Physics Reports vol.300

Physics Reports 300 (1998) 1—111

New developments in D-dimensional conformal quantum field theory E.S. Fradkin!,",*, M.Ya. Palchik# ! Theoretical Division, CERN, CH-1211 Geneva 23, Switzerland " Lebedev Physical Institute, Moscow 117924, Russia # Institute of Automation and Electrometry, Novosibirsk 630090, Russia Received October 1997; editor: A. Schwimmer

Contents 1. Introduction 1.1. Preliminary remarks 1.2. Conformal symmetry in D dimensions 1.3. Conformal partners and amputation conditions 1.4. Conformal partial wave expansions in Minkowski space 1.5. Conformal partial wave expansions of Euclidean Green functions 2. Conformally invariant solution of the Ward identities 2.1. Definition of conserved currents and energy—momentum tensor in Euclidean conformal field theory 2.2. The Green functions of the current 2.3. The solution of the Ward identities for the Green functions of irreducible conformal current 2.4. Green functions of the energy—momentum tensor and conditions of absence of gravitational interaction 2.5. The algorithm of solution of Ward identities in D-dimensional space 2.6. Conformal Ward identities in twodimensional field theory

4 4 6 9 12 13 23

23 25

30

33 38 41

2.7. Ward identities for the propagators of irreducible fields jI and ¹I k kl 3. Hilbert space of conformal field theory in D dimensions 3.1. Model-independent assumptions. Secondary fields 3.2. Green functions of secondary fields 3.3. Dynamical sector of the Hilbert space 3.4. Null states of dynamical sector 4. Examples of exactly solvable models in D-dimensional space 4.1. A model of a scalar field 4.2. A model in the space of even dimension D54 defined by two generations of secondary fields 4.3. Primary and secondary fields 4.4. A model of two scalar fields in D-dimensional space 4.5. Two-dimensional conformal models 5. Conformal invariance in gauge theories 5.1. Inclusion of the Gauge interactions 5.2. Conformal transformations of the gauge fields 5.3. Invariance of the generating functional of a gauge field in a non-Abelian case

* Corresponding author. e-mail: [email protected]

0370-1573/98/$19.00 Copyright ( 1998 Elsevier Science B.V. All rights reserved PII S 0 3 7 0 - 1 5 7 3 ( 9 7 ) 0 0 0 8 5 - 9

45 47 48 53 54 58 62 62

65 70 73 75 78 78 81 82

NEW DEVELOPMENTS IN D-DIMENSIONAL CONFORMAL QUANTUM FIELD THEORY

E.S. FRADKIN!,", M.Ya. PALCHIK# ! Theoretical Division, CERN, CH-1211 Geneva 23, Switzerland " Lebedev Physical Institute, Moscow 117924, Russia #Institute of Automation and Electrometry, Novosibirsk 630090, Russia

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111 5.4. Conformal QED in D"4 5.5. Linear conformal gravity in D"4 6. Concluding remarks 6.1. Conformal models of non-gauge fields 6.2. The propagators of the current and the energy—momentum tensor for even D54 6.3. The propagators of irreducible components of the current and the energy—momentum tensor

88 95 98 98

6.4. The equivalence conditions for higher Green functions of the current and the energy—momentum tensor Appendix References

3

106 108 109

101

103

Abstract A review of recent developments in conformal quantum field theory in D-dimensional space is presented. The conformally invariant solution of the Ward identities is studied. We demonstrate the existence of D-dimensional analogues of primary and secondary fields, the central charge, and the null vectors. The Hilbert space is shown to possess a specific model-independent structure defined by the 1 (D#1)(D#2)-dimensional symmetry and the Ward identities. In 2 particular, there exists a sector H of the Hilbert space related to an infinite family of “secondary” fields which are generated by the currents and the energy-momentum tensor. The general solution of the Ward identities in D'2 defining the sector H necessarily includes the contribution of the gauge fields. We derive the conditions which single out the conformal theories of a direct (non-gauge) interaction. We examine the class of models satisfying these conditions. It is shown that the Green functions of the current and the energy-momentum tensor in these models are uniquely determined by the Ward identities for any D52. The anomalous Ward identities containing contributions of c-number and operator analogues of the central charge, are discussed. Closed sets of expressions for the Green functions of secondary fields are obtained in D-dimensional space. A family of exactly solvable conformal models in D52 is constructed. Each model is defined by the requirement of vanishing of a certain field Q , s"1,22. The fields Q are constructed as definite superpositions of secondary fields. After s s that, one requires each field Q to be primary. The latter is possible for specific values of scale dimensions of fundamental s fields (a D-dimensional analogue of the Kac formula). The states Q D0T are analogous to null vectors. One can derive s closed sets of differential equations for higher Green functions in each of the models. These results are demonstrated on examples of several exactly solvable models in D'2. The approach developed here is based on the finite-dimensional conformal symmetry for any D52. However the family of models under consideration does have the structure identical to that of two-dimensional conformal theories. This analogy is discussed in detail. It is shown that when D"2, the above family coincides with the well-known family of models based on infinite-dimensional conformal symmetry. The analysis of this phenomenon indicates the possibility of existence of D-dimensional analogue of the Virasoro algebra. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 11.25.Hf Keywords: High-energy physics; Conformal field theory

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E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

1. Introduction 1.1. Preliminary remarks The present review describes a family of exactly solvable models of quantum field theory in D-dimensional space. The studies of conformal models by the authors was initiated as far back as 1970s [1,2] and approached its essential developments in the recent works [3—6]. The conformal symmetry is usually treated as a non-perturbative effect pertinent to the description of physical phenomena in the asymptotic region. Any dimensional parameters entering the bare Hamiltonian are supposed to be immaterial in this region. Regardless of the specific character of physical phenomena under discussion (the critical behaviour of statistical systems, late turbulence or interactions of elementary particles, etc.), the systems in this region may exhibit certain fundamental properties which are independent on the structure of the initial Hamiltonian. Adopting this hypothesis, the following formulation of the problem becomes natural: one aims to find a set of axiomatic principles which would completely fix the effective interaction in the asymptotic region. It is straightforward to expect the result to be independent of the choice of initial Hamiltonian. Formulated in such manner, the problem has been repeatedly discussed both by physicists and mathematicians. As the most popular candidates to the role of the above principles, the field algebra hypothesis and the hypothesis of the scale and conformal symmetry were considered (see, for example, [7—11]). In the works [1,2,12] we have studied additional restrictions which follow from generalized Ward—Fradkin—Takakhashi identities [13,64], provided that the latter are completed by the requirement of conformal symmetry. As a result, a closed set of conditions defining a family of exactly solvable models in D-dimensional space was formulated. Each model is determined [3,6] by a certain condition on the states generated by the currents and the energy—momentum tensor. These states are analogous to the null-vectors of two-dimensional conformal theories. This fact was first demonstrated on the example of the Thirring model in late seventies, see Refs. [2,14]. A complete and detailed solution of several aspects of this approach was given in the book [15] as well as in the works [5,6]. When concerning exactly solvable models, we imply the following feature: one may derive a closed set of differential equations for any higher Green function. In addition one can deduce algebraic equations for scale dimensions of fields and massless parameters analogous to the central charge. An important approach to obtain exactly solvable models in two-dimensional space was developed in the works [16—19]. However, the case of D"2 is exceptional since the conformal group of two-dimensional space is infinite dimensional. The method developed in Refs. [16—19] does not allow a straightforward generalization until the proper D-dimensional analogue of the Virasoro algebra is found. What is essential in our approach is that the 1(D#1)(D#2)-dimen2 sional conformal symmetry is assumed in the space of any dimension D. For the case of D"2 this symmetry is 6-parametric. Its generators ¸ , ¸ , ¸M , ¸M 0 B 0 B compose the algebra of the group SL(2, R)]SL(2, R)

(1.1)

(1.2)

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

5

which is the maximal finite-dimensional subalgebra of the Virasoro algebra. However, all the conformal models found in the works [16—19] may be derived in the framework of our approach (see Section 4). Note that the form of the Ward identities for the Green functions of the energy—momentum tensor is prescribed by the symmetry under the group (1.2). The Ward identities include the complete information on the commutators between the components of the energy—momentum tensor, as well as between these components and other fields. Thus, in the approach based on the symmetry (1.2) and the Ward identities for D"2, the infinite-dimensional symmetry arises as an auxiliary result which is not presumed initially: adopting the formulation described here to the case of D"2 implies one to act as if an infinite-dimensional symmetry were unknown. A similar situation is likely to be realized in the case D'2 as well. The structure of Ddimensional models discussed below is analogous to the structure of two-dimensional conformal models. We shall demonstrate the existence of certain analogues of primary and secondary fields, the central charge and the null vectors. Moreover, we shall show that each of the models possesses an infinite set of self-consistency conditions which provides the analogy with conformal symmetry of two-dimensional models. Hence, one can expect that a definite analogue of the Virasoro algebra should exist in D-dimensional space, the realization of the very analogue being observed in the above models. The principal difference between the conformal models in D53 and two-dimensional ones consists in the following. The general solution of conformal Ward identities necessarily includes the contribution of gravitational interaction, while the conformal solution of the Ward identities for conserved currents includes the contribution of gauge interactions (see Section 2). Since in the present article we restrict ourselves solely to the discussion of the models of direct (non-gauge) interaction, the problem which arises is: how to eliminate gauge interactions from the general solution of conformally invariant Ward identities? This problem is solved in Section 2. To facilitate the understanding of these results, Section 5 contains the discussion of gauge interactions (gravitational included) in four-dimensional space. In its course, all the main results of Section 2 are reproduced on a slightly different standpoint. Moreover, a possibility of introduction of gauge interactions into a family of models under consideration is also discussed in Section 5. For the sake of illustration, the solution of several non-trivial models is presented in Section 4. Guided by methodical considerations we restricted the discussion to the study of models which demanded technically simple calculations. The latter models were meant to serve as illustration of the principal ideas as well as the features of calculation technique developed in the paper. Besides that, the class of models considered is the one that allows for the most evident analogy between the structure of our approach and that of known two-dimensional theories. The most physically interesting models require that the operator analogue of the central charge should be introduced (the field PD~2(x) of scale dimension d "D!2, see Section 3). In particular, P we believe that the three-dimensional Ising model is contained in this larger class. Such models demand a certain modification of the technique. Though up to that time the principal investigations had been already completed, we decided to refrain from surveying these new results in the present review due to the two reasons: firstly, it appeared to be methodically inexpedient — since the modified formulation mentioned above clouded the analogy with two-dimensional theories. Secondly, the analysis of the more complex models called for cumbersome calculations which were rather appropriate for the separate publications.

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E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

The present review primarily deals with the Euclidean formulation of quantum field theory [20,64]. Discussions concerning the structure of the Hilbert space of conformal theory will be based on the formulation in Minkowski space with the metric g "(!#2#). kl A number of questions involved is sketched rather briefly: the more detailed discussion of those is found in Ref. [15]. Especially the latter concerns the solution of conformal Ward identities for D52, and the analysis of two-dimensional conformal models in the framework of the approach developed in this paper. The work [15] includes also a consistent review of the main principles of conformal quantum field theory in D52, see also [2,21,22]. 1.2. Conformal symmetry in D dimensions The conformal group of D-dimensional space is a 1(D#1)(D#2)-parametric group. It includes 2 1D(D#1) transformations of the Poincare´ group, as well as scale transformations and special 2 conformal transformations j jxk, xk P a xk P

xk#akx2 , 1#2ax#a2x2

(1.3)

where j'0, ak is an arbitrary D-dimensional vector. Rather than handling special conformal transformations it proves helpful to use the transformation of conformal inversion R R Rx"xk/x2. xk P

(1.4)

A special conformal transformation may be derived as a sequence of three transformations: conformal inversion, translation by the vector bk"ak/a2, conformal inversion again. Each conformal field in D-dimensional space is characterized by its scale dimension which determines its transformation properties under scale and special conformal transformations. Let u(x) be a scalar conformal field of scale dimension d, j jdu(jx) , u(x) P R (x2)~du(Rx) . u(x) P

(1.5)

The condition of invariance of the theory under transformations (1.5) leads to the following coordinate dependence of two- and three-point invariant Green functions (see reviews [2,21,22] and references therein): Su(x )u(x )T&(x2 )~d , (1.6) 1 2 12 Su(x )u(x )u(x )T"g(x2 )~(d1`d2~d3)@2 (x2 )~(d1~d2`d3)@2 (x2 )~(d2`d3~d1)@2 , (1.7) 1 2 3 12 13 23 where u , u , u are scalar conformal fields with dimensions d , d , d , and g is an arbitrary 1 2 3 1 2 3 constant. Scale dimensions are the most fundamental parameters of conformal theory. Its values determine the character of physical phenomena described by the conformal field theory. For example, the quantities measured experimentally in the statistical systems near the 2nd-order phase transition point are the critical indices. They govern the singular behaviour of the correlators of a free energy, magnetization, etc. in the critical region. The above parameters are expressed through the scale

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

7

dimensions of fundamental fields with the help of known relations [11]. Thus, scale dimensions are experimentally observable in statistical systems. Viewing the conformal symmetry as an asymptotic symmetry of the theory of elementary particles, the scale dimensions control the powers of growth and decay of the effective potential in the asymptotic region. One can show (see Ref. [2,22,23] and references therein) that the positivity axiom restricts the possible values of scale dimension to the interval (in the case of the scalar field) d5D/2!1 .

(1.8)

The lower bound in this inequality coincides with the dimension of a free massless field d

"D/2!1 , (1.9) #!/ and is called the canonical dimension. The latter is fixed by the quantization rules. In the presence of interaction we face up with renormalized fields u (x)"z1@2 u (x) . (1.10) 3%/ 2 #!/ The renormalization constant is dimensional; hence, the dimension of the renormalized field takes an anomalous value dOd . Under the shifting of the regularization #!/ z P0 , (1.11) 2 provided that no negative-norm states are present in the theory. One has, in accordance with Eq. (1.8): d'd . #!/ This expresses the well-known fact that the singularity of the propagator in the presence of interaction is more sharp than in the case of free theory (assuming that the states with negative norms are absent). Besides Eqs. (1.6) and (1.7), the conformal symmetry conditions lead to several strong restrictions on the higher Green functions. In particular, one has for the Green function of four fields (see, for example, Refs. [2,22] and references therein): Su (x )u (x )u (x )u (x )T"(x2 )~d1(x2 )~d2F (m, g) , (1.12) 1 1 1 2 2 3 2 4 12 34 12 where F is an arbitrary function of variables 12 m"x2 x2 /x2 x2 , g"x2 x2 /x2 x2 (1.13) 12 34 13 24 12 34 14 23 known as harmonic ratios. Thus the problem of the construction of the exact solution of conformal theory consists in the evaluation of scale dimensions of the fields and “coupling constants” such as the parameter g in Eq. (1.7), as well as functions of harmonic ratios entering the conformally invariant representations of the type (1.12), which define higher Green functions. Consider the operator product expansion of the pair of scalar fields u (x )u (x ) at neighbouring 1 1 2 2 points. The tensor fields U (x) together with all their derivatives contribute to this expansion. Each k field U is a traceless symmetric tensor k U (x)"Ulkk(x)"Ulk12 k(x) , (1.14) k s k k

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E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

where l is the scale dimension and s is the tensor rank. Under the coordinate transformations (1.3) k k and (1.4) the fields (1.14) transform as follows: j jlUl R (x2)~lg (x) g (x)Ul (jx), Ul 12 s(x) P (Rx) , Ul 12 s(x) P k12ks k k k1l1 2 ksks l12ls k k

(1.15)

where g (x)"d !2 x x /x2 . (1.16) kl kl k l Denote the mth order derivative of the field U (x) as U(m)(x). Thus the operator expansion of the k k product u u is written as 1 2 u (x)u (0)"+ A(m)(x)U(m)(0) , (1.17) 1 2 k k k,m where A(m)(x)&(x2)~(d1`d2~lk~m)@2. For example, the contribution of the scalar field Ul (x) and its k 0 derivatives into Eq. (1.17) is [U ]"g (x2)~(d1`d2~l)@2 MU (0)#a x ­ U(0)#a x x ­ ­ U(0)#a x2hU(0)#2N , (1.18) 0 0 0 1 k k 2 k l k l 3 where g and a are some constants. 0 i The operator equality (1.17), like any other relation between Euclidean fields, should be understood as a symbolic notation representing the asymptotic expansion of Euclidean Green functions Su (x)u (0)U (x )2U (x )TD K + A(m)(x)SU(m)(0)U (x )2U (x )T , (1.19) 1 2 1 1 n n x?0 k k 1 1 n n k,m where U 2U are arbitrary conformal fields. The words “neighbouring points” mean that 1 n x2@x2 , 14k, r4n . kr one can show [24] (see also Refs. [2,22] and references therein) that the invariance under the transformations (1.15) fixes all the coefficients a in the expansion series of the type (1.18) uniquely m and allows one to take an explicit sum of all the terms with derivatives of the field U (x). The result k has the form

P

I 2lk (x, 0Dy)U [U ]"g dy Qd11d2 (y) , k k k12ksk k ksk

(1.20)

I 2lk is a known function, which expression where g is the coupling constant of the field U and Qd11d2 k ks k k k can be found in Refs. [2,10,21,22] and in the references therein. In what follows the operator expansions of the type (1.17) will be written in a symbolic form

u (x)u (0)"+ [U ] , 1 2 k k where [U ] is given by expression (1.20). k

(1.21)

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

9

One can show [2,10] that the problem of calculation of conformally invariant Green functions (1.12) is equivalent to the problem of calculation of coupling constants and dimensions g ,l (1.22) k k of the fields U . The examples of such calculations (in the approximation of skeleton graphs) may be k found in Refs. [12,15,58], see also references therein. 1.3. Conformal partners and amputation conditions Let us introduce a unified notation p for the pair of quantum numbers l, s. Each field (1.14) transforms by irreducible representation of the conformal group [2,15,21,22] of the Euclidean space. Denote this representation as ¹ . The fields U "U k and U "U m, transforming by p k p m p non-equivalent representations ¹ k and ¹ m, are orthogonal: p p SU k(x )U m(x )T"0 if l Ol or s Os . (1.23) p 1 p 2 k m k m This may be checked directly using the transformation laws (1.15). The pair of fields U (x)"Ul (x) , U J (x)"UlI (x) , lI "D!l , p s p s where p"(l, s) , pJ "(lI , s)"(D!l, s)

(1.24)

(1.25)

is an exception. These fields transform by equivalent representations [2,21,22] ¹ &¹ J . p p One has for the fields (1.24) (see Ref. [15] for the details):

(1.26)

SU (x )U J (x )T"symMd 1 12d s sNd(x !x ) , (1.27) p 1 p 2 kl kl 1 2 where the notation “sym” stands for the symmetrization and subtraction of traces performed in each group of indices k 2k , l 2l . Below we call fields (1.24) the conformal partners. 1 s 1 s Equivalence condition (1.26) is expressed by the following operator equality (see Ref. [15] for more details):

P

Ul 12 s(x)" dy Dl 12 s 12 s(x!y)UD~l (y) , k k k kl l l12ls

(1.28)

which will be used below in a shorthand notation

P

U (x)" dy D (y)U J (y) . p p p The intertwining operator D coincides [15] with the conformally invariant propagator p D (x )"D 12 s 12 s(x )"SUl 12 s(x )Ul 12 s(x )T p 12 k k l l 12 k k 1 l l 2 "(2p)~D@2n(p)(1 x2 )~l symMg 1 1(x )2g s s(x )N , 2 12 k l 12 k l 12

(1.29)

(1.30)

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E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

where g (x ) is given by expression (1.16), the notation “sym” has the same sense as in (1.27), and kl 12 n(p) is the normalization factor. Consider the invariant function of three fields: Gd1d2(x x x )"Su (x )u (x )U (x )T"g Cd1d2(x x x ) , (1.31) p 1 2 3 1 1 2 2 p 3 p p 1 2 3 where g is the coupling constant, while the invariant function Cd1d2 is calculated using Eqs. (1.5) p p and (1.15) and has the following form [2,21,22]: Cd1d2(x x x )"(2p)~D@2N(pd d )(1x2 )~(d1`d2~l`s)@2 p 1 2 3 1 2 2 12 ](1x2 )~(d1~d2`l~s)@2 (1x2 )~(d2~d1`l~s)@2 jx312 s(x x ) , 2 13 2 23 k k 1 2 where

(1.32)

(x ) (x ) (1.33) jx312 s(x x )"jx31(x x )2jx3s (x x )!traces, jx3(x x )" 13 k! 23 k , k k 1 2 k 1 2 k 1 2 k 1 2 x2 x2 13 23 and N(pd d ) is the normalization factor. 1 2 When applied to the Green functions (1.30) and (1.31), the operator Eq. (1.28) means that under the suitable choice of normalization factor the following relations should hold: D~1(x )"D J (x ) , p 12 p 12

(1.34)

GdJ 1d2(x x x )" dx Gd1d2(x x x)D~1(x!x ) , 1 2 3 p p 1 2 p 3

(1.35)

(x !x)Gd1d2(xx x ) , dI "D!d , GdI 1d2(x x x )" dx G~1 d1 1 p 1 2 3 p 2 3 1 1

(1.36)

P P

and in the x argument by analogy. Here G 1 stands for the propagator of the field u : 2 d 1 C(d) 1 ~d x2 G (x )"Su (x )u (x )T"(2p)~D@2 . d 12 d 1 d 2 C(D/2!d) 2 12

A B

(1.37)

In what follows, conditions (1.34)—(1.36) will be called the amputation conditions. As one can easily check by a direct calculation, the amputation conditions will hold provided one chooses [2,15] C(D!l!1) C(l#s) , n(p)" C(D/2!l) C(D!l#s!1)

(1.38)

N(pd d ) 1 2

G

C

"

C

A

A

B A

B A B A

B A B A

l#d #d #s!D d #d !l#s l!d #d #s l#d !d #s 1 2 2 1 2 1 2 C 1 C C 2 2 2 2

B A

B

2D!l!d !d #s D#l!d !d #s D!l!d #d #s D!l#d !d #s 1 2 1 2 1 2 1 2 C C C 2 2 2 2

H

1@2

B

(1.39)

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

11

for the normalization factors in Eqs. (1.30) and (1.32). The calculations use conducted using relations (A.1)—(A.3). One can show that the Green function Gd1d2 is expressed through the function p Qd1d2p which defines the operator product expansions of fields, (see Eq. (1.20)): p Gd1d2(x x x )"!g p 1 2 3 p sin p(l!D#s) 2

G

P

H

] Qd1d2p(x x Dx )! dx D (x !x)Qd1d2pJ (x x Dx) . 1 2 3 p 3 1 2

(1.40)

The amputation condition (1.35) is the natural consequence of such representation. Fields (1.24) have the same status. The latter means that when the field U is present in the p theory, the field U J must also exist, their Green functions being related by Eq. (1.29): p

P

SU J (x)U (x )2U (x )T" dy D~1(x!y)SU (y)U (x )2U (x )T , p 1 1 n n p p 1 1 n n

(1.41)

where U 2U are any conformal fields. One can prove [2,15] that the skeleton and bootstrap 1 n equations are invariant under the change: lPD!l or d PD!d . (1.42) 1,2 1,2 This symmetry is also present in the system of the renormalized Schwinger—Dyson equations [25,64] as long as its conformally invariant solution [1,2] is concerned. The transition from the fields U , u p d to the conformal partners U J , u I is equivalent to the transition from the formulation of the conformal p d theory in terms of Green functions to the formulation in terms of vertices and propagators. However, the symmetry (1.42) is broken in the Ward identities. The latter selects one of the fields (1.24) to be a physical field. It is essential that the current j and the energy- -momentum tensor k ¹ belong to the class of fields with canonical dimensions kl l "D!2#s (1.43) s and do not literally satisfy the equivalence conditions (1.29) (or (1.41)), since the corresponding representations of the conformal group are undecomposable [21,26]. The same is also true for the conformal partners of the fields U with the integer dimensions s lI "D!l "2!s . (1.44) s s The Euclidean fields j ,¹ k kl have the canonical dimensions

(1.45)

l "D!1, l "D . (1.46) j T Its conformal partners are the electromagnetic potential A and the traceless part of the metric k tensor h (in linear conformal gravity). The Euclidean fields kl A ,h (1.47) k kl have the dimensions (1.44) l "1, l "0 . A h

(1.48)

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E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

In the next section we discuss the analogue of equivalence relations (1.29) between the fields (1.45) and (1.47). When D"4, these relations coincide with the Maxwell equations and the equations of linear conformal gravity. Let us remind that any relations between Euclidean fields are understood as if they were placed inside the averaging symbols. In what follows we examine the Euclidean Green functions Gj (xx 2x )"S j (x)u(x )2u`(x ) T , k 1 2n k 1 2n

(1.49)

GT (xx 2x )"S¹ (x)u (x )2u (x ) T , kl 1 m kl 1 1 m m

(1.50)

GA(xx 2x )"SA (x)u(x )2u`(x ) T , k 1 2n k 1 2n

(1.51)

Gh (xx 2x )"Sh (x)u (x )2u (x ) T , kl 1 m kl 1 1 m m

(1.52)

where U is a charged field of dimension d, and u 2u are neutral fields of dimensions d 2d , 1 m 1 m respectively. The Green functions (1.49) and (1.50) satisfy the conformally invariant Ward identities

C C

D

n 2n ­xGj (xx 2x )"! + d(x!x )! + d(x!x ) G(x 2x ) , k k 1 2n k k 1 2n k/1 k/n`1 m d n ­xGT (xx 2x )"! + d(x!x )­xk!­x + k d(x!x ) G(x 2x ) , k k k 1 m l kl 1 m k D k/1 k/1 where

D

G(x 2x )"Su(x )2u`(x )T, G(x 2x )"Su (x )2u (x )T . 1 2n 1 2n 1 m 1 1 m m

(1.53)

(1.54)

(1.55)

1.4. Conformal partial wave expansions in Minkowski space Consider the fields u , u and U k in Minkowski space. Let Q` be the positive frequency 1 2 p p representations of the conformal group of Minkowski space. On can show that for any values of quantum numbers p "(l , s ) different from Eqs. (1.43) and (1.44), the states k k k U k(x)D0TLM`k p p

(1.56)

form a basis of the space M`k of the representation Q`, see Ref. [2] and, for more details, [15]. The p p Wightman functions ¼ (x )"S0Du(x )u(x )D0T, ¼ k(x )"S0DU k(x )U k(x )D0T d 12 1 2 p 12 p 1 p 2

(1.57)

represent invariant scalar products of the states (1.56). The spaces M`k of different irreducible p representations are mutually orthogonal S0DU k(x )U m(x )D0T"0 if p Op . p 1 p 2 k m All the states (1.56) form the basis in the Hilbert space of the conformal theory.

(1.58)

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Consider the states u (x )u (x )D0T. These states transform by an infinite direct sum of irredu2 2 1 1 cible representations and can be decomposed through an infinite set of states (1.56) (see Refs. [2,15] and references therein):

P

S0Du (x )u (x )"+ A dx Qd1d2pJ k (x x Dx ) S0DU k(x ) , 1 1 2 2 k 3 1 2 3 p 3 k

(1.59)

where A are unknown constants, Qd1d2pJ k is the same function as in Eq. (1.20) but written for the case k of Minkowski space. Decomposition (1.59) is called the vacuum operator decomposition [22]. It is analogous to the operator decomposition of the Euclidean fields product, see Eq. (1.21). Taking into account Eq. (1.58), we have from (1.59)

P

S0Du (x )u (x )U k(x )D0T"A dx Qd1d2pJ k (x x Dx)¼ k(x!x ) . 1 1 2 2 p 3 k 1 2 p 3

(1.60)

The l.h.s. of this equality is a conformally invariant Wightman function. Any higher conformally invariant Wightman function, for example ¼(xyx 2x )"S0Du (x)u (y)U (x )2U (x )D0T , 1 m 1 2 1 1 m m

(1.61)

where U 2U are arbitrary conformal fields in Minkowski space, may be represented as an 1 m invariant scalar product of the states u (y)u (x)D0T and U (x )2U (x )D0T . 2 1 1 1 m m

(1.62)

Using Eq. (1.59) we find the conformal partial wave expansion of the ¼ightman functions (see Refs. [2,15,22] and references therein):

P

¼(xyx 2x )"+ A dz Qd1d2pJ k (xyDz)S0DU (z)U (x )2U (x )D0T 1 m k k 1 1 m m k

(1.63)

1.5. Conformal partial wave expansions of Euclidean green functions The Euclidean conformal partial wave expansion may be derived from the expansions (1.63) by analytic continuation into Euclidean coordinates [2,15], see also Refs. [10,21—24] and references therein. Let us present several results. The notations and normalizations are chosen as in Refs. [2,15], see Eqs. (1.38) and (1.39). Let u, s be neutral scalar fields. The Green function

(1.64)

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E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

may be written in the form of expansion which represents an Euclidean analogue of expansion (1.63)

(1.65)

where

P

1 D@2`*= dl k(p) , +" + 2pi p s D@2~*= 1 C(D/2#s) k(p)" 2s(2p)~D@2 n(p)n(pJ ) , 4 C(s#1)

(1.66) (1.67)

n(p) is the normalizing factor (1.38), C is the conformally invariant function (1.32) satisfying the p amputation conditions (1.35) and (1.36), which is normalized by the condition

(1.68) The dots on internal lines mean the amputation of argument, see Eq. (1.36); d-symbols in the r.h.s. are defined by the condition + d f (p@)"f (p) . pp{ p{ Condition (1.68) holds for functions (1.31) when g "1, see Refs. [2,15]. The kernels of partial wave p expansions G are determined, on account of Eq. (1.68), by the relation p

(1.69)

On the internal lines d-functions are placed. To a contribution of each field U into expansion (1.21) corresponds a pole of the kernel G at the m p point p"p "(l , s ) , m m m res G (xx 2x ) , G(n) U (xx 2x )"SU (x)u(x )2u(x )T"K m 1 n m 1 n m p 1 n p/pm where K is known constant [2,21,22]. m

(1.70) (1.71)

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15

Note that due to amputation conditions, the kernels G are invariant under the change pPpJ p G (xx 2x )"G J (xx 2x ) . p 1 n p 1 n Each term [U ] in the operator expansion (1.21) may be derived by the shift of the integration m contour in Eq. (1.65) to the right up to the intersection of the pole in the point (1.70). Under this, one should use the symmetry of the integration contour in Eq. (1.71), the integrand under the changes pPpJ and the representations of functions C in the form of Eq. (1.40). As the result, we p have (see Refs. [2,21,22] for more details)

A

BP

2pg m p G(xyx 2x )K + ! dz Qd DpJ m (xyDz)G(n) U (zx 2x ) , m 1 n 1 n sin p(l !D/2#s ) m m lm,sm where g m is the coupling constant: p Su(x )s(x )U (x )T"g mC m(x x x ) . 1 2 m 3 p p 1 2 3 The contribution of each pole into partial wave expansion may be written as [2,21]

(1.72)

(1.73)

The r.h.s. of this formula contains the Green functions G "Su(x )s(x )U (x )T , G(n) U "SU (x)u(x )2u(x )T , m 1 n m 1 2 m 3 with the inner line corresponding to an inverse propagator D~1 of the U field m m D (x )"SU (x )U (x )T . m 12 m 1 m 2 The partial wave expansion of the function including four fields

may be written as

(1.74)

where the function o(p) is defined by G (xx x )"o(p)C (xx x ) . p 3 4 p 3 4

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The function o(p) satisfies the equation o(p)"o(pJ ) and has the poles in the points (1.70). Eq. (1.73) in this case can be rewritten in the form

(1.75)

Consider the partial wave expansion of Green functions which include a single vector or tensor field besides the scalar ones. To proceed we need explicit formulas for the other pair of conformally invariant three-point functions: SUl uUl{T and SUl uUl{ T . s k s kl The Green function which includes the tensor Ul 12 s and the conformal vector Ul{, has the form: k k k SUl 12 s(x )u (x )Ul{(x )T k k 1 d 2 k 3

G

" Ajx3(x x )jx112 s(x x ) k 1 2 k k 2 3

C

DH

1 s (1.76) + g k(x )jx112 L k2 s (x x )!traces Dll{(x x x ) , 1 1 2 3 kk 13 k k k 2 3 x2 13 k/1 where A and B are unknown constants, kL means the omission of the index, k Dll{(x x x )"(x2 )~(l`d~l{~s`1)@2 (x2 )~(l`l{~d~s~1)@2 (x2 )~(l{`d~l`s~1)@2 . (1.77) 1 1 2 3 12 13 23 To derive Eq. (1.76), Eqs. (1.5) and (1.15) were used. The expression in curly braces provides an example of typical tensor structures. It does not depend on field dimensions. We will use the notation MA, BN for such structures: #B

(1.78) SUl 12 s(x )u(x )Ul{(x )T"MA, BNDll{(x x x ). 2 k 3 1 1 2 3 k k 1 As another example, consider the Green function that includes the tensors Ul 12 s and Ul{ . It kl k k contains three terms SUl 12 s(x )u(x )Ul{ (x )T k k 1 2 kl 3

G

C

A

B

s 1 " Ajx3 (x x )jx112 s(x x )#B jx3(x x ) + g k(x )jx112 L k2 s(x x )!traces lk 13 k k k 2 3 kl 1 2 k k 2 3 k 1 2 x2 13 k/1

D

#(k%l)!trace in k, l #C

C

DH

1 s + g k(x )g r(x )jx112 L k2 L r2 s (x x )!traces kk 13 lk 13 k k k k 2 3 (x2 )2 13 k,r/1

Dll{(x x x ) , 2 1 2 3

(1.79)

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

17

where A, B, C are unknown constants, Dll{(x x x )"(x2 )~(l`d~l{~s`2)@2 (x2 )~(l`l{~d~s~2)@2 (x2 )~(l{`d~l`s~2)@2 . 2 1 2 3 12 13 23

(1.80)

It is convenient to introduce the notation analogous to Eq. (1.78) SUl 12 s(x )u(x )Ul{ (x )T"MA, B, CNDll{ (x x x ) . 2 kl 3 2 1 2 3 k k 1

(1.81)

Consider partial wave expansion of the Green function

(1.82)

including the vector field Ul1 of dimension l . In this case the partial wave expansion is made up by k 1 a pair of terms of the type (1.65), since there are two independent invariant functions Cp "Cp 12 s (x x x )"MA(1), A(1)NDldl1 (x x x ) , 1 2 1 1 2 3 1 1k,k k 1 2 3

(1.83)

Cp "Cp 12 s (x x x )"MA(2), A(2)NDldl1 (x x x ) . 1 2 2 2k,k k 1 2 3 1 1 2 3 It is useful to choose coefficients A(i) , i"1, 2 in such a way as to make the functions (1.83) 1,2 mutually orthogonal

(1.84)

and normalized by the condition (1.68).

(1.85)

Then the partial wave expansion for the Green function (1.82) may be written in the form [2,15]

(1.86)

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E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

where the kernels Gp are determined, according to Eqs. (1.84) and (1.85), by the equations 1,2

(1.87)

Each pole of kernels Gp corresponds to a field which contributes to operator product expansion 1,2 U (x)u(0). k ¹here are two mutually orthogonal sets of fields MU N and MU N 1m 2m{

(1.88)

contributing to this operator expansion U (x)u(0)"+ [U ]#+ [U ] . k 1,m 2,m{ m m{

(1.89)

The Green functions of the fields U and U are determined by the equations 1,m 2,m{ SU (x)u(x )2u(x )T"K res Gp (xx 2x ) , i,m 1 n i,m i 1 n p/pm

i"1,2 .

(1.90)

The orthogonality of the fields U and U may be proved by an analysis of the expansion for 1,m 2,m{ 4-point Green function:

(1.91)

Let us put its expansion in the form

(1.92)

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

where the functions o

19

(p) are determined by the equations 1,2

(1.93)

Note that the diagonality condition for the expansion (1.92) together with the orthonormality conditions (1.84) and (1.85) fix the coefficients A(i) in Eq. (1.83) up to SO(2) transformation. Thus, 1,2 the choice of combinations of independent invariant structures in Eq. (1.83) is possible only on the base of analysis of the Green function (1.91). The meaning of this result [2,3,15] may be easily understood, considering the Wightman function S0Du(x ) Ul1(x ) Ul1(x ) u(x )D0T , 2 k 1 l 3 4 which represents an invariant scalar product of the states

(1.94)

Ul1(x )u(x )D0T , Ul1(x )u(x )D0T . (1.95) k 1 2 l 3 4 Apparently, all the fields, contributing into the vacuum operator expansion of these states, are represented by the sets (1.88). If among these fields there is a pair of fields UI and UI with 1,m 2,m{ identical dimensions and tensor ranks, one can always choose such their combinations U "aUI #bUI and U "cUI #dUI , that the states 1,m 1,m 2,m{ 2,m{ 1,m 2,m{ U (x )D0T and U (x )D0T (1.96) 1,m 1 2,m{ 2 will be orthogonal to each other: S0DU (x ) U (x )D0T"0 . 1,m 1 2,m{ 2 Thus the states (1.95) represent a direct sum of mutually orthogonal subspaces:

(1.97)

H =H (1.98) 1 2 which belong to the total Hilbert space of the theory. The states U (x)D0T span the basis of 1,m H space 1 U (x)D0TLH for all m . (1.99) 1,m 1 In analogy, U (x)D0TLH for all m@ . (1.100) 2,m{ 2 Correspondingly, in Euclidean version of the operator product expansion, the orthogonality of H and H manifests itself in the diagonal form of the partial wave expansion (1.92), with the 1 2 Euclidean Green functions SU U T being equal to zero 1,m 2,m{ SU (x )U (x )T"0 1,m 1 2,m{ 2 even when quantum numbers of fields U and U coincide. 1,m 2,m{

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E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

By analogy, one can consider partial wave expansions for the Green functions which include the tensor field Ul1 . It contains three different terms, since there exist three independent invariant kl functions (see Eq. (1.81)): Cp"Cp 12 s (x x x )"MA(i), A(i), A(i)NDldl1 (x x x ) , i i kl,k k 1 2 3 1 2 3 2 1 2 3

(1.101)

where i"1, 2, 3. It is useful to choose the coefficients A(i) from the condition 1,2,3

(1.102)

(1.103)

This does not yet fix the A(i) coefficients uniquely. Let us demand the expansion for the Green 1,2,3 function

(1.104)

to be diagonal [3,15]

(1.105)

This fixes the choice of orthonormal basis of functions Cp, i"1, 2, 3, up to a SO(3) transformation. i To make the choice of these functions unique, additional physical arguments are necessary, see Section 2.

