anchoring of biomolecules is still an underdeveloped art. Here we will see the advent of new approaches

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stemming from classical chemistry and genetic engineering. The suitability for the purpose of single molecule force spectroscopy of many established immobilization procedures remains to be explored. Another important line of development will be the genetic engineering of molecules to be studied. Using such an approach `ideala molecules can be produced to answer speci"c questions. The study of the Fernandez group on one speci"c titin domain [28] is a good example for the potential of such an approach. We may look forward to many surprising discoveries that will deepen our understanding of the action of mechanical forces on single molecules and thus on some of the very basic phenomena in living organisms.

Acknowledgements I am most grateful to my scienti"c teachers, Erich Sackmann and Evan Evans, who started my interest in this "eld of research. My own work was funded by the Deutsche Forschungsgemeinschaft and the Alexander von Humboldt Stiftung. I thank my students Christian Maier and Melanie Hohenadl for many walks to di!erent, and in some cases remote, libraries. The insightful remarks of Berenike Meier, Valentin Kahl, Zeno Guttenberg, Melanie Hohenadl, Udo Seifert, and Erich Sackmann greatly helped to improve this review.

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Colossal-magnetoresistance materials: manganites and conventional ferromagnetic semiconductors E.L. Nagaev* Institute for High Pressure Physics of Russian Academy of Sciences, 142092 Troitsk, Moscow Region, Russia Received June 2000; editor: A.A. Maradudin Contents 1. Introduction 2. Structural and magnetic properties of the basic materials 2.1. Conventional ferromagnetic semiconductors 2.2. Crystallographic and magnetic properties of undoped lanthanum manganites 2.3. Theories for the undoped manganites 3. Nondegenerate ferromagnetic semiconductors, s}d-model and double exchange 3.1. Optical and transport properties, experimentally 3.2. The s}d-model for the simple and double exchange 3.3. The charge carrier energy spectrum 3.4. The resistivity peak and CMR in nondegenerate semiconductors. A general picture 3.5. Donor indirect exchange Hamiltonian 3.6. Simple exchange. Spinwave region 3.7. Simple exchange. Paramagnetic region 3.8. The negative and positive magnetoresistance of nondegenerate semiconductors 3.9. The double exchange case and discussion of the results

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397 399 403 403 405 410

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4. Degenerate ferromagnetic semiconductors and magnetoimpurity theory 4.1. Resistivity and magnetoresistance of degenerate ferromagnetic semiconductors 4.2. The Mott transition at ¹"0 4.3. Physical background for the resistivity peak and CMR 4.4. Dielectric and magnetoelectric response functions in magnetic semiconductors far from the Curie point 4.5. Spin-wave region for a simple exchange 4.6. Paramagnetic region for a simple exchange 4.7. Magnetoelectric response functions for the double exchange 4.8. Magnetoimpurity resistivity and colossal magnetoresistance 4.9. Reentrant or nonreentrant temperatureinduced Mott transition? 5. Ferromagnetic lanthanum manganites 5.1. Crystallographic properties of doped manganites 5.2. Magnetic properties of ferromagnetic manganites 5.3. Resistivity of lanthanum manganites

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* Tel.: #7(095)1383211; fax: #7(095)3340012. E-mail address: [email protected] (E.L. Nagaev). 0370-1573/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 1 1 - 3

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E.L. Nagaev / Physics Reports 346 (2001) 387}531 5.4. Negative isotropic colossal magnetoresistance of crystals 5.5. Negative isotropic colossal magnetoresistance of thin "lms 5.6. Hall e!ect and thermopower 5.7. Electron spectra and problem of the double exchange 6. Underdoped and overdoped LaMnO  based manganites 6.1. Underdoped manganites: instability of the canted antiferromagnetic ordering 6.2. Bound magnetic polarons (ferrons) 6.3. Electronic ferro-antiferromagnetic phase separation and record CMR 6.4. The impurity and mixed impurityelectron phase separations. Spinglass } ferromagnetic state 6.5. Overdoped samples, charge ordering, and stripes 6.6. Polaron lattice? 7. Other manganites 7.1. Pr-based manganites 7.2. Nd-based and less-common manganites

450 453 456 459 460 460 464 466

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7.3. Photoinduced insulator}metal transition in manganites 7.4. Theoretical problems of Pr(Nd)-based manganites 7.5. Layered manganite crystals 8. Isotope-e!ect and electron}phonon interaction 8.1. The oxygen isotope e!ect in the manganites 8.2. Oxygen nonstoichiometry as an origin for the isotope e!ect 8.3. Conventional polarons in the ferromagnetic semiconductors 8.4. The problem of the Jahn}Teller polaron in the manganites 9. Theories of transport and magnetotransport properties. Concluding remarks 9.1. State of theory for the transport and magnetic properties of manganites 9.2. Problems of physics of the manganites Acknowledgements Note added in proof References

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Abstract Experimental data and their theoretical interpretation are presented for the colossal magnetoresistance (CMR) materials to which conventional ferromagnetic semiconductors and manganites belong. The similarity between the transport and magnetotransport properties of the ferromagnetic semiconductors and manganites is stressed. Di!erent theoretical points of view on the physics of the phenomenon are presented, including the magnetoimpurity theory and `magnetization#Jahn}Teller phonona theory. Of the manganites, the La-based underdoped, normally doped, and overdoped manganites are discussed in detail, including the dilemma `canted antiferromagnetic ordering or phase separationa and di!erent models of charge ordering. Radical di!erence between the Pr-based and La-based manganites is explained. Layered manganites are considered, too.  2001 Elsevier Science B.V. All rights reserved. PACS: 75.50.Pp; 75.50.Ss; 75.70.Pa Keywords: Colossal magnetoresistance; Resistivity peak; Phase separation; Charge ordering

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1. Introduction Current problems of microelectronics make it highly desirable to construct devices with large isotropic negative magnetoresistance functioning at room temperatures. In particular, they are required for magnetic recording and reading heads, for reliable storage of information, etc. In recent years the main line of investigation comprised multilayered magnetic "lms and granulated magnetic systems. The maximal value of the relative magnetoresistance
"[(H)!(0)]/(H), & taken at a "eld strength of 6 T, was found in Fe}Cr "lms at 4.2 K: it amounts to !150% [1]. This line may turn out to be nonoptimal, as there exist magnetic conductors with colossal magnetoresistance (CMR), by many orders of magnitude exceeding giant magnetoresistance of multilayered "lms and granulated systems (it is customary to speak about the giant magnetoresistance in the multilayered "lms and about the colossal magnetoresistance in the materials of the manganite type). These materials are classical degenerate ferromagnetic semiconductors and manganites. They are characterized with a high resistivity peak in the vicinity of the Curie point ¹ ! (Figs. 7, 9 and 15). The height and the width of the peak diminish with decreasing zero-temperature resistivity. This peak can be suppressed or reduced by an external magnetic "eld which is the manifestation of the isotropic negative magnetoresistance. Hence, the CMR is intimately associated with this resistivity peak, and both these speci"c features are inseparable. Below, when I speak about CMR I often shall mean also the resistivity peak. The classical magnetic semiconductors are the transition-metal or rare-earth compounds. Examples of the former are CdCr Se or HgCr Se compounds, examples of the latter EuO,     EuS. All these materials, in the absence of donor or acceptor impurities, are insulating. The superexchange between d-spins in the former and between the f-spins in the latter establishes the ferromagnetic or antiferromagnetic ordering (all the materials presented above are ferromagnetic). If these materials are doped with donor (acceptor) impurity then at su$ciently large doping degrees the Mott transition in the system of donors takes place: the donor electrons become delocalized, and the samples become highly conductive. The electron delocalization leads to the e!ective exchange between the localized d- or f-spins via conduction electrons (holes) whose nature is as follows. A speci"c feature of the magnetic semiconductors is a strong dependence of the conduction electron energy on the magnetization of the crystal due to the exchange interaction between the mobile electron and localized d(f )-spins. The minimal electron energy is achieved at the complete ferromagnetic ordering (this statement can be proved in a general form [11]). For this reason the electrons tend to establish and support this ordering if their density is not very high. As a result, an initially insulating antiferromagnetic crystal can be transformed into highly conductive ferromagnetic by the doping. Such a transformation was observed, e.g., in EuSe. The existence of the resistivity peak does not depend on the initial magnetic ordering. The same occurs in the La-based manganites LaMnO as a result of their doping by divalent Ca,  Sr and so on. These impurities function as acceptors transforming the initially antiferromagnetic and insulating crystal LaMnO into ferromagnetic and highly conductive one. Strictly speaking,  one may talk about these manganites as the degenerate ferromagnetic semiconductors only at relatively small dopings. At larger dopings (30% and more) their properties become still more complicated.

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But there are other manganites whose properties di!er drastically from degenerate ferromagnetic semiconductors at any doping, e.g., PrMnO . These materials are insulating at ¹"0  independent of the doping degree, but exhibit an insulator-to-metal transition with increasing temperature or under an external magnetic "eld. Perhaps, they are not very promising as the CMR materials. In talking about the manganites, "rst of all, we shall keep in mind the La-based ones. A shortcoming of the standard magnetic semiconductors for applications is the fact that their CMR is observed at low temperatures. For this reason, a special interest is attached to the LaMnO manganites, "rst synthesized by Jonker and van Santen in 1950 [2]. The reason for this is  the fact that they can function at room temperature. For example, in La Ca MnO "lms the     W relative magnetoresistance
was found to amount to !127 000% at 77 K and to !1300% at & room temperature [3]. Certainly, these CMR values greatly exceed those obtained for multilayered "lms and granulated systems, but they are not the upper bound for the CMR values which can be achieved in heavily doped magnetic semiconductors. The champion is the heavily doped antiferromagnetic semiconductor EuSe in which quite fantastic values of
of order 10% were & observed at liquid-helium temperatures [4]. But in the La}Ca}Mn}O system the magnetoresistance at low temperatures (57 K) can reach a value of the order of 10% which is very impressive, too [5]. The main shortcoming of the manganites is the fact that they manifest the CMR only at su$ciently large "eld strengths. For applications, they should be highly "eld sensitive for small "eld strengths. Meanwhile, not only the manganites but also other ferromagnetic semiconductors have Curie points at room temperatures, e.g., CuCr Te I (Section 3.1). For this reason, it would be   useful to extend investigations of the CMR materials to other ferromagnetic semiconductors, for example, to the Cu-based ones, which, in essence, are the relatives of HTSC. All the ferromagnetic semiconductors and manganites are very interesting from the physical point of view as they belong to systems with strong electron correlations. First of all, one should speak about correlations between the mobile electrons and the localized electrons of the d- or f-partially "lled shells of the transition or rare-earth ions (the s}d-exchange). These correlations take place on the background of the random electrostatic potential produced by the donor or acceptor impurities, which modi"es them not only quantitatively but also qualitatively. Certainly, as the ferromagnetic semiconductors are partially polar crystals, the interaction of the charge carriers with the optical phonons may in#uence the electronic state. In the manganites one should also take into account the fact that they are the Jahn}Teller (JT) systems. Here a qualitative description of the magnetoimpurity theory [118}136] of the CMR materials will be given which was initially formulated for the classic ferromagnetic semiconductors but, in my opinion, is also applicable to not very heavily doped manganites. First of all, it is necessary to point out that the resistivity peak cannot be explained by the destruction of the magnetic ordering close to ¹ alone (see Refs. [79,80,78]). According to the magnetoimpurity theory it should be related ! also to the in#uence of the impurity. Just the impurity determines the resistivity at ¹"0, and the fact that the height of the resistivity peak in the vicinity of ¹ depends on the zero-temperature ! resistivity means that it depends on the impurity density. At ¹"0 the charge carriers are scattered by the screened Coulomb potentials of randomly distributed ionized donors (acceptors). This scattering was investigated long time ago in the theory of the degenerate nonmagnetic semiconductors, and generalization of the corresponding BrooksHerring expression for the relaxation time to ferromagnetic semiconductors is trivial (see Ref. [11]).

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In addition to the scattering, the conductivity decreases due to formation of the band tail below the band bottom. This tail consists of the localized states of the electrons, but they di!er drastically from the localized states in the nondegenerate semiconductors. Namely, in the latter each electron is trapped by a single donor. In the former the localized electron is trapped by a group of ionized donors which, due to randomness in the impurity distribution, are located closer to each other than in the average over the crystal (the impurity cluster). As these clusters are of arbitrary size, the energy levels below the band bottom produced by them form a quasicontinuous spectrum. As the electrons screen the impurity potential, their density in the vicinity of the impurities is higher than the averaged value. It is especially high in the vicinity of the impurity clusters. Meanwhile, the indirect exchange via charge carriers is more intense the higher the charge carrier density is. For this reason the ferromagnetic coupling between magnetic atoms in the vicinity of a donor or of their cluster is stronger than that far from them. When temperature increases, the magnetization is reduced to a larger extent the weaker the ferromagnetic coupling is. As this coupling is maximal close to donors, the local magnetization there turns out to be higher than far from them. Hence, the impurity atoms acquire an e!ective local magnetic moment which is absent at ¹"0 when the magnetization per atom is equal to the d(f)-spin magnitude S for any magnetic atom in the crystal independent of its location with respect to impurity atoms. This impurity moment increases up to temperatures of the order of ¹ . At higher temperatures not only the local magnetization far from the ! impurity atoms but also that close to them reduces. As a result, the local moment of an impurity (more accurately, that of the region around the impurity) passes a maximum as a function of temperature. It should be noted that not only do spatial #uctuations of the magnetization appear at "nite temperatures but also the #uctuations of the electron density increase. This is related to the fact that the electrons are attracted by regions of enhanced magnetization as their energy is lower here. Hence, as the magnetization close to donors is enhanced, in addition to the electrostatic force, the electrons are attracted to a donor by its e!ective magnetic moment. On the other hand, additional increase in the local electron density around the donor increases its magnetic moment, so that the problem of magnetic and electron density #uctuations is self-consistent. Since at "nite temperatures the electron is attracted to the donor not only by the electrostatic force but also by the magnetic one, its scattering by the donors increases with increasing donor e!ective moment. Hence, the relaxation time decreases and the number of electrons in the band tail increases with increasing e!ective moment. As already mentioned, this moment is maximal not far from the Curie point. Hence, the resistivity should be maximal there. An external magnetic "eld H suppresses the magnetization #uctuations (in the limit HPR the moments of all atoms are equal to S). This is the reason for the CMR. The considerations presented above correspond to a not very high peak when the resistivity remains relatively low even in its peak. But in many cases the peak resistivity is many orders of magnitude larger than the zero-temperature resistivity. In this case one can speak about the temperature-induced metal-to-insulator phase transition occurring with increasing temperature. Its existence can be explained qualitatively as follows. The Mott delocalization of the donor electrons occurs when the radius of their orbit (the e!ective Bohr radius a ) is comparable with the
mean donor separation n\. In the highly conductive state the inequality a n'C is met
+

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where C is of order 1. However, with increase in the donor magnetic moment, the attraction of the + electron to the donor increases, and the e!ective Bohr radius a decreases. Hence, if for ¹"0 the
inequality just pointed out is met, it cannot be met at su$ciently high temperatures, and the highly conductive state becomes unstable. On further increase in temperature, this inequality is encountered again and the temperature-induced transition from the insulating state to the highly conductive state takes place. Perhaps, if the zero-temperature resistivity is su$ciently large, the crystal can remain insulating up to very high temperatures. This is caused by lowering of the energy of the localized electrons due to their interaction with the random #uctuations of the magnetization. The magnetoimpurity theory is formulated in the framework of the s}d-model commonly used for the magnetic semiconductors. In it all the electrons of the crystal are separated into mobile s-electrons and localized d-electrons. In reality, the s-electrons can model the p- and even d-electrons if they are mobile. They can also model the holes. The d-electrons can model both the immobile d- and f-electrons of the inner partially "lled shells. It should be stressed that the magnetoimpurity theory explains in a uni"ed manner the speci"c features of both the rare-earth semiconducting compounds and transition metal compounds including manganites with su$ciently low charge carrier density when they can be treated as degenerate semiconductors. This seems to be its plausible feature as the qualitative properties of the system do not depend on the ratio of the s}d-exchange energy AS to the charge carrier band width = though, quantitatively, such dependence certainly exists (A is the s}d-exchange integral, S the d-spin magnitude). This ratio is certainly small for the rare-earth compounds as the charge carriers in them are of s- or p-type with large overlapping of the external orbits of the neighboring atoms and with relatively weak exchange interaction with the internal f-electrons. The inequality AS/=;1 is used in the RKKY theory of the indirect exchange developed for the rare-earth metals. I shall call this limiting case the simple exchange. But many investigators believe that in the manganites the opposite case AS/=<1 is realized which is called the double exchange. The reason for this is the fact that the charge carriers (holes) are presumably of the same d-type as the localized spins of the magnetic atoms. Though this case can be described within the framework of the s}d-model, paradoxically, as it may seem, the role of an s-electron is played by a d-electron (in the manganites, in the zeroth approximation appearance of a hole means decrease in the number of d-electrons of a Mn ion: their number becomes three instead of four). As the overlap of the inner d-orbits is relatively small, and the intra-d-shell exchange integral A is large, the inequality AS<= is likely to be quite realistic. (Strictly speaking, the transition between neighboring Mn ions occurs via adjacent O ions but these virtual states can be excluded from the total Hamiltonian of the system by a canonical transformation leaving only the Mn d-orbitals in the transformed Hamiltonian.) Certainly, the states of the charge carriers in the cases of the simple and double exchanges are quite di!erent. In the former case the s-electron states can be characterized by their spin projections  which are conserved in the zeroth approximation in AS/=. Very crudely, the energy of the s-electron di!ers from the usual band energy proportional to = and not depending on M by the term !AM where M is the magnetization per atom. The basic feature of the double exchange is the fact that in zeroth approximation the d-spin and the s-spin on the same atom form a united spin of the magnitude S#1/2 for A'0 or S!1/2 for A(0, and only the projection of this total spin has a physical meaning. The s}d-exchange energy

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corresponds to an s-electron "xed at a certain atom and, hence, does not depend on directions of the spins of other atoms. For this reason it plays the role of an additive constant, and the transport properties of a double-exchange system do not depend on A. The magnetic ordering manifests itself in the kinetic energy of the system. For example, if A'0 and SPR, the state S, 1/2 of the atom with the spin projection SX"S and with the s-electron on it having the spin projection 1/2 corresponds to the energy !AS/2. If the electron spin projection is (1/2), but the d-spin projection is !S, then the state !S, 1/2 corresponds to the energy AS/2 which is too large for its realization under the normal conditions. Now let us consider the transition of an s-electron between the atoms 1 and 2. It is very important that the term describing the hopping process, arises from the electrostatic energy of lattice which cannot change the direction of the spin. Hence, if the atomic spins are parallel, then the transition should take place between the states S, 1/2S and SS, 1/2. In this case the energies of the initial and "nal states coincide, and the transition occurs without hindrance as in a nonmagnetic crystal. But if the atomic spins are antiparallel, then the transition between the states S, 1/2!S and S!S, 1/2 increases the energy by a very large quantity AS. Hence, such a transition is forbidden. Keeping in mind these considerations, one may assume that at SPR the e!ective hopping integral should be proportional to cos /2 where  is the angle between the spins. For quantum spins, this quantity depends strongly on S, remaining for the antiferromagnetic ordering only (2S#1 times less than that for the ferromagnetic one. As for the paramagnetic state, obviously, its energy is higher than the energy of the ferromagnetic state by AS/2 for =
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mathematical procedure [78] cannot be considered su$ciently rigorous to give reliable predictions for this extremely di$cult problem, nevertheless, qualitatively, the "rst statement [78] does not cause objections, all the more, because it was con"rmed recently by Monte-Carlo simulations for the Kubo formula in the case of the double exchange system with classical spins [263]. The second statement of Ref. [78] which, in fact, is simply an assumption and not a result of a theoretical calculation, should be checked by a direct comparison with the experiment. First, the vast majority of ferromagnetic semiconductors displaying qualitatively the same properties as the manganites do not belong to the double-exchange systems (EuO, EuS and so on). Hence, the double exchange is not a necessary condition for the CMR. Second, the points of view of di!erent authors on the role of the JT polarons di!er very strongly. There are arguments that the charge carrier interaction with the JT phonons cannot determine the speci"c features of the manganites. Namely, of several dozens of ferromagnetic semiconductors with similar properties (the CMR and resistivity peak), only the manganites are the JT systems (the list of the ferromagnetic semiconductors is presented in Ref. [11], their transport properties are described in Sections 3.1 and 4.1). Hence, the JT e!ect is not a necessary condition for the speci"c features mentioned. Keeping in mind that, apart from the JT e!ect, all other properties of the manganites are qualitatively the same as those of the rest of the ferromagnetic semiconductors, one can conclude that the mechanism of the CMR acting in the latter, should act in the LaMnO  manganites, too. If the CMR e!ect in the manganites had greatly exceeded those in other ferromagnetic semiconductors, possibly, it would have been necessary to "nd another mechanism of CMR which would enhance its value drastically. A candidate for such a mechanism could be the interaction of the holes with the JT phonons. But the real situation is quite the opposite. The resistivity peak in the vicinity of ¹ and CMR are much more pronounced in the non-JT systems than in the JT ! systems. For example, as will be discussed in Section 4.1, the height of the resistivity peak in the vicinity of ¹ in the non-JT EuO is up to 17 orders of magnitude higher than in the JT manganites. ! Hence, there is no need to invoke the JT phonons for qualitative explanation of the transport properties of the ferromagnetic manganites though these phonons, certainly, in#uence their transport properties quantitatively. Supporters of the decisive role of the hole interaction with the JT phonons believe that the manganites di!er drastically from other ferromagnetic semiconductors. For this reason, a special mechanism for CMR of the manganites should exist, and it should be based on the joint action of the magnetization and the JT phonons. In their opinion, joint action of the magnetization and the impurities is irrelevant. These supporters refer mainly to numerous experimental facts evidencing that the insulator}metal transition is accompanied by marked changes in the lattice state (Section 8.4). But the point is that these changes can be not an origin but a consequence of the metal}insulator transition. For example, ferromagnetic disordering is always accompanied by magnetostriction. But this does not mean that the magnetostriction is a reason for the ferromagnetic disordering: it is simply its consequence. Both points of view on the role of JT phonons will be presented in detail in the present review article. It remains to add that the charge carrier interaction with the lattice polarization cannot be very strong in ferromagnetic semiconductors, as, unlike NaCl, they are only partially polar crystals with the weight of the covalent binding being quite considerable. Hence, the large polarons cannot cause

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the speci"c properties of the manganites. On the other hand, due to the same reason, the small polarons are impossible in them (Section 8.3). Returning to the statement that charge carrier interaction with the magnetic subsystem alone cannot ensure the resistivity peak, I would like to remember that all the semiconductors, including manganites, are imperfect systems. Just joint interaction of the charge carriers with the magnetization and lattice imperfections can ensure the resistivity peak in the vicinity of ¹ . ! A sort of compromise is possible between these `polaronica and `impuritya points of view: when the charge carriers are trapped by impurities, they change the lattice state in their vicinity. Hence, the bound polarons arise. If they are large polarons, the trapped charge carriers realize the indirect exchange in the vicinity of the impurities increasing the local ferromagnetic ordering. Thus, a complex bound lattice-magnetic polaron arises which combines the local lattice and magnetic polarizations. Keeping in mind the massive amount of publications on the colossal magnetoresistance materials in the last four years, it seems necessary to order them in some way and to relate the manganites to other materials with similar properties. The present review article is aimed at presentation of the main experimental and theoretical results on the manganites already established, and at pointing out the problems which are not yet solved. I admit that I may have omitted some interesting papers only because I missed them in the ocean of information on the subject.

2. Structural and magnetic properties of the basic materials 2.1. Conventional ferromagnetic semiconductors At present, more than 30 ferromagnetic semiconductors are known. The "rst ferromagnetic semiconductor CrBr had been synthesized in 1960 [6], this being an answer to the question as to  compatibility of the semiconducting and ferromagnetic properties. Shortly afterwards the ferromagnetic semiconductors EuO [7] were synthesized, which were popular with investigators in the 1960s and 1970s on account of the simplicity of their crystallographic structure (cubic, of the NaCl-type) and due to the absence of an orbital angular momentum of electrons ¸ of partially "lled f-shells of the Eu>> ions (those f-shells are in S states with ¸"0, S"J"7/2). EuS crystals  possess similar properties [16]. The Curie point ¹ of EuO is very high for the rare-earth ! semiconducting compounds } about 69 K, whereas that of EuS is only 16 K. But the Curie point of EuS can be strongly raised by the pressure } up to 150 K at p"20 GPa [19]. According to Ref. [8] the magnetic dipole}dipole energy of ferromagnetically ordered spins in the cubic lattice is zero. Those are the reasons for the crystallographic anisotropy of such crystals being quite small: the anisotropy "eld in EuO is equal to 190 Oe, that in EuS being less than 30 Oe [9,10]. As EuO and EuO are almost ideal Heisenberg magnetic systems, they were used for determination of the critical exponents, data on which are collected in Ref. [11]. The magnetostriction d ln ¹ /d ln < (< is the volume) amounts to !6 for EuO and to !5.5 for EuS [12,13]. ! Another important group of ferromagnetic semiconductors is composed of chrome spinels of which CdCr Se and HgCr Se have been studied most extensively [14]. Their crystallographic     structure is much more complicated than that of EuO and EuS, and because of this they are less suitable for testing physical theories. But, to make up for this, they have much higher Curie points

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at about 100 K. There are also the Cu-based ferromagnetic semiconductors with still higher Curie points [488]. For CuCr Te I it even reaches room temperature (¹ "294 K), which, probably, is   ! related to the presence of the magnetic ions Cu>>. In some sense, these materials are predecessors of the Cu-containing HTSC. A special group of the ferromagnetic semiconductors consists of highly anisotropic materials (e.g., Dy(OH) , Ho(OH) ) with very low Curie points (a few degrees K) [15]. In them the   ferromagnetic ordering is either completely or partly the result of the spin}dipole}dipole interaction. A highly anisotropic ferromagnetic semiconductor is also (CH NH ) CuCl with ¹ "    ! 8.9 K which in the immediate vicinity of ¹ behaves like a three-dimensional magnetic system but ! already 1 K above ¹ its behaviour corresponds to that of a two-dimensional system [17]. ! Some ferromagnetic semiconductors can be obtained from the antiferromagnetic semiconductor EuSe which is almost ferromagnetic in the sense that it can be converted into a ferromagnet by a weak pressure. According to Ref. [18], a pressure of 1 kbar lowers its NeH el point ¹ by 1 K. But, , according to Ref. [19] pressure "rst does not a!ect ¹ . Nevertheless, at 5 kbar, the system becomes , ferromagnetic, and at 20 GPa its ¹ reaches 60 K. The same occurs in the alloys EuS Se with ! \V V x"0.87 which behaves like a mixture of two ferromagnets with di!erent intensities of the exchange interaction [20]. It should be stressed that some antiferromagnetic systems become ferromagnetic as a result of their doping introducing charge carriers. First of all, the above-mentioned EuSe should be pointed out which will be discussed in detail in Section 6.3. Large interest is attracted by EuB . The sample  with ¹ "15.3 K displays the spin reorientation phase transition at 12.5 K [23]. Possibly, the ! ferromagnetic EuB is a degenerate semiconductor with the Curie point between 11 and 15.3 K  [21}23]. Perhaps, it can be an intrinsic ferromagnetic semimetal [153]. Of the quite recently discovered ferromagnetic semiconductors, the pyrochlore Tl Mn O with    ¹ "124 K should be mentioned with ¹ about 150 K. Its low-temperature resistivity  amounts ! ! to 10\  cm, and (¹ ) is an order of magnitude larger and continues to increase with increasing ! temperature. The steepest increase in  takes place close to ¹ . The neutron studies revealed ! well-de"ned ferromagnetic magnons in them. The authors [24}26] stressed that this system is a non-JT one. Perhaps, all the ferromagnetic semiconductors mentioned above are not, too. Information about other ferromagnetic semiconductors is presented in Ref. [11]. 2.2. Crystallographic and magnetic properties of undoped lanthanum manganites Now we consider the JT manganite LaMnO which is a parent system for numerous manganite based compounds. The ion Mn> entering it is degenerate with respect to the d-orbitals in a cubic "eld. It splits the atomic d-level into the two-fold degenerate e and three-fold degenerate t levels. E E The former lies higher than the latter, so that 4 d-electrons of the Mn> ion occupy the t level E completely and the e level only partially. The eigenstates corresponding to the e level are as E E follows: d  ,d    " 0 and d   "(1/2)(2#!2) where lX is the state with a "xed X \V \W V \W X value of the orbital moment. It is important that the doubly degenerate e levels are only partly "lled. This is the reason for the E co-operative Jahn}Teller e!ect: it reduces the energy of such a degenerate system by lowering its symmetry which lifts the degeneracy of the e electronic levels, so that the electron can occupy the E lowest of them.

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Usually, ferromagnetic manganites are obtained from LaMnO by partial replacement of La by  a divalent metal, e.g., Ca. Then in the limiting case of the complete replacement it goes over into CaMnO . Both these materials have the perovskite structure [27,28]. The structure of an ideal  perovskite ABO may be presented as a set of the regular octahedrons BO , which make contact   with each other by their vertices [29]. Here A is the large cation which occupies the centre of the cubooctahedron, B the small cation which occupies the centre of the octahedron. At vertices of the polyhedrons the oxygen ions are located. In the case of LaMnO the large cation is La>, its radius  being 122 pm, and the small one Mn>, its radius being 70 pm. The Ca> radius is 106 pm and the Mn> radius is 52 pm [30]. Compounds of cubic perovskite structure are seldom met. Usually, the crystalline lattice is distorted in some way. The distortion may be separated in two groups: (1) those caused by discrepancy between sizes of cations and sizes of pores which they occupy; (2) those caused by the Jahn}Teller e!ect [29,31]. In the "rst case the minimum of the free energy is achieved by rotation of the BO octahedrons about a single axis or several axes of the initial lattice. If the octahedrons  rotate about the [1 0 0] axis, a tetragonal distortion takes place, about the [1 1 0] axis the orthorhombic one (aObOc, " " "
/2), and about the [1 1 1] axis the rhombohedral (trigonal) one (a"b"c, " " O
/2). In the second case the lowering of the symmetry splits the partially "lled e level (as already pointed out, the t levels are completely "lled). E E Both these structures were found in stoichiometric samples, depending on their preparation procedure [27,28] which shows that the crystallographic structure is highly sensitive to the state of a sample. In Ref. [33] the compound was found to be orthorhombic, space group Pnma. A fairy precise determination of the oxygen positions made it possible to establish that the MnO are not  only distorted, but also rotated from the ideal perovskite orientation. The orthorhombic structure was found also in Ref. [31] where conclusion was made that the lower of the two e levels is the E d  sublevel. Such d  orbitals display the antiferrodistorsional ordering which just manifests itself X X in the O-orthorhombic structure of the crystal. The conclusion about the orthorhombic structure was con"rmed in Ref. [33], where the following lattice parameters (at 300 K) were presented: a"0.57385, b"0.77024, c"0.55378 nm. According to Ref. [34], above 750 K, the antiferrodistorsional ordering disappears. The observed average cubic lattice is probably the result of dynamic spatial #uctuations of the underlying orthorhombic distortion. In Ref. [32] it was concluded that the structure below 875 K is strongly distorted orthorhombic. According to Refs. [35,36] the structure at 300 K is monoclinic though very close to orthorhombic: the angle between the a and b axes di!ers from 903 by less than 13. Resonant X-ray scattering was used to investigate the orbital ordering in LaMnO [194].  Dipole resonant X-ray scattering from ordered Mn 3d-orbitals was observed near the Mn K-absorption edge. The polarization of the scattered phonons is rotated from  to
 and the azimuthal angle dependence shows a characteristic twofold symmetry. A theory for the reasonant scattering mechanism has been developed by considering the splitting of the Mn 4p levels due to the Mn 3d orbital ordering. The order parameter of the orbital ordering decreases above ¹ "140 K and disappears at ¹ "780 K concomitant with the structural phase , transition. In Ref. [145] the optical phonon spectrum of LaMnO was investigated in the Brillouin zone  points
and R. There are 3A #5E infrared active branches, A #4 E Raman active branches, S S E E

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and 2A #3A silent branches. The frequencies of several modes are very high, e.g., one S E of the A modes has the frequency 716 cm\, and three modes have frequencies exceeding E 640 cm\. Now we shall consider the magnetic structure of the unsubstituted lanthanum manganites, which was established by neutron studies in Ref. [28]. It is represented by an antiferromagnetic lattice consisting of ferromagnetic layers of Mn ions. But the alternating (1 0 0) planes have opposite spin directions (the so-called A antiferromagnetic structure). If one chooses the antiferromagnetism vector as the z axis, then the moments should lie in the x}y-plane or very close to it. According to Ref. [36], the NeH el temperature ¹ of this material is 141 K, and it displays a weak , ferromagnetism due to the Dzyaloshinskii "eld, arising as a result of the JT distortions of the cubic crystalline structure. Though this weak ferromagnetism may be easily confused with the ferromagnetism appearing as a result of nonstoichiometry of the sample, there are evidences that it really exists in this case: in the most perfect samples the spontaneous magnetization vanishes at just the same temperature at which the susceptibility displays a kink typical of antiferromagnetic systems. An additional evidence is the hysteresis along the magnetization axis at the sample cooling typical of the weak ferromagnets. The magnon spectrum in LaMnO was found by neutron studies [53]. The whole spin-wave  spectrum is well accounted for with a Heisenberg Hamiltonian and a single-ion anisotropy term which induces a gap of 2.7 meV. The ferromagnetic exchange integral (I "0.83 meV) coupling the  spins within the ferromagnetic basal plane (a, b) is larger by a factor of 1.4 than the antiferromagnetic I "!0.58 meV coupling spins belonging to adjacent MnO plane along the c axis.   In Refs. [28,34] the magnetic structure of CaMnO with ¹ "131 K was determined. In this  , material each Mn> ion is surrounded by six neighbouring Mn> ions whose spins are antiparallel to the spin of the ion given. Such a structure may be represented as two interpenetrating face-centred cubic lattices with the opposite spin directions. The same structure is realized also in SrMnO with ¹ "260 K.  , 2.3. Theories for the undoped manganites The manganites are highly interesting systems from the theoretical point of view due to interplay between the structure, magnetic and electronic properties. The undoped manganites, in reality, behave like the insulators due to a large gap in their spectrum. We begin with the exchange interaction between the localized atomic moments, modi"ed for the case of the JT systems. To describe the state of the e subshell of the atom f, it is convenient to introduce the E two-component pseudospin operators Pf determined by the relationships between the wave functions of this subshell: PXd   "d   , PXd    "!d    , V \W V \W X \V \W X \V \W (2.1) PVd   "d    , PVd    "d   . X \V \W X \V \W V \W V \W Certainly, as the invariance with respect to rotations of the pseudospins is absent, the Hamiltonian of the orbital}orbital interaction should be anisotropic. Nevertheless, to make the problem tractable, at least semiqualitatively, the following isotropic spin}orbital Hamiltonian was proposed

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in Refs. [54}57]: 1 H"! I(u!f )(Su Sf )!HSXu 2 1 ! [¸(u!f )(Pu Pf )#R(u!f )(Su Sf )(Pu Pf )] 2

(2.2)

with Su being the operator of the d-spin for the atom u for the transition metal compounds and the total moment of the f-shell for the rare-earth compounds. Obviously, the "rst two terms in Hamiltonian (2.2) are of the same type as for the systems without orbital degeneracy, the third term corresponds to an e!ective interaction between orbitals via the lattice and the fourth term describes change in the exchange interaction between spins related to a change in the orbital state of the interacting electrons. Strictly speaking, Eq. (2.2) was deduced from a many-electron Hamiltonian in which, in addition to the Hubbard on-site repulsion, the Hund exchange between the electrons of the same shell was taken into account. But, to obtain such a simple form, some very strong assumptions were used, so that it seems better to consider Eq. (2.2) as a model Hamiltonian. In taking into account the interaction between the atomic spins and electron partially "lled orbitals, one is able to obtain important qualitative results using Hamiltonian (2.2). For example, one can explain [54}57] the metamagnetic phase transitions described in Sections 7.1 and 7.2. The ground state of the system is considered under assumption I(R(¸ when for H"0 it is antiferromagnetic for spins S and ferromagnetic for pseudospins P. At HO0 the energy of the system is given by E"!zI cos 2!z¸ cos !zR cos 2 cos !H cos 

(2.3)

where  is the angle between the magnetic "eld direction and the sublattice spin. As no external "eld acts upon the pseudospins, cos  can assume the values #1 (orbital ferromagnetism) and !1 (orbital antiferromagnetism), z is the coordination number. The spin and pseudospin operators being considered as classical vectors are normalized in such a manner that their moduli are 1. Comparison of the energies corresponding to both signs of cos  shows that, on increase in H, the transition takes place between the orbital ferromagnetic and orbital antiferromagnetic phases at the "eld strength H determined by the relationships A







¸ R H "2z 2 1! (I!R) for ( , A R I H "!4z(I#R) 1! A 1#¸/R " . 3!¸/R

R#¸ R#I



for

R ' , I

(2.4)

(2.5)

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Eq. (2.5) remains valid also for R(I. If R/I(, then a continuous canting of the sublattice moments occurs with increasing "eld, and at H an abrupt change in the angle 2 between A moments takes place as H . cos "! 4z(I#R cos ) For R/I', the transition from the orbital ferromagnetic phase to the orbital antiferromagnetic phase is accompanied by the spin-#op transition. In Ref. [428] a similar metamagnetic transition was obtained, if one accounts for the biquadratic term in the magnetic Hamiltonian arising due to the spin interaction with the JT phonons. It was pointed out there that an abrupt change in the orbital state should result in the change of the lattice structure which may explain the magnetic-"eld-induced structural changes described in Section 5.1. By the way, the same should occur in the model [54}57]. Many theoretical papers are devoted to the microscopic theory of the manganites. If one considers LaMnO as a cubic crystal and uses the standard band approach, one must conclude  that the crystal should be ferromagnetic and metallic as the number of e states exceeds twice the E number of e electrons. Hence, a more sophisticated approach is necessary to justify the insulating E antiferromagnetic state of these crystals. In some papers cited below the JT interaction is assumed to be the origin of such a state. But it should be pointed out that the Coulomb on-site repulsion can also be responsible for this. In fact, when it is very strong, the ground state of the system is insulating with one electron per atom. The orbital degeneracy becomes inessential in this limit. Obviously, in a consequent theory both the JT e!ect and the on-site Coulomb interaction should be taken into account. In papers [48,49] the structure and energy band spectrum of the crystals were calculated. In them a local spin density approximation is used. In Ref. [48], the very di!erent structural and antiferromagnetic symmetries for the ground states of LaMnO and CaMnO are obtained.   In Ref. [49] a band theory for ground state properties and excitation spectra of perovskites was developed. The local spin-density approximation describes well the electronic structures for LaMnO . The importance of accounting for the distortion of a cubic structure was stressed:  only this makes it possible to describe the magnetic structure and insulating state of LaMnO  correctly. A similar conclusion was made in Ref. [50] where the electronic structures of LaMnO and  CaMnO were studied using the density-functional method. Antiferromagnetic insulating solu tions were obtained for both compounds within the local-density approximation (LDA). For LaMnO , the JT distortion, found necessary for the antiferromagnetic insulating solutions,  produces occupied Mn orbitals along the long basal-plane MnO-bonds. The large on-site Coulomb ; and exchange I energies obtained from constrained LDA calculations, ; about 8}10 eV and I about 0.9 eV, indicate important correlation e!ects, and yield large redistribution of the spectral weight within the LDA#; approach used. In order to obtain a gap at the Fermi surface, one should account for the lattice distortion due to the Jahn}Teller e!ect. As in Ref. [48], in Ref. [50] a considerable degree of the Mn(d)}O(p) hybridization was found. But this paper was criticized in Ref. [420] where it was pointed out that the value of I found there is an order of magnitude larger than its realistic value.

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Similar qualitative results were obtained in Ref. [467] where the energy of the JT distortion was assumed to exceed the width of the bare-hole band. An approach to the A-type antiferromagnetic structure in the JT systems resembling that of Ref. [467] was developed in Ref. [61]: the proof of its existence is given in the limiting case of a very large JT splitting. In my opinion, the requirement of a very large JT energy is a consequence of the fact that the Coulomb on-site interaction was not included in the Hamiltonian of the system. In Ref. [62], and e!ective Hamiltonian was constructed to investigate the electronic structure in the manganites of the perovskite structure. In this Hamiltonian, the doubly degenerate e orbitals E and the electron}electron interactions in these orbitals are taken into account. Also, Hund coupling and interaction between the nearest neighbour t spins are introduced. The Hamiltonian E is represented by the conventional spin operators for spins, the pseudospin operators for orbitals, and the restricted electron operators due to the electron}electron interactions. The product of the spin and orbital parts in this model implies the strong coupling between the orbital and spin structures. The competition and interplay between the spin and orbital degrees of freedom bring about rich phenomena in this system. The results of calculations in the undoped system and their implications are summarized as follows. The phase diagrams and the transition temperatures for the spin and orbital ordered states were calculated in the mean-"eld approximation. The A-type antiferromagnetic structure comes out by the competition between the antiferromagnetic interaction between t spins and the E anisotropic ferromagnetic interaction induced by the virtual electron hopping in the e orbitals. It E is stressed that the two-sublattice (3x!r/3y!r)-type orbital ordering of di!erently polarized d  orbitals and the orbital dependence of the electron transfer are very important for the X stabilization of the A-type antiferromagnetic ground state. The excitations in the spin and orbital degrees of freedom were calculated in the spin-wave approximation. The calculated dispersion relation of the spin waves well reproduced the experimental results. The anisotropic nature of the spin excitations re#ects the strong ferromagnetic interaction in the ab plane and weak antiferromagnetic one in the c direction due to the orbital ordering. The characteristic features in the dispersion relation and the energy gap of the orbital waves, which originate from the (3x!r/3y!r)-type orbital ordering and A-type antiferromagnetic spin ordering were pointed out. In Ref. [58] the stability of the A-type antiferromagnetic ordering and of the canted antiferromagnetic structure in LaMnO is explained in the itinerant-electron picture  based on the local-spin-density approximation. The crucial role of the lattice distortion was demonstrated which strongly a!ects the magnetocrystalline anisotropy, as well as the anisotropic and isotropic exchange interactions in this compound. But the extent to which such an approach is accurate for a system with strong correlations such as the manganite is unclear. An e!ect of anomalous X-ray scattering was studied in relation to its role as a detector of the orbital orderings and excitations in the orbital-ordered manganites [63]. The scattering matrix is given by virtual electron excitations in Mn from the 1s level to the unoccupied 4p level. It was found that the orbital dependence of the Coulomb interaction between 3d and 4p electrons is essential to the anisotropy of the scattering factor near the K edge. The calculated results in MnO clusters explain the forbidden re#ections observed in LaMnO and Sr-replaced   manganites.