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

21

In the general case, there exist three mutually orthogonal sets of fields [3,15] MU N , MU N , MU N , 1,m 2,r 3,k contributing to the operator product expansion Ul1 u: kl

(1.106)

Ul1 (x)u(0)"+ [U ]#+ [U ]#+ [U ] . kl 1,m 2,r 3,k m r k The orthogonality conditions

(1.107)

SU (x ) U (x )T"SU (x )U (x )T"SU (x )U (x )T"0 1,m 1 2,r 2 1,m 1 3,k 2 2,r 1 3,k 2 hold even when quantum number of two fields from different sets coincide. The states

(1.108)

Ul1 (x )u(x )D0T (1.109) kl 1 2 span the sector of a Hilbert space which can be represented as a direct sum of three orthogonal subspaces: H =H =H , (1.110) 1 2 3 U (x)D0TLH for all m, U (x)D0TLH for all r, U (x)D0TLH for all k. (1.111) 1,m 1 2,r 2 3,k 3 The expansion of higher Green functions could be discussed no sooner than all the three independent sets of functions (1.101) are found. Let us stress that the requirement of diagonality of expansion (1.105) should be treated as one of the conditions that fix the form of these functions. Resultantly, we have [3,15]:

(1.112)

It is essential that the expansions of Green functions SUl1 uU U 2U T (1.113) kl 1 2 n with any number of fields U 2U of any tensor structure, also have the form (1.112). The latter is 1 n quite apparent from the above analysis. Besides that, one can show that this property is the consequence [2,12,15] of exact solution to renormalized Schwinger—Dyson system. Finally, let us consider several consequences of operator product expansions for the Green function Su(x )s(x )U (x )U (x )T , 1 2 1 3 2 4

(1.114)

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E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

where U , U are any fields. Its asymptotic behaviour in each of the two regions 1 2 1. x P0, i.e. x2 @x2 , x2 , 12 12 13 14 2. x P0, i.e. x2 @x2 , x2 , 34 34 13 14 is determined by one of the operator produce expansions u(x )s(x )D 12 K+ [U ] , m 1 2 x ?0 m U (x )U (x )D 34 K+ [U@ ] . 1 3 2 4 x ?0 k k On the other side, two sets of fields

(1.115) (1.116)

(1.117) (1.118)

MU N and MU@ N (1.119) r k do not generally coincide. Each field U(x), contributing to the asymptotic region (1.115) should also contribute to the asymptotics (1.116). The latter is caused by the orthogonality property of conformal fields SU (x)U@ (x@)TO0 only if l "l@ , s "s@ . r k r k r k Hence two sets of fields (1.119) should have an intersection MUI N which consists of the fields UI and UI @ with the same quantum numbers: r k (1.120) MUI N3MU N, MUI N3MU@ N . r k If such an intersection is empty, then the Green function (1.114) is zero. The states of a Hilbert space u(x )s(x )D0T , U (x )U (x )D0T 1 2 1 3 2 4 are orthogonal: S0Du(x )s(x )U (x )U (x )D0T"0 . 1 2 1 3 2 4 This statement can be formulated in terms of partial wave expansions in the following manner: the poles of kernel o(p) of the expansion

(1.121) correspond to the fields UI that belong to an intersection of sets (1.119). The most interesting consequences of this statement may be obtained from the analysis of Green functions for the energy—momentum tensor or the current. Examining the Green functions S¹ u¹ u`T or S j uj u`T , kl op k l

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

23

one can find the operator product expansions ¹ (x )u(x ) or j (x )u(x ) . (1.122) kl 1 2 k 1 2 Suppose that the fields contributing to these expansions are found. Then one can expect that the latter contribute, as well, to the operator product expansion of fundamental fields u(x )s(x ) , 1 2 since the Green functions

(1.123)

Susu`j T and Susu`¹ T k kl are non-zero. Thus from the analysis of the conformally invariant solution of ¼ard identities one can find some of the operator contributions into operator product expansions of fundamental fields. This is done in Sections 2—4. 2. Conformally invariant solution of the Ward identities 2.1. Definition of conserved currents and energy—momentum tensor in Euclidean conformal field theory As already mentioned in Section 1.3 to the current, energy—momentum tensor and their conformal partners, undecomposable representations of the conformal group correspond. The latter belong to the so-called representations in exceptional points (1.43), (1.44), studied in Refs. [21,26]. Denote any undecomposable representation as Q. Let M be a space of such representation. The characteristic property of these representations is that in the space M there exists a subspace which is invariant under the action of group transformations: M LM . (2.1) 0 In the case of current and energy—momentum tensor the subspace M consists of transversal 0 Euclidean conformal fields j53, ¹53 k kl ­ j53(x)"0 , ­ ¹53 (x)"0 . (2.2) k k l kl Indeed, consider the transformation laws of the fields j (x) and ¹ (x) with respect to the conformal k kl inversion (see Eqs. (1.15), (1.43)): R j@ (x)" 1 j (x)P g (x) j (Rx) , k k l (x2)D~1 kl

(2.3)

R ¹@ (x)" 1 g (x) g (x) ¹ (Rx) . ¹ (x)P kl kl lp op (x2)D ko

(2.4)

As may be easily demonstrated by a direct check, after transformations (2.3) and (2.4), the fields j53(x) , ¹53 (x) k kl beget the fields j@53, ¹@53 that are also transversal: k kl R ­x j@53(x)"0 , ­x¹ (x)"0P R ­x¹@53 (x)"0 . ­x j (x)"0P k k k k l kl l kl

(2.5)

(2.6)

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An analogous result is valid for the conformal partners (1.47). The invariant subspace M in this 0 case consists of the longitudinal fields A-0/'(x)"­ u (x) , k k 0

(2.7)

2 h-0/'(x)"­ h (x)#­ h (x)! d ­ h (x) , kl k l l k D kl j j

(2.8)

where u (x) is the scalar fields of dimension d "0, h (x) is the conformal vector of dimension 0 A l d "!1. As may be easily checked, under the action of conformal inversion h R A@ (x)"(1/x2)g (x) A (Rx) , A (x)P k k kl l

(2.9)

R h@ (x)"g (x) g (x) h (Rx) h (x)P kl kl ko lp op

(2.10)

the conformal fields (2.7) and (2.8) transform to longitudinal fields A@-0/'(x)"­x u@ (x) , k k 0

(2.11)

2 h@-0/'(x)"­x h@ (x)#­ h@ (x)! d ­ h@ (x), kl k l l k D kl j j

(2.12)

where u@ (x)"u (Rx) , h@ (x)"x2g (x)h (Rx) . 0 0 k kl l To prove the results (2.6), (2.11) and (2.12), we have employed the relation

(2.13)

­x"(1/x2) g (x)­Rx . (2.14) k kl l More detailed calculations may be found in Ref. [15]. Thus in the case of fields A , h the invariant k kl subspace M consists of longitudinal fields (2.7) and (2.8), respectively. 0 As it is known from the group theory, any undecomposable representation may be coupled to a pair of irreducible representations: Q , QI . (2.15) 0 The representation Q acts in the invariant subspace M , while the other, in the quotient space 0 0 MI "M/M . (2.16) 0 Correspondingly, one could consider a pair of different conformal fields, the first being transformed by the representation Q , and the second, by the representation QI . Hence, there exist two types of 0 conformal fields with canonical dimensions (1.43), and two types of conformal partners with dimensions (1.44). This means that Euclidean conformal field theory comprises two types of currents and two types of potentials: j53, jI and k k

AI , A-0/' . k k

(2.17)

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

25

Each of these fields transforms by an irreducible representation of the conformal group. Similarly, one has a pair of types of irreducible conformal fields for each of ¹ and h : kl kl ¹53 , ¹I and hI , h-0/' . (2.18) kl kl kl kl Furthermore, one can consider the fields j , ¹ or A , h which transform by direct sums of k kl k kl irreducible representations (2.15): Q = QI . (2.19) 0 It is important that for the fields with canonical dimension the conditions of equivalence of representations are different from Eq. (1.29). The latter are substituted by the conditions of partial equivalence [21,26] which manifest themselves independently among the representations of the type Q , and among the representations of the type QI . This means that the relations of type (1.29) 0 hold independently both in the transversal and in longitudinal sectors. From the above arguments it follows that different possible definitions of the conserved currents and the energy—momentum tensor in the conformal theory may be given. A more comprehensive discussion of these definitions is given in Ref. [15] (see also the next sections of the present article). 2.2. The Green functions of the current For simplicity we shall discuss the case of an Abelian theory in the space of dimension D53. All the results obtained below are easily generalized to the case of non-Abelian theories. Introduce the notation Q for an irreducible representation defined by the transformation law j (2.3) and acting in the space M . According to Eq. (2.15), there exists a pair of irreducible j representations QI and Q53 j j corresponding to irreducible conformal fields

(2.20)

jI (x) and j53(x) , ­x j53(x)"0 . k k k k The space of irreducible representations (2.20) will be denoted as

(2.21)

MI "M /M53, M53LM . (2.22) j j j j j By analogy, Q denotes an undecomposable representation defined by the transformation law (2.9) A and acting in the space M . For a pair of irreducible representations A Q-0/' and QI (2.23) A A the pair of irreducible conformal fields A-0/'(x)"­ u (x) , AI (x) k k 0 k correspond. The notation

(2.24)

M-0/'LM , M I "M /M-0/' A A A A A will stand for the spaces of irreducible representations (2.23).

(2.25)

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The representations (2.20) and (2.23) are pairwise equivalent [21,26], see also Refs. [27—29]: QI &Q-0/' , Q53&QI . j A j A The equivalence conditions are expressed by the operator relations

P

A-0/'(x)& dy D-0/' (x!y) jI (y) , kl l k

P

j53(x)" dy D53 (x!y)AI (y) , k kl l

(2.26)

(2.27)

where the kernels D-0/' and D53 coincide with the invariant propagators kl kl D-0/'(x )"SA-0/'(x )A-0/'(x )T"C g (x ) (1x2 )~1"C ­x1­x1 ln x2 , (2.28) kl 12 k 1 l 2 A kl 12 2 12 A k l 12 C is the normalization constant, and A (2.29) D53 (x )"S j53(x ) j53(x )T , ­x1D53 (x )"0 . l kl 12 kl 12 k 1 l 2 The explicit form of the expression of the conformally invariant propagator D53 depends on the kl dimension of the space. Introduce the regularized conformal fields je and Ae with the dimensions k k le"D!1#e , le "D!le"1!e . (2.30) j A j The regularized propagator of the current 1 g (x ) , Dj e (x )"S je (x ) je (x )T"CI kl 12 k 1 l 2 j (1x2 )D~1`e kl 12 2 12 where CI is the normalization constant is divergent when eP0 for even values of D: j 1 Dj e (x )&(d h!­ ­ ) #O(1) . kl 12 kl k l (1x2 )D~2`e 2 12 The factor (x2 )~D`2~e is singular for even D54. Thus we set 12 1 (d h!­ ­ ) , D-odd , kl k l (x2 )D~2 D53 (x )" 12 kl 12 lim eDje (x )&(d h!­ ­ )h(D~4)@2 d(x ), D-even . e?0 kl 12 kl k l 12 Here we utilized the relation [30]

G

(2.31)

(2.32)

(2.33)

K

1 pD@2 4~k K! hkd(x) . (2.34) e C(D/2#k) C(k#1) e?0 Thus, for even D54 one has a pair of invariant functions Dje and D53 . We shall show in Section 6 kl kl that these functions are related to a pair of irreducible representations QI and Q53 and define the j j propagators of irreducible fields jI and j53 . In the case of odd D the situation is analogous, and will k k not be discussed here. Note that when D"4, Eq. (2.27) coincide with the equations for electromagnetic field in the a-gauge: 1 (x2)D@2`k`e

P

(d h!­ ­ ) AI (x)"j53(x) , AI -0/'(x)"a­x dy ln(x!y)2­ jI (y) . kl k l l k k k l l

(2.35)

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27

Let us stress that in conformal theory these equations are the consequences [27—29] of the equivalence conditions for irreducible representations (2.26). The usual equivalence conditions may be written in any of the forms:

P

P

U (x)" dy D (x!y)U J (y) or U J " dy D J (x!y)U (y) . p p p p p p

(2.36)

Unlike this, Eq. (2.27) cannot be inverted since the kernels D-0/' , D53 are degenerate. The groupkl kl theoretic reason behind this consists in the fact that the elements of the spaces Mj/M53 and M /M-0/' are the equivalence classes. Each equivalence class MjI N, j A A k M jI NLM /M53 (2.37) k j j includes a set of functions with different transversal parts. In particular, if the field jI LMjI N, then k k the field jI @ "jI #jI 53 also belongs to the same class k k k jI LM jI NPjI @ "jI #jI 53LM jI N , (2.38) k k k k k k where jI 53 is an arbitrary transversal field. Analogously, each equivalence class k MAI NLM /M-0/' (2.39) k A A consists of the set of functions with different longitudinal parts: if AI LMAI N, then AI #AI -0/'LMAI N , (2.40) k k k k k where AI -0/' is an arbitrary longitudinal field. Thus any representatives of the equivalence classes k MjI N and MAI N can enter the corresponding r.h.s. of Eq. (2.27), see Section 6 for more details. k k As an example let us consider the Green function of the current Gj (x x x x )"Su(x )u`(x )s(x ) j (x )T , (2.41) k 1 2 3 4 1 2 3 k 4 where v(x) is a neutral scalar field of dimension D. The general conformally invariant solution of the Ward identity ­x4Gj (x x x x )"![d(x )!d(x )]Su(x )u`(x )s(x )T k k 1 2 3 4 14 24 1 2 3 can be written as [2,15]

(2.42)

Gj (x x x x )"Kx4(x x )Su(x )u`(x )s(x )T#G53(x x x x ) , k 1 2 3 4 k 1 2 1 2 3 k 1 2 3 4 where G53 is an arbitrary transversal conformally invariant function k ­x4G53(x x x x )"0 , k k 1 2 3 4 and the function K has the form k D x2 (D~2)@2 1 12 Kx4(x x )" p~D@2 C jx4(x x ) , k 1 2 k 1 2 2 x2 x2 2 14 24 ­x4Kx4(x x )"!d(x )#d(x ) . k k 1 2 14 24

(2.43)

A BA

B

(2.44)

(2.45)

28

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

The set of functions differing by their longitudinal parts may be viewed as a definite equivalence class (2.37). Introduce the conformally invariant function [5] GA-0/' (x x x x )"Su(x )u`(x )s(x )A-0/'(x )T"g jx4(x x )Su(x )u`(x )s(x )T k 1 2 3 4 1 2 3 k 4 A k 1 2 1 2 3 x2 1 " g ­x4 ln 14 Su(x )u`(x )s(x )T , A 1 2 3 k x2 2 24 where g is the coupling constant. According to the first equality in Eq. (2.27) we have A

P

GA-0/' (x x x x )" dx D-0/' (x !x)Gj (x x x x) . k 1 2 3 4 kl 4 l 1 2 3

(2.46)

(2.47)

On the other hand, the set of transversal conformally invariant functions G53), see Eq. (2.44), forms k a space of irreducible representation Q53 j G53LM53 . (2.48) k j Due to the second equality in Eq. (2.27) one has

P

G53(x x x x )" dx D53 (x !x) GI A(x x x x) , kl 4 l 1 2 3 k 1 2 3 4

(2.49)

where GI A(x x x x)"Su(x )u`(x )s(x )AI (x)T (2.50) l 1 2 3 1 2 3 l is an arbitrary conformally invariant function of the field AI (x). k It is natural that the general conformally invariant solution of the Ward identities includes the contribution of electromagnetic interaction described by the field AI .What is problematic is how k this contribution could be extracted explicitly. The latter is especially important when one concerns the models neglecting electromagnetic interaction. This problem is a complicated task in the case of conformally invariant theory. The longitudinal part of the first term in Eq. (2.43) is not conformally invariant. The invariance could be achieved only after a certain transversal (also non-invariant) correction to the longitudinal part is added. The latter is defined up to an arbitrary conformally invariant function which could be added to the second term in Eq. (2.43). So the question on the electromagnetic contribution into the first term remains open. Similarly, extracting the contribution of the gravitational interaction poses the problem in the case of Green functions of the energy—momentum tensor, see Section 2.5. The mathematical origin of this problem is concealed in the fact that the representations due to the transformation laws (2.3) are undecomposable. Hence the first step should consist in the transition to the direct sum of irreducible representations (2.15): QPQI = Q . (2.51) 0 The representations of the type Q are related to the contributions of gauge fields, while the 0 representations of the type QI are related to the contributions of the matter fields. However, the representations QI are defined in the space of equivalence classes. Thus, from the mathematical

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

29

viewpoint, the problem of extracting of the unique solution to the Ward identities (in the absence of gauge interactions) is equivalent to the formulation of definite prescription which would uniquely fix the choice of representative in each equivalence class. The above prescription could be formulated on the ground of the following arguments. Represent the Euclidean current j (x) which transforms by the direct sum of representations (2.51), as k a sum of two terms j (x)"jI (x)#j53(x) . (2.52) k k k This decomposition is conformally invariant. It becomes unique if the following condition is satisfied: the states of the Hilbert space generated by the fields jI and j53 are mutually orthogonal. k k The physical meaning of the first term is discussed in detail in Section 6.1. The second term in Eq. (2.52) corresponds to the contribution of electromagnetic interaction and is related to the field AI due to the second of Eq. (2.27). One can show that the above requirement of the orthogonality k fixes the choice of representative in each of the equivalence classes M jI N and MAI N uniquely, i.e. each k k representation QI , QI becomes realized in the space of functions. Simultaneously, Eq. (2.27) j A become invertible. A more detailed discussion of this construction may be found in Section 6, see also [15] as well as the next sections of this paper. Let us emphasize that all the above is valid for the space of any dimension D53. The case D"2 is exceptional and will be discussed separately in the end of this section. To put forth the program described above, let us consider the operator product expansion j (x )u(x ). According to Eq. (2.52), it includes two types of contributions k 1 2 j (x )u(x )"jI (x )u(x )#j53(x )u(x ) , (2.53) k 1 2 k 1 2 k 1 2 where jI (x )u(x )"+ [Pj ] , j53(x )u(x )"+ [Rj ] . k 1 2 k k 1 2 k k k Let us require that two sets of fields

(2.54)

MP j N and MR j N k k should be mutually orthogonal

(2.55)

SP j R j T"0 for all k, m (2.56) k m even if their quantum numbers (dimensions and tensor ranks) coincide. This ensures the orthogonality of the states (see the end of Section 1): jI (x )u(x )D0T and j53(x )u(x )D0T , (2.57) k 1 2 k 1 2 Su(x )u`(x ) jI (x ) j53(x )T"0 , (2.58) 1 2 k 3 k 4 as well as the unambiguity of the decomposition into the sum (2.52) for the solution of the Ward identities. Below we demonstrate, in particular, that the expression (2.43) for the Green function Gj satisfies conditions (2.56) and (2.58). k

30

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

2.3. The solution of the Ward identities for the Green functions of irreducible conformal current All the fields (2.55) that arise in the operator expansion j (x )u(x ) can be found analysing the k 1 2 Green functions S j uj u`T. Consider its connected part k l G j (x x x x )"S j (x )u(x ) j (x )u`(x )T . (2.59) kl 1 2 3 4 k 1 2 l 3 4 #0// The Ward identity for the latter has the form ­x1G j (x x x x )"![d(x )!d(x )]Su(x )u`(x ) j (x )T , k kl 1 2 3 4 12 14 2 4 l 3 where

(2.60)

(2.61) Su(x )u`(x ) j (x )T"Kx3(x x )Su(x )u`(x )T , l 1 2 1 2 1 2 l 3 and Kx3(x x ) is the function (2.45). l 1 2 In accordance to Eqs. (2.52) and (2.58), represent the Green function (2.59) as a sum of the pair of terms Gj (x x x x )"GI j (x x x x )#G53 (x x x x ) , (2.62) kl 1 2 3 4 kl 1 2 3 4 kl 1 2 3 4 where G53 "S j53uj53uT is the transverse function: kl k l (2.63) ­x1G53 (x x x x )"­x3G53 (x x x x )"0 . k kl 1 2 3 4 l kl 1 2 3 4 The first term in Eq. (2.62) is uniquely determined from the Ward identity and only contains the contributions of the fields P j. As shown in Refs. [2,3], see also [15] and below, it has the form k 1 GI j (x x x x )"Kx (x x )Kx3(x x )Su(x )u`(x )T . (2.64) kl 1 2 3 4 k 2 4 l 2 4 1 2 The second term in Eq. (2.62) only contains the contributions of the fields R j . k To derive these results and to formulate condition (2.56) in terms of higher Green functions, it proves useful to apply conformal partial wave expansion. Let us start with the Green function (2.59). Its expansion has the form (1.92) after one sets Ul{(x)"j (x) in this formula. The two terms in k k Eq. (1.92) can be identified with the two terms in Eq. (2.62) if the transversal functions are chosen for Cp 2 Cp "Cl,53 12 s (x x x )"SUl 12 s(x )u(x ) j53(x )T 2 k,k k 1 2 3 k k 1 2 k 3 D!2!l#d#s , 1 Dl (x x x ) , (2.65) & s j 1 2 3 l!d

G

H

where Dl (x x x )"(1x2 )~(l`d~s~D`2)@2 (1x2 )~(l~d~s`D~2)@2 (1x2 )~(d`s~l`D~2)@2 . j 1 2 3 2 12 2 13 2 23 The notation (1.78) has been used in Eq. (2.65). One can show [2,12] that

(2.66)

­x3Cl,53 12 s (x x x )"0 for all l . (2.67) k k,k k 1 2 3 Such a choice of the functions Cp guarantees the transversality condition (2.63) to hold identically. 2 Conditions (2.56) and (2.58) also turn out to be fulfilled due to Eqs. (1.97) and (1.84). Thus the poles of the kernels o (p) and o (p) in the expansion (1.92) determine the contributions of the fields 1 2 P j and R j, respectively. k k

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

31

The total set of the normalized functions Cp and Cp is determined by the orthogonality 1 2 conditions (1.84) and (1.85), the transversality condition, and the diagonality condition of the expansion (1.92). The more detailed calculations are presented in Ref. [2], see also [3,15]. It is useful to represent the result in the following form. Introduce the functions CI p "CI l 12 s (x x x )"SUl 12 s(x )uJ (x )A-0/'(x )T , 1 1k,k k 1 2 3 k k 1 2 k 3 CI p "CI l 12 s (x x x )"SUl 12 s(x )uJ (x )AI (x )T , k k 1 2 k 3 2 2k,k k 1 2 3 where uJ is the conformal partner of the field u. These functions can be derived from the functions Cp and Cp through the amputation by the arguments x , x . Under the normalization (1.84), (1.85), 1 2 2 3 these functions read Cp "Cl 12 s (x x x )"(2p)~D@2N (p, d)M(D!2#l!d!s), 1NDl (x x x ) , 1 j 1 2 3 1 1k,k k 1 2 3 CI p "CI l 12 s (x x x )"(2p)~D@2 2(s~1)@2 NI (p, d)M(l#d!D!s), 1NDI l (x x x ) , 1 j 1 2 3 1 1k,k k 1 2 3 where 2(s`3)@2 N (p, d)" 1 (d#s!l)(D!2#l!d#s)

G

]

A A

B A B A

B A B A

B A

l#d#s!D D#l!d#s D!l#d#s l#d#s C C C 2 2 2 2 d#s!l 2D!l!d#s l!d#s D!l!d#s C C C C 2 2 2 2

C

G

A A

B A B A

B A B A

B A

B A B A

B

H

B

DI l (x x x )"(1x2 )~(l~d~s`D)@2 (1x2 )~(l`d~s~D)@2 (1x2 )~(l`d~s~D)@2, j 1 2 3 2 12 2 13 2 23 Cp &Cl,53 12 s (x x x ) , 2 2k,k k 1 2 3 3D!l!d!4#s CI p & !s , 1 DI l (x x x ) . 2 j 1 2 3 2D!l!d!2

G

H

(2.69)

1@2 ,

(2.70)

H

B B

l!d#s#D l#d#s!D l!d#s D!l!d#s C C C 2 2 2 2 NI (p, d)" 1 2D!l!d#s d#s!l D!l#d#s l#d#s C C C C 2 2 2 2 C

(2.68)

1@2 ,

(2.71) (2.72)) (2.73) (2.74)

The expressions for the renormalization factors in the last pair of functions are presented in Refs. [2,3,12]. Their explicit form is irrelevant for what follows. The higher Green functions of the fields P j, Rj k k SP j(x )u(x )2u`(x )T , SR j(x )u(x )2u`(x )T (2.75) k 1 2 2n k 1 2 2n are given by the expressions (1.87) for i"1, 2, respectively, after one substitutes the functions (2.69) and (2.74) into them. Below we consider the models with no regard to electromagnetic interaction, having the property that G53 (x x x x )"0, or kl 1 2 3 4 (2.76) SR j(x )u(x )2u`(x )T"0 . 2 2n k 1

32

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

In virtue of Eq. (1.87) with i"1, 2, the condition of vanishing of the fields R j can be written in the k form

P

dy dy CI l 12 s (x y y )S j (y )u(y )u(x )2u`(x )T"0 . 1 2 2k,k k 1 1 2 k 2 1 2 2n

(2.77)

Condition (2.77) will hereinafter be referred to as the irreducibility condition of the conformal current, or the condition of absence of the electromagnetic interaction. Note that Eq. (2.77) select out the class of theories in which all the Green functions of the current are uniquely determined by the ¼ard identities; see also Section 6 for details. The Green functions of the fields P j are determined by the Ward identities. As an example, let us k consider the Green functions (2.41) and (2.59). The solution of the Ward identities satisfying the conditions (2.77) reads [2,3,12,15]: Su(x )u`(x )s(x ) j (x )T"Kx4(x x )Su(x )u`(x )s(x )T , (2.78) 1 2 3 k 4 k 1 2 1 2 3 S j (x )u(x ) j (x )u`(x )T"Kx1(x x )Kx3(x x ) Su(x )u`(x )T . (2.79) k 1 2 l 3 4 k 2 4 l 2 4 1 2 Conditions (2.77) for these functions mean that the kernels os (p) and o j (p) of the second term in the 2 2 partial expansion of the type (1.92) should vanish for both of these functions. As may easily be demonstrated by a direct calculation (with the help of the integral relations (A.2) and (A.4)), o s(p)"o j (p)"0 ; (2.80) 2 2 see Refs. [2,15] for more details. So the fields R do not contribute to Eqs. (2.78) and (2.79). One can k find the expressions for the kernels o s(p) and o j (p) corresponding to these functions [2,3,15] 1 1 D!D C 1 (2p)D 1 2 o s(p)" (D!2#l!d#s)~1 1 D J2 C2(D/2) (d#s!l) C 2

A

A

B A B A

AB

B

B

l#d#s!D l!d#D#s C 2 2 ][N(pdD)N (p, d)]~1 , 1 2D!l!d#s D!l#d!D#s C C 2 2 C

A

B

(2.81)

where D is the dimension of the field s, N(pdD) is the normalization factor (1.39), 1 (2p)D2s (N (p, d))~2 o j (p)"(!1)s`1 1 (d#s!l) C2(D/2) 1 C ](D!2#l!d#s)~1

A

B A B A

B

l#d#s!D D!2#l!d#s C 2 2 . d!l#s#2 2D!l!d#s C C 2 2

A

B

(2.82)

In the course of calculations we have used the relations (A.2) and (A.4). The quantum number of the fields P j are determined by the poles of the partial wave expansion, see, for example, Eq. (1.75). k

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

33

It follows from Eqs. (2.81) and (2.82) that there exists an infinite set of fieldsPj of rank s and s dimension l "d#s , s which contribute to the operator product expansions j u and us: k

(2.83)

j (x )u(x )"+ [Pj] , u(x )s(x )"+ [Pj] . (2.84) k 1 2 s 1 2 s s s The invariant three-point Green functions of the fields Pj have the form [2,15]: s SPj(x )u`(x ) j (x )T"gs (1 x2 )~(D~2)@2 ­on x3 [(1 x2 )~(D~2)@2 jx112 s (x x )] s 1 2 k 3 j 2 23 k 2 13 k k 3 2 (2.85) ](1 x2 )(D~2)@2 Su(x )u`(x )T , 1 2 2 12 where gs are known coupling constants. These Green functions may be shown to satisfy [2,3,12,15] j the anomalous Ward identities C(s) ­x3SPj 12 s(x )u`(x ) j (x )T"gs(!2)~s`1 C(d) 2 k 3 j k k k 1 C(D/2#s!1) s 1 ]+ [­x3 ­x3 d(x )­x1k`12­x1k #(2)!traces]Su(x )u`(x )T , (2.86) k 13 k 1 2 C(k)C(d#s!k) k12 kk k/1 where (2) stands for the sum of terms which arise after the transmutations of indices which enter asymmetrically. Higher Green functions of the fields Pj may be evaluated from the Ward identities using s Eq. (1.87) for i"1. The explicit form of these expressions will be presented in the next section. 2.4. Green functions of the energy—momentum tensor and conditions of absence of gravitational interaction Consider the Green functions of the energy-momentum tensor (1.50). Let Q be an undecomposT able representation defined by the transformation law (2.4) and M the space of this representation. T Introduce two types of irreducible tensors ¹I (x) and ¹53 (x), ­ ¹53 (x)"0 , kl kl k kl transforming by the irreducible representations

(2.87)

QI and Q53 , (2.88) T T respectively. The representation Q53 is defined by the transformation law (2.4) on the invariant T subspace M53LM T T of transversal tensors ¹53 LM53. The representation QI acts in the quotient space kl T MI "M /M53 . T T T

(2.89)

(2.90)

34

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

Denote the space of metric fields h as M . The space of longitudinal fields (2.8) which is invariant kl h under (2.10) will be denoted as M-0/'. Introduce the quotient space h MI "M/M-0/' . (2.91) h h Consider the irreducible representations Q-0/' and QI h h acting in the spaces

(2.92)

M-0/' and M I , h h respectively. Let us introduce two types of irreducible fields

(2.93)

h-0/' and hI , kl kl transforming by irreducible representations (2.92). Consider the fields

(2.94)

¹ (x)"¹I (x)#¹53 (x), h (x)"h-0/'(x)#hI (x), kl kl kl kl kl kl which transform by the direct sums of irreducible representations

(2.95)

QI = Q53 , Q-0/' = QI . (2.96) T T h h Each of the Green functions (1.50) and (1.52) may be represented as the sum of two terms (2.95): GT (xx 2x )"GI T (xx 2x )#GT 53 (xx 2x ) , kl 1 m kl 1 m kl 1 m Gh (xx 2x )"Gh -0/' (xx 2x )#GI h (xx 2x ) , kl 1 m kl 1 m kl 1 m where GT 53 is the conformally invariant transversal function kl ­x GT 53 (xx 2x )"0 , k kl 1 m while Gh -0/' is the conformally invariant longitudinal function kl 2 Gh -0/' (xx 2x )"­xGh (xx 2x )#­xGh (xx 2x )! d LxGh (xx 2x ) , m m m l k 1 m k l 1 kl 1 D kl j j 1 where

(2.97) (2.98) (2.99)

(2.100)

Gh (xx 2x )"Sh (x)u (x )2u (x )T (2.101) k 1 m k 1 1 m m and h (x) is the conformal vector of dimension d "!1. k h Note that the decompositions (2.97) and (2.98) are not unique since both representations QI and T QI are defined in the space of equivalence classes M I and M I , respectively, see Section 6 and [15] h T h for more details. Similar to the case of current we shall formulate the prescription which fixes decompositions (2.95) uniquely. This prescription will be given in terms of certain constraints on the Green functions GT which allow one to conduct an unambiguous evaluation of the Green kl functions of GI T from the Ward identities, and to relate GT 53 to gravitational degrees of freedom. kl kl Let us demand that the groups of the states of the Hilbert space generated by the conformal fields ¹I (x) and ¹53 (x) should be mutually orthogonal. The fields ¹53 and hI are related to gravitational kl kl kl kl degrees of freedom. It is convenient to formulate the above orthogonality condition [4,15] in terms of the states generated by operator product expansion ¹ u. Represent the latter expansion in the form: kl ¹ (x )u(x )"¹I (x )u(x )#¹53 (x )u(x ) , (2.102) kl 1 2 kl 1 2 kl 1 2

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

35

where ¹I (x )u(x )"+ [PT] , ¹53 (x )u(x )"+ [RT] . kl 1 2 k kl 1 2 k k k We require the two sets of fields

(2.103)

MPTN and MRTN k k to be mutually orthogonal

(2.104)

S0DPT(x ) RT (x )D0T"0 for all k, k@ , (2.105) k 1 k{ 2 even when the quantum numbers (dimension and ranks) of some of these fields coincide. The latter guarantees the orthogonality of the states ¹I (x )u(x )D0T and ¹53 (x )u(x )D0T , kl 1 2 kl 1 2 which is expressed by the condition

(2.106)

S¹I (x )u(x )¹53 (x )u(x )T"0 . (2.107) kl 1 2 op 3 4 The above remarks on the role of the fields (2.87) and (2.95) are based on the following arguments. Note first that the irreducible representations (2.88) and (2.92) are partially equivalent [21,26], see also Ref. [31]: QI &Q-0/', Q53&QI . (2.108) T h T h The equivalence conditions are expressed by the following relations between independent irreducible components of (2.97) and (2.98)

P

Gh -0/' (xx 2x )& dy Dh (x!y) GI T (yx 2x ) , kl 1 m kl,op op 1 m

(2.109)

where Dh is the conformally invariant propagator of the longitudinal field h-0/' kl,op kl Dh (x )"Sh-0/' (x ) h-0/' (x )T kl,op 12 kl 1 op 2 2 "C g (x )g (x )#g (x )g (x )! d d h ko 12 lp 12 kp 12 lo 12 D kl op

C

D

where C is a constant. Expression (2.110) may be represented in the form [15,31] h 2 Dh (x )"­x1D (x )#­x1D (x )! d ­x1D (x ) , kl,op 12 k l,op 12 l k,op 12 D kl j j,op 12

(2.110)

(2.111)

where

C

D

2 1 Dh (x)" C x g (x)#x g (x)# d x . o kp k,op D op k 2 h p ko From Eqs. (2.109) and (2.111) the representation of GI -0/' in the form (2.100) follows. kl

(2.112)

36

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

The second condition of equivalence is given by the relation

P

GT 53 (xx 2x )" dy DT,53 (x!y) GI h (yx 2x ) , kl 1 m kl,op op 1 m

(2.113)

where DT,53 is the transversal propagator. The general conformally invariant expression for the kl,op propagator of the energy-momentum tensor has the form DT (x )"S¹ (x )¹ (x )T kl,op 12 kl 1 op 2

G

H

1 2 "CI g (x )g (x )#g (x )g (x )! d d . (2.114) T ko 12 lp 12 kp 12 lo 12 D kl op (x2 )D 12 In the space of odd dimension this expression is given by a well-defined function and may be represented as [15,31] 1 , DT,53 (x )&H53 (­x1) ko,op 12 kl,op (x2 )D~2 12 where

(2.115)

G

1 (D!2) ­ ­ ­ ­ ! (d ­ ­ #d ­ ­ #d ­ ­ #d ­ ­ )h H53 (­x)" kp l o lo k p lp k o kl,op (D!1) k l o p 2 ko l p 1 (d ­ ­ h#d ­ ­ h) # op k l (D!1) kl o p

H

1 1 d d h2 , # (d d #d d )h2! ko lp kp lo (D!1) kl op 2

(2.116)

H53 "H53 "H53 , H53 "0 . kl,op lk,op op,kl kk,op H53 (­x)H53 (­x)"h2H53 (­x) , (2.117) kl,jq jq,op kl,op ­x H53 (­x)"0 . (2.118) k kl,oq In the even-dimensional space the expression (2.114) diverges due to the singularity of the factor (x2 )~D. Let us redefine this propagator as follows. Introduce the conformally invariant regulariz12 ation by an addition of a small anomalous correction to the dimension l of the field ¹ : T kl l "D P le "D#e . (2.119) T T The regularized propagator DTe results from Eq. (2.114) after the substitution of the factor kl,op (x2 )~D~e for the factor (x2 )~D. Define a new propagator for the space of even dimension D54 by 12 12 DT,53 (x )"lim eDTe (x ) . kl,op 12 kl,op 12 e?0 Resolving the ambiguity with the help of relation (2.34) one gets DT,53 (x )&H53 (­x1)h(D~4)@2 d(x ) , for even D54 . 12 kl,op 12 kl,op

(2.120)

(2.121)

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

37

Thus, there are two invariant kernel DTe and D53 for even D54. In Section 6 we show that kl,op kl,op these kernels are related to a pair of irreducible representations QI and Q53, and define the T T propagators of irreducible fields ¹I and ¹53 . In the case of odd D the situation is analogous, and kl kl will not be discussed here. It is convenient to introduce the projection operators P53 and P-0/' onto the transversal and longitudinal sectors. Owing to Eq. (2.117) the latter is done in a manner which is natural to conformal theory, by setting 1 P53 (­x)" H53 (­x) , kl,op h2 kl,op

A

(2.122)

B

2 1 . P-0/' (­x)#P53 (­x)" d d #d d ! d d kp lo D kl op kl,op kl,op 2 ko lp

(2.123)

One can easily check that thus defined operator P-0/' has the following properties: kl,op P-0/' (­x) P-0/' (­x)"P-0/' (­x) , kl,jq jq,op kl,op 2 P-0/' (­x)"­xP (­x)#­xP (­x)! d ­x P (­x) , kl,op k l,op l k,op D kl j j,op

(2.124) (2.125)

where

C

1 D!2 1 1 1 1 P (­x)"! ­­­ !(d ­ #d ­ ) # d ­ k,op k o p ko p kp o op k 2 D!1 h2 h D!1 h

D

.

(2.126)

Furthermore, one can explicitly check that the longitudinal propagator (2.110) satisfies the relation P-0/' (­x1) Dh (x )"Dh (x ) . (2.127) kl,op op,jq 12 kl,jq 12 As follows from Eqs. (2.115), (2.121) and (2.122) the transversal propagator of the energy—momentum tensor satisfies the relation P53 (­x1)DT,53 (x )"DT,53 . op,jq 12 kl,jq kl,op Using these relations one finds from Eqs. (2.109) and (2.113)

(2.128)

P53 (­x) GT 53(xx 2x )"GT 53(xx 2x ) , (2.129) kl,op op 1 m kl 1 m P-0/' (­x) Gh -0/'(xx 2x )"Gh -0/'(xx 2x ) . (2.130) kl,op op 1 m kl 1 m Note that the remaining pair of irreducible functions, GI T and GI h do not satisfy similar relations. Each of these functions has both the transversal and the longitudinal components, and only the whole sums possess the property of conformal invariance. As shown in Ref. [15], the requirement of conformal invariance allows to reconstruct the transversal part of the function GI T uniquely from kl the longitudinal part which is known from the Ward identities provided that one chooses a certain realization of the representation QI . The choice of the realization in this case is imposed by the T orthogonality condition (2.107). As shown in Section 6, see also Ref. [15], the latter allows to separate out the contribution of the gravitational interaction into the Green function (1.50) in an explicit manner; see below.

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A resume consists in the following. The general solution of the Ward identities (1.54) represents a sum of the two conformally invariant terms (2.97). The first one, GI T is uniquely defined by the kl Ward identity and the requirement of the conformal symmetry. The second term is transverse, and may be expressed through the Green function of the metric field by Eq. (2.113). In the space of even dimension this equation takes the form: GT 53(xx 2x )"hD@2P53 (­x)Sh (x)u (x )2u (x )T . (2.131) kl 1 m x kl,op op 1 1 m m In four-dimensional space it coincides [31] with the equation of linear conformal gravity: h2h (x)!h(­ ­ h (x)#­ ­ h (x))#2­ ­ ­ ­ h (x)#1d h­ ­ h (x)"¹53 (x) . 3 kl p j pj kl kl k p lp l p kp 3 k l p j pj (2.131a) The longitudinal part Gh -0/' of the Green function Gh "Sh u 2u T does not contribute to kl kl kl 1 m Eq. (2.131). It is determined from Eq. (2.109) and may be calculated directly from the Ward identities:

P

Gh -0/' (xx 2x )"!2 dy D (x!y)­y S¹ (y)u (x )2u (x )T , kl,o p op 1 1 m m kl 1 m

(2.132)

where D (x) is the function (2.112) kl,o D (x!y)"P (­x)Dh (x!y) . kl,o o,jq kl,jq Thus, the functions (2.133) GI T and Gh -0/' kl kl are determined by the Ward identities, the function GI h remains arbitrary, and the function GT 53 is kl kl expressed through it by Eq. (2.131) (or a similar one for odd D). To this pair of functions (2.133) a pair of equivalent irreducible representations QI &Q-0/' corresponds. T h According to Ref. [15], see also Section 6, the Green functions (2.133) describe the contribution of matter fields into energy—momentum tensor, while the function GI h , as well as the transversal kl function GT 53 which expresses through it, are related to gravitational interaction. To this pair of kl functions, GT 53 and GI h , another pair of equivalent irreducible representations QI 53&QI h correskl kl ponds. Due to this, the theories which are free of gravitation interaction are selected by the following condition: the energy—momentum tensor transforms by the irreducible representation QI T. Its Green functions coincide with GI T : kl S¹ (x)u (x )2u (x )T"GI T (xx 2x ) , (2.134) kl 1 1 m m kl 1 m and, in virtue of the above arguments, are uniquely determined by the ¼ard identities. 2.5. The algorithm of solution of Ward identities in D-dimensional space Condition (2.105) or (2.107) select out irreducible contributions of the energy—momentum tensor into the Green functions (1.50). Let us formulate these conditions in terms of conformal partial

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

39

wave expansions. Consider the Green function GT (x x x x )"S¹ (x )u(x )¹ (x )u(x )T . klop 1 2 3 4 kl 1 2 op 3 4 #0// On account of Eq. (2.107), the latter may be represented as

(2.135)

GT (x x x x )"GI T (x x x x )#GT 53 (x x x x ) , klop 1 2 3 4 klop 1 2 3 4 klop 1 2 3 4 where

(2.136)

GI T "S¹I u¹I uT , GT 53 "S¹53 u¹53 uT . (2.137) klop kl op klop kl op The partial wave expansion of the Green function (2.135) has the form (1.105), if one sets Ul{ (x)"¹ (x) in that formula. The invariant functions Cp,Cp entering this expansion may be kl kl i ikl chosen so that the last term in Eq. (1.105) becomes transverse. The general expression for the functions Cp has the form (1.101) after one sets l "D. We have ikl 1 Cp (x x x )"SUl 12 s(x )u(x )¹ (x )T"MA(i), A(i), A(i)NDl (x x x ) , (2.138) ikl 1 2 3 k k 1 2 kl 3 1 2 3 T 1 2 3 where A(i) are the constants; i, k"1,2,3, k (2.139) Dl (x x x )"(1x2 )~(l`d~s~D`2)@2 (1x2 )~(l~d~s`D~2)@2 (1x2 )~(d`s~l`D~2)@2 . 2 13 2 23 T 1 2 3 2 12 The coefficients A(i) are determined by the orthogonality conditions (1.102) and (1.103) up to an k SO(3) transformation. Using this ambiguity, one can make the function Cp (x x x ) transversal: 3kl 2 2 3 Cp (x x x )"SU (x )u(x )¹53 (x )T , ­x3 Cp (x x x )"0 . (2.140) 3kl 1 2 3 p 1 2 kl 3 k 3kl 1 2 3 One can show [3,15] that this condition leads to the following equations for the coefficients A(3), k k"1,2,3: D!2 1 [(D!1)(l!d)#s]A(3)!s (D!l#d#s)A(3)"0 , 1 2 D D

A

B

2s 1 ! A(3)# l!d! A(3)#(s!1)(l!d!s!D#2)A(3)"0 . 1 3 D 2 D

(2.141)

Note that these equations have non-empty solutions only for s52, when all three coefficients A(3) are non-zero. When s"0, 1, no transversal function exists, see Eqs. (1.79) and (1.81). Under k this choice of Cp the partial wave expansions for the Green functions (2.137) take the form, see 3kl Eq. (1.105): 2 GI "+ + klop i/1 p

P

P

dx dy o (p) Cp (xx x )D~1(x!y) Cp (yx x ) , i ikl 1 2 p iop 3 4

(2.142)

(2.143) G53 "+ dx dy o (p) Cp (xx x )D~1(x!y) Cp (yx x ) , 3 3kl 1 2 p 3op 3 4 klop p One can show [3,15] that the function GI is determined uniquely from the Ward identity. klop Substituting expression (2.142) into this identity and using the orthogonality conditions (1.102) and (1.103), one can evaluate the kernels o (p) and o (p) as well as the coefficients A(1) and A(2), on 1 2 k k which the functions Cp and Cp depend. The problem consists in the diagonalization of the 1kl 2kl

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partial wave expansion of the general solution of the Ward identity (similar to the procedure conducted in Section 3 for the Green function Gj ). It is discussed to a greater extent in Ref. [15], kl see also Ref. [3]. In principle, this problem is not much complicated, though it calls for quite cumbersome calculations to be published separately. The fields PT(x) are determined by the poles of the kernels o (p) and o (p), while the field RT(x), k 1 2 k by the poles of o (p). From what has been said it follows that the Green functions of the fields 3 PT are calculated from the Ward identities, while the Green functions of the fields RT are k k determined by the metric field hI . kl The higher Green functions of the fields PT(x) and RT are expressed through the residues of the k k kernels Gp "Gp (x 2x ) , Gp "Gp (x 2x ) , Gp "Gp (x 2x ) (2.144) 1 1 1 m 2 2 1 m 3 3 1 m of conformal partial wave expansions (1.112) for the Green functions S¹ u 2u T. The first two kl 1 m terms in Eq. (1.112) coincide with the function GI T , see Eq. (2.97), while the third term, with the kl function GT 53 . To calculate the kernels (2.144) it proves helpful to introduce the functions kl CI p amputed by two arguments. The latter enter into orthogonality conditions (1.102) and (1.103) ikl and into the equations of the type (1.87), CI p (x x x )&SUl (x )uJ (x )h (x )T , (2.145) ikl 1 2 3 s 1 2 kl 3 i where uJ is the conformal partner of the field u. In particular, one has for the kernel Gp 3

P P

Gp (x 2x )" dx dy CI p (x xy)S¹53 (y)u (x)u (x )2u (x )T 3 1 m 3kl 1 kl 1 2 2 m m " dx dy CI p (x xy)S¹ (y)u (x)u (x )2u (x )T . 3kl 1 kl 1 2 2 m m

(2.146)

The last equality is the consequence of orthogonality of the function CI p to the functions Cp and 3kl 1kl Cp : 2kl

P

P

dx dy CI p (x xy) Cp{ (x xy)" dx dy CI p (x xy) Cp{ (x xy)"0 3kl 1 1kl 2 3kl 1 2kl 2

(2.147)

for all p, p@. The irreducibility condition (2.134) means that the kernels o (p) and Gp vanish 3 3 o (p)"0, Gp (x 2x )"0 . (2.148) 3 3 1 m Taking into account Eq. (2.146), one can rewrite these conditions as the following equations on the Green functions of the energy—momentum tensor:

P

dx dy CI p (x xy)S¹ (y)u (x)u (x )2T"0 . 3kl 1 kl 1 2 2

(2.149)

In the models where these conditions are fulfilled, all the Green functions of the energy—momentum tensor are uniquely determined by the Ward identities.

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41

The Green functions of the fields PT(x) in such models are calculated from the equations of the k type (1.71)

P

SPT(x )u (x )2u (x )T"KT res dx dy CI p (x xy)S¹ (y)u (x)2u (x )T k 1 2 2 m m k kl 1 kl 1 m m p/pk where CI p is the function (2.145) orthogonal to the transversal functions Cp : kl 3kl

P

dx dy CI p (x xy) Cp{ (x xy)"0 for all p, p@. kl 1 3kl 2

(2.150)

(2.151)

As will be shown in the next section, the only fields which may exist in the models satisfying conditions (2.149) are the fields PT with the quantum numbers p "(d#s, s). We will consider an k k infinite set of such fields MPlsN where l "d#s . s s

(2.152)

2.6. Conformal Ward identities in two-dimensional field theory Two-dimensional space is specific by the property that both the current and energy—momentum tensor are irreducible fields. When D"2, there is no problem in the decoupling of Euclidean transversal field ¹I 53 (x), just because this field is zero. Gravitational interaction in this case is trivial kl and has no influence on the dynamics of matter fields. The representation of conformal group1, given by the transformation law R ¹ @ (x)" 1 g (x)g (x)¹ (Rx) ¹ (x)P lp op kl kl (x2)2 ko

(2.153)

is irreducible. The energy—momentum tensor, being the traceless symmetric tensor, has two independent components ¹ #¹ "0, ¹ "¹ . 11 22 12 21 The transversality condition

(2.154)

­ ¹53 (x)"0 k kl is equivalent to a pair of equations on these components, having the unique solution ¹53 (x)"0 . kl The projection operator introduced above to utilize the decoupling of the subspace M53 also T vanishes for D"2, while the longitudinal projector P-0/' (­/­x) is unity kl,op ­ ­ "0, P-0/' "I for D"2 . P53 kl,op ­x kl,op kl,op ­x

A B

A B

1 The six-parameter conformal group is assumed.

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Thus any traceless symmetric tensor ¹ (x) is longitudinal kl ­ ¹ (x)"P-0/' ¹ (x)"¹-0/'(x) kl kl,op ­x op kl

A B

and may be represented in the form ¹ (x)"­ ¹I (x)#­ ¹I (x)!d ­ ¹I (x) , kl k l l k kl j j ­l 1 ¹I " ¹ (x)" ¹ (x) , k h kl h k

(2.155) (2.156)

where ¹ (x)"­ ¹ (x) is the conformal vector of scale dimension D#1. Thus the irreducible k l kl representation, given by the transformation law (2.153), is the analogue of the representation QI , T which corresponds for D'2 to the models where the gravitation is neglected. From the above it is clear that the Green functions S¹ (x)u (x )2u (x )T, S¹ (x)¹ (y)u (x )2u (x )T (2.157) kl 1 1 m m kl op 1 1 m m are uniquely determined by Ward identities. For the case of D'2 this property is proved for the conformal theories satisfying the condition (2.151), which fixes the realization of the representation QI and simultaneously drops down the gravitational interaction. We have already mentioned the T similarity between such theories and two-dimensional models. In the next sections we expand this analogy to a greater extent. For this reason, in the present section we keep the component form of Ward identities (though, the complex variables are more useful). The conformally invariant solution to Ward identities is given by Eqs. (2.155) and (2.156). Consider the Ward identities for the Green functions (2.157) for D"2:

C

D

m m 1 ­xS¹ (x)u (x )2u (x )T"! + d(x!x )­xk! ­x + d d(x!x ) l kl 1 1 m m k k k k k 2 k/1 k/1 ]Su (x )2u (x )T , (2.158) 1 1 m m where d are scale dimensions of the scalar fields u , k"1,2, m. The r.h.s. represents the Green k k functions for the vector ¹ (x) k S¹ (x)u (x )2u (x )T k 1 1 m m Using Eqs. (2.155) and (2.156), and the relation 1 1 d(x)"! ln x2 , h 4p

(2.159)

we find

G

m 1 [(x!x ) ­xk#(x!x ) ­xk!d (x!x ) ­xk] + kl k kl kj j kk l (x!x )2 k k/1 m d k !+ g (x!x ) Su (x )2u (x )T . (2.160) k 1 1 m m (x!x )2 kl k k/1

1 S¹ (x)u (x )2u (x )T" kl 1 1 m m 2p

H

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43

The anomalous Ward identity considered in the next section takes the form for D"2 ­x1S¹ (x )¹ (x )u(x )u(x )T k kl 1 op 2 3 4

G

H

d "! d(x )­x3#d(x )­x4#d(x )­x2! ­x1[d(x )#d(x )] 13 l 14 l 12 l 13 14 2 l ]S¹ (x )u(x )u(x )T#­x1d(x ) S¹ (x )u(x )u(x )T op 2 3 4 o 12 lp 2 3 4

#­x1d(x )S¹ (x )u(x )u(x )T!d ­x1 d(x )S¹ (x )u(x )u(x )T p 12 lo 2 3 4 op j 12 lj 2 3 4

G

1 1 C ­x1(­x1­x1! d h 1)d(x ) ! 12 l o p 2 op x 24p

H

1 ! (d ­x1#d ­x1!d ­x1 ) h 1 d(x ) Su(x ) u(x )T , 12 4 x lp o op l 3 4 lo p

(2.161)

where C is the central charge. Its solution may be also easily written down using Eqs. (2.155), (2.156) and (2.159). For the sake of convenience in juxtaposition of some of D-dimensional theory results with those of known two-dimensional models, let us list several formulas concerning the transition to complex variables for D"2 1 xB"x1$ix2, ­ " (­ Gi­ ) . B 2 1 2

(2.162)

Any traceless symmetric tensor in two-dimensional space has two independent components. Define the complex components of the tensor » 12 s by the relations k k

(2.163) The contraction of a pair of traceless symmetric tensors » and ¼ has the form 2s~2» 12 s ¼ 12 s"» ¼ #» ¼ . (2.164) k k k k ` ~ ~ ` In particular, we will use complex components of the tensor fields P , with dimensions d#s s PT(x)"PT12 s(x) . s k k They read

(2.165)

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Denote the components of the tensor jx1(x x )"jx112 s(x x ) s 2 3 k k 2 3 as jx1 (x x ). One can show, that sB 2 3

A

B

1 1 xB s 32 [jx1(x x )]s" jx1 (x x )" sB 2 3 2s~1 xB xB 2s~1 B 2 3 12 13

(2.166)

where jx1(x x )"j x1(x x )Gij x1(x x )"xB /xB x B . B 2 3 1 2 3 2 2 3 32 12 13

(2.167)

The components (1/x2)g (x) of the tensor (1/x2)g (x) are B kl 1 1 1 g (x)" [g (x)Gig (x)]"! . B 11 12 x2 x2 (xB)2 Let us rewrite the Ward identities using the complex variables. Introduce ¹ components of the B energy-momentum tensor ¹ (x)"¹ (x)Gi¹ (x) . B 11 12

(2.168)

The Ward identities have the form ­x S¹ (x)u(x )2u (x )T Y B 1 m m

G

H

m m 1 "! + d(x!x )­xk ! ­x + d d(x!x ) Su (x )2u (x )T , k k k B 2 B 1 1 m m k/1 k/1

(2.169)

­ x1S¹ (x )¹ (x )u(x )u(x )T Y B 1 B 2 3 4

G

H

d "! d(x )­x3#d(x )­x4! ­x1 [d(x )#d(x )] S¹ (x )u(x )u(x )T 13 B B 2 3 4 14 B 2 B 13 14 ![d(x )­x2!2­x1d(x )] S¹ (x )u(x )u(x )T 12 B B 12 B 2 3 4 C ­x1 ­x1 ­x1d(x )Su(x )u(x )T . ! 3 4 12p B B B 12 Eqs. (2.155) and (2.156) take the form ¹ (x)"4­ ¹I (x), h¹I (x)"­ ¹ (x) , B B B B Y B where ¹I (x)"1 (¹I (x)Gi¹I (x)) . B 2 1 2

(2.170)

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45

The solution of the Ward identities (2.169) and (2.170) reads, in complex variables S¹ (x)u (x )2u (x )T ` 1 1 m m m m 1 2 d k #+ ­xk Su (x )2u (x )T , + " 1 1 m m 2p (xB!xB) B (xB!xB)2 k k k/1 k/1 d d 4 1 4 1 # # #2 + ­xk S¹ (x )¹ (x )u(x )u(x )T" B 1 B 2 3 4 2p (xB )2 (xB )2 (xB )2 xB B 13 14 12 k/2 1k C 1 ]S¹ (x )u(x )u(x )T# Su(x )u(x )T . B 2 3 4 1 2 8p2 (xB )4 12

G

H

G

(2.171)

H

(2.172)

2.7. Ward identities for the propagators of irreducible fields2 jI and ¹I k kl The conformally invariant propagators should be defined according to the assumptions on the contractions of Euclidean fields

P

dx A (x)j (x), k k

P

dx h (x)¹ (x) . kl kl

(2.173)

In particular, such contractions enter to the equivalence relations (2.27), (2.109) and (2.113). The definition of the propagators depends on the choice of representations (2.15) of the conformal group, which are connected to the fields A , j and h , ¹ . Consider the models satisfying k k kl kl conditions (2.77) and (2.149). Only irreducible components (2.174) jI (x), A-0/'(x), ¹I (x), h-0/'(x) k k kl kl are non-zero in such models, while the components AI (x) and hI vanish: k kl AI (x)"hI (x)"0 . k kl The propagators of the current and the energy—momentum tensor in these models may be chosen from the requirement of finiteness for the contractions

P

dx dy A-0/'(x)Dj (x!y)A-0/'(y), k kl l

P

dx dy h-0/'(x)DT (x!y)h-0/'(y) . kl klop op

(2.175)

As follows from Eqs. (2.31) and (2.114), the latter property depends on the dimension of the space. Here we restrict ourselves to the case of even D52. A somewhat modified recipe may be proposed for the case of odd D. It will be discussed in the other paper. Now consider the propagators of the current. The first contraction is finite, provided that one utilizes the expression (2.32) for e"0. The propagator (2.32), though divergent, is formally transversal for e"0. Its contraction with longitudinal fields enters the first expression (2.175). As 2 The total conformal propagators of current and energy-momentum tensor, see Eqs. (2.52) and (2.95), are discussed in Section 6.

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the result, the integral in Eq. (2.175) has an ambiguity 0]R. To resolve it, let us make use of the regularization (2.30). As eP0, we get from Eq. (2.32): ­x1 Dje (x )D "!e(D!2)~1CI ­x1 (1 x2 )~D`1~e . (2.176) k kl 12 e?0 j l 2 12 It is seen that in the limit e"0 propagator (2.30) contains a finite longitudinal part. Using relation (2.34) for calculation of the limit (2.176) we get the following Ward identity: ­x1 Sj (x )j (x )T"C ­x1h(D~2)@2 d(x ) , (2.177) k k 1 l 2 j l 12 where C is the independent parameter analogous to the central charge, and j CI "1 p~D@2 C(D)C(D/2) C . j 2 j Note that one cannot use the limiting expression (2.177) for the calculation of the contraction (2.175). The latter would break down the conformal invariance. One should calculate the integral of the regularized expression, taking the limit e"0 only after that. Let us stress that such a definition of the propagator is admissible in the theories which are free of electromagnetic interaction, where AI (x)"0. k The condition of equivalence of representations QI and Q-0/' may be written in any of the two j A forms

P

P

jI (x)" dy Dj (x!y)A-0/'(y), A-0/'(x)" dy DA (x!y)jI (y) . k kl l k kl l

(2.178)

The conformally invariant propagators D "S jI jI T and DA "SA A T satisfy kl k l kl k l

P

Sj (x )A (x )T" dx Dj (x !x)DA (x!x )"d d(x ) . k 1 l 2 ko 1 ol 2 kl 12

(2.179)

The integrals in Eqs. (2.178) and (2.179) are calculated with the help of regularization (2.30). The normalization constant C in Eq. (2.28) is calculated from Eq. (2.179) and is equal to A 1 1 (4p)~D@2 1 C " (!1)D@2`1(2p)~D@2 C(D) "(!1)D@2`1 . (2.180) A 2 CI C(D/2) C j j In the course of calculation we have used the integral relation (A.1). It follows from Eqs. (2.178) and (2.179) that in the class of models under discussion one can pass from the Green functions of the current to the Green functions of the field A-0/'. This results in k a number of technical facilities, see Section 4. The Green functions which include several fields A-0/' satisfy the following Ward identities [5]: k C h(D~2)@2 ­xSu(y )u`(y )A (x)A 1(x )2A k(x )T j x k 1 2 k k 1 k k "­xSu(y )u`(y )j (x)A 1(x )2A k(x )T k 1 2 k k 1 k k "![d(x!y )!d(x!y )] Su(y )u`(y )A 1(x )2A k(x )T 1 2 1 2 k 1 k k k # + ­xrrd(x!x ) Su(y )u`(y )A 1(x )2AK r2A k(x )T (2.181) k k r 1 2 k 1 k k r/1

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47

where the notation AK r means that the field A r(x ) is dropped. Here and below A is assumed to be k k r k a longitudinal field A (x),A-0/'(x) . k k The solution to the Ward identity has the form [5]

(2.182)

Su(x )u`(x )A 1(x )2A k(x )T"(g )k jx11(y y )2jx1k (y y ) Su(y )u`(y )T , k 1 2 k k A k 1 2 1 2 1 2 k 1 where

(2.183)

(4p)~D@2 1 g "2C "2(!1)D@2`1 . (2.184) A A C(D/2) C j Consider the propagator of the energy-momentum tensor. The second contraction (2.175) can be made meaningful introducing the dimensional regularization (2.119) and l Ple "D!le "!e. h h T All the above statements concerning the propagator of the current, as well as relations (2.176)—(2.180) are easily generalized to the case of the fields ¹K , h-0/'. In particular, one has for the kl kl regularized propagator:

G

(D!1#e) ­x1DTe (x )"eCI [(D!1#e)(D#1!e)]~1 ­ ­ ­ ! (d ­ #d ­ )h k l o kp o l kl,op 12 T 2(D#2e) ko p

H

1 d ­ h (x2 )~D`1~e . ! 12 (D#e)2 op k

(2.185)

Transition to the limit e"0 with the help of relation (2.34) will result in the following Ward identity: ­x1 S¹ (x )¹ (x )T l kl 1 op 2 D!1 1 "C ­ ­ ­ ! (d ­ #d ­ )h! d ­ h h(D~2)@2 d(x ) , T k o p ko p kp o 12 2D D2 op k

G

H

(2.186)

where C is the analogue of the central charge of two-dimensional conformal models. T When using partial wave expansions, both formulations, either in terms of the fields ¹ or in terms kl of h-0/', are equivalent. However, this transition is possible no sooner than the Ward identities for the kl Green functions of two or more ¹ fields are solved, due to the non-Abelian character of the latter. kl 3. Hilbert space of conformal field theory in D dimensions Let M be a Hilbert space. The requirement of conformal symmetry together with several assumptions of quite general character leads to rigid constraints on its structure. Any field model may be formulated in terms of a certain consistent conditions imposed on the states of the Hilbert space. In this section we study possible types of these conditions and consider the class of conformal models issued by the latter. We demonstrate the existence of the subspace H of the Hilbert space HLM in which the conditions fixing each model are formulated.

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3.1. Model-independent assumptions. Secondary fields Consider a theory described in terms of a definite (finite) sets of fields. The latter fields will be treated as fundamental. In the Lagrangian approach they are the very fields the Lagrangian is constructed from. In our approach no model-dependent assumptions are made. The number of these fields as well as their spin-tensor structure is supposed to be given. For the simplicity, let us discuss the theory including one or two fundamental scalar fields u(x), s(x) having scale dimensions d, *. The u(x) field may be either charged or neutral, and s(x) is the neutral field. Then we make two model-independent assumptions: 1. ¹he existence of the field algebra: U (x )U (x )"+ [U ] . (3.1) i 1 k 2 m m It may be shown [2,12,15,21,22] that the latter result, being indifferent to a choice of bare interactions, is a general consequence of renormalized Schwinger—Dyson system. 2. It is supposed that the field algebra includes the energy—momentum tensor and the conserved currents (in the presence of internal symmetries). The latter satisfy the following conservation laws (in Minkowski space): ­ ¹ (x)"0, ­ j (x)"0 (3.2) k kl k k and have the canonical dimensions l "D, l "D!1. T j The generators of the conformal group are expressed through the components of the energy—momentum tensor. If a symmetry higher than the conformal one will appear to be present in the model, then its generators will also be representable in terms of energy—momentum tensor moments. Analogously, the internal symmetry generators are expressed in terms of local currents. For the simplicity, the Abelian symmetry will be considered here (though the most interesting models arise in non-Abelian case). Let us stress that no model-dependent assumptions on the structure of either the energy-momentum tensor or the currents in terms of fundamental fields are made. Being the “local symmetry generators” of the theory, the current and energy—momentum tensor define the transformation properties of the fields. Equal-time commutators of their components with the fields d(x0!y0) [¹ (x), u(y)], d(x0!y0) [ j (x), u(y)], d(x0!y0) [¹ (x), s(y)] (3.3) 0l 0 0l are considered to be given. Moreover, the algebra of the conformal group fixes the equal-time commutators of energy—momentum tensor components up to gradient terms. An internal symmetry algebra fixes the commutators of currents components (the only terms admissible in an Abelian case are the gradient terms). Thus the equal-time generators [¹ (x ), ¹ (x )], [ j (x ), j (x )] 0k 1 op 2 0 1 k 2

(3.4)

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49

are considered to be known up to gradient terms. Their choice defines different types of models (see below). These two statements, supplied with the requirement of conformal symmetry, would result in a quite specific structure of a Hilbert space resembling that of two-dimensional conformal theories. There exists a special subspace H in a total Hilbert space, which is begot by the energy—momentum tensor and the current. We call this subspace the dynamical sector (see Sections 3.3 and 3.4). The next step consists in a formulation of the dynamical principle that fixes a model. As we show below, under a definite choice of anomalous (gradient) terms in commutators (3.4) the dynamical sector includes special states, which may be set to zero with no contradiction to Ward identities. These states are analogous to null vectors of two-dimensional conformal theories. Each of them defines an exactly solvable model. The simplest models of scalar fields discussed in Refs. [4—6], belong to this class of models, see also Section 4. This program is conducted in three steps. At first, the total Hilbert space of the conformal theory is constructed in the framework of the first statement. As described in Section 1, the Hilbert space can be represented as an infinite direct sum of mutually orthogonal subspaces M` k M"+ =M` . (3.5) k k Each subspace M` is spanned by the states [2,15] k U (x)D0T , (3.6) k where U (x) is any field entering the algebra (3.1). The states (3.6) for each k form a basis of the space k of irreducible representations of the conformal group. From this fact the orthogonality of subspaces M` follows k S0DU (x )U (x )D0T"0 for mOk . (3.7) m 1 k 2 The subspace of states u(x)D0T, s(x)D0T

(3.8)

also enter the sum (3.5). Thus the set of states (3.6) for all k may be considered as a basis of a total Hilbert space M. All the states of fields u and s of the type u(x )u(x )D0T, u(x )s(x )u(x )D0T, u(x )u(x )u(x )D0T, 2 1 2 1 2 3 1 2 3 as well as the states

(3.9)

U (x )u(x )D0T, U (x )U (x )D0T (3.10) m 1 2 m 1 k 2 are decomposed into this basis owing to the statement (3.1) on the conformal field algebra, see Eq. (1.59). At the next step the second statement, that is, the existence of the energy—momentum tensor and the current, is studied. The principal result consists in the fact that the choice of commutators (3.3) and (3.4) together with the conditions of conformal symmetry makes possible to find the states

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of the type j (x )u(x )D0T, j (x )j (x )u(x )D0T, 2, (3.11) k 1 2 k 1 l 2 3 ¹ (x )u(x )D0T, ¹ (x )¹ (x )u(x )D0T, ¹ (x )s(x )D0T, 2, (3.12) kl 1 2 kl 1 op 2 3 kl 1 2 and to obtain their expansion in the basis (3.6). This result is justified rigorously under the following additional constraint: a current and energy—momentum tensor in Euclidean space transform by irreducible representations of the conformal group. This constraint singles out a class of models of direct (non-gauge) interaction of the matter fields for D52 and is expressed by Conditions (2.77) and (2.149) (for D"2 the latter hold identically). As shown in the Section 2, the Green functions S j u2u`T, S j j u2u`T, S¹ u2sT, S¹ ¹ u2sT, 2, (3.13) k kl kl kl op and, therefore the states (3.11) and (3.12) are uniquely determined by the Ward identities, provided that Conditions (2.77) and (2.149) hold. The expansion of the states (3.11) and (3.12) in the basis (3.6) is formulated in terms of Euclidean operator product expansions. In particular, when applied to the states j (x )u(x )D0T, ¹ (x )u(x )D0T k 1 2 kl 1 2 these expansions read (see Section 2 and [2,3,15] for more detail) j (x )u(x )"+ [Pj] , k 2 1 s s

(3.14)

¹ (x )u(x )"+ [PT] , (3.15) kl 2 1 s s where Pj and PT are symmetric traceless tensor fields of the rank s with the scale dimensions s s dj"dT"d#s . s s This result is the consequence of the Ward identities, see below. In what follows we use the unified notation (3.16) P (x)"MPj(x), PT(x)N"P 12 s(x), d "d#s . s s s s k k The fields Pj and PT may be either orthogonal to each other (i.e. SPj(x )PT(x )T"0) or not, s s s 1 s 2 depending on the model. When s"0 both these fields coincide with the fundamental field P (x)D "u(x) . (3.17) s s/0 The fields PT exist only for a definite type of gradient terms in commutators (3.4); see below. s The fields PT and Pj have the transformation properties similar to those of secondary fields of s s two-dimensional conformal theories. The part way evidence to this is the presence of non-zero Green functions SPju`j T, SPTu`¹ T, sO0 , s k s kl which satisfy anomalous Ward identities. The Green functions SPju`j T as well as the Ward s k identities for there functions are given above, see Eqs. (2.85) and (2.86), and they are discussed in more detail in Refs. [2,3,15], where the explicit expressions for the Green functions SPTu`¹ T s kl

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

51

may be also found. The commutators of fields Pj with the component j are expressed through s 0 the fields Pj in the usual manner and have, beyond that, anomalous contributions of the s fields Pj (x), where s@"0, 1, 2, s!1. The corresponding Ward identities also have anos{ malous contributions of these fields. In Section 4 we give the examples of such anomalous Ward identities. The commutators of fields PT with energy—momentum tensor and current components include s anomalous operator terms d(x0!y0)[¹ (x), PT(y)] 0k s s "id(D)(x!y)­ PT(y)# + HK s,m(­o , ­y)d(D)(x!y)PT (y), k"0, 1,2, D!1 . (3.18) k s k x s~m m/1 The analogous terms enter the commutator [j (x), Pj(y)]. The HK s,m operators in Eq. (3.18) are the 0 s k differential operators of rank s#1 ­ ­ ­o " , i"1, 2, 2, D!1; ­y" , k"0, 1, 2, D!1 . x ­xi ­yk The ­: derivatives act on the argument of a field PT (y) only. The HK operators are the sums of s~m terms of the type (­o )m~r`1 d(x!y) (­y)s~m`r, r"0, 1, 2, m , x which, in principle, could be calculated for any class of models. In the same manner, conformal ¼ard identities for the Green functions GPs 12 s(xx 2x )"S¹ (x)PT12 s(x )u(x )2u (x )T kl,k k 1 m kl k k 1 2 m m

(3.19)

satisfy the anomalous Ward identities

C

m d m ­xGPs 12 s(xx 2x )"! + d(x!x )­xk! + ­xd(x!x ) l kl,k k 1 m k k k D k k/2 k/1 #HT (­x, d(x!x ); ­x1) SP (x )u(x )2u(x )T s,0 1 s 1 1 m s # + HT (­x, d(x!x ), ­x1) SP (x )u(x )2u(x )T , s,k 1 s~k 1 1 m k/1 where HT are the differential operators which consist of the terms s,k

(3.20)

(­x)k`1~r d(x!x ) (­x1)r, r"0, 1, 2, k . 1 The explicit form of these differential operators may be derived from the Ward identities for the conformally invariant Green functions S¹ (x)P (x )P (x )T, s@"0, 1, 2, s!1 . kl s 1 s{ 2

(3.21)

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These expressions are in general very cumbersome. As an example consider the Green functions S¹ uP T. The corresponding formulae were found explicitly in Refs. [3,15]. They read: kl s

G

SP (x )u(x )¹ (x )T"gT (x2 )~(D~2)@2 (x2 )~(D~2)@2 s 1 2 kl 3 s 12 23

C

D

2 ] ­on x3 ­on x3! (0 ­x3 ­o x3#0 ­x3 ­o x3)!trace k l l k (D!2) k l

H

](x2 )~(D~2)@2 jx112 s (x x ) Su(x )u(x )T#2 , 13 1 2 k k 3 2

(3.22)

where ­on "­o !0 ­ , gT is the coupling constant, and dots stand for quasilocal terms, see Refs. [3,15]. s The Green functions (3.22) satisfy the Ward identities ­x3SP 12 s(x )u(x )¹ (x )T"HT (­x3, d(x ), ­x1) Su(x )u(x )T . 2 kl 3 s,s 13 1 2 l k k 1

(3.23)

The general expression for the operators HT for s52 depends on two parameters [3,15]. One has s,s for s"1; D!2 HT & ­x31­x3d(x )#d 1 h 3d(x ) 1,1 13 k k 13 kk x D

C

D

D 2 ­x31d(x )­x1#d 1 ­x3d(x )­x1! ­x3d(x )­x11 . ! 13 k k kk l 13 l 13 k d D k

(3.24)

The fields having anomalous commutators of the type (3.18) will be called secondary fields generated by the primary field u(x). A complete set of secondary fields will be studied below. Let us remark that the origin and transformation properties of these fields are analogous to those of the secondary fields in two-dimensional conformal theories. Note also that no field other than fields (2.15) can enter the operator expansions (2.14) and (2.15). Indeed, suppose that some field U "Ul0 with l Od#s is present in the expansion (2.15). Then 0 s 0 there exists a non-vanishing Green function SU u¹ T. The Ward identity for this function has the 0 kl form: ­x3SU (x )u(x )¹ (x )T"M!d(x )­x1!d(x )­x2#2N SU (x )u(x )T"0 . k 0 1 2 kl 3 13 l 23 l 0 1 2

(3.25)

In the last equality the orthogonality condition SU (x )u(x )T"0 if l Od, sO0 0 1 2 0

(3.26)

has been used. Consequently, the Green function SU u¹T is either transversal, or vanishes. 0 However, the transversal functions cannot exist in theories satisfying the condition (2.149). Hence it follows that SU (x )u(x )¹ (x )T"0 if l Od, sO0 . 0 1 2 kl 3 0

(3.27)

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

53

The fields PT is an exception since its Green functions satisfy anomalous Ward identities. The r.h. s sides of the latter contain the terms &Su(x )u(x )T besides the “usual” terms accounted for in 1 2 Eq. (3.25). Though, such anomalous terms are admissible only for the fields with dimensions d#s, see Ref. [3,15] for details. 3.2. Green functions of secondary fields Consider the higher Green functions of the fields Pj. In the theories satisfying the conditions s (2.77) the latter are defined by Eq. (1.87): SPj(x )u(x )2u`(x )T s 1 2 2n

P

dx dy CI l 12 s(x xy)Sj (y)u(x)u(x )2u`(x )T , (3.28) "Kj res 1k,k k 1 k 2 2n s l/dBs where the functions CI l 12 s are given by the expressions (2.69) and (2.72), and may be represented 1k,k k as CI l 12 s(x x x )"SUl 12 s(x )uJ (x )A-0/'(x )T"­x3CI l 12 s(x x x ) , k k 1 2 k 3 k k k 1 2 3 1k,k k 1 2 3 where

(3.29)

CI l 12 s(x x x )"(2p)~D@2 2(s~1)@2 NI (p, d) jx112 s (x x )DI l (x x x ) . 1 k k 3 2 j 1 2 3 k k 1 2 3 Substitute Eq. (3.29) into Eq. (3.28) and bring it into the form

(3.30)

SPj(x )u(x )2u`(x )T s 1 2 2n

P

dx dy CI l 12 s(x xy)­y Sj (y)u(x)u(x )2u`(x )T . (3.31) "!Kj res k k 1 k k 2 2n s l/dBs The integral in the r.h.s. is taken as follows. Applying the Ward identity, represent integrand expression as a sum of terms, each term containing the factor d(y!x ). The term containing i d(x!y) does not contribute since it is multiplied by the power of the difference (x!y). Resultantly, the integral over y evaluates due to the factors d(y!x ). As for the integral over x, note that the i function DI l (x xy) entering Eq. (3.30) contains the factor [(x !x)]~(D`l~d~s)@2, see Eq. (2.72). This j 1 1 factor is singular in the limit lPd#s. Represent the expression (3.30) as the sum of terms containing this factor and then thread the symbol res through the integral sign. Calculating l/d`s the pole of the integrand, we obtain the sum of terms each containing derivatives of d(x !x). After 1 evaluation of the integral over x, Eq. (3.31) may be rewritten in the form [5,15] SPj 12 s(x )u(x )2u`(x )T"PK j 12 s (x, ­ 1) Su(x )2u`(x )T , (3.32) k k 1 2 2n k k x 1 2n where PK j 12 s (x, ­ 1) is known polynomial of rank s in derivative the ­ 1, its coefficients being k k x x dependent on the differences (x !x ), k"2, 2, 2n. As an example we display its expression for 1 k s"1:

C

D

2n (x ) n (x ) 1k k . 1k k#2d + PK j (x, ­ 1)& ­x1!2d + k x k x2 x2 1k 1k k/n`1 k/2 Next section contains expressions in more complicated cases.