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3. Nondegenerate ferromagnetic semiconductors, s}d-model and double exchange 3.1. Optical and transport properties, experimentally Of numerous speci"c properties of ferromagnetic semiconductors, here only the giant red shift of the optical absorption edge and the resistivity peak in the vicinity of ¹ leading to their CMR, will ! be discussed. As these phenomena survive transition of semiconductors from the nondegenerate to the degenerate state, they demonstrate the universality of speci"c features and their origins in ferromagnetic semiconductors. Moreover, the resisitivity peak and CMR are directly related to the red shift which can be used for diagnostics of materials. It also provides independent information necessary for a semiempirical calculation of the magnetoresistance. All ferromagnetic semiconductors display a giant shift of the absorption edge with the temperature, which exceeds by two or three orders of magnitude that in nonmagnetic semiconductors (in the latter it is of order of 10\ eV/3). As the temperature is lowered from the high-temperature side, this abnormally large shift begins even before the Curie point is reached, and continues down to ultralow temperatures. The shift occurs in the long-wave direction as the temperature is lowered, whereas in the nonmagnetic semiconductors it occurs in the opposite direction (there is a red shift in the ferromagnetic semiconductors instead of the blue one in the nonmagnetic semiconductors). An external magnetic "eld shifts the absorption edge in the same direction as a decrease in temperature, this e!ect being especially pronounced in the vicinity of ¹ . At very low and very high ! temperatures, the shift of the absorption edge caused by an external magnetic "eld disappears. All these regularities will be seen in Fig. 1 depicting the position of the absorption edge E E is EuS. The absorption edge in EuSe is also presented which in the magnetic "eld is ferromagnetic, too, and in its absence, is very close to ferromagnetic [16]. The red shift in EuO amounts to 0.25 eV [67]. The greatest red shift of all ferromagnetic semiconductors was observed in the ferromagnetic spinel HgCr Se : it is as high as 0.5 eV with the optical band gap diminishing by a factor of three. It   is worth noting that, for this material, the portion of the red shift in the paramagnetic region is larger than that in the ferromagnetic [68]. The giant red shift was also discovered in doped manganites La Sr MnO . The di!erence      between the absorption edge positions between 180 and 140 K amounts to 0.3 eV. The position of the edge is the highest at about 155 K, and the di!erence between the absorption edge positions between 180 and 140 K amounts to 0.4 eV [482]. Now the transport and magnetotransport properties of nondegenerate ferromagnetic semiconductors will be described. First, it should be pointed out that, as a rule, the behaviour of the electrons and holes is di!erent in them. More often, the p-type samples behave like nonmagnetic semiconductors, and the speci"cs of the ferromagnetic semiconductors manifest themselves in the n-type samples, though the opposite situation is possible. These speci"cs are as follows: the resistivity  vs. temperature ¹ curve is N-shaped for them. At very low temperatures, the resistivity diminishes with increasing ¹ like in nonmagnetic semiconductors. Then it begins to grow, and increases by several orders of magnitude. After passing a maximum in the vicinity of the Curie point, it diminishes again, following the standard exponential law. Such a behaviour was observed, for example, in In-doped n-CdCr Se [70,71], in EuO  

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Fig. 1. Temperature- and "eld-induced shift of optical absorption edge in EuSe, EuS [16].

and EuS with a moderate donor density [72,73] and so on. A typical  vs. ¹ curve for nondegenerate ferromagnetic semiconductors is presented in Fig. 2 (the Cd Ga Cr Se [74])       (see also Fig. 3 where the resistivity for both nondegenerate and degenerate EuO samples with di!erent oxygen de"ciencies is presented. The resistivity drops with increase in the oxygen de"ciency). Certainly, the resistivity peak in the vicinity of ¹ is related to the conduction electron density ! minimum. This follows directly from the temperature dependence of normal Hall e!ect constant R [70] (it should be recalled that in the ferromagnetic conductors the Hall "eld is the sum of the normal Hall "eld RBj and the anomalous Hall "eld R Mj where j is the current density,  B"H#4
M the magnetic induction, M the magnetic moment per unit volume). The activation energy in the spin-wave region is considerably less than that in the paramagnetic region. For example, for CdCr Se doped with In and Ga, the former, being of order 0.001 eV, is two orders of   magnitude less than the latter [74,75]. The nondegenerate ferromagnetic semiconductors with the resistivity peak display a negative isotropic magnetoresistance presented, in particular, in Fig. 2. (It is absent from samples without the resistivity peak.) The relative magnetoresistance
"[(H)!(0)]/(H) reaches 20 000% & at a "eld of 50 kOe. As a function of temperature it is sharply peaked in the vicinity of ¹ (Cd Ga Cr Se , Fig. 4). The
peak is located at 130 K which is slightly lower than !       & the temperature of the resistivity peak which shifts with the "eld, increasing up to 50 kOe by 20}40 K towards higher temperatures. It should be noted that samples doped with In or with Se de"ciency display large positive magnetoresistance in a certain temperature range well above

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Fig. 2. A Cd Ga Cr Se : a temperature dependence of the logarithm of the speci"c resistivity  in di!erent "elds:       1}H"0, 2}5 kOe, 3}10 kOe, 4}20 kOe, 5}30 kOe, 6}50 kOe. The dotted line denotes the temperature at which the ratio " / is maximum. The inset represents the  dependence at the magnetic "eld at di!erent temperatures: the & & lowest curve corresponds to 124.5 K, the nest curve to 128 K and the upper curve to 132 K [76,77].

¹ if the "elds are su$ciently low. In strong "elds the magnetoresistance again becomes negative ! (Fig. 5) [76,77]. 3.2. The s}d-model for the simple and double exchange The theory of the ferromagnetic semiconductors presented below is based on the s}d-exchange Hamiltonian H , in which the d-electrons model the immobile d- or f-electrons of partially "lled  shells and the s-electrons model the mobile conduction electrons or holes (they can be in the currentless states, e.g., be trapped by donors or acceptors. To be considered as mobile, it is su$cient

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Fig. 3. Resistivity vs. temperature for several EuO specimens, containing oxygen vacancies [72].

Fig. 4. The temperature dependence of the quantity " / for di!erent magnetic "elds (in kOe) for & & Cd Ga Cr Se [76,77].      

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Fig. 5. Temperature dependence of magnetoresistance in di!erent magnetic "elds (in kOe) in Cd In Cr Se       [76,77].

for them to go over from one atom to the other): H(s}d)"H #H #H ,    a , H "Ek aH k N kN  A H "!  (sSu ) exp[i(k!k)u]aH , k ak  N YNY NNY N I H "!  (Su Su
)!H SXu (3.1) >  2u
 where aH k , ak are the s-electron operators corresponding to the conduction electrons or holes with N N the quasimomentum k and spin projection , s its spin operator, Su that of the d- or f-spin of atom u,
vector connecting the nearest neighbours, their number being z, N the number of atoms. The "rst term in Eq. (3.1) is the s-electron kinetic energy with 1 Ek "!zt k ; k "  exp(i
k), "1 , z

(3.2)

the second term is the s}d-exchange energy and the third term is the d}d-exchange energy and the d-spin energy in a magnetic "eld H. The direct interaction of the s-electrons with the magnetic "eld is omitted as the indirect interaction with it via d-spins is much stronger. Hamiltonian (3.1) includes only magnetic ions as the nonmagnetic ones can be eliminated from the exact many-electron Hamiltonian by a canonical transformation. The typical parameters for the free s-electrons are their band width = (="12t for a simple cubic lattice), the s}d-exchange energy AS and the d}d-exchange energy zIS where S is the d-spin magnitude. The d}d-energy is always small compared with the s- and s}d-exchange energies. As for

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the latter, the relationship between them can be arbitrary. The case =<AS is realized, e.g., in the rare-earth compounds. In EuO the conduction electron moves over the external Eu> shells. Their overlapping for nearest neighbours is su$ciently large (3}4 eV), and the exchange with the f-shells relatively weak (a few tenths eV). As there is no special name for this case, it will be called `the simple exchangea below, opposite to the term `double exchangea introduced in Refs. [83,84] for the case =;AS. The latter can be realized in the transition-metal compounds under the condition that the charge carrier moves over the d-shells of transition metal ions. For example, many investigators believe that in the manganites a hole moves over the Mn> ions, converting the ion at which it is located at the moment, into the Mn> ion. Then the exchange between the charge carrier and the localized d-electrons is especially strong (up to 10 eV) whereas the overlapping is relatively small (=&1 eV). But there exist experimental data contradicting this picture (Section 5.7) so that the question whether or not the double exchange realizes in the manganites waits for a "nal answer. In the spin-wave region the structure of the electron-magnon Hamiltonian does not depend on the ratio AS/= if the spin is not too small. One can construct it using the inverse spin magnitude 1/S as a small parameter for the expansion in its powers [82,11,133]. First, the Holstein}Primako! transformation for the d-spin operators S\ SXu "S!bH S> u "(2Sbu , u "(2SbH u, u bu

(3.3)

will be carried out in the Hamiltonian (3.1) with bH u , bu being the d-magnon operators. After their Fourier transformation and eliminating the terms linear in the magnon operators, it takes the form (only the diagonal terms corresponding to the up spins are retained as A assumed to be positive): A ak # C(k#q, k)aH a bHb # q bHq bq , H "[Ek !AS/2]aH k k   2N  k q q CK Ek q !Ek > C(k#q, k)" , Ek q !Ek #AS > q "H#zIS(1! q ) .

(3.4)

Outside the spin-wave region it is convenient to use an e!ective double exchange Hamiltonian. First, the problem of the double exchange was solved for two d-spins and one s-electron in Ref. [84]. For classical spins, the double-exchange Hamiltonian was "rst introduced in a simpli"ed form appropriate for ¹"0 [85]. Its general form was constructed in Refs. [86,11]. As the leading term in Hamiltonian (3.1) is H , in the zeroth approximation in AS/= the spin of the s-electron,  dwelling at atom u at a moment, should be parallel (A'0) or antiparallel (A(0) to the spin of this atom. It should be recalled that, in reality, the role of the s-electron is played by an extra or missing d-electron in the d-shell which can make interatomic transitions. Hence, the direction of the s-electron spin changes, on the s-electron transition between atoms. Introducing the polar angles for the directions of the d-spins, one can write the sum H #H in the "rst approximation in   =/AS u u
I AS  > exp($i u u
)cH cH H (de)"! u cu !tcos u cu
! Su Su
> > > A 2 2 2

(3.5)

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where cH u , cu are the spinless fermion operators, corresponding to an s-electron with the spin bound to the d-spin of atom u, the sign in the exponent corresponds to the sign of A. Further, cos u u
"cos u cos u
#sin u sin u
cos(u !u
) > > > >

(3.6)

where u u
is the angle between the spins of the u and u#
atoms, and the expressions for the > phases are





cos u u
 > tan u u
,

u u
"tan\ > > cos u u
> u !u
 #u
 !u
> , u u
" u > , u u
" u > u u
" > > > 2 2 2

(3.7)

In Refs. [86,11] terms are written down that make it possible to carry out calculations in higher orders in =/AS. The Hamiltonian deduced in Ref. [85] di!ers from Eqs. (3.5)}(3.7) by the phases being put equal to zero. Recently, Hamiltonian (3.5)}(3.7) was rederived in Ref. [96] where attention was drawn to the fact that the phases may play an important role in the physics of the double-exchange systems. Now the double-exchange Hamiltonian in the form (3.5)}(3.7) is commonly used. By analogy with the Berry phases, the same term is used for the phases . Now an e!ective double exchange Hamiltonian for quantum spins will be presented [87,11]. Unlike two-atom results of Ref. [84], it is valid for an arbitrary large number of atoms. Below the s}d-exchange integral A is assumed negative. Under these conditions, in the zeroth approximation in =/AS, the charge carrier is "xed at one of the magnetic atoms, their total spin being S!1/2. This just corresponds to the situation in the manganites, where the appearance of a hole at a regular ion Mn> reduces its spin by 1/2. Hence, the motion of the s-electrons over the crystal is tantamount to the motion of the S!1/2 spins among the S spins. The e!ective Hamiltonian for A'0 is presented in Refs. [87,11]. A special version of the perturbation theory in =/AS was developed in Refs. [87,11]. To construct the e!ective Hamiltonian, an e!ective spin of the magnitude S is ascribed to each magnetic atom, independent of its being free or occupied by an s-electron. In the last case, it formally increases the number of spin degrees of freedom by 1 as compared with its real number. But the structure of the e!ective Hamiltonian ensures vanishing of the contribution from the extra degree of freedom. As for the charge carriers, they are again treated as spinless fermions with the operators cH u , cu . Then one obtains: A(S#1) I H (de)" cH u cu !tF(Su , Su
)cH u cu
! Su Su
, O > > > 2 2



1 1 1 F(Su , Su
)" ((S#SXu )(S#SXu
)# S> u S\ u
> > > 2S#1 (S#SXu (S#SXu
>



.

(3.8)

In Eq. (3.8) small terms of order IS/t describing a change in the d}d-exchange of the atom bearing the charge carrier are omitted. An explicit expression for them is presented in Ref. [87]. In the

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classical limit Eq. (3.8) goes over into Eqs. (3.5)}(3.7) and in the spin-wave region it goes over into Eq. (3.4). 3.3. The charge carrier energy spectrum In this section the ¹ and H dependences of the charge carrier energy in ferromagnetic semiconductors will be investigated. Most clearly, these dependences manifest themselves in the giant optical red shift. As already mentioned, of the two types of the charge carriers, usually only one is magnetization sensitive. For this reason the magnetization dependence of the optical absorption edge is determined by the corresponding dependence of the band bottom position for the magnetization-active charge carrier type. The reason, for which the band bottom becomes lower with lowering temperature, is of a quite general character. As is shown in Ref. [11], just the ferromagnetic ordering ensures the minimum of the s-electron energy in a system described by the s}d-Hamiltonian independent of its parameters. Hence, both temperature decrease and application of an external magnetic "eld must result in the band bottom lowering and red optical shift for all ferromagnetic semiconductors. According to Eq. (3.4), in the spin-wave region the renormalized s-electron energy is given by the expression A Ek "Ek # gq C(k#qk)   2N

(3.9)

where gq is the magnon distribution function, gq "[exp( q /¹)!1]\ . As is seen from Eq. (3.4), in the case of the simple exchange AS;=, for small k the energy is nonanalytical in AS/= [82]. A characteristic wave number q "(2mAS exists. For temperatures  that are not very low, when the essential magnon wave numbers greatly exceed q , the renor malized energy is given by Ek "Ek !AM/2 ,  1 M"S! gq N

(3.10) (3.11)

where M is the mean magnetization per atom. In this temperature region the nonanalyticity of the energy with respect to AS/= is masked in the "rst approximation and manifests itself only in terms of higher orders in this small parameter. But it is clearly seen at low temperatures, when the essential magnon wave numbers are small compared with q . Then the temperature-dependent part of the s-electron energy is proportional to  ¹, whereas the temperature-dependent part of magnetization (3.11) is proportional to ¹. Moreover, the electron energy becomes independent of A. The ¹ law for the band bottom temperature shift was con"rmed in Refs. [94,95]. The nonanalyticity in AS/= has a deep physical meaning [82]. If Eq. (3.10) had been valid up to ¹ , this would have meant that the electron energy is determined by the long-range order which is !

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impossible for a microscopic particle. Then the red shift should begin only after the appearance of the ferromagnetic ordering, which contradicts the experiment (Fig. 1). The nonanalyticity becomes of principal importance in the vicinity of the Curie point and above it. The electron energy is determined not by the long-range order but by the long-range ferromagnetic correlations to which the s-electron spin adjusts following change of the local magnetization direction in the crystal. Rather complicated calculations [88,89] show that the downward s}d-shift of the electron energy Z should be of the order of AS(AS/=) at ¹ . Practically, even for =



A =  . E "!zt!Z!<M, <&  S AS

(3.12)

Hence, for small "elds and ¹'¹ , using the Curie}Weiss magnetic susceptibility, one "nds ! dE AS(=/AS)H  &! (3.13) dH (¹!) where  is the paramagnetic Curie point. Obviously, this quantity is very large as compared with the corresponding quantities at ¹P0 and ¹PR, which should tend to zero. This means, in accordance with the experimental data in Fig. 1, that the rate of the "eld-induced band-bottom shift is maximal close to  which, in the Heisenberg ferromagnets, is close to ¹ . ! In the simple exchange case, each state of the band shifts with temperature by the same quantity, and the shape of the conduction band does not change. Only the centre of the band changes its position. The picture in the double exchange case is quite di!erent: here the band centre remains "xed but the band becomes narrower with increasing temperature. This is directly seen from Eq. (3.4) for =;AS and S<1: the s-electron band is given by Refs. [86,11]: AS !zt(¹) k , Ek "! 2





1 t(¹)"t 1!  (1! )gq . O 2SN q

(3.14)

The main di!erence between the simple and double exchange consists in the fact that, though in both cases the position of the band bottom depends on temperature, in the former case the e!ective mass is temperature independent and in the second case it is temperature dependent. The temperature dependence of the e!ective mass modi"es the temperature dependence of all the physical quantities whose expressions include this mass. Now we shall analyse the temperature dependence of the charge carrier spectrum in the paramagnetic region, when it is generally impossible to diagonalize Hamiltonians (3.5) and (3.8) even approximately. For this reason, a very useful approach to the problem is the method of moments, which does not require the prior diagonalization of these Hamiltonians. The moments of the electron part of Hamiltonians (3.5) and (3.8), in which the constant terms &A are omitted, are

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determined by the expressions 1 M " Tr[H(de)!H ]L L N B

(3.15)

where the trace is calculated for a "xed set of the spin projections, while 2 denotes the temperature averaging over spins and H is given by Eq. (3.1). B Only even central moments are nonzero for these Hamiltonians, i.e., the energy band is symmetrical about its centre. Then from the classical Hamiltonian (3.5) [86,11], using correlation functions from Ref. [90], one obtains the numerical values of the ratio r(¹)"M/M:   r(0)"1.25, r(¹ )"1.23, and r(R)"1.23. ! As is seen from these "gures, the ratio r changes very little with temperature, which suggests that the shape of the density of states remains close to that for the simple cosine dispersion law (3.2). This agrees with Eq. (3.14), from which the validity of this dispersion law follows for the spin-wave region. Hence, the density of states becomes essentially nonzero only for energies exceeding the e!ective band bottom !zt(¹) (strictly speaking, there should be a tail of the density of states below this e!ective band bottom, but the number of states in it should be very small). For the classical spins, the band width at ¹"¹ is by 18% and at ¹PR by 29% less than at ! ¹"0. It is worth noting that in this case, too, the red shift in the paramagnetic region is quite sizeable. Now we consider the case of the quantum spins [91]. Using Eq. (3.8), one obtains for the e!ective hopping integral t: M "z(t) ,  zt (S#SX )(S#SX )#(S#SX )S>S\(S#SX )\ M "        (2S#1) #(S#SX )S>S\(S#SX )\#[S(S#1)#SX !(SX )][S#SX ]\        ;[S(S#1)!SX !(SX )][S#SX #1]\ . (3.16)    As Eq. (3.16) is very complicated, only the case of ¹PR will be considered when all the correlations are absent, and the average spins vanish. Then one obtains, e.g., for S"2 (like for the manganites): t(0)"0.8t and t(R)"0.632t (the former "gure follows from the fact that, for A(0 the band narrowing occurs even at ¹"0, and then t"2t/2S#1. For A'0 such a narrowing is absent). One sees from these "gures that the total band narrowing amounts to 21%. Hence, the narrowing decreases with the spin magnitude. 3.4. The resistivity peak and CMR in nondegenerate semiconductors. A general picture Keeping in mind the similarity of properties of the conduction electrons and holes, I shall speak about the electrons assuming that their s}d-interaction is su$ciently strong. Cases where the charge carrier sign is essential will be speci"ed. The red shift of the conduction band bottom should lead to temperature dependence of the donor level depth E . This should result in the resistivity peak located at a temperature where 

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d(E /¹)/d¹"0 [92,93,82,97]. Another root of this equation corresponds to the resistivity min imum preceding the peak. The reason why the activation energy passes a maximum at a certain temperature is as follows. At "nite temperatures, the electron levels are the lower, the higher the magnetization is. But the local magnetization in the vicinity of an unionized donor is higher than the average magnetization over the crystal which was "rst pointed out in Refs. [98,99] in calculating the paramagnetic Curie point of the doped ferromagnetic semiconductors. In fact, as the indirect exchange via free charge carriers is exponentially small in nondegenerate semiconductors, the average ferromagnetic ordering is supported only by the superexchange. Unlike this, the magnetization close to the donor is supported also by the indirect exchange via the electron of the donor. Hence, at "nite temperatures, the donor magnetization is destroyed to a lesser degree than the average magnetization. But for both ¹"0 and ¹PR the di!erence between the averaged and local magnetizations absent (they both are equal to S at ¹"0 and 0 at ¹PR). Hence, at intermediate temperature this di!erence should be maximal (strictly speaking, the above-made statement about the situation at ¹PR is valid only for large-radius bound states which does not change the conclusion made above). The donor overmagnetization means that, with increasing temperature, the donor level depth "rst increases, as the conduction band bottom rises much more rapidly than the donor level. But, on further temperature increase, the local ordering begins to disappear, too. Then the level depth decreases with increasing temperature. As a result, the charge carrier density is minimal, and the resistivity is maximal at a certain temperature in the vicinity of the Curie point. A magnetic "eld decreases the depth of the donor level and, hence increases the charge carrier density which manifests itself in an isotropic negative magnetoresistance. In calculating the donor electron state, one should keep in mind that its orbital radius must depend on the magnetization, as the overmagnetized region close to the donor is a potential well for the donor electron. Hence, the electron is attracted to the donor not only by the Coulomb potential but also by the magnetic potential well. As a consequence, the orbital radius must be magnetization dependent and should be found by a self-consistent calculation similar to that carried out for the antiferromagnetic semiconductors in Ref. [100]. In full analogy with the donors in the antiferromagnetic semiconductors, where magnetized regions arise close to the unionized impurities, one can use the term `the bound magnetic polarona, or `the bound ferrona, for the overmagnetized donors. In addition to the already-known low- and moderate temperature bound ferron, a quite new type of the bound ferron will be considered which exists at ¹PR and can be called the paramagnetic bound polaron (ferron) [479,481]. Whereas the ferrons investigated so far are related to a self-consistent enhancement of the ferromagnetic correlations in the region of the electron localization, the correlations are absent here, and only the #uctuating magnetization of the region increases with decreasing size, being of order 1/N where N is the number of magnetic atoms over which the electron of the donor is spread. Though the mean local magnetization remains zero, the electron spin adjusts to the #uctuating magnetization, #uctuating jointly with it and, thus, ensuring the gain in the exchange energy between the electron and magnetic atoms. This means that the tendency arises for the electron to be concentrated inside a region with a size as small as possible. This tendency jointly with the Coulomb interaction competes with the kinetic energy in determining the equilibrium orbital radius. The shrinking of the electron orbit caused by the magnetization #uctuations can lead to

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a considerable lowering of the donor level as compared with its depth at ¹"0. Hence, the low-temperature activation energy for the resistivity is less than the high-temperature one, in full agreement with the experimental results presented in Section 3.1. 3.5. Donor indirect exchange Hamiltonian In what follows, it will be proved that the charge carrier density in nondegenerate ferromagnetic semiconductors depends on temperature in the manner corresponding to Fig. 2. A simpli"ed treatment of this problem was carried out in Ref. [11]. As some essential features of this phenomenon were lost at this treatment, a special treatment of the problem was carried out specially for the present review article. It will be presented below. Mainly the simple exchange case will be investigated in detail. For the double exchange case, only some model results will be obtained which agree qualitatively for the simple exchange case. To analyse the magnetic properties of the unionized donors for AS;=, it is advisable to begin with the magnetic Hamiltonian describing the indirect exchange via the electron of the donor [478,11]. It must di!er strongly from the RKKY indirect Hamiltonian as the latter assumes that the conduction electrons are completely spin depolarized to the zero approximation. Certainly, the situation with the sole electron of the donor is quite the opposite. In the case of the simple exchange =


 w(u)w( f )(Su Sf ) (3.17)  where w(u) is the probability for the s-electron to be located at atom u. The double sign in Eq. (3.17) corresponds to two possible spin projections of the conduction electron upon the total moment L of the d-spins at which the s-electron is located. Below, for A'0 only the lower sign will be used. We begin with the low-temperature limit. With account taken of the d}d-exchange, the total magnon Hamiltonian is obtained by the transformation (3.3) in Eqs. (3.1) for H and (3.17) for H  K' A (3.18) H "IS (bH u bu !bH u bu
)#  w(u)w( f )(bH u bu !bH u bf ) . > KE 2uf u
  The last term &bH u bf in Eq. (3.18) is basically important to ensure the absence of the gap in the magnon spectrum. But it does not in#uence the bulk of the magnon frequencies. For example, if w(u)"1/N , only the q"0 magnon has the zero frequency. In the absence of the d}d-exchange, (N !1) magnons with other wave vectors have the same frequency A/(2N ). Hence, in calculating the free energy, one can use the following Hamiltonian for the magnon frequencies: u f

H "IS (bH u bu !bH u bu
)#H(u)bH u bu "0 , > KE u
 Aw(u)a H(u)" , w(u)"(u) . 2

(3.19)

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But Hamiltonian (3.19) is still too complicated to be diagonalized exactly. One cannot also use the perturbation theory. To carry out an approximate calculation, the magnon potential hump H(u) of a complicated shape entering Eq. (3.19) is replaced by a rectangular potential hump with the height h and radius R equal to the mean height of the hump (3.19) and the mean radius of the electron wave function. In the variational procedure for the free energy used below, the trial ground-state electron wave function will be taken in the form




(r)"




x  xr  exp ! , a "  a a me

(3.20)

where x is the variational parameter to be found from the condition of the minimal free energy,  the dielectric constant, m e!ective electron mass, and u is considered as a continuous variable r.  Then Ax , h"H(g)(g)" 16b

(3.21)

3a a R"g(g)" , b" . 2x a

(3.22)

This means that the magnon frequency in the region close to an unionized donor is given by 'q (x)"q #h , q "J(1! q ), J"zIS .

(3.23)

Now the high-temperature limit will be considered. First, expression (3.17) will be analysed in the limit ¹PR. Though the correlations between the d-spins are absent, the s}d-exchange energy remains nonzero in the "rst order in AS/=. Let us take w(u)"1/N . In this case A E "$ K' 2



S(S#1) . N

(3.24)

The physical meaning of result (3.24) becomes clear if one remembers that, according to the mathematical statistics, a system of N noninteracting spins should possess the total moment of the order of N\ of their maximal moment NS. The direction of this moment is not "xed but #uctuates freely, so that its mean value should vanish. But the spin of the s-electron adjusts to the direction of the #uctuating moment and #uctuates jointly with it, ensuring the maximum gain in the s}d-exchange energy for the energetically favoured direction of the s-electron spin relative to the total spin of its localization region. This gain should be of the order of the total moment per atom, i.e., &AS/N as is just the case for Eq. (3.24). Certainly, the term of order AS/= is essential only for orbital radii su$ciently small. For larger radii, the terms of the next order in AS/= should be taken into account. Now we consider the bound ferron at ¹<¹ , taking into account the #uctuation lowering of ! the energy just discussed. When the correlations between the d-spins are weak, the donor magnetic Hamiltonian (3.17) jointly with the direct d}d-exchange Hamiltonian (3.1) can be represented in the

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Heisenbergian form as A H "! (P#H , H "!(1/2)  I (u, f )(Su Sf ) , & & R' K. 2 u f $ A I (u, f )" w(u)w( f )#I(u!f ), P"S(S#1)w(u) . R' 2(P

(3.25)

3.6. Simple exchange. Spinwave region To calculate the low-temperature density of the conduction electrons in a nondegenerate semiconductor, it is necessary to write down the spin-wave Hamiltonian with allowance for the conduction electrons. The relative number of the donors  is assumed to be small. One can divide all regular magnetic atoms into those which enter spheres of a radius R surrounding donors and those which are outside these spheres (the number of the latter greatly exceeds the total number of the former). Using an expression for the conduction}electron}magnon Hamiltonian H (3.19), KE (3.21) and (3.23), one can represent the total electron}magnon Hamiltonian in the form of an operator functional of the trial wave function (3.20): H"n (E # 'q mq )# (1!n )q mq #Ek nk G 'G G 'G ' # Bq nk mq # q mq !  (n #nk ) , E "(x!2x)E , (3.26) 'G ' where mq and mq are the magnon operators for the ith sphere and outside the spheres surrounding G donors, respectively. As the magnon number operators for di!erent donor regions and outside them are constructed of magnon operators bHu , bu with di!erent u, all the operators mq and mq are G independent. Further, n and nk are the operators for an electron in the localized state at the donor i and for 'G the delocalized electrons with the quasimomentum k, respectively. The spin index is absent from the electron operators as the electrons are completely spin-polarized in the spin-wave region. For the same reason the s}d-exchange energy (!AS/2) is the same for all the electronic states considered and, hence, can be omitted as an additive constant. The quantity  is the chemical potential. The energy E (x) of an electron at the donor is an average value of the hydrogen-like atom ' Hamiltonian calculated with wave function (3.20). At low temperatures one can set x"1 in Eq. (3.20), so that E "!E "!e/2 a . The quantity Bq describing the s}d-interaction of the '  delocalized electrons with magnons, for electron quasimomenta that are small compared to the magnon quasimomenta has the form (3.4) Aq , q "2mAS . (3.27) Bq "  2N(q #q)  With allowance for mutual independence of mq and mq , the mean number of electrons at G a donor is given by the expression (the index of the donor is omitted): n "1#exp[(E #
F !)/¹]\ , ' ' K'
F "F !F "N ( f !f ) K' K' K' * ' 

(3.28)

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with F and F being the magnon free energies for a region of radius R containing the unionized K' K' and ionized donors, respectively,

 

a dq ln[1!exp(! 'q /¹)] , f "¹ ' (2
)

(3.29)

a f "¹ dq ln[1!exp(!q /¹)] ,  (2
) 4
R . N " * 3a

(3.30)

Hence,
F is the change in the magnon free energy resulting from the donor ionization. K' A similar calculation is carried out for the mean number nk  of electrons with the quasimomentum k, keeping in mind that the electrons with only the #1/2 spin projection are present: nk "1#exp[(Ek #
F !)/¹]\ , K!
F "F !F "N( f !f ) , K! K! K! !  a dq ln[1!exp(!q /¹!Bq /¹)] . f "¹ ! (2
)



(3.31) (3.32) (3.33)

Keeping in mind the fact that Bq &1/N, one sees that
F coincides with the second term in K! Eq. (3.9) describing the temperature shift of the band bottom for k"0. Equating the number of ionized donors with the total number of conduction electrons, one "nds an expression for the charge carrier density n for Ek "k/2m:  n "(nn ) exp[(!E #
F /2¹)] , (3.34)   K (m¹) ,
F "
F !
F (3.35) n " K K' K!  2(2
where n is the e!ective density of states in the conduction band, n"/a the donor density.  One can check that the activation energy in Eq. (3.44) increases with temperature in the spin-wave region. For example, for A"2, J"0.02 (in the E units)
F "!0.022 at ¹"0.01, K but
F "!0.214 at ¹"0.03. K The fact that the activation energy increases with temperature in the spin-wave region, points to the resistivity peak at elevated temperatures caused by a minimum in the charge carrier density. The ferron e!ect omitted when x was set equal to 1 decreases the depth of the donor level and increases the stability of the unionized donor. Using Eqs. (3.26) and (3.28), the total free energy of a separated unionized donor will be written down in E units as F (x)"(x!2x)#
F (x) . (3.36) ' K' If one considers the term
F (x) in Eq. (3.36) as a small perturbation, the optimal value of K' x delivering a minimum to F is equal to K' 1 d
F K' (1) (3.37) x"1! 2 dx

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and, to the "rst approximation, the optimal free energy is F"!1#
F (1) . (3.38) ' K' As
F (x) (3.28)}(3.30) decreases with decreasing x, and, hence, the last term in Eq. (3.37) is K' negative, the parameter x increases with temperature, and, hence the electronic radius decreases. This is a manifestation of the ferron e!ect: the electron is dragged in by the region of the enhanced magnetization and simultaneously supports it, realizing the indirect exchange inside it. 3.7. Simple exchange. Paramagnetic region Now we consider the in#uence of #uctuation trapping on resistivity of nondegenerate semiconductors in the far paramagnetic region. The free energy of the system is obtained with use of Eqs. (3.25) and (3.26) by the high-temperature expansion to the "rst order in 1/¹: Q(x) F.'(x)"(x!2x)E !¸x! , ¹ A ¸" 2



S(S#1) a b\, b" 8
a

(for z"6) ,




(3.39)



xb 4xb ¸x #AIS(S#1) 1#2xb# exp(!2xb) . Q(x)" 32
3 12

(3.40)

The entropy term !N¹ ln(2S#1) is omitted from the free energy here and in what follows. Minimizing the free energy (3.39) with respect to x, one obtains its optimal value and inverse orbital radius for ¹PR (in E and 1/a units, respectively): 8 8 F "! l[l#(1#l]! l(1#l!4l!1 ,  3 3

(3.41)

3¸ AS x "[l#(1#l], l" & .  8E (=e/a)

(3.42)

With a "a, for AS/E varying from 1 to 5, F varies from !1.104 to !1.659 and x from 1.077   to 1.1445. Hence, the electron interaction with random (uncorrelated) magnetization #uctuations leads to a marked decrease in the donor ionization energy and in the orbital radius, and this is true for any type of magnetic ordering at ¹"0. The corresponding electron state can be called the bound paramagnetic #uctuation polaron (ferron). For ¹PR, one obtains the following equation for the charge carrier density similar to Eq. (3.34):




F n "(nn exp    2¹

.

(3.43)

As is seen from comparison of Eqs. (3.43) and (3.34), the high-temperature activation energy of the conductivity (!F /2) exceeds the low-temperature activation energy, which agrees with the  experimental data in Section 3.1.

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Instead of a direct calculation of the temperature dependence for the conductivity activation energy, a proof will be given that the stability of an unionized donor increases, on decrease in temperature. This just means an increase in the activation energy. For this, it is su$cient to calculate the free energy of the donor. At "nite temperatures, from Eqs. (3.40) one obtains the total free energy of a system of N magnetic atoms and n< donors: n
(3.44)

and for the donor orbital radius: 1 dQ x(¹)"x # (x ) (3.45)  2¹(1!lx\) dx   where x is given by Eq. (3.42). One can persuade oneself that the second term on the right-hand  side of Eq. (3.45) is positive at actual values of parameters. This points to the tendency of the electron localization at lower temperatures if the electrons were delocalized at ¹PR. 3.8. The negative and positive magnetoresistance of nondegenerate semiconductors The calculations presented in this section can be easily used for an analysis of the magnetoresistance. The fact that it should be negative in the spin-wave region, is obvious from Eq. (3.34). Actually, to "nd the "eld dependence, it is su$cient to shift all the magnon frequencies in Eqs. (3.29), (3.30) and (3.33) by a quantity H. Then, according to Eq. (3.34), the relative change in the charge carrier density, and hence in the resistivity, in weak "elds H is H dF (H) n (H)!n (0) K  " . (3.46)
"   2¹ dH & n (0)  This equation will be concretized for the case of large spins at temperatures exceeding the magnon frequencies:






H A (HN
" ! .  2 2
qaJ
(2J

(3.47)

The "rst term in Eq. (3.47) exceeds the second term by a factor of order (=/J. Hence, the latter can be discarded, and the magnetoresistance is certainly negative as the "eld increases both the charge carrier density and conductivity. The quantity
can be called colossal, as it is proportional  not to H/J but to (A=H/J with (A=/J<1. As the leading term in Eq. (3.47) originates from the H-dependence of the red shift, the relationship between these two quantities is obvious (see Figs. 1, 2 and 4). As for the paramagnetic region, the results obtained above correspond to the case ¹PR. Meanwhile, the most interesting is the region not very far from ¹ . The corresponding calculation ! was carried out in Ref. [140] using a quite di!erent procedure. Not only the charge carrier density but also their mobility was calculated, and it was proved that only the "rst factor is essential. To calculate the density, the Landau method of the nonequilibrium statistical potentials [141] was used. The donors are separated into ionized and unionized ones. In the latter the electron binds

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spins of the magnetic atoms embraced by its orbit into a cluster with a changeable magnitude of its total moment. Minimizing the thermodynamical potential of the system with respect to occupancies of the di!erent states, one obtains an expression for the number of ionized donors n : G !E \ " (3.48) n "n 1#R(¹, H) exp G ¹








where E is the energy of the donor ionization in the absence of the s}d-exchange, the quantity " R(¹, H) with a very complicated structure is determined by the magnetic state of both ionized and unionized clusters. It takes into account the tendency of the moments of the clusters to be aligned with the "eld and increase in moment of the cluster caused by the "eld. After deducing Eq. (3.48), further calculation is carried out by a standard procedure. For the charge carrier density to diminish under a "eld at its small strength H(¹/N S, the * following condition should be met: AS(S#1) N S' * 2(3(¹!)

(3.49)

where  is the paramagnetic Curie point, and N is the number of atoms in a cluster. With larger * "elds, it can only increase in the "eld. Hence, this result explains why, at small "elds, the magnetoresistance can be positive or negative, depending on the properties of a concrete system, but is always negative in strong "elds (Fig. 5, Section 3.1). 3.9. The double exchange case and discussion of the results It remains to demonstrate that for the double exchange case the donor level depth at ¹PR is larger than that at ¹"0 due to the band narrowing. To be sure of this, it is su$cient to consider a system with a point defect of strength ; described by the Hamiltonian H "!;cHc #H (3.50) G   CD where for H the classical expression (3.5) will be used. For ; su$ciently large, the s-electron is CD distributed between the impurity atom 0 and its nearest neighbours . The ground-state energy is given by



; E "! ! G 2



; #t cos  /2  4 

.

(3.51)

According to Eqs. (3.16) and (3.51), the temperature-dependent depth of the impurity level is



; ; D(¹)"!zt (¹)# # #zt (¹) . CD CD 2 4

(3.52)

One sees from Eqs. (3.52) that at ¹"0 a discrete level exists for ;/t'5, and the di!erence D(0)!D(R) is always negative, tending to =(0)/2!=(R)/2 with ;PR. Correspondingly, the

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radius R of the localized state diminishes with increasing temperature: za , R" z#(v#(v#z)

; v" . t (¹) CD

(3.53)

The main results of the present treatment for the case of the simple exchange can be formulated as follows. For the nondegenerate semiconductors, it is established that the activation energy of the conductivity in the spin-wave region is determined not only by the depth of the donor level but also by the di!erence in the magnon free energies for a delocalized electron and for a localized electron. As this di!erence increases with temperature, the activation energy E increases, too. In the  paramagnetic region, the activation energy decreases with temperature. Qualitatively, the activation energy behaves like the di!erence between the local magnetization in the vicinity of an unionized donor and the mean magnetization over the crystal: with increasing temperature, it "rst increases and then decreases, passing a maximum at a temperature comparable with the Curie point. A very important result consists in the fact that the high-temperature activation energy exceeds its low-temperature value. This is a consequence of the #uctuation lowering of the donor level which is caused by the fact that the moment of a region in which the localized electron dwells, remains "nite even at ¹PR. The direction of this moment #uctuates in the space so that its mean value vanishes. But the spin of electron adjusts to the direction of the moment of the region and #uctuates jointly with the moment. For this reason, the gain in the s}d-exchange energy remains "nite for the localized electron, though it diminishes with increasing size of the region. The situation for the double exchange case is qualitatively similar.

4. Degenerate ferromagnetic semiconductors and magnetoimpurity theory 4.1. Resistivity and magnetoresistance of degenerate ferromagnetic semiconductors We begin with the classical degenerate ferromagnetic semiconductors. This term denotes that, at ¹"0, the materials possess a resistivity typical of degenerate semiconductors, both magnetic and nonmagnetic, i.e., about 10\!10\  cm. A speci"c feature of the ferromagnetic semiconductors is the resistivity peak in the vicinity of ¹ . In some cases it is so high that the sample remains ! insulating up to the highest temperatures. In other cases the metal}insulator transition is reentrant which manifests itself in the resistivity peak in the vicinity of ¹ . ! The most impressive is the metal}insulator transition in EuO investigated in Refs. [102}107]. It was observed only in samples with the excess Eu atoms acting as donors (Fig. 3, [102]). Then the resistivity increases by 13}19 orders of magnitude, which, apparently, is a record value among the systems displaying the transition. It is so high that it is di$cult to even measure it exactly. The transition takes place considerably below the Curie point which even in undoped samples amounts to 69 K, and in the doped samples should be still larger due to the indirect exchange via the conduction electrons. In the insulating phase the resistivity drops exponentially with inverse temperature, its activation energy at ¹<¹ being equal to 0.3 eV. In external magnetic "elds ! the transition point shifts to higher temperatures, with the (¹) curve becoming less steep

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Fig. 6. E!ect of magnetic "eld on the metal}insulator transition in EuO containing oxygen vacancies [104]. Fig. 7. Temperature dependence of resistivity of EuO with excess Eu. The electron density is 3.4;10 cm\ at 4.2 K and 5.5;10 cm\ at 298 K, the mobility 290 and 19 cm/V, respectively [104].