(3.33)

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The Green functions of the fields PT are defined by Eq. (2.150) upon the substitution s p "(d#s, s). The functions CI p entering Eq. (2.150) are orthogonal to the functions Cp , see k kl 3kl Eq. (2.151), and can be represented in the form (see Refs. [3,6,15] for more details) CI p "CI l 12 s(x x x )"SUl 12 s(x )uJ (x )hI -0/'(x )T kl kl,k k 1 2 3 k k 1 2 kl 3 2 "­x3 Bl 12 s(x x x )#­x3 Bl 12 s(x x x )! d ­x3 Bl 12 s(x x x ) , k l,k k 1 2 3 l k,k k 1 2 3 D kl j j,k k 1 2 3

(3.34)

where Bl 12 s is the conformally invariant function k,k k Bl 12 s(x x x ) k,k k 1 2 3 "SUl 12 s(x )uJ (x )h (x )T k k 1 2 k 3 1 s & jx3(x x ) jx112 s(x x )#aJ + g k(x )jx112 L k2 s(x x )!traces k 1 2 k k 3 2 s x2 kk 13 k k k 3 2 13 k/1 ]DI l (x x x ) , (3.35) h 1 2 3 uJ is the conformal partner of the field u, h is the conformal vector of dimension d "!1, aJ is the k h s constant,

G

C

DI l (x x x )"(x2 )~(l~d~s`D`2)@2 (x2 )~(l~d~s`D~2)@2 (x2 )~(l`d~s~D~2)@2 . h 1 2 3 12 13 23 Substitute Eq. (3.34) into Eq. (2.150) and bring it to the form

DH

(3.36)

SPT12 s(x )u(x )2u(x )T k k 1 2 m

P

"!2KT res dx dy Bl (x xy)­yS¹ (y)u(x)u(x )2u(x )T . (3.37) s k 1 l kl 2 m l/d`s The integral in the r.h.s. is calculated in the same manner as the integral (3.31). One can show that the resulting expression takes the form (3.38) SPT12 s(x )u(x )2u(x )T"PK T12 s (x, ­ 1) Su(x )2u(x )T , 2 m k k x 1 m k k 1 where PK T12 s (x, ­ 1) is known polynomial of rank s#1 in derivatives. Its expression is presented in k k x the fourth section for the case s"1, m"4. 3.3. Dynamical sector of the Hilbert space Consider the states j (x )j (x )u(x ) D 0T, ¹ (x )¹ (x )u(x ) D 0T . (3.39) k 1 l 2 3 kl 1 op 2 3 According to Eq. (3.1), the basis vectors of a Hilbert space may be found from expansion of the states j (x )Pj(x ) D 0T, ¹ (x )PT(x ) D 0T . k 1 s 2 kl 1 s 2

(3.40)

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55

It is possible to bring the result into the form of operator expansions3 j (x )Pj(x )"+ [Pj ]#+ [Pj,s] , k 1 s 2 s{ s{ s{ s{

(3.41)

¹ (x )PT(x )"+ [PT]#+ [PT,s] , kl 1 s 2 s{ s{ s{ s{ where

(3.42)

Ps (x)"MPj,s, PT,sN"Ps 12 s{(x) s{ s{ s{ k k is a new family of symmetric traceless tensor fields with dimensions

(3.43)

ds "d#s@. s{ The fields Ps have the same properties as the P fields, see Eqs. (3.18) and (3.20). s{ s Repeating the above steps again and again, we end up with a family of fields (3.44) P , Ps1, Ps1s2, 2, Ps12sk, 2, s #s #2#s 4s s 1 2 k s s s spanning the basis of a subspace of total Hilbert space, to which all the states (3.11) and (3.12) belong. The dynamical principle which governs the effective interaction, is formulated in this subspace, called below the dynamical sector of a Hilbert space. The states ¹ (x )s(x ) D 0T, ¹ (x )¹ (x )s(x ) D 0T,2 kl 1 2 kl 1 op 2 3 also belong to this sector. Since the dynamical sector is generated by energy-momentum tensor and the current, an introduction of any consistent model-fixing condition on the states of this sector may be regarded as a means of specification of an effective Hamiltonian. For a number of simplest models the constraint on the states of the dynamical sector can be obtained directly from the initial Lagrangian [4,15]. For this purpose one introduces the conformally invariant regularization. The renormalization constants z , z , z (in models with two fields) remain finite as long as the 1 2 3 regularization is kept up. In the renormalized Schwinger—Dyson equations the term z c is held, 1 leading to the greater transparency of the equations [4,15]. The above discussion is a subject of extensive studies in Refs. [2—6,15]. The fields P were first s found in our works [12,32] as a consequence of conformally invariant solution to Ward identities and later they were discussed in Refs. [2,33,34]. Note that the fields (3.44) have the properties analogous to those of the secondary fields in two-dimensional theory. In the next section we show that for D"2 the fields (3.44) literally represent covariant combinations of secondary fields. Adopting the terminology of two-dimensional theories [16], these fields can be viewed as the conformal family of fields generated by a primary field u(x).

3 These expressions are written in a formal style. In fact, for each pair of values s,s@'s there may exist several fields Pss{ orthogonal with each other. The number of those depends on s, s@.

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The dynamical sector might be completely fixed only when the anomalous contributions to commutators (3.4) are given. The assignments of definite values to commutators (3.3) and (3.4) may be thought as the way by which the quantization rules are taken into account effectively, leading to a consistent definition of renormalized Schwinger—Dyson system, see Refs. [5,15]. For this purpose it proves necessary to define operator product expansions j (x )j (x ), ¹ (x )¹ (x ), j (x )¹ (x ) . (3.45) k 1 l 2 kl 1 op 2 k 1 op 2 It is apparent that the contributions to commutators (3.4) are solely due to C-number terms of these expansions, or to operator terms comprising the fields having integer dimensions (which cannot exceed dimensions of a current or energy—momentum tensor) and a definite tensor structure (different for spaces with even and odd dimensions). In Refs. [3,4,15] the following anomalous contributions are considered: j (x )j (x )"[C ]#[P ]#[¹ ]#2, (3.46) k 1 l 2 j j jq ¹ (x )¹ (x )"[C ]#[P ]#[¹ ]#2, (3.47) kl 1 op 2 T T jq ¹ (x )j (x )"[ j ]#2, (3.48) kl 1 o 2 p where [C ] and [C ] are the C-number contributions to expansions. The constants C and j T j C define the normalization of Euclidean Green functions T S j (x )j (x )T, S¹ (x )¹ (x )T . (3.49) k 1 l 2 kl 1 op 2 In the class of theories under consideration, the dependence on C and C appears only in spaces j T with even dimensions, see Eqs. (2.177) and (2.186). The second terms in Eq. (3.46) and in Eq. (3.47) denote anomalous operator contributions of the scalar fields P (x) and P (x), for which the following unified notation is useful: j T P(x)"MP (x), P (x)N . (3.50) j T Both fields have the same dimension d j"d T"D!2 . P P The conformally invariant Green functions of the fields (3.50) have the form SP (x )P (x )T&SP (x )P (x )T&(x2 )~D`2 , j 1 j 2 T 1 T 2 12

(3.51)

(3.52)

A

B

x2 (D~2)@2 12 Su(x )u`(x )P (x )T&Su(x )u`(x )P (x )T"gP Su(x )u`(x )T , (3.53) 1 2 j 3 1 2 T 3 T x2 x2 1 2 13 23 where gP is the coupling constant. T In two-dimensional space the fields P and P become constants j T P (x)D "gP, P (x)D "gP , (3.54) j D/2 j T D/2 T SP(x )P(x )TD "const., Su(x )u`(x )P(x )TD &Su(x )u(x )T . 1 2 D/2 1 2 3 D/2 1 2 The two leading contributions in Eqs. (3.46) and (3.47) become c-numbers, their sum coinciding with the term proportional to the central charge of two-dimensional theory.

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

57

Note that for even D54 the fields P (x) and ¹ (x) are secondary fields generated by the T kl c-number contribution C . Similarly, the fields P (x) and j (x) are secondary fields generated by C . T j k j In conclusion we list the anomalous Ward identities based on the expansions (3.46)—(3.48). They are derived from the conditions of conformal invariance of the r.h.s. and have the following form [3,4,15] (see Ref. [35] for more details) ­x1Sj (x )j (x )u(x )u`(x )T k k 1 l 2 3 4 "![d(x )!d(x )] Sj (x )u(x )u`(x )T 13 14 l 2 3 4 #C ­x1h(D~2)@2 d(x )Su(x )u`(x )T#­x1d(x )SP (x )u(x )u`(x )T , (3.55) j l 12 3 4 l 12 j 2 3 4 ­x1S¹ (x )¹ (x )u(x )u(x )T k kl 1 op 2 3 4 "C M­ ­ ­ !((D!1)/2D) (d ­ #d ­ )h!(1/D2) d ­ hN h(D~2)@2 d(x )Su(x )u(x )T T l o p lo p lp o op l 12 1 2 3 4 1 !Md(x )­x #d(x )­x !(d/D)­x [d(x )#d(x )]N S¹ (x )u(x )u(x )T 13 l 14 l l 13 14 op 2 3 4 #MFK (­x1, ­x2) SP (x )u(x )u(x )TN T 2 3 4 l,op #M[!d(x )­x2#(1!(2/D) a)­x1d(x )] S¹ (x )u(x )u(x )T l 12 op 2 3 4 12 l #1 (1#a) [­x1 d(x ) S¹ (x )u(x )u(x )T 2 o 12 lp 2 3 4 #­x1d(x ) S¹ (x )u(x )u(x )T!(2/D) d ­x1 d(x ) 12 p 12 lo 2 3 4 op j 1 ]S¹ (x )u(x )u(x )T]#1 (a!1) [d ­x d(x ) 12 lo j jl 2 3 4 2 ]S¹ (x )u(x )u(x )T#d ­x1 d(x ) S¹ (x )u(x )u(x )T jp 2 3 4 lp j 12 jo 2 3 4 !(2/D) d ­x1 d(x ) S¹ (x )u(x )u(x )T]N (3.56) op j 12 lj 2 3 4 where a as a free parameter and FK is the following differential operator: l,o,p 1 1 (2#f )­x1­x1 d(x )­x2 FK (­x1, ­x2)" (1#f ) ­x1­x1­x1 d(x )# l o p 12 l p 12 o l,op 2 (D!2)

C

1 1 ! [ f D#(D#2)]­x1­x1 d(x )­x2# d ­x1h 1d(x ) 12 o p 12 l 2(D!2) 2 lo p x

A

B

D2#2D!2 1 Df# d ­x1d(x )h 2 ! lo p 12 x D!1 (D!2)2 1 D # d h 1d(x )­x2! d ­x1­x1d(x )­x2 12 p 12 q (D!2) lo x 2(D!2) lo p q D 1 ! ­x1d(x )­x2­x2# ­x1d(x )­x2­x2 p 12 l o 12 o q 2(D!1)(D!2) (D!1)(D!2) l

D

D ­x1d(x )d ­x2­x2 #(o % p)!trace in o, p . ! 13 lo p q 2(D!1)(D!2) q

(3.57)

Here (o % p) stands for the expression in square brackets with o, p indices interchanged, f is a free parameter.

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There are four groups of terms (each placed between curly brackets) of different origin in this identity. The second group is due to common contribution of the commutator d(x0!y0)[¹ (x), u(y)]. The third group is the PK field contribution. The fourth group consists of 0l T anomalous terms due to anomalous gradient corrections to the commutator [¹, ¹], which contain the energy—momentum tensor components. On account of Eq. (3.57), four free parameters enter this Ward identity (for even D): C , f, gP , a . T T

(3.58)

One can show that conformally invariant Ward identities corresponding to the expansion (3.48) have the following form [15,35]: ­x1S¹ (x )j (x )u(x )u`(x )T k kl 1 p 2 3 4

G

H

d "! d(x )­x3#d(x )­x4! ­x1[d(x )#d(x )] 13 l 14 l 13 14 D l

]Sj (x )u(x )u`(x )T!d(x )­x2 Sj (x )u(x )u`(x )T p 2 3 4 12 l p 2 3 4

A

B

2 # 1! b ­x1 d(x ) S j (x )u(x )u`(x )T l 12 p 2 3 4 D #b­x1 d(x ) Sj (x )u(x )u`(x )T#(b!1) d ­x1 d(x ) Sj (x )u(x )u`(x )T , p 12 l 2 3 4 lp q 12 q 2 3 4

(3.59)

where b is the free parameter, ­x2S¹ (x )j (x )u(x )u`(x )T p kl 1 p 2 3 4 "![d(x )!d(x )] S¹ (x )u(x )u`(x )T 23 24 kl 1 3 4

G

#b ­x2d(x ) Sj (x )u(x )u`(x )T#­x2d(x ) Sj (x )u(x )u`(x )T k 12 l 1 3 4 l 12 k 1 3 4

H

2 ! d ­x2d(x ) Sj (x )u(x )u`(x )T . 12 q 1 3 4 D kl q

(3.60)

The free parameter enters both identities in such a way as to produce the same results after taking the derivatives ­x2 in Eq. (3.59) and ­x1 in Eq. (3.60). p k 3.4. Null states of dynamical sector Let us show that any dynamical model may be defined by a certain self-consistent constraint on the states of dynamical sector. To do this, let us compose the superpositions of the fields (3.44) having equal scale dimensions, see Eq. (3.85). Denote these superpositions as Q . Let us choose the s coefficients in the superpositions in a way that ensures the “normal” form of its commutators with j and ¹ , i.e. the absence of anomalous terms (which are, for example, present in Eq. (3.18)). Then 0 0l

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

59

the field Q (x) transforms as a primary field: s d(x0!y0) [ j (x), Q (y)]"!d(D)(x!y)Q (y) , 0 s s d(x0!y0) [¹ (x), Q (y)]"id(D)(x!y)­ Q (y)#2 . (3.61) 0k s k s In the last commutator the dots stand for the gradient term &­xd(D)(x!y)Q (y). k s Such a form of the commutators is guaranteed by the following self-consistency conditions: (3.62) SQ j Ps12skj T"0 or SQT Ps12sk¹ T"0 k s s{ kl s s{ for all s@"0, 1, 2, s!1. These conditions ensure the cancellation of anomalous contributions into the commutators (3.61) and, simultaneously, lead to the system of algebraic equations on the coefficients of the superposition Q . This is discussed in detail in the next section using particular s examples. Note that on account of operator product expansions of the type (3.42) the self-consistency conditions (3.62) may be replaced by the following equations: SQju`j 12j nT"0, n"1, 2, 2, s k k

(3.63)

or SQTu¹ 1 12¹ n nT"0, n"1, 2, 2, kl s kl which are equivalent to the set of conditions (3.62) for all values of s@:

(3.64)

s@"0, 1, 2 Suppose that conditions (3.62) are satisfied. Then the equation Q (x)"0 (3.65) s gives a self-consistent condition on the states of dynamical sector. Any such equation defines a certain exactly solvable model. The Lagrangean models also belong to this family, see Refs. [2—4,15]. Consider the simplest model. It is defined by the requirement of vanishing of the field P "Pd`1 k k having the scale dimension d "d#1 1 P (x)"P (x)D "0 . (3.66) k s s/1 This equation means that the states P (x) D 0T disappear in the dynamical sector k P (x) D 0T"0 . (3.67) k The Euclidean Green functions Su(x )2u`(x )s(x ) s(x )T 1 2n 2n`1 2 2n`m will satisfy the following system of differential equations: SP (x )u(x )2u`(x )s(x ) s(x )T"0 , k 1 2 2n 2n`1 2 2n`m

(3.68)

(3.69)

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or, owing to Eq. (3.32) PK (x, ­x)Su(x )2u`(x )s(x ) s(x )T"0 . (3.70) k 1 2n 2n`1 2 2n`m The latter is a vector equation. Thus one has a system of differential equations for each Green function. Consider the equations PK (x, ­x) Su(x )u`(x )s(x )T"0 , (3.71) k 1 2 3 PK (x, ­x) Su(x )u`(x )j (x )T"0 , (3.72) j,k 1 2 l 3 PK (x, ­x) Su(x )u`(x )¹ (x )T"0 . (3.73) T,kl 1 2 op 3 The PK and PK operators depend on parameters of anomalous terms entering Ward identities. j,k T,kl Since the coordinate dependence of three-point functions is known, we get the equations on the free parameters of the theory, i.e. the scale dimensions d, D

(3.74)

and the parameters entering anomalous Ward identities. (One can show that the additional constraints on the parameters appear during the solution.) Depending on the choice of anomalous terms in Ward identities (3.55) and (3.56) the following three variants of a model are possible:4 Pj (x)#bPT(x)"0, Pj (x)"0, PT(x)"0 , (3.75) k k k k where b is a constant. All these variants are dealt with in Refs. [4,5,15], see also Section 4. In the second and third cases one of the two equations (3.72) and (3.73) survives. A more complicated model is given by the equation Q (x)"P (x)#aPM (x)"0 , kl kl kl where a is unknown parameter and

(3.76)

. P (x)"P (x) D , PM (x)"Ps1(x) D s s/2,s1/1 kl s s/2 kl Eq. (3.76) means vanishing of corresponding states in dynamical sector: Q (x) D 0T"0 . kl The Green functions (3.68) satisfy differential equations

(3.77)

QK (x, ­x)Su(x )2u`(x )s(x ) s(x )T"0 , kl 1 2n 2n`1 2 2n`m where

(3.78)

QK (x, ­x)"PK (x, ­x)#aPMK (x, ­x) , kl kl kl

(3.79)

4 In two-dimensional space the PT(x) field is absent. For D53 it appears under the definite choice of anomalous k operator terms.

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61

and PK (x, ­x) is the operator which defines the field P (x), see Eq. (3.32) or Eq. (3.38); PMK (x, ­x) is kl kl kl analogous operator, defining PM (x). The free parameters (3.74) as well as the parameters in kl anomalous Ward identities are, similar to an above model, calculated from the equations QK (x, ­x) Su(x )u`(x )s(x )T"0 , kl 1 2 3 QK (x, ­x) Su(x )u`(x )j (x )T"0 , j,kl 1 2 j 3 QK (x, ­x) Su(x )u`(x )¹ (x )T"0 . T,kl 1 2 jp 3 Moreover, the following self-consistency conditions are present in this model:

(3.80) (3.81) (3.82)

QK 1 (x, ­x) Su(x )P`(x )j (x )T"0, QK 1 (x, ­x) Su(x )P`(x )¹ (x )T"0 , (3.83) P ,kl 1 o 2 j 3 P ,kl 1 o 2 jp 3 fixing the value of the a parameter. These conditions are non-trivial due to Eq. (3.18) and the anomalous Ward identities for Green functions (3.19). As before, the three variants of a model are possible: Qj (x)#bQT (x)"0, Qj (x)"0, QT (x)"0 . (3.84) kl kl kl kl As an example, in Section 4 we discuss the solution of the model defined by equation Qj (x)"0. kl Finally, let us consider a general case s~1 Q (x)"P (x)# + + a 1 2 k Ps1,2,sk(x) , s s s , ,s s k/1 s1,2,sk where

(3.85)

s 4s 424s , s #s #2#s 4s!1 . 1 2 k 1 2 k The coefficients a 1 2 k are determined by consistency conditions (3.62) for each of the fields s , ,s 2 Ps1, ,sk(x) where s #2#s 4s@(s . 1 k s{ No more principal differences with the previous model exist. Some of the models are equivalent to Lagrangian models [15]. A possible conjecture is that the three-dimensional Ising model corresponds to one of the solutions of a model QT "0 . kl The constraints on the states of dynamical sector, viewed as a means to define a model, were first studied by the authors in 1978. It was shown [2,14,36] that the solutions of trivial models (the Thirring model and gradient model for D"4) are defined by these conditions. Later [33] this problem was examined in a slightly different context. However, the far better understanding of this scheme, especially its features related to the necessity of introduction of fields Ps1,2,sk [3] into s dynamical spectrum and to the role of self-consistency conditions, has come to us after the works [16,18]. The scheme described above has a striking resemblance to the structure of two-dimensional models. The states Q (x) D 0T s

(3.86)

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are analogous to null vectors of two-dimensional models. As will be shown in the next section the states (3.86) for D"2 literally coincide with null vectors, and all the two-dimensional models known to the present time may be solved by the method described in this work without any reference to Virasoro algebra, as if we were completely unaware of its existence. Having in mind the explicit analogy between two-dimensional models and those described here, it looks quite probable that the latter models should correspond to yet unknown realization of D-dimensional analogue of Virasoro algebra.

4. Examples of exactly solvable models in D-dimensional space We consider a solution of several models discussed above under assumption that only a cnumber analogue of the central charge exists, while the operator ones are absent: P (x)"P (x)"0 . (4.1) j T In the space of even dimension D54 it proves useful to work in terms of the potential A (x),A-0/'(x) rather than Abelian current j , see Section 2.7. The self-consistency conditions k k k (3.63) take the form SQju`A 12 A kT"0, k"1, 2, 2 . s k k

(4.2)

4.1. A model of a scalar field To illustrate the main ideas and calculation specifics we start with the simplest model in the space of even dimension D54 defined by equation (see also [5,15]) Pj (x)"0 . (4.3) k This model is a scalar version of the pure gauge model discussed in Ref. [36]. It is presented here due to methodical reasonings. Due to Eq. (3.31), the Green functions of the field Pj are calculated from the equation k SPj (x )u(x )2u`(x )T k 1 2 2n

P

dy dy CI l (x y y )­y2S j (y )u(y )u(x )2u`(x )T , res 1 2 k 1 1 2 l l 2 1 2 2n l/d`1 where " is a constant and 1 CI l (x x x )"jx1(x x )(x2 )~(l~d~1`D)@2 (x2 )~(l`d~1~D)@2 (x2 )(l`d~1~D)@2 . k 3 2 12 13 23 k 1 2 3 Using the Ward identities (1.53) one can transform the r.h.s. of Eq. (4.4) to get "!K

1

!K res 1 l/d`1

P C dy

1

D

n 2n + CI l (x y x )! + CI l (x y x ) Su(y )u(x )2u`(x )T . k 1 1 r k 1 1 r 1 2 2n r/2 r/n`1

(4.4)

(4.5)

(4.6)

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63

Note that the term containing d(y !y ) is omitted: it is multiplied by the power factor 1 2 [(y !y )2](l`d~1~D)@2 and does not contribute to the integral. Let us thread the residue symbol 1 2 res through the integral sign and make use of relation l/d`1 (x ) 1 pD@2 (D#2) !­x2#2d 13 k d(x ) , (4.7) res CI l (x x x )" k 12 k 1 2 3 x2 2 D#4 13 l/d`1 C 2

A

C

B

D

which follows from Eq. (2.34) for k"0. As the result we obtain SPj (x )u(x )2u`(x )T k 1 2 2n

C

D

n (x ) 2n (x ) 1r k#2d + 1r k Su(x )2u`(x )T , "K (D#2)c ­x1!2d + 1 k 1 2n x2 x2 1r 1r r/2 r/n`1 where we introduced 1 c" 2

pD@2 . D#4 C 2

A

(4.8)

(4.9)

B

Eq. (4.3) leads to differential equations of the first order for all the Green functions of the model. For n"2 we get the equation

G

C

DH

(x ) (x ) (x ) ­x1#2d ! 12 k# 13 k# 14 k k x2 x2 x2 12 13 14 It has the solution

Su(x )u(x )u`(x )u`(x )T"0 . 1 2 3 4

A

(4.10)

B

x2 x2 d (4.11) Su(x )u(x )u`(x )u`(x )T"Su(x )u`(x )T Su(x )u`(x )T 12 34 , 1 2 3 4 1 3 2 4 x2 x2 14 23 where Su(x )u`(x )T"(1x2 )~d. 1 2 2 12 Consider the self-consistency conditions (4.2) for the model (4.3). One must find all the Green functions SPj u`A 12A kT, k"1, 2, 2. Consider the simplest case k"2 first. We have k k k SPj (x )u`(x )A 1(x )A 2(x )T k 1 2 k 3 k 4

P

dy dy CI l (x y y )­y2Sj (y )u(y )u`(x )A 1(x )A 2(x )T . res 1 2 k 3 k 4 1 2 k 1 1 2 l l 2 l/d`1 Employ the Ward identity: "!K

1

­x Sj (x)u(x )u`(x )A 1(x )A 2(x )T l l 1 2 k 3 k 4 "[!d(x!x )#d(x!x )] Su(x )u`(x )A 1(x )A 2(x )T 1 2 1 2 k 3 k 4 #­x1d(x!x ) Su(x )u`(x )A 2(x )T#­x2 d(x!x ) Su(x )u`(x )A 1(x )T . 3 1 2 k 4 k 4 1 2 k 3 k

(4.12)

(4.13)

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As before, the first term &d(x!x ) does not contribute to the integral (4.12), 1

P

SPj (x )u`(x )A 1(x )A 2(x )T"K res dy M­x31 CI l (x yx ) Su(y)u`(x )A 2(x )T k k 1 3 k 1 2 k 3 k 4 1 2 k 4 l/d`1 #­x42 CI l (x yx ) Su(y)u`(x )A 1(x )T 2 k 3 k k 1 4 !CI l (x yx ) Su(y)u`(x )A 1(x )A 2(x )TN . (4.14) k 1 2 2 k 3 k 4 Let us substitute the explicit expressions for the Green functions Suu`A T, Suu`A 1A 2T from k k k Eq. (2.183): Su(x )u`(x )A 1(x )A 2(x )T"Su(x )u`(x )T SA 1(x )A 2(x )T 1 2 k 1 k 2 1 2 k 1 k 2 3 #(g )2jx1(x x )jx42(x x ) Su(x )u`(x )T . (4.15) A k 1 2 k 1 2 1 2 Note that disconnected component of this Green function does not contribute to Eq. (4.14) due to Eq. (4.3) for n"1:

C

D

(x ) ­x1#2d 12 k Su(x )u`(x )T"0 . 1 2 k x2 12 Recalling also that due to Eq. (4.7)

(4.16)

1 g (x )d(x ) , ­x31 res CI l (x x x )"!c(D#2)2d k 1 2 3 12 k x2 kk1 13 13 l/d`1 we find from Eq. (4.14) SPj (x )u`(x )A 1(x )A 2(x )T k 1 2 k 3 k 4

C

(4.17)

1 "!K c(D#2) (2d#g ) g (x ) Su(x )u`(x )A 2(x )T 1 A x2 kk1 13 1 2 k 4 13 1 # g (x ) Su(x )u`(x )A 1(x )T . (4.18) 1 2 k 3 x2 kk2 14 14 Making analogous calculations in the general case k'2 will lead to the following result:

D

SPj (y )u`(y )A 1(x )2A k(x )T k 1 2 k 1 k k "!K c(D#2) (2d#g ) 1 A

C

k 1 + g (y !x ) r (y !x )2 kkr 1 r r/1 1

D

]Su(y )u`(y )A 1(x )2AK r(x )2A k(x )T , k r k k 1 2 k 1

(4.19)

where the symbol AK r denotes the omission of the field A r(x ) in this expression. In the derivation of k k r this equation we have used the explicit form of the Green functions Suu`A 12A kT, see k k Eqs. (2.183) and (4.15).

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65

According to the general program described in the end of the previous section, the dependence of dimension d on the parameter C is determined by the equation j (4.20) SPj (x )u`(x )A 1(x )T"0 , k 1 2 k 3 while the vanishing of all the higher Green functions (4.21) SPj (y )u`(y )A 1(x )2A k(x )T"0, k"2, 32 k k k 1 2 k 1 is interpreted as the infinite set of self-consistency conditions of the model. The Green function SP u`A 1T has the form k k 1 SPj (x )u`(x )A 1(x )T"!K c(D#2) (2d#g ) g (x ) Su(x )u`(x )T . k 1 2 k 3 1 A x2 kk1 13 1 2 13 Eq. (4.20) implies (2d#g )"0, or, owing to Eq. (2.184) A D ~1 1 d"(!1)D@2 (4p)D@2C . (4.22) 2 C j Let us remark that the general expression (4.19) contains the factor (2d#g ) which is independent A on the value of k. So if Eq. (4.22) is taken into account, self-consistency conditions for the model hold identically. The model described here is the simplest D-dimensional analogue of two-dimensional exactly solvable models: the Thirring model and Wess—Zumino—Witten model. The first one was solved by the authors of the present article using the method discussed here as far back as in 1978 in the works [2,14], and the second, in the work [37], see also Refs. [3,5,15]. Both methodically and technically, the method of solution of these models is analogous to the solution of pure gauge model. Not surprisingly, the results of the work [18] are reproduced in discussion therein.

C

A BD

4.2. A model in the space of even dimension D54 defined by two generations of secondary fields Consider the second of the models (3.84) Qj (x)"Pj (x)#aPM j (x)"0 , (4.23) kl kl kl where a is unknown parameter, and the fields Pj and PM j are those appearing in the operator kl kl product expansion of j u and j Pj : k k l j (x )u(x )"[u]#[Pj ]#[Pj ]#2 , k 1 2 l op (4.24) j (x )Pj (x )"[u]#[Pj ]#[PM j ]#2 . k 1 l 2 j op Both tensor fields have the same scale dimension l "lM "d#2 . 2 2 Three dimensionless parameters exist in this model: d, C , a . j

(4.25)

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According to Eq. (3.62), these parameters are related by a pair of algebraic equations which may be derived from the system SQj u`j T"SPj u`j T#aSPM j u`j T"0 , (4.26) kl q kl q kl q SQj P`jj T"SPj P`jj T#aSPM j P`jj T"0 . (4.27) kl o q kl o q kl o q It is also necessary to convince oneself that the self-consistency conditions hold. Below the solution of the model (4.23) will be found. We will obtain a pair of algebraic equations for the parameters (4.25) and the closed set of differential equations for higher Green functions. 4.2.1. Self-consistency conditions We must prove that SQj (y )u`(y )A 1(x )2A k(x )T"0 where k"1, 22. kl 1 2 k 1 k k Similar to the previous model, we will concurrently find the parameters (4.25). Consider the Green functions of the field Pj . According to Eq. (3.31) we have kl SPj (y )u`(y )A 1(x )2A k(x )T kl 1 2 k 1 k k "!K

2

res l/d`2

P

dz dz CI l (y z z )­z2S j (z )u(z )u`(y )A 1(x )2A k(x )T , 1 2 kl 1 1 2 o o 2 k k 1 2 k 1

(4.28)

(4.29)

where CI l (x x x )"jx1 (x x )(x2 )~(l~d~2`D)@2 (x2 /x2 )(l`d~2~D)@2 . kl 1 2 3 kl 3 2 12 23 13 The result is, see Ref. [5] for more details

(4.30)

SP (y )u`(y )A 1(x )2A k(x )T k k kl 1 2 k 1 "cK

2

G

(g2 #4(d#1) g ) A A

C

k 1 1 + g r (y !x ) g (y !x ) k k 1 r t (y !x )2 (y !x )2 ktl 1 1 r 1 t r,t/1 rOt

D

]Su(y )u`(y )A 1(x )2AK r2AK t2A k(x )T!trace in k,l k k k k 1 2 k 1

A C

k #(4d(d#1)!2g ) + A r/1

1 g (y !x )jy1(y x ) r l 2 r (y !x )2 krk 1 1 r

DBH

]Su(y )u`(y )A 1(x )2AK r2A k(x )T#(k%l)!trace in k,l k k 1 2 k 1 k

,

(4.31)

where c is the parameter (4.9), and hats on AK r and AK t imply dropping of the fields A r(x ), A t(x ). k k r k t k Note that for k"1 the first term in Eq. (4.31) is absent: SPj (x )u`(x )A 1(x )T"cK (4d(d#1)!2g ) g kl 1 2 k 3 2 A A 1 ] g (x ) jx1(x x )#(k%l)!trace in k,l Su(x )u`(x )T . 1 2 x2 k1k 13 l 2 3 13

C

D

(4.32)

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

67

Consider the Green functions of the field PM j . Recall that this field enters the operator product kl expansion of j Pj . Hence the analogue of Eq. (3.31) reads k l SPM j(x )u(x )2u`(x )T s 1 2 2n

P

"KM j res dy dy CI l 12 s (x y y )S j (y )Pj (y )u(x )2u`(x )T , s 1 2 k k ,k,o 1 1 2 k 2 o 1 2 2n p/ps where KM j are some constants, and s

(4.33)

CI l 12 s (x x x )"SPl 12 s(x )PI (x )A-0/'(x )T 3 k k 1 o 2 k k k ,k,o 1 2 3

(4.34)

is the longitudinal conformally invariant function. Here PI denotes the conformal partner of the o field P . Its dimension equals to D!d!1. Note that the general conformally invariant expression o for the function of the type SPl 12 sP A T includes five independent structures. One can check that k k o k only two independent combinations of the latter are longitudinal. The function (4.34) is longitudinal due to equation of the type (2.77) for the Green functions Sj P 2T. The general conformally k o invariant expression for the longitudinal function may be shown to have the form CI l 12 s (x x x )"­x3 CI l 12 s (x x x ) , k k k ,o 1 2 3 k k ,k,o 1 2 3 where CI l 12 s (x x x ) k k ,o 1 2 3

G

C

1 s + g r(x )jx112 L r2 s(x x )!traces in k 2k " jx2(x x )jx112 s(x x )#b o 1 3 k k 3 2 1 s ok 12 k k k 3 2 x2 12 r/1 x2 (l`d~D~s`2)@2 ](x )~(D`l~d~s~2)@2 23 , 12 x2 13

A B

DH (4.35)

where b is arbitrary parameter. Substituting Eq. (4.35) into Eq. (4.33) we get SPM j(x )u(x )2u`(x )T s 1 2 2n

P

dy dy CI l 12 s (x y y )­y2 S j (y )Pj (y )u(x )2u`(x )T . "!Kj res (4.36) 1 2 k k ,o 1 1 2 k s k 2 o 1 2 2n p/ps Thus the Green functions of the fields PM j which arise in the operator product expansions P j may s o k be calculated from the Ward identities as well. The Green functions of the field PM j containing the fields A have a similar representation kl k SPM j (y )u`(y )A 1(x )2A k(x )T k k kl 1 2 k 1 "!KM

2

res l/d`2

P

dz dz CI l (y z z )­z2 S j (z )Pj (z )u`(y )A 1(x )2A k(x )T , j 2 o 1 2 k 1 1 2 kl,o 1 1 2 j k k

(4.37)

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where CI l (x x x ) kl,o 1 2 3

G

H

1 " jx2(x x )jx1 (x x )#b [g (x )jx1(x x )#(k%l)!trace in k, l] o 1 3 kl 3 2 ok 12 l 3 2 x2 12 x2 (l`d~D)@2 ](x2 )~(D`l~d~4)@2 23 . 12 x2 13 The calculation of the r.h.s. of Eq. (4.37) is presented in Ref. [5]. The result is

A B

(4.38)

SPM j (y )u`(y )A 1(x )2A k(x )T"2c2(D#2)K KI (2d#g ) kl 1 2 k 1 k k 1 2 A 1 k 1 g (y !x ) g (y !x ) ] ( g #2(d#2)) + A r (y !x )2 lkt 1 t (y !x )2 kkr 1 1 t 1 r r,t/1 rOt

G

C

D

]Su(y )u`(y )A 1(x )2AK r2AK t2A k(x )T!trace in k,l k k k k 1 2 k 1

A C

k ! + r/1

1 g (y !x )jy1(y x ) Su(y )u`(y )A 1(x )2AK r2A k(x )T 1 2 k 1 k k r l 2 r k (y !x )2 kkr 1 1 r

#(k%l)!trace in k,l

DBH

,

(4.39)

where KI "KM [1#b(D!2)] . (4.40) 2 2 The symbols AK r, AK t mean that the fields A r(x ), A t(x ) are omitted. For k"1 the first term in k k k r k t Eq. (4.39) is absent SPM j (x )u`(x )A 1(x )T"!2c2(D#2)K KI (2d#g ) 1 2 A kl 1 2 k 3 1 ] g (x ) jx1(x x )#(k%l)!trace in k,l Su(x )u`(x )T . (4.41) 1 2 x2 k1l 13 k 2 3 13 Consider the self-consistency conditions (4.28). The Green functions of the Qj have the form kl SQj (y )u`(y )A 1(x )2A k(x )T kl 1 2 k 1 k k k 1 1 g r (y !x ) g (y !x ) + "cK N 1 2 r (y !x )2 ktl 1 t (y !x )2 k k 1 1 r 1 t r,t/1 rOt

C

D

G A

B

]Su(y )u`(y )A 1(x )2AK r2AK t2A k(x )T!trace in k,l k k k k 1 2 k 1 #N

2

A C k + r/1

1 g (y !x )jy1(y x ) Su(y )u`(y )A 1(x )2AK r2A k(x )T k k r l 2 r k 1 2 k 1 (y !x )2 kkr 1 1 r

#(k%l)!trace in k,l

DBH

,

(4.42)

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

69

where 2pD@2 K KI 1 2 (2d#g ) [g #2(d#2)] a , N "g2 #4(d#1)g # A A 1 A A K D#2 2 C 2

A

B

(4.43)

B

(4.44)

K KI 2pD@2 1 2 (2d#g ) a . N "4d(d#1)!2g ! A 2 A K D#2 2 C 2

A

All the Green functions (4.42) vanish if one sets N "N "0 . 1 2 This leads to the following equations: g2 #4(d#1)g #aN [g #2(d#2)]"0 , A A A 2g !4d(d#1)#aN"0 . A where

(4.45) (4.46)

2pD@2 K KI 1 2 (2d#g ) . N" A K D#2 2 C 2

A

(4.47)

B

Let us remind that the coupling constant g is expressed through the parameter C , see Eq. (2.184). A j Eliminating the factor aN from the system (4.45), (4,46) we obtain the equation which expresses dimension d in terms of parameter C : j g2 !4(d2#d!1)g !8d(d#1)(d#2)"0 . (4.48) A A One easily see that this equation has a solution satisfying physical requirements d'D/2!1, C '0 j for any (even) space dimension D54.

(4.49)

4.2.2. Differential equations for Green functions of fundamental fields Consider the equation SQj (x )u(x )2u`(x )T"0 . (4.50) kl 1 2 2n It represents a source to differential equations for Green functions Su 2u` T. The Green 1 2n functions of the fields Pj and PM j are calculated from Eqs. (3.31) and (4.36) for s"2. Applying the kl kl Ward identities to these equations, we get

P C

n + CI l (x y x )Su(y )u(x )2u`(x )T dy SPj (x )u(x )2u`(x )T"K res kl 1 1 r 1 2 2n 1 kl 1 2 2n 2 r/2 l/d`2 2n (4.51) ! + CI l (x y x )Su(y )u(x )2u`(x )T , kl 1 1 r 1 2 2n r/n`1

D

70

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P C

n SPM j (x )u(x )2u`(x )T"KM res dy + CI l (x y x )SPj (y )u(x )2u`(x )T kl 1 2 2n 2 1 kl,o 1 1 r o 1 2 2n l/d`2 r/2 2n (4.52) ! + CI l (x y x )SPj (y )u(x )2u`(x )T . kl,o 1 1 r o 1 2 2n r/n`1 As the result we obtain, see Ref. [5] for more details

D

SPj (x )u(x )2u`(x )T"PK j (x, ­x1) Su(x )2u`(x )T , 1 2n kl 1 2 2n kl where

G

(4.53)

C

D

n (x ) 2n (x ) 1r k ­x1# + 1r k ­x1#(k%l) PK j (x, ­x)"cK ­x1­x1#2(d#1) ! + kl 2 k l x2 l x2 l 1r r/2 r/n`1 1r n (x ) (x ) 2n (x ) (x ) 1r k 1r l# + 1r k 1r l !trace , #4d(d#1) ! + (x2 )2 (x2 )2 1r 1r r/2 r/n`1 SPM j (x )u(x )2u`(x )T"PMK j (x, ­x1) Su(x )2u`(x )T , kl 1 2 2n kl 1 2n where

C

D

PMK j (x, ­x1)"MPK j (x, ­x1)PK j (x, ­x1)#(k%l)!traceN . 1k l kl Here PK j and PK j are the differential operators of the first order 1k l n (x ) 2n (x ) 1r k# + 1r k , ­x1#2(d#2) ! + PK j "!cKI 1k 2 k x2 x2 1r r/2 r/n`1 1r n (x ) 2n (x ) 1r l# + 1r l . PK j "cK (D#2) ­x1#2d ! + l 1 l x2 x2 r/2 1r r/n`1 1r Note that the operator PK j is given by Eq. (4.8). l Introduce the differential operator

G

G

C

C

DH

DH

H

(4.54) (4.55) (4.56)

(4.57) (4.58)

QK j (x, ­x1)"PK j (x, ­x1)#aPKM j (x, ­x1) . (4.59) kl kl kl where PK j and PMK j are defined by Eqs. (4.54) and (4.56). From Eq. (4.50) one finds kl kl (4.60) QK j (x, ­x1) Su(x )2u`(x )T"0 . 1 2n kl Thus we obtain a closed set of differential equations for any Green function of the model. Note that QK j is a tensor operator. One can show that each Eq. (4.60) is equivalent to a set of several kl equations of the second order in variables m "x2 x2 /x2 x2 . It can be shown that the tensor a ik mn im kn equation (4.60) written in these variables is equivalent to a system of the three differential equations of the second order. The derivation of these equations and their solution was published in Ref. [38]. 4.3. Primary and secondary fields The commutators of fields with the zero components of current or energy—momentum tensor determine the transformation properties of the fields. The gradient terms in commutators might turn out to be significant provided that a higher symmetry like the D-dimensional analogue of the Virasoro algebra would be found. Above we have introduced the concepts of primary and

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

71

secondary fields for D53. The fundamental field is primary by definition. The commutator of the field u(x) with j has the standard form 0 d(x0!y0) [ j (x), u(y)]"!d(D)(x!y)u(y) . (4.61) 0 Similarly, the Ward identities in this case also have the standard form. Note that the primary fields have the following property. Let Ul (x) be any primary field of s dimension l and tensor rank s. It is known that invariant three-point functions SUl u`j T of these s k fields are either zero or are transverse. The latter follows from the Ward identity ­x3 SUl (x )u`(x )j (x )T"![d(x )!d(x )] SUl (x )u`(x )T"0 , k s 1 2 k 3 13 23 s 1 2 due to orthogonality of conformal fields SUl (x )u`(x )T"0, if sO0, lOd. The transversal s 1 2 invariant functions may not appear in the theory if the condition (2.77) holds. Thus one has SUl (x )u`(x )j (x )T"0 if sO0, lOd (4.62) s 1 2 k 3 for any primary field Ul . The Green functions s SPju`j T, SPTu`¹ T s k s kl are nonzero owing to the presence of anomalous terms in the Ward identities. The anomalous Ward identities are pertinent to any generation of secondary fields. The fields Qj and QT are constructed as superpositions of secondary fields satisfying the usual s s commutation relations (3.61). Due to the discussion above, see Eq. (4.62), the latter is guaranteed by the self-consistency conditions (3.62) which are equivalent to Eqs. (3.63) and (3.64). It is essential that the fields with such properties arise only for specific dependence of dimension on the central charge (i.e. the parameter C or C ). The situation resembles the one in two-dimensional conformal j T models, and the dependence mentioned above is analogous to the Kac formula. To illustrate what has been said let us find the anomalous Ward identities for the case of fields Pj , Pj , PM j and demonstrate that the dependence of dimension on the parameter C is the k kl kl j consequence of the requirement for Qj and Qj to be primary fields. k kl Consider the Ward identities in the case of the field Pj . The most general form of anomalous k Ward identities reads: ­x S j (x)Pj (x )u(x )2u`(x )T"!d(x!x ) SPj (x )2u`(x )T j j k 1 2 2n 1 k 1 2n #a­xd(x!x ) Su(x )2u`(x )T#2 , (4.63) k 1 1 2n where the dots stand for all the “usual” contributions. To find the constant a we consider the Ward identity for the Green function Sj Pj u`A 1T. Using Eqs. (2.181) and (4.19) we get: j k k 4 ­x2 SPj (x )u`(x )A 1(x )j 2(x )T k k 1 2 k 3 k 4 ­x42 SPj (x )u`(x )A 1(x )A 2(x )T "C h(D~2)@2 k 1 2 k 3 k 4 j x4 k 1 "!K c(D#2)(2d#g ) g (x )C h(D~2)@2 ­x42 Su(x )u`(x )A 2(x )T 1 A x2 kk1 13 j x4 1 2 k 4 k 13 1 4 #C h(D~2)@2 ­x g (x ) Su(x )u`(x )A 1(x )T . (4.64) j x4 1 2 k 3 k2 x2 kk2 14 14

C

D

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Finally we obtain, see Ref. [5] for more details ­x S j (x)Pj (x )u`(x )A 1(x )T"![d(x!x )!d(x!x )] SPj (x )u`(x )A 1(x )T j j k 1 2 k 3 1 2 k 1 2 k 3 (4.65) #a­xd(x!x ) Su(x )u`(x )A 1(x )T , k 1 1 2 k 3 where a is the same constant as in Eq. (4.63). The derivation makes use of the fact that the Green function SP u`A 1T is defined by Eq. (4.19) for k"1. The constant a turns out to be k k D#2 . (4.66) a"!K (2d#g )C (!1)(D`2)@2 (4p)D@2 C 1 A j 2

A B

Setting a"0 we obtain the previous result (4.22): g "!2d. So this result is a consequence of the A requirement for Qj "Pj to be a primary field. k k Consider the anomalous Ward identities for the fields Pj and PM j . Each of them involves a pair kl kl of independent parameters, one of the two being related to anomalous contribution of the field Pj , k and the other, with a contribution of the field u. Evaluating the quantity C h(D~2)@2 ­x42 SPj (x )u(x )A 1(x )A 2(x )T j x4 kl 1 2 k 3 k 4 k and making use of the results of Section 4.2, we get, see Ref. [5] for more details ­x S j (x)Pj (x )u`(x )A 1(x )T j j kl 1 2 k 3 "![d(x !x)!d(x !x)] SPj (x )u`(x )A 1(x )T 1 2 kl 1 2 k 3 #b [­xd(x!x ) SPj (x )u`(x )A 1(x )T#(k % l)!trace] 1 l 1 k 1 2 k 3 (x ) 12 k ­xd(x !x)#(k%l) !trace # b ­x­xd(x !x)#b 2 k l 1 3 x2 l 1 12 ]Su(x )u`(x )A 1(x )T . 1 2 k 3 One obtains the following expressions for the constants:

G

C

D

H (4.67)

b "!1 b (g2 #4(d#1)g ) [K (D#2)(2d#g )]~1 , 1 2 0 A A 1 A b "!1 b "1 b [4d(d#1)!2g ] , A 2 2 3 4 0 where

AB

D C cK . b "(!1)D@2`1(4p)D@2 C j 2 0 2 The coefficients in anomalous terms of the Ward identity for the Green function S j PM j u`A 1T are j kl k calculated analogously. This Ward identity coincides with Eq. (4.67) after the change Pj PPM j ; kl kl b PbM , i"1, 2, 3 in the latter identity. The coefficient bM turn out to be i i i bM "!1 bM (2(d#2)#g ) [K (D#2)(2d#g )]~1, bM "!1 bM "1 bM , 1 2 0 A 1 A 2 2 3 4 0 where bM "Nb , 0 0 and the coefficient N is given by the formula (4.47).

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

73

Consider the field Qj "Pj #aPM j . Let us demand this field to be primary. It means that the kl kl kl Ward identity for the Green function S j Qj u`A T j kl k should comprise “usual” terms only. This requirement leads to the equations b #abM "0, b #abM "0 , 1 1 2 2 which are easily seen to coincide with Eqs. (4.45) and (4.46). Thus Eq. (4.48) which relates the parameters d, C j is the consequence of the fact that the field Qj is a primary field. In Ref. [6] we show, see also below kl Eqs. (4.89) and (4.94), that for D"2 such an approach leads to the well-known [16,19] results. In particular, the Kac formula [39] arises as a consequence of the second equation from Eq. (3.62). This feature may prove useful in the derivation of the analogue of the Kac formula in Ddimensional space. 4.4. A model of two scalar fields in D-dimensional space Let us consider the model defined by the first equation from Eq. (3.75): Pj (x)#bPT(x)"0 . (4.68) k k Analysing partial wave expansions of the Green functions S¹ u¹ u`T and S¹ uu`sT one can kl op kl show that after the setting d(D!2)2!D(D2!2D#D) a"! (D!2) [d(D!4)!D(D!2)] in Eq. (3.56) there exists a pair of mutually orthogonal fields PT and PT , 1k 2k ¹ (x )u(x )"[u]#[PT ]#[PT ]#2 , (4.69) kl 1 2 1o 2o with one of them, say PT , having a negative norm. Due to that the contributions of the fields 2k PT and PT into each of the Green functions mentioned above, compensate each other. On the 1k 2k other hand, the contribution of the field PT,PT into the partial wave expansion of the function k 1k S¹ uj u`T is still uncompensated, if the parameter b in the Ward identities (3.59) and (3.60) which kl o determine the latter function, equals to zero. One can show that SPj (x )PT(x )TO0 only if b"0 , (4.70) k 1 l 2 but SPj PT T"0 for all b. The calculations needed to prove these statements and to find the k 2l solution of the model (4.68), are analogous to those in the previous sections (though are more cumbersome), and will be published separately, see also Refs. [3,15]. Here we just present a final result.

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Consider the equations SPj u`j T#bSPTu`j T"0, SPj u`¹ T#bSPTu`¹ T"0 , (4.71) k l k l k lo k lo SPj u`sT#bSPTu`sT"0 , (4.72) k k SPj (x )u`(x )u(x )u`(x )T#bSPT(x )u`(x )u(x )u`(x )T"0 . (4.73) k 1 2 3 4 k 1 2 3 4 One can show that for C "0 the first three equations lead to the following algebraic relations: j D(2d!D) (d!D/2#1) D" , b"4D , (4.74) D!1 D2!d(d#1) 2d2(D2!2D!4)!dD(4D2!5D!6)#D3(2D!1)"0 . ¹he solutions read: dK2.7, D"3.6 for D"3, dK3.65, D"4.4 for D"4 , dK6.24, D"6.24 for D"6 . Eq. (4.73) gives PK j (x, ­ 1) Su(x )u`(x )u(x )u`(x )T#PK T(x, ­ 1) Su(x )u`(x )u(x )u`(x )T"0 , (4.75) k x 1 2 3 4 k x 1 2 3 4 where PK j is the operator (3.33), while PK T is defined by Eq. (3.37) for s"1. It is convenient to k k represent the general conformally invariant expression for the four-point Green functions in the form G(x x x x )"(x2 x2 )~d U(m, g) , 1 2 3 4 12 34 where

(4.76)

x2 x2 x2 x2 m"ln 12 34 , g"ln 12 34 x2 x2 x2 x2 13 24 14 23 are the conformally invariant variables. Written in these variables, Eq. (4.75) may be shown to have the form: Mjx1(x x )GK (m, g, ­ , ­ )#jx1(x x ) GK (m, g, ­ , ­ )N U(m, g)"0 , k 2 3 1 m g k 2 4 2 m g where

(4.77)

G

GK (m, g ­ , ­ )" (e~m!e~g!1)­2#2(e~m!e~g#1)­ ­ m m g 1 m g

A

B

D!3 ­ #(e~m#eg~m!2e~g#2)­2#2d e~g!e~m# m g 2D

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

A

75

B

1 #2d(e~g!e~m!1)­ #d e~m!e~g# g D

A

B

H

1 D ~1 # d! #1 (d#­ ) , m 4b 2

G

(4.78)

GK (m, g ­ , ­ )" (e~g#em~g!2e~m#2)­2#2(e~g!e~m#1)­ ­ #(e~g!e~m!1)­2 2 m g m m g g

A

B

A

D!3 1 ­ #d e~g!e~m# #2d(e~m!e~g!1)­ #2d e~m!e~g# g m 2D D

A

B

H

1 D ~1 # d! #1 (!d#­ ) . g 4b 2

B

(4.79)

Thus the Green function (4.76) is determined by the pair of differential equations: GK (m, g, ­ , ­ )U(m, g)"0, GK (m, g, ­ , ­ )U(m, g)"0 . 1 m g 2 m g

(4.80)

4.5. Two-dimensional conformal models Let u be a neutral scalar field of dimension d. Let us pass to the complex components ¹"¹ B and P "P , see Eqs. (2.165) and (2.168). As already explained in Section 2.6, when D"2, all the s sB Green functions of the energy—momentum tensor satisfy Eq. (2.149) identically, and consequently are completely determined by the Ward identities, see Eq. (2.171). Write down the operator product expansion ¹u as = ¹(x )u(x )"[u]# + [P ] . (4.81) 1 2 s s/2 The Green functions of the fields P are uniquely determined by Eq. (3.37) for D"2. Taking the s integrals in the r.h.s., see Refs. [41,6,15] for more details, we get (for s52): 1 SP (x)u (x )2u (x )T& (s!1)(s#1)(d#s!2) (­ )s Su(x)u (x )2u (x )T s 1 1 m m x 1 1 m m 2 s`1 C(s#2)C(d#s) !+ C(k#1)C(s!k#2)C(d#s!k#1) k/3 1 d m r ] + (x!x )~(k~2) ­ r# (k!2) (­ )s~k`1 x 2 x r x!x r r/1 ]Su(x)u (x )2u (x )T . (4.82) 1 1 m m Here we have used the complex variables (2.162), x"xB and ­ "­ for the component x B P "P , u(x)"u(x`, x~), P "P (x`, x~), s sB s s xB 2 xB s 12 23 SP (x)u(x)¹ (x)T"gT(d, C) Su(x )u(x )T , (4.83) sB B s 1 2 xB xB xB xB 13 23 12 13

G

C

A

BA

D

B

H

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where Su(x )u(x )T"(x2 )~d, gT(d, C) is the coupling constant: 1 2 12 s d(d!1)(d!2) 1 C d#s!1 # ! gT(d, C)& s 12 (d#s)(d#s!1)s(s!1) d#s!2 s#1

G

#(!1)s`1 C(s!1)

C

D

C

DH

C(d#1) 1 1! (s#1)d(d#s!2) C(d#s#1) 4

,

(4.84)

C is the central charge. The same way as it was done in Section 3, one can introduce a complete family of secondary fields P , Ps1, Ps1,s2, 2 , (4.85) s s s which begets a basis of the dynamical sector. This sector comprises the states of the type ¹(x )2¹(x )u(x)D0T where k"0, 1, 2, 2 . 1 k Any exactly solvable conformal model is defined by the equation, see Eq. (3.85) s~1 Q (x)"0 where Q (x)"P (x)# + + a 12 k Ps12sk (x) , s s sB sB sB sB k/1 s12sk where 24s 4s 424s , s #s #2#s 4s!2. 1 2 k 1 2 k The simplest models are defined by the equations Q (x),P (x)"0 or Q (x),P (x)"0 . 2 2 3 3 Setting the Green function (4.83) to zero for s"2 or s"3, and using Eq. (4.84), we find d(5!4d) 3d2!14d#8 C" or C"! . d#1 2(d#2)

(4.86)

(4.87)

(4.88)

(4.89)

For the first model one has [41,34], see Eq. (4.82)

C C

A

BD

1 1 1 d D 3 ­21! ­ 2! ­ 3! # Su(x )u(x )s(x )T"0 , (4.90) x x 1 2 3 x x 2 x2 x2 2(d#1) x 12 13 13 12 4 1 d 1 D 1 1 3 ­21! + ­ k! ! # Su(x )u(x )s(x )s(x )T"0 . (4.91) x 1 2 3 4 2 x2 2 x2 x2 2(d#1) x x 14 12 13 k/2 1k Substituting the first equation into the conformally invariant expression for the function SuusT, we get [41,34]

A

d"3 D!1 . 8 4 The more complex model is defined by the equation Q (x)"P (x)#bPM (x)"0 , 4 4 4

BD

(4.92)

(4.93)

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

77

where the field PM arises in the operator product expansion ¹(x )P (x ). One can prove that the 4 1 2 2 equation SQ u¹T"0 has two solutions 4 (d!2)(33!4d) , (4.94) C"1!4d, C" 5(d#3) see Refs. [6,15] for details. The results (4.89)—(4.94) coincide with known the results derived in Refs. [16,19] on the basis of infinite dimensional conformal symmetry. To compare, one must factorize all the Green functions (i.e., to make transition to the fields which solely depend either on x` or on x~), and also introduce the new quantum numbers d "(l#s)/2, dM "(l!s)/2 in place of (l, s). For one-dimensional fields s s u(x`), P (x`) the latter means s d"d/2, d "d/2#s , (4.95) s see Refs. [6,15] for more details. One easily checks that the values of the central charge (4.89) and (4.94) coincide with those following from the Kac formula [39,40]. Eqs. (4.90) and (4.91) coincide with the equations for the Ising and Potts models. Let us stress that in our consideration its derivation is solely based on the six-parametric symmetry (1.2). The solution to the Ising model in this formulation is studied in detail in Refs. [41,34], see also Ref. [15], while the solution of the Wess—Zumino—Witten model is studied in Refs. [37,15]. One can demonstrate [5,15] that the whole list of results known in two-dimensional conformal theories is reproduced in the framework of the approach developed herein. It is readily seen that the infinite-dimensional symmetry is implicitly present in this formulation. Indeed, the Ward identities are completely determined by the symmetry (1.2). On the other hand, knowledge of Ward identities for D"2 amounts to a definition of the commutators [¹ (x ), ¹ (x )] which the Virasoro B 1 B 2 algebra (or its central extension, to be exact) follows from. Thus the two approaches do coincide in principle, differing only by technical details. One can expect that the family of models defined in Section 3 is also related to a certain D-dimensional analogue of the Virasoro algebra, see Refs. [42,43] for example. When D"2, the dynamical sector coincides with the representation space of the Virasoro algebra, and the states Q (x)D0T coincide with the null vectors. Indeed, consider a family of s secondary fields [16] u(k1,2,km)(x)"¸ 1(x)¸ 2(x)2¸ m(x)u(x) , (4.96) k k k where ¸ are the generators of the Virasoro algebra. The two families (4.85) and (4.96) are easily k seen to be isomorphic. Both of them arise as the result of operator product expansions (4.86). For example, consider the expansion = = ¹(x )u(x )" + (x )~2`ku(~k)(x )" + (x )~2`k ¸ (x)u(x) . (4.97) 1 2 12 2 12 ~k k/0 k/0 Passing to one-dimensional fields P , compare the above with the expansion (4.81). The latter is s realized by the combinations of secondary fields (4.96) covariant under the transformations of the group SL(2, R): P "[¸ #a ¸ ¸ #a (¸ )2 ¸ #2#a (¸ )s] u , s ~s 1 ~1 ~s`1 2 ~1 ~s`2 s~1 ~1

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which satisfy the conditions

A B

d ¸ P "­P , ¸ P " #s P , ¸ P "0 . s 1 s ~1 s s 0 s 2

(4.98)

If one makes use of the commutation relations of the Virasoro algebra and the identity ¸ (x)u(x)"0, the coefficients a ,2,a are determined from the last condition. For example, one 1 a s has

C

D

3 3 (¸ (x))2 u(x)"u(~2)(x)! ­2u(x) , P (x)" ¸ (x)! ~2 2 2(2d#1) ~1 2(2d#1) x 1 2 ­ u(~2)(x)# ­3u(x) , P (x)"u(~3)(x)! 3 (d#1)(d#2) x d#2 x

(4.99)

et cetera. Here d"d/2. Each of the fields Q (x) may also be expressed either through the fields (4.85), or the fields (4.96). s The Green functions containing any of the fields (4.85) satisfy the anomalous Ward identities (3.20) for D"2. Using the relations of the type (4.99), the anomalous terms may also be expressed through the anomalous terms in the case of the fields (4.96). By definition, each field Q is s constructed as a combination of secondary fields (4.85) which represents a primary field, see Eq. (3.61). It may be expressed through the secondary fields (4.96) as well. The condition of cancellation of anomalous terms for the case of Green functions of the field Q leads to s identical results independently on the choice of the type of secondary fields which the field Q s is expressed through. The cancellation of anomalous terms in the Ward identities for the Green functions S¹Q 2T is guaranteed by Eq. (3.62) which determine the dependence of s dimension of the field u on the central charge (and the coefficients of the superposition (3.85) as well). The latter is demonstrated in Section 4.3 on an example of the Green functions Sj Qj2T. k s It is evident from the above that the Kac formula in our approach results as the consequence of the second group of Eq. (3.62) for D"2. The latter is demonstrated to a greater extent in Refs. [6,15].

5. Conformal invariance in gauge theories 5.1. Inclusion of the gauge interactions This section has two goals. First we are going to discuss how the gauge interactions could be taken into account. Our second aim is to present a new viewpoint on the irreducibility conditions (2.77) and (2.149) for the current and energy—momentum tensor, which define the models discussed above. Formally, these conditions are the ones allowing to derive a unique solution to the conformal Ward identities in D53. According to Section 2, a general solution of the Ward identities may be uniquely represented as a sum of the two conformally invariant terms, see Eqs. (2.52) and (2.102). The second term is transversal and is caused by gauge interactions, while the first one contains the information on equal-time commutators of the j and ¹ components with 0 0k the matter fields, see Ref. [15] for more details. The fields j and ¹ , being determined by the k kl

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Ward identities, transform by direct sums of irreducible representations5 QI =Q53, QI =Q53 . (5.1) j j T T The conformal partners A , h are, correspondingly, transformed by the direct sums k kl Q-0/'=QI , Q-0/'=QI . (5.2) A A h h (Recall that the initial representations Q , Q and Q , Q defined by the transformation laws (2.3), j T A h (2.4) and (2.9), (2.10), are undecomposable). The importance of these results is due to the following reason. The renormalized Schwinger—Dyson equations contain the integrals over internal lines of gauge fields. Let DA (x )"SA (x )A (x )T be the propagator of the gauge field. Consider the integrals kl 12 k 1 l 2

P

dx dy G (x2) (DA )~1(x!y)G (y2)" dx dy C (x2)DA (x!y)C (y2) , k kl l k kl l

P

P

dx dy A (x)Dj (x!y)A (y), k kl l

P

dx A-0/'(x)jI (x)# dx AI (x)j53(x) , k k k k

P

dx AI (x)jI (x) . k k

where G (x2)"SA (x)2T are the Green functions, and C (x2) are the corresponding vertices. k k k Analogous integrals appear in the approach developed herein (which is based on the solution [1,2] of Schwinger—Dyson equations). As shown in Sections 5.4 and 5.5, the calculation of these integrals in the case of conformal field theory is reduced to a calculation of contractions of Euclidean fields

P

dx dy j (x)DA (x!y)j (y) , k kl l

where Dj is the propagator of the current. The claim that the fields A , j transform by the direct kl k k sums of representations (5.1) and (5.2) manifests in a quite specific structure of the contractions. Below we shall show that due to Eqs. (5.1) and (5.2) each of these contractions could be represented in the following form:

P

where A-0/', AI and jI , jtr are the conformal fields transforming by irreducible representations k k k k Q-0/', QI and QI , Q53, respectively. Note that this expression does not include the formally invariant A A j j “cross-term”

This means that the transversal part of the current jI does not contribute to the gauge interaction. k Similarly, the longitudinal part of the field AI decouples from the gauge sector. In other words, the k 5 Let us remind that the conformal partial wave expansion of the current Green functions contains two terms, each being identified unambiguously. The first term is an expansion into invariant three-point functions Cl 2 , see 1k,k1 ks Eq. (2.68), which correspond to the direct irreducible representation QI . The second term is decomposed into invariant j transversal functions Cl, 53 2 , see Eq. (2.65), which correspond to the irreducible representation Q53. An analogous k,k1 ks j situation arises in the case of the energy—momentum tensor. More comprehensive comments may be found in Ref. [15], see also Section 6.

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“transversal current” jI 53"(d !­ ­ /h)jI and the “longitudinal field” AI -0/'"(­ ­ /h)AI do not k kl k l l k k l l contribute to physical phenomena.6 From the mathematical point of view it implies the orthogonality of subspaces generated by the conformal currents jI and j53 (or the fields A-0/', AI ). The k k k k orthogonality conditions (2.56) and (2.58) derived in Section 2 are the reflections of this feature. All the above admits a straightforward generalization to the case of the fields h (x) and ¹ (x), see kl kl Section 5.5. Due to importance of these results, in the latter sections we reproduce them explicitly from the requirement of conformal invariance for the generating functional of the gauge theory. Basing on this requirement we shall show that the fields A and h transform by direct sums of representak kl tions (5.2), while the invariant contractions : A j and : h ¹ have the structure described above. k k kl kl It provides one with a new standpoint concerning the irreducibility conditions (2.77) and (2.149), which select out the models of direct (non-gauge) interaction of the matter fields. A more general family of models may be obtained by the introduction of gauge interactions into the models discussed above. To achieve this, it is necessary to give up the irreducibility conditions for the current and the energy—momentum tensor which have been accepted in Section 2. After that the dynamical sector acquires the states of the kind j53(x )u(x )D0T, ¹53 (x )u(x )D0T, 2 , k 1 2 kl 1 2 where j53 and ¹53 are the transversal conformal fields (2.5) corresponding to representations of the k kl type Q . Let us pass to reducible fields transforming by the representations of the type (2.51): 0 jI PjI #j53, ¹I P¹I #¹53 . (5.3) k k k kl kl kl The transversal components j53 and ¹53 are generated by gauge interactions. The electromagnetic k kl interaction leads to the appearance of transversal parts Sj532T in all the Green functions, k Eqs. (2.43) and (2.62) in particular. Analogously, the gravitational interaction begets a non-zero transversal part G53 in Eq. (2.136) and non-vanishing kernels (2.146) in the case of higher Green klop functions S¹ u2uT. We stress that these transversal parts are to be “found” from the electrokl magnetic and gravitational interactions, meaning that one should evaluate the asymptotic operator product expansions j53(x )u(x ) and ¹53(x )u(x ). The latter is sufficient for the setting up all the k 1 2 k 1 2 states of the dynamical sector. In the same manner as in Section 3, the dynamical sector is defined as the set of states of the type j 1(x )2j k(x )u(x)TD0T, ¹ 1 1(x )2¹ r r(x )u(x)D0T , k, r"0,1,2 , k 1 k k kl 1 kl r where the current and the energy—momentum tensor are defined as in Eq. (5.3). After that one considers a family of the models (3.65). The primary fields Q still represent combinations of the 4 secondary fields (3.44). However the latter are calculated taking into account electromagnetic and gravitational interactions. In particular, the simplest model (3.66) is generalized as Pj (x)#Rj (x)"0 , k k provided that non-vanishing field Rj exists in the expansion (2.54). k 6 Recall that the conformal fields jI and AI are representatives of the equivalence classes, see Eqs. (2.38) and (2.40). k k The fields jI !jI 53 and AI !AI -0/' are different representatives of the same classes and thus are physically equivalent to the k k k k fields AI and jI . k k

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5.2. Conformal transformations of the gauge fields As should be clear from Section 2, the formal transition of usual conformal transformation laws to the case of gauge theories poses a number of obstacles. The problem consists in the following. When considering the gauge field A as a conformal vector with the transformation law (2.9), the k requirement of conformal invariants leads to the purely longitudinal expression (2.28) of the propagator SA A T. Note that this expression arises as long as the dimension of the field A is k l k canonical (l "1). The canonical dimension in D"4 conformal QED results from linearity (in the A field A ) of the Maxwell equations. In (non-linear) non-Abelian theories one could expect that k anomalous corrections to the dimension should appear. As shown in Ref. [44] (see also Ref. [15]), the latter is not the case in D"4. This is the tensor field F kl F (x)"­ A (x)!­ A (x)#[A (x), A (x)] (5.4) kl k l l k k l that acquires anomalous dimension, while the dimension of the field A (x) remains canonical. Thus k the difficulty mentioned above persists in non-Abelian theories. The group-theoretical source of this difficulty is rooted in undecomposability of the representation Q given by the transformation law (2.9). The invariant longitudinal propagator (2.28) is A related to the irreducible representation Q-0/' which acts in the space of gauge degrees of freedom. A In non-Abelian case the latter are described by the field A (x)"g(x)L g~1(x) . (5.5) k k Below we show that in the Lagrangean approach, the invariance with respect to transformation (2.9) is possible only on the space of fields (5.5), i.e. under the condition F "0. kl When F O0, the transformation law (2.9) needs to be modified. Let us remind that non-trivial kl (i.e., having a transversal part) field AI (x) is a certain representative of the equivalence class which k the irreducible representation QI is defined on. The transformation (2.9) relates different equivaA lence classes. The fields entering the same class are connected by gauge transformations. Hence one easily concludes that the more consistent transformation law for the field AI must be a combinak tion of the transformation (2.9) and a gauge transformation of definite sort. The combined transformation law is discussed in the next three subsections. In what follows we utilize the infinitesimal form of special conformal transformations dU(x)"e KxU(x) , j j where e are the parameters while K are the generators of conformal transformations. For j j the scalar and vector fields of dimension l one has (see, example Refs. [2,22,15] and references therein): KxUl(x)"(x2­ !2x x ­ !2x l )Ul(x) , j j j q q j KxUl (x)"(x2­ !2x x ­ !2x l )Ul (x)#2x U (x)!2d x A (x) . j k j j q q j k k j jk q q The invariance condition for the propagator of the field A of dimension l"1 has the form k (Kx1#Kx2) SA (x )A (x )T"0 , (5.6) j j k 1 l 2

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where K A (x)"(x2­ !2x x ­ !2x )A (x)#2x A (x)!2d x A (x) . j k j j q q j k k j jk q q The solution of Eq. (5.6) is the longitudinal function (2.28). One easily checks that Eq. (5.4) and the action 1 S " 0 4

P

(5.7)

dx Sp F2 kl

are invariant under the transformations dA (x)"e K A (x) . (5.8) k j j k Consider the gauge term. There exists a unique choice of Lorentz- and scale invariant gauge term, namely

P

S & dx Sp (L A )2 . k k '!6'%

(5.9)

It is not difficult to see that in general this term breaks the invariance under Eq. (5.7) dS &e '!6'% j

P

dx Aa ­ Aa O0 . j k k

(5.10)

However, on the pure gauge space (5.5) its invariance revives. Considering g(x) as a scalar field of zero dimension dg(x)"e K g(x)"e (x2­ !2x x ­ )g(x) , j j j j j q q we have

P

dS & dx Sp [­ (­ gg~1)­ gg~1] k k j '!6'% "!e j

P

dx Sp [hg~1­ g#­ g~1­ ­ g]"0 . j k j k

The result (5.10) is clearly understandable from the viewpoint of the above discussion: as far as the proper choice of a representative in the equivalence class of the field AI had not been made, the k latter should contain an uncontrollable longitudinal part breaking down the gauge choice in the form (5.9). 5.3. Invariance of the generating functional of a gauge field in a non-Abelian case Let us consider the generating functional of a non-Abelian theory 1 Z(J)" N

P

GP C

dA dC dCM exp k

1 1 dx ! Fa Fa # (­ Aa )2!CM ­+C#Aa Ja kl kl k k 4 2a k k

DH

,

(5.11)

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83

where C, CM are ghost fields, + is a covariant derivative k

+ "­ #[A ,2], ­+"­ + . k k k k k Invariance of the functional the under linear the conformal transformations (5.8) is violated by the gauge and ghost terms. So, the term : dx (Fa )2 is invariant under the direct product of conformal and gauge groups, kl while the term

GP C

dA dC dCM exp k

dx

DH

1 (­ Aa )2!CM ­+C 2a k k

(5.12)

breaks either of the symmetries. The idea of the approach proposed is as follows [44—46]. We shall show that a complete change of the term (5.12) under special conformal transformations (5.7) and (5.8) can be compensated by a certain gauge transformation whose parameter depends in a special way on the field A . ¹his will thus prove k the existence of combined transformations under which the term (5.12) remains invariant. They consist of special conformal and compensating gauge transformations. Then we shall show that these combined transformations form a non-linear representation of the conformal group. Let us consider the variation of the first term in Eq. (5.12) under special conformal transformations:

P

P

dx (­ Aa )2P dx (­ Aa #4e Aa )2 . k k k k j j

(5.13)

We shall find the variation of the ghost term. The ghost fields C and CM are transformed as conformal scalars with scale dimensions d and d M , and from scale invariance it follows that C C d #d M "2. It will be shown that the property of compensation of conformal and gauge transC C formations occurs under the condition d "0, C

d M "2 . C

(5.14)

Special conformal transformations corresponding to these values are dC(x)"e K C(x), dCM (x)"e K CM (x) , j j j j

(5.15)

where K C(x)"(x2­ !2x x ­ )C(x) , j j j q q K CM (x)"(x2­ !2x x ­ !4x )CM (x) . j j j q q j It can be readily verified that the quantity + C(x) is transformed as A (x), while ­+C(x) is k k transformed as ­ A (x). As a result we have k k d

P

dx CM (x)­+C(x)"e

j

P

dx M4CM + C#(x2­ !2x x ­ !8x )CM ­+CN . j j j q q j

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The second term in braces can be omitted since it is a total derivative. Combining this result with Eq. (5.13), we find dA dC dCM exp k

GP C GP C dx

DH

1 (­ Aa )2!CM ­+C 2a k k

PdA dC dCM exp k

dx

DH

1 (­ Aa #4e Aa )2!CM (­+#4e + )C k k k k 2a k k

.