(Fig. 6, [104]). However, even in "elds of 140 kOe the resistivity rises in the range 70}120 K by six orders of magnitude [104]. According to Ref. [104], the metal}insulator transition is observed at low-temperature charge carrier densities of about 1 to 2;10 cm\ but is replaced by the reentrant transition at 3;10 cm\. Other donors, e.g., Gd, cause only the resistivity peak. If the EuO#Eu samples do not display the record metal}insulator transition but only the resistivity peak of a few orders of magnitude, the behaviour of these samples in the "eld is rather traditional: their resistivity drops with the "eld strength. In Fig. 7 the resistivity is presented of a sample with charge carrier densities of 3.4;10 cm\ at 4.2 K and 5.5;10 cm\ at 298 K, the mobility is equal to 290 and 19 cm/V s, respectively [104]. In some EuO#Eu specimens a maximum of the negative magnetoresistance well below ¹ is observed in addition to the one in ! the vicinity of ¹ (Fig. 8) [102]. ! In specimens displaying a metal}insulator transition the Curie point is not shifted with respect to that of the undoped crystal. But it is greatly displaced in specimens with larger electron densities which are not insulating at elevated temperatures. In them the resistivity peak is somewhat observable above ¹ , the relative height of the peak diminishing as the density rises. ! In contrast to EuO, EuS crystals containing excess Eu do not experience a transition to the insulating state, although in this case, too, a sharp resistivity peak is observable in the region of ¹ ! [115] (Fig. 9). As seen in this "gure, in the vicinity of the resistivity peak the magnetoresistance
amounts to a colossal value exceeding 10% at a "eld of 10 kOe. Some EuS samples display an & additional singularity of the resistivity: as the temperature rises, a rapid increase in the resistivity by about an order of magnitude takes place close to 8 K, but no corresponding decrease in the resistivity takes place as the temperature drops again. The high-resistivity state is metastable and decays in several minutes. The metastable state can also be destroyed by a magnetic "eld of several kOe, the decrease in the resistivity being accompanied by a decrease in the Hall constant [116,117].

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Fig. 8. Relative magnetoresistance of a EuO sample with excess Eu [72]. Fig. 9. Temperature dependence of the resistivity of the EuS crystals with excess Eu [115].

4.2. The Mott transition at ¹"0 In what follows, the magnetoimpurity theory of colossal magnetoresistance materials will be explored. In discussing the CMR theory, it is advisable to begin with the Mott transition from the insulating state to the highly conductive state that occurs with increasing impurity density at ¹"0. Obviously, it should occur when the impurity atom separation is comparable with the orbital radius of the electron a , i.e. the delocalization criterion should be of the form a n"C , C &1 (4.1) K + + where n is the critical donor density. For a system of one-electron donors in nonmagnetic K semiconductors, Mott [137] obtained a value of C from the condition of appearance of a discrete + level in the potential well corresponding to the screened Coulomb potential of a donor. The following single-electron Hamiltonian was considered by Mott e 6
en  exp(!r), " . (4.1a) H "! ! C   2m  r   The trial wave function was taken in the form (3.20). The average electron energy, calculated using Eqs. (4.1a) and (3.20), is minimized with respect to the variational parameter x, and one searches for the condition at which its minimum exists at "nite x, i.e., that a discrete level appears in the potential well. This means that the metallic state loses its stability. This condition takes the

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form a "1. Finding the corresponding value of density n from it, one arrives at Eq. (4.1) with C "0.25. + It should be stressed that, strictly speaking, the Mott criterion is not a criterion of the metal}insulator transition: it is only a criterion for the loss of stability by a uniform metallic state in the screened Coulomb "eld. One cannot even judge from it whether the transition is continuous or abrupt in a periodic system of atoms. Furthermore, one cannot make conclusions about the in#uence of the atomic disorder on the metal}insulator transition. Nevertheless, it is very useful for some estimates. If one uses the same approach for a ferromagnetic semiconductor, one will obtain C "0.41 + (in the case =
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the experimental data evidence that the perfect ferromagnetic conductors display not the resistivity peak but the peak of its temperature derivative in ¹ [113]. Hence, the resistivity peak should be ! related to the imperfections in the crystal which is con"rmed by the fact that it is strongly impurity dependent (Section 4.1). On the other hand, it cannot be explained also only by the standard electrostatic impurity}carrier interaction. In degenerate semiconductors charge carriers interact with randomly distributed ionized impurity atoms (donors or acceptors) which causes scattering of charge carriers and transition of a part of them from delocalized states to localized ones with a large localization length. In this case an electron is trapped not by a separate donor atom but by their complex with decreased spacing between donors, arising due to the randomness in the impurity distribution (appearance of the valence or conduction band tails). Both the scattering and the number of carriers in the tails increase with increasing strength of the impurity potential. But the resistivity of the nonmagnetic degenerate semiconductors in which such a purely electrostatic interaction occurs is virtually temperature independent. Here the qualitative considerations leading to the magnetoimpurity theory of CMR will be presented [118}136]. It is based on the fact that the ionized donors not possessing a magnetic moment at ¹"0, acquire it at "nite temperatures, and this moment is maximal close to ¹ . As ! a result, the conduction electron is scattered not only by the electric charge of the donor but also by its magnetic moment. This enhances the total strength of the scattering and, hence, increases the number of electrons in the currentless tail states and simultaneously diminishes the relaxation time for the electrons in the current-carrying states. Both these factors increase the resistivity of the system. The magnetic moment of the donor is maximal in the vicinity of ¹ , and the resistivity ! behaves in the same manner. Physically, the magnetic moment of the donor is nothing else but the excess magnetic moment of a region surrounding the ionized donor, and in this sense one can speak about an overmagnetized donor. The nature of its overmagnetization is as follows. The electron density in the vicinity of the donor is higher than that far from it as the electron screens the charge of the donor. The conduction electrons realize an indirect exchange which is stronger, the higher the electron density is. In the range of the semiconducting charge carrier densities, when all the conduction electrons are spin-polarized, this indirect exchange tends to establish the ferromagnetic ordering in agreement with the general theorem [11]. Hence, the total ferromagnetic coupling induced by the d}d-exchange and indirect exchange is the strongest close to donors. On increase in temperature, the ferromagnetic ordering becomes thermally destroyed close to the donors considerably less than far from them. The di!erence between the donor vicinity magnetization and the mean is just the magnetic moment of the donor. It is maximal in the vicinity of ¹ and diminishes with both decreasing and increasing temperature: in the former case it tends ! to zero and in the latter case it remains "nite but small at ¹PR. This is the reason for which the resistivity is maximal close to ¹ . ! Another possible physical interpretation of the joint action of the donor charge and magnetization is a screened Coulomb potential in which the true charge is replaced by a magnetizationdependent e!ective charge passing through a maximum close to the Curie point. Depending on the e!ective charge, one can speak about two limiting cases. If the e!ective charge di!ers from the true charge, but not very strongly, one can speak about increasing number of the charge carriers in the band tail. This does not mean a transition to the insulating state but simply an increase in the

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resistivity. In particular, such a situation is realized in the wings of the resistivity peak. A similar idea was expressed in Ref. [139]. If the e!ective charge is considerable, the Mott localization of the charge carriers can take place. As a result of this temperature-induced Mott transition, the crystal becomes a real insulator. The Mott localization di!ers from the Anderson localization in the band tails by the fact that here each electron is trapped by its own donor whereas in the wings of the resistivity peak the electron is trapped by a complex of the donor atoms. Hence, after the Mott localization, a degenerate semiconductor transforms into a nondegenerate semiconductor. On further increase in temperature, two situations are possible. If the charge carrier density is su$ciently large, the reverse transition to the highly conductive state is possible, and then the metal}insulator transition is reentrant. At lower densities, the crystal remains in the insulating state up to very high temperatures. The reason for this, in essence, is the same as the increase in the high-temperature activation energy as compared with the low temperature one (Section 3.7). The origin of the giant negative magnetoresistance is obvious from the aforesaid: everywhere the magnetic "eld moves the local magnetization closer to its saturation value, which decreases the di!erence between the local magnetic ordering in the vicinity of impurity and that far from it. In other words, it diminishes the s}d-exchange force attracting the carrier to the ionized impurity. Certainly, the magnetoresistance should be negative and isotropic, and it should be maximal in the temperature range where the resistivity passes the maximum. 4.4. Dielectric and magnetoelectric response functions in magnetic semiconductors far from the Curie point The problem which is necessary to solve to consider this magnetoimpurity mechanism consists in description of the impurity electrostatic potential e!ect on the local magnetization. For this, the magnetoelectric response functions will be introduced here. As is well known, a uniform electrostatic "eld can cause the magnetization only in antiferromagnetic crystals with a special crystalline symmetry [480]. But for nonuniform "elds, there are no restrictions for the magnetoelectric e!ect. The Hamiltonian of the system under consideration is the sum of the s}d-Hamiltonian (3.1), the Hamiltonian of the electron}electron interaction, and the Hamiltonian of the electron interaction with an external "eld  (i.e., with the "eld of ionized impurities): H"H(s}d)#H #H , GLR 1 2
e ak q ,  aH H " k ak q aH k GLR q N > N NY \ NY   O$ e H " (q)aH (q)"H(!q) . (4.2) k ak q , N > N   The q"0 component in H is omitted due to the presence of the compensating charge. GLR In this section the degenerate semiconductors with the simple exchange, =
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will be used, with  being the Fermi energy and  the dielectric constant of the crystal. The "rst of  them expresses a relatively small number of charge carriers in a degenerate semiconductor and ensures the complete spin polarization at ¹"0, the second is the standard condition for a semiconductor to be degenerate [149]. The condition of the complete spin polarization will be assumed as being met at all temperatures under consideration. Only temperatures and magnetic "elds will be considered when the equality EH(k, )"Ek !AM

(4.3)

is valid, where M is the mean crystal magnetization per atom. This is the spin-wave region, excluding the lowest temperatures, (cf. (3.10)) and the paramagnetic region in a magnetic "eld strong enough to ensure the complete electron spin polarization. In imperfect samples the static magnetization M #uctuates, being a function of the coordinate r. Then one may use Eq. (4.3) in the Born}Oppenheimer approximation, treating the quantity !AM(r) as an e!ective potential related to the magnetization #uctuations. It should be added to the conventional electrostatic potential to obtain the total impurity potential. Such an approach is justi"ed in the long-wave limit when the magnetization changes considerably over a length greatly exceeding the lattice constant. This is just the case here: the magnetization-#uctuation length should be of the order of the screening length which in semiconductors considerably exceeds the lattice constant [149]. The problem consists in "nding the impurity potential with account taken of the s}d-exchange energy (the second term in Eq. (4.3)) modi"ed by the electric "eld of this impurity. To describe this modi"cation, it is convenient to introduce the magnetoelectric response function in addition to the conventional dielectric function. The latter, (q) as usual, relates an internal electric "eld (q) to an external electric "eld (q) with the wave vector q. Thus, (q)"\(q)(q) .

(4.4)

The magnetoelectric response function, (q), relates the magnetization to the external "eld, which changes the magnetization M(q) at "nite temperatures by changing the indirect exchange via charge carriers. In case (4.3) is valid, one may introduce a magnetoelectric response function by means of the linear expression M(q)"(q)(q) .

(4.5)

(Under conditions when the linear relationship (4.4) is valid, the relationship (4.5) must be linear, too.) Hence, the e!ective "eld  (q) acting on the electron with the energetically favoured spin  projection is equal to e (q)"e(q)!AM(q)/2,e\(q)(q)  with `the e!ective dielectric functiona \(q)"\(q)!A(q)/2e .

(4.6)

(4.7)

To "nd it, one should use the constancy condition of the electrochemical potential over the sample in the form following from Eq. (4.2): !AM/2# (n)"!AM(r)/2#e(r)# [n(r)] N N

(4.8)

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where n and M are the mean values of n(r) and M(r), respectively,  is the Fermi energy of N spin-polarized electrons,  (n)"[6
n]/2m . N The r-dependence of the local magnetization M(r) is caused by the same dependence of the local carrier density n(r) on which M(r) depends directly. Hence, after linearization of Eq. (4.8) with respect to
n(r)"n(r)!n one obtains: n(q)"![e(q) dn/d ]/(1!
) N where

(4.9)


"(A/2)[dM(n)/dn] dn/d (4.10) N will be called the magnetoelectric constant. In calculating the dielectric function one should keep in mind that the electrostatic "eld acting in the crystal is the sum of the external "eld (q) (i.e., the "eld produced by ionized impurities) diminished by  times due to the polarization of the crystal, and the "eld
(q) produced by the  charge carriers:
(q)"(q)!(q)/ "[1!(q)/ ](q) . (4.11)   Using Eq. (4.9) makes it possible to represent the Poisson equation for the electron part of the electrostatic "eld in the form q
(q)"!4
e(dn/d )(q)/[ (1!
)] . (4.12) N  Then from Eqs. (4.11) and (4.12), one can deduce the following expression for the dielectric function of the conducting magnetic crystal: (q)" # /q ,    "/(1!
), "4
e dn/d . (4.13)  N N N In the long-wave limit the magnetoelectric response function (q) (6) is found from Eqs. (4.5), (4.8) and (4.9) to be (q)"!2e
/[A(1!
)(q)] .

(4.14)

In writing (4.14) the derivative dM(q)/dn(q) is replaced by dM/dn as its q-dependence is much weaker than that of (q). Using Eqs. (4.13) and (4.14), one obtains the e!ective dielectric function (4.7) in the form (q)"(q)(1!
)" (1!
)#/q . (4.15)  N According to Eqs. (4.15), (4.13) and (4.10), at A"0 both (q) and (q) coincide with the true dielectric function of a nonmagnetic crystal with completely spin-polarized charge carriers. As seen from Eq. (4.15), the e!ective dielectric function (q) di!ers from the true one (q) in replacement of the true dielectric constant  by the e!ective dielectric constant   " (1!
) . (4.16)  

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As follows from Eqs. (4.6) and (4.15), the electron potential energy in the e!ective "eld of an ionized donor is e (4.17) e (r)"!  exp (!kr) ,   r  4
e (4.18) "  dn/d , N   e (4.19) e "  (1!
where $e is the e!ective charge of the donor and of the electrons.  It is necessary to make the following important remark. As usual, the response functions are calculated for a perfect crystal, and then they are used for description of the state of the system perturbed by impurities. Such an approach suggests that the impurities do not in#uence very considerably the state of the bulk of the conduction electrons. It is commonly used, for example, for description of the screening of the ionized impurity "elds in degenerate nonmagnetic semiconductors. For the ferromagnetic semiconductors, it is applicable if the height of the resistivity peak in the vicinity of ¹ is not very large, or, if this is not the case, for the low- and high-temperature wings of ! the peak. 4.5. Spin-wave region for a simple exchange According to Eq. (4.10), to determine
, and, hence,  and e , it is su$cient to determine the   n-dependence of M. This program can be carried out for the spin-wave region, when Eq. (3.11) is valid. But the magnon frequency entering the distribution function should be renormalized by taking into account the indirect exchange via the charge carriers. It is easy to perform using Eq. (3.4) [82,11,133]: (4.20) (q)"H#J(1! q )#Aq/(q#q )  where "na, q "2mAs. Eq. (4.20) di!ers from the well-known RKKY result drastically which is  quite natural as the condition for the applicability of the RKKY theory 
(4.22)

¹ 1 3A
" , " . ¹
¹
256
 Jaq N 

(4.23)

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More general expressions for the response functions valid for any characteristic length are obtained in Refs. [82,142] using a diagram procedure for the temperature-dependent Green's functions expressing the Kubo formulae. As seen from Eqs. (4.22) and (4.23), at small densities,
increases with n like n, and at large densities, it decreases like n\. Hence, at a certain density  &J/A, it passes a maximum of the

height




A  ¹ .
&

J =

(4.24)

Like the inequality ¹'J used in deducing Eq. (4.21), extrapolating Eq. (4.23) to ¹ one sees that ! the magnetoelectric constant can be of the order of (AS)/=J. Keeping in mind the inequalities =


(H#zIM#A/2)S ¹



where B (x) is the Brillouin function. One obtains from Eqs. (4.10) and (4.24) 1 ASB 1 ,
"(3/8) [ (¹!zISB )] N 1 dB (x) (H#zIM#A/2)S B " 1 , x" . 1 dx ¹

(4.24)

(4.25)

Unfortunately, deciphering Eq. (4.25) requires numerical calculations, which were not carried out so far. Only the limiting case of very large H was analysed. The situation is quite di!erent in weak "elds. Below, the magnetic "eld will be assumed to be nonexistent and, hence, the magnetization in the paramagnetic region is zero. This means that one cannot use the magnetoelectric response function (4.5), and one should introduce its analogue in another way.

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In the paramagnetic region, but not very far from ¹ where the magnetic correlations are still ! signi"cant, the s-electron energy can be considered as analytical in AS/=. Hence, the electron energy renormalized due to the s}d-exchange (4.2) is given by the expression Ek "Ek #;k ,

(4.26)

with ;k " Su Su

>h

 Gk (h) ,

A exp(iqh) , (4.27) Gk (h)"  4N Ek !Ek q !i0 > where Kh (u)"Su Su h  is the correlation function of the d-spins, which in an ideal periodic system > does not depend on u. But in a real degenerate semiconductor, where the donor or acceptor impurity is distributed randomly over the sample, the electron density #uctuations change over the screening length r . Hence, the correlation function which depends on the indirect exchange  via charge carriers should change over r , too. As before, one may use the Born}Oppenheimer  approximation, considering Ek as a function of `the electron coordinatea u. On the other hand, the electron momenta are small compared with 1/a where a is the lattice constant, and ; is essentially  nonzero. This makes it possible to put ;k "; ,; in Eq. (4.27).  As the s-electron density #uctuates over the crystal due to the #uctuating electrostatic "elds of the randomly distributed impurities, one can relate #uctuations of the magnetic correlation functions to the #uctuations of these "elds. Going over from the coordinate representation to the wave-vector representation, one can introduce for qO0 a linear relationship between the Fourier component of the correlation function [143] Kh (q) and that of the external electric "eld (q): Kh (q)"h (q)(q)

(4.28)

where the coe$cient h (q) may be called the correlation magnetoelectric response function (CMERF). The term `correlationa is used to distinguish the response function introduced here by Eq. (4.28) from the magnetoelectric function (4.5). Obviously, for h"0, CMERF vanishes since the corresponding correlation function is simply S(S#1) independent of the electric "eld strength. Using CMERF, one may relate the energy of the s}d-exchange interaction to the external "eld as follows: ;(q)"(q)(q); (q)" h G (h)  h

(4.29)

where (q) will be called the energy magnetoelectric response function (EMERF). The EMERF, like the standard dielectric response function (q), should be determined from the condition of the constant electrochemical potential and the Poisson equation following the same pattern as in the preceding section. For purposes of further calculations, it is su$cient to write down the e!ective dielectric constant (q). It relates the e!ective "eld ; ";#e acting on C the electron and consisting of the internal electric "eld  and the s}d-exchange "eld ; to the external "eld : e(q) ; (q)" C (q)

(4.30)

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where the e!ective dielectric function (q) is given by an equation similar to (4.15) and the e!ective dielectric constant  exactly by Eq. (4.16):   (q)" 1# , (4.31)  q





4
e dn " ,  d  (3
n) (n)" , ( "1) . 2m

(4.32) (4.33)

The magnetoelectric constant
describing the coupling between the charge carriers and magnetic subsystem is given by d; dn
"! . dn d

(4.34)

Unfortunately, at present there is no universal rigorous theory of the indirect exchange under the condition (AS, typical of degenerate semiconductors. The spin-wave theory di!ers greatly from results of the RKKY theory which is a consequence of the complete spin polarization. But in the paramagnetic region the electrons are completely spin depolarized, and one may hope that the RKKY theory is valid in this case. If the assumption made above about its universal validity is wrong, nevertheless, this calculation remains, certainly, valid for  " 1# $ ln $ (4.35) I (q)"!   8 4k q 2k !q 2N Ek !Ek q > $ $ where nk is the s-electron distribution function, a the lattice constant, k "(2m). $ The correlation functions are found in the paramagnetic region using the high-temperature expansion. One obtains to the lowest order:



S(S#1)I (h) R+ (1!
h0 ) , Su Su h "S(S#1)
h0 # > 3¹ 1 I (h)"  I (q) exp(iqh) , R+ R+ N



(4.36)

where
uf is the three-dimensional Kronecker symbol. Hence, with allowance for Eqs. (4.35) and (4.36), the quantity
(4.34) is given by the expression AS(S#1)mak n $ (1!3k a/4
) .
" $ 4
¹

(4.37)

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As the magnetoelectric constant
(4.34) is positive, the magnetoelectric e!ect manifests itself in an enhanced value of the e!ective charge eH"e(1!
)\ as compared with the true charge. The magnetoelectric e!ect diminishes with increasing charge carrier density. It also disappears as 1/¹ with increasing temperature. Keeping in mind that in the spin-wave region it increases with temperature, one should conclude that it is maximum at intermediate temperatures of order ¹ . ! 4.7. Magnetoelectric response functions for the double exchange In the case of the double exchange the s}d-exchange energy is "xed, being equal to !AS/2 or A(S#1)/2, depending on the sign of the s}d-exchange integral. As the energies of two states with the moments S#1/2 and S!1/2 di!er by a very large quantity AS, it is su$cient to consider only the lower of these states. Only the e!ective hopping integral depends on the magnetization here (see Eqs. (3.5), (3.14) and (3.16)). Hence, the magnetoelectric response function should be indirect, relating the external electric "eld to the e!ective hopping integral [146,123,134]: t (q)"(q)(q) . 

(4.38)

Strictly speaking, a "eld-induced change in t causes changes both in the local conduction band  bottom position and in the local value of the e!ective mass. But the latter factor can be neglected as it manifests itself only in the kinetic energy of the s-electrons which is of order = and, hence, small compared to the band width at small . Thus, the e!ective "eld acting on the electron is e (q)"e(q)!zt (q),\(q) (q)   

(4.39)

where z is the coordination number. Using again the condition of constancy for the electrochemical potential and the Poisson equation, one arrives at the following expression for the magnetoelectric constant:
"z

dt (0) dn  , dn d

t (0),t .  

(4.40)

In calculating
in the spin-wave region an expression for the magnon frequencies under condition of the double exchange is necessary. It was "rst derived in Refs. [147,148] (for a more detailed discussion see Section 5.2). Here, like in Ref. [133], it will be obtained directly from Eq. (3.4): zt 1 q "H#(J#J )(1! q ), J "  nk k ,   2S N

(4.41)

where nk is the charge carrier distribution function. This theoretical result was con"rmed by the experimental data [165]. Using Eqs. (4.40), (4.41) and (3.14) for t , one obtains the following equation for the magneto electric constant: dgq 3zt 1  (1! q )
"! d q 8(6
)S N

(4.42)

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where  is the number of electrons per atom, gq the magnon distribution function. In particular, at H"0, z"6, for ¹ <¹<¹ /S ! ! one has 1 48t ¹ " .
" , ¹
¹
(6
)(2SI#6t)

(4.43)

The behaviour of
for the double exchange (4.43) strongly resembles that of
for the simple exchange (4.21)}(4.23). The constant
(4.43) "rst increases with n like  and then decreases like \. It reaches maximum at  "SJ/15t where it is about 0.2t¹(IS)\. Hence, formally, on

increase in temperature, the magnetoelectric constant can approach unity and even exceed it. Unfortunately, an expression for
in the paramagnetic region is not obtained yet. But, keeping in mind the similarity between the behaviour of the magnetoelectric constants for the simple and double exchanges just mentioned one can hope that in the paramagnetic region they behave in the same manner, too. 4.8. Magnetoimpurity resistivity and colossal magnetoresistance The results obtained earlier in this section, make it possible to calculate the magnetoimpurity resistivity which is caused not only by the scattering of the charge carriers but also by the Anderson localization in the band tails. In the spin-wave region the calculation is valid for both the simple and double exchanges. As follows from Eq. (4.17), the total magnetoimpurity potential of the Coulomb centre exhibits the same coordinate dependence as the standard electrostatic potential in degenerate semiconductors. For this reason the expressions found for nonmagnetic semiconductors remain valid here, too. The only di!erence consists in replacement of the true dielectric constant  by the e!ective dielectric constant  . One should also take into account that the   fermions here are spinless which is tantamount to their complete spin polarization. Then one arrives at a modi"ed Brooks}Herring expression for the relaxation time (see, e.g., Refs. [11,149]):

4k 2
enm F( ); F( )"ln(1# )! ; " . (4.44) \" I 1# 

k  Eq. (4.44) is valid when k greatly exceeds the inverse screening length 1/r "/, i.e., <1. As   follows from Eq. (4.44), on increase in temperature, the scattering probability increases as ln  / .   As for fermions in the band tail, for the case when the Fermi level is well above the subband bottom, generalizing results presented in Ref. [149] for nonmagnetic semiconductors, one "nds the density of states in the tail to be given by the expression




(m E G g(E)"  (2


(4.45)

with



1 V G(x)" exp(!y)(x!y dy; (
\

e " (4
nr .   r  

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According to (4.45), the total number of states in the tail is

 

 V 1 Q" (4.46) dx exp(!y)(x!y dy . (
\ \ Hence, on increase in temperature, the number of tail-localized fermions which do not participate in the charge transport increases as \.  The corresponding equations for the paramagnetic regions di!er from (4.44) and (4.45) by the fact that the spin polarization is absent, and these expressions should be taken in the standard form which means changes in some factors of order one. A direct comparison of the magnetoimpurity scattering with the magnon scattering of charge carriers was carried out in Ref. [134]. It shows that, at typical parameter values, the temperaturedependent part of the magnetoimpurity scattering is, at least, an order of magnitude larger than the magnon scattering. The role of the magnetoimpurity scattering increases due to increase in the number of charge carriers localized in the band tails. On increase in temperature in the spin-wave region, the e!ective dielectric constant  decreases  which re#ects a more intensive attraction of the electrons by ionized donors. This attraction can become so strong that the Mott transition from a highly conductive state to the insulating state takes place. The corresponding calculation can be carried out using the Mott pattern described in Section 4.2. The only di!erence consists in the replacement of the true dielectric constant by the e!ective one in Hamiltonian (4.1a). Similarly, the trial wave function is taken in the form (m)Q N "  (2








x  xr exp !  (r)" 
a a @ @  a "  "(1!
)a @ me

(4.47)

where x is the variational parameter, a a formal quantity characterizing the highly conductive @ state of the crystal (to avoid misunderstanding, note that this is not the radius of the Bohr orbit at "nite temperatures). With Eqs. (4.1a) and (4.47), one obtains the following analogue of the Mott condition (4.1) for the temperature-induced metal}insulator transition: na (¹)"C . (4.48) K @ + The constant C is equal to 0.25 above ¹ (the electrons are completely spin depolarized) or 0.41 + ! below ¹ (the electrons are completely spin polarized). ! In writing the temperature-dependent Mott criterion (4.48) one should keep in mind that, "rst, the Anderson localization in the band tails takes place before the Mott transition, so that the charge carrier density is smaller and the indirect exchange is weaker close to the Mott transition temperature ¹ than at ¹"0. Hence, ¹ found from Eq. (4.48) is an upper bound for its + + real value. Second, as already pointed out in Section 4.2, the Mott treatment establishes only conditions under which the uniform metallic state ceased to be stable. But it does not follow from it that the metal}insulator transition should be abrupt. According to Ref. [81], it should be continuous. If this is true then ¹ represents the temperature at which a rapid resistivity rise + begins.

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Let us consider a state with the density n exceeding n at ¹"0 (the latter will be denoted as n ). K  Then this state is highly conductive at ¹PR. But, according to Eqs. (4.13) and (4.48), it loses its stability at the temperature at which the magnetoelectric constant attains the value




n  ,
(n)"1!  + n

(4.49)

which for the case ¹ /S;¹;¹ reduces to equality ! !




n ¹ (n)"¹
1!  + n



(4.50)

where ¹
is given by Eqs. (4.22), (4.23) and (4.43). As is seen from Eq. (4.50), the temperature of the metal}insulator transition can di!er strongly from ¹ in accordance with experimental data for ! EuO presented in Section 4.1. This situation di!ers strongly from that observed in the manganites which should be ascribed to the fact that the d}d-exchange in them is not ferromagnetic but antiferromagnetic. In fact, the metal-to-insulator transition switches o! the indirect exchange in them. But this means that the ferromagnetic long-range order should disappear simultaneously, as it is established only by the indirect exchange. As seen from Eq. (4.49), at n/n P1 the critical
value, at which the highly conductive state  + loses its stability, tends to zero. But, for nPR, the critical
value reaches unity which can + be unattainable in real systems. Hence, in materials with high donor densities, the transition to the insulating state will not take place. The theory developed makes it possible to explain qualitatively the colossal magnetoresistance of the ferromagnetic semiconductors. A basis for this explanation is provided by Eq. (4.49). In fact,
depends not only on ¹ but also on H. Let us assume that, at ¹"0 the crystal is highly conductive, but at H"0 and a temperature ¹,
(¹, H"0) exceeds
(n) (4.49), and, hence, the + system is in the insulating state. Under a certain "eld, at ¹ "xed,
(¹, H) becomes less than
, and + the system again becomes highly conductive. One can write:
(¹, H)"
(¹, 0)#H d
/dH which implies that the "eld H necessary for the insulator-to-metal transition, should be +
!
(¹, H"0) . H " + + d
/dH

(4.51)

According to Eqs. (4.22), (4.23) and (4.24), for large spins in the region of the linear
dependence on ¹, this equation can be rewritten as ¹ H & ! (¹!¹ ) . + ¹ + + If
(¹, H"0) is close enough to
, or ¹ to ¹ , a very small "eld causes the transition from the + + insulating state to the metallic. The magnetoresistance should be colossal even in such small "elds. The quantity [(0)!(H )]/H tends to in"nity at the temperature of the metal}insulator + + transition ¹ determined by (4.49) and (4.50). Certainly, randomness in the impurity distribution + should spread this peak.

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4.9. Reentrant or nonreentrant temperature-induced Mott transition? The Mott-like approach used in the preceding section, does not allow us to answer a very important question: what occurs after the transition into the insulating state, on further increase in temperature. Experimental data of Section 4.1 show that, depending on the charge carrier density, the crystal can return to the highly conductive state or remain in the insulating state. To solve this problem, it is necessary to take into account the interaction of the bound electrons with random #uctuations (Section 3.7), which is impossible to implement within the framework of the magnetoelectric approach. This is one reason why another approach will be used to solve this problem. Another reason for it is that there exists a large di!erence between critical densities for a ferromagnetic crystal found by the Mott approach and the approach from the insulating state (Section 4.2). The procedure used below consists in direct comparison of the free energies of the metallic and insulating states. It is di$cult to hope that the procedure used below will give an accurate value for n . But it can give  a rather accurate value for n !n .   The procedure used below makes it possible to prove that the Mott critical density n at  ¹PR is higher than n at ¹"0. For this reason, if the charge carrier density n lies between  n and n , then, on increase in temperature, the crystal goes over from the highly conductive state   to the insulating state and remains in the latter up to the highest temperatures. But, if n exceeds both n and n , then the crystal, after passing the resistivity maximum, returns to a highly   conductive state at higher temperatures (the reentrant metal}insulator phase transition). We begin with the phase transition in the spin-wave region under the condition (AS. Using expressions for the energy of the electron gas from Refs. [150,151], one "nds the following expression for the donor metal energy per donor atom (unlike the `magnetica index m, the index M denotes metal): 3(6
n) #E (n) E$ "E(k"0)#  + 10m

(4.52)

where E(k"0) is the electron energy at the conduction band bottom, E the exchange energy  between conduction electrons, n the electron (or donor) density. The term !AS/2 is omitted. Under condition of the complete spin polarization, E (n) is easily obtained by generalization of the  corresponding Bloch expression for the completely spin-depolarized electron gas presented, e.g., in Ref. [150]:




3 6n  e . E (n)"!   4


(4.53)

To calculate the energy E(k"0), the Wigner}Seitz procedure will be used (see, e.g., Ref. [151]). Each ionized donor is surrounded by a sphere of radius ¸"[3/(4
n)]. Inside each Wigner}Seitz cell, the electron wave function  corresponding to k"0, satis"es the wave equation






 e ! ! !E(k"0) (r)"0 2m r

(4.54)

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with the boundary condition d (¸)"0 . dr

(4.55)

As is well known from the theory of the metallic cohesion, the wave function  should be almost constant with the boundary condition (4.55). A special analysis shows that, for relative densities "na between 0.001 and 0.1, the "Const approximation ensures an accuracy in the energy higher than 1%. For this reason, with a su$cient accuracy, one can put




E(k"0)"!3

4
n  E . 3

(4.56)

With allowance for Eqs. (4.53) and (4.56), energy (4.52) in E units takes the form:




3 3 6  E$ " (6
)b!(36
)! b + 5 2


(4.57)

with b"a /a, E "e/2 a , a " /me.   At "nite temperatures, the free energy of the donor metal with the volume < is given by the expression G$(n)"n<E$ (n)#Nf , + + a f "¹ dq ln[1!exp(! q /¹)] , + (2
)



(4.58)

where the magnon frequencies q are given by Eq. (4.20). Equating the energy E$ (n) (4.57) with the + donor energy E "!E , one "nds that the density n , at which the electron delocalization takes '  place at ¹"0, obeys the relationship na "C with C "0.208, which is slightly less than the  + + value of 0.25 found by Mott. To "nd the transition temperature from the highly conductive state to the insulating state for a material with n exceeding n , one should equate the metal free energy G$ (4.58) with the free  energy of the localized state found with the use of Eqs. (3.29) and (3.30): F'"N(E #N f )#Nf (1!N ) . ' * '  * Then, for n su$ciently close to n , one obtains the following implicit expression for the transition  temperature: (! ) 

d(nE$ ) + "N ( f !f )#( f !f ),  "n a . * '   +   dn

(4.59)

Numerical calculations based on Eq. (4.57) show that the quantity d(nE$ )/dn is negative for + (0.2. This does not mean an instability of the system as this derivative is not the electron Fermi energy, but the change in the energy of the donor metal resulting from the change of the number of donor atoms by unity. The expression on the right-hand side of Eq. (4.59) is also negative for x close to 1, which is seen from numerical calculations. The proof of this statement is especially simple in the case S<1, if one considers the region ¹ /S;¹;¹ and uses Eqs. (3.21), (3.22), (3.27), (3.29), ! ! (3.30) and (4.58) (¹ is the Curie point). This means that equality (4.59) can be met for  exceeding !

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 not very strongly, and the transition from the metallic state to the insulating state should take  place with increasing temperature. But for large densities '0.2 this transition is prohibited, at least, in the spin-wave region, which agrees with experimental data cited in Section 4.1. Now we investigate in more detail the temperature-induced transition from the metallic to insulating state which can occur with decreasing temperature in the paramagnetic region. In the high-temperature limit, the total free energy of the donor metal is given by G.+"n<E.+#
G.+ . (4.60) + The energy of a nonmagnetized crystal per donor atom, instead of Eq. (4.57) is given by the following expression, which includes the correlation contribution [150]:




0.113b 3 3  3 b! . E.+" (3
)b!(36
)b! + 0.1216#b 2
5

(4.61)

By using the high-temperature expansions one "nds the magnetic free energy to be given by S(S#1)I (q) R+
G.+"!N¹ ln(2S#1)! 12¹

(4.62)

where I is given by Eq. (4.35). R+ First, it will be proved that a sample which was in the highly conductive state at ¹"0, can become insulating at elevated temperature and remain nonmetallic up to arbitrary high temperatures. This is related to the fact that the #uctuations lower the donor level strongly, whereas, for delocalized electrons, such a lowering is absent. As a result, according to Eq. (4.57), at ¹"0, the delocalization of the donor electrons occurs at the density n corresponding to the Mott-like  equality na "0.208. But, equating the energy (4.61) to the energy F (3.41), one "nds that the   delocalization density n at ¹PR exceeds the ¹"0 value n , if the ratio AS/E exceeds 1.27.   Normally, this ratio is essentially larger, and, for AS/E "5, the Mott-like relation takes the form na "0.378. Hence, normally, n considerably exceeds n .    This fact results in a nontrivial temperature dependence of the electric properties of a degenerate ferromagnetic semiconductor. For the donor density n in the range between n and n , at low   temperatures the system behaves like a metal, but remains insulating up to arbitrarily high temperatures after its transition from the metallic state to the insulating state. If the density n exceeds n , then the reentrant metal}insulator transition takes place with increasing temper ature. This implies a high resistivity peak at elevated temperatures of the order of the Curie point. Using Eqs. (3.44), (4.60) and (4.62), one can "nd the expression for the temperature at which the temperature-induced metal}insulator transition occurs when the donor density n exceeds n :  (! ) d(nE.+) 1  + "! Q!R dn ¹



S(S#1) R" dq I (q) . GB 12(2
)

(4.63)

In writing Eq. (4.63), the fact of the d(nE.+)/dn being negative is taken into account. This fact is + established by numerical calculations, which evidence that, at least up to "0.2, this derivative is about (!2) in E units.

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Numerical calculations show also that, for I"0 and " , the denominator in Eq. (4.63) for  1/¹ is positive which guarantees the positive transition temperature ¹ . It decreases with  increasing density  and depends also on the direct exchange integral I. For example, for AS/E "5, Q!R it is equal to 0.008 for I"0.02, to 0.005 for I"0 and 0.001 for I"!0.02 (a negative I value corresponds to initially antiferromagnetic systems such as the manganites which means that in them the transition from the metallic state to the insulating state is possible, too). The existence of not only a low-temperature, but also a high-temperature, transition point means that the metal}insulator transition is reentrant. The problem of the two-electron donors was not investigated in detail yet though qualitative considerations about them were presented in Refs. [138,152]. As already stated in Section 4.2, in the ferromagnetic semiconductors their ground state can be (1s)(2s) due to the molecular "eld of the crystal AS. As the radius of the 2s orbit is large, the conditions for delocalization of the 2s electrons are very favourable. But, on increase in temperature, the molecular "eld of the crystal diminishes at AS;=. As a result, the inversion of the (1s) and (1s)(2s) terms can occur. Since the 1s-orbital radius is considerably less than the 2s-radius, this should cause transition to the insulating state, in which the crystal will remain up to the highest temperatures. But the donor (1s)(2s) state can remain stable at high temperatures due to the electron interaction with the random magnetization #uctuations. In this case the behaviour of the two-electron donors resembles the behaviour of the one-electron-donors: again reentrant metal}insulator transitions become possible. The same is true for the two-electron donors at the double exchange. 5. Ferromagnetic lanthanum manganites 5.1. Crystallographic properties of doped manganites In the following sections a description of the properties of manganites will be given. It is advisable to begin with the crystallographic properties of the doped manganites keeping in mind that these properties for the undoped manganites have already been described in Section 2.2: their structure di!ers from cubic due to JT distortions. As will be seen from the following, the doping changes character of the distortions and diminishes their magnitude making the lattice cubic or almost-cubic at x su$ciently large. The crystallographic structure changes on doping and, due to its being easily variable, results of various experimental groups di!er. According to Ref. [31], at small x in La Ca MnO , like in \V V  pure LaMnO , the O-rhombohedral phase occurs when b/(2(c(a and with the static JT  e!ect. The La Ca MnO phase diagram at x less than 0.15 is characterized by a wide \V V  coexistence range of the O- and O-orthorhombic phases. On increase in temperature, the O-phase is changed to the O-phase continuously. At x"0.4 an additional phase transformation occurs and the nature of this is still not established. According to Refs. [35,36], on increase in x, the La Ca MnO structure goes from mono\V V  clinic "rst to orthorhombic (about x"0.05) and then to cubic (about x"0.15). At xO0 the temperature rise causes transitions of the monoclinic phase into the orthorhombic and of the orthorhombic phase into the cubic. According to Refs. [37,52], the La Sr MnO room\V V  temperature structure is orthorhombic at x less than 0.175. At larger x it is rhombohedral. But

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Fig. 10. Structural as well as electronic phase diagram of La Sr MnO with the composition x. ¹ , ¹ , and ¹ stand \V V  1 , ! for structural, antiferromagnetic and ferromagnetic transition temperatures, respectively. Abbreviations: antiferromagnetic insulator, AFI; ferromagnetic insulator, FI and ferromagnetic metal, FM [52]. Fig. 11. Magnetostructural phase diagrams obtained from the ¹-sweep experiments at a constant "eld for x"0.160, 0.170, and 0.180. The open and closed symbols stand for the transition from the O to the R phase, and reverse transition, respectively. The region surrounded by the two phase boundaries is a hysteretic region [52].

in the x value range between 0.175 and 0.25 the rhombohedral structure appears as a result of the phase transition from the orthorhombic structure occurring with increasing temperature (Fig. 10 [52]). The O-orthorhombohedral phase was observed also in La Sr MnO with x40.17 [42]. But \V V  in the same system, the pseudocubic phase O with b/(2+c+a exists. At x"0.125, the reentrant phase transition OPOPO is observed. For 0.104x40.17 a ferromagnetic ordering strongly suppresses lattice distortions at and below the Curie point. The same conclusion was made in Ref. [42]: the JT distortion of the structure is sensitive to the ferromagnetic ordering, appearing in the crystal at doping: it suppresses these distortions [42]. The magnetic "eld causes an orthorhombic to rhombohedral structural phase transition in La Ca MnO [43] and in La Sr MnO [40,52]. According to single crystal neutron studies \V V  \V V  [40] carried out on a La Sr MnO , with a transition temperature from the orthorhombic      phase to the rhombohedral that is about 290 K, a "eld of 30 kOe causes a similar transition at 290.5 K. On decrease in the "eld strength, below 20 kOe the sample partially transforms into an orthorhombic structure. This transition is especially pronounced for x"0.170 (Fig. 11 [52]) where the hysteresis is clearly seen.