(5.16)

In the right-hand side there exist additional terms 4 e ­ Aa Aa !4e CM + C j j a j k k j expressing symmetry breaking. We can easily make sure that to compensate these terms it is necessary to make the following BRST transformation: 1 1 dA (x)"!+ C(x)e, dCa(x)"! tabcCb(x)Cc(x)e, dCM a(x)"! ­ Aa (x)e , k k 2 a k k

(5.17)

where e is a parameter of BRST transformation. To compensate the additional terms, the parameter e should be chosen in the form e"4e j

P

dy CM a(y)Aa (y) . j

Specific for these transformations is a non-linear dependence of the parameter on the fields. Such BRST transformations were studied in Refs. [47—50]. It is of importance that these transformations affect the measure of integration in the functional integral (5.11). This should necessarily be taken into account in a verification of invariance of the term (5.12) under combined transformations. So, we substitute in Eq. (5.12) the fields transformed according to Eqs. (5.7), (5.8), (5.15), (5.16) and (5.17) and examine the first-order terms in e . For the constant parameter e the term (5.12) is j invariant under BRST transformations, and therefore it is necessary to trace only their contribution from the measure variation. It can be easily verified that this contribution compensates the variation (5.16). We have thus provided that the generating functional is invariant under the following combined transformations [44,45] dA (x)"e KI A (x), dC(x)"e KI C(x), dCM (x)"e KI CM (x) , k j j k j j j j

(5.18)

KI A (x)"K A (x)!4+ C(x) j k j k k

(5.19)

P

dy CM b(y)Ab (y) , j

KI Ca(x)"K Ca(x)!2tabdCb(x)Cd(x) j j 4 KI CM a(x)"K CM a(x)! ­ Aa (x) j j a k k

P

P

dy CM f(y)Af(y) , j

dy Ab (y)CM b(y) . j

(5.20) (5.21)

In the next section we check that the new operators KI are actually generators of special conformal j transformations.

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These results can be represented in an alternative form where the ghost fields are integrated. To this end, we integrate Eq. (5.11) over the ghost fields to obtain7 1 Z(J)D " j/0 N

P

dA det D ­+ D exp k

GP C

1 1 dx ! Fa Fa # (­ Aa )2 4 kl kl 2a k k

DH

.

(5.22)

Combined transformations of a gauge field, which consist of conformal transformations and compensating gauge transformations have the form [45,46]: dA (x)"e KI A (x) , k j j k where

(5.23)

1 A (x) . KI A (x)"K A (x)!4+ k ­+ j j k j k

(5.24)

We note that the generator KI is obtained from Eq. (5.19) by a formal substitution j 1 C(x)CM (y)P d(x!y) . ­+ Invariance of the generating functional (5.22) under the transformations (5.23) is readily proved by a direct verification. Consider the factor det D­+ D exp

GP

dx

H

1 (­ Aa )2 . 2a k k

Let us make a change A PA #e K A . It may be shown that k k j j k det D­+ DPdet D­+#e ­ K A D&det D­+#4e A D . j k j k j j The factor independent of A is omitted. As a result, we find k 1 1 dx (­ Aa )2 Pdet D­+#4e A D exp dx(­ Aa #4e Aa )2 . det D­+ D exp k k j j k k k k 2a 2a

C P

D

C P

D

We shall now make a compensating gauge transformation. If we choose it in the form of the second term in Eq. (5.24) 1 A d@A "!4e + k j k ­+ j

(5.25)

then we again come to the initial expression for the generating functional Z(J). For this purpose it is convenient to represent det D­+#4e A D as the value of the functional D(A) determined j j

7 For the sake of simplicity, we carry out all further calculations as if the operator ­+ had no zeros. In the presence of zeroes these expansions become much more complicated. But we do not consider this case here.

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by the relation D(A)

P

du d(­ Au(x)#4e Au(x)!g(x))"1 , k k k k

where Au(x)"A !+ u, on the surface ­ A (x)#4e A (x)"g(x), and then use the standard k k k k k k k technique. Let us present a new law of conformal transformations of the tensor field F . Supplementing the kl previous transformation law with the gauge transformation (5.25), we find

C

D

1 A (x), F (x) , dF (x)"e KI F (x)"e K F (x)#4e kl kl j j kl j j kl j ­+ j where K F (x)"(x2­ !2x x ­ !2d )F (x)#2x [R , F (x)] . j kl j j q q F kl o jo kl Here d is an anomalous dimension of the field F . As distinguished from d , there occur no F kl A restrictions on its value. Let us consider the invariance conditions of the Green functions under the combined transformations 0"dSA A T"dSF A T"dSF F T"dSCCM T k l kl o kl oq "dSA A A T"dSF A A T"2 k l o kl o q and so forth. Formally they can be obtained in a usual way in an analysis of the generating functional with allowance for its invariance. Substitution of the field variations (5.18)—(5.21), (5.23) and (5.24) gives the relations that link different Green functions. As distinguished from the usual conformal Ward identities, these relations are non-linear in the field due to a non-linear character of the transformations. Consider the first of these identities: dSA A T"SdA (x )A (x )T#SA (x )dA (x )T k l k 1 l 2 k 1 l 2

C A+

"e SK A (x )A (x )T#SA (x )K A (x )T!(6!2d ) j j k 1 l 2 k 1 j l 2 A ] S

BD

1 1 A (x )A (x )T#SA (x )+ A (x )T k 1 l ­+ j 2 k ­+ j 1 l 2

"0 .

(5.26)

Here we have used an extension of the transformation (5.23), (5.24) to the case of anomalous dimension d of a gauge field. The condition d "1 is a consequence8 of the group law [15]. As will A A be shown right now, formal invariance of the Green functions holds for any d values. A

8 This can be proved directly from the equations, see Refs. [44,15].

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We shall analyse Eq. (5.26) in the transverse gauge and show [46] that its solution is a transverse power-law function

A

B

­­ 1 . D53 (x )& d ! k l kl kl 12 h (x2 )dA 12 Let us examine the two last terms in Eq. (5.26). We take the notation

T

(5.27)

U

1 D (x )" + A (x )A (x ) . kjl 12 k ­+ j 1 l 2 In the transverse gauge we have ­ D (x)"D (x), ­ D (x)"0 . k kjl jl l kjl The most general scale-invariant solution of these equations is ­x1 ­x1 D (x )" k D (x )#b j D (x ) , jl 12 kjl 12 h1 h 1 kl 12 x x where b is an unknown constant. When substituted in Eq. (5.26), the terms &b are cancelled, and we come to the linear integro-differential equation

C

D

­x2 ­x1 SK A (x )A (x )T#SA (x )K A (x )T!(6!2d ) k D (x )# l D (x ) "0 . j k 1 l 2 k 1 j l 2 A h jl 12 h 2 kj 12 x1 x Its solution is the function (5.27). We shall analyse the Green function DFA (x )"SF (x )A (x )T . kl,q 12 kl 1 q 2 We shall show that the general conformally invariant expression for this function is 1 , (5.28) DFA (x )"C(d ­ !d ­ ) kl,q 12 kq l lq k (x2 )dF@2 12 where d is an anomalous dimension of the field F . Note that due to a non-linear character of F kl conformal transformations of the fields F , A this function is non-zero for d O2. The correkl q F sponding Ward identity for it is of the form

T

U TC

1 A (x ) #4 (Kx1#Kx2)DFA (x )!4 F (x )+ kl 1 q­+ j 2 j j kl,q 12

D U

1 A (x ), F (x ) A (x ) "0 , q 2 ­+ j 1 kl 1

where the action of the operator Kx1 is defined above. In the Abelian case this equation becomes j linear ­x2 (Kx1#Kx2)DFA (x )!4 q DFA (x )"0 . j j kl,q 12 h 2 kl,j 12 x One can readily make sure that the expression (5.28) satisfies this equation only under the condition d "2 . F

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In a non-Abelian case for d O2 we have from Eq. (5.28) F ­x1 ­x2 (Kx1#Kx2)DFA (x )"!2(d !4) q DFA (x )#(d !2)(d !4) j DFA (x ) F F j j kl,q 12 F h 2 kl,j 12 h 1 kl,q 12 x x #2(d !2)C(d d !d d )(x2 )~dF@2 . F kq lj kj lq 12 Substitution of this expression into the Ward identity yields some restrictions on the form of non-linear terms, for more details see Ref. [44]. Thus, anomalous dimensions d O2 do not F contradict the Ward identity. One can similarly examine the Green function SF F T and show kl oq that the most general expression for it, compatible with the Ward identity, is DFF (x )"SF F T"f [g (x )g (x )!g (x )g (x )](x2 )~dF kl,oq 12 kl oq 1 ko 12 lq 12 kq 12 lo 12 12 #f (d !2)(d d !d d )(x2 )~dF , 2 F ko lq kq lo 12 where f are arbitrary constants. 1,2 5.4. Conformal QED in D"4 Let us analyse the group-theoretical structure of the transformations (5.23) and (5.24) in Abelian case, where they take the form: ­ dA (x)"e K A (x)!4e kA (x) . k j j k jh j

(5.29)

Consider the transformations of the form

C

D

1 dA (x)"e KI A (x)"e K A (x)!4­ A53(x) , k j j k j j k kh j

(5.30)

where A53(x)"(d !­ ­ /h)A (x). They differ from Eq. (5.29) by the gauge transformation j jq j q q ­­ d A (x)"e ­ j q A (x) , g k j k h2 q which leaves the gauge term (5.9) invariant. Thus for the modified conformal transformations one may choose either Eq. (5.29) or Eq. (5.30). For the discussion in this section it is convenient to utilize Eq. (5.30). Introduce the projection operators

A

B

­­ ­­ P53A (x)" d ! k l A (x), P-0/'A (x)" k lA (x) . k kl l k h h l The generator KI entering the transformations (5.30) may be represented in the form j KI A (x)"P53K P53A (x)#K P-0/'A (x) . j k j k j k

(5.31)

(5.32)

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To prove this, let us make use of the relations ­ K A (x)"(x2­ !2x x ­ !4x )­ A (x)#4A (x) , k j k j j q q j k k j 1 1 ­ [(x2­ !2x x ­ !4x )­ A (x)]"(x2­ !2x x ­ ) ­ A (x)!4 j ­ A (x), j j q q j k k j j q qh k k h h2 k k P-0/'K P-0/'"K P-0/' , j j and put the r.h.s. of Eq. (5.30) into the form 1 KI A (x)"K A (x)!4 ­ A53(x)"P53K P53A (x)#K P-0/'A (x) . j k j k j k j k h k j

(5.33)

So the modified special conformal transformations in QED have the form [27—29]: dA (x)"e [P53K P53A (x)#K P-0/'A (x)] . k j j k j k The modified global conformal transformations have the similar structure:

(5.34)

ºI A (x)"P53º P53A (x)#º P-0/'A (x) , (5.35) g k g k g k where º are the operators in the Hilbert space which generate a vector representation of the g conformal group (for d"1). In particular, the modified transformations of conformal inversion have the form [27—29]:

A

B

A

B

R ­x­x 1 ­Rx­Rx A (x)P A@ (x)" d ! k l g (x) d ! o q A (Rx) k k kl oq q h x2 lo h x Rx 1 ­Rx­Rx # g (x) o q A (Rx) . q x2 ko h Rx One easily checks that the operators ºI satisfy the group law g ºI ºI "ºI if º º "º . g1 g2 g1g2 g1 g2 g1g2 To demonstrate this one employs the relations

(5.36)

P53º P-0/'A (x)"0, P-0/'º P-0/'A (x)"º P-0/'A (x) . (5.37) g k g k g k The transformation law (5.35) defines a reducible representation of the conformal group: QI =Q-0/' . (5.38) A A Indeed, the representation Q-0/' is realized on the space of longitudinal fields A-0/' and corresponds A k to second terms in each of Eqs. (5.34), (5.35) and (5.36). The first terms correspond to the representation QI . Recall that the latter acts on the space of equivalence classes MAI N, each class A k comprising all the fields with a given transversal part A53"P53AI k k MAI N : AI "A53(x)#­ u (x) , k k k 0

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where A53 is fixed, and u (x) is any scalar field. The transformations º map different equivalence k 0 g classes into one another. To fix a definite realization of the transformations ºI , it is necessary to g select a certain representative in each equivalence class. In the Eqs. (5.34), (5.35) and (5.36), the role of such a representative is played by the transversal part of the field AI : k A (x)"A53(x)#A-0/'(x) where A53"P53AI (x) . (5.39) k k k k k As the result, the irreducible representation QI is herein realized (unlike the realization in A Section 2) on the space of transversal functions. Any transformation in this realization may be represented as a sequence of the three transformations: 1. Gauge transformation inside within equivalence class: AI PAtr"PtrAI "PtrA ; k k k k 2. Conformal transformation to a new equivalence class: MAI NPMAI @N (or A53PAI @ ); k k 3. Gauge transformation within a new equivalence class: AI @ PP53AI @ "P53A@ . k k k Finally, the r.h.s. of the transformation (5.35) (and, similarly, Eqs. (5.34) and (5.36)) may be represented in terms of the fields AI , A-0/' dealt with in Section 2: k k A@ "P53A@ #A {-0/'"P53º P53AI #º A-0/' . k k k g k g k Consider the conditions of invariance with respect to the new transformations. An infinitesimal form of these conditions is derived from Eq. (5.30). One gets for the propagator of the field A : k 1 2 ­x ­x (Kx1#Kx2) SA (x )A (x )T!4 k SA53(x )A (x )T!4 l SA (x )A53(x )T"0 . (5.40) j 1 l 2 k 1 j 2 j j k 1 l 2 h1 h2 x x The general solution of these equations reads (up to a normalization) [27—29]: 1 DA (x )"SA (x )A (x )T& kl 12 k 1 l 2 4p2

CA

B

D

­­ 1 a d ! k l # ­ ­ ln x2 , kl 12 h x2 4 k l 12

(5.41)

where a is the gauge parameter, ­x1hSA (x )A (x )T"a­ d(x ) . k k 1 l 2 l 12 The result (5.41) may also be derived from the condition of R-invariance

(5.42)

SA (x )A (x )T"SA@ (x )A@ (x )T , k 1 l 2 k 1 l 2 where A@ is given by the expression (5.36). k The photon and the current propagators appear on internal lines in the Schwinger—Dyson equations (exact ones or those in the skeleton approximation9),as well as in the equations of the formalism presented in this paper. So let us examine the invariant contractions

P

dx dy j (x)DA (x!y)j (y), k kl l

P

dx dy A (x)Dj (x!y)A (y), k kl l

P

dx j (x)A (x) , k k

(5.43)

9 The skeleton approximation of conformal theories is described in the reviews [2,26,15], see also references therein.

E.S. Fradkin, M.Ya. Palchik / Physics Reports 300 (1998) 1—111

91

where Dj is the propagator of the current, see below. Unlike the discourse of Sections 2—4, the kl calculations of these contractions should now involve the electromagnetic interaction. Let us remind that there exists a pair of types of conformal currents: j53(x) and jI (x), transforming by k k irreducible representations Q53 and QI , see Section 2. The field jI (and also the field AI k) is nothing j j k but a certainly fixed representative of the equivalence class MjI N. The physics resulting does not k depend on the choice of representative. Above we have already passed from the field AI to the field k A53. The results of calculations should coincide no matter which of these fields had been used. In the k case of the current the situation is analogous. To examine the contractions (5.43) it proves more useful to utilize the representatives A53 and j-0/'"P-0/'jI instead of the fields AI and jI . Recall that k k k k k the field jI is the representative of the equivalence class MjI N which includes all the fields jI with k k k a fixed longitudinal part. Formally, the transition jI Pj-0/'"P-0/'jI (5.44) k k k leads to the change of the current’s transformation law, see below. On account of Eq. (5.44) the total current should be represented as j (x)"j53(x)#j-0/'(x)"j53(x)#P-0/'jI (x) . (5.45) k k k k k This differs from Eq. (2.52): the transversal part of the field jI is now omitted. According to k Section 2, this part does not contribute into electromagnetic interaction. The current (5.45) corresponds to a new realization of the reducible representation Qtr=QI . j j The transformation law of the current in the new realization reads [27—29]:

(5.46)

dj (x)"e [K P53j (x)#P-0/'K P-0/'j (x)] , k j j k j k or in the case of global transformations

(5.47)

j@ "[º P53#P-0/'º P-0/'] j (x) . (5.48) k g g k The first and the second terms describe the transformations of transversal and longitudinal parts of the current, respectively. Note that the second term manifests itself (alike the case of AI field, see k above) as a combination of the three transformations: a transformation within equivalence class towards a longitudinal representative, conformal transformation to a new equivalence class, and a transformation towards a longitudinal representative within a new class. The Green function S j j T, which is invariant under these transformations, may be shown [27—29] to have the k l following form: Dj (x )"S j (x )j (x )T"f (d h!­ ­ )d(x )#C ­ ­ d(x ) , (5.49) kl 12 k 1 l 2 j kl k l 12 j k l 12 where f is some constant and C is the central charge introduced above. j j Let us remind that the representations (5.38) and (5.46) are equivalent in virtue of Eq. (2.26), while the propagators DA and Dj are the kernels of intertwining operators. Fix the free parameters kl kl entering Eqs. (5.41) and (5.49) by the constraint

P

dx DA (x )Dj (x )"d d(x ) . 3 ko 13 ol 32 kl 12

(5.50)

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Then the equivalence conditions for the representations in the realizations (5.39) and (5.45) are written as

P

P

A (x)" dy DA (x!y)j (y), j (x)" dy Dj (x!y)A (y) . k kl l k kl l

(5.51)

The second condition leads [28,29] to the Maxwell equations.10 Let us come back to the contractions (5.43). From Eqs. (5.50) and (5.51) one gets

P

P

P

dx dy j (x)DA (x!y)j (y)" dx dy A (x)Dj (x!y)A (y)" dx j (x)A (x) . k kl l k kl l k k

(5.52)

The invariant propagators DA and Dj coincide with the kernels of invariant scalar products on the kl kl direct sums of subspaces (see Section 2) M53=M I and M I =M-0/' , (5.53) j j A A respectively. Each contraction in Eq. (5.52) may be represented as a sum of the two conformally invariant terms correspondent to two terms in the sums (5.53), both being independent on the choice of realization of the spaces M I and M I (i.e., on the fixing of representatives in equivalence j A classes). For example, consider the second contraction. Decompose it into a sum of transversal and longitudinal contributions

P

dx dy A53(x)Dj, 53(x!y)A53(y)# dx dy A-0/'(x)Dj, -0/'(x!y)A-0/'(y) . k kl l k kl l

P

P

dx dy A53(x)Dj, 53(x!y)A53(y)" dx dy AI (x)D53 (x!y)AI (y) . k kl l k kl l

(5.54)

The transversal part Dj, 53 coincides with invariant propagator D53 "Sj53 j53T introduced in Seckl kl k l tion 2, see Eq. (2.33). Considering the first term, it is useful to pass to conformal fields AI belonging k to the same equivalence class as A53: k

P

The second term includes conformal fields A-0/'3M-0/'. Hence one can add a singular longitudinal k A part to the kernel Dj, -0/' and make transition to a singular invariant propagator Dj (see Eq. (2.32) kl kl and, for more details, Section 2.7):

P

dx dy A-0/'(x)Dj, -0/'(x!y)A-0/'(y)" dx dy A-0/'(x)Dj (x!y)A-0/'(y) . k kl l k kl l

P

P

dx dy A (x)Dj (x!y)A (y) k kl l

Finally the expression (5.54) may be rewritten as

P

P

" dx dy AI (x)D53 (x!y)AI (y)# dx dy A-0/'(x)Dj (x!y)A-0/'(y) . k kl l k kl l

(5.55)

10 Introduce the tensor of the field F which transforms by irreducible representation Q of the conformal group. kl F According to Refs. [21,26], we have Q &QI , Q &Q53. Calculating the intertwining operators for these equivalence F A F j conditions one can show that F (x)"­ AI (x)!­ AI (x), ­ F (x)"j53(x). The second condition in Eq. (5.51) is equivakl k l l k l kl k lent to Maxwell equations in a-gauge.

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93

It is essential that both terms in this expression are conformally invariant in usual sense (i.e., as was understood in Sections 1 and 2). Let us also stress that the “cross-term”

P

dx dy AI (x)Dj (x!y)AI (y) k kl l

(5.56)

is absent. This term is formally invariant, but is divergent. The third contraction may be represented analogously:

P

P P

P P

dx j (x)A (x)" dx j53(x)A53(x)# dx j-0/'(x)A-0/'(x) k k k k k k " dx j53(x)AI (x)# dx j (x)A-0/'(x) . k k k k

(5.57)

Here the r.h.s. is also expressed through the conformal fields AI , jI introduced in Section 2. k k Moreover, the cross-term

P

dx jI (x)AI (x) k k

(5.58)

is also absent. The absence of terms (5.56) and (5.58) is equivalent to the orthogonality condition, Eqs. (2.56) and (2.58). This condition means that the transversal part of the current jI is expelled k from the interaction with the field Atr. The final result is that through a direct analysis of k electromagnetic interaction we have managed to get the former orthogonality condition, Eqs. (2.56) and (2.58). Let us remind that Euclidean fields are understood as if they were placed inside the averaging symbols. All the above relations for the fields A , j , including Eqs. (5.54), (5.55), (5.56), (5.57) and k k (5.58), should be treated as the relations between the Green functions. In Section 2 we have discussed two types of conformally invariant Green functions G53(x,2)"S j53(x)u2u`T, GI j (x,2)"S jI (x)u2u`T k k k k together with the two types of conformally invariant Green functions of the potential GI A(x,2)"SAI (x)u2u`T, G-0/'(x,2)"SA-0/'(x)u2u`T , k k k k where the currents j53, jI and the fields AI , A-0/' have the usual transformation laws (2.3) and (2.9). k k k k Relations (5.55) and (5.57) viewed in terms of these Green functions mean that the integrals over the internal photon line have the form:

P

dx dy Su2u`AI (x)TD53 (x!y)SAI (y)u2u`T k kl l

P P

# dx dy Su2u`A-0/'(x)TDj (x!y)SA-0/'(y)u2u`T k kl l

P

" dx Su2u`j53(x)T SAI (x)u2u`T# dx Su2u`jI (x)T SA-0/'(x)u2u`T . k k k k

(5.59)

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The divergent cross-terms

P P

dx dy Su2u`AI (x)TDj (x!y)SAI (y)u2u`T , k kl l dx Su2u`jI (x)T SAI (x)u2u`T k k

are absent. Recall that in order to evaluate the second terms in both sides of the equality (5.59) one should introduce a regularization, see Section 2.7. As an example, let us consider the case of spinor QED. The three-point functions invariant under Eqs. (2.3) and (2.9) have the form (see Refs. [22,15] and references therein): xL xL 1 1 xL 12 ! jx3(x x ) , G53(x x Dx )"St(x )tM (x )j53(x )T& 13 c 32 k k 1 2 3 1 2 k 3 (x2 )2 (x2 )2 (x2 )d~3@2 (x2 )d~1@2 x2 x2 k 1 2 12 23 12 13 23 13 1 xL 12 jx3(x x ) . (5.60) GI j (x x Dx )"St(x )tM (x )jI (x )T& k 1 2 3 1 2 k 3 (x2 )d~1@2 x2 x2 k 1 2 13 23 12 The analogous Green functions for the fields AI , A-0/' read: k k xL xL 1 13c 32 , GI A(x x Dx )"St(x )tM (x )AI (x T& k 1 2 3 1 2 k 3 (x2 )d~1@2 x2 k x2 23 13 12 xL x2 12 . (5.61) G-0/'(x x Dx )"St(x )tM (x )A-0/'(x T&­x3 ln 23 k 1 2 3 1 2 k 3 k x2 (x2 )d`1@2 13 12 These functions satisfy the following invariance conditions:

C

D

(Kx1#Kx2#Kx3) St(x )tM (x )AI (x )T"(Kx1#Kx2#Kx3) St(x )tM (x )A-0/'(x )T"0 . j j j 1 2 k 3 j j j 1 2 k 3

(5.62)

The infinitesimal transformations of the spinor field, i.e., the form of K t(x) may be found in j reviews [22,2,15], see also references therein. Introduce the invariant under Eqs. (5.47) and (5.33) Green functions of currents and potentials. The Green functions G53(x x Dx ) and G-0/'(x x Dx ) k 1 2 3 k 1 2 3 remain unchanged, while the functions GI j and GI A, as may be shown [27—29], are replaced by the k k following expressions:

C

D D

xL 1 ­x3­x3 12 jx3(x x ) , (5.63) GI j, -0/'(x x x )"St(x )tM (x )j-0/'(x )T& k l 1 2 3 1 2 k 3 h 3 (x2 )d~1@2 x2 x2 l 1 2 x 12 13 23 ­x3­x3 xL 1 xL 13c 32 . GI A, 53(x x x )"St(x )tM (x )A53(x )T& d ! k l (5.64) 1 2 3 1 2 k 3 kl l (x2 )d~1@2 x2 x2 h3 12 13 23 x Unlike Eq. (5.62), the function (5.64) satisfies the following condition of infinitesimal invariance:

A

BC

A

B

­x3­x3 1 d ! j q St(x )tM (x )A (x )T"0 . (Kx1#Kx2#Kx3) St(x )tM (x )A (x )T!4­x3 jo 1 2 o 3 j j j 1 2 k 3 kh h3 x3 x (5.65)

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95

As before, the integral over internal photon line may be written through the functions (5.63) and (5.64), and then expressed through the functions (5.60) and (5.61):

P

dx dy St(x )tM (x )A (x)TDj (x!y)SA (y)t(x )tM (x )T 1 2 k kl l 3 4

P P

P

" SttM AI TD53 SAI ttM T# SttM A-0/'TDj SA-0/'ttM T k kl l k kl l

P

" SttM j53T SAI ttM T# SttM jI T SA-0/'ttM T . k k k k

(5.66)

These results, as well as the conformal QED in skeleton approximation based on them, were obtained by authors in Refs. [27—29], see also Ref. [15]. The conformal bootstrap in spinor QED was discussed in Ref. [27,15]. Let us remark that the different version of conformal QED was examined independently by several authors in Refs. [51—54]. In conclusion we note that Eq. (5.65) only describe the linear part of exact non-linear invariance conditions discussed in the previous subsection. Indeed, the modified transformations of the spinor field include a gauge transformation:

A

B

1 dt(x)#e Kxt(x)!ie4e Atr(x) t(x) , j j j h j where e is the electric charge. As the result, one has 1 (Kx1#Kx2#Kx3) St(x )tM (x )A (x )T!4­x3 St(x )tM (x )A (x ) 1 2 j 3 j j j 1 2 k 3 kh x3 1 1 A (x ) t(x )tM (x )A (x ) #ie t(x ) A (x ) tM (x )A (x ) "0 . !ie 1 2 k 3 1 h j 2 2 k 3 h j 1

TA

B

U T A

B

U

(5.67)

Eq. (5.65) may be viewed as an approximation of these equations when e@1. It is evident that Eq. (5.67) are linearized when passing to gauge invariant combination

CP

W(x x )"tM (x )exp ie 1 2 1

C

D

dx A (x) u(x ) . k k 2

The analysis of linearized equations and the evaluation of averages SW(x x )T, SW(x x )A (x )T 1 2 1 2 k 3 was conducted in Refs. [55,56], see also Ref. [15]. The formulation of conformally invariant gauge theories in terms of such string averages looks more natural and deserves further investigations. 5.5. Linear conformal gravity in D"4 The metric field h is twinned to a pair of irreducible representations: QI and Q-0/'. Section 2 kl h h dealt with the two types of fields hI (x) and h-0/'(x) which transformed by these representations. As kl kl

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in the case of QED, the representation QI acts in the space of equivalence classes. The modified h conformal transformations of the field hI may be derived through an addition of gauge transkl formations inside the equivalence class MhI N. Applying the arguments presented in the previous kl subsection literally, one introduces the field h (x)"P53hI (x)#h-0/'(x) kl kl kl possessing the following transformation law [31]:

(5.68)

dh (x)"e [P53K P53#K P-0/']h (x) , kl j j j kl where

(5.69)

P53h (x)"P53 (­x)h (x), P-0/'h (x)"P-0/' (­x)h (x) , kl kl,op op kl kl,op op while the projection operators P53 and P-0/' are defined in Eqs. (2.122), (2.123), (2.124), (2.125) kl,op kl,op and (2.126). As in the previous section, this transformation law corresponds to a direct sum of irreducible representations QI =Q-0/' . h h By analogy, introduce the irreducible energy—momentum tensor

(5.70)

¹ (x)#¹53 (x)#P-0/'¹I (x) , kl kl kl which transforms by the direct sum of representations

(5.71)

Q53=QI . T T Its transformation law reads [31]

(5.72)

d¹ (x)"e [K P53#P-0/'K P-0/']¹ (x) . (5.73) kl j j j kl One easily deduces that the propagators, which are invariant under Eqs. (5.69) and (5.73), have the form Dh (x )"Sh (x )h (x )T&P53 (­x) ln x2 #D-0/' (x ) , (5.74) kl,op 12 kl 1 op 2 kl,op 12 kl,op 12 where the expression for D-0/' is given by Eq. (2.110): kl,op DT (x )"S¹ (x )¹ (x )T"DT, 53 (x )#DT, -0/'(x ) . (5.75) kl,op 12 kl 1 op 2 kl,op 12 kl,op 12 where DT, 53 is given by Eq. (2.121) for D"4, and the longitudinal part equals to [31] kl,op DT, -0/'(x )"­ H (x )#­ H (x )!1 d ­ H (x ) , (5.76) kl,op 12 k lop 12 l kop 12 2 kl j jop 12 H (x)&[22­ ­ ­ !9(d ­ h#d ­ h)!d ­ h]d(x ) . (5.77) kop k o p ko p kp o op k 12 This function may be directly evaluated from the invariance conditions. Though in practice it proves more convenient to make use of the equivalence property for the representations (5.70) and (5.72), see Ref. [31] for details.

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97

It is useful to normalize the propagators by the condition

P

dx DT (x )Dh (x )"1(d d #d d !1d d )d(x ) . 3 kl,qj 13 qj,op 23 2 ko lq kp lo 2 kl op 12

(5.78)

The equivalence conditions are expressed by the relations

P

P

¹ (x)" dy DT (x!y)h (y), h (x)" dy Dh (x!y)¹ (y) , kl,op op kl kl,op op kl

(5.79)

The first of these conditions coincides with the equations of linear conformal gravity in a conformally invariant gauge (its explicit form is presented in Ref. [31]). Note that there exists another equivalence condition, which has not been mentioned before. Introduce the Weyl tensor (in linear approximation) C "R !1 (d R #d R !d R !d R )#1 (d d !d d )R , (5.80) klop klop 2 ko lp lp ko kp lo lo kp 6 ko lp kp lo where R "1 (L L h #L L h !L L h !L L h ) is the linear part of the Riemann curval o kp k o lp l p ko klop 2 k p lo ture. The Weyl tensor transforms by irreducible representation of the conformal group. Let us denote it as Q . According to Refs. [21,26], there exist the following equivalence relations: C Q &Q53, Q &Q53 . C h C T Calculating the intertwining operators for these representations one can check that the first condition results in the constraint relating the fields C ,h and coinciding with Eq. (5.80), while klop kl the second leads to the equation ­ ­ C &¹ , o p klop kl which coincides with Eq. (2.131a). Thus the equations of conformal gravity in linear approximation follow from the equivalence of corresponding representations of the conformal group. Consider the invariant contractions

P

P

dx dy h (x)D T (x!y)h (y)" dx ¹ (x)h (x) . kl kl,op op kl kl

(5.81)

The first one may be brought into the form

P

P

dx dy h53 (x)DT,53 (x!y)h53 (y)# dx dy h-0/'(x)DT, -0/'(x!y)h-0/'(y) kl kl,op op kl kl,op op

P

P

" dx dy hI (x)D53 (x!y)hI (y)# dx dy h-0/'(x)DT (x!y)h-0/'(y) , kl kl,op op kl kl,op op

(5.82)

where D53 and DT are conformally invariant (in usual sense) propagators (2.121) and (2.115), kl,op kl,op hI (x) and h-0/'(x) are conformal fields transforming by the standard law (2.10). It is important that kl kl the (singular) “cross-term”

P

dxdy hI (x)DT (x!y)hI (y) kl kl,op op

(5.83)

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is absent. Analogously, the second contraction is also written as the sum of terms which are invariant under the usual transformation laws (2.4) and (2.10):

P

P

P

dx ¹ (x)h (x)" dx ¹53 (x)hI (x)# ¹I (x)h-0/'(x) . kl kl kl kl kl kl

(5.84)

Here the cross-term

P

dx ¹I (x)hI (x) kl kl

(5.85)

is also absent. It is not hard to understand that the absence of terms (5.83) and (5.85) in invariant contractions is equivalent to the orthogonality condition (2.105). Here it means that the transversal part of the field hI does not interact with the transversal part of the tensor ¹I which was related to kl kl a direct (non-gravitational) interaction of the matter fields in Section 2.

6. Concluding remarks The approach developed here is based on a number of general principles of quantum field theory. we have inspected axiomatic hypotheses which select out definite classes of conformal models, in the same way as it is done in two-dimensional theory. The conformally invariant Ward identities for the energy—momentum tensor and the current play a principal role in our approach. Effectively, the latter contain the information on the quantization rules, and, to some extent, are equivalent to the definition of Hamiltonian. The conformal symmetry leads to a highly specific structure of the Hilbert space, essentially of its sector begotten by the current and the energy—momentum tensor. This Section presents a detailed review and a more comprehensive commentary concerning the properties of these fields, which are indispensable in conformal theory and altogether imperative in our approach. We investigate, in particular, the propagators and Green functions of the fields j , ¹ and also of their irreducible components jI , j53, ¹I , ¹I 53 in different realizations of the QI -type k kl k k kl kl representations; see Ref. [65] for more detailed considerations. 6.1. Conformal models of non-gauge fields Consider the Euclidean fields j (x) and ¹ (x). The Green functions S j 2T and S¹ 2T are the k kl k kl Euclidean analogues of ¹-ordered vacuum expectation values in Minkowski space. Here we treat the Euclidean fields j (x) and ¹ (x) as the symbolic notation for the complete sets of Green k kl functions S j 2T and S¹ 2T. Correspondingly, the derivatives of the Euclidean fields ­ j (x) kk kl k and ­ ¹ (x) denote the derivatives of Green functions ­ Sj 2T and ­ S¹ 2T. Calculating these k kl k kl k k derivatives, one encounters the two types of terms of different nature. Consider those terms on an example of the conserved current in Minkowski space. One gets for the propagator of the current: ­ S0D¹ Mj (x)j (0)ND0T"d(x0)S0D[j (x),j (0)]D0T#S0D¹ M­ j (x)j (0)ND0T . k k l 0 l kk l The second term vanishes due to the conservation law ­ jM*/,(x)"0 . kk

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To ensure the covariance of the ¹-ordered average, one should add quasilocal terms to the first term of the expression. The form of the total contribution of these terms and the commutator is imposed by conformal invariance, and reads ­ S0D¹ Mj (x)j (0)ND0T"C ­ h(D~2)@2d(x) . k k l j l In Euclidean conformal theory, one associates the above pair of contributions with the irreducible components of the Euclidean current j (x), so that k ­ j (x)"­ jI (x), ­ j53(x)"0 . kk kk kk In particular, the total propagator of the current for even D54 may be represented as (see Section 5 and below): S j (x )j (x )T"S jI (x )jI (x )T#S j53(x )j53(x )T , k 1 l 2 k 1 l 2 k 1 l 2 ­ S j (x )j (x )T"­ S jI (x )jI (x )T"C ­ h(D~2)@2d(x ) , k k 1 l 2 k k 1 l 2 j l 12 ­ S j53(x )j53(x )T"0 . k k 1 l 2 An analogous expansion for the higher Green functions was dealt with in Sections 2 and 4. As will be shown below, the irreducible components jI and j53 have different physical meaning, k k and hence the different group-theoretic structure. According to Section 5, see also the discussion below, only the current jtr(x), but not jI (x), induces k k a non-trivial contribution to the electromagnetic interaction. The Green functions of the current jI satisfy non-trivial Ward identities and contain the information on the (postulated) communicak tion relations of the total current: [j (x), j (0)] 0 , [j (x),u(0)] 0 ,2 0 k x /0 0 x /0 As shown in Section 2, all the Green functions SjI 2T are uniquely determined by the condition k of conformal invariance and by the Ward identities. Note that the fact that the fields P arise in the operator expansion s jI (x)u(0)"+ [P ] , k s s is a necessary consequence of the Ward identities (see Section 2), independent of the choice of dynamical model. Therefore the existence of the fields P , generated by the current jI , follows from s k the two statements taken as postulates: the requirement of conformal symmetry and the definition of the equal-time commutator [ j (x), u(0)]. The dynamical requirement is the choice of null 0 vectors. The latter depends on how the commutator [ j (x), j (0)] 0 is defined. In Euclidean 0 k x /0 version of the theory, this commutator is determined by the type of the operator product expansion jI (x)jI (0). k l The expansion jI (x)jI (0)"[C ]#[P ]#2 was considered in Section 3, while in Section 4 we k l j j have examined the models with C O0, P (x)"0. j j One should remark that the current jI (x) arises as a representative of an equivalence class k M jI NLM I "M /M53. The conformal transformations of the current jI depend on the type of k j j j k representatives chosen in each class, see Sections 2 and 5. Thus the transversal parts of the Green functions S jI 2T may be redefined by performing a different choice of representatives. Particularly,

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in the non-local realization of conformal transformations (for D"4), considered in Section 5, these Green functions are longitudinal. This realization is useful for conformal QED. However, it is essential that the local realization of conformal transformations of the current jI considered in k Section 2, is needed for the analysis of the operator product expansions jI u and jI jI . Below we l kl discuss both realizations to a greater extent, and describe the relations between them. In a local realization, the Green functions S jI 2T have quite definite transversal parts, which do not k contribute to the electromagnetic interaction since an irreducible component jI of the total current k only appears in contractions of the type : dx jI (x)A-0/'(x), see below. As shown in Section 5, the k k interaction with the irreducible field AI is caused by the component jtr of the total current, and has k k the from : dx j53(x)AI (x). k k All what has been said is equally valid for the case of the energy—momentum tensor in conformal theory. The two irreducible components ¹I and ¹53 have the same meaning as explained in the kl kl case of the current above. The component ¹53 leads to a non-trivial contribution to the gravikl tational interaction (with the fields hI ). The component ¹I appears only in the contractions kl kl : ¹I (x)h-0/'(x) and describes the postulated commutation relations of the total tensor ¹ (x) with kl kl kl the fields, and with itself. Though the conservation laws ­ ¹M*/,(x)"0 k kl are satisfied in Minkowski space, for the Euclidean propagator (and for the ¹-averages in Minkowski space) we have S¹ (x)¹ (0)T"S¹I (x)¹I (0)T#S¹53 (x)¹53 (0)T , kl op kl op kl op where ­ S¹53 (x)¹53 (0)T"0. In the case of C O0, we get l kl op T ­ S¹ (x)¹ (0)T"­ S¹I (x)¹I (0)TO0 . l kl op l kl op All the Green functions of the irreducible component ¹I (x) are determined from the requirement kl of conformal symmetry and from the Ward identities, as it is shown in Section 2. Let us consider the structure of the Hilbert space in more detail. We have shown in the previous sections that the Hilbert space of conformal field theory contains two orthogonal sectors HI =H , 0 which are generated by irreducible fields

(6.1)

jI , ¹I and j53, ¹53 . (6.2) k kl kl The subspace HI is related to irreducible representations of the type QI , while H , to representations 0 of the type Q , see Eq. (2.15). The orthogonality of the subspaces HI and H means the vanishing of 0 0 the Green functions Sj53jI T"Suj53 jI u`T"0, S¹53 ¹I T"Su¹53 ¹I u`T"0 . (6.3) kl k l kl op kl op Due to the conditions of equivalence for the representations (2.27), (2.131), the subspace H includes electromagnetic and gravitational degrees of freedom (while the subspace HI includes 0 gauge degrees of freedom only). The appearance of non-zero conformal fields j53 and ¹53 necessarily k kl leads to the appearance of electromagnetic and gravitational fields AI and hI . The Green functions k kl

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of current and energy—momentum tensor, which are calculated from Ward identities, generally include the total fields j (x)"jI (x)#j53(x), ¹ (x)"¹I (x)#¹53 (x) . (6.4) k k k kl kl kl Thus the general conformally invariant solution of the Ward identities contains contributions of electromagnetic and gravitational interactions. To derive the models of non-gauge interactions considered in Sections 3 and 4, one must impose special conditions on the Green functions of current and energy—momentum tensor, see Eqs. (2.77) and (2.149). These conditions restrict the total space (6.1) to the subspace HI , setting transversal fields to zero: j53"¹53 "0 on HI . k kl We arrive at the following picture. The conformal symmetry, which arises as a non-perturbative effect, may take place in a special class of models (not necessarily Lagrangean). Starting with the structure of Hilbert space described above, it is natural to assume that gauge interactions are present initially (for D'2) in conformal models based on Ward identities; both the components (6.2) contributing to the total expressions for current and energy—momentum tensor. A “true” conformal theory must contain the fields (6.4), and, consequently the fields, AI , hI due to k kl Eqs. (2.27), (2.113), and (2.131) (see also Eqs. (6.33) and (6.42)). If we choose to consider the “approximate” models without gauge interactions, the solution is to be looked for in the restricted class of Green functions Sj 2T, S¹ 2T corresponding to irreducible representations QI , QI . k kl j T This class of Green functions is singled out by conditions (2.77) and (2.149). A theory with such set of conditions is non-trivial if the operator product expansions j (x)j (0) and ¹ (x)¹ (0), where k l kl op J (x)"jI (x),¹ (x)"¹I (x) are irreducible fields, include anomalous terms [C] and [P(x)], see k k kl kl Eqs. (3.46) and (3.47). The conformal Green functions S jI (x)u2u`T, S¹I (x)u2u`T k kl are uniquely determined from the Ward identities (Section 2). As shown in Sections 3 and 4, the latter feature allows one to derive a D-dimensional analogue of the family of exactly solvable two-dimensional models. In this Section we examine the propagators and higher Green functions of the total fields (6.4). Each of the Green functions will be represented as a sum of two terms, the first one corresponding to the subspace HI , and the second, to the subspace H . These terms have different group0 theoretical structures and different partial wave expansions. Note that the results will be technically different for the spaces of even and odd dimensions. Here we restrict ourselves to the case of even D. 6.2. The propagators of the current and the energy—momentum tensor for even D54. As shown in Section 5, the conformal transformation of the fields (6.4) have the following form: R »j j (x)"ºj P53j (x)#P-0/'ºj P-0/'j (x) , j (x)P k R k R k R k

(6.5)

where P53"d !­ ­ /h, P-0/'"­ ­ /h , kl k l k l R »T¹ (x)"ºTP53¹ #P-0/'ºTP-0/'¹ (x) , ¹ P R kl R kl R kl kl

(6.6)

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where the projection operators P53 and P-0/' are given by the expressions (2.122) and (2.125). The action of the operators ºj and ºT is given by the r.h.s. of the transformations (2.3) and (2.4). The R R conformally invariant propagators are determined by the conditions Sj (x )j (x )T"S»j j (x )»j j (x )T, S¹ (x )¹ (x )T"S»T¹ (x )»T¹ (x )T . k 1 l 2 R k 1 R l 2 kl 1 op 2 R kl 1 R op 2 For even D54 these equation have exceptional solutions which may not be obtained by the limiting transition from anomalous dimensions. One can check that the solutions have the form: Dj (x )"Sj (x )j (x )T"f (d h!­ ­ )h(D~4)@2d(x )#C ­ ­ h(D~4)@2d(x ) , (6.7) kl 12 k 1 l 2 j kl k l 12 j k l 12 DT (x )"S¹ (x )¹ (x )T"f H53 (­x)h(D~4)@2d(x ) kl,op 12 kl 1 op 2 T kl,op 12 (6.8) #C H-0/' (­x)h(D~4)@2d(x ) , T kl,op 12 where f and f are some constants, H53 is given by expression (2.166), and H-0/' has the form j T kl,op kl,op 2 H-0/' (­x)"­ H (­x)#­ H (­x)! d ­ H (­x) , kl,op k lop l kop D kl j jop where D!1 1 2D2!3D#2 ­­­! (d ­ #d ­ ) h! d ­ h. H (­x)" k o p kp o kop 2D ko p 2D(D !1) op k 2D(D!1) Propagators (6.7) and (6.8) satisfy the Ward identities (2.177) and (2.186). In Section 2, the Ward identities were obtained using the kernels invariant under (2.3) and (2.4):

K

G

CI 1 1 j " Dje (x )"CI g (x ) (d h!­ ­ ) kl k l kl 12 j(x2 )D~1`e kl 12 @ 2(D!1#e)(D!2#e) (x2 )D~2`e 12 12 e 1 e 1 ! d h (6.9) D!2#2e kl (x2 )D~2`e @ 12 e 1 1 CI j (d h!­ ­ ) #C ­ ­ h(D~4)@2d(x )#O(e) , " k l (x2 )D~2`e j k l 12 2(D!1#e)(D!2#e) kl 12 (6.10)

C

HK

DK

1 2 (6.11) DTe (x )"CI g (x )g (x )#g (x )g (x )! d d kl,op 12 T(x2 )D`e ko 12 lp 12 kp 12 lo 12 D kl op @ 12 e 1 CI @ 1 " TH53 (­x) #CI A H53 (­x)h(D~4)@2d(x ) T kl,op 12 e kl,op (x2 )D~2`e 12 #C H-0/' (­x)h(D~4)@2d(x )#O(e) , (6.12) T kl,op 12 where CI @ and CI A are some constants. The longitudinal terms in Eqs. (6.10) and (6.12) are derived T T with the help of relations (2.34) for k"(D!4)/2. Notice that their form coincides with longitudinal terms of the propagators (6.7) and (6.8). The kernels (6.9) and (6.11) define in the limit eP0 the

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propagators of irreducible fields jI and ¹I ; they were used in calculations in Sections 3 and 4. The k kl transversal parts of these kernels, singular at e"0, do not contribute to conformally invariant graphs, see Eqs. (5.55), (5.59) and (5.82), and do not change the results. Technically, the latter manifests in the fact that the representations of the type QI are defined on equivalence classes (see Sections 2 and 5), not on the fields. The kernels (6.9) and (6.11) for e"0 are related to longitudinal terms in Eqs. (6.10) and (6.12) by transformations inside the equivalence class, corresponding to the transition to the other realization of representations QI and QI . The more detailed discussion is j T presented in the next subsection. Another approach to a definition of propagators was recently considered in Refs. [59—61]. These works introduce a special regularization of the non-integrable function (x2 )~D`2 in Eqs. (6.10) and 12 (6.12) at e"0, which breaks the conformal symmetry. Moreover, the longitudinal terms11 are dropped in Eqs. (6.10) and (6.12). After that, it is shown that such breakdown of the conformal symmetry may be related to conformal anomalies (the physical motivation and the methods of derivation of conformal anomalies are discussed in Refs. [62,63]). Note that the possibility of the breakdown of the symmetry also remains in our approach. Though, our principal aim is the analysis of models with exact conformal symmetry. 6.3. The propagators of irreducible components of the current and the energy—momentum tensor Consider the irreducible currents j53 and jI . The pair of invariant propagators (for even D54) k k S j53(x )j53(x )T S jI (x )jI (x )T k 1 l 2 k 1 l 2 may be related to the kernels of invariant scalar products on the spaces MI and M-0/', see A A Eq. (2.23). These kernels Dj, 53(x ) Dje (x ) (6.13) kl 12 kl 12 are invariant under the transformation (2.3) and were examined above for the case D54. The first one is transversal and may be identified with the propagator of the current j53 k Dj, 53(x )"Sj53(x )j53(x )T&(d h!­ ­ )h(D~4)@2d(x ) . (6.14) kl 12 k 1 l 2 kl k l 12 The second kernel is singular at e"0, see Eq. (6.10), and gives a certain realization of the propagator S jI jI T, see below. kl 11 The right-hand sides of the Ward identities (2.177) and (2.186) are non-zero owing to contributions of equal-time commutators of the current and energy—momentum tensor components between themselves, see Eqs. (3.46) and (3.47). Let us remind that the vacuum expectation values of ¹-ordered products of the fields (in Minkowski space) are defined up to quasilocal terms, allowing one to make a transition to transversal propagators. However, such a redefinition breaks the conformal symmetry, which fixes longitudinal parts uniquely. In a well-known example of two-dimensional theory, the conformal propagator is given by the expression (6.11) for D"2, eP0 and may be written as S¹ (x)¹ (0)TD &C [(­ ­ !1 d h)(­ ­ !1 d h)!1 (d d #d d !d d )h2]1/h d(x). This propagator kl op D/2 T k l 2 kl o p 2 op 8 ko lp kp lo kl op satisfies the Ward identity (2.186) for D"2. Passing to the complex variables z"x #ix , ¹ "¹ #¹ in this 1 2 zz 11 22 formula, we get the well-known results S¹ (z)¹ (0)T&C (­ )4(­ ­ N )~1d(2)(z,zN )&C /z4,­ N S¹ (z)¹ (0)T zz zz T z z z T z zz zz &C (­ )3d(2)(z,zN ), S¹ N (z,zN )¹ N (0,0)T"0. This propagator differs by quasi-local terms from the non-invariant transversal T z zz z,z expression (­ ­ !d h)(­ ­ !d h) ln x2, which has a non-zero trace. k l kl o p op

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As already mentioned in Sections 5 and 2, the Green functions of the current jI depend on the k choice of realization of the representation QI . Choosing different representatives in the equivalence j class MjI N, one can obtain different realizations of the Green functions S jI 2T. In the Section 5.4 we k k reflect upon the special realization (5.44) of the representation QI : there, the current jI is longitudij k nal, and its propagator reads Dj, -0/'(x )"S jI -0/'(x )jI -0/'(x )T"C ­ ­ h(D~4)@2d(x ) . (6.15) kl 12 k 1 l 2 j k l 12 Conformal transformations are non-local in this realization, see Eq. (6.5), and differ from the transformations of the current j53. The propagator of the total current j , on account of Eq. (6.3), k k equals to the sum of the terms (6.14) and (6.15). In Sections 3 and 4 we studied another realization of the representation QI . Its conformal j transformations are local, coincide with transformations of the current j53, and have the form (2.3). k The propagator SjI jI T in this case also demand regularization and coincides with the Kernel Dj kl kl 1 Dje (x )"S jI (x )jI (x )T "C ­ ­ h(D~4)@2d(x )#A(d h!­ ­ ) #O(e) . (6.16) kl 12 k 1 l 2 e j k l 12 kl k l (x2 )D~2`e 12 The physical significance has, however, the equivalence class MjI NMjI , j-0/' rather than the current k k k jI by itself. The framework of the conformal theory described in the preceding sections is k constructed in a manner which prevents the transformations inside an equivalence class from having an influence on the results. In particular, the transformation jI PjI -0/' does not change the k k values of invariant contractions (5.52). The latter may be written either in the form (5.54), or in the form (5.55). The divergent at e"0 transversal component of the kernel Dj in Eq. (5.55) does not kl contribute, since the conformal Green functions SA-0/'2T are longitudinal in the leading order in e (the conformally invariant regularization is used for calculation of such integrals, see Section 6.4). All that was said above is equally valid for the field A , which transforms by the representation k QI =Q-0/'. The irreducible fields AI and A-0/' correspond to the kernels A A k k DAe (x ) and DA, -0/'(x ) , kl 12 kl 12 which are invariant under the transformation (2.9). The kernel DAe is singular at e"0 kl 1 1 DAe (x )& (x2 )~1`eg (x )D @ K DA, -0/'(x )#DA, 53(x )#2 (6.17) 12 kl 12 e 12 kl 12 kl 12 1 e e kl and defines the scalar product on an invariant subspace M53

P

. Mj53, j53N " dx dx j53,e(x )DAe (x )j53,e(x )D 1 2 k 1 kl 12 l 2 e/0 1

(6.18)

The invariant regularization of the fields j53Pj53,e is described below. Consider the longitudinal k k kernel DA, -0/'. It has the form (2.28) and defines the invariant scalar product on the space M I of kl equivalence classes MjI N, see Eq. (2.22), k

P

MjI , jI N " dx dx jI (x )DA, -0/'(x )jI (x ) . 1 2 k 1 kl 12 l 2 2

(6.19)

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All the Green functions of the field A-0/' are longitudinal, while the Green function of the field k AI depend on the choice of realization of the representation QI . To different equivalence classes k A MAI NLMI , different realizations correspond. In the Section 5.4 we have dealt with the realization k A which corresponds to a transversal representative: AI (x)PA53(x). Its propagator coincides with the k k transversal component of the kernel (6.17): DA, 53(x )"SAI 53(x )AI 53(x )T&(d h!­ ­ )ln x2 . (6.20) kl 12 k 1 l 2 kl k l 12 The conformal transformations in this case are non-local and different from the transformations of the field A-0/'. k Another realization of the representation QI , in which the transformations are local and have the A usual form (2.9), has been studied in Sections 2—4. In this case, the propagator SAI AI T coincides k l with the kernel (6.16) 1 DAe (x )"SAI (x )AI (x )T& (x2 )~1`eg (x ) , kl 12 kl 12 k 1 l 2 e 12

(6.21)

singular at e"0. Unlike Eq. (6.20), it has a longitudinal part which is singular for e"0 and does not contribute to conformally invariant contractions (6.18). Naturally, the invariant scalar product on M53 does not depend on the choice of realization j

P

P

dx dx j53,e(x )DAe (x )j53,e(x )D " dx dx j53(x )DA, 53(x )j53(x ) . 1 2 k 1 kl 12 l 2 e/0 1 2 k 1 kl 12 l 2

(6.22)

The regularization (6.21) is more preferable technically. The energy—momentum tensor and the metric field may be considered analogously. Let us couple the pair of irreducible fields ¹tr and ¹I to the pair of invariant propagators kl kl S¹53 (x )¹53 (x )T and S¹I (x )¹I (x )T . kl 1 op 2 kl 1 op 2 They are identified with the kernels of invariant scalar products on the spaces M I and M-0/', see h h Eq. (2.91). Accordingly, for even D54 one has a pair of kernels (6.23) DT, 53 (x ) and DTe (x ) , klop 12 klop 12 invariant under the transformation (2.4). The propagator of the tensor ¹53 coincides with the first kl of them DT, 53 (x )"S¹53 (x )¹53 (x )T&H53 (­x)h(D~4)@2d(x ) . (6.24) klop 12 kl 1 op 2 kl,op 12 The propagator of the field ¹I depends on the realization of the representation QI , which acts on kl T equivalence classes M¹I NLM I , see Eq. (2.90). In a non-local realization of the previous section kl T this propagator is longitudinal DT, -0/'(x )"S¹I -0/'(x )¹I -0/'(x )T"C H-0/' (­x)h(D~4)@2d(x ) . (6.25) klop 12 kl 1 op 2 T kl,op 12 The propagator of total energy—momentum tensor ¹ (x) equals to sum of these two expressions. kl In the local realization (2.4) the propagator of the field ¹I demands regularization kl (6.26) DTe (x )"S¹I (x )¹I (x )T klop 12 kl 1 op 2 e

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and is given by the expression (6.12). The transition from Eq. (6.25) to Eq. (6.26) is performed through the transformation inside the equivalence class ¹I -0/', ¹I LM¹I N. Both kernels kl kl kl DT, -0/'(x ) and DTe (x ) (6.27) klop 12 klop 12 define the same scalar product on an invariant subspace M-0/'LM (see (2.8)): h h

P P

(h-0/',h-0/')" dx dx h-0/'(x )DT, -0/'(x )h-0/'(x ) 1 2 kl 1 klop 12 op 2 " dx dx h-0/',e(x )DTe (x )h-0/',e(x )D . 1 2 kl 1 klop 12 op 2 e/0

(6.28)

The irreducible fields hI and h-0/' correspond to the kernels kl kl Dhe (x ) and Dh, -0/'(x ) (6.29) klop 12 klop 12 which are invariant under the transformation (2.10). The first one is singular at e"0 and is given by the expression

C

2 1 Dhe (x )& (x2 )e g (x )g (x )#g (x )g (x )! d d ko 12 lp 12 kp 12 lo 12 klop 12 D kl op e 12

D

1 K P-0/' (­x)ln x2 #2 12 e kl,op

(6.30)

In analogy with Eqs. (6.18) and (6.19), the kernels Dhe and Dh, -0/' define invariant scalar products klop klop on the spaces M53 and M I , respectively (see Eqs. (2.89) and (2.90)). The propagator of the field hI in T kl the local realization coincides with Eq. (6.30), while in the realization of the previous section, with the transversal part of this expression, finite for e"0; see Eq. (6.22) for comparison. 6.4. The equivalence conditions for higher green functions of the current and the energy—momentum tensor Consider the higher Green functions of the fields (6.4). These functions can be expressed through the green functions of irreducible fields SAI u2u`T, SA-0/'u2u`T and S j53u2u`T, S jI u2u`T . (6.31) k k k k Here we assume the invariance of the Green functions SAI u2u`T, S jI u2u`T under the local k k transformations (2.9) and (2.3). Such a realization of the representations QI and QI was used in A j Sections 2—4. In this realization, the Green functions SA u2u`T and Sj u2u`T may be represented as k k sums of pairs of terms SAI (x)u2u`T#SA-0/'(x)u2u`T and S j53(x)u2u`T#S jI (x)u2u`T . (6.32) k k k k Single components of these sums have different partial wave expansions. For example, the functions Sjtr2T are decomposed into the set of transversal functions (2.65), while the functions k SjI 2T, into the set (2.68). These sets are mutually orthogonal. Hence each term in the second sum k

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(6.32) may be readily identified in terms of partial wave expansions. It was used in Section 2 for the derivation of the conditions (2.77) which retain only the second term in the sum (6.32). In non-local realization, Eqs. (6.5) and (6.6), the total Green functions of the fields (6.4) may also be expressed through the functions (6.31). Taking into account that the functions SA-0/'u2u`T k are longitudinal and Sj53u2u`T are transversal, we obtain k ­­ SA (x)u2u`T" d ! k l SAI (x)u2u`T#SA-0/'(x)u2u`T k kl l k h

A

B

­­ Sj (x)u2u`T"Sj53(x)u2u`T# k l SjI (x)u2u`T . k k h l Consider the equivalence conditions (2.26). The operator relations (2.27) may be rewritten in terms of kernels Dje and DAe which are singular at e"0: kl kl

P P

SAI (x)u2u`T" dy Dje (x!y) Sj53,e(y)u2u`TD , l kl l e/0

(6.33)

SjI (x)u2u`T" dy DAe (x!y) SA-0/',e(y)u2u`TD , l kl l e/0

(6.34)

Let us represent the Green functions (6.31) inn terms of partial wave expansions. The expansion of the function SA-0/'u2u`T includes a set of longitudinal three-point functions which are derived : from Eq. (2.69) by the change dPD!d. Denote it as Bl,-0/' k,k12k4 Bl,-0/' (x x x )"SUl 12 4(x )u(x )A-0/'(x )T&M(l!d!s),1NDl (x x x ) k,k12k4 1 2 3 k k 1 2 k 3 A 1 2 3 x2 (l~d~s)@2 &­x3 jx112 4(x x )(x2 )~(l`d~s)@2 23 . (6.35) k k k 2 3 12 x2 13 The functions SAI u2u`T are decomposed into the set of functions k Bl 12 4(x x x )"SUl 12 4(x )u(x )AI (x )T"MA, BN Dl (x x x ) , (6.36) 1k,k k 1 2 3 k k 1 2 k 3 A 1 2 3 where the notation MA, BN was introduced in Eq. (1.83),

C

A B

D

(6.37) Dl (x x x )"(x2 )~(l`d~s)@2(x2 )(l~d~s)@2(x2 )~(l~d~s)@2 . A 1 2 3 12 23 13 Under a suitable choice of coefficients A, B in Eq. (6.36) these functions are related by the equivalence relation with the functions (2.65)

P

Bl 12 4(x x x )" dx DA e (x )Cl,53,e12 4(x x x )D . 2k,k k 1 2 3 4 kl 34 2l,k k 1 2 4 e/0

(6.38)

Here we have used the conformally invariant regularization of the function (2.65). This is done through the substitution l Ple"D!1#e, which is equivalent to introducing of the factor j j (x2 )~e@2(x2 x2 )e@2 into the expression (2.66). Analogously, the functions (6.35) are related by the 12 13 23 equivalence condition with the functions (2.68):

P

(x x x )D . Cl 12 4(x x x )" dx Dj e (x )Bl,-0/',e 4 kl 34 l,k12k4 1 2 4 e/0 1k,k k 1 2 3

(6.39)

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The regularized functions Bl,-0/' are derived by the substitution l Ple "1!e, which is k,k12k4 A A equivalent to introducing of the factor (x2 )e@2(x2 x2 )~e@2 into Eq. (6.37). 12 13 23 Relations (6.38) and (6.39) are the analogues of the amputation conditions (1.35) for the case of pairs of fields AI , j53 and jI , A-0/', respectively. Taking into account that partial wave expansions of k k k k the functions (6.31) include just one term (of the two terms of Eq. (1.86)), as well as the relations (6.38) and (6.39), one can conclude that the equivalence conditions (6.33) and (6.34) are reduced to the equality of kernels for corresponding conformal partial wave expansions. Finally, let us write down the relations inverse to the relations (6.38) and (6.39):

P P

Cl,53 12 4(x x x )" dx Dj, 53(x )Bl 12 4(x x x ) , 2k,k k 1 2 3 4 kl 34 2l,k k 1 2 4

(6.40)

Bl,-0/' (x x x )" dx DA, -0/'(x )Cl 12 4(x x x ) . k,k12k4 1 2 3 4 kl 34 1l,k k 1 2 4

(6.41)

They are related with the equivalence conditions for the Green functions (6.31) written in the form (2.27). All that has been said evidently admits a generalization to the case of the energy—momentum tensor and the metric field. In particular, the equivalence condition (2.108) may be written either in the form given by Eqs. (2.109) and (2.113), or, in analogy with Eqs. (6.33) and (6.34), in the form

P P

hI (x)" dy Dhe (x!y)¹53,e(y)D , kl klop op e/0

(6.42)

¹I (x)" dy DTe (x!y)h-0/',e(y)D . kl klop op e/0 Acknowledgements This work is partially supported by RFBR grant No. 96-02-18966.

Appendix A.

P P A B A B A B p~d

C(d)C(D!d) C(D/2!d)C(d!D/2)

dx (x2 )~d(x2 )~D`d"d(x ) . 3 13 23 13

(A.1)

1 ~d1 1 ~d2 1 ~d3 dx x2 x2 x2 jx112 4(x x ) k k 4 2 4 2 14 2 24 2 34 C(D/2!d )C(D/2!d #s)C(D/2!d ) 1 2 3 C(d #s)C(d )C(d ) 1 2 3 1 ~D@2`d3 1 ~D@2`d2 1 ~D@2`d1 ] x2 x2 x2 jx112 4(x x ) , k k 3 2 2 12 2 13 2 23

"(2p)D@2

A B

where d #d #d "D. 1 2 3

A B

A B

(A.2)

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109

P A B A B A B

1 ~d1 1 ~d2 1 ~d3 dx x2 x2 x2 qx412 4(x Dx x ) k k 1 2 3 4 2 14 24 34 2 2 "(2p)D@2

C(D/2!d )C(D/2!d )C(D/2!d )C(D!d #s!1) 1 2 3 1 C(d #s)C(d #s)C(d #s)C(D!d !s) 1 2 3 1

A B

]

A B

A B

1 ~D@2`d1 1 ~D@2`d2`s 1 ~D@2`d3`s jx112 4(x x ) , x2 x2 x2 23 13 12 k k 2 3 2 2 2

(A.3)

where d #d #d #s"D, 1 2 3 q

k12ks

"q 12q s!traces, qx4(x Dx x )"g (x )jx4(x x ) . k 1 2 3 kl 14 l 2 3 k k

P A B A B A B

1 ~d1 1 ~d2 1 ~d3 dx x2 x2 x2 jx2(x x )jx112 4(x x ) 4 2 34 24 14 k 3 4 k k 3 4 2 2 "(2p)D@2

GA DHA B

C(D/2!d #s)C(D/2!d )C(D/2!d ) 1 2 3] C(d )C(d #1)C(d #s) 1 2 3

C

1 s # + g k(x )jx112 L k2 s(x x )!traces kk 12 k k k 3 2 2x2 12 k/1

B A B

D !d jx2(x x )jx112 s(x x ) 1 l 3 1 k k 3 2 2

A B

1 ~D@2`d1 1 ~D@2`d2 1 ~D@2`d3 x2 x2 x2 , 12 13 23 2 2 2 (A.4)

where d #d #d "D. 1 2 3 To derive the relations (A2)—(A3) we have used the result of Ref. [57].

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Physics Reports 300 (1998) 215—347

Explicitly covariant light-front dynamics and relativistic few-body systems J. Carbonell!,*, B. Desplanques!, V.A. Karmanov", J.-F. Mathiot# ! Institut des Sciences Nucle& aires1, 53 avenue des Martyrs, F-38026 Grenoble Cedex, France " Lebedev Physical Institute, Leninsky Prospekt 53, 117924 Moscow, Russia # Laboratoire de Physique Corpusculaire2, Universite´ Blaise Pascal, F-63177 Aubie% re Cedex, France Received October 1997; editor: G.E. Brown Contents 1. Introduction 1.1. The relevance of relativity in few-body systems 1.2. Why light-front dynamics? 2. Covariant formulation of light-front dynamics 2.1. Transformation properties of the state vector 2.2. Covariant light-front graph technique 2.3. Simple examples 3. The two-body wave function 3.1. General properties of the wave function 3.2. Equation for the wave function 3.3. Relation with the Bethe—Salpeter function 3.4. Application to the Wick—Cutkosky model 3.5. Angular momentum and angular condition 4. The nucleon—nucleon potential 4.1. Mesonic degrees of freedom in nuclei 4.2. The non-relativistic NN potential 4.3. Meson exchange interaction on the light front 5. Applications to the nucleon—nucleon system 5.1. The deuteron wave function 5.2. Connection with the Bethe—Salpeter amplitude 5.3. Two nucleon wave function in the Jp"0` scattering state 5.4. Numerical results

218 218 219 221 221 228 238 244 244 252 255 257 260 261 262 263 268 273 274 277 279 282

6. The electromagnetic amplitude 6.1. Factorization of the electromagnetic amplitude 6.2. Extracting the physical form factors 6.3. Light-front electromagnetic vertex 7. Electromagnetic observables 7.1. Electromagnetic form factors in the Wick—Cutkosky model 7.2. Applications to the quark model 7.3. Application to the nucleon—nucleon systems 8. Concluding remarks Appendix A. Notations Appendix B. Relation to other techniques B.1. Relation to the Weinberg rules B.2. Relation between the Feynman amplitudes and the Weinberg rules B.3. Relation between the Feynman amplitudes andthe covariant light-front graph technique Appendix C. Relation between the deuteron components Appendix D. Two-body kinematical relations D.1. One-loop diagram D.2. Two-loop diagram Appendix E. Three-body kinematical relations References

* Corresponding author. Fax: 334 76 284 004; e-mail: [email protected]. 1 Unite´ mixte de recherche CNRS-Universite´ Joseph Fourier. 2 Unite´ mixte de recherche CNRS-Universite´ Blaise Pascal. 0370-1573/98/$19.00 Copyright ( 1998 Elsevier Science B.V. All rights reserved PII S 0 3 7 0 - 1 5 7 3 ( 9 7 ) 0 0 0 9 0 - 2

285 285 286 288 305 306 309 316 326 327 329 329 331 333 338 340 340 341 343 344

EXPLICITLY COVARIANT LIGHT-FRONT DYNAMICS AND RELATIVISTIC FEW-BODY SYSTEMS

J. CARBONELL!, B. DESPLANQUES!, V. A. KARMANOV", J.-F. MATHIOT# !Institut des Sciences Nucle& aires, 53 avenue des Martyrs, F-38026 Grenoble Cedex, France "Lebedev Physical Institute, Leninsky Prospekt 53, 117924 Moscow, Russia #Laboratoire de Physique Corpusculaire, Universite& Blaise Pascal, F-63177 Aubie% re Cedex, France

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

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Abstract The wave function of a composite system is defined in relativity on a space—time surface. In the explicitly covariant light-front dynamics, reviewed in the present article, the wave functions are defined on the plane u ) x"0, where u is an arbitrary four-vector with u2"0. The standard non-covariant approach is recovered as a particular case for u"(1, 0, 0,!1). Using the light-front plane is of crucial importance, while the explicit covariance gives strong advantages emphasized through all the review. The properties of the relativistic few-body wave functions are discussed in detail and are illustrated by examples in a solvable model. The three-dimensional graph technique for the calculation of amplitudes in the covariant light-front perturbation theory is presented. The structure of the electromagnetic amplitudes is studied. We investigate the ambiguities which arise in any approximate light-front calculations, and which lead to a non-physical dependence of the electromagnetic amplitude on the orientation of the light-front plane. The elastic and transition form factors free from these ambiguities are found for spin 0, 1 and 1 systems. 2 The formalism is applied to the calculation of the relativistic wave functions of two-nucleon systems (deuteron and scattering state), with particular attention to the role of their new components in the deuteron elastic and electrodisintegration form factors and to their connection with meson exchange currents. Straightforward applications to the pion and nucleon form factors and the o!p transition are also made. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 03.65.Pm; 21.45.#v; 25.30.!c Keywords: Relativity; Light-front dynamics; Few-body systems

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1. Introduction 1.1. The relevance of relativity in few-body systems The relevance of a coherent relativistic description of few-body systems, both for bound and scattering states, is now well recognized in nuclear as well as in particle physics. This is already clear in particle physics for the understanding of the wave functions of the valence quarks in the nucleon or in the pion, as revealed for instance in exclusive reactions at very high momentum transfer. The typical example of such a reaction is elastic scattering, and the extraction of the electromagnetic form factors, with their well known behavior at very high momentum transfer, the so-called scaling laws expected from QCD. The need for a coherent relativistic description of few-body systems has also become clear in nuclear physics in order for instance to check the validity of the standard description of the microscopic structure of nuclei in terms of mesons exchanged between nucleons. In this case, electromagnetic interactions play also a central role in “seeing” meson exchanges in nuclei. The forthcoming experiments at Thomas Jefferson National Accelerator Facility (former CEBAF) at momentum transfer of a few (GeV/c)2 are here of particular importance. In both domains, it is obligatory to have, first, a relativistic description of the bound and scattering states. It is also necessary to have a consistent description of the electromagnetic current operator needed to probe the system. This is mandatory in order to have meaningful predictions for the various cross-sections. A few relativistic approaches have been developed in the past ten years in order to meet these goals. Among them, two have received particular attention in the last few years. The first one is based on the Bethe—Salpeter formalism [1] or its three-dimensional reductions. The Bethe—Salpeter formalism is four-dimensional and explicitly Lorentz covariant. The calculational technique to evaluate electromagnetic amplitudes is based on Feynman diagrams and associated rules. Three-dimensional reductions result in equations of the quasi-potential type. The second one is light-front dynamics (LFD) [2]. In this case, the state vector describing the system is expanded in Fock components with increasing number of particles. The state vector is defined on a surface in four-dimensional space—time which should be indicated explicitly. The Fock components — the relativistic wave functions in this formalism — are the direct generalization of the non-relativistic wave functions. In the non-relativistic limit (cPR), the wave function is defined at time t"0, and the time evolution is governed by the Schro¨dinger equation, once the Hamiltonian of the system is known. This is the “instant” form of dynamics. Physical processes are thus calculated according to old fashioned (time ordered) perturbation theory. This form of dynamics is however not very well suited for relativistic systems, since the interaction of a probe (the electron, for instance) with the constituents of the system is not separated from its interaction with the vaccum fluctuations. Moreover, the plane t"cte is not conserved by a Lorentz boost. In the standard version of LFD, the wave function is defined on the plane t#z/c"cte [2]. It is also equivalent to the usual equal time formalism in the infinite momentum frame. From a qualitative point of view, all the physical processes become as slow as possible because of time dilation in this system of reference. This greatly simplifies the description of the system. The investigation of the wave function is equivalent to make a snapshot of a system not spoiled by vacuum fluctuations.