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Judging from the measurements of the thermal expansion in La Sr MnO [41], the      pressure, increasing the ferromagnetic coupling, lowers the stability of the low-temperature orthorhombic phase: at p"0 the Curie point ¹ amounts to 260 K and the structural transforma! tion point 290 K. But at p"9 kbar the Curie point becomes higher, reaching 290 K, whereas the structural transformation point becomes lower, amounting to 180 K. Changes in the structure of the basic manganite occur also on increase in the oxygen content by annealing in an oxygen atmosphere (nonstoichiometric LaMnO ) [33]. First, a second ortho>W rhombic phase of the same Pnma space group appears. Then a monoclinic phase of the space group P112 /a appears. At last, the oxidation can result in the appearance of a rhombohedral phase of the  space group R3 c which exists only at elevated temperatures. The results of Ref. [33] strongly suggest the progressive development of cation vacancies in equal numbers on the La and Mn sites as the oxygen content is increased. Earlier the fact that, on oxidation, the oxygen sublattice remains fully occupied, and no interstitial oxygen atoms appear, whereas vacancies occur on both the Mn and La sites, was established in Refs. [44}47]. The more complicated La Sr MnO system was investigated in Ref. [38], using the \V V >W neutron powder di!raction as a function of both Sr doping and oxygen partial pressure during the synthesis (from &0.0002 atm to 1 atm at 1000 K with subsequent rapid cooling). Here, too, lattice distortions were observed up to x"0.225. Depending on x and the oxygen pressure, three phases were found at room temperature: rhombohedral, with space group R3 c, orthorhombic with space group Pbnm, and monoclinic with space group P2 /c. The former is ferromagnetic  and the latter antiferromagnetic. The orthorhombic phase is ferromagnetic for x exceeding 0.125. A speci"c character of the structural phase transition in the manganites is evidenced by sound velocity anomalies in the transition points [65,66]. In Ref. [66], unlike [65], the samples were single crystalline. According to Ref. [66], in La Sr MnO a strong jump-like increase (up to 30%) in \V V  both the longitudinal and transverse sound velocities which attests to sizeable hardening of acoustic branches of the phonon spectrum, is observed near the temperatures of the structural phase transitions between the rhombohedral and orthorhombic phases (for x"0.175, 0.2, 0.25) as well as another phase transition the nature of which is not established yet (for x"0.1). Hence, the results [52] on the existence of the temperature structural transformations are con"rmed in [66]. It is interesting to note that, according to Ref. [66], the crystallographic anisotropy does not manifest itself in the acoustic modes. As indirect evidence of disappearance of the static JT e!ect with doping, the data [39] should be presented: the Raman spectra of La M MnO with M"Sr, Ca and x from 0.1 to 0.3 do not \V V  reveal a structure like those for a cubic lattice. According to Ref. [144], in these crystals local JT distortions of the structure are possible despite the absence of the total JT distortion of the crystalline structure. In Ref. [64] the problem of absence of the orbital ordering in La Sr MnO system with \V V  x'0.2 was considered theoretically. For this, the #uctuations in the orbital degrees of freedom are investigated. Mathematically, this means studying the e!ects of the degeneracy of the e orbitals in E the limit of the strong repulsive electron}electron interaction. The isospin is introduced to describe the orbital degrees of freedom and is represented by the boson with a constraint. The dispersion of this boson is #at along the direction (
/a,
/a, k ) and the other two equivalent directions in the X k-space. This enables the orbital disordered phase down to low temperatures, though the amplitude

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Fig. 12. Magnetization vs. concentration of Mn> ions for La Ca MnO . Solid line from data of [28], dashed line \V V  from [2]. Straight line, top, denotes the maximum magnetization for the specified composition (M in Bohr magnetons per Mn ion).

of the short-range order is large. Some of the anomalous experiments in the low-temperature ferromagnetic phase of La Sr MnO are interpreted in terms of this orbital liquid picture. \V V  5.2. Magnetic properties of ferromagnetic manganites The initially antiferromagnetic manganites become ferromagnetic as a result of their doping making them degenerate semiconductors or metals. The ferromagnetic ordering is established due to the indirect exchange via charge carriers. As an example, La Ca MnO can be considered. As \V V  at both the x"0 and x"1 ends this compound is antiferromagnetic, it is clear that it can be ferromagnetic only inside a range of intermediate x values. Experimental investigations show that the magnetization is close to complete beginning from about x"0.17 [2] or 0.3 [28], and at about x"0.5}0.6 it vanishes (Fig. 12). As seen from this "gure, the notion of the completely ferromagnetic ordering is rather conventional as the experimental values of the magnetization reach the maximal possible magnetization at x given only for an only x value. For other x, strictly speaking, the magnetization is unsaturated, so that the choice of the boundary x value at which the complete ferromagnetic ordering begins is rather arbitrary when one uses the magnetization value as a criterion.

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The possibility to establish this boundary exists due to the fact that, at relatively small x values, the doped manganites display features of the ferromagnetic and antiferromagnetic systems simultaneously. According to the neutron studies [255,256] the transition to the pure ferromagnetic state in La Sr MnO should take place near 0.175, when antiferromagnetic peaks disappear and only \V V  ferromagnetic ones remain. The same criterion was used in the NMR investigations of the La Sr MnO high-quality single crystals [154]. Below x"0.1 two lines were observed, one of \V V  which corresponds to the antiferromagnetic ordering and the other to ferromagnetic ordering. The former disappears at x"0.1, and hence, the saturated ferromagnetism appears at this x value which is considerably lower than that obtained from Refs. [255,256]. The authors [256] explain such a large di!erence in results by high quality of the samples used by them. But it is impossible to conclude from the data [154] whether the ferromagnetism is saturated above x'0.1. It can occur that the ferromagnetic ordering is incomplete. The reason for this is, possibly, a magnetic inhomogeneity of single phase manganite samples discovered in Ref. [221] by the FMR measurements and in Ref. [172] by the muon spin relaxation. The inhomogeneity should result from inhomogeneous charge carrier density distribution due to presence of ionized acceptors in the samples (Sections 4.3 and 4.7). Unlike the boundary x value for the saturated ferromagnetism, the Curie point of the spontaneously magnetized manganites is a well-de"ned quantity. It depends strongly on the composition. For example, ¹ of La Ca MnO with x"0.3 is 250 K [35]. For La Sr MnO , in the range ! \V V  \V V  from x"0.25 to 0.5 the Curie point is virtually x-independent, being close to 350 K [37] and for La Pb MnO it attains the maximum value of 370 K at x"0.4 [155]. Completely ferromag\V V  netic materials may also be fabricated without doping through production of nonstoichiometric compositions. Thus, LaMnO is a ferromagnet with ¹ of about 160 K [156,157]. The fer  ! romagnetism can also be brought into existence by the La de"ciency: epitaxial thin La MnO \V \W deposited on the SrTiO substrate are ferromagnetic already at x"0. But, on increase in the La  de"ciency, the Curie point of as-grown "lms increases from 135 K at x"0 to 265 K at x"0.33. Annealing in the oxygen atmosphere raises the Curie point almost up to room temperature [158]. Investigations of the ¹ dependence on the pressure P carried out on La Sr MnO single ! \V V  crystals [159,160] yielded that in the range 0.154x40.5 the Curie temperature increases with pressure. But the shift of ¹ depends on x: the quantity d ln ¹ /dP amounts to 0.065 GPa\ at ! ! x"0.15 and only to 0.005 GPa\ at x"0.4}0.5. The pressure e!ect on ¹ of more complicated ! systems was investigated in Ref. [177]. In all of them ¹ increases with pressure. ! In the spin-wave region the ferromagnetic manganites display the standard behaviour. In Ref. [161], magnetically uniform La Ba MnO systems were investigated as being in the      completely ferromagnetic state. At low temperatures the magnetization reduces with increasing temperature according to the Bloch ¹-law. This gives evidence that magnons are well-de"ned elementary excitations in them. Similar results were obtained from magnetization measurements in La Sr MnO [163] and from spin-resonance measurements [164].      The same conclusion was made in Ref. [162] in investigating La Ca MnO by the inelastic \V V  neutron scattering. At x"1/3 (¹ "250 K) the long-wave magnon frequencies are given by the ! usual quadratic relationship for isotropic ferromagnetic systems q "Dq with D"170 meV As . This D value is close to that found in [163] (154 meV As ). At x"1/3, in La Ca MnO an \V V  anomalous strongly "eld-dependent di!usive component develops above 200 K and dominates the #uctuation spectrum as ¹P¹ . This component is not present at lower x. !

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Fig. 13. Magnetic moment of La Ca MnO with x"0.42 on cooling and warming for samples containing di!erent \V V  oxygen isotopes: (a) O, (b) O. Hysteretic behaviour is observed at the charge-ordering transition, but is more prominent in the O sample [168].

In Ref. [165] inelastic neutron scattering was used to measure the spin wave dispersion throughout the Brillouin zone of La Pb MnO . Magnons with energies as high as 95 meV are      directly observed. A very simple Heisenberg Hamiltonian with solely a nearest-neighbour coupling of 8.8 meV accounts for the entire dispersion relation. The calculated Curie temperature for this local moment Hamiltonian overestimates the measured Curie point (355 K) by only 15%. Raising the temperature yields unusual broadening of the high-frequency spin waves, even within the ferromagnetic phase which is likely to resemble the results of Ref. [162]. Some samples display a highly unusual behaviour: with increasing temperature instead of a monotone decrease a peak in the magnetization whose height exceeds its ¹"0 value was discovered in La Ca MnO with x about 0.4 [168] (Fig. 13) and in La Sr MnO with \V V  \V V  x"0.1 [169]. This peak displays a thermal hysteresis [168].

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In all the papers the disappearance of the magnetization was found to be continuous. In some papers the critical exponents are determined. According to Ref. [170], in La Sr MnO with      ¹ "354 K the critical exponent for the spontaneous magnetization below ¹ is equal to 0.37 ! ! and the critical exponent for the magnetic susceptibility above ¹ to 1.22. But, according to ! Ref. [171], the exponent is equal to 0.45. Zero-"eld spin relaxation experiments on La Ca MnO yield "0.345 [172]. Critical phenomena in more complicated La-based      manganites were investigated in Refs. [173}176,191]. If the situation in the spin-wave region seems to be rather simple, the behaviour of the system close to ¹ is much more complicated. A ferromagnetic ordering is established in the manganites as ! a result of the indirect exchange via the charge carriers. But in them the Curie point coincides or is very close to the temperature at which the resistivity of the samples increases by several orders of magnitude (cf. Section 5.3). This means that no delocalized charge carriers remain to implement the ferromagnetic indirect exchange though localized charge carriers can establish the short-range ferromagnetic order. But this order should di!er from that which is realized in the Heisenberg ferromagnets above ¹ . For this reason the critical exponents for these doped manganites and ! Heisenberg ferromagnets hardly coincide. In Ref. [193] evidence was obtained for collective spin dynamics above the ordering temperature in La Ca MnO from the temperature dependence of the EPR lines at 9.2 and 35 Ghz from \V V >W 270 to 800 K. The intensity of the EPR line decreases exponentially for ¹'¹ , with an activation ! energy which varies with x and y. EPR data below 600 K cannot be attributed to an individual Mn ion of any valency MnL> for all annealing conditions. The maximum EPR intensity corresponds to correlated moments with an e!ective spin of 30 at 280 K. In Refs. [277,278] formation of small magnetic clusters was observed in (La-Y/Tb) Ca MnO    above ¹ as shown by sharply increased small-angle-neutron scattering and change in the sample ! volume. Both these e!ects are suppressed by a magnetic "eld. It is assumed that above ¹ small ! lattice polarons appear and become centres of magnetic clusters of about 1.2 nm size (Fig. 14). But such an explanation meets di$culties: if the mobile holes transform into small polarons, they cannot realize the indirect exchange, as the small polaron mobility is very low. Hence, there would be no reason for formation of the ferromagnetic clusters. One can propose another explanation for this e!ect without invoking small lattice polarons. In a crystal ionized acceptors exist, and they, and not the lattice polarization, capture the holes by their Coulomb "elds at elevated temperatures, being unable to do that at ¹"0 (the hole Mott delocalization (Section 4.2)). Thus, in reality, hole capture means a temperature-induced transition of a degenerate semiconductor into a nondegenerate one. Certainly, such a change in the electron states from the delocalized ones to the localized ones should lead to a change in the lattice state which manifests itself in a change of its volume. In Ref. [276] MoK ssbauer investigations of La Ca MnO established formation of the      spin clusters below as well as above ¹ . A magnetic "eld results in a coalescence of small ! spin clusters into larger clusters with more oriented spins. One can talk about superparamagnetic particles of order 10 nm with a size distribution. A fraction of the ferromagnetic material remains intact and survives a breakup into superparamagnetic clusters up to ¹ . ! In my opinion, one can explain such behaviour by localization of holes at acceptors. As the crystal after hole localization becomes again mainly nonmagnetic, the radius of the trapped hole orbit is less than that in the ferromagnetic state (the bound ferrons, Section 6.2).

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Fig. 14. (a) Experimental volume thermal expansion,

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A magnetic "eld increases the orbital radius and unites several clusters into a ferromagnetic microregion. Similar results were obtained in Ref. [291] for La Sr MnO in studying the small-angle      neutron scattering close to ¹ . Nanometre magnetic inhomogeneities were observed which do not ! correspond to magnetic polarons. The average magnetic coherence extends over several Mn spins with an unusual spin correlation function. Possibly, this is a result of a hole trapping by a complex of Mn ions in a state corresponding to the band tail (Section 4.3). As for comparison with the theory, the experimental data [165] completely correspond to Eq. (4.41) for the magnon frequencies in the double exchange ferromagnetic systems and can be considered as con"rmation of the theory [147,148] leading to this equation. Theoretically, Eq. (4.41) was reproduced in the latest investigations [166,167]. Attempts to calculate the Curie point of the double exchange systems were performed in an earlier paper [185] using the mean-"eld approximation, and in the latest papers [186}188]. Though it is obvious that ¹ should be of order ! zt, numerical results of these papers di!er ( is the number of charge carriers per atom). In particular, in Ref. [188] the classical double-exchange Hamiltonian (3.5) and the Monte-Carlo method were used for numerical estimates. 5.3. Resistivity of lanthanum manganites Now the transport properties of the lanthanum manganites will be discussed. A typical resistivity vs. temperature curve is presented in Fig. 15 for La Sr MnO [37] (earlier quite similar curves \V V  were presented in Ref. [2]). As seen from this "gure, at x(0.15 the crystals are characterized by the conductivity of the semiconducting type. But, unlike the nondegenerate ferromagnetic semiconductors (Fig. 2), the manganites do not display the resistivity peak. This means that the antiferromagnetic ordering is dominating in them (the nondegenerate antiferromagnetic semiconductors, unlike the ferromagnetic semiconductors, do not display the resistivity peak [11]). The behaviour of the system considered is of the same type as that of the nondegenerate ferromagnetic semiconductors only at x"0.15. At x50.175 their conductivity is of the metallic type with values of the order of 10\}10\  cm at low temperatures which correspond to degenerate semiconductors or poor metals. But after passing a maximum close to the Curie point, their resistivity becomes too high for the metals, and the corresponding compounds should unambiguously be classi"ed as degenerate semiconductors. It should be stressed that the behaviour of samples with x50.175 is qualitatively the same as that of the conventional ferromagnetic semiconductors (Section 4.1). In particular, the height of the  peak diminishes with increasing charge carrier density. But one should note that the resistivity peak in the vicinity of ¹ in the ! manganites is many orders of magnitude lower than in EuO and EuS (Figs. 4 and 9). Such behaviour of the resistivity of the manganites corresponds to the results of the magnetoimpurity theory at least, for x not very far from 0.175 (Section 4). On the other hand, supporters of the polaronic theory (Sections 8.3, 8.4 and 9.1) explain it by polaron formation at elevated temperatures. In the literature one often calls the increase in the resistivity close to ¹ as a metal}insulator ! transition. In my opinion, such a terminology leads to misunderstanding as, even for x"0.175, the resistivity increases only by two orders of magnitude, and for larger x the increase is still less. The real metal}insulator transitions are presented in Figs. 4 and 9 for EuO and EuS.

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Fig. 15. Temperature dependence of resistivity of La Ca MnO crystals. The arrows denote the Curie points. \V V  Anomalies denoted by light triangles are related to the structural transformations [37].

Thus, between x"0.15 and x"0.175, the type of electroconductivity changes drastically. In Ref. [37] the resistivity is correlated with the crystalline structure of samples. As seen from Fig. 15, there are no marked singular features of resistivity at the points where the structural phase transition takes place. Other examples of the transport properties of manganites will be given here. According to Ref. [172], below ¹ the zero-"eld resistivity  of La Ca MnO (x"0.33) obeys the relation! \V V  ship ln  proportional to M where M is the magnetization measured by muon spin relaxation. The resistivity peak was discovered in the La Pb MnO single crystals with 0.25(x(0.5 [190]. \V V  At the same temperature which is close to the Curie point of 290 K, the thermopower peak is located, and the magnetothermopower changes its sign. If one accepts that the low-temperature side of the resistivity peak corresponds to the metallic conductivity, and the high-temperature side to the semiconducting conductivity, then the activation energy should be about 0.05 eV. A similar temperature dependence of the resistivity was observed in La Pb MnO single crystals in \V V  Ref. [155]. The resistivity of La Ba MnO is similar to that of Sr-substituted samples for small \V V  x and slightly lower for larger x [195]. The resistivity of the manganites is very sensitive to the pressure. The corresponding investigations were carried out on the La A MnO system (A"Na, K, Rb, Sr) with the rhombohedral \V V  structure. Below ¹ a pressure up to 1.1 GPa reduces resistivity by more than 60%, but the e!ect is !

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less pronounced above ¹ [207]. A pressure of 5 kbar decreases twice the peak resistivity of ! La Ca MnO [196] (see also Ref. [197]). \V V  A very important problem is the mutual location of Curie point ¹ and the temperature of the ! resistivity maximum ¹ . The experimental data on this subject are contradictory. An essential + di!erence exists between results of papers [37,155,191] and results of other investigations on conventional degenerate ferromagnetic semiconductors described in Section 4.1: in the former the resistivity peak is found to coincide with the Curie point, whereas in the latter these temperatures are close to each other but do not coincide. In Ref. [172] the direct statement is made that the Curie point (274 K) and the resistivity peak temperature coincide to within 1 K. But there are experimental results on the manganites where these points do not coincide, e.g. papers [159,160,198] where the La Sr MnO single crystals were investigated, too, as well as in \V V  Refs. [85,89]. In Ref. [89] it is pointed out specially that ¹ in La Sr MnO is higher than the ! \V V  temperature of the resistivity peak ¹ . A similar result was obtained in Ref. [159]. But an opposite + result was obtained for La Sr MnO where ¹ and ¹ are found to be 302 and 318 K,      ! + respectively [189]. Commenting on these contradictory results, one should note that ¹ is unlikely to be below ¹ . + ! In fact, the ferromagnetic long-range order in initially antiferromagnetic manganites is established by the indirect exchange via delocalized charge carriers, and in the region of the peak they are localized. Hence, there is no origin for existence of this order above ¹ . The value of the critical + exponent close to this, 0.35}0.37 presented in Section 5.1, which is a normal value for the ferromagnets of the Heisenberg type, points to the fact that the long-range ferromagnetic order disappears before the electrons become localized. Hence, one can expect that the magnetic disordering should take place below the temperature ¹ of the metal}insulator transition. + 5.4. Negative isotropic colossal magnetoresistance of crystals This section will be devoted to the property most interesting for technical applications of the materials considered } to their colossal negative isotropic magnetoresistance (CMR) "rst discovered in Ref. [200]. Its systematical investigation began with the papers [182}184]. In discussing these materials one used to take the quantity
"[(H)!(0)]/(H) as a characteristic of the & relative magnetoresistance. But many authors use the conventional relative magnetoresistance
"[(H)!(0)]/(0) instead of
. Under typical conditions, the quantity
may exceed
by  & &  2}3 orders of magnitude (usually one takes H equal to 6 T). Typical results were obtained in investigating the La Sr MnO system in papers [37,201]. \V V  The main feature of the CMR consists in the fact that it is maximal at temperatures where the resistivity displays the peak at H"0, i.e., close to ¹ . Here it is directly related to suppression of ! this peak by the magnetic "eld. In reality, all the samples display the CMR maximum in the vicinity of ¹ (Fig. 16). At x"0.2, when the sample remains highly conducting also at ¹P0, the CMR is ! very small at low temperatures. But in the vicinity of the concentration metal}insulator transition, CMR is considerable already at 78 K. These results strongly resemble the corresponding results for conventional ferromagnetic semiconductors (Section 4.1) and may be described by the magnetoimpurity theory (Section 4). Qualitatively similar results were obtained also in investigating the same system in Ref. [202]. At x50.13 it displays the CMR cusp with a typical value of the maximal magnetoresistance

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Fig. 16. Temperature dependence of resistivity in La Sr MnO crystals in various magnetic "elds: (a) x"0.15; \V V  (b) x"0.175; (c) x"0.20. The light circles correspond to !
in a magnetic "eld of 15 T; the light triangles represent the  structural transitions [37].

R/R(10¹)"10. For a sample with x"0.1 the CMR retains a considerable value far below the Curie point which is close to 140 K. A considerable low-temperature CMR was observed in this material also at x"0.25 when the sample should possess the metallic conductivity at ¹P0. At low temperatures it increases with decreasing temperature, approaching
"!40 value which is only half the maximal value in the  vicinity of ¹ [203]. A similar behaviour was observed also in some "lms (see next section). ! Possibly, suppression of the domain structure by the magnetic "eld plays an important role here.

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Fig. 17. Temperature dependence of resistivity and magnetoresistance
for the La Pb MnO single crystal. In the       inset, the temperature dependence of the magnetization M in a 1 T "eld is presented for the same sample [155].

An analysis of the temperature and "eld dependence of the resistivity carried out by the authors of Refs. [37,201] has led them to the conclusion that the resistivity depends only on the magnetization M independent of the way in which a given value of magnetization is achieved } by change of temperature or of the magnetic "eld. As a result, the following expression is proposed which describes CMR at relatively small magnetization values:




M
"!C(x)  M 

where M is the saturation magnetization. The constant C is close to 4 in the vicinity of the  metal}insulator boundary, but decreases down to 1 at x"0.4. A very large magnetoresistance was found in the La Pb MnO single crystals: its maximal \V V  value of
reaches !20% at a "eld strength of 1 T [190]. It was also observed in Refs. [203}205].  In Ref. [155] the CMR was investigated in La Pb MnO single crystals, where the magneto    \W resistivity retains a large value also at room temperature (Fig. 17). As already mentioned above, the CMR was observed not only in lanthanum manganites, but also in other manganites. For example, it was found in the Nd Pb MnO single crystals where      at 11 T the quantity
reaches !99% [206]. The magnetoresistance peak in the vicinity of ¹ can  ! be observed at microwave frequencies, as it was shown in Ref. [221] carrying out measurements on a Nd Sr MnO at a frequency of 10 GHz.      Not only the resistivity, but also the magnetoresistivity of the manganites is very sensitive to pressure. The corresponding investigations on the La A MnO system (A"Na, K, Rb, Sr) \V V  show that the pressure of 1.1 GPa reduces the CMR down to !10% [207]. As was shown by

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investigations of the Nb Sr Pb MnO single crystals, the pressure reduces both the        resistivity and magnetoresistivity of this manganite [208]. Measurements on polycrystals lead, in general, to results which are close to those obtained on single crystals. For example, in Ref. [209] La Ca MnO polycrystals were investigated. \V V  The quantity
in them increases with decreasing temperature ¹ at which the CMR peak is  + located: it amounts to !90% at ¹ "100 K (this corresponds to x"0.5), and to !72% at + ¹ "240 K (x"0.1). + It remains to add that recently a positive CMR was discovered in La Sr MnO crystals with \V V  x about 0.1. It is about 400% at 120 K and H"5 T [210,211]. This e!ect can be compared with the corresponding e!ect in conventional ferromagnetic semiconductors (Section 3.4). 5.5. Negative isotropic colossal magnetoresistance of thin xlms Investigations of CMR in the manganite "lms are much more numerous. Their properties strongly depend on their composition and preparation procedure. In this respect the paper [212] is very instructive. In it the La Sr MnO "lms were prepared by quite identical methods, but they      were deposited on di!erent substrates. As a result, some of them were epitaxial and others polycrystalline, and their resistivities di!er by two orders of magnitude. Their temperatures of magnetic ordering and of resistivity peaks di!er very strongly, too. On the other hand, FMR investigations on the La Ba MnO "lms show that as-grown      "lms are magnetically nonuniform, and they become uniform only after annealing in the oxygen atmosphere [179]. How sensitive are properties of the "lms to their treatment, one can see from Ref. [182] where the epitaxial "lms of the same composition were investigated. In the as-grown samples the Curie temperature and the saturation magnetization are essentially less than those in the bulk samples. In Figs. 18 and 19 the resistivity and magnetoresistivity of these "lms are presented as functions of ¹ and H. The quantity
at 300 K amounts to !60%.  In Ref. [213] the La Sr MnO epitaxial "lms of thickness about 400 nm were investigated at \V V  0.164x40.33. In the as-grown samples, ¹ is considerably lower and the magnetic transition is ! much more spread than in the bulk samples. But the annealing in the N atmosphere leads to  growth of their Curie points, saturation moments, conductivity and CMR as compared with the as-grown "lms. For the annealed "lms at x"0.2 and a "eld strength of 5 T, the relative magnetoresistance
reaches its maximum value of !60% at ¹ "260 K. At x"0.33 the  + quantity
reaches its maximum value of !35% at ¹ "330 K. These temperatures are slightly  + lower than the temperatures of the resistivity peaks, but they are higher than ¹ determined from ! the maximal slope of dM/d¹. As seen from Fig. 20, in a "lm with x"0.16, in addition to the CMR peak in the vicinity of ¹ , ! a low-temperature increase in CMR is observed, on decrease in temperature. A similar CMR rise was observed in the La Sr MnO single crystals (cf. the preceding section). Such an e!ect was \V V  also observed in some other "lms. For example, in the epitaxial "lms La}Ca}Mn}O a large magnetoresistance exists even at 5 K [214]. In the absence of the magnetic "eld, the resistivity of La Sr MnO with x"0.16 in the \V V  temperature range above the resistivity peak is described by an exponential temperature dependence with an activation energy of about 70 meV for the annealed "lms and of about 100 meV for the as-grown "lms which are close to values obtained for ceramic samples.

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Fig. 18. Temperature-dependent resistivity of the La Ba MnO "lms in zero "eld and in a 5 T "eld (a) for the     X as-grown "lms deposited at 6003, and (b) after annealing at 9003 for 12 h. The dot-and-dash lines represent the magnetoresistance !
[182].  Fig. 19. The resistivity vs. "eld (in T) at 300 K for the same sample as in Fig. 18.

Fig. 20. The temperature dependence of the resistivity and magnetoresistance (!
) of annealed thin La Sr MnO  \V V  "lms in zero "eld and in a 5 T "eld (x"0.16) [213].

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The data on the La Pb MnO "lms obtained by radiofrequency sputtering and deposition      on Si substrates in the mixed Ar}O atmosphere, are as follows: their ¹ is equal to 325 K, the  ! magnetization at 4.2 K is by 14% lower than its saturation value. The zero-"eld resistivity peak is located at 250 K. Their CMR
is characterized by a maximal value of !22% at a "eld strength  of 2 T which is achieved at the peak. In the whole temperature range from 4.2 to 300 K the absolute value of
remains higher than !15%, and even at 380 K it reaches !10% [215]. The  La Pb MnO "lms with the composition di!ering very little from the former are characterized      by
"!40% at 300 K and a "eld strength of 6 T [216].  As already pointed out above, the absolute magnetoresistivity record in the manganites was achieved in the La}Ca}Mn}O system [5]: values of
amounting to !830 000% at 125 K and to & !10% at 57 K have been found. Results of other authors on the same system are as follows. In Ref. [217] the magnetoresistance is investigated of the epitaxial La Ca MnO "lms of a thickness of 100 nm grown by the laser      evaporation on the LaAlMnO substrate and then subjected to a thermal treatment. Such "lms  have a cubic structure with a lattice constant value of 0.389 nm. The temperature dependence of CMR for as-grown "lms is characterized by the
peak of a height of !460% at about & 100 K (H"6 T). Subsequent annealing at 7003C for 30 min in the oxygen atmosphere shifts the CMR peak to 200 K and increases its height up to !1400%. Further optimization processes lead to enhancement of
up to !127 000% at 77 K, which is equivalent to
"!93.3%. This &  is accompanied by a drop in the resistivity from 11.6  cm at zero "eld to 9.1 m  cm at H"6 T. The larger part of this drop occurs in weak "elds below 6 T which is important for practical applications. It should be pointed out that the properties of "lms can be considerably improved if superlattices are produced in them. For example, the CMR of the La Ca MnO "lms can be increased      three times, and their Curie point can be increased simultaneously [317]. If the CMR in crystals is isotropic, in "lms it can be anisotropic. The low-"eld magnetoresistance of molecular-beam-epitaxy-grown tetragonal La Ca MnO "lms as a function of temperature      and both magnitude and direction of the applied magnetic "eld was investigated [318]. Low-"eld anisotropic hysteresis was observed that depends on the direction of the applied "eld in the plane of the "lm. The hysteresis e!ect can result in a sharp drop in resistance during magnetization reversal which is more than 10 times steeper than the already colossal magnetoresistance. In Ref. [183] the CMR of the La Ca MnO "lms was found in still weaker "elds: at      H"1 T and 200 K the quantity
amounts to !53%. Here also the zero-"eld resistivity peak is  found which is located at a higher temperature than the CMR peak arranged at 190 K. The CMR peak usually lies on the left-hand side of the resistivity peak at the temperature, at which the resistivity value is half its peak value, and where the magnetization is still considerable. This result agrees qualitatively with that of Ref. [213]. The La ions in the manganite under discussion may be replaced by other trivalent rare-earth ions. An analysis of these materials leads to the conclusion [218] that, on decrease in the radius of an ion replacing La, the temperatures of both the magnetic disordering and the resistivity peak drop, and the height of the latter increases. In stating this, one should keep in mind that the La radius exceeds the Pr radius, and the latter exceeds the > radius. In principle, nondoped materials with the La de"ciency possess a high magnetoresistance, too, for example La MnO (04x40.33). All the "lms deposited on the SrTiO substrate \V W 

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are ferromagnetic [219]. After annealing in the oxygen atmosphere, the quantity
in a "eld & of 4 T amounts to !130% at room temperatures. In some "lms the CMR memory e!ect was observed. In Nd Sr MnO the conductivity increases linearly, on initially applying     W the "eld. At temperatures essentially lower than the resistivity peak temperature, the "lm goes over into the highly conductive state (!
'10). This state persists even after remov& ing the "eld. To return the "lm into the initial state, one should heat the sample up to the resistivity peak temperature. Unlike the conductivity, the magnetization displays only a weak hysteresis [220]. 5.6. Hall ewect and thermopower One of the most important problems in the physics of manganites of the type La Di MnO \V V >W is the determination of the charge carrier density in them. The divalent Di atoms or excess oxygen function as acceptors so that the holes play the role of the charge carriers here. Some estimates can be made even without detailed investigations. First of all, the manganites can be considered as degenerate ferromagnetic semiconductors, at least, for 0.2(x(0.3 or 0.1(y(0.3. In reality, this is possible if the number  of holes per atom over which they move is small compared to one. Experimental data on the location of the holes in manganites are contradictory as will be discussed in the next section. Most of the authors believe that they move over Mn ions but there are experimental evidences that, in actual reality, they move over the O ions (cf. Section 5.7). If this is the case then the conditions for the inequality ;1 are still more favourable as the number of O ions is three times larger than the number of Mn ions. It is clear that  should be less than x or y as impurity atoms can form clusters. The impurities entering the clusters are not electroactive. Only separate impurity atoms can function as acceptors but nobody can tell how many impurity atoms are separate. As is well known, in degenerate semiconductors the number of donors or acceptors can be several orders of magnitude less than the total number of impurity atoms. But, even if one assumes the worst case for degenerate semiconductors "x, nevertheless, results obtained for degenerate semiconductors remain in force, at least semiqualitatively, up to x"0.3. Strictly speaking, the fact that at x about 0.2}0.3 the manganites are degenerate semiconductors, is directly seen from Fig. 15 where a continuous change from a nondegenerate semiconductor to a metallic-like conductor takes place. But with increasing x, a nondegenerate semiconductor should "rst transform into a degenerate one, and only then it can convert into a metal, if the latter will occur at all. The possibility to consider the manganites with x up to 0.3 as a degenerate semiconductor is also con"rmed experimentally by the facts that the speci"c features of the Raman scattering in the manganites strongly resemble those of the ferromagnetic semiconductor EuB [234], and that they display a giant red shift of the optical absorption edge  (Section 3.1). The most popular way to "nd the charge carrier density is using the Hall e!ect. But this procedure is very complicated in ferromagnetic conductors due to the anomalous Hall e!ect. As is well known (cf. Ref. [11]) the nondiagonal component of the resistivity tensor in them is given by  "R (¹)B#R M VW & 1

(5.1)

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where R and R are, respectively, the normal and anomalous Hall coe$cients, B the induction & 1 and M the spontaneous or induced magnetization. An exact expression for the anomalous coe$cient is not known. Hence, experimentally, the problem reduces to separation of the normal and anomalous Hall e!ects which is quite nontrivial as very often the former is small comparing with latter [11]. For this reason number of experimental papers devoted to the Hall e!ect in the manganites is very small. In Ref. [224] where the Hall e!ect was measured in (La Gd ) Ca MnO in the \V V      paramagnetic region, its separation into the normal and anomalous parts was not carried out, and hence, physical conclusions made in this paper are unconvincing. An attempt to carry out such measurements was made in Ref. [25] where both the longitudinal and transversal resistivity of La Ca MnO "lms were measured in a perpendicular "eld up to 2 T. The former displays      a sharp peak in the temperature range from 230 to 270 K. This peak is shifted by the "eld, leading to the magnetoresistance, the peak of which is located at a temperature slightly lower than the resistivity peak. The transversal resistivity displays a part uneven in the "eld, and it is interpreted as the Hall resistivity. The sign of the Hall e!ect corresponds to holes. Their Hall density diminishes from 290 K down to the temperature of the resistivity peak where it amounts to 0.01 hole per unit cell. On further decrease in temperature, the Hall density increases again, reaching a value of 0.3 holes per unit cell at very low temperatures. The Hall mobility is also temperature dependent but weaker than the Hall density. In Ref. [225] where the Hall e!ect in La Sr MnO was investigated the separation into the \V V  normal and anomalous Hall e!ect was carried out. If one uses the standard expression for the one type of the charge carriers R "1/nec, then one obtains astonishing results. Namely, at 4.2 K the & charge carrier density  is independent of x and amounts to 1 hole per Mn site in the range of x values between 0.18 and 0.5. Similar results were obtained on epitaxial "lms La Ca MnO with x"1/3 and ¹ "265 K \V V  ! [249]. The Hall voltage below ¹ may be decomposed into a positive term R B and a negative ! & anomalous term that is strongly temperature dependent. Using the same expression for the normal Hall e!ect, than before, one again arrives at an astonishing result: the number of holes per Mn site is equal to 1.5. In both cases the number of holes exceeds considerably the number of the doped Sr or Ca atoms, which are their sources, and for this reason it is hardly possible to interpret these results in terms of a standard single-band model. In Ref. [249] an explanation for this phenomenon is proposed: the manganites at su$ciently large x are intrinsic semimetals with two types of charge carriers. Their densities can be considerably smaller than 1.5. This explanation seems to be quite reasonable keeping in mind the negative sign of the thermopower instead of the normal positive sign for hole-doped samples observed at large x (see below). Going over to the thermopower S now, one should "rst remember the fact which was already mentioned in Section 5.2: this quantity in La Pb MnO is peaked close to ¹ [190]. A similar \V V  ! result was obtained in Ref. [229] for La Ca MnO where the resistivity peak is about 270 K      and the thermopower peak at about 250 K. For more complicated compounds the thermopower peak was discovered in Ref. [228]. A strong temperature dependence of the thermopower was observed in La Sr MnO in the vicinity of the metal}insulator transition. The measured      relative change of S was found to be proportional to that of  [226]. According to Ref. [227], the

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Fig. 21. Temperature dependence of Seebeck coe$cient S and electrical resistivity  under magnetic "elds of 0 and 7 T for La Sr MnO : (a) low-doped (x"0.18), (b) moderately doped (x"0.35), and high-doped (x"0.40) crystals \V V  [230].

thermopower of La Ca MnO above ¹ is considerably less than the nominal ratio \V V  ! Mn>/Mn> so that the determination of the charge carrier density from its value is unreliable. Nevertheless, as usually the thermopower decreases with increasing charge carrier density, the thermopower peak evidences that the density should be minimal there. Very interesting results were obtained in Refs. [230}232]: the thermopower S in the manganites can be of the anomalous negative sign. In Ref. [230] La Sr MnO with 0.154x40.50 was \V V  investigated (Fig. 21). For the relatively low-doped (x"0.18) sample the S above ¹ is character! istic of semiconductors, namely, S increases with lowering of temperature. For the samples with x"0.24 and x"0.4 the S above ¹ is characteristic of metals, i.e., it decreases with lowering of ! temperature. For all the samples, the rapid decrease in S around ¹ , irrespective of its sign, ! corresponds to that in the resistivity  upon the phase transition to the ferromagnetic state. Application of a magnetic "eld (7 T) also decreases S considerably near ¹ , suggesting that the ! decrease in S should be compared with the magnetoresistance. As seen from Fig. 21, for x"0.18 the thermopower is nonmonotone but of constant positive sign. For x"0.25, the thermopower is positive below ¹ but negative above ¹ . Finally, for ! ! x"0.4 at all temperatures, S is negative. In Ref. [231] the thermopower of La Ca MnO      crystals and in Ref. [232] that of La Ca MnO "lms were found to be negative. In the latter      case a colossal magnetothermopower e!ect was discovered: [S(8¹)!S(0¹)]/S(8¹)"14% was found. It remains to add that the phase separation discussed in Section 6 can explain the anomalous sign of the thermopower observed in some cases (Sections 6.3 and 6.4). In this case for x(0.17 when the crystal is semiconducting, the low-temperature thermopower is negative. On increasing temperature, it changes sign to the normal one. If the phase separation really occurs at low temperature then the total thermopower can be determined by the thermopower of the semiconducting phase. If its conductivity  is intrinsic, its thermopower can be either positive or negative depending on the sign of charge carriers (e or h) which contribute to S

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predominantly, S  #S  F F . S" C C  # C F Hence, it can be negative if these are the conduction electrons. At elevated temperatures the phase-separated state is destroyed and the doped holes are distributed uniformly over the crystal. For this reason the high-temperature S is positive, and the change of its sign with temperature is reproducible. Certainly, these results on the Hall e!ect and thermopower make the properties of the manganites still more enigmatic. 5.7. Electron spectra and problem of the double exchange Now we shall discuss the data which can be used for elucidation of the nature of the hole band in manganites. As the replacement of the trivalent La by bivalent atoms leads to the appearance of holes in the crystal, it seems quite natural that these holes correspond to the appearance of Mn> ions instead of Mn> ions, as only the former exist in the CaMnO system which is the limiting  case of doped La-based manganites La Di MnO . In this case the hole moves over the Mn> \V V >W ions, converting them into the Mn> ions when the latter are occupied. As overlapping of the Mn d-orbits for neighbouring ions is small, the hole band width = should be relatively small, too (about 1}2 eV). On the other hand, the energy of the Hund exchange between the electrons in the d-shell AS/2 can reach 10 eV [233]. The situation AS<= is called double exchange (apparently as a consequence of the fact that the "rst theoretical treatment of the problem was carried out for the two-atomic system [84]). In this case the upper hole band is so far from the lower band that the former can be omitted from the consideration, and the holes can be considered as spinless fermions. There are hints that the double-exchange case can really exist in manganites. First, the dispersion relation for the magnons in the manganites (4.41) observed experimentally [165] corresponds to that obtained for the case of the double exchange [147,148] (cf. Section 5.1). Optical investigations [238] lead to the same conclusion: the e band is higher than the 2p oxygen band. E But investigation of the electronic structure of La Sr MnO at 0.24x40.6 by the photo\V V  emission and X-ray absorption spectroscopy shows that the holes are in the states of the 2p-oxygen type [239]. The holes are coupled with the high-spin con"guration of Mn> ions. Similarly, excitation of the pair electron}hole means the charge transfer from the oxygen p-level to the Mn d-level. Investigations of the oxygen K-edge electron-energy-loss spectra [236] lead to the same conclusion. The authors [236] have measured them for La Sr MnO (04x40.7) and \V V  La Sr MnO thin "lms as functions of x and z. The spectra show a prepeak at the Fermi level,    V X corresponding to transitions to empty states in the oxygen 2p band, at the threshold of the K edge around 529 eV. This prepeak systematically increases with an increase in conductivity through divalent doping (x) or oxygen content (z). This con"rms that these materials are charge-transfertype oxides with carriers having signi"cant oxygen 2p hole character. Study of the photoemission spectra of La Di MnO reveals that they are strongly temper\V V  ature dependent [237]. The spectral line shape dramatically changes in the entire valence-band

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region, particularly, for x"0.2 and 0.4. It is suggested that these changes are related to degree of hole localization on oxygen p orbitals, which is in#uenced by the electron-lattice coupling and magnetic correlations. The temperature-dependent redistribution of the spectral weight in the photoemission spectra of La Di MnO was observed in Ref. [240].      These results agree with results of band structure calculations for the La}Ca}Mn}O system carried out by the self-consistent local-spin-density method [48,235]. In these papers a strong hybridization of the Mn d-levels with the oxygen p-levels is found, too. But quite di!erent results were obtained in Ref. [241] where optical studies were carried out to establish the structure of the electron bands of CaMnO and LaMnO . In the latter the valence   oxygen 2p band is found to be closer to the Fermi level than the valence Mn t band, but the E e valence band is still more closer. The t }e exchange energy was found to be about 3.4 eV, and E E E the JT stabilization energy was estimated to be less than 0.5 eV. A value of 3.4 eV for the d}d-exchange interaction, apparently, points to a regime intermediate between the double and simple exchanges (=&AS). Thus, keeping in mind the controversy in results of di!erent authors, it would be premature to make de"nite conclusions about the reality of the double exchange in manganites. Possibly, an intermediate regime should be realized. Other important results were obtained using the photoemission studies. In Ref. [245] resonant photoemission spectroscopy was used to investigate changes in screening and electron density across the coupled metallic-magnetic phase transition of La Ca MnO with x"0.35 at 250 K. \V V  There is a pronounced shoulder of about 1.75 eV below the Fermi level at 173 K, which is absent at 293 K. This is an indicator of a change in the metallicity. A well-de"ned Fermi level is not observed. The intensity at the Fermi level decreases continuously with increasing temperature near ¹ . ! In Ref. [246] high-resolution photoemission Mn 2p resonant photoemission and O 1s X-ray absorption spectroscopies in La Ca MnO and La Pb MnO were investigated \V V  \V V  (0.24x40.5). Above ¹ a band gap exists which collapses below ¹ , and the density of states at ! ! the Fermi level increases with cooling. It is assumed that the small polaron e!ects are responsible for the insulating behaviour above ¹ , though the very fact of the energy gap is su$cient to explain ! this behaviour. Finally, the paper [473] should be pointed out, in which La Pb MnO was investigated \V V  using the scanning-tunnelling microscopy and spectroscopy. It is established that below the Curie point the density of states at the Fermi level is nonzero. But near ¹ a hard gap opens near the ! Fermi level. Its width is somewhat larger than the transport gap.

6. Underdoped and overdoped LaMnO3-based manganites 6.1. Underdoped manganites: instability of the canted antiferromagnetic ordering As was pointed out in Sections 2.2 and 5.2, the undoped La-based manganite is antiferromagnetic (strictly speaking, the antiferromagnetic ordering is canted due to the Dzyaloshinskii}Moriya "eld in the cubic structure disturbed by the JT e!ect but this relativistic e!ect is small). On increasing the doping, the magnetization of the manganite increases until it becomes ferromagnetic [2]. Compounds intermediate between antiferromagnetic and ferromagnetic and with nonsaturated magnetization will be called underdoped.

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Fig. 22. Neutron scattering intensity (in arbitrary units) vs. scattering angle 2 in La Ca MnO with x"0.18 at 4.2 K \V V  minus the intensity at 300 K. Shaded peaks are antiferromagnetic, unshaded } ferromagnetic. Dashed lines depict the same spectrum taken in the 4.5 kOe "eld [28].

As was established in Ref. [2], replacement of the trivalent La ions by the bivalent Ca, Ba or Sr ions leads to the appearance of the spontaneous magnetization in the LaMnO crystals. At small  doping degrees x it is nonsaturated, and becomes saturated only at x"0.2}0.3. In the vicinity of x"0.5 the magnetization disappears again (Fig. 12). The nature of the nonsaturated magnetized state in La Ca MnO was elucidated in Ref. [28] by neutronographic studies of this compound. \V V  In the x range from which the complete ferromagnetic ordering is absent, the neutron scattering spectra at 4.2 K are a superposition of spectra corresponding to both the ferromagnetic and antiferromagnetic orderings. In Fig. 22 the spectrum of a sample with x"0.18 is presented. The antiferromagnetic peaks are shaded, the remaining peaks are ferromagnetic. Principally, such spectra may be related both to the two-phase state of the sample, when it is a mixture of the ferromagnetic and antiferromagnetic regions, and to a single-phase two-sublattice state with a nonzero total moment. As examples of the latter, the canted antiferromagnetic state or collinear state of the ferrimagnetic type may be pointed out. A unique choice between the two-phase state and a single-phase state may be made by investigation of the dependence of the neutron scattering spectra on the external magnetic "eld. If the "eld is aligned with the unit neutron scattering vector q then the ferromagnetic peaks must become lower and disappear completely in the high-"eld limit. This follows from the fact that the intensity of the ferromagnetic scattering I is proportional to [1!(qm)] where m is the unit $+ vector aligned by the magnetic moment (see, e.g., [253]). The stronger the "eld, the closer is the direction of m to the "eld H and the less is the intensity which vanishes as HPR. On the other hand, the intensity of the antiferromagnetic scattering I is proportional to $ [1!(ql)] where l is the unit vector aligned with the di!erence between the moments of the sublattices. If the system is phase separated, then the vectors m and l are not connected to each other, and the magnetic "eld of a strength less than or comparable to the anisotropy "eld H will  rotate only the vector of magnetism m, not a!ecting the vector of antiferromagnetism l. In reality, if the "eld is aligned with the anisotropy axis, then the vector l changes its orientation abruptly only

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at a su$ciently large "eld strength } of the order of (H H ) where H is the exchange "eld  # # (cf., e.g., Ref. [11]). Usually, H is 2}4 orders of magnitude less than H , so that the spin-#ip "eld  # (H H ) exceeds H greatly. Thus, in weak "elds the antiferromagnetic scattering should not change.  #  But, if the system is single phase, the vector m will rotate together with the vector l, and for this reason, if the ordering is ferrimagnetic (m  l), the antiferromagnetic scattering should also decay, and, if the ordering is canted antiferromagnetic (mNl), it should be enhanced. Fig. 22 shows clearly that a "eld of 4.5 kOe reduces the ferromagnetic scattering by more than a half and has no e!ect whatsoever on the antiferromagnetic scattering. For this reason the authors [28] conclude that the crystal is in the two-phase ferro-antiferromagnetic state. Despite the conclusion [28], subsequent experimentalists preferred to interpret their results in terms of canted antiferromagnetic ordering. Possibly, it is associated with a misunderstanding: most of the authors are convinced that Wollan and Koehler [28] proved the existence of canted antiferromagnetic ordering in the manganites, whereas their conclusions were just opposite. The nonsaturated state was investigated in Refs. [255,256] by neutrons in the La Sr MnO \V V  systems, respectively. In the former for x"0.04 the ferromagnetic and antiferromagnetic peaks appear simultaneously at the same temperature 136 K, which is likely to evidence the canted antiferromagnetic structure with a total moment of 3.7 per Mn atom and spontaneous magnetization 0.28 . But this value seems to be unusually large for the canting of the relativistic origin. Normally, it leads to the magnetization of the order of 10\}10\ of the maximal magnetization [113]. One may assume that the x"0 sample is unintentionally doped, and the unionized acceptors contribute to the total moment. According to Ref. [297], the spontaneous magnetization of the undoped LaMnO does not exceed 0.1 . The picture is quite di!erent for x"0.125. The  ferromagnetic peaks appear at 230 K and the antiferromagnetic peaks only at 150 K. The ferromagnetic moment amounts to 3.7 and antiferromagnetic moment only to 0.23 . For x"0.17, the antiferromagnetic peaks are completely absent [255]. In Ref. [256] it is stated that the x"0.15 system is ferromagnetic below 235 K, but below 205 K the antiferromagnetic peaks appear. The most natural explanation of the di!erence between temperatures of ferromagnetic and antiferromagnetic peak appearance is as follows: the ferromagnetism is not related to the antiferromagnetism as the corresponding regions are spatially separated. In other words, these results may be considered as agreeing with the conclusions of Ref. [28]. But in Refs. [255,256] a hypothesis was advanced that, on decrease in temperature, "rst a ferromagnetic ordering is established, and then it is replaced by a canted antiferromagnetic one. Investigations in a magnetic "eld similar to those carried out in Ref. [28] could answer the question of whether the hypothesis [255,256] is adequate. In Ref. [257] the structural, magnetic, and transport behaviour of orthorhombic, insulating La Sr MnO samples (x"0.125) were studied as a function of temperature from 20 to 300 K. \V V >W The magnetization appears at 220 K, as follows from neutron di!raction. The sample displays spontaneous magnetization at low temperatures and paramagnetic behaviour above 360 K. At intermediate temperatures, the sample shows complex magnetic behaviour. The neutron di!raction pattern was interpreted in terms of the canted antiferromagnetic ordering. The canting angle should be saturated at 183 for ¹(150 K. This angle should rapidly increase between 150 and 220 K, where it should be close to 903. A theory of the charge-carrier-induced canted antiferromagnetic ordering in the manganites was "rst developed by De Gennes [254]. He considered this ordering as a result of competition between

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the Heisenbergian direct d}d-exchange and indirect non-Heisenbergian exchange via the delocalized charge carriers whose angular dependencies di!er strongly. Hence, this theory is applicable only to highly conductive samples (it should be noted that many experimentalists, e.g., [257], used this theory for insulating samples to which it is, certainly, inapplicable). In addition, one should keep in mind that the applicability of the double exchange model to the manganites is not proved yet: some believe that the holes move over the oxygen atoms (Section 5.7). De Gennes used the classical Hamiltonian of the double exchange (3.5) with "0, adding the d}d-exchange Heisenberg Hamiltonian (3.1) to it. He presented the ground-state energy for the canted layered A-type antiferromagnetic ordering in the form E"!NSI cos !2tn cos

 2

(6.1)

where N is the total number of magnetic atoms, n the number of charge carriers, "n/N, t the hopping integral for the nearest neighbours. The energy (6.1) after minimizing with respect to the angle 2 between moments of the sublattices gives the equilibrium canting: t  . cos " 2 2IS

(6.2)

This procedure means that the state with angle (6.2) has energy (6.1) minimal among all other canted antiferromagnetic structures. But this does not mean that this energy is absolutely minimal. A necessary condition for this is the positive de"niteness of the magnon frequencies, but they are not determined for the double-exchange case yet. Meanwhile, the magnon instability for the canted antiferromagnetic ordering in the simple exchange case, =
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magnetization #uctuations accompanying them. This points to the tendency for the system to go over into the nonuniform state, i.e., to the phase separation (see the next section). Certainly, the imaginary screening length is a su$cient but not a necessary condition for the instability of the uniform state. The corresponding calculation was done in Refs. [264,259] for the staggered antiferromagnetic ordering and in Ref. [262] for the A-type layered antiferromagnetic structure. It consists in a calculation of the e!ective dielectric function (q) which is related to the magnetoelectric constant
(Eq. (4.16)). Let us illustrate it by the case of the staggered ordering in the system of classical spins. In the case of the double exchange at complete charge carrier spin polarization it is given by t (M) dM dn , (6.3)
"z  dM dn d N where  "t(6
) (cf. Eq. (4.40)). Then, using Eq. (6.2), one obtains N  2IS 9 ;  " . (6.4)
" $ t 2(6
)  $ Obviously, the density  at which the crystal becomes ferromagnetic is small with allowance for $ the fact that t
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wave functions of the bound ferrons overlap slightly, the bound ferrons can have the same direction orthogonal to the moments of the sublattices [489]. In some sense, such a system imitates the canted antiferromagnetic ordering. The experimental data on strongly underdoped manganites agree well with the picture of the bound ferrons. Recently, it was con"rmed that in the R Sr "lms with the oxygen de"ciency     (R"Nd, Pr) below 30 K ferromagnetic regions exist with a diameter of 7}10 nm [267]. These spin clusters lead to the appearance of superparamagnetic properties in these "lms which manifest themselves in (1) an enhanced di!erence between magnetization measured at sample cooling with or without magnetic "eld; (2) a drastic decrease of the latter at low temperatures; (3) a strongly enhanced value of the paramagnetic Curie temperature as compared with the magnetic ordering temperature; (4) a sharp decrease of the coercive force with increasing temperature at its low values. Still earlier, it was established [36,37] that for small doping degrees x the paramagnetic temperature  of the La}Ca}Mn}O system sharply increases, on increase in x. But the NeH el temperature ¹ remains practically unchanged. The author of Refs. [39,40] points out with good , reason that this proves the existence of ferromagnetic regions inside the crystal which do not interact with each other. In reality, as was shown in Refs. [265,11], the bound ferrons enhance  strongly but do not a!ect ¹ virtually. The constancy of ¹ established in papers [36,37] shows , , that the magnetic ordering in the main part of the crystal remains collinear antiferromagnetic (neglecting the inherent weak ferromagnetism), though in Refs. [36,37] it is considered as canted antiferromagnetic with the moment increasing with the hole density. It should be pointed out that other data of magnetic investigations exist, which certainly cannot be explained within the framework of the canted antiferromagnetic ordering, but possibly may be explained by phase separation. For example, a study of NMR on Mn [36,37] yields that at x(0.175 the spectrum consists of two parts, and for samples with x"0.2 and 0.3 of only one part, which is identi"ed with a line narrowed due to the translational motion of charge carriers over the Mn ions and for this reason is related to the high conductivity of these samples. If so, then the question arises as to why two peaks arise in the region of the assumed canted antiferromagnetic ordering. This region should just appear as a result of the translational motion of the carriers over the crystal and for this reason its properties should be of the same type as those of the ferromagnetic region. Quite similarly, it is di$cult to explain results of the NMR studies on La nuclei [268,272], in which a large "eld was observed on them in La Na MnO and LaMnO even after the      >W disappearance of the crystal magnetization (these papers developed studies carried out in [269]). According to the papers cited, these "elds evidence the presence of ferromagnetic inclusions into the antiferromagnetic or paramagnetic host. Inelastic neutron scattering in semiconducting La Ca MnO with x"0.05, 0.08 reveal that \V V  there are two excitation branches: spin waves and hole hoppings. Ferromagnetically ordered regions with a length of 0.8}1 nm exist [270]. But, probably, these ferromagnetic microregions correspond not to moving, but to bound ferrons, whose number is considerably higher. In the subsequent paper of the same group [271] a liquid-like distribution was revealed by neutron scattering in the same material. The canting of the relativistic origin is characterized by an angle of 103. On its background, the ferromagnetic droplets are seen with a diameter of 1.7 nm, which corresponds to 4}5 lattice constants. But the magnetization of these droplets is relatively small: the di!erence between the cantings inside the droplets and those outside them is only 103. The number

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of droplets is 1/60 of the assumed number of holes which makes one believe that each droplet contains several holes. As already discussed in this section, the relativistic canting angle seems to be overestimated. Mn experiments were carried out at 1.3}4.2 K on six samples of the lanthanum manganites, with the hole doping concentration ranging from the antiferromagnetic insulator to the ferromagnetic conductor region of the phase diagram [273]. The dependence of the resonance frequencies on external "elds is as expected either for ferromagnets or for antiferromagnets. No indication of a canted phase could be detected. The ferromagnetic and antiferromagnetic resonances were found to coexist in all samples and are grouped into three broad bands, whose frequencies do not depend in the "rst order on hole concentration. Strong evidence of interaction between the antiferromagnetic and ferromagnetic regions was provided by longitudinal relaxation experiments in several samples. An explanation of this behaviour is o!ered in terms of intrinsic phase separation of the holes into ferromagnetic microdomains. Further evidences of magnetic inhomogeneities of the samples due to the presence of magnetic small polarons are given in Ref. [274]. A La NMR study revealed the coexistence of the ferromagnetic and antiferromagnetic phases only in a lightly doped sample with x"0.05 [154]. 6.3. Electronic ferro-antiferromagnetic phase separation and record CMR Certainly, the bound ferrons discussed in the previous section cannot lead to such well-de"ned ferromagnetic and antiferromagnetic peaks in the elastic neutron scattering as were observed in papers [28,255}257]. In the best case, they will lead to small-angle neutron scattering. One may expect that, close to the density of the metal}insulator transition, a real phase separation will take place with large-size regions of both phases and with all the charge carriers concentrated inside the ferromagnetic phase and antiferromagnetic phase insulating. There is direct evidence that such states really exist in manganites. For example, FMR data were obtained that con"rm the fact that the epitaxial manganite "lms MnO (A"Ln, Nd; B"Ca, Sr, Ba) are in a nonuniform state: they consist of highA B V \V \W conducting regions with the resistivity less than 10\ and low-conducting regions with the resistivity 100 times or more higher [279]. In Ref. [275], in investigating the critical behaviour of La Sr MnO close to the metal}insulator transition (x"0.16) the presence of the spin\V V  polarized anomalous metallic phase is established. On passing the transition density, the holes display very small mass renormalization. Recently, experimental evidence was obtained [296] showing that, above the Curie point, long-lived antiferromagnetic clusters coexist with ferromagnetic critical #uctuations in ferromagnetic La Sr Mn O . This makes it necessary to investi      gate the phase separation in manganites. Before analysing the nature of the phase separation in lanthanum manganites, it is advisable to analyse the phase separation as a general physical phenomenon in degenerate antiferromagnetic semiconductors. First of all, it should be pointed out that there exists a trivial phase separation caused by nonuniformity of the chemical composition of a sample, for example, due to the nonuniformity of the impurity distribution over a sample, which arises at the moment of its synthesis at elevated temperatures and remains frozen after its cooling. Such a phase separation is not sensitive to external factors (temperature or magnetic "eld) and for this reason it is an individual property of each speci"c sample.

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Fig. 23. The phase-separated state of a degenerate antiferromagnetic semiconductor: (a) insulating state; (b) conducting state (hatched is the ferromagnetic part and nonhatched is the antiferromagnetic part of the crystal) [281].

But a nontrivial thermodynamic equilibrium phase separation can be changed under external factors, and for this reason it is controllable. Two main mechanisms of such a reversible phase separation are known in degenerate magnetic semiconductors: the electronic phase separation, occurring at frozen impurity positions (it was "rst proposed and investigated in Refs. [280}285,11]), and magnetoimpurity phase separation, occurring through the impurity atom di!usion [284,286}289]. Both these mechanisms are related here to the fact that the ferromagnetic ordering is more energetically favoured for charge carriers than the antiferromagnetic one [11]. For this reason they tend to establish the ferromagnetic ordering. But, in order to do this in the entire crystal, the carrier density must be su$ciently high. If it is insu$cient for this aim, all the carriers may concentrate in a portion of the crystal and establish the ferromagnetic ordering there. The remaining part of the crystal is antiferromagnetic and insulating. In the case of the electronic phase separation [280}285,11] the calculations were carried out for the simple exchange, excluding the paper [285] where the double exchange was considered, too. The di!erence between the simple exchange and double exchange is only qualitative. In papers [284,285] the total phase diagram of phase-separated crystal at "nite temperatures and magnetic "elds was constructed. In these calculations, Coulomb "elds are taken into account. Concentration of the charge carriers in a portion of the crystal, where they cause the appearance of the ferromagnetic ordering, leads to the mutual charging of both the phases. This is a consequence of the fact that the ionized donor or acceptor impurity with the charge, opposite to that of charge carriers, is distributed, unlike them, uniformly over the crystal. Thus, strong Coulomb "elds arise which tend to intermix regions of the ferromagnetic and antiferromagnetic phases. As a result, the minor-volume phase becomes separated into nanometre droplets. At relatively small carrier densities, the highly conductive ferromagnetic regions form separated, not making contacts with each other, droplets inside the insulating antiferromagnetic host (Fig. 23a). As they are separated from each other by the insulating layers, the crystal as a whole is an insulator at ¹"0 (if one neglects the tunnel currents between droplets). On increase in the carrier density, the volume of the ferromagnetic phase also increases, and, beginning from a certain critical density value n , the ferromagnetic droplets begin to make contact N with each other, i.e., percolation of the ferromagnetic ordering, as well as of the charge carrier liquid, occurs. This means that the concentration insulator-to-metal transition takes place. On further increase in density, the geometry of the two-phase state changes drastically: the ferromagnetic region transforms from multiply connected to simply connected. In other words, the

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Fig. 24. Resistivity  in di!erent "elds H vs. temperature for EuSe specimens with two charge carrier densities n at GLR 297 K [4].

ferromagnetic portion of the crystal consists of separated droplets inside the ferromagnetic host (Fig. 23b). Finally, at still higher carrier density, the entire crystal becomes ferromagnetic. The typical size of the ferromagnetic droplets in the antiferromagnetic host amounts to several nm. Obviously, such droplets cannot lead to the neutron peaks, and can manifest themselves only in the small-angle neutron scattering. But close to n su$ciently large ferromagnetic and antiferN romagnetic regions can arise, and then the appearance of the corresponding peaks becomes possible. It should be noted that the compound investigated in Ref. [28] just corresponds to the percolation density as it is very close to the insulator}metal boundary. Such an electronic phase separation occurs in heavily doped antiferromagnetic semiconductors EuSe and EuTe (see experimental veri"cation of this statement in Refs. [11,283,284]). In particular, it leads not to a simply colossal magnetoresistance but to a supercolossal magnetoresistance. This e!ect is especially pronounced in EuSe which is an isotropic metamagnetic system with a very small "eld for the transition to the ferromagnetic state [4,11]. At the mean conduction electron density somewhat below 10 cm\ the main portion of the crystal is antiferromagnetic at temperatures below 5 K, and all the conduction electrons are concentrated into separated ferromagnetic droplets (the con"guration of Fig. 23a). The resistivity of such a crystal exceeds 10  cm. But already a "eld of 10 kOe reduces the resistivity to 10\  cm, i.e., by 9 orders of magnitude (Fig. 24a). This happens due to the transition of the entire crystal in the ferromagnetic state. As a result, the conduction electrons that were locked inside the droplets spread over the entire crystal. Evidently, in this case the absolute record of the isotropic negative magnetoresistance is achieved

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(
&10%). In the case of the con"guration of Fig. 23b, being realized at larger electron & densities, both the resistivity and magnetoresistivity are typical of a degenerate ferromagnetic semiconductor: the resistivity displays a peak in the vicinity of ¹ , and a "eld of 10 kOe reduces the ! peak by a factor of "ve (Fig. 24b). Unfortunately, neutron studies of EuSe were not carried out. The colossal magnetoresistivity of the phase-separated EuTe with the geometry of Fig. 23a manifests itself already in weak "elds: at 4.2 K a "eld of less than 200 Oe twice reduces the resistivity of a sample [290]. The e!ect can be explained as follows: the magnetic "eld increases the size of ferromagnetic droplets and, hence, it facilitates the electron tunnelling between droplets. In addition, the "eld aligns moments of droplets which also facilitates tunnelling. Finally, the "eld tends to destroy the ferromagnetic droplets [11]. For this reason it increases the energy of electrons trapped in the droplets and diminishes the energy of their delocalization (transition to the conduction band of the antiferromagnetic portion of the crystal). The tunnel transitions and the magnetoresistance related to them do not vanish even at ¹"0. It should be stressed that such a phase-separated state is the crystal ground state and it is destroyed by the temperature rise. Then the crystals which were in the insulating state of the Fig. 23a type at ¹"0 go over into a highly conductive state as the insulating regions cease to exist. In Ref. [285] a reentrant phase separation was predicted. If at ¹"0 the hole density exceeds the value, beginning from which the entire crystal becomes uniformly ferromagnetic, then, with increasing temperature, the crystal can "rst go over into the phase-separated state and then becomes again uniformly ferromagnetic. This result can explain the appearance of the antiferromagnetic correlations at "nite temperatures observed in the ferromagnetic La Sr Mn O       [296]. On the other hand, in a high-temperature paramagnetic state of the antiferromagnetic insulator Pr Ca MnO , ferromagnetic spin #uctuations with small energy were detected. They \V V  decrease in intensity at ¹ , but persist below this temperature and vanish only at ¹ [331]. These !, data, too, can point to a high-temperature phase separation. In Ref. [475] a state intermediate between the lattice polaronic and magnetically phaseseparated ones is proposed for interpretation of the experimental data obtained there. Lowtemperature La and Mn NMR line shape measurements in La Ca MnO for \V V  0.1254x40.5, and La (Ca Ba ) MnO for 04y41 are shown to directly re#ect   \W W    changes of the structure and of the magnetic state at a local magnetic level. For low x values in La Ca MnO evidence is given about the presence of the polaronic lattice distortions. By \V V  increasing x, structurally undistorted conductive ferromagnetic regions are formed indicating a percolative metal-to-insulator transition, whereas for x"0.5 the system is shown to break into the ferromagnetic and antiferromagnetic regions in accordance with the predictions of the electronic phase separation model. I would like only to note that as the bound hole radius at small x amounts to or exceeds the lattice constant, the indirect exchange should exist via the hole. It should increase ferromagnetic correlations in the vicinity of the acceptor, so that a lattice-magnetic polaron should appear (such a quasiparticle was "rst considered in Ref. [469]). In papers [483}485] anomalies of the volume magnetostriction and thermal expansion in La Sr MnO and Eu Ca MnO were discovered and related to the magnetic two-phase \V V  \V V  state. Theories of the phase separation were developed by other theoreticians recently [292}295] in which the case of the double exchange was considered. Though models used in them are rather

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simpli"ed as compared with those discussed above (e.g., the Coulomb interaction is not taken into account, and only the case ¹"0 was considered) they all con"rm the tendency to the electron phase separation. It should be noted that the electronic phase separation can be not only of magnetic origin but also of lattice origin, if the charge carrier energies in the crystallographic phases di!er considerably, and crystallographic energies di!er weakly [476,477]. 6.4. The impurity and mixed impurity-electron phase separations. Spin-glass } ferromagnetic state An alternative to the electronic phase separation is the magnetoimpurity phase separation which can be realized when the acceptors are mobile at actual temperatures, e.g., the excess oxygen atoms. This type of phase separation is caused jointly by two factors: (1) Interaction between impurity donor or acceptor atoms, as a result of which an impurity metal arises which consists of them. In other words, delocalization of the charge carriers occurs according to the well-known Mott mechanism. In full analogy with the conventional metal, the tendency appears to establish such interatomic spacing which provides the minimum to the total energy of the system. If the mean impurity density over the crystal is less than this optimal density, then it is energetically favoured for the impurity atoms to concentrate in a certain portion of the crystal. Then the remaining part of the crystal will not contain them at all. (2) The same tendency of charge carriers to establish the ferromagnetic ordering, as at the electronic phase separation, makes the electrons or holes to assemble in a portion of the crystal where they jointly establish this ordering. But, together with them, in this portion the impurity atoms also assemble which engender them. As essential di!erence between the impurity phase separation and the electronic phase separation is the absence of the mutual charging of the phases here, as the electron or hole charges are compensated by the charges of impurity ions di!using together with carriers. As a result, the ferromagnetic region size can be su$ciently large even at small carrier densities. But here this region size is limited by the elastic forces which do not allow, for example, that a crystal of macroscopic size be separated only in two regions of di!erent phases. For this reason here, on increase in the carrier density, the topology of the highly conductive ferromagnetic portion should change at a certain percolation density n , which occurs jointly with the concentration phase N transition from the insulating state to the highly conductive one. In principle, at the impurity phase separation the magnetic "eld should also increase the size of the ferromagnetic regions, facilitating the electron tunnelling between them. Hence, here the very fact of the phase separation is one of the possible mechanisms of the negative magnetoresistance, too. On increase in temperature, the phase-separated state should be broken, and the impurity distribution over the crystal should become uniform. Thus, the mean impurity density will be less than the local density in impurity-rich regions. Meanwhile, the delocalization of electrons belonging to the donor impurity or holes belonging to the acceptor impurity can take place only at su$ciently large impurity density. Hence, the situation is possible in principle, when, at destruction of the insulating impurity-phase-separated state of the Fig. 23a type, the crystal does not go over into the highly conductive state. In fact, if the impurity density in the impurity-rich regions was su$ciently large for the electron (hole) delocalization, after destruction of the phase-separated state

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it may turn out to be insu$cient for this aim. Perhaps, it may also be su$cient for the metallization, and then destruction of the phase-separated state will lead to the crystal transition to the highly conductive state. In the case of the Fig. 23b type geometry, destruction of the phase-separated state does not change the metallic type of the conductivity or make the crystal insulating. The latter is possible if the mean impurity density is lower than its Mott value. It remains to add that there are magnetites which contain the mobile and immobile acceptors simultaneously, e.g., excess O and Ca. In this case a mixed electron-impurity phase separation is possible, when inside the same ferromagnetic region simultaneously oxygenand Ca-originating holes concentrate jointly with the oxygen ions, the density of the Ca ions being "xed [289]. In this case ferromagnetic droplets turn out to be charged although to a smaller degree than at the pure electronic phase separation. The volume of the ferromagnetic droplet inside the antiferromagnetic host is (1#n /n ) times larger than in the pure -  electronic phase separation where n and n are the numbers of holes delivered by O and  Ca ions, respectively. Hence, the observation of the ferromagnetic and antiferromagnetic peaks will be easier in this case. Certainly, in order to realize the impurity-phase separation, the impurity atom di!usion coe$cient must be su$ciently large at actual temperatures. There is direct experimental evidence that the oxygen di!usion coe$cient is high in many perovskites: at room temperatures it may reach values of the order of 10\ cm/s which ensure its rapid di!usion at these temperatures. Moreover, in the La CuO perovskite the crystal separation into the oxygen-rich and oxygen-poor regions  >W occurs even at 265 K. There exist many other HTSC in which the impurity-phase separation was discovered (see Refs. [283,284]). Possibly, the impurity phase separation is realized in LaMnO with y ranging from 0 to 0.15 >W [297]. All these samples are insulating, and changes in their crystallographic state do not in#uence their electric and magnetic properties. The y"0 sample displays small spontaneous magnetization (&0.1 ) which, apparently, is of the relativistic origin and corresponds to the sublattice moment canting (Section 2.2). In the y"0.025 sample the magnetization is larger: the ferromagnetic component is about 1.48 and the antiferromagnetic component about 2.52 . The fact that both components appear at the same temperature makes one believe that the canted antiferromagnetic ordering exists. For y"0.07, only a small antiferromagnetic component of 0.25 was found. A large ferromagnetic component was observed below ¹ "160 K which reaches a value of 3.25 ! at the lowest temperature. But it is only about 2.86 for y"0.1 and 0.97 for y"0.15. The small-angle neutron scattering is maximal just for y"0.07. The results for y"0.07 compound evidence that it is in a mixed ferro-antiferromagnetic state. The absence of an antiferromagnetic component at y"0.15 and other features suggest the existence of a disordered cluster glass state with magnetic clusters entering the compound. Such a state was investigated both experimentally and theoretically in Ref. [299] using a nonmanganite system. According to Ref. [298] in the series (La Tb ) Ca MnO , the magnetic ground \V V    state at low temperatures evolves from ferromagnetic for x below 0.25, to spin glass between 0.33 and 0.75 and "nally antiferromagnetic for x"1. Muon spin relaxation experiments for x"0.33 sample con"rm the absence of the microscopic local magnetic order and give evidence for the existence of static local "elds randomly oriented below 44 K which should correspond to magnetic clusters.

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Fig. 25. (a) [0 0 1] zone-axis electron di!raction pattern obtained at 95 K for La Ca MnO . The fundamental Bragg      peaks labelled a, b, and c can be indexed as (2 0 0), (0 2 0), and (1 1 0), respectively. The presence of superlattice spots with the modulation wave vector (1/2, 0, 0) or (0, 1/2, 0) is evident. (b) Schematic charge ordering picture of Mn> and Mn>. Open and closed circles represent Mn> and Mn> ions, respectively. The orientational order of d  orbitals Mn> ions X which results in the cell doubling along the a axis is also indicated [302].

6.5. Overdoped samples, charge ordering, and stripes We shall call those samples overdoped which contain too many divalent impurities to remain ferromagnetic. First of all, the most popular is La Ca MnO system with x close to or exceeding \V V  0.5. In this material at x"0.50 the ground state changes from ferromagnetic and highly conductive to antiferromagnetic and insulating, leading to a "rst-order antiferromagnetic transition with large supercooling and strongly hysteretic behaviour in resistivity. On decreasing the temperature, the x"0.5 compound "rst undergoes a ferromagnetic transition and then follows a "rst-order transition to an antiferromagnetic charge ordered state at about 135 K (about 180 K on warming) [300}302]. According to Ref. [302], the antiferromagnetic-to-ferromagnetic transition in La Ca MnO      is structurally associated with a commensurate to incommensurate charge ordering phase transition. Investigations carried out by electron di!raction lead to conclusions about the existence of charge ordering in these materials when a charged sublattice arises inside the crystalline lattice. The di!raction evidence of the charge ordering is shown in Fig. 25a which represents a [0 0 1] zone-axis electron di!raction pattern obtained at 95 K. Sharp superlattice spots are evident in

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addition to the fundamental Bragg re#ections. Furthermore, the wave vector of the superlattice modulation was found to be nearly commensurate at this temperature and can be written as q"(2
/a) (1/2!, 0, 0) with incommensurability parameter "0.013, varying slightly from grain to grain. In La Ca MnO , there are just as many Mn>(3d) ions as Mn>(3d) ions. It is      assumed that the charge ordering corresponds to the ordering of these ions of the Verwey type (an alternative explanation of the superlattice will be presented below but the commonly used term `charge orderinga will be used instead of the more rigorous term `superlatticea here). In Fig. 25b a schematic picture is shown of ionic ordering of Mn> and Mn> consistent with the ordering vector q"(2
/a) (1/2, 0, 0). Previously, it was proposed that the cell doubling along the a-axis could be achieved by the orientational ordering of the d  orbitals (Section 2.2) in Mn>, X in addition to the Mn>}Mn> ordering [303,304]. A modulated structure with a similar modulation wave vector to that in Ref. [302] was found below ¹ [308]. The paper [300] should , also be mentioned in which for 0.634x40.67 charge ordering was observed. In Ref. [300] the electron di!raction was used for investigation of a series of compounds La Ca MnO (Fig. 26). For 0.634x40.67 charge ordering was observed at 260 K and was \V V  accompanied by a dramatic (exceeding 10%) increase in the sound velocity. At 300 K the di!raction pattern is consistent with the known Pbnm orthorhombic structure. Below the charge ordering temperature ¹ , additional superlattice spots develop with a modulation wave vector !q"(2
/a) (y, 0, 0) with y about 0.3. Hence, like in the previous case, the modulation occurs only along one of the axes. Charge ordering was established also for x"0.65 in the paramagnetic phase. It does not occur with any lattice e!ects [312]. The existence of charge ordering in La Ca MnO with x"0.65 was con"rmed in Ref. [399]. \V V  It is assumed there that from 700 K down to room temperature the electrical conduction takes place through thermally activated hopping of polarons with an activation energy of 0.045 eV, which seems rather smaller than that assumed for small polarons [11]. At the charge ordering transition temperature ¹ "275 K, pronounced anomalies in the resistivity and the lattice state !were observed. The neutron thermodi!ractogram clearly establishes that the charge ordering state occurs in the paramagnetic phase and is accompanied by a large anisotropic lattice distortion of the MnO octahedra. The antiferromagnetic phase appears at 160 K, and is not accompanied by  a lattice e!ect. The charge ordered-paramagnetic and charge ordered-antiferromagnetic ground states are stable under an applied magnetic "eld up to 12 T. A more detailed study of the charge ordering in La Ca MnO was carried out in Ref. [313], \V V  mainly with x"2/3, where it was shown that it is associated with stable pairs of contracted Mn>O stripes separated by stripes of non-distorted Mn>O octahedra. The electron di!rac  tion and electron microscopy yield images which are dominated by lattice distortions. The charge ordering is commensurate with a wave vector of (2
/a) (1/3, 0, 0) and a"0.55 nm. The most intriguing feature of the images (Fig. 27) is the presence of doublet lattice fringes with 0.55 nm spacing in each of the dark stripes. The regular array of the paired dark fringes exhibits a 3a ("1.65 nm) periodicity. The enhanced contrast of the paired dark fringes is mainly due to the strain induced by the JT distortion of the Mn>O octahedra. Therefore, the charge ordered  structure is constituted of two heavily JT-distorted diagonal Mn>O stripes which always appear  in pairs. The nondistorted Mn>O exhibit a much lighter contrast.  The Mn>O stripes are separated by 0.55 nm, and the new periodicity of this charge-ordered  state is now 3a. The coherence length of the 3a structure for x"2/3 exceeds 50 nm with the

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Fig. 26. (a) C/¹ of La Ca MnO for x"0.33 and 0.63 with lattice contribution (inset) substracted from each. \V V  (b) Sound velocity v for x"0.33 and 0.63. The dashed lines are v in a "eld of 4 T. (c) Insets. Electron di!raction patterns for x"0.67 material. Charge ordering is seen as superlattice spots at 200 K (left) in addition to the main spots seen at 300 K (right). (b) Intensity of a representative Bragg spot at
"0.33 vs. temperature where the modulation wave vector is (2
/a)(
, 0, 0) [300].

longitudinal coherence length being much greater (of the same order as the sample thickness). The pairing of the JT stripes is also con"rmed by investigation of a sample with x"3/4. The samples with x"4/5 reveal the same con"guration as x"3/4 samples, but with reduced overall contrast and coherence length. For samples with arbitrary charge carrier densities, charge ordering appears as proper mixtures of the two adjacent distinct commensurate con"gurations according to the lever rule. Although the phase correlation is strong along the longitudinal direction of the paired stripes, amplitude #uctuations along the stripes are commonly observed with the length scale of 10}30 nm. Small regions without charge ordering were found, and it is tempting to associate these regions with ferromagnetic metallic clusters. In Ref. [314] the La Ca MnO compound with x"1/2 \V V  was investigated. An inhomogeneous spatial mixture of uncommensurate charge ordered and

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Fig. 27. Pairing of charge-ordered stripes in La Ca MnO [33]. I. (a) High-resolution lattice image for x"0.67 at \V V  95 K showing 3a periodicity with the stripe pairing. (b) Schematic model in the a}b plane. (c) Inverted intensity scan. II.  The same for x"0.75 and the periodicity 4a . 

ferromagnetic microdomains of about 20}30 nm was found. The former are mixtures of paired and unpaired JT distorted Mn>O stripes. In Ref. [315] the stripes in the manganites are compared to  stripes in the HTSC. According to Ref. [316], for the La Ca MnO compounds closely spaced in doping near \V V  x"0.50, the resistivity data can be quantitatively described by a model based on the coexistence of two carrier types: nearly localized carriers in the charge-ordered state exhibiting variable-range hopping and a parasitic population of free carriers which is tunable by either magnetic "eld or stoichiometry. It is important to note that according to Ref. [302], the x"0.33 sample does not manifest the charge ordering. In many cases experimentalists declare that they observed the charge ordering at x less than 0.5. But, in fact, they observed only metal}insulator transition which can be due to quite di!erent reasons, e.g., by a transition into the phase-separated state. Only electron or neutron di!raction studies can con"rm that charge ordering really occurs. The complicated structure of the La Ca MnO spectrum is seen from its behaviour in     >W a magnetic "eld [305]. This is a metamagnet which undergoes a transition from the antiferromagnetic state to the ferromagnetic state and vice versa under a magnetic "eld or by change of

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temperature. It is assumed in Ref. [305] that the low-temperature ground state is canted antiferromagnetic, though this statement seems to be doubtful (cf. Section 6.1). Nevertheless, a spontaneous magnetization is present, though it amounts to only 5.7% of the saturation magnetization. At a su$ciently high H denoted as H\$, a "eld-induced "rst-order A phase transition occurs, bringing the system from the antiferromagnetic to the ferromagnetic state. At 4.2 K, after switching o! the "eld, the system remains in the insulating state, but at about 50 K, the system collapses back to the antiferromagnetic state. Now the CMR will be discussed. With increasing H, there appear four distinctive electronic states at low temperatures. The zero-"eld-cooled state is an insulator with a large  of 3.12;10  cm. At H "4.52 T, H "5.98 T, and H "11.80 T, three consecutive precipitous    drops in  cascade the system to three high-H states. The state H reached at the highest "eld  which coincides with H\$ of the antiferromagnetic}ferromagnetic transition, corresponds to an A insulator-to-metal transition. At 4.2 K, once in the trapped ferromagnetic state,  remains at a low value regardless of any subsequent applied "eld. A maximum 10-fold magnetoresistance ratio has been observed at 4.2 K between the least and the most conductive phases (see also Ref. [449]). In Ref. [306] La NMR in a zero external magnetic "eld has been used as a probe in order to study the Mn spin ordering of La Ca MnO . For ¹(¹ two di!erent spin con"gurations are      ! detected, assigned to a ferromagnetic and an antiferromagnetic one, which persists down to the lowest measured temperature ¹"90 K. Close to ¹ , strong coherent spin #uctuations are ! detected that presumably drive the paramagnetic-to-ferromagnetic phase transition. The magnetization as a function of temperature displays a maximum and then steeply drops, disappearing at ¹ . ! In this respect the behaviour of the system closely resembles data obtained by other experimental groups (Fig. 13). General conclusions are as follows: the low-temperature antiferromagnetic phase consists of stacked antiferromagnetic and partially ferromagnetically ordered octants. In Ref. [446] the properties of charge ordered La Ca MnO were investigated. The charge    ordering transition around 270 K is con"rmed by discovered anomalies of the resistivity, magnetic susceptibility, speci"c heat, and lattice parameters. A structural model of the charge ordered state is developed with the crystallographic unit cell given by (3a;b;c) of the high-temperature unit cell. At 170 K an antiferromagnetic phase transition is clearly seen in the magnetic susceptibility, resistivity and speci"c heat. Neutron di!raction shows that the Mn moments order antiferromagnetically with a propagation vector k"(0, 1.2, 0). The observed magnetic con"guration belongs to one of the two bidimensional representations of the group Pbnm associated with this k. NMR studies of La Ca MnO [311] showed the existence of antiferromagnetic and ferro     magnetic domains, with the former assigned to an insulating charge ordered state and the latter to a metallic state, respectively. The antiferromagnetic}ferromagnetic transition is of the "rst order. Properties of La Sr MnO di!er from those of La Ca MnO . The La Sr MnO \V V  \V V       compound below x"0.54 is a ferromagnetic metal, but at larger x it becomes a layered antiferromagnetic system. Nevertheless, in the vicinity of x"0.54 it displays a very low resistivity of 4;10\  cm at 5 K. It goes over into an antiferromagnetic insulator only at about x"0.58 [309,307]. In Ref. [310] partial replacement of Ca ions by Sr ions of larger radius was done, so that the manganites La Ca Sr MnO were investigated. The system undergoes structural    \V V  transitions with increasing x. Both ferromagnetic and antiferromagnetic phases were observed. ¹ increases monotonically with x. ¹ increases with x at small values, followed by a decrease and ! , disappearance at x"0.4.

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Now some alternative ideas concerning the nature of the superlattices for x'1/2 will be put forth. The situation with the classical charge ordered system Fe O is relatively simple: there are   only two charged species Fe> and Fe> which really can order in the periodic electrostatic potential of the oxygen ions. But in the manganites the situation is much more complicated. In addition to the assumedly ordering Mn> and Mn> ions, there is a compensating charge of the ionized donors La>. If the La ions are distributed randomly over the manganite crystal (a disordered La}Ca alloy), and the main interaction between the ions is electrostatic, La and Ca ions create #uctuating Coulomb "elds in addition to the periodic potential of other ions. These "elds tend to destroy charge ordering of the Mn ions. As an alternative to charge ordering, the presence of the superlattice spots in the manganites could be explained by the fact that the La}Ca alloy is ordered with the La ions forming the superlattice. Hence, not the ordering of Mn> and Mn>, as it is customary to think, but the ordering of the Ca and La ions should occur. An electron should be located in the vicinity of each La ion. For ordered Ca and La ions these #uctuating "elds are absent, and they do not prevent their ordering. Hence alloy ordering can be an alternative to charge ordering. The fact of the existence of the ordered alloys is well known, and the thermally induced destruction of the atomic ordering is one of the examples of the second-order phase transitions [141]. At disordering of the La donor atoms, the crystal can either remain insulating or become highly conductive. The latter can occur if the La ion threads with small spacing arise as a result of disordering. Some authors believe that the charge ordering phenomena are related to the elastic energy which exceeds the Coulomb energy. E.g., in Ref. [313], in discussing the stripe pairing, the following considerations were presented. The physical origin of pairing of the Mn>O stripes is caused by  the minimization of the strain energy associated with the JT distorted Mn>O octahedra. Each  nonpaired Mn>O stripe will cause signi"cant lattice distortions in its immediate neighbourhood  and the overall strain energy can be lowered by forming periodic array of pairs of Mn>O stripes  separated by undisturbed regions of Mn> ions. The gain in lowering the elastic energy of the JT stripes should outweigh the cost of Coulomb energy. Unfortunately, at present, it is di$cult to compare the elastic and Coulomb energies. 6.6. Polaron lattice? Very interesting results were obtained in neutron studies of La Sr MnO with x"0.10 and \V V  0.15 [463]. For both samples a series of the satellite re#exions was found. As neutron di!raction does not make it possible to make a choice between di!erent models, electron di!raction observations are carried out using single-domain samples. Comparing results of both investigations, the conclusion was made that the true index of the satellite re#ections is (h, k$1/2) (O symbolizes the orthorhombic structure). The positions of Bragg re#ections within the pseudocubic Brillouin zone are as follows: the main Bragg re#ections due to the average perovskite structure appear at
points. The superlattice re#ections due to the small distortions appear at M or X point, and the newly observed satellites are at  points. Temperature dependences of the satellite intensities as well as the main Bragg re#ections and superlattice re#ections are investigated (Fig. 28) and compared with the resistivity of the same samples. The following important characteristics of the experimental results should be pointed out.

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Fig. 28. `Polaron latticea. Temperature dependencies of the intensities of various types of re#ections: (a) x"0.15, (b) "0.10. The important characteristics of each type of re#ection for both cases are described in Section 6.6 [463].

Fig. 28a gives (the results on the sample with x"0.15): (i) The satellite re#ection (2, 2,!0.5) sets in at ¹ "190 K, exactly corresponding to the point of resistivity upturn. (ii) The main Bragg N re#ection (2, 0, 0) starts to increase at ¹ "240 K corresponding to the onset of ferromagnetic ! order. (iii) The superlattice re#ection (2, 1, 0) shows a break at ¹ and increases below ¹ . (iv) The N N temperature variation of the satellite and the superlattice re#ections at ¹4¹ is rather gradual N indicating that the transition at ¹ is of the second-order type. N

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Fig. 28b gives (the results on the sample with x"0.10): (v) The satellite re#ection (2, 2,!0.5) sets in at ¹ "100 K, where the resistivity curve shows a similar upturn. (vi) The main Bragg N re#ection (2, 2, 0) starts to increase at ¹ "120 K, corresponding to the onset of a ferromagnetic , component of magnetic order. (vii) The superlattice re#ection (1, 0, 0) shows an abrupt decrease at ¹ . (viii) Temperature variation of the satellite and superlattice re#ections shows clear cut-o! N indicating that the transition at ¹ is of the "rst-order type. N These experimental results are explained in Ref. [463] by the following picture. It is postulated that the state of an isolated hole is a small polaron which is given by a localized hole in the 3d   V \W orbital surrounded by inverse JT distortion. The local distortion is postulated as above because the hole site (Mn> site) is JT inactive whence the electron}phonon energy is lowered by restoring higher symmetry around the hole. The system is considered to undergo successive transition as polaron liquid (insulator) P Fermi liquid (`metala) P polaron lattice (insulator). The transition at ¹ is due to the polaron ordering. The lattice distortion in the polaron ordered phase (P phase) is N characterized by the order parameter .(r)"-(r)# O exp(ik.r) ,   k."(1/2, 1/2, 1/4) (6.5)  where the "rst term in Eq. (6.5) is the order parameter of the orthorhombic (O) phase. Thus, the atomic arrangement in one of the successive two layers is identical with that of the JT ordered phase, while the other layer which contains holes recovers a higher symmetry similar to that of the JT disordered orthorhombic phase. The recovery is caused by the postulated JT distortion around the holes. The hole concentration is 1/8. It seems that the commensurate polaron order is so strongly stabilized that the polaron ordering tends to lock into this commensurate structure even when the x value is slightly o! from the stoichiometric value of x"0.125. The excess or de"cient holes due to o!-stoichiometry should leave some disorder in the hole con"guration. In fact, a strong di!use scattering was observed even at 10 K around main Bragg re#ections in both samples of x"0.10 and x"0.15. As for the order of the phase transition, the transition from the O phase to P phase is a disorder}order transition in terms of the JT ordering, and, hence, continuous. But the transition from JT phase to the P phase is that from an ordered phase [speci"ed by the wave vector k"(1/1, 1/2, 0)] to another ordered phase [speci"ed by the wave vector k"(1/2, 1/2, 1/4)] and, hence, discontinuous. This explanation is very elegant, but leaves unanswered some questions. First, as seen from Fig. 2, a quite similar behaviour of the resistivity is found in nondegenerate non-JT semiconductors. Hence, it is not necessarily related to the JT e!ect. Furthermore, the role of the JT e!ect is impossible to con"rm by estimates of the JT-phonon-hole interaction, which are absent at present. Second, the polaron ordered phase, being an analogue of the Wigner crystal, can arise only as a result of the Coulomb repulsion between the polarons. But the Wigner crystal is formed if the compensating charge is uniformly distributed over the crystal. Meanwhile, in addition to the Coulomb repulsion between the polarons, there is the Coulomb attraction between the hole polarons and ionized acceptors Sr>. If the acceptors are distributed randomly over the crystal, the ordered polaron phase is unlikely to form. Hence, to justify the existence of the superstructure, it is necessary to assume that the acceptors are ordered. But, if the Sr sublattice exists, there is no need

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to assume the existence of the JT polarons. Disappearance of the superlattice in the relatively highly conductive state can be attributed to the thermal ionization of the acceptors changing the lattice distortions around the Sr ions. This idea is close to that put forward in the preceding section: here an ordering of La}Sr alloy takes place, and the ordering should take place even if the JT e!ect is completely absent. Perhaps, the existence of the Sr sublattice also requires justi"cation.

7. Other manganites 7.1. Pr-based manganites In this section properties of those manganites will be described in which La is replaced by Pr, Nd, Sm, Eu or Tb. A striking di!erence between the properties of the Pr-manganites and those of the La-based manganites should be pointed out. The properties of Pr Ca MnO (or Pr-manganites \V V  doped with Sr) are investigated only for x40.3 at present. The La Ca MnO (0.24x40.45) \V V  samples are metallic and ferromagnetic at ¹"0 but become insulating at the Curie point. In the Pr-based materials with 0.35x50.5 ferromagnetism was not observed in zero "eld, and they are antiferromagnetic and insulating. On increase in temperature, an abrupt decrease in resistivity takes place at the NeH el point simultaneously with the "rst-order phase transition from the antiferromagnetic state to the ferromagnetic. Their most striking feature is a metamagnetic transition under an external magnetic "eld which is accompanied by a transition from the insulating state to a highly conductive state which is a speci"c manifestation of CMR. As a result of this transition, the resistivity decreases the more the farther the temperature is from ¹ . , In our discussion of the Pr-based manganites we begin with almost-metallic Pr Sr MnO    [324,325]. At low temperatures it is antiferromagnetic, and at about 140 K it displays a "rst-order phase transition in the ferromagnetic state (Fig. 29). In the low-temperature phase its conductivity increases with temperature though its value is of the same order of magnitude as in degenerate semiconductors. At the transition point the conductivity increases abruptly by about 10 times and then decreases with increasing temperature, approaching its low-temperature value again. According to Ref. [321], the x"0.5 compound, on decrease in temperature at this "eld, "rst decreases its resistivity, and then abruptly increases it by three orders of magnitude at 160 K. As a result, at lower temperatures it attains the same value as at zero "eld. At temperatures below 140 K the antiferromagnetic phase may be transformed into the ferromagnetic phase by an external magnetic "eld (Fig. 30). If this "eld exceeds 7 T, then the crystal is ferromagnetic at any temperature. At ¹P0 a "eld of 7 T increases the conductivity by two orders of magnitude (
&!10%). According to Ref. [321], a "eld of 6 T does not change the & semiconducting character of the increase in resistivity for the sample with x"0.5, when the temperature decreases down to 100. In the low-temperature phase, it shows no clear sign of charge ordering [339]. The situation with the Pr-based manganites resembles that described in Section 6.3: the magnetic "eld causes a sharp transition from the insulating state to the highly conductive state in some EuSe samples. The resistivity jump in a "eld of 1 T can attain nine orders of magnitude, and the ordering in the crystal changes from antiferromagnetic to ferromagnetic.

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Fig. 29. Temperature dependence of the resistivity of Pr Sr MnO and Nd Sr MnO . Thick and thin arrows       denote transition temperatures for Pr and Nd manganites, respectively [339].

The R Ca MnO system (R"Pr, Nd, x"0.5, 0.45) was investigated in the "elds up to 40 T \V V  [330]. The magnetization, CMR and striction were measured. It is established that the "eld destroys charge ordering and causes a metal}insulator transition at all temperatures below 250 K. Destruction of charge ordering is accompanied by a structural phase transition in addition to the magnetic phase transition. In R Ca MnO the charge ordered state is transformed into      a ferromagnetic state by H"6.5 T at 30 K. This is manifested in a huge change of optical spectra over a wide photon-energy region (0.05}3 eV) [328]. For an arbitrary x, at ¹"0 these materials are insulating but their resistivity drops with increasing temperature, approaching values of order 10\  cm at 300 K. A magnetic "eld can cause a transition to a highly conductive state at considerably lower temperatures [304,318}323]. At x"0.3 several phase transitions were observed in the material [320]. At 200 K the symmetry of the crystalline lattice changes, at 140 K the collinear antiferromagnetic ordering is established of the so-called CE type: the ferromagnetic and antiferromagnetic atomic planes alternate, and the moments of the nearest-neighbouring ferromagnetic planes are opposite. Below 110 K the crystal acquires a nonsaturated spontaneous magnetization which is interpreted in Ref. [320] as a transition to a canted antiferromagnetic structure. But such an interpretation causes objections (Section 6.1). These conclusions were based on an analysis of the neutron and magnetic data. The re#ections with the scattering vector q"(h, k, 0) correspond to the ferromagnetic long-range order, with h and k being integer and h#k even. On decrease in temperature, the (1, 1, 0) peak intensity diminishes below 200 K. This is explained by the authors of Ref. [320] in terms of the formation of a Mn> ion superlattice, which occurs simultaneously with a weak distortion of the crystal symmetry as a whole. As a consequence of this distortion, peaks appear with the indices slightly di!ering from

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Fig. 30. Temperature dependence of resistivity of Pr Sr MnO in a magnetic "eld. Results are obtained on sample    cooling from 330 to 4.2 K in the corresponding magnetic "elds (in kOe). For a 30 kOe "eld, both the curves are presented which correspond to the cooling in the "eld (FC) and to the heating of samples cooled in the "eld (FCW) [324].

integers. The ferromagnetic peaks are sharply ampli"ed below about 110 K when the crystal acquires magnetization. As for the antiferromagnetic peaks for the CE structure, the neutron scattering vectors [(2n#1)/2, (2m#1)/2, 0] and [n, (2m#1)/2, 0] correspond to them where n and m are integers. Their intensities increase sharply at 140 K and continue to grow down to the lowest temperatures. But if one assumes that the canted antiferromagnetic ordering is established over the entire crystal, then the magnitude of the magnetic moment per unit cell should be only 2.36 whereas its expected magnitude for the Mn ions is 3.7 . In order to explain this discrepancy, it is assumed in this paper that, due to the inequality of the Mn> and Mn> densities, a portion of the crystal is in the spin-glass state, i.e., the crystal is in a nonuniform state. But it is still simpler to interpret this state as a phase-separated ferro-antiferromagnetic state. At 4.2 K a "eld of 4 T establishes the saturated ferromagnetic ordering with the simultaneous transition from the insulating state to the conducting state. A study of the behaviour of the neutron spectra in the "eld yields that the intensity of the ferromagnetic (1, 1, 0) peak increases with the "eld

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strength attaining saturation at 4 T. On the contrary, the antiferromagnetic (1/2, 1/2, 0) peak disappears in this "eld. The (2.1, 5, 0) peaks corresponding to the distorted crystallographic structure also weaken. The lattice e!ects in Pr Ca MnO were investigated also in Ref. [333].    The resistivity and magnetoresistivity of the Pr Ca MnO compounds in the range \V V  0.34x40.5 was investigated in more detail in Ref. [321] where it was established that the magnetic "eld causes a well-pronounced insulator-to-metal transition. The conductivity of all the samples investigated is of the order of 10\  cm at 300 K, but, on decrease in temperature, it increases by 6}7 orders of magnitude. In samples with x"0.35 and x"0.3 this "eld causes the transition to the highly conductive state at temperatures below 100 K. The jump in the conductivity amounts to six orders of magnitude at x"0.35 and to four orders of magnitude at x"0.3. Both these compounds are metallic at all temperatures in a "eld of 12 T. All these compounds display a strong hysteresis in the magnetic "eld. Similar results were obtained in Refs. [322,323,443]. In Ref. [322] it is shown that, for a range of compositions (0.34x(0.5) there is a "rst-order magnetic-"eld-induced metal insulator transition. For the x"0.4 composition the magnetic and charge ordering e!ects are decoupled with antiferromagnetic ordering, ¹ "170 K, developing at considerably lower temperatures than , the charge ordered state, ¹ "250 K. Below this temperature, metamagnetic transitions exist !that transform the magnetic correlations from either paramagnetic or anti-ferromagnetic to ferromagnetic. Metastable conducting states are formed below 25 K. Linear-thermal-expansion and magnetostriction measurements up to 14 T show that in Pr Ca MnO with x"1/3 for temperatures below ¹ , about 400 K, a charge localization is \V V  N present, ensuring the semiconducting properties as in the case of La Ca MnO with x"1/3. \V V  Below ¹ "210 K, a charge ordering is observed. A magnetic "eld suppressed the charge ordered !state below ¹ giving rise to the "rst-order structural transition. A continuous change in volume !is measured between ¹ . The electrical behaviour is similar to the structural behaviour. !Both internal chemical pressure by ion substitution and external hydrostatic pressure induce the metal}insulator transition with destruction of the charge ordered state [457]. At low pressures the system is always insulating. In the intermediate pressure region the system undergoes a metal}insulator transition with large hysteresis. At large pressures the charge ordered state is completely suppressed; the system directly transforms from a paramagnetic insulator to a ferromagnetic metallic state. Application of external pressure and resultant enhanced carrier itineracy suppresses the charge ordering transitions also for Pr Sr MnO . For the x"0.30 crystal, application of pressure \V V  induces a metallic phase from the low-temperature side in the charge ordering insulating phase. The pressure}temperature phase diagrams relating to the charge ordering transitions or to the concurrent metal}insulator transition were shown to scale well with the magnetic-"eld-temperature phase diagrams [341]. The NMR on Mn nuclei was measured in Pr Ca Sr MnO and Pr Ba MnO             [354]. The former is metallic below ¹ , and the spectrum consists of a single, motionally narrowed ! line. In the latter, lines corresponding to localized Mn> and Mn> ions were detected at 77 K. In this compound the increase in temperature leads to an increased rate of the electron hopping. As a consequence, motional narrowing of the NMR line was observed, resulting in a single line for ¹'140 K. In both compounds the hyper"ne "elds on Mn nuclei remain "nite up to ¹ , ! indicating the "rst-order magnetic order}disorder transition.

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Information about the nature of the charge carriers in the manganites R Sr MnO (R"Pr, Nd, 1/44x41/2) was obtained by measurements of the resistivity and \V V  thermopower [474]. In the paramagnetic region the electronic transport is realized by charge carriers in narrow bands. Evidence was found for a gap in the density of states. The thermopower is found to be positive below the Curie point. The structural properties of Pr Ca MnO were studied by X-ray synchrotron and neutron     powder di!raction as a function of temperature (15(¹(300 K), and as a function of X-ray #uence at 15 and 20 K [335]. The temperature evolution of the lattice parameters and of the superlattice re#ections is consistent with the development of charge and orbital ordering below ¹ &180 K, followed by antiferromagnetic ordering below ¹ &140 K, similar to what was !, previously observed for La Ca MnO . Below ¹ &120 K, the magnetic structure develops      ! a ferromagnetic component along the a-axis on the Mn ions. At low temperatures, a small ferromagnetic moment of 0.45 oriented in the same direction appears on the Pr ions as well. The observation of the signi"cant lattice strain developing below ¹ , as well as the development of !the ferromagnetic component at ¹ &120 can be interpreted in terms of the presence of the ! ferromagnetic clusters with an associated lattice distortion from the average structure. Still earlier, the Pr (Ca Sr ) MnO was investigated [351]. For x40.15 the system is   \V V    charge ordered below 150 K. For x"0 the system is antiferromagnetic with ¹ "170 K. On , increase in ¹, a magnetization appears. For x'0.15 the magnetic structure is simple ferromagnetic, and ¹ increases with x. ! The stability of the charge ordered state against variation of temperature and external magnetic "eld has been investigated for Pr (Ca Sr ) MnO crystals with controlled one-electron   \W W    band width. For y"0.3 charge- and orbital-ordering appears below 200 K as in the y(0.3 crystals but collapses into the metallic state below 100 K. Under the magnetic "eld, such a reentrant feature is observed in a wider range of y [400] (possibly, the charge ordering takes place not among the Mn ions but among the Pr ions). According to Ref. [352], the change in the transport and magnetic properties of Pr La Ca MnO is related to a change in the atomic structure. The phase transition into        a metallic state takes place simultaneously with the transition into the ferromagnetic state. Static distortions of the oxygen octahedra observed before the transition, are absent from the ferromagnetic state. In the case of the Pr-based manganites, there are evidences of the existence of the ferromagnetic clusters, too. Thin, c-axis-oriented "lms of Pr Sr MnO were prepared by dc-magnetron      sputtering, structurally characterized by X-ray di!raction and their physical properties investigated by magnetization and electric transport measurements. A ferromagnetic ordering appears in zero "eld at 263 K, followed by a second transition into an antiferromagnetic state at 160 K. The zero-"eld resistivity has a semiconducting behaviour above the Curie and below the NeH el temperature. Quasimetallic behaviour is observed within the ferromagnetic state, whose temperature range can be extended by applying an external magnetic "eld. The negative magnetoresistance e!ect increases systematically with decreasing temperature and reaches a value of 700% at 1.5 K in a "eld of 12 T. In the ferromagnetic and antiferromagnetic phases, the "eld-induced resistivity decrease scales with the Brillouin function, suggesting the existence of magnetically polarized regions with a total magnetic moment which is reduced upon entering the antiferromagnetic state.

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Neutron studies in ferromagnetic Pr Sr MnO with ¹ "300.9 K and Nd Sr MnO      !      (¹ "197.9 K) revealed that in both these materials the spin dynamical behaviour is very similar at ! low temperatures. But around ¹ it is drastically di!erent. While in the Nd sample formation of the ! spin cluster of size about 2 nm is observed, the magnetic disordering transition in the Pr sample is more conventional. Formation of the spin clusters about ¹ was observed also in La manganites ! (Section 5.2). According to Ref. [329], spin dynamics in the Pr-sample is incompatible with the double exchange. As follows from EPR investigations [332], in R B MnO with \V   >W R"La, Pr and B"Ca, Sr the spin clusters exist over a wide range of temperatures exceeding ¹ . ! In a high-temperature paramagnetic state of the antiferromagnetic insulator Pr Ca MnO \V V  ferromagnetic spin #uctuations with small energy were detected. They decrease in intensity at ¹ , !but persist below this temperature and vanish only at ¹ [331]. , The optical spectra of Pr Sr MnO manganites were investigated in Ref. [334] with the aim      of determining density of states. The transition from the metallic ferromagnetic state into the insulating antiferromagnetic state is accompanied by an abrupt spectral weight transfer over 1.2 eV, which is large compared to the gap (about 0.1 eV). In Ref. [448] the resistive relaxation in Pr Ca MnO was investigated. At low temper     atures, a magnetic "eld induces the insulator-to-metal transition, and the reverse relaxation is studied after its switching o!. At 41}45 K a sharp increase in the resistivity occurs after 100}10000 s, whereas the magnetization decreases monotonically without singularities at that time when the resistivity exerts a jump. One might think about the phase separation into the highly conductive ferromagnetic and insulating antiferromagnetic phases. Then the jump into the insulating state corresponds to the disappearance of percolation with the magnetization not altering after this. Other Pr-based manganites were prepared, e.g. Pr K MnO , whose properties strongly \V V  resemble those of other Pr-based manganites [337]. 7.2. Nd-based and less-common manganites To understand the speci"c features of Nd-based manganites, it is convenient to begin with consideration of the La Nd Ca MnO system [353]. An increase in x leads to the \V V   transition of the compound from the metallic into the semiconducting state at x"1.5. But already at x"1.0 the resistivity peak and signi"cant magnetoresistance are observed. Now results obtained in investigating Pr Sr MnO and Nd Sr MnO with x about 1/2 by \V V  \V V  neutron di!raction [339] will be presented. Despite the similarity of these samples, they behave di!erently. The samples with x"1/2 exhibit a transition from a ferromagnetic metal to an antiferromagnetic nonmetal (Fig. 29). In the low-temperature phase, Nd Sr MnO has    a CE-type antiferromagnetic structure with charge ordering, while Pr Sr MnO and    Nd Sr MnO exhibit an A-type layered antiferromagnetic structure, but show no clear sign      of charge ordering. Investigations of the latter compound carried out in Ref. [450] show that it is an antiferromagnetic metal, and its magnon spectrum re#ects a strong magnetic anisotropy. Above ¹ ferromag, netic short-range order exists. Magnetic phase transitions with a short range order above the transition point di!ering from the long-range order below it were investigated in Ref. [453]. They arise due to a non-Heisenbergian exchange interaction.

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In Nd Sr MnO a magnetic-"eld-induced metal}insulator transition was observed which    should be classi"ed as a "rst-order phase transition. If a "eld of 0.5 T remains "xed, then, on temperature increase up to 150 K, the magnetization changes abruptly from 0 to 2.5 , and the resistivity from 0.1  cm down to 0.001  cm. Simultaneously, discontinuous changes in the lattice parameters take place. A hysteresis is observed. At 2.3 K the corresponding transition occurs in a "eld of 12 T. Investigations [440] of the same compound show evidence for the "rst-order phase transition due to the charge ordering around 220 K, accompanied by a large decrease in volume (0.45%). Valence sum calculations show the presence of two di!erently charged Mn atoms (3.3 and 4). The conductivity in the charge ordered state is rather low. This state melts on application of a magnetic "eld. We consider the system Nd K MnO now. Direct evidence of charge ordering has been \V V  observed by electron di!raction in the x"0.35 sample. They demonstrate charge ordering with the modulation wave vectors of q"(2
/a)(1/2, 0, 0) and q"(2
/a)(0, 1/2, 0). Thus, charge ordering occurs only in the ab-plane, but is frustrated along the c-axis. Upon cooling the sample in zero "eld, it appears at ¹ "270 K, while the material remains paramagnetic. An antiferromagnetic !ordering takes place at ¹ "160 K, below which CMR occurs. The magnetization displays , hysteresis. After reaching a certain "eld strength H , a metamagnetic transition occurs with a sharp D increase in the magnetization. With decreasing "eld, the magnetization "rst remains unchanged and disappears only when the "eld becomes su$ciently low [338]. Investigations of Nd Sr Pb MnO [69] were carried out in record magnetic "elds        up to 50 T. It was concluded that the Mott charge carrier hopping is the origin of CMR. As investigations of Nd Sr Pb MnO have shown, both resistivity and magnetoresistance are strongly        suppressed upon application of pressure [340]. The peak resistivity of (La Nd ) Ca MnO          is decreased by up to three orders of magnitude by a pressure of 5 kbar [417]. This problem was investigated in more detail in [341]. E!ects of chemical and external pressures have been investigated on the two types of charge ordering systems of perovskite manganites with the use of single-crystal specimens. One is Nd Sr MnO with moderate charge ordering \V V  instability (i.e. with ¹ (¹ ) occurring only near x"1/2 and the other is Pr Sr MnO with !! \V V  stronger charge ordering instability extending over a wide x region 0.34x40.7 (this means that ferromagnetic phase is no longer present, and ¹ (¹ ). The Nd ions of Nd Sr MnO were , !   partly substituted with larger La ions or an external pressure was applied to them with the aim of destabilizing the charge ordered state via an increase of the 3d-electron hopping interaction. An electronic phase diagram relevant to the charge ordering transition was derived for (Nd La ) Sr MnO by such a control of the one-electron band width =. With an increase of \X X    = the enhanced ferromagnetic double-exchange interaction increases ¹ and suppresses the ! charge ordered state with a concomitant antiferromagnetic charge-exchange spin ordering (antiferromagnetic CE). In a narrow window of z (0.44z40.6), or in the pressurized state for z"0.4, another type of the antiferromagnetic phase (perhaps, the S-type) replaces the antiferromagnetic CE state. The metal}insulator transitions relevant to charge ordering have been investigated for the perovskite-type (Nd Sm ) Sr MnO crystals, in which the one-electron bandwidth = is \W W    systematically controlled by varying the average ionic radius of the A site and by application of quasihydrostatic pressure P [342,343]. Competition between the ferromagnetic double-exchange

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and the antiferromagnetic charge ordering interactions gives rise to a complex metal (M)}insulator (I) diagram with temperature and = (y and/or P) as the parameters. The M}I phase boundaries are associated with critically =- and ¹-dependent hystereses, which results in unique appearance of the metastable state. The pressure induced phase transition from the metastable ferromagnetic metal to the thermodynamically stable charge ordered insulator for y"0.875 crystal locating near the critical M}I phase boundary. With decrease of =, the charge ordering instability accompanying the antiferromagnetic spin correlations subsists even above the ferromagnetic transition temperature and enhances the electron-lattice coupling. Consequently, the lattice-coupled "rstorder I}M transition is observed at ¹ in the small = region of y50.5. Application of magnetic ! "eld also induces the phase transition from the insulator with antiferromagnetic spin-correlations to the ferromagnetic metal, which is accompanied by lattice structural change. The manganites R Sr MnO with R"Nd Sm or Nd La have been investigated \V V  \W W \X X under hydrostatic pressure [177]. Positive pressure coe$cient d ln ¹ /dp is observed for all the ! compounds, and its absolute value increases with decrease of =, or, equivalently, ¹ . ! Epitaxial Nd Sr MnO "lms, displaying the resistivity peak and ranging in peak resistiv    \W ity from 1000  cm to less than 1  cm at a temperature ¹ , exhibit the following features [345]: (1) N Although the peak resistivities span a wide range, the temperature dependence of the zero "eld resistivity at ¹'¹ is consistent with activated behaviour and a characteristic energy of 0.115 eV, N in every case. (2) The virgin magnetoconductivity at ¹ well below ¹ is linear in the applied "eld B, N up to "eld values well beyond those required for technical saturation. (3) For ¹ close to ¹ and N above, the conductivity depends on B quadratically. Now we consider other A-cations. The Tb Ca MnO compound has been studied by means      of a.c. magnetic susceptibility, X-ray and neutron di!raction from 1.5 up to 330 K [346]. This compound shows a structural transition at 299.7 K. Below this temperature a charge ordered state was observed by both X-ray and neutron di!raction measurements. The structure is orthorhombic at temperatures higher than 300 K, but at lower temperatures the patterns show new re#ection peaks that can be accounted for in a monoclinic unit cell with Mn> and Mn> stripes in the ac-plane. Moreover, the system orders antiferromagnetically at around 120 K, giving rise to a magnetic structure of the CE type. The magnetic susceptibility measurements clearly show an anomalous behaviour at both transition temperatures. The charge ordering temperature for the Tb Ca MnO compound is the highest reported in the R Ca MnO series           (R"rare earth). This is argued to be a consequence of the larger orthorhombic distortion produced in the unit cell by the lower size of the Tb> ion. CMR has been observed in polycrystalline Gd Sr MnO perovskite [347]. Irreversibility      and sharp anomalies in the magnetostriction, magnetization and magnetoresistance isotherm take place below 90 K, which have been attributed to the establishment of charge ordering. For temperatures lower than 42 K, the charge ordered state coexists with a cluster-glass state, as deduced from the magnetic and speci"c heat behaviour and neutron scattering experiments. The linear-thermal-expansion measurements show an abrupt drop for ¹ about ¹ when a su$ciently !high magnetic "eld is applied. The appearance of this anomaly has been attributed to the partial delocalization of the carriers by the "eld. Investigations of Dy Ca MnO [451] and Gd Ca MnO [452] show that not only           the Mn ions are ordered ferromagnetically, but also the Dy or Gd ions. But the moment of the Dy or Gd sublattice is antiparallel to the moment of the Mn sublattice.

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In Ref. [348] magnetic and electric properties of R Sr MnO (R"Sm, Eu, Gd, Tb, Dy, Y)     \W compounds have been studied. All the samples have the same pseudo-cubic perovskite structure with small unit-cell distortions. Decrease in the rare-earth radii leads to a destruction of both the long-range ferromagnetic and charge order and stabilizes a spin-glass-like state. The Sm Sr MnO ferromagnetic compound exhibits a resistivity peak and an anomalous behav     iour of elastic properties around ¹ "125 K, whereas the resistivity and magnetoresistance of the ! R Sr MnO (R"Eu, Gd, Tb) spin glass increase gradually with decreasing temperature.      The idea that some manganites are spin-glasses is supported by other investigations, too. In Ref. [349] magnetic and magnetoresistance measurements were carried out on a distorted perovskite Eu Sr MnO . In the absence of an applied magnetic "eld, this compound is an      insulator down to 5 K. Results of the AC- and DC-susceptibility measurements suggest the spin-glass ground state. An applied "eld of 7 T drives the insulator ground state to a ferromagnetic metallic state below ¹ "120 K. For a range of intermediate "elds, the compound undergoes ! insulator}metal}insulator transitions which resemble those of 1 : 1 Mn>/Mn> charge ordered compounds. A temperature-"eld phase diagram is established which depicts di!erent magnetic and electronic states. A thorough study of the low-temperature magnetic state of polycrystalline (Tb La ) Ca MnO has been carried out [350]. This perovskite-like compound is an      insulator at low temperatures and shows an insulator}metal transition under magnetic "eld. All the measurements of its magnetic state (magnetization, susceptibility, and neutron di!raction) are consistent with spin-glass behaviour. 7.3. Photoinduced insulator}metal transition in manganites It seems rather obvious that a photoinduced increase in the charge carrier density can transfer a magnetic material from the insulating to metallic state. Simultaneously, the magnetic state of the crystal can change. These photomagnetic phenomena were observed in the magnetic semiconductors long ago (see the review articles [401,402]). All of what is said above for conventional ferromagnetic semiconductors, remains true for the manganites. The corresponding e!ect was observed in investigating Pr Ca MnO illuminated      by X-rays below 40 K [403]. As was already pointed out in Section 7.2, this compound is metallic at high temperatures, but exhibits a transition to the insulating charge ordered state at about 200 K. The low-temperature magnetic structure is antiferromagnetic with a weak uncompensated moment due to the spin canting. The material can be driven back into the metallic state by applying a magnetic "eld of 4 T at low temperatures. The metallic state is ferromagnetic. The "eld-induced transition is associated with large hysteresis. The neutron di!raction was measured under X-ray illumination and is compared to that in the absence of the illumination. The X-ray di!raction was also measured which is only sensitive to the lattice structure and not to the magnetic moments. The intensity of the (4, 1.5, 0) re#ection characteristic of the charge ordered structure, decreases continuously with X-ray illumination. No change is observed when the X-ray beam is switched o!. The diminution of the charge order is associated with a dramatic change in transport properties. The conductivity is essentially nonohmic. The resistivity value after prolonged X-ray exposure is identical to the resistivity measured without X-rays above a critical magnetic "eld of 4 T. Hence, photoinduced and "eld-induced

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metallic phases are identical. It is assumed that the insulating state in Pr Ca MnO should be      regarded as a polaron lattice. As was already mentioned in Section 7.1, the Pr Ca MnO compound displays ferromag     netic clusters at low temperatures. At low temperatures exposure to the X-ray beam produces a phase-segregation phenomenon, whereby the ferromagnetic droplets coalesce into larger aggregates. Further exposure results in a gradual melting of the charge ordered phase and formation of the second phase, shown to be a ferromagnetic metallic phase [405]. The ferromagnetic phase has a signi"cantly smaller a lattice parameter and unit-cell volume than that of the charge ordered phase. The insulator-to-metal transition can be caused by the light illumination, too [404]. An optical parameter oscillator excited by a pulsed YAG laser was used for optical excitation. The photocurrent is a highly nonlinear function of the applied electric "eld and of the light intensity. The dependence of the anomalous photocurrent on the excitation photon energy and the temperature excludes the laser heating as the cause of the e!ect, suggesting the photocarrier-mediated collapse of the charge ordered state. The photoinduced insulator-to-metal transition in the same compound was also investigated in Ref. [405]. Time evolution of the signal form, irradiation power and applied voltage dependence of the photocurrent after irradiation with laser pulse was measured and analysed. Furthermore, irradiation patterns were controlled and sample length and width were varied in order to investigate the state after the transition. As a result it was shown that the photocurrent observed comprises two components, fast and slow, both of which relax exponentially. The fast component was assigned to be typical photocarrier generation and relaxation. The slow component which was concluded to be critical for the transition, was suggested to originate from small metallic islands being created within the gap by the irradiation. Measurements done on samples with the controlled band width gave proof of the shrinking of the conduction path with decreasing current, supporting the picture of the conduction path being under stress from the surrounding charge ordered, antiferromagnetic bulk and being sustained by the current. Lack of dependence of the resistance on the sample length was observed, indicating that the major contribution to the voltage drop is determined by some neck in the conduction path. In Ref. [407] a photoinduced insulator-to-metal transition was found in a set of Pr Ca MnO \V V  samples from x"0.31 to x"0.50 [407]. In addition to the photoinduced ferromagnetism discussed above, photoinduced demagnetization was observed in (Nd Sm ) Sr MnO thin "lms. Photocurrent passing Nd Sr MnO               in the charge ordered state can cause a nontransient drop in resistivity [408]. 7.4. Theoretical problems of Pr(Nd)-based manganites First, it is necessary to explain why, e.g., the lightly doped La Ca MnO behaves like a doped \V V  magnetic semiconductor with Ca> functioning as acceptors, and why, e.g., Pr Ca MnO \V V  behaves quite di!erently. Usually, the speci"c properties of the Pr-based manganites are explained by charge ordering. But this explanation meets certain di$culties. First, the metamagnetic transition occurs at any dopings between x"0.3 and 0.5 and not only at those when the number of Mn> ions is a multiple of the number of Mn> ions. Second, the superstructure is one dimensional

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in them: alternating atomic planes assumedly consisting of only Mn> ions and of a mixture of Mn> and Mn> ions were discovered in Pr Ca MnO with x"1/3 [486]. Certainly, the \V V  former should be insulating and the latter highly conductive. But no conductivity is found along them yet. Another explanation may be proposed for the di!erence between the La-based and Pr-based manganites [487]. It is based on the fact that only La> ions can exist in the former, and La> cannot. Meanwhile, Pr can enter chemical compounds in the form of Pr> [30]. If this is the case for the manganites, then the holes in the Pr-based manganites which arise due to their doping by Ca or Sr can be "xed at a certain number of Pr ions transforming them from Pr> to Pr>. Such holes are not electroactive. The Pr-hole energy depends on an external magnetic "eld rather weakly. Meanwhile, the bottom of the Mn-hole band lowers with increasing "eld considerably more rapidly than the discrete Pr levels. For this reason, if at H"0 the discrete levels are below the Mn-hole band bottom, at "nite H they can occur above the bottom, and then the holes go over from these levels to the Mn-band. Then they participate in the indirect exchange which tends to establish the ferromagnetic ordering. The joint action of the external magnetic "eld and the "eld-induced indirect exchange causes the metamagnetic transition jointly with the insulator-tometal transition. Correspondingly, the Pr-based manganite transforms into a doped ferromagnetic semiconductor. Formally, Pr-based manganites di!er from La-based manganites in a smaller value of the e!ective hopping integral which is related to smaller values of their radii (the radii of the threefold ionized atoms Pr, Nd, Sm, Eu or Tb are, respectively, 106, 104, 111, 98 or 93 pm as compared with the La> radius of 122 pm [30]). Instead of the ionic radius, the tolerance factor is often used, which is determined as r #r tol" (2(r #r )  where r , r and r are the ionic radii of the A site, B site, and oxygen ions, respectively. The A site  corresponds to La, etc. Hence, the tolerance factor is larger in them than in the La-based manganites. Concerning the theoretical interpretation of metamagnetic insulator-to-metal transition, the scheme proposed in the beginning of this section (the reaction Pr> to Pr>#hole), can explain it qualitatively. But there are other models with the corresponding properties. As already pointed out in Section 2.3, in JT antiferromagnetic systems a metamagnetic phase transition is possible [54}57]. The same is true for the systems with biquadratic exchange interaction which, in particular, can result from the interaction between the spins and JT phonons [428]. Thus, both the static and dynamic JT e!ects can cause a metamagnetic transition, but it does not assume the simultaneous transition into a highly conductive state. On the other hand, one can explain magnetic-"eld-induced transition from the insulating state to the metallic state using the model of doped magnetic semiconductors [429]. In this model an impurity density range exists inside which the Mott delocalization in the antiferromagnetic state is impossible but it is possible in the ferromagnetic state. The reason for this is the fact that in the antiferromagnetic state the localized electron is in the bound-ferron state (Section 6.2) when a ferromagnetic microregion arises around the impurity, and the electron is attracted to the

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impurity not only by its Coulomb "eld but also by the ferromagnetic potential well. For this reason the electron orbital radius for the antiferromagnetic state is less than that for the ferromagnetic state and, hence, in the former case the impurity density may be insu$cient and in the latter case su$cient for the Mott delocalization. Keeping in mind that at the metamagnetic transition the magnetization appears abruptly, one can expect a simultaneous abrupt transition into the highly conductive state. But this argumentation is valid when the doping is not too large. The corresponding calculation for the double-exchange case was done in Ref. [135]. Recently, a calculation [114] was carried out in which the possible magnetic structures in the manganites were considered, and a phase diagram was constructed for them. Materials of the Ca R MnO are considered with very small y (R stands for a rare-earth element). In this case \W W  the charge carriers are the doped electrons which move over double degenerate e orbitals causing E the indirect exchange. The approach [114] takes into account simultaneously the orbital degeneracy of the e orbitals over which the doped electrons move, the Heisenberg exchange interaction E between these electrons and the spins of the localized t electrons, as well as the Heisenberg E exchange interaction between the localized spins. The situation corresponds to the double exchange with the classical treatment of the d-spins. The phase diagram shows that, depending on the electron doping concentration and the ratio of the e band width and t superexchange, the E E following structures are possible: ferromagnetic and antiferromagnetic of A or C type. One of the ways of increasing the charge carrier density is through photogeneration. From the point of view of the Mott theory, an increase of the charge carrier density due to the photoelectrons diminishes the screening radius and, hence, favours the delocalization of charge carriers. In turn, an increase in the charge carrier density makes the indirect exchange via them stronger. As was discussed in Sections 3.1}3.4, the latter tends to support the ferromagnetic ordering. Hence, illumination can cause the appearance of magnetization of the crystal if it was absent in the dark. The photoinduced magnetization was observed in many magnetic semiconductors and is described in detail in the review articles [401,402]. As this phenomenon is nonequilibrium, this magnetization cannot be found from the condition of the minimum free energy. On the other hand, it cannot be found from the condition of the extremal entropy production often used in the nonequilibrium thermodynamics as this condition has very limited applicability range and does not take into account possible magnetic ordering. In Refs. [401,402] an extremal principle is proposed which takes into account both the generation of the entropy and establishing the magnetic ordering. This result can be applied to manganites, too. 7.5. Layered manganite crystals In outlining this section, I partly follow Ref. [368]. Here studies will be described in the so-called Ruddlesden}Popper (RP) structure series for manganese oxides which are characterized by the chemical formula (R, Di) Mn O (R and Di being trivalent rare earth or divalent alkaline L> L L> earth ions, respectively). The basic structure in this homologous series is based on alternate stacking of rock-salt-type block layers (R, Di) O and n MnO -sheets along the c-axis [355}360].    The n"R RE AE MnO compound was described in preceding sections (Fig. 30). \V V  The n"2 member of the RP series, RE AE Mn O also shows a wide variety of \V >V   physical properties like perovskite analogues, including large magnetoresistance [356,374]

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and magnetostriction [356,361]. In La Sr Mn O the crystallographic structure shows       a very small JT e!ect, but a large local JT distortion [375]. The La Ca Mn O with \V >V   0.224x40.5 is a ferromagnetic metal with the two-dimensional (2D) ferromagnetic alignment between the 2D- and 3D-ordering temperatures. This compound consists of bilayers of metallic MnO sheets separated by insulating material. With x"0.30, it displays anisot ropic magnetoresistance at ¹;¹ "90 K. A magnetic "eld of 1.5 T switches from antifer! romagnetic stacking to ferromagnetic stacking (spin-valve devices) [381]. According to Ref. [383], in La Sr Mn O spins in neighbouring layers within each bilayer are strongly       canted with the canting angle depending on H and ¹. The in-plane correlation length does not diverge at ¹ . ! Detailed investigation of the magnetic structure of the layered manganites was performed. According to Ref. [382], critical components of La Sr Mn O exhibiting the metal-insulator       transition simultaneously with a magnetic transition at ¹ "116 K correspond to the 2D X> ! model. According to Ref. [395] the same compound exhibits a transition between paramagnetic state with "10  cm and ferromagnetic state with "10\  cm. In the paramagnetic state the compound contains long-lived antiferromagnetic clusters coexisting with ferromagnetic critical #uctuations. The principal implication is that at least in two dimensions Anderson localization e!ects should be taken into account for CMR which agrees with the magnetoimpurity theory. According to ESR data [384], in La Sr Mn O , having the Curie point 100 K, the 2D       ferromagnetic correlations exist up to 300 K. They coexist with long-lived antiferromagnetic correlations. Localized t states are coupled into intralayer ferromagnetic clusters. The e carriers E E form magnetic polarons with an e!ective spin growing from 1/2 to 7. The 3D order originates from the onset of coherence of the FM clusters. One of the most distinctive features for the bilayered manganite is its anisotropic characters in charge transport (the resistivity ratio  / '100) and in magnetic interaction (the exchange ratio A ?@ I /I '100) due to the layered structure. Another remarkable feature for the bilayered manganite ?@ A is the interplane tunnelling magnetoresistance e!ect which can provide a novel approach to the large magnetoresistance attainable at low "elds [363]. Several kinds of ferromagnetic tunnelling junctions with use of perovskite manganites have so far been investigated, such as trilayer junctions [364], grain-boundary junctions in polycrystalline samples [365] and in thin-"lms on bicrystal substrates [366]. The bilayered compound is composed of ferromagnetic-metallic MnO bilayers  with intervening insulating (La, Sr) O blocks. In other words, the bilayered manganite intrinsi  cally contains an in"nite array of FM/I/FM tunnelling junctions in its crystal structure. In such a quasi-two-dimensional ferromagnetic metal (FM), the interplane as well as inplane charge dynamics (and, hence, the magnetoresistance characteristics) are expected to critically depend on the interlayer magnetic coupling between the ferromagnetic MnO bilayers.  Fig. 31 shows the temperature dependence of the  in series of RP phases (n"1, 2 and R) ?@ with the nominal hole concentration "xed at x"0.3. The  as well as its temperature dependence ?@ change from metallic to semiconducting, with decreasing the number n of the MnO sheets. In the  n"R compound, the steep drop of the resistivity at 360 K corresponds to the onset of the ferromagnetic ordering. The n"1 phase with an isolated MnO sheet, on the other hand, does  not undergo the ferromagnetic-metallic transition and remains insulating down to the lowest temperature.

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Fig. 31. Temperature dependence of inplane resistivity ( ) for n"1, 2, and R members of Ruddlesden}Popper ?@ phase (La, Sr) Mn O . The nominal hole concentration is "xed at x"0.3. Inset: the crystal structure L> L L> (La, Sr) Mn O (n"1, 2, and R). Shaded planes represent MnO layers [368]. L> L L> 

In the n"2 compound,  shows a semiconducting temperature dependence above room ?@ temperature. With decreasing temperature,  shows a broad maximum around 270 K, and then ?@ shows a metallic temperature dependence in contrast to the semiconducting behaviour of  . This A result implies that the short-range ferromagnetic correlation which extends only within MnO  bilayers evolves with decrease of temperature below 270 K. With further decreasing temperature the long-range spin ordering takes place around ¹ "90 K. Below ¹ the spin correlation extends ! ! over the adjacent MnO bilayers, and reduces the spin scattering of the conduction electrons in the  transport process along the c-axis as well as the ab-plane, which is re#ected in steep drops of  and  around ¹ . ?@ A ! Now the e!ect of the pressure will be discussed. Above ¹ both  and  monotonically ! ?@ A decrease with increasing pressure. But below ¹ both these quantities are remarkably increased by ! applying pressure. The interplane magnetoresistance is much larger than inplane magnetoresistance. The magnitude of the former is drastically enhanced up to 4000% on applying pressure of 0.7 GPa. But the resistivity is weakly temperature dependent at high magnetic "elds (Fig. 31). According to Ref. [369], the application of pressure reduces the JT distortion in the MnO  octahedron. Study of La Ca Mn O showed that a pressure up to 9 kbar causes the "rst-order       transition from the paramagnetic insulator to ferromagnetic metal at 242 K. The pressure increases ¹ by 1.1 K/kbar [377]. In Ref. [379] the metal}insulator transition under pressure was studied in !

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La Sr Mn O . In Ref. [382] the pressure-induced interplane tunnelling magnetoresistance \V >V   was investigated in this compound [382]. In Ref. [383] the La (Sr Ca ) Mn O compound   \V V     was studied under both external pressure p and internal pressure (the latter is produced by replacement of Sr with Ca). The metal}insulator transition temperature and ¹ increase with p, but ! decrease with the Ca content though the latter creates an internal pressure. A theory of the pressure e!ect was developed in Ref. [370]. Account was taken of the anisotropic change in the Mn}O bond length. Considering the energy di!erence between 3d   and 3d   V \P V \W in the pressure-deformed lattice, the authors [370] showed that the applied pressure can weaken the interlayer charge and spin couplings through the stabilization of 3d   orbital. The change in V \W the orbital character may thus be responsible for the observed pressure e!ect on the interlayer coupling in the bilayered manganite. An increase in the nominal charge carrier density above x"0.3 changes the temperature dependence of the resistivity drastically. Whereas the x"0.3 compound displays metallic behaviour, the x"0.4 compound displays resistivity peak in the vicinity of ¹ [368]. This peak can be ! suppressed by an external magnetic "eld. With further increase in x toward x"0.5, the ferromagnetic highly conductive state disappears in bilayered manganites. This is closely related to the stabilization of the charge ordered state at a commensurate value of the nominal charge carrier density. The charge ordering accompanies, in general, a remarkable change in the lattice parameters as well as magnetic and transport properties. In the n"1 phase [356,371,376,380] the  steeply ?@ increases around 220 K, which corresponds to the onset of charge ordering. Resonant X-ray di!raction measurements demonstrated that the charge ordering accompanies the orbital ordering. For n"2 phase the resistive jump signalling the onset of the charge ordering also takes place at 220 K as in the n"1 phase. Recent electron di!raction measurements [368] showed the appearance of superlattice spots indicating (1/4, 1/4, 0) and (3/4, 3/4, 0) structural scattering below ¹ which is the same pattern observed for the n"1 phase. Hysteresis was observed during the !cooling and warming runs at the temperature region between 50 and 160 K. It is reported on the basis of the neutron powder di!raction [359], that the low-temperature phase for x"0.5 is the layered antiferromagnet with antiferromagnetic coupling between each ferromagnetic MnO layer  within a bilayer. Transmission-electron-microscopy measurements characterizing the structure and charge ordered states in LaSr Mn O are presented in Ref. [373]. The crystal structure of this phase has    been identi"ed as a well-de"ned 3}2}7 layered structure by high-resolution electron spectroscopy and image simulation. Electron di!raction reveals at low temperatures the presence of additional superstructure spots along the [1 1 0] direction. This superstructure is interpreted in terms of charge ordering of Mn> and Mn> associated with the d X orbital ordering of Mn>. B Highly anisotropic optical conductivity spectra and their temperature variation have been investigated for a layered ferromagnetic (¹ "121 K) crystal of La Sr Mn O (x"0.4). ! \V >V   The in-plane spectra show conspicuous spectral-weight transfer over the energy region of 0}3 eV with evolution of spin polarization below ¹ and in the ferromagnetic, highly conductive ! ground state form a prominent peak structure around 0.4 eV with minimal Drude weight. By contrast, the c-axis spectra show little change with temperature and remain insulator-like. The results indicate highly incoherent dynamics of the fully quasi-two-dimensional spin-polarized carriers [372].

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The behaviour of the system depends strongly on the additional doping. For example, the La Y Ca Mn O is a ferromagnetic metal with ¹ "170 K and a tetragonal structure. For  \V V     ! x"0 and 0.3 the metallicity is suppressed [396]. The k-dependent electronic structure of a low-temperature ferromagnetic system La Sr Mn O was measured using angle-resolved photoemission spectroscopy and calculated       in the local spin density approximation. Spectral features are too broad to be well-described as Fermi-liquid-like quasiparticles, and there is pseudogap in the excitation spectrum [398]. Magnetic and transport properties have been investigated in single crystals of layered-type doped manganites R Sr MnO with systematic variation of the average radius r of the      0 trivalent rare-earth ions R> [454]. For R"La, a charge ordering transition exists at ¹ "230 K. With a decrease in the in-plane Mn}O bond length, ¹ decreases and eventually !!the transition disappears for R"Nd. Lattice e!ects on the anisotropic magnetic and transport properties have been investigated for single crystals (La Nd ) Sr Mn O with a layered structure [455]. The Curie point is \X X       suppressed from 130 K for z"0 to 80 K for z"0.2, and eventually, the transition disappears beyond z"0.4. From a signi"cant change of the magnetic anisotropy, it is found that the suppression originates from the increasing d  character in the occupied e state due to E X a Jahn}Teller distortion of the MnO octahedra. These observations indicate that the e!ects of the  chemical substitution are qualitatively di!erent between the cubic and layered doped manganites. The pressure-induced variations of the anisotropic charge transport and magnetic properties have been investigated for a bilayered manganite crystal La Sr Mn O (x"0.3) [456]. \V >V   The magnitude of the magnetic coupling between the adjacent bilayers can be weakened by application of pressure which makes the charge dynamics more two dimensional but highly incoherent. As a result, the restoration of the interlayer hopping of the spin-polarized carriers by application of magnetic "eld, namely, the "eld-induced 2D}3D crossover, gives rise to an extremely high ratio of interplane tunnelling magnetoresistance, say, up to  / &4000% at 4.2 K under A A 10 kbar.

8. Isotope-e4ect and electron+phonon interaction 8.1. The oxygen isotope ewect in the manganites This section is devoted to the electron}lattice interaction and to phenomena which are usually related to this interaction although their origin can be quite di!erent. I mean the isotope e!ect which turned out to be very prominent in the manganites. It manifests itself in changes of electric and magnetic properties of the manganites produced by replacement of the light oxygen O with the heavy oxygen O. Traditionally, beginning with the conventional BCS superconductors, one tried to explain the isotope e!ect by a strong electron}phonon interaction. Meanwhile, examples are known when the isotope e!ect has nothing in common with the electron}phonon interaction. For example, the thermodynamic equilibrium concentration of deuterium D in metallic hydrogenabsorbing Pd di!ers from that of the protium H several times, though the electron}phonon interaction is irrelevant here [409]. As will be shown below, a similar nonelectron}phonon mechanism can function in manganites, too.

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We begin with a description of the experimental results on the isotope e!ect. It was "rst discovered in La Ca MnO [410]. For what follows, it is very important to describe the \V V >W procedure of the sample preparation. The O and O samples were prepared from the same batch of the starting material, and were subjected to the same thermal treatment in closed ampoules (one was "lled with O gas, and another with the O gas.) The di!usion was carried out for 48 h at   9503C and in an oxygen pressure of about 1.0 bar. The di!erence between the oxygen partial pressures of the O and O ampoules was less than 4%. The cooling was very slow, with a rate of 303C/h. The O samples had about 95% O and about 5% O. For x"0.2, the O samples have lower Curie points than the O samples by about 21 K, i.e., the oxygen isotope exponent  "!d ln ¹ /d ln M "0.85 (M is the oxygen isotope mass). For x"0.1 the exponent is found  ! to be 0.7. These exponents are considerably larger than in La Sr MnO ( "0.19 for x"0.10 and \V V >W  0.07 for x"0.3, respectively). From SrRuO investigated simultaneously, the isotope e!ect is  practically absent. As in the latter the JT e!ect is absent, too, the conclusion is made in Ref. [410] that the giant isotope e!ect in La Ca MnO is related to this e!ect. The possibility that the \V V >W e!ect is associated with di!erent oxygen contents is dismissed in Ref. [410]. The same e!ect in La Ca MnO was investigated in Ref. [168] at larger x between 0.41 and \V V >W 0.5. The values obtained are considerably less than those in Ref. [410]. For x"0.41 the oxygen isotope exponent turned out to be equal to 0.191 but for x"0.43  "0.14. The isotope exponent  is essentially a function of the tolerance factor (6.1). As already pointed out in Section 5.2 (Fig. 13), the magnetization M vs. ¹ curve displays a maximum before the vanishing of M in the Curie point. This maximum is hysteretic and is much more pronounced in the case of O than in the case of O. The fact that a hysteresis takes place close to the second-order ferromagnetic-paramagnetic phase transition makes the authors assume that it is related to the charge ordering which occurs in the form of the "rst-order phase transition. But a similar M peak was observed in Ref. [169] at small dopings where the charge ordering should be absent. The argument [168] that in the range 0.41(x(0.5 the crystal is in a two-phase ferro-antiferromagnetic state (Section 6.3) with ¹ slightly less than ¹ seems more convincing. , ! This should lead to superposition of two temperature dependences of magnetization of the type observed experimentally. In [411] the oxygen-isotope e!ect in La Ca MnO was investigated from x"0.2 to 0.43. \V V >W The isotope exponent  decreases from 0.36 to 0.14 with increasing Ca concentration. A large  value of  "0.83 was found for x"0.2. It is likely to be connected with excess oxygen content  judging from the fact that annealing of these samples at a high temperature in the argon atmosphere reduces the isotope e!ect sharply. This important conclusion points to an explanation of the e!ect di!erent from that proposed in Ref. [410]. The isotope e!ect decreases with increasing tolerance factor. At x"0.35 it is independent of the magnetic "eld. The EPR signal in La Ca MnO was investigated in the paramagnetic regime for O and \V V >W O isotope substituted compounds [435]. The characteristic di!erences observed in the EPR intensity and linewidth for the two isotope samples can be explained by a model in which a bottleneck spin relaxation takes place from the exchange-coupled constituent Mn> ions via the Mn> JT ions to the lattice. For x"0.2 the ferromagnetic exchange energy exhibits an oxygen isotope e!ect of about 10%.

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Fig. 32. Temperature dependence of inplane resistivity ( ) and interplane ( ) resistivity under magnetic "elds of 0 T (a) ?@ A and 3 T (b) with the "eld along the c-axis at pressures of 0, 0.4, and 1.1 GPa in the La Sr Mn O (x"0.3) \V >V   crystal [368].

Other manifestations of the isotope e!ect were discovered. In Ref. [412] the resistivity and magnetization of the (La Nd ) Ca MnO were studied. This material when containing          O behaves like a metal with the ¹"0 resistivity of order 10\  cm. But isotope replacement transforms the material into a semiconductor increasing its resistivity by about eight orders of magnitude at 70 K. Simultaneously, the ferromagnetic sample transforms into the nonmagnetic one (Fig. 32). Similar results were obtained for La Pr Ca MnO in Refs. [413,414]: isotope replacement        transforms a ferromagnetic metal into a semiconductor. In addition, measurements in an external magnetic "eld were performed. They made it possible to evaluate the stability of the semiconducting state which is interpreted as charge ordered. A "eld of 2 T melts the charge ordering state. In Ref. [415] the oxygen isotope e!ect in (La Nd ) Ca MnO was investigated, and          essentially new results as compared to Ref. [412] were obtained. Neutron di!raction, resistivity, magnetostriction and thermal-expansion measurements make it possible to con"rm transition of a metallic O sample into a semiconducting state. In more detail, both isotope samples segregate into a charge-ordered and a paramagnetic state below the charge ordering temperature of 210 K, which does not depend on the oxygen mass. But the relative percentage of both phases strongly depends on the isotope mass. At low temperatures, the charge ordered phase orders antiferromagnetically whereas the remaining phase orders ferromagnetically. Replacement of O by O diminishes the volume of the ferromagnetic phase and increases that of the antiferromagnetic phase. These investigations are supplemented by isotope investigations of Pr Ca MnO [416].    Both samples exhibit semiconducting conductivity and can be transformed into a metallic state by a magnetic "eld. Its strength is isotope dependent: for O sample it is 6 T but for O sample the critical "eld amounts to 12 T. In Ref. [417] the resistivity and thermopower of (La Nd ) Ca MnO with x between 0.7 \V V      and 0.85 were investigated. In particular, the thermopower as a function of temperature displays

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a maximum for both isotopes, but its value for O is lower than that for O. In Ref. [418] thermal expansion and magnetic susceptibility measurements were performed on oxygen-substituted (La Ca ) Ca MnO where a colossal oxygen-isotope e!ect on thermal-expansion coe$c\V V \W \W  ient was observed jointly with the isotope ¹ lowering by 10 K. ! In [419] the isotope e!ect in La Ca MnO was investigated. Replacing the light oxygen by      the heavy oxygen enhances the charge ordering temperature by 9 K at H"0. This shift increases under an external magnetic "eld, so that it reaches 40 K at a "eld strength of 5.4 T. In all the papers cited above the isotope e!ect is associated with the electron}phonon interaction. Some theoretical publications appeared supporting this point of view. In Ref. [420] a simpli"ed model is considered consisting of two Mn ions, an oxygen ion playing the part of a bridge between these ions, and one electron. The dynamical JT e!ect is assumed to consist of two minima in the potential energy curve for the oxygen ion. The electron transfer from one Mn ion to the other via the oxygen ion is accompanied by the oxygen ion transitions between these minima. The change in energy of the system due to the electron transition is assumed to give an estimate for the Curie point. As this change depends on the oxygen mass due to oxygen transitions between minima, the same should be true for the Curie point. By an appropriate choice of a value of the Franck}Condon factor, a value of the isotope exponent can be obtained close to unity, i.e., close to the experimental value. The authors of Ref. [420] criticize [50] for strongly overestimated Curie temperature amounting to 2000 K (they themselves obtained a value an order of magnitude less). Virtually the same model was used in Ref. [431] but conclusions turned out to be quite di!erent: the isotope exponent being equal to only 0.2. Another approach was proposed in Ref. [414] which is based on the assumption that the samples are close to the ferromagnetic-charge-ordering boundary. A model of a crystal with one electron per two atoms is used. The standard mean-"eld treatment shows that the charge ordered state occurs if the hopping integral t is less than a certain critical value, t(t "



t  (8.1) ¹ "





where the interatomic distance r"r #u, t "t(r ), and &1/r .     The mean-square displacement u, and hence t , depend on the ionic mass M even at ¹"0  due to the zero-point vibrations. In the case of a dominant contribution by zero-point vibrations, one has 1 1 " u" 2M " 2(BM

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where B is the bulk modulus, thus u
M . (8.3)
t "!t   r M  The corresponding change in t is generally not large, especially taking into account that one has  to use the reduced mass instead of pure ionic mass. But this change may be su$cient to shift the system from one state to another if it is close to the phase boundary at the phase diagram. Then t is close to t and, as follows from Eq. (8.1), 




t  . (8.4) t  As, typically,
1!

8.2. Oxygen nonstoichiometry as an origin for the isotope ewect A quite di!erent mechanism of the isotope e!ect was proposed in Refs. [421}423], which cannot occur in the BCS superconductors but is inherent in the oxide compounds. It consists of an isotope dependence of the thermodynamic equilibrium number of the excess or de"cient oxygen atoms (certainly, the nonequilibrium numbers of the heavy and light oxygen atoms can di!er, too, due to di!erence in their di!usion coe$cients, but this nonequilibrium di!erence is rather trivial). This dependence should lead to an isotope dependence of the equilibrium charge carrier density and volume of the lattice. As the excess or de"cient oxygen atoms are acceptors or donors, respectively, it implies the isotope dependence of the hole density, which manifests itself in the electroconductivity and in the intensity of the indirect exchange via holes. On the other hand, the oxygen nonstoichiometry causes contraction (or dilatation) of the lattice. Hence, it acts as an e!ective pressure. As the electric and magnetic properties of the manganites are highly pressure sensitive, the isotope dependence of the nonstoichiometry degree should lead to the isotope dependence of these properties. When one studies the isotope e!ect in the conventional superconductors of the Hg type, the di!erence in the number of imperfections (vacancies and so on) for di!erent isotopes is inessential. In fact, the electric properties of metallic samples are determined by regular atoms, the number of which is many orders of magnitude larger than the number of imperfections. But the electric properties of semiconductors are determined by a very small number of electroactive impurities, and this is the principal di!erence between the isotope e!ect in a semiconductor and that in a metal. Usually, to transform manganites into doped semiconductors, one uses divalent ions Di (Ca, Sr and so on). But also the excess oxygen can function as an acceptor. Very small quantities can change the material properties drastically. The undoped LaMnO is an antiferromagnetic insula tor, but LaMnO is, at least, partly a ferromagnetic metal [33,38,156,157]. Hence, only 4% of   the excess oxygen is su$cient to cause such a transformation. Similar experimental results

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concerning the role of the oxygen nonstoichiometry were found in Refs. [297,424,425] and others. On the contrary, the oxygen de"ciency in the manganite samples doped by another acceptor impurity (e.g., Ca) causes a sharp drop in their conductivity as it makes the p-type sample compensated. In reality, the La Ca MnO is ferromagnetic and highly conductive at y"0,     >W but becomes antiferromagnetic and insulating at y"!0.37 [426]. The nonstoichiometry of a crystal is a direct consequence of the thermodynamical laws: each sample should be, to some degree, nonstoichiometric. A change in the oxygen isotope leads, inevitably, to a change in the nonstoichiometry, even if the samples are prepared at the same temperature and pressure under the thermodynamical equilibrium condition between them and the surrounding oxygen gas (at lower temperatures the number of imperfections is frozen). Below, a situation will be considered which reproduces the general idea of the calculations [421}423]: the excess oxygen occupying interstices in the crystal. The role of the phonons will not be taken into account. This situation is simpler than the real situation in the manganites: the excess oxygen manifests itself not in the interstitial oxygen but in the cation vacancies with equal numbers of La and Mn vacancies (Section 2.2). Second, the phonons play a very important role at elevated temperatures. Namely, only due to the phonons a well-known theorem of the statistical physics is ful"lled: in the classical limit (¹PR) the isotope e!ect should be absent [427]. Here only temperatures that are low enough will be considered when the phonons can be neglected. In order to elucidate the actual temperature range, it is advisable to return to the experimental procedure [410] of the isotope replacement in the manganites depicted in the previous section. It is carried out at about 1250 K during 48 h when almost all the oxygen sites become occupied by the corresponding isotope. Then the isotope-replaced samples are cooled very slowly in the atmosphere of the corresponding isotope at a rate of 303/h, so that room temperature is achieved in more than 30 h. The concentration of the excess oxygen n must change on cooling, following the temperature of the environment. But the possibility to correspond it exactly is determined by the oxygen di!usion at this temperature. For this reason, in reality n corresponds to the equilibrium at a certain temperature, intermediate between 1250 and 300 K. In a more accurate approach, one should take into account a nonuniform distribution of the excess oxygen over the sample. It is worth noting that establishing the equilibrium oxygen density in the isotope-replaced sample via the di!usion can be essentially more rapid than replacing the light isotope by the heavy one, as the latter includes the replacement reaction in addition to the di!usion. The di!usion can depend essentially on the content of the acceptor impurity, so that for some compositions it can be more rapid than for the others. Though in the absence of accurate data on the oxygen di!usion, it is impossible to determine the freezing temperature for the excess oxygen accurately, one can present some estimates. As well known, the oxygen di!usion in some perovskites, e.g., in La CuO is possible even at 230}250 K   [283,284], and, at room temperature, the di!usion coe$cient D in them amounts to 10\ cm/s. The activation energy for the di!usion Q usually amounts to several tenths eV and can even exceed 1 eV. The larger the value of Q, the more rapidly D increases with temperature. If one takes, possibly, a conservative value of 0.5 eV for Q, then, with the sample size ¸ of 0.1 cm, the typical di!usion time (¸/D) is equal to 1 h at 450 K. As at higher temperatures, the di!usion coe$cient is still larger, one may expect that the "nal nonstoichiometry reached with the sample cooling will correspond to 400}600 K.

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This temperature should be compared with the typical phonon frequencies in manganites presented in Section 2.2. According to Ref. [145] (Section 2.2), they are rather high (one mode even has a frequency exceeding 1000 K). Hence, the classical approach is certainly not valid in the case considered, and the isotope e!ect does exist. To carry out the calculation with accounting for phonons, the phonon spectrum should be determined from the neutron studies not only for stoichiometric but also for nonstoichiometric crystals. Unfortunately, such data are absent for manganites at present. To illustrate the role of phonons, it is advisable to mention theoretical results [423] for Pd:H where such data are known, and where the same theoretical approach as for the manganites is applicable. The theoretical temperature dependence of the isotope e!ect in Pd:H di!er from the experimental data by less than 2% in a very wide temperature range from 100 to 1500 K. One may hope that without accounting for the phonons, this approach is valid for manganites, too, but inside a much more narrow temperature range including actual temperatures. In fact, the number of excess or lacking oxygen atoms in the crystal is determined from the condition of the equal chemical potentials for oxygen in the crystal and gas at the temperature of the sample preparation. But the chemical potential of the oxygen in the gas depends on the mass of the atom M via their translational and rotational motion energies, as well as via the dissociation energy of the O molecule. On the other hand, the chemical potential in the crystal depends on the  number of oxygen imperfections. (It must depend also on the oxygen mass via phonon frequencies, but, as already pointed out, this fact will not be taken into account.) This leads to the oxygen mass dependence of the numbers of oxygen vacancies and excess atoms. If the numbers of regular oxygen atoms and interstices are N and N , respectively, and the numbers of vacancies and excess atoms are n and n , respectively, then the free energy of the   crystal is given by F (¹)"F#n E !n E #¹N [ ln  #(1! ) ln(1! )]       -     # ¹N [ ln  #(1! ) ln(1! )]    

(8.5)

where F is the free energy of the crystal in the absence of imperfections, ¹ is the temperature of the  sample preparation, E is the work function for the oxygen, E the oxygen a$nity of the crystal,    "n /N ,  "n /N . Formally, the calculation will be done with an accuracy of  ln ;1. This   -   justi"es omitting from Eq. (8.5) terms describing the in#uence of the imperfections on the phonon spectrum as its contribution to the free energy is of order  (in reality, for the isotope e!ect at elevated temperatures such an accuracy is insu$cient). On the other hand, the free energy of the oxygen molecules in the gas surrounding the crystal is given by the expression [427]







n F (¹)"!nD#¹n ln !ln !1 ,  n  n (2S#1)J¹ M  1 , p"¹  , " 2
p <

 n "(N !N )/2,   

n"n #n /2!n /2 .   

(8.6)

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Here N is the total number of oxygen atoms in the system crystal#gas, n the number of  molecules in the gas, D the dissociation energy for the O molecule, S"1 the spin of the molecule,  M the atom mass, J"Ml/2 the inertia moment of a molecule with the interatomic distance l, p the pressure of the oxygen gas. Though the dissociation energy depends on the oxygen mass due to the zero-point oscillation energy, this dependence will not be taken into account as the corresponding dependence for the lattice phonons was omitted (as shown in Ref. [423] for Pd:H, these two factors almost compensate each other). The thermodynamical equilibrium numbers n and n are determined from the   condition of the minimal total free energy F #F with respect to them, which gives at n , n ;n      the densities of extra atoms






E !D/2 1 exp  .  "  ( ¹

(8.7)

As follows from Eq. (8.7) the relative di!erence in the excess atom content is  (O)! (O)  "!0.156 . (8.8)

"    (O)  Obviously, this di!erence is quite considerable. Nevertheless, as well known to chemists, an accurate determination of the excess or lacking oxygen is a very complicated problem. Probably, this explains the di!erence in conclusions made in Refs. [410,411]: in the former it is stated that the nonstoichiometry is the same for both the isotopes and in the latter that a di!erence does exist. The fact that a di!erence should exist is without question. The question is whether the experimental devices can determine this di!erence. It should only be noted that, in addition to the holes produced by the excess oxygen atoms, there are holes produced by divalent atoms entering the crystal. If the number of the latter exceeds greatly the number of the former, this mechanism of the isotope e!ect will be inactive. If the relative number of holes produced by the excess oxygen is considerable, the di!erence in the numbers of excess oxygen atoms is su$cient to explain the sign and magnitude of the ¹ isotope shift observed ! in the manganites (Section 8.1). One should still remember that the excess atoms create an internal pressure, and this mechanism may contribute to the isotope e!ect, too. 8.3. Conventional polarons in the ferromagnetic semiconductors Now we shall discuss the in#uence of the electron}phonon interaction on the state of the charge carriers in the CMR materials. As the ferromagnetic semiconductors are polar crystals, the charge carriers interact not only with the magnetic subsystem but also with the lattice polarization. In discussing the polaron problem, it is instructive to present numerical estimates of the polaron energy and compare it with the typical energy of the electron interaction with the magnetic subsystem. As it will be demonstrated here, the polaron energy is quite moderate and can hardly in#uence the properties of the ferromagnetic semiconductors drastically. First, it should be pointed out that the chemical binding in rare-earth and transition metal compounds is less polar than in NaCl and other alkaline-halide crystals. For this reason, one can expect that in the former the large polaron energy should be less than that in the latter. But even in

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NaCl it is quite moderate. In fact, the energy of the large polaron E and its e!ective mass m can * * be calculated using the long-wave Froehlich Hamiltonian. Then one "nds that they are given by the expressions ensuring a su$cient accuracy up to the constants of the electron}phonon coupling  as high as 10 (see, e.g., Ref. [108]):




 E "! , m "m 1# , * * 6 e(2m ) " , 1/H"1/ !1/ ,   2 H 

(8.9) (8.10)

where  and  are the static and high-frequency dielectric constants of the crystal, respectively,   the longitudinal optical phonon frequency, m the bare electron e!ective mass. For NaCl, the electron}phonon coupling constant  is 5.0 (see Ref. [108]) and "4.87;10 s\ (taken from Ref. [109]) so that E is only !0.15 eV, and m exceeds m only * * twice. As for the magnetic semiconductors, detailed information is available for EuS ( "11.1,  "5.0, "5.02;10 s\) and EuSe ( "9.4,  "5.0, "3.43;10 s\) (see     Ref. [110]). In Eu chalcogenides the electron e!ective mass is close to the true that. Then one obtains from Eqs. (8.9) and (8.10): "2, E "!0.065 eV for EuS, "2.5, E "!0.06 eV * * for EuSe. Hence, the polaron mass is only a few tens percent higher than that of the bare electron. As seen from these estimates, there is no sense in relating speci"c properties of CMR materials to the large polarons, as, according to Eqs. (8.9) and (8.10) under typical conditions, its e!ective mass is comparable or even close to the e!ective mass of the bare electron. The role of the large polarons in the metallic state of the crystal should be especially small, as then the charge carrier density is high, and the charge carriers screen their interaction with the lattice polarization. This strongly diminishes the polaron gain in the energy and its e!ective mass. As for small polarons, when in the zeroth approximation the electron is localized inside a unit cell [111], their mobility must be very small and exponentially increase with temperature. Hence, in principle, they could be responsible for the insulating state of the ferromagnetic semiconductors. But, "rst, it is necessary to prove that they can exist in such materials, in particular, in manganites. Meanwhile, an accurate calculation of the small polaron energy is hampered by the fact that one cannot use the Froehlich Hamiltonian for this aim, and the electron}phonon Hamiltonian valid everywhere inside the Brillouin zone is necessary. But the latter is not known at present. The reason for the fact that the Froehlich Hamiltonian is insu$cient for a treatment of the small polaron problem is as follows. Despite the common belief that the electron interaction with the lattice polarization should be long range, in reality, for the small polaron it is short range. Actually, the density of the interaction energy between the lattice dipole moment P and the electronproduced "eld D is (!PD/2). Using the well-known equations of the macroscopic electrodynamic and extracting contribution of the electronic polarization, one obtains for the small polaron energy (see [112,11]):



e e dr "! E "! 1 2Hb 8
H r

(8.11)

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where b is a quantity about the lattice constant a arising due to the necessity to cut o! the integral (8.11) at small r. As follows from obvious physical considerations, or estimates with accuracy of several tens % it is su$cient to put b"a. It is instructive to evaluate E from Eq. (8.11). Setting  "10,  "5, b"0.5 nm, one obtains 1   E "!0.14 eV. This estimate corresponds to EuS. As for the manganite LaMnO , one can use 1  the values  "5 and  "3.4 established in Ref. [60] by an analysis of the experimental data   [447]. Taking again b"0.5 nm, one obtains the same value !0.14 eV for the small polaron energy in the manganite, i.e., it is really moderate. One can "nd an upper bound for the small polaron energy using the dielectric constants of the strongly polar crystal NaCl presented above: it is 0.4 eV. The fact that the polarization is absent or reduced at the distances about the lattice constant must reduce E  additionally. 1 Some investigators use the Froehlich Hamiltonian everywhere inside the Brillouin zone and obtain the small polaron binding energy, E , an order of magnitude larger than the values 1 presented above. For example, for SrTiO a calculation [430] yields an E  value of 1.26 eV.  1 Meanwhile, using the experimental values  "5.0 and  "300 [108] and a value of b"0.5 nm,   from Eq. (8.11) one "nds E "0.27 eV. As was already mentioned, the cut-o! used in Eq. (8.11) 1 produces an uncertainty in the E value but it is small compared with the error produced by the 1 Froehlich Hamiltonian in the short-wave region where it can di!er very strongly from the true electron}phonon Hamiltonian which remains unknown at present. Meanwhile, it is easy to prove that the small polaron energies close to or exceeding 1 eV, are incompatible with the stability of the crystal. In fact, according to Eqs. (8.11a), (8.11b), 1 1 ( ,  "  #1/  

(8.11a)

E  " 1 . e/2b

(8.11b)

As the quantity e/2b is close to 1 eV for typical b values of about 0.5 nm, it follows from (8.11a) that for E "1 eV the dielectric constant  should be smaller than one in this case. To ensure not only 1  the stability condition  '1, but also realistic values of  '5, the small polaron energy cannot   exceed a few tenths eV. The typical polaronic energies are considerably less than the energy of interaction of the electrons with the magnetic subsystem. For the ferromagnetic semiconductors, the latter can be determined from the di!erence between the optical absorption edges at ¹"0 and ¹PR. As discussed in Section 3.1, in EuS the red shift amounts to 0.19 eV, i.e., it is three times larger than the large polaron energy. There are ferromagnetic semiconductors, for which the red shift is considerably larger, e.g., HgCr Se , where it amounts to 0.5 eV.   Certainly, it is impossible to expect small polaron formation in the simple exchange materials of the EuS type, as their wide band does not become more narrow at elevated temperatures. But the temperature band narrowing in the double-exchange systems discussed in Section 3.9, might give physical grounds for the hypothesis of the formation of small polarons at elevated temperatures, if they were nonexistent at ¹"0. A necessary condition for the small polaron formation is as follows: the small polaron energy is lower than the quantity (!=/2#E ). Hence, in principle, it is *

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possible that this inequality is not met at ¹"0 but is met with increasing temperature which implies a transition of a crystal into the insulating state. Perhaps, with this scenario, the temperature of the small polaron formation should not be related to ¹ , which contradicts experimental data on the manganites where these temperatures ! are very close for all samples investigated (cf. Sections 5.2 and 5.3). This already casts doubts upon its applicability to the manganites. A more detailed investigation of this problem can be carried out by estimation of the band width in the manganites. One can use the fact that the hole band states are constructed of hybridized O p-states and Mn d-states (Sections 2.2 and 5.7). But the former atomic states are of a large radius and, hence, the O (p) band width normally amounts to several eV which corresponds to the e!ective mass of the order of the true electron mass. Keeping in mind the hybridization with the highly localized d states, the real band width should be more narrow. Nevertheless, one can safely put it as equal to 1}2 eV. Another way to estimate the hole band width is as follows. One considers a separate acceptor in the manganites and takes its radius coinciding with the distance between the Ca(Sr) ion and the surrounding Mn ions which is of the order of the lattice constant a. On the other hand, this radius should be of the order of the Bohr radius a " /me. Putting a &a, =&10/ma and  "5,   one "nds =&5 eV. Keeping in mind the estimate for the E obtained above, one sees that the small polaron could 1 exist if the band width = is less than 0.3 eV. But, in reality, = should be several times or even an order of magnitude larger than this value which excludes the small polaron existence in the manganites at ¹"0. It is highly improbable that the temperature band narrowing will make it possible for the small polaron to exist at ¹PR if it is impossible at ¹"0. In fact, as shown in Section 3.9, for the spin S"2 this narrowing amounts only to 20%. Perhaps, even this estimate of the band narrowing is su$cient to state that if, experimentally, the small polarons do not exist at ¹"0, they can hardly exist at elevated temperatures. There is another reason to reject the role of the small polarons in the manganites. Some small-polaron supporters believe that the small polarons exist even at ¹"0 and ensure their metallic properties at heavy doping. Let us take their value of E  and see whether such a situation 1 is possible. A necessary condition for this is the inequality =
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polarons are incompatible with metallic properties. The same is true for other magnetic semiconductors. The bound polarons which are unionized donors and acceptors are quite di!erent. If the electron is trapped by the donor it polarizes its vicinity independent of the electron}phonon strength. Hence, participation of the bound polarons on the insulating side of the metal}insulator transition is beyond doubt. 8.4. The problem of the Jahn}Teller polaron in the manganites A speci"c feature of the manganites di!erentiating them from other magnetic semiconductors, is the presence of JT phonons in them. A special interest in them is caused by the publication [78] in which it was claimed that double exchange alone cannot explain the CMR and resistivity peak in the vicinity of ¹ , and one should invoke some other interaction to ensure these features. The "rst ! statement [78] coincides with the theoretical conclusions made by numerous authors earlier (e.g., [79,80]) and is supported by the fact that the ferromagnetic metals with a perfect lattice do not display the resistivity peak at ¹ , although one could expect it because of the critical scattering of ! the charge carriers [113]. Explanation of the fact that there is no resistivity peak in the vicinity of ¹ in perfect ferromagnetic conductors despite the charge carrier critical scattering lies in the ! suppression of the features of the critical scattering by the multiple scattering. A new feature of Ref. [78] is the fact that there, unlike in the preceding theoretical papers, the double exchange was considered. The mathematical procedure [78] cannot be considered su$ciently rigorous to give reliable predictions for this extremely di$cult problem, as the Kubo formula for the conductivity was deciphered inconsistently. But I would not like to reproduce my evaluation of this approach given in Ref. [134] here as, in my opinion, the qualitative conclusion of Ref. [78] is, nevertheless, right. Furthermore, it was con"rmed recently by Monte-Carlo simulations for the Kubo formula in the case of the double exchange system with classical spins [263]. Hence, a problem arises to "nd an additional mechanism which should combine with the exchange interaction to ensure the main features of the manganites. As such, the interaction of holes with the JT phonons was proposed in Ref. [78]. A real trouble consists in the fact that nobody knows an estimate of the JT electron}phonon coupling constant. One can only express some qualitative arguments. For example, if one rejects the double exchange picture and assumes the motion of the hole over the oxygen atoms, then this interaction can hardly be signi"cant. If one cannot calculate or measure this coupling constant, one can try to obtain information about the role of the JT phonons from comparison of the JT manganites with the non-JT ferromagnetic semiconductors. There are arguments that the charge carrier interaction with the JT phonons cannot determine the speci"c features of the manganites. Namely, of several dozens of ferromagnetic semiconductors with similar properties (the CMR and resistivity peak), only the manganites are JT systems (a list of the ferromagnetic semiconductors is presented in Ref. [11] and Section 2.1, their transport properties are described in Sections 3.1 and 4.1). Hence, the JT e!ect is not a necessary condition for the speci"c features mentioned. Keeping in mind that, apart from the JT e!ect, all other properties of weakly or moderately doped LaMnO manganites are qualitat ively the same as those of the rest of the ferromagnetic semiconductors, one can conclude that the mechanism of the CMR acting in the latter, should act in manganites, too.

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This non-JT mechanism, based on the joint action of the magnetization and ionized donors or acceptors (the magnetoimpurity mechanism [118}136]), is su$cient to ensure the resistivity peak and colossal magnetoresistance in the LaMnO -based manganites, too. If the CMR e!ects in  them had greatly exceeded those in other ferromagnetic semiconductors, possibly, it would have been necessary to "nd another mechanism of CMR which would enhance its value drastically. A candidate for such a mechanism could be the interaction of the holes with the JT phonons. But the real situation is quite the opposite. The resistivity peak in the vicinity of ¹ and CMR are much ! more pronounced in the non-JT systems than in the JT systems. For example, as discussed in Section 4.1, the height of the resistivity peak in the vicinity of ¹ for the non-JT EuO is up to 17 ! orders of magnitude higher than that in the JT manganites. For this reason, I believe that there is no need to invoke the JT phonons for qualitative explanation of transport properties of the ferromagnetic manganites though these phonons, certainly, in#uence their transport properties quantitatively. Supporters of the decisive role of the JT phonons believe that the numerous experimental evidences of changes in the lattice state accompanying the metal}insulator transition are the best proof of this. Certainly, there are numerous con"rmations of the JT distortions in the insulating phase (see below). But the point is that these distortions can be not the origin but only a consequence of the metal}insulator transition. For example, the ferromagnetic disordering is always accompanied by the magnetostriction. But this does not mean that the magnetostriction is a reason for the ferromagnetic disordering: it is simply its consequence. This point of view is shared by some of the experimentalists whose results will be described below. In addition, many investigators believe that, if a crystal displays a semiconducting behaviour this means that the charge carriers are in the free polaronic states. Meanwhile, the carriers can be trapped by impurity atoms or by their clusters like in conventional nondegenerate semiconductors. Certainly, the carrier trapping should be accompanied by a change in the lattice structure in the vicinity of the impurity. Then one can talk about a bound polaron which, in addition, should be a bound ferron simultaneously, as the trapped carrier enhances the local magnetization. The existence of such bound ferron}polarons is self-obvious, and the aim of the experimentalists is to establish their nature. Of experimental works, supporting the subordinated role of the JT e!ect, I must point out the following two papers. First of all, the giant isotope e!ect is pointed out as the main proof of the decisive role of the JT phonons. But, as was proved in Section 8.2, other explanations of this e!ect are possible. There are experimental indications [411] that the e!ect is related to the excess oxygen, in agreement with the theory put forth in Section 8.2. Second, the problem of the JT polarons was also discussed in Ref. [242] in investigating the electronic structure of La Di MnO with Di"Ca or Ba. For Ca-doped manganites, a change \V V  in the valence band structure was discovered across the metal}insulator transition. Their occupied states are extremely localized, with band widths of less than 200 meV. The screening parameter of this material changes dramatically across the coupled magnetic-metallic transition, and is a signature of changes in the local screening of the Mn ions. On the other hand, the bands of the Ba-doped manganites are highly itinerant. One band was observed to disperse by as much as 1.25 eV. Their bands do not shift, nor do their dispersions change, across the magnetic phase transition. Changes in the screening parameter were observed, but were found to be less dramatic than those for the Ca-doped system.

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It was concluded that the shifts of bands in Ca-doped samples are a consequence of large changes in the screening parameter, and the existence of the JT distortion is a consequence and not the cause of the change in the electronic states. In the examples presented below, the authors themselves believe in the polaron nature of the CMR. But they do not take into account the fact that other explanations for their data are possible. In Ref. [247] the Mn K-edge extended X-ray absorption "ne structure spectra were investigated on La Ca MnO across the metal}insulator transition. Results are interpreted in terms      of the JT polarons. There are large JT polarons in the metallic phase (¹(170 K). Above the metal}insulator transition they are replaced by the small JT polarons. Investigation of La Ca MnO by extended X-ray absorption "ne structure reveals increase in \V V  nonthermal (non-Debye) disorder at the metal}insulator transition temperature [244]. A substantial amount of residual disorder was observed in the metallic regime which gradually decreases with decreasing temperature. An unusual decrease of the spectral weight at the Fermi level with increasing temperature was observed in the paramagnetic region. The maximal excess of the Debye}Waller factor was found at the metal}insulator transition point. The e!ects observed are related to the formation of lattice polarons in the paramagnetic region. X-ray absorption "ne-structure studies of La Ca MnO were carried out [433] which made \V V  it possible to "nd the root-mean-square variation in the Mn}O bond length. Large-amplitude local distortions are found of a size comparable to that in the undoped material. As temperature is lowered through ¹ , the distortion drops to a value typical of the undistorted materials. ! Similar results were obtained in Ref. [144]. The atomic pair-density function of La Sr MnO (04x40.4) obtained by pulsed neutron di!raction indicates that their local \V V  atomic structure signi"cantly deviates from the average structure, and that the local JT distortion persists even when the crystallographic structure shows no distortion. In the paramagnetic insulating phase doped holes form one-site small polarons, represented by the local absence of JT distortion. The polarons become more extended at low temperatures, but local distortions are found even in the metallic phase. Apparently, the authors [144] interpret the word `polarona otherwise than one usually does. In Ref. [439] the local atomic structure of La Ca MnO (x"0.12, 0.21, and 0.25) has been \V V  studied using pair-distribution function analysis of neutron powder di!raction data. A change is seen in the local structure which can be correlated with the metal}insulator transition in the x"0.21 and 0.25 samples. This local structural change is modelled as an isotropic collapse of oxygen towards Mn of magnitude 0.012 nm occurring on one in four Mn sites. The authors argue that this is direct observation of lattice polaron formation associated with the metal}insulator transition in these materials. Similar results were obtained in Ref. [470] in investigations of La Ca MnO (04x4) by \V V  pulsed neutron di!raction. JT distortions are locally present in the insulating phase (allegedly, the small polarons). But they were observed even in the metallic phase up to x"0.35 suggesting that the charge distribution in the metallic phase just above the metal}insulator transition may not be spatially uniform. Possibly, phase separation occurs in the metallic phase. In La Sr MnO with x"0.12 and 0.15 measurements of the temperature dependence of the \V V  resistivity and thermopower under several hydrostatic pressures were performed [434]. For zero pressure, the resistivity tends to in"nity when ¹P0. At x"0.12 this behaviour survives a pressure of 12.6 kbar, but becomes replaced by a metallic behaviour at x"0.15. A conclusion is made that

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the transition from the polaronic to an itinerant electronic state takes place under pressure. The polaronic state is modelled as a mobile three-manganese cluster within a matrix of dynamic JT deformations at Mn> ions [434]. Meanwhile, the very fact that results of [434] resemble results obtained for non-JT semiconductors shows that the polaronic explanation is not the only one possible. According to Ref. [437], comparison of thermopower and resistivity measurements in (La, Ca)MnO above the metal}insulator transition indicates a transport mechanism not domin ated by spin disorder, but by small polaron formation, although it was impossible to distinguish between lattice polarons and magnetic polarons. The JT polarons are mentioned also in many other papers (e.g., [435,403,436]). But in Ref. [442] it is stated that the distortions (polarons) in the trigonal La Sr MnO are of non-JT      nature. In paper [257] another type of the lattice distortion was detected. The structural, magnetic and transport behaviour of orthorhombic, insulating La Sr MnO were studied     >W as a function of temperature from 20 to 300 K. Upon cooling from room temperature, the rapid development of a large breathing-mode distortion coincident with large positive and negative thermal expansions for the c and b axes, respectively. It is assumed, that the canted antiferromagnetic ordering is established at 220 K. Its onset is correlated with the relaxation of this distortion, which reaches values similar to those at room temperature when the magnetic ordering saturates. Results of optical investigations of the manganites [441,444] are interpreted in terms of the JT polarons. In particular, in the former an anomalous temperature variation of the 1.5 eV absorption band in R Sr MnO (R"La, La Lnd , Nd Sm , Nd Sm , Sm) was found. On                  approaching the metal}insulator transition from the high-temperature side, the band increases in intensity and then merges into a Drude-like ferromagnetic metallic state. This band is attributed to the optical excitations from the small polarons existing in the paramagnetic phase. Its disappearance in the ferromagnetic state is interpreted as a result of changing small polarons in large polarons or bare electrons. In the latter a large transfer of spectral weight from high energy to low energy in Nd Sr MnO was found as this material is cooled into the ferromagnetic phase, or as      the magnetic "eld is increased near the phase boundary. A new model, though still lacking a theoretical justi"cation, was proposed in Ref. [434]. It is assumed there that a polaronic phase separation into hole-rich and hole-poor regions takes place in the manganites instead of separation into di!erent magnetic phases with di!erent hole densities described in Section 6.3. But a change in the hole density should inevitably lead to a change in the magnetic properties of these domains as the indirect exchange should be di!erent in them. One paper [471] is devoted to the optical conduction spectra of La Sr MnO , in which,    according to Ref. [463], a polaron lattice should exist (see Section 6.6). On cooling, the phonon bending mode starts to split near a structural phase transition temperature. With further cooling, the strength of a midinfrared small polaron peak which is located at 0.4 eV increases. These temperature-dependent changes in the phonon spectra and phonon absorption peak were explained by a phase separation and a percolation transition. One can interpret the role of the JT phonons di!erently, but one cannot deny that they make the picture of the metal}insulator transition in the manganites highly unusual.

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9. Theories of transport and magnetotransport properties. Concluding remarks 9.1. State of theory for the transport and magnetic properties of manganites In analysing the state of the theory of CMR materials, one should separate them into the conventional ferromagnetic semiconductors, underdoped and moderately doped La-based quasicubic manganites, and complicated manganites (overdoped La-, Pr- and Nd-based quasicubic manganites, layered manganites and so on). The properties of the conventional ferromagnetic semiconductors are described quite satisfactorily by the magnetoimpurity theory [11,118}136] presented in Sections 4 and 5. On the other hand, there are no theories for the complicated manganites at present. Possibly, considerations presented in Ref. 7.4, could be a basis for their future theory, but so far the main attention of theorists is devoted to the La quasicubic nonoverdoped manganites. There are two main approaches to this problem. The "rst is based on the fact that not too heavily doped La manganites are, in essence, degenerate magnetic semiconductors and, hence, the magnetoimpurity theory is applied to manganites, too. This point of view is supported not only by the present author, but also by other authors though, evidently, they were not aware of papers [11,118}136]. In semiqualitative papers [139], it is assumed that spatial #uctuations in the Coulomb and spin-dependent potentials localize the e electrons in wave pockets that E are large on the scale of the Mn}Mn separation. Transport in the magnetically localized state involves zero-point assisted hopping or tunnelling of electrons from one weakly localized wave pocket to another. In Ref. [458] the metal}insulator transition is ascribed to a modi"cation of the s}d-exchange energy associated with the onset of the magnetic order at ¹ . Above ¹ , the e electrons are localized by the random spin-dependent potential, and ! ! E conduction is by variable-range hopping. Over the whole temperature range, the resistivity varies as ln / "¹ [1!(M/M )]/¹, where M/M is the reduced magnetization,   Q Q ¹ "18/N(E), 1/ the radius of the wave function and N(E) the density of available states.  The temperature and "eld dependence of the resistivity deduced from the molecular-"eld theory of the magnetization reproduces the experimental data over a wide range of temperatures and "elds. In Ref. [460] it is found that the sharp resistivity peak in the vicinity of ¹ is closely correlated to ! the residual resistivity  , suggesting that nonmagnetic randomness plays an important role in  determining their anomalous properties. Using the one-parameter scaling theory to study the electronic localization due to both the nonmagnetic randomness and the double exchange spin disorder, it was shown that the resistivity peak is caused by the Anderson metal}insulator transition and that  ' (a critical value) is a prerequisite for the occurrence of the   metal}insulator transition. ¹ as a function of  has also been calculated. !  In paper [459] of the same authors a localization model comprising spin disorder and nonmagnetic randomness is presented. Localization length of electrons as a function of magnetization is determined by means of the transfer matrix method. Including the Coulomb interaction between electrons, the variable-range hopping resistivity was calculated as a function of temperature and magnetic "eld. The resulting resistivity peak in the vicinity of ¹ and, in particular, the CMR ! magnitude are in good agreement with experimental measurements. A similar approach based on the Anderson localization was also applied in Refs. [48,461].

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A quite di!erent approach ignoring the presence of the charge impurities in a crystal and taking the polaronic e!ects as basic was adopted in another group of theoretical papers. This line of inquiry was initiated by the papers [78,432] and the former work had already been discussed in this article (Sections 1 and 8). This idea was elaborated in more detail in Ref. [432], where the total Hamiltonian of the system is taken as a sum of the s}d-Hamiltonian and the JT Hamiltonian in which the kinetic energy term of the JT phonons is omitted. The d-spins and the phonons are considered to be classical, and the s}d-exchange integral in"nitely large. A special version of the mean "eld approximation is used ensuring an exact solution for the dimensionality DPR. The procedure leads to an e!ective renormalization of the hopping integral making it magnetization dependent. In the limit ¹P0 the ground state is metallic and ferromagnetic. Results at "nite temperatures depend on the electron}phonon coupling constant . At small values of  and ¹(¹ , the ! resistivity  is small and has a ¹-independent piece due to the spin disorder and a ¹-linear piece due to the electron}phonon scattering. As ¹ decreases through ¹ ,  drops as the spin scattering is ! frozen out and the phonon contribution changes slightly. For larger , a gap opens in the electron spectral function at ¹'¹ and  rises as ¹ is lowered to ¹ . Below ¹ ,  drops sharply, as the ! ! ! gap closes and metallic behaviour is restored. Finally, at still stronger coupling, insulating behaviour occurs on both sides of the transition, although there is still a pronounced drop in  at ¹ . The same treatment evidences the existence of the CMR. ! These results were obtained for the electron density n"1 and agree well with experiment. But the resistivity calculated for nO1 is in poor agreement with data. But one should keep in mind that the approach of this paper does not take into account the on-site Coulomb repulsion. If it is strong, the crystal with n"1 should be insulating and not metallic (only one charge carrier can be located at each d-spin, and, hence, each of them is locked at his own atom and cannot go to another atom despite the fact that, formally, one orbital is empty at the latter). Now other papers of the same line will be discussed. In Ref. [462], superexchange and higher-order contributions to the exchange interaction energy between Mn layers in LaMnO are  studied using the linear-mu$n-tin-orbital method as generalized to treat noncollinear magnetic con"gurations. The degree of the internal lattice distortion f and the angle  between moments of the ferromagnetically ordered Mn layers are considered as simulation parameters. Both global and internal lattice distortions dramatically in#uence the character of the exchange interaction: global distortions associated with variations of the apical Mn}O bond promotes the double exchange contribution; the bending of the Mn}O bonds in the (a}b) plane suppresses the double exchange and promotes the antiferromagnetic superexchange contribution to the interlayer exchange energy. Overall, the character of exchange interactions is determined by superexchange contributions: in a hypothetical ferromagnetic phase the double exchange interactions mediated by itinerant electrons are present but are still rather small compared with the superexchange. In the antiferromagnetic phases this double exchange is negligible and non-Heisenberg terms are small compared with superexchange. The metal}insulator transition (in the sense of the band-gap opening) occurs in the range of parameters of magnetic and lattice distortion variations given by 51203 and f50.7. One should only keep in mind that the canted antiferromagnetic structure is unstable against the charge carrier density #uctuations (Section 6.1). In Ref. [60] a model of localized classical electrons coupled to lattice degrees of freedom and via Coulomb interaction to each other has been studied to gain insight into the charge and orbital

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ordering observed in lightly doped manganese perovskites. Expressions are obtained for the minimum energy and ionic displacements caused by given hole and electron orbital con"gurations. The expressions are analysed for several hole con"gurations including that experimentally observed in La Sr MnO (see Ref. [463] and Section 6.6). It was found that, although the    preferred charge and orbital ordering depend sensitively on parameters, there are ranges of the parameters in which the experimentally observed hole con"guration has the lowest energy. For these parameter values it is also found that the energy di!erences between di!erent hole con"gurations are of the order of the observed charge ordering transition temperature. The e!ects of additional strains are also studied. In many papers the JT polarons are studied theoretically. As an example, the paper [464] can be presented, according to which, moving electrons are accompanied by the JT phonon and spin-wave clouds may form composite polarons in ferromagnetic narrow-band manganites. The ground state and "nite-temperature properties of such composite polarons are studied. By using a variational method, it is shown that the energy of the system at zero temperature decreases with formation of the composite polaron and the composite polaron behaves as a JT phononic polaron. The energy spectrum of the composite polaron at "nite temperatures is found to be strongly renormalized by the temperature and magnetic "eld. It is suggested that the composite polaron contributes signi"cantly to the transport and thermodynamical properties in ferromagnetic narrow-band metallic manganese oxides. It should be pointed out that for the "rst time a composite latticemagnetic polaron was investigated in Ref. [469] but there an antiferromagnetic crystal was considered, and the charge carrier was assumed to interact mainly with the optical phonons. A di!erent line was chosen in Ref. [465]. The paramagnetic insulator to ferromagnetic metal transition in lanthanum manganites and the associated magnetoresistive phenomena are treated by considering the localization due to random hopping induced by slowly #uctuating spin con"gurations and electron}electron interactions. The transition temperature and its variation with composition is derived. The primary e!ect of the magnetic "eld on transport is to alter the localization length, an e!ect which is enhanced as the magnetic susceptibility increases. Expressions for the conductivity, its variation with magnetic "eld, and its connection with magnetic susceptibility in the paramagnetic phase are given. 9.2. Problems of physics of the manganites As seen from this review article, numerous problems in the physics of the CMR materials still remain unresolved. To begin with, the role of the JT phonons remains unclear. Though there is a wealth of experimental data con"rming the appearance of the JT lattice distortions in the insulating state, this does not mean that these distortions are the true origin of the metal}insulator transition. They can be only concomitant phenomena. It is necessary to "nd out which of the explanations of the giant isotope e!ect is adequate. It is di$cult to explain properties of the La-based manganites with x'0.3 in terms of the hole conductivity. In particular, one cannot explain the anomalous sign of the thermopower and Hall charge carrier densities exceeding the number of Mn atoms. Problems arise with magnetic properties of the La-based manganites: in the ferromagnetic state, on increase in temperature, they display the magnetization peak preceding its vanishing. No explanations of this phenomenon exist yet.

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Other problems are related to superstructures in the manganites. If one considers the divalent ions as randomly distributed, it is impossible to explain the charge ordering as the random electrostatic "elds of these ions should destroy this ordering. One may assume that, e.g., La Ca MnO is an alloy with ordered locations of the La and Ca ions for certain x, and this can \V V  explain the appearance of the superstructures. On the other hand, the existence of the stripes might evidence the dominating role of the lattice distortions in the formation of these stripes and other superstructures. In addition, there are contradictions between di!erent experimental groups concerning the charge ordering in weakly or moderate-doped La-based manganites. In my opinion, additional experiments are necessary to con"rm the existence of the polaron lattice in them. As for Pr- and Nd-based manganites, the "rst question which arises is the drastic di!erence in their properties as compared with the La-based manganites. Their doping does not lead to the appearance of conductivity, and they become highly conductive only under pressure or external magnetic "eld. The explanation of this fact proposed in the present paper (formation of the hole in the inner shell of the Pr or Nd ions instead of the d-shell of the Mn ions) needs veri"cation. Other problems with the Pr- or Nd-based manganites are the same as those for the La-based manganites. Certainly, other problems exist, in addition to those pointed out here. Thus, one may see that very much remains to be done in the physics of the CMR materials. Additional information about the properties of the manganites and magnetic semiconductors one can "nd in the papers [51,59,101,178,180,181,192,199,222,223,243,248,250}252,326,327,336,344,362,367,378,385}394,397, 406,438,445,466,468,481].

Acknowledgements This article was partly supported by Grant No 98-02-16148 of the Russian Foundation for Basic Investigations, Grant 97-1076 (072) of the Russian Ministry of Sciences, Grant HTECH LG 972942 of NATO, and Grant INTAS 9730253.

Note added in proof Papers published in 1999}2000. I would like to mention some papers which appeared after my review article was already written. As the manganite science develops very dynamically, this is necessary to do in order to avoid this review article being out of date still before its publication. 1. The isotope ewect and the electron}phonon coupling in the manganites. The experimental paper [490] is likely to solve the problem of the mechanism of the giant isotope e!ect completely. As is shown in it, the oxygen o!-stoichiometry mechanism described in Section 8.2, is realized at very slow cooling of the La Ca MnO samples and is su$ciently powerful: after extracting it the      electron}phonon contribution to the isotope e!ect turns out to be twice less than the value assumed in Ref. [410]. The experimental data on the oxygen o!-stoichiometry agree with the theoretical estimates of Refs. [421}423]. The quenching from temperatures of about 1500 K, at

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which the samples are prepared, to the room temperature leads to the disappearance of the oxygen o!-stoichiometry e!ect. The paper [490] strongly contrasts with the paper [491] in which an attempt of the experimental refutation of papers [421}423] was made. Strange as it might seem, no measurements of the oxygen excess were carried out in the paper [491], which makes conclusions of this paper highly doubtful. A further development of the theory [421}423] is proposed in the paper [492] in which the e!ect of the quenching is explained by the fact that the oxygen equilibrium is established at very high temperatures in this case. But in the classical limit the isotope e!ect should be absent. Using a realistic model of the manganite with excess oxygen entering the oxygen cage and with the La and Mn vacancies in equal numbers, I obtained the 25% di!erence in the equilibrium numbers of the O and O atoms which agrees with results [490]. Nevertheless, even after extracting the o!-stoichiometry contribution, the electron}phonon contribution to it remains rather considerable. On the other hand, as was shown in the paper [493], the electron}optical phonon coupling in the manganites is rather small as the chemical bond is not completely polar in them. The strong covalency of the manganites is con"rmed in Ref. [494]. To explain the considerable electron}phonon isotope e!ect without invoking a giant electron}phonon coupling, several theoretical papers were published recently [495,496]. As is shown experimentally in paper [497], in samples with the heavy oxygen the anharmonicity is very high which leads to the phonon softening. This in#uences the hopping integral. Hence, contrary to the opinion expressed in Ref. [410], a considerable isotope e!ect does not necessarily evidence a strong electron}phonon coupling in the manganites. 2. Problem of the small polaron. The idea that the electron}phonon coupling in the manganites is so strong that the charge carriers in the metallic manganites are the small polarons is propagated in Ref. [498]. But this paper is based on two basic errors. First, instead of an obvious necessary condition for the small polaron existence E 'z=/2, the condition E '=/[2(2z)] taken 1 1 from Ref. [112] is assumed to be met by the authors [498] (E is the small polaron energy, = 1 the band width, z the coordination number). But this condition is absurd, as it allows existence of the small polaron even in the cases when E is higher than the minimum bare electron energy 1 (!=/2). Second, in calculations of E [498] the long-wave Froehlich is used Hamiltonian whereas the 1 lattice polarization only over a distance comparable with the lattice constant is essential for the small polaron. The Froehlich Hamiltonian is invalid in such a situation. Really, it leads directly to Eqs. (8.11a), (8.11b). If one goes over to the wave vector q&1/r in these equations, then the integrand in the integral over q is q-independent. In other words, the contribution to E of the 1 q;1/a range where the Froehlich Hamiltonian is valid is very small, and its use makes it possible to estimate only the order of magnitude of E . 1 If one corrects both these errors, one should conclude that the small polarons are impossible in the manganites and other magnetic semiconductors as the small polaron energy is an order of magnitude smaller than the band widths in the manganites [493]. In the answer [499] to the Comment [493], the authors [499] avoided any arguing with Ref. [493] and limited themselves by calculations of E  for a large number of materials, using their erroneous procedure. But they 1 did not insist in Ref. [499] that the small polarons exist in the manganites and magnetic semiconductors, keeping silence on this subject.

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In the paper [500] the idea was advanced that the small polaron can be very mobile at ¹"0. Even if one puts aside the problem of the small polaron existence, this paper contains other errors. To begin with, according to it the Froehlich Hamiltonian in the site representation behaves asymptotically as 1/r and not as 1/r as should be the case according to all the textbooks on electrodynamics. Perhaps, as was already pointed out, this Hamiltonian, which allegedly leads to a high small polaron mobility, has nothing to do with the small polarons. But even one assumes for a moment that the condition for the small polaron existence is met, one sees that the ¹"0 small polaron band = width should be extremely narrow, so that its trapping  by the impurity potential should be inevitable. Really, due to the decisive role of the short-range polarization the well-known Holstein expression = " = exp(!E / ) is valid where is the  1 optical phonon frequency. This expression should be valid for the allegedly high mobile small polarons also for another reason: their screening of small polaron-phonon interaction. As = and are of order of 1 eV and 0.01, respectively, with accounting for E '=/2, = should be of order 1  of 10\* eV with ¸ amounting to several tens. Meanwhile, the impurity potential amounts to several tenths eV. As a result, the interaction of the small polarons with the electrostatic potential of the donor or acceptor impurities makes delocalized states of the small polarons impossible. More speci"cally, the Mott delocalization of the small polarons is impossible which is seen from Eq. (4.1) in which the electron e!ective mass, which enters a , must be replaced by the small polaron that J1/= . Then,  formally, the Mott density (4.1) should greatly exceed the density of the atoms in the lattice. Thus, one sees that, if the small polaron had existed at all, it would have been in the bound state, i.e., would not have been able to take part in the electronic transport. Even the bound polaron, thermally excited to the free state, cannot move according the band theory because of the Anderson localization inevitable in heavily doped manganites due to the electrostatic #uctuations of the potential. For the reasons pointed out above one should take with great reservation claims of some experimentalists that they observed the small polaron band conductivity, e.g., Ref. [501]. In it the temperature dependence of the conductivity in a metallic manganite at ¹P0 is explained uniquely by the small polaron band picture. But the impurity scattering which dominates in this system is temperature-dependent (see the book [11]) and makes it possible to explain the results of Ref. [501] without invoking the small polarons. 3. An insulating ferromagnetic phase in the manganites. Does it exist? This idea was expressed in several experimental papers, e.g., in the papers [502,503]. But as far as I know nobody proved that the allegedly insulating ferromagnetic state is saturated. For example, in the paper [504] where the observation of such a state is claimed, the ¹"0 magnetization values for La Sr MnO with \V V  x"0.1 and 0.14 di!er more than twice. In addition, changes in the resistivity interpreted as a transition to the insulating state are relatively small (according to Ref. [505], only about 5 times). One might assume that this state can be a nonuniform partially magnetized state, e.g., a mixture of the usual high-conductive ferromagnetic state and of a spin-glass-like state with all the charge carriers concentrated in the former. Nevertheless, one cannot discard the possibility of the charge ordering as an origin of the ferromagnetic insulating state pointed out in the paper [506] and in other publications cited here. In the paper [507] a theoretical foundation for the existence of the insulating ferromagnetic state in La Sr MnO is proposed. The ferromagnetic ordering is related to a very complicated   

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orbital ordering in the absence of the charge carriers. But the energy of such a ferromagnetic state is found to be very close to the energy of the antiferromagnetic ordering. Keeping in mind that a simpli"ed model was used in this paper, and that the accuracy of the numerical calculations is absolutely unclear, one cannot consider this paper to be a convincing proof for the existence of the insulating ferromagnetic state. Another explanation for them is presented in Section 6.6. It should be noted that the ferromagnetism of the bound ferrons proposed in Ref. [489] can be confused with the insulating ferromagnetic state. The ferromagnetic ordering of bound ferrons arises at a small overlap of their orbitals as a result of joint establishing the ferromagnetic ordering by a pair of the bound electrons in the region of their orbital overlapping. 4. The phase separation in the manganites. The ferro-antiferromagnetic phase separation in the manganites was an object of numerous investigations. Neutron studies of the bound ferrons in the La-manganites slightly doped by Sr which were carried out in Ref. [508] were interpreted as an evidence for the existence of the many-hole magnetic polarons (ferrons) with a plate-like shape. Their magnetization is not presented, and the authors do not insist on the conclusion made earlier in Refs. [270,271] for the Ca-doped manganites according to which the magnetic moment of the bound ferrons is very small (see Section 6.2). The existence of the ferromagnetic polarons in La Ca MnO with x(0.2 close to ¹ was observed in Ref. [509]. In the theoretical paper \V V  ! [510] calculations of the magnetic susceptibility above ¹ with the use of the classical limiting form ! of the double exchange Hamiltonian show that its temperature dependence does not follow the Curie}Weiss law. This fact is related to the formation of the magnetic polarons. The phase separation is observed also in heavily doped manganites. Of the phase separation phenomena in highly conductive states, a special attention is paid to the half-doped manganites of the La Ca MnO with x&0.5. As such a material is on the border between the insulating \V V  antiferromagnetic phase and the highly-conductive ferromagnetic state, the phase separation should manifest itself. According to Ref. [511] these crystals are highly conductive and ferromagnetic at 265 K. Below 160 K the insulating antiferromagnetic phase with the CE ordering appears which can be suppressed by a magnetic "eld. According to Ref. [512], the zero-"eld sample cooling results in the antiferromagnetic state. A magnetic "eld about 4 T drives most of the sample into a ferromagnetic phase. In the ferro-antiferromagnetic Pr Ca Sr MnO the microdomains of        the minority ferromagnetic phase are &20}30 nm in size [513]. Other half-doped manganites display metamagnetic properties [514]. But the phase separation was observed also far from the half-doping. In particular, in Ref. [564] scanning tunnelling spectroscopy applied to La Ca MnO with x&0.3 made it possible \V V  to discover coexistence of the insulating paramagnetic and metallic ferromagnetic regions above ¹ . Insulating regions are found to persist far below ¹ where they are intermixed with ! ! the ferromagnetic metallic regions. As the system as a whole is insulating above ¹ and metallic ! below ¹ , the metal-to-insulator transition is related to the percolation of the ferromagnetic ! phase. The phase separation is observed not only in the p-type manganites but also in the n-type manganites La Ca MnO with x exceeding 0.8 [515] and with x"0.65 where the coexistence \V V  of the ferromagnetic and antiferromagnetic phases was discovered [516]. An idea of a dynamic phase separation was advanced in an attempt to explain a giant random telegraph noise in this manganite with x"1/3 [517].

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In addition to the traditional ferromagnetic}antiferromagnetic phase separation in the manganites, new ideas about the possibility of the highly conductive ferromagnetic}insulating ferromagnetic state appeared [518]. It is assumed to exist right below ¹ and causes the resistivity peak in ! the vicinity of ¹ . ! In some manganites of a complicated composition which display a sharp transition from the antiferromagnetic state to the ferromagnetic state with changing composition a speci"c large-scale ferro}antiferromagnetic phase separation is observed close to the phase boundary. The size of the clusters is several orders of magnitude larger than for the electronic phase separation where the cluster size is limited by the Coulomb interaction. The appearance of the ferromagnetic}antiferromagnetic phase separation with large clusters was observed in investigating the isotope e!ect in a mixed La}Pr manganite [519]. Other experimental data are presented in Ref. [520] where a model was proposed to explain this phenomenon. Obviously, #uctuations in the exchange and hopping amplitudes are inevitable due to the randomness in local density of various ions. Such #uctuations should lead to the stabilization of one of the phases. Though these #uctuations change over the lattice constant scale, the ordering cannot follow such changes. For this reason the size of the clusters of a certain phase turns out to be large. Apparently, an electric-"eld induced transition from the insulating state to the highly conductive state which was observed in Pr Ca MnO is related to the phase separation [521]. It is \V V  accompanied with an increase in the magnetization. The phase separation is likely to take place also in the layered manganites. According to Ref. [522], it occurs in ferromagnetic La Sr Mn O with x"0.32 under pressure which \V >V   causes appearance of the antiferromagnetic phase. The situation is highly unusual as the antiferromagnetic phase is electron-rich and the ferromagnetic phase electron-poor. According to Refs. [523,524] in this material with x"0.3 ferromagnetic domains arise around the impurities under weak magnetic "elds. Concerning theoretical papers on the phase separation, one should point out that in the droplet model described in Section 6.3 an abrupt phase transition takes place with change in the geometry of the phase-separated state, on increase in temperature. In particular, the transition from the state with the dominating ferromagnetic phase to the completely ferromagnetic state is possible with increasing magnetization. This result of the paper [525] can explain experimental data on the magnetization [525] peak observed experimentally in some manganites [168,169] (e.g. Fig. 13). According to the paper just mentioned, the topological phase transitions from the insulating state to the highly conductive state are also possible. A layered geometry of the electronic phase separated state in the isotropic degenerate semiconductors was investigated in detail recently [526]. In this model the ferromagnetic and antiferromagnetic layers alternate. Unlike the droplet model discussed in Section 6.3, there is no electronic and ferromagnetic percolation, and the resistivity should be highly anisotropic. It is shown that in many important cases the energies of the layered and droplet structures are very close to each other. For this reason very weak forces can change the geometry of the phase separation with a radical change in the electric properties. Under certain conditions, the layered structure reduces to the ferromagnetic stripes inside the antiferromagnetic host. The role of the lattice polaron e!ects in the phase separation was investigated also in this paper. It is shown that these e!ects enhance the stability of the phase separated state and increase the size

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of the ferromagnetic and antiferromagnetic regions, as well as the density of the charge carriers in them. One should also mention other theoretical papers on the stripe structures [527}532]. In Ref. [565] an attempt was undertaken to construct a theory of the separation into the metallic ferromagnetic and charge-ordered antiferromagnetic phase. 5. The charge ordering. First, some experimental papers are devoted to the phase diagrams in which the charge ordering state is presented for L A MnO with L"Pr, Sr and A"Ca, Sr [533]; and \V V  for Nd Sr MnO [534]. In the former it is pointed out that for x close to 1 the spin glass state \V V  replaces the charge ordered antiferromagnetic state. But the statement that for small x the insulating ferromagnetic state should be realized seems doubtful. A special investigation [535] is devoted to Sm Ca MnO for x between 0.3 and 1. \V V  The charge ordering is observed in the layered manganites, e.g., in the single-layered La Sr MnO [536]. According to Ref. [537], it takes place also in the bilayered LaSr Mn O .         Degree of the charge and orbital ordering in the quasi-two-dimensional systems is lower than in the isotropic manganites. In other papers the charge ordering is investigated in more detail. According to Ref. [538], in La Ca MnO with x"0.667 a Wigner crystal is observed with the a lattice constant tripled \V V  and c constant doubled. Along the b axis the stacking of equal charges occurs. This structure competes with the `bi-stripea model. The conclusion about the charge ordering in this material is con"rmed also in the paper [539]. Possible models for this phenomenon are discussed in Ref. [540]. Melting of the charge ordering in La Ca MnO and Pr Ca MnO with x \V V  \V V  between 0.5 and 0.667 was investigated in Ref. [541]. In Pr Sr MnO with x " 0.4 or 0.5 the charge and orbital ordering establish at the same \V V  temperature, but the long-range orbital ordering is never achieved. Rather an orbital domain state is formed [542]. Uniaxial pressure raises the temperature of the charge and orbital ordering [543]. Theoretically, a close correlation between the stability of the e orbitals and the charge ordering E transition is established in Ref. [544]. As was already pointed out in Section 6.5, the nature of the charge ordering remains enigmatic, keeping in mind that, in addition to Mn> and Mn> ions which are able to be ordered, there are the doped divalent ions Ca>, Sr> which are located randomly. Their Coulomb potential tends to destroy the charge ordering. The present theoretical papers ignore this fact. For example, such an approach is used in Ref. [545] where a reentrant charge ordering transition is found in the Hubbard model with on-site and next site repulsion. In Ref. [546] an attempt was made to "nd the conditions of the stability of the charge and orbitally ordered half-doped manganites. It is assumed that there are quasi-one-dimensional zigzag ferromagnetic d   /d   orbitally-ordered chains in the a!b planes with opposite spin V \P W \P directions in the adjacent zigzag chains. An exactly solvable model is proposed which makes it possible to "nd the stability boundaries for the zigzag ordering. Certainly, this approach is interesting. But it cannot be considered as a theory of the CE antiferromagnetic structure. First, the model neglects the Coulomb "eld e!ects, in particular, those related to the randomly distributed impurities. Second, only conditions for the relative stability are found whereas other uniform structures can be more stable than the zigzag ones. Third, this theory is valid only for the half-doping whereas this structure is also observed at quite

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di!erent x values. Fourth, the possibility of the nonuniform states (phase-separated one and so on) is not investigated. A similar problem of zigzag chains was considered in Ref. [547]. It was shown there that for dopings less than 1/2 the phase separation into the ferromagnetic metallic and charge ordered insulating phases should occur. In Ref. [548] numerical calculations con"rm existence of insulating charge- and orbital ordered antiferromagnetic state of the CE type for x"1/2. The zigzag structure for x"1/2 was also investigated in Ref. [549] using the extended double exchange orbitally degenerate model. Meanwhile, in some cases the phase separation really occurs between the highly-conductive ferromagnetic phase and insulating charge ordered phase. For example, this happens in Nd Sr MnO with x"0.5 where at low temperatures the charge ordering of the a;2b;c \V V  takes place. At elevated temperatures the metallic ferromagnetic state coexists with the charge ordered microdomains [550]. Another point of view is expressed in Ref. [551]: the CE charge ordered state coexists with the antiferromagnetic metal. In Ref. [552] the phase separation is denied at all. Instead, a very complicated charge ordered structure above 150 K is assumed to be realized with the CE type antiferromagnetic ordering. Existence of the ferromagnetic microdomains in Pr Sr MnO with x " 0.5 is discovered \V V  also in Ref. [553]. In Ref. [554] the presence of the incommensurate structure in the paramagnetic insulating phase between 180 K and 260 K was established. 6. Some other theoretical problems. Some papers are devoted to the types of the magnetic ordering in the manganites. Though I am not an admirer of numerical calculations for the concrete materials, it is my duty to point out several papers of such a type. In the paper [555] the phase diagram of La Sr MnO as a function of x and the tetragonal distortion. The transitions FMPA\V V  AFPC-AFPG-AF should occur with increasing x. The A-type antiferromagnetic and C-type orbitally ordered states using the scenario of the cooperative Jahn}Teller (JT) phonon were investigated in Ref. [556]. Also the Pr Sr MnO was investigated in Ref. [566]. It is established \V V  there that the energy di!erence between the ferromagnetic and charge- and orbitally-ordered state is very small which is the origin of the metamagnetic properties of this compound. The double exchange via degenerate electronic d-orbitals was investigated in Ref. [557]. But, according to Ref. [558], the double exchange is insu$cient to explain the experimental data on the optical conductivity of La Sr MnO . \V V  The spin wave spectrum and attenuation were for the systems with the double exchange was investigated in Ref. [559] where, unlike results obtained earlier in the leading order in the inverse spin (Eq. (4.41)), the next order in this small parameter was considered. In Ref. [560] a magnon theory is constructed with account taken of the orbital degrees of freedom. The problem of the cooperative JT interaction in the manganites was investigated in Ref. [561]. In Ref. [562] phase transitions in the manganites are investigated with accounting for the orbital degrees of freedom. Orbital order}disorder transition is of the "rst order in a wide range of the hole densities. In Ref. [563] orbital polarons are predicted: the holes polarize the e states on the sites  neighbouring to the sites where holes are located. Such polarons should in#uence the metal-toinsulator transition.

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CONTENTS VOLUME 346 J. Manjavidze, A. Sissakian. Very high multiplicity hadron processes M. Chaichian, W.F. Chen, C. Montonen. New superconformal "eld theories in four dimensions and N"1 duality

1

89

R. Merkel. Force spectroscopy on single passive biomolecules and single biomolecular bonds

343

E.L. Nagaev. Colossal-magnetoresistance materials: manganites and conventional ferromagnetic semiconductors

387

PII: S0370-1573(01)00036-9

534

FORTHCOMING ISSUES K. Michielsen, H. De Raedt. Integral-geometry morphological image analysis E. Nielsen, D.V. Fedorov, A.S. Jensen, E. Garrido. The three-body problem with short-range interactions C. Song, Dense nuclear matter: Landau Fermi-liquid theory and chiral Lagrangian with scaling D.R. Grasso, H.R. Rubinstein. Magnetic "elds in the early Universe D. Atwood, S. Bar-Shalom, G. Eilam, A. Soni. CP violation in top physics V.K.B. Kota. Embedded random matrix ensembles for complexity and chaos in "nite interacting particle systems V.M. Loktev, R.M. Quick, S. Sharapov. Phase #uctuations and pseudogap phenomena T. Nakayama, K. Yakubo. The forced oscillator method: eigenvalue analysis and computing linear response functions S.O. Demokritov, B. Hillebrands, A.N. Slavin. Brillouin light scattering studies of con"ned spin waves: linear and nonlinear con"nement I.M. Dremin, J.W. Gray. Hadron multiplicities C.O. Wilke, C. Ronnewinkel, T. Martinez. Dynamic "tness landscapes in molecular evolution V. Narayanamurti, M. Kozhevnikov. BEEM imaging and spectroscopy of buried structures in semiconductors J. Richert, P. Wagner. Microscopic model approaches to fragmentation of nuclei and phase transitions in nuclear matter H.J. Drescher, M. Hladik, S. Ostapchenko, T. Pierog, K. Werner. Parton-based Gribov}Regge theory D.G. Yakovlev, A.D. Kaminker, O.Y. Gnedin, P. Haensel. Neutrino emission from neutron stars V.A. Zagrebnov, J.-B. Bru. The Bogoliubov model of weakly imperfect Bose-gas G.E. Volovik. Super#uid analogies of cosmological phenomena H. Heiselberg. Event-by-event physics in relativistic heavy-ion collisions C.N. Likos. E!ective interactions in soft condensed matter physics C. Ronning, E.P. Carlson, R.F. Davis. Ion implantation into gallium nitride

PII: S0370-1573(01)00037-0