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It is thus very natural in the description of high energy experiments like deep inelastic scattering. The calculational technique is here based on the Weinberg rules [3]. This formulation has however a serious drawback since the equation of the plane t#z"cte (we take here and in the following c"1) is still not invariant by an arbitrary rotation in space and time (Lorentz boost). This plane breaks rotational invariance. As we shall see later on, this fact has many important consequences as far as the construction of bound or scattering states of definite angular momentum is concerned, or in the calculation of electromagnetic amplitudes. 1.2. Why light-front dynamics? We shall present in this review a covariant formulation of LFD which provides a simple, practical and very powerful tool to describe relativistic few-body systems, their bound and scattering states, as well as their physical electromagnetic amplitudes. In this formulation, the state vector is defined on the plane characterized by the invariant equation u ) x"0, where u is an arbitrary light-like four vector u"(u0, x), with u2"0 [4,5]. With the particular choice u"(1, 0, 0,!1), we recover the standard LFD defined on the plane t#z"0. The covariance of our approach is realized by the invariance of the light-front plane u ) x"0 under any Lorentz transformation of both u and x. This implies in particular that u cannot be kept the same in any system of reference, as it is the case in the usual formulation of LFD with u"(1, 0, 0,!1). There is of course equivalence, in principle, between these approaches. Any exact calculation of a given electromagnetic process should give the same physical cross-section, regardless of the relativistic formalism which is used. In practical calculations however, one can hardly hope to carry out an exact calculation of physical observables starting from “first principles”, although some simple theoretical systems can of course be used as test cases (the Wick—Cutkosky model for instance). From a practical point of view, one may thus be led to choose a particular formalism depending on the system one is interested in. This choice is of course dependent on various criterions, as well as function of personal taste. Among these criterions, let us mention those we think are the most important: (i) The formalism should enable us to have physical insights into the various processes under consideration, at each step of the calculation. (ii) It should have a direct and transparent non-relativistic limit in order to gain more physical intuition from our present knowledge in the non-relativistic domain. This is particularly important in nuclear physics where a lot is already known from the non-relativistic description of the microscopic structure of nuclei and their electromagnetic interactions at the scale of 1 GeV or less [6]. (iii) It should provide a as simple as possible calculational procedure to evaluate physical processes and compare them with experimental results. We would like to show in this review that the covariant formulation of LFD is, according to these criterions, of particular interest. As we shall see extensively in the following, it has definite advantages as compared to both the Bethe—Salpeter formalism, and the usual formulation of LFD. Let us recall the most important ones below. (i) The calculational formalism — time ordered graph technique (in the light-front time) in contrast to old fashioned perturbation theory — does not involve vacuum fluctuation contributions. This strongly simplifies a priori the physical picture and calculations.

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(ii) In this approach, the wave functions — the Fock components of the state vector — satisfy a three-dimensional equation, and have the same physical meaning (probability amplitudes) as the non-relativistic wave functions. Their non-relativistic limit is thus explicit from the general structure of the wave function. This enables a transparent link with non-relativistic approaches in first 1/m order and in particular with the contribution of the dominant meson exchange current (the so-called pair term) in the two-nucleon systems. (iii) The explicit covariance of LFD is very important in practical applications: it allows to construct states with definite total angular momentum, to separate contributions of relativistic origin from non-relativistic ones, and simplifies very much the calculations in the framework of a special graph technique. In these respects, the advantages of the covariant formulation in comparison to ordinary LFD are the same as the advantages of the Feynman graph technique in comparison to old fashioned perturbation theory. (iv) A very important property of relativistic wave functions and off-shell amplitudes is their dependence on the orientation of the light-front plane. It takes place both in non-covariant and covariant approaches. In the covariant approach this dependence is parametrized explicitly in terms of the four-vector u. On the contrary, exact on-shell physical amplitudes should not depend on the orientation of light-front plane. However, in practice, this dependence survives due to approximations. The covariant representation of the electromagnetic amplitudes allows to separate the physical form factors from the unphysical contributions. As already mentioned, the present approach differs from the standard ones by the parametrization of the current choice of the z direction by an arbitrary one n. A second major difference is the reference to a particular field theory inspired dynamics, which implies that our approach describes states with an unfixed number of particles. This is a source of many problems, especially related to the mass dependence of the interaction. These problems, which will receive a particular attention, have a strong relationship with those encountered in nuclear physics when using an energydependent nucleon—nucleon interaction (Bonn-E potential [7]), or with the contribution of recoil and norm corrections to meson-exchange currents [8]. Our aim in this review is to present the various facets of the covariant formulation of LFD, and compared them to the standard formulation. For completeness, we shall also make contacts with the BS formalism. The relevance of light-front quantization in quantum field theory is the subject of an intense present activity. Among others, the non-perturbative problems of how the condensates present in many theories appear on the light-front, or how renormalization should be implemented, are not yet completely solved [9]. However, directions of research are clearly identified (zero modes in LFD in particular). These problems are outside the scope of this review, although the explicit covariance of our formalism may be of particular interest in solving these problems. We believe that the real advantages of the covariant formulation of LFD can be apprehended in practice, i.e. in applications to relativistic nuclear and particle physics and field theoretical problems. The first applications to relativistic nuclear and particle physics are reviewed in the present paper, with particular attention to two-nucleon systems for which many experimental data exist at present or are expected in the near future. Many extensive or review papers have been devoted to the description of few-body systems in relativistic approaches. Some of them are based on different versions of the quasi-potential equations [10,11]. Many applications were made using the three-dimensional Gross equation

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[12—16]. Two- and three-nucleon systems were investigated in the framework of the Bethe—Salpeter equation in Refs. [17—20] (see for a review Ref. [21]). The solution of the Bethe—Salpeter equation was also found and applied to deep inelastic scattering on deuteron and to the deuteron electrodisintegration in Refs. [22—24]. The standard version of LFD and its applications to few-body systems was reviewed in Refs. [25—31]. The first results on the covariant formulation of LFD are given in Ref. [5]. This approach has been investigated and developed also in Ref. [32]. The content of this review is the following. We develop in Section 2 the general properties of the covariant formulation of LFD, and derive the graph technique associated to it. Section 3 is devoted to the properties of the two-body wave function: spin structure, equation for the wave function, as well as its connection to the Bethe—Salpeter amplitude. We pay particular attention to the construction of the angular momentum operator. We apply our formalism to the two-nucleon system in Section 4 by first deriving the nucleon—nucleon potential in this formalism. The twonucleon wave function (deuteron and 1S0 wave function) is then constructed in Section 5. The general structure of the electromagnetic amplitude is detailed in Section 6, where we show how to extract the physical form factors for spin 0, 1/2 and 1 systems, as well as transition form factors. We discuss in Section 7 the electromagnetic form factor of the simplest states with J"0 and 1 of the Wick—Cutkosky model, as well as a few hadronic systems (pion, nucleon, etc.). We apply our formalism to the two-nucleon system (deuteron form factors and electrodisintegration crosssection). We also discuss, in a 1/m expansion, the relationship between our formalism and first-order relativistic corrections taken into account as meson exchange currents (the so-called pair term) in non-relativistic approaches. The Sections 2,3 and 6 are quite general and apply to any system. Sections 4,5 and 7 are direct applications to the two-nucleon systems, as well as three quark and quark-antiquark systems.

2. Covariant formulation of light-front dynamics We detail in this section the general properties of the covariant formulation of LFD. In contrast to the standard approach, the transformations of the coordinate system and of the light-front plane can be done independently from each other. These two types of transformations entail the corresponding transformation properties of the state vector and its Fock components. Particular attention is paid to the angular momentum operator. This formal field-theoretical introduction (Section 2.1) can be omitted by a reader not interested in these details. We then present the graph technique associated to this formulation, for particles of spin 0, 1/2 and 1. We illustrate peculiarities of this graph technique by a few simple examples. 2.1. Transformation properties of the state vector The state vector is defined in general on a hypersurface in space—time and therefore depends dynamically on its position. For example, the non-relativistic wave function t(t) depends dynamically on translations of the plane t"cte, i.e. on time t (by the trivial phase factor exp(!iEt) for

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a bound state, where the binding energy E is determined by the dynamics). In ordinary LFD, the wave function is defined on the plane t#z"t`"cte. It depends dynamically on any translation of this plane. However, some Lorentz transformations and rotations change the orientation of this plane [2]. These transformations are thus also dynamical ones. That means lack of explicit covariance (i.e. the impossibility to transform the state vector from one reference system to another one without knowledge of the dynamics). This does not mean the absence of Lorentz covariance at all, since there exist anyhow a closed system of generators of the Poincare´ group, and the observable amplitudes calculated exactly would be covariant. In practical approximate calculations, however, the covariance is lost. In the covariant formulation of LFD, the wave function is defined on the general plane u ) x"p, where u is an arbitrary four-vector restricted to the condition u2"0, and p is the “light-front time”. When there is no need to refer to the p evolution, we shall take p"0. The kinematical transformations of the system of reference are thus separated from the dynamical transformations of the plane u ) x"p. The dynamical dependence of the wave function on the light-front plane results in that case in their dependence on u. This separation of kinematical and dynamical transformations has a definite advantage in the sense that it provides a definite prescription for constructing bound and scattering states of definite angular momentum. Since the total angular momentum of a composite system is determined by the transformation properties of its wave function under rotation of the coordinate system, the construction of systems with definite J is now purely kinematical, in contrast to the formulation on the plane t#z"t`. The dynamical part of this problem, resulting from the presence of the interaction in the generators of the Poincare´ group which change the position of the light-front plane, is separated out and replaced by the so-called angular condition. 2.1.1. Kinematical transformations Let us first specify the transformation properties of the state vector with respect to transformations of the coordinate system. We will use for this purpose a field-theoretical language. The operators associated to the four-momentum and four-dimensional angular momentum are expressed in terms of integrals of the energy-momentum ¹ and the angular momentum Mo tensors kl kl over the light-front plane u ) x"p, according to:

P P

PK " ¹ uld(u ) x!p) d4x"PK 0#PK */5 , k kl k k

(2.1)

JK " Mo u d(u ) x!p) d4x"JK 0 #JK */5 , kl o kl kl kl

(2.2)

where the 0 and int superscripts indicate the free and interacting parts of the operators, respectively. For generality, we consider here the light-front time pO0. The description of the evolution along the light-front time p implies a fixed value of the length of x, or, equivalently, of u . This is necessary in order to have a scale of p. However, the most 0 important properties of the physical amplitudes following from covariance do not require to fix the scale of u and will be invariant relative to its change.

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We work in the interaction representation in which the operators are expressed in terms of free fields. For example, for a scalar field u(x), the free operators PK 0 have the form: k

P P

PK 0" as(k)a(k)k d3k , k k

A

(2.3)

B

­ ­ JK 0 " as(k)a(k)i k !k d3k , kl k­kl l­kk

(2.4)

where as and a are the usual creation and destruction operators, with [a(k), as(k@)]"d3(k!k@). The operators PK */5 and JK */5 contain the interaction Hamiltonian H*/5(x):

P

PK */5"u H*/5(x)d(u ) x!p) d4x , k k

P

JK */5" H*/5(x)(x u !x u )d(u ) x!p) d4x . k l l k kl

(2.5) (2.6)

For the particular applications considered in this review, we do not need to develop the field theory in its full form. We therefore do not pay attention here to the fact that the field-theoretical Hamiltonian H*/5(x) is usually singular and requires a regularization. The regularization of amplitudes in our formulation will be illustrated by the example of a typical self-energy contribution at the end of this section. Eq. (2.5) is consistent with the expectation that only the component P along the direction u of k the light-front “time” has a dynamical character in the light-front formalism [2]. Under translation x P x@"x#a of the coordinate system A P A@, the equation u ) x"p takes the form u ) x@"p@, where p@"p#u ) a. The state vector is transformed in accordance with the law: / (p) P /@ (p@)"º 0(a)/ (p) , (2.7) u u P u where º 0(a) contains only the operator of the four-momentum (2.3) of the free field: P (2.8) º 0(a)"exp(iPK 0 ) a) . P The “prime” at /@(p) indicates that /@(p) is defined in the system A@ on the plane u ) x@"p in contrast to /(p) defined in the system A on the plane u ) x"p (the value of p being the same). The state vector /@(p@) is defined in A@ on the plane u ) x@"p@, which coincides with u ) x"p. Therefore no dynamics enters into the transformation (2.7). This is rather natural, since under translation of the coordinate system the plane u ) x"p occupies the same position in space while it occupies a new position with respect to the axes of the new coordinate system, as indicated in Fig. 1. The formal proof of Eqs. (2.7) and (2.8) can be found in Ref. [33]. In the case of infinitesimal four-dimensional rotations x P x@ "gx "x #e xl, the result is k k k k lk similar [33]: / (p) P /@ (p)"º 0(g)/ (p) , u u{ J u where u@ "u #e ul and k k lk º 0(g)"1#1JK 0 ekl . 2 kl J The operator JK 0 is given by Eq. (2.4). kl

(2.9)

(2.10)

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Fig. 1. Translation of the reference system along the light-front time.

2.1.2. Dynamical transformations The properties of the state vector under transformations of the hypersurface are determined by the dynamics and follow from the Tomonaga—Schwinger equation [34]: id//dp(x)"H*/5(x)/ .

(2.11)

From the definition of the variational derivative in Eq. (2.11) we obtain id/"H*/5(x)/d»(x) , where d»(x) is the volume between the initial surface and the surface obtained from the original one by the variation dp(x) around the point x. Under the translation p P p#dp of the plane, the total increment of the state vector is obtained through the increment at each point of the surface:

P

id/" H*/5(x)d(u ) x!p) d4x /dp .

(2.12)

This relation gives the Schro¨dinger equation. In the interaction representation in the light-front time, we have i­//­p"H(p)/(p) ,

(2.13)

where

P

H(p)" H*/5(x)d(u ) x!p) d4x , u

(2.14)

and H*/5(x) may differ from H*/5(x) because of singularities of the field commutators on the light u cone. This point is explained below in Section 2.2.1. Similarly, in the case of a rotation of the light-front plane, u P u@ "u #du , du "e ul, k k k k k lk we find

A

B

1 ­ ­ / (p) P / (p)"/ #d/ , d/ " e uk !ul / (p) . u u`du u u u 2 kl u ­u ­u l k

(2.15)

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225

The increment of the volume over the point x is d»"e xk ul d(u ) x!p) d4x , kl and it follows from Eq. (2.12) that [33]

(2.16)

JK */5/ (p)"¸K (u)/ (p) , kl u kl u where

(2.17)

A

B

­ ­ ¸K (u)"i u !u , kl k­ul l­uk

(2.18)

and JK */5 is given by Eq. (2.6). kl Eq. (2.17) is called the angular condition. It plays an important role in the construction of relativistic bound states, as we shall explain below. The transformation of the coordinate system and the simultaneous transformation of the light-front plane, which is rigidly related to the coordinate axes, correspond to the successive application of the two types of transformations considered above (kinematical and dynamical). Thus, under the infinitesimal translation x P x@"x#a of the coordinate system, A P A@, and of the plane, we have / (p) P /@ (p)"(1#iPK ) a)/ (p) . (2.19) u u u Note that for the state with definite total four-momentum p (i.e., for an eigenstate of the four-momentum operator), Eqs. (2.7) and (2.19) give exp(iPK 0 ) a)/(p)"exp(ip ) a)/(p#u ) a) .

(2.20)

This equation determines the conservation law for the four-momenta of the constituents, given in Section 3. 2.1.3. Role of the angular condition We are interested here in the state vector of a bound system. It corresponds to a definite mass M, four-momentum p, total angular momentum J with projection j onto the z axis in the rest frame, i.e., the state vector forms a representation of the Poincare´ group. This means that it satisfies the following eigenvalue equations: PK /Jj(p)"p /Jj(p) , k k PK 2 /Jj(p)"M2 /Jj(p) ,

(2.21)

SK 2 /Jj(p)"!M2 J(J#1) /Jj(p) ,

(2.23)

SK /Jj(p)"M j/Jj(p) , 3 where SK is the Pauli—Lubanski vector: k SK "1e PK l JK oc . k 2 kloc

(2.22)

(2.24)

(2.25)

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The state vector Dp, jT for a given J is normalized as follows: u Sj@, p@Dp, jT "2p d(3)(p!p @)dj{j . u u 0 For convenience we introduce here another notation for the same state vector:

(2.26)

Dp, jT ,/Jj(p) . u u We omit in /Jj(p) the argument p, but show explicitly the argument p. For simplicity, we have left u out the u subscript when not absolutely necessary. We can now use the angular condition (2.17) and replace the operator JK */5, which is contained in kl JK in Eq. (2.25), by ¸K (u). Introducing the notations: kl kl MK "JK 0 #¸K (u) , (2.27) kl kl kl ¼ K "1e PK l M K oc , (2.28) k 2 kloc we obtain instead of Eqs. (2.23) and (2.24) ¼ K 2/Jj(p)"!M2J(J#1) /Jj(p) ,

(2.29)

¼ K /Jj(p)"M j /Jj(p) . (2.30) 3 ¹hese equations do not contain the interaction Hamiltonian, once / satisfies Eqs. (2.21) and (2.22). The construction of states with definite angular momentum becomes therefore a purely kinematical problem. At the same time, the state vector must satisfy the dynamical equation (2.17). In terms of the operators JK and MK , the angular condition (2.17) can be rewritten as kl kl MK (u)/ (p)"JK / (p) . (2.31) kl u kl u The commutation relations between the operators PK , JK ,M K have the form: k kl kl [PK , PK ]"0 , (2.32) k l 1 [PK , JK ]"g PK !g PK , (2.33) ko i ki o i k io 1 [JK , JK ]"g JK !g JK #g JK !g JK , ko lc lo kc lc ko kc lo i kl oc

(2.34)

1 [PK , M K ]"g PK !g PK , io ko i ki o i k

(2.35)

1 [M K ,M K ]"g M K !g M K #g M K !g M K , kl oc ko lc lo kc lc ko kc lo i

(2.36)

1 [JK , M K ]"g JK !g JK #g JK !g JK . oc ko lc lo kc lc ko kc lo i kl

(2.37)

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Eqs. (2.32), (2.33) and (2.34) are the standard commutation relations of the Poincare´ group. Eqs. (2.35), (2.36) and (2.37) simply reflect the tensor nature of the operators. This set also corresponds to the Poincare´ group transformations. The derivation of these equations is explained in Ref. [5]. We have thus shown that under condition (2.17) the problem of constructing states with definite angular momentum can be formulated in terms of the kinematical operators M K and ¼ K in exactly kl k the same way as in terms of the operators JK and S which depend on the interaction. This kl k naturally reflects the fact that the angular momentum of a system determines the kinematical properties of the wave function relative to transformations of the coordinate system. The dynamics is involved in the composition of the angular momentum of the system from the spins of its constituents. As already mentioned, the angular momentum generators in ordinary LFD contain the interaction. To avoid any misunderstanding, note that the interaction Hamiltonian does not disappear in our approach. It is moved into the angular condition (2.31). However, it is very convenient to first construct on purely kinematical grounds the general form of the light-front wave function for a given angular momentum. One can then find from dynamics the coefficients in front of all the spin structures. This way of separating dynamical from kinematical transformation properties is of particular interest since the interaction Hamiltonian is often approximate. We emphasize that by introducing the operator ¸K (u), Eq. (2.18), containing the derivatives kl over u, we enlarge the Hilbert space where the state vector is defined. It is now the direct sum of the Hilbert space where the “normal” field-theoretical operators act and of the Hilbert space, where the operator ¸K (u) acts. Hence, in this enlarged space the scalar product and, correspondingly, the kl normalization condition contains integration over u with the appropriate measure dk : u

P

Sj@, p@Dp, jT dk "2p d(3)(p!p @) dj{j . u u 0 u

(2.38)

The particular form of the measure dk corresponds to integration over the directions of x in u a particular system of reference. The angular condition, Eqs. (2.17) and (2.31), just ensures the equivalence of the approach developed in the enlarged Hilbert space to the ordinary approach. In particular, the orthogonality condition (2.38), which contains integration over u, has to be equal to the orthogonality condition (2.26), where u is a fixed parameter. This means, of course, that the product Sj@, p@Dp, jT does not depend on u at all (though for an arbitrary vector D2T from the u u u enlarged Hilbert space, the scalar product S2D2T depends on u). However, any separate u u contribution of the Fock component to Sj@, p@Dp, jT depends on u, while the u-dependence u u disappears in the sum over all Fock components. In the enlarged Hilbert space, the angular momentum operator is given in terms of derivatives on the momenta and u and do not contain the interaction. The construction of states with definite angular momentum becomes therefore very simple. In practice, it is convenient first to solve the kinematical part of the problem — the construction of states with definite angular momentum — and then satisfy the angular condition (2.17). The transformation properties of a state with definite angular momentum are discussed in Section 3. We will see that they completely determine the structure of the wave function, and, in particular, the number of its spin components (six in the case of the deuteron). These components

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will be found below by solving the light-front generalization of the Schro¨dinger equation. At first glance, this procedure is unambiguous and nothing remains to be determined by the angular condition. What is then the role of this condition? Without solving the angular condition, there is an ambiguity in finding states with definite angular momentum. To show that, let us construct the operator: AK "(¼ K ) u)2 ,

(2.39)

where ¼ K is the kinematical Pauli—Lubansky vector (2.28). It is readily verified that this operator k commutes with PK ,MK ,¼ K and also with the parity operator. It seems therefore that the state k kl kl vector must be characterized not only by its mass, momentum and angular momentum, but also by the eigenvalue a of the operator A: AK / "a / . (2.40) a a One can show that, for instance, for the total angular momentum J"1, there are only two states with a"0 and a"1. However, these states / are degenerate. Since the commutators of M K and a kl of JK with PK ,JK are equal to each other (see Eqs. (2.32), (2.33), (2.34), (2.35), (2.36) and (2.37)), the kl k kl operator DJK "M K !JK "¸K (u)!JK */5 (2.41) kl kl kl kl kl commutes with PK and with JK , but [DJK , AK ]O0. The state /@"DJK / is therefore not an k kl kl kl a eigenvector of the operator AK , i.e., it can be represented in the form /@"+ b / . But it corresponds a a a to the same mass as / . We would thus always get for the deuteron and for any state with J"1 two a degenerate states with a"0 and 1, in evident contradiction with reality. The angular condition (2.17) is just distinguishing a definite superposition of states / , a /"+ c / , (2.42) a a a which is such that DJK /"0. This equation eliminates the problem of the “spurious” states of kl relativistic composite systems. This procedure will be illustrated in Section 3.5 by a simple example. 2.2. Covariant light-front graph technique The light-front graph technique is a method for calculating the S-matrix. In the framework of perturbation theory, the on-shell amplitude given by this graph technique coincides with the one given by the Feynman graph technique. However, the methods to calculate the amplitude drastically differ from each other, as we shall see in this section. The most important difference lies in the fact that in the LFD all four-momenta are always on the mass shell. This three-dimensional form of the theory has enormous advantages in solving several problems. In applications to relativistic few-body systems, it provides a direct and close connection between the relativistic wave functions and the non-relativistic ones, as we shall detail in the next sections. Moreover, the diagrams corresponding to vacuum fluctuations are absent. This simplifies very much the theory as compared to other three-dimensional approaches (like the old fashioned time ordered perturbation theory). Another important advantage results from the explicitly covariant formulation of the light-front plane defined by u ) x"0 as compared to the standard formulation on the plane t#z"0.

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We shall derive in this section the rules pertinent to the covariant light-front graph technique by transforming the standard expression for the S-matrix. We explicitly show in Appendix B the relations between the covariant light-front amplitudes and the amplitudes given by the Weinberg and Feynman rules. 2.2.1. General derivation The graph technique described below was developed by Kadyshevsky [35] (see Ref. [36] for a review) and applied to LFD in Ref. [4]. It is manifestly covariant, like the Feynman graph technique, and retains all the positive features of the old fashioned perturbation theory developed by Weinberg [3]. Following Ref. [35], we start from the standard expression for the S-matrix:

C P P

D

S"¹ exp !i H*/5(x) d4x "1

#+ (!i)nH*/5(x )h(t !t )H*/5(x )2h(t !t )H*/5(x ) d4x 2 d4x , (2.43) 1 1 2 2 n~1 n n 1 n n where H*/5(x) is the interaction Hamiltonian. The sign of the ¹-product (and 1/n!) are omitted, since the time ordering is made explicit by means of the h-functions. The expression (2.43) is then represented in terms of the light-front time p"u ) x:

P

S"1#+ (!i)nH*/5(x )h(u ) (x !x ))H*/5(x )2h(u ) (x !x ))H*/5(x ) u 1 1 2 u 2 n~1 n u n n ]d4x 2d4x . 1 n

(2.44)

The index u at H*/5 indicates that H*/5 and H*/5 may differ from each other in order to provide the u u equivalence between Eqs. (2.43) and (2.44). The region in which this can happen is a line on the light cone. Indeed, if (x !x )2'0, the signs of u ) (x !x ) and t !t are the same and hence 1 2 1 2 1 2 H*/5"H*/5. If (x !x )2(0, the operators commute: u 1 2 [H*/5(x ), H*/5(x )]"0 , 1 2

(2.45)

and their relative order has no significance. On the light cone, i.e., if (x !x )2"0, u ) (x !x ) 1 2 1 2 can be equal to zero while t !t may be different from zero. If the integrand has no singularity at 1 2 (x !x )2"0, this line does not contribute to the integral over the volume d4x. However, if the 1 2 integrand is singular, some care is needed. To eliminate the influence of this region on the S-matrix, we have introduced in Eq. (2.44) a new Hamiltonian H*/5, such that expressions (2.43) and (2.44) be u equal to each other. The form of H*/5, which provides the equivalence between Eqs. (2.43) and u (2.44), depends on the singularity of the commutator (2.45) at (x !x )2"0. For the scalar fields, 1 2 the singularity is weak enough, and expressions (2.43) and (2.44) are the same, so that H*/5"H*/5. u For fields with spins 1 and 1 or with derivative couplings, the equivalence is obtained with 2 H*/5 differing from H*/5 by an additional contribution (counterterm) leading to contact terms in the u propagators (or so called instantaneous interaction). We shall come back to this point later on in this section.

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Introducing the Fourier transform of the Hamiltonian:

P

HI (p)" H*/5(x) exp(!ip ) x) d4x , u u

(2.46)

and using the integral representation for the h function:

P

1 `= exp(iqu ) (x !x )) 1 2 dq , h(u ) (x !x ))" 1 2 2pi q!ie ~= we can transform expression (2.44) to the form:

(2.47)

S"1#R(0)"1!iHI (0) u

P

dq dq 1 n~1 # + (!i)n HI (!uq ) HI (uq !uq )2 HI (uq ) . (2.48) u 1 2pi(q !ie) u 1 2 2pi(q !ie) u n~1 1 n~1 nz2 The q variable appears here as an auxiliary variable, as defined in Eq. (2.47); uq has the dimension of a momentum. The S-matrix (2.48) gives the state vector /(p)"S(p)/ for asymptotic states, i.e., for an infinite 0 value of the light-front time p"R. It determines the on-energy-shell amplitude. The off-energy shell amplitude is determined by S(p) at finite p. Similarly, introducing another h-function h(p!u ) x ) in Eq. (2.44), one can find easily that the S-matrix on a finite light-front plane p is 1 represented in the form:

P

S(p)"1#

= exp(iqp) R(uq) dq , 2pi(q!ie) ~=

(2.49)

where

P

dq dq 1 n~1 R(uq)"+ (!i)n HI (uq!uq ) HI (uq !uq )2 HI (uq ) . u 1 2pi(q !ie) u 1 2 2pi(q !ie) u n~1 1 n~1 n (2.50) These formulae give the iterative solution of Eq. (2.13). The matrix elements of the operator R(uq) for qO0 correspond to the off-energy shell amplitudes, whereas at q"0 they give the on-energy shell amplitude. We shall precise the difference between off-energy shell and off-mass-shell amplitudes below in Section 2.3.1. At p P R, the only residue in Eq. (2.49) for q"ie P 0 survives, and we recover Eq. (2.48). We emphasize that despite the presence of the four-vector u in Eq. (2.48), the S-matrix and any physical amplitude do not depend on u, since Eq. (2.48) gives the same S-matrix, as the initial one given by Eq. (2.43). Similarly, the off-shell matrix R(uq) in Eq. (2.50), depends on u and off-shell light-front amplitude does not coincide with the Feynman one. This is natural since R(uq) determines the S-matrix at a given light-front plane u ) x"p in the interaction region. As we shall see in Section 3, the same is also true for light-front wave functions. The latters depend on u since they are always off-shell objects and are defined not as asymptotic states, but at any finite time (at any given light-front plane in our case).

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2.2.2. Spin 0 system The covariant light-front graph technique arises when, as usual, one represents the expression (2.48) in normal form. Let us consider for example the simple case of an interaction Hamiltonian of the form H"!gu3(x), where u is a scalar field. We introduce the Fourier transform uJ of the field u given by

P P

1 u(x), uJ (k) exp(ik ) x) d4k (2p)3@2 1 d3k " [a(k) exp(!ik ) x)#as(k) exp(ik ) x)] . (2p)3@2 J2e k We thus have uJ (k)"[a(!k)h(!k )#as(k)h(k )]J2e d(k2!m2) . 0 0 k When the S-matrix (2.48) is reduced to normal form, we obtain the contractions:

(2.51)

(2.52)

"uJ (k)uJ (p) !:uJ (k)uJ (p) " : h(p )d(p2!m2)d(4)(p#k) . (2.53) 0 It is convenient to replace in the following h(p ) in the propagator (2.53) by h(u ) p). This is always 0 possible, since p2"m2'0. We would like to emphasize at this point that propagator (2.53) contains the delta-function d(p2!m2), and therefore all particles are always on their mass shells. The reason of this property, which drastically differs from the Feynman approach, is the absence of the T-product operator in Eq. (2.43). In this graph technique, the four-vectors uq in Eq. (2.48) are associated with a fictitious particle j — called spurion — and the factors 1/(q !ie) are interpreted as the propagator of the spurions j responsible for taking the intermediate states off the energy shell. This spurion should be interpreted as a convenient tool in order to take into account off-energy shell effects in the covariant formulation of LFD (in the absence of off-mass shell effects), and not as a physical particle. It is absent, by definition, in all asymptotic, on-energy shell, states. We shall show below on simple examples how the spurion should be used in practical calculations. The general invariant amplitude M of a transition m P n is related to the S-matrix by nm m n M nm S "1#i(2p)4d(4) + k ! + k@ , (2.54) nm i i ((2p)32e 2(2p)32e (2p)32e 2(2p)32e )1@2 1 n 1 m k{ k k k{ i/1 i/1 where, e.g., e 1"Jm2#k2. The cross-section of the process 1#2 P 3#2#n is thus expressed as: 1 1 k (2p)4 d3k d3k 3 2 n d(4)(k #k !k !2!k ) , dp" DMD2 (2.55) 1 2 3 n 4je 1e 2 (2p)32e 3 (2p)32e n k k k k where j is the flux density of the incident particles:

A

B

je 1e 2"1[s!(m #m )2]1@2[s!(m !m )2]1@2 , s"(k #k )2 . k k 2 1 2 1 2 1 2 To find the matrix element M of order n one must proceed as follows [35—37,4,5]:3 3 In order to stick to conventional notations, the normalization of the amplitude given by the standard formulae (2.54) and (2.55) and some factors in the rules of the graph technique (mainly, the degrees of 2p) differ from the ones previously used in Refs. [5,35—37]. Besides, the numbering of vertices here and in [5,35—37] has opposite order.

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Fig. 2. The vacuum vertices.

1. Arbitrary label by a number the vertices in the Feynman graph of order n. Orientate continuous lines (the lines of physical particles) in the direction from the smaller to the larger number. Initial particles are oriented as incoming into a graph, and final particles as outgoing. Connect by a directed dashed line (the spurion line) the vertices in the order of decreasing numbers. Diagrams in which there are vertices with all incoming or outgoing particle lines (vacuum vertices, as indicated in Fig. 2) can be omitted. Associate with each continuous line a corresponding four-momentum, and with each jth spurion line a four-momentum uq . j 2. To each internal continuous line with four-momentum k, associate the propagator h(u ) k)d(k2!m2), and to each internal dashed line with four-momentum uq the factor j 1/(q !ie). j 3. Associate with each vertex the coupling constant g. All the four-momenta at the vertex, including the spurion momenta, satisfy the conservation law, i.e., the sum of incoming momenta is equal to the sum of outgoing momenta. 4. Integrate (with d4k/(2p)3) over those four-momenta of the internal particles which remain unfixed after taking into account the conservation laws, and over all q for the spurion lines from j !R to R. 5. Repeat the procedure described in 1—4 for all n! possible numberings of the vertices. We omit here the factorial factors that arise from the identity of the particles and depend on the particular theory. The vacuum vertices indicated in Fig. 2 disappear for a trivial reason: it is impossible to satisfy the four-momentum conservation law for them. Indeed, the conservation law for the vertex of Fig. 2 has the form k #k #k "u(q !q ). Since the four-momenta are on the mass shell: 1 2 3 1 2 k2 "m2'0, so that the left-hand side is always strictly positive: (k #k #k )258m2, whereas 1~3 1 2 3 the right-hand side is zero since u2"0. However, it will be seen that the vacuum contributions that vanish in the light-front approach leave their track in a different way, making in the cases discussed below the light-front interaction H (x) in Eq. (2.44) different from the usual interaction H(x) in u Eq. (2.43). The light-front diagrams can be interpreted as time-ordered graphs. As soon as the vertices are labelled by numbers, any deformation of a diagram changing the relative position of the vertex projections on the “time direction” from left to right does not change the topology of the diagram and the corresponding amplitude. Therefore, it is often convenient to deform the diagram so that the vertices with successively increasing numbers are disposed from left to right. This just corresponds to time-ordered graphs.

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The light-front amplitudes can be also obtained by direct transformation of a given Feynman amplitude. This transformation is given in Section B.3 of Appendix B. 2.2.3. Spin 1 system 2 The rules of the graph technique for spin 1 particles are similar to those given above except for 2 the fact that one has to worry about contact interactions we already mentioned in Section 2.2.1. To see this from a more practical point of view, let us first consider the diagrams of Fig. 3a and Fig. 3(b) for scalar particles and consider for a moment the space-like plane j ) x"p with j2"1, as developed by Kadyshevsky [35]. In accordance with the rules given above, the amplitude for the diagram of Fig. 3a gets the form:

P

dq 1 . M "g2 d[(k#p#jq )2!m2]h[j ) (k#p)#q ] a 1 1 q !ie 1 Integrating over dq by means of the d-function, we get 1 g2 M" , a 2q J[j ) (k#p)]2#m2!(k#p)2 1 q "!j ) (k#p)#J[j ) (k#p)]2#m2!(k#p)2 . 1 Similarly, we obtain the following expression for the amplitude of Fig. 3b:

(2.56)

(2.57)

g2 , M" b 2q J[j ) (k#p)]2#m2!(k#p)2 2 q "j ) (k#p)#J[j ) (k#p)]2#m2!(k#p)2 . 2

(2.58)

Fig. 3. Exchange of a particle in the s channel; (a) contribution from intermediate state containing one particle; (b) contribution from antiparticle intermediate state; (c) same as (b) but rewritten as a time ordered diagram.

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The sum of Eqs. (2.57) and (2.58) gives the usual Feynman amplitude in the s channel: g2 . M #M " a b m2!(k#p)2 In the case where j"(1,0), we recover the contribution of the usual time ordered graphs in the old fashioned perturbation theory: g2 M" , a 2ek p[ek p!ek!ep] ` ` g2 . (2.59) M" b 2ek p[ek p#ek#ep] ` ` The diagram of Fig. 3b and the amplitude M correspond to a vacuum contribution. This is clear if b we draw it as a time ordered diagram, as it is done in Fig. 3c. The light-front case can be obtained by introducing first the four vector u with u2"d2, replacing in the above formulae j by u/d and then taking the limit d P 0, i.e., u2 P 0. This limit is, however, a delicate issue. For the scalar case under consideration above, it can be checked that the amplitude M disappears. In this limit, the system of reference where j"(1, 0) moves with b a velocity close to c. The graph technique for u2"0 is therefore naturally related to the old fashioned time ordered perturbation theory in the infinite momentum frame, corresponding to the particular case u"(1, 0, 0,!1), that defines the “standard” light front t#z"0. The full consideration of spin 1 particles shows that the above result has to be completed. In such 2 a case, indeed, one should associate to each propagator the factor (kK #m) for a fermion and (m!kK ) for an antifermion, where kK "k ck (see for instance Section B.3.2 in Appendix B). k The amplitude M , in Eq. (2.57), is thus multiplied by (pL #m)"(pL #kK #jK q #m) and M in a 1 1 b Eq. (2.58), is multiplied by (m!pL )"!(jK q !pL !kK !m) (for shortness we omit the initial and 2 2 final spinors). Replacing again j by u/d with u2"d2, we get in the limit d P 0: kK #pL #m uL M "g2 #g2 , a m2!(k#p)2 2u ) (k#p)

(2.60)

uL M "!g2 . b 2u ) (k#p)

(2.61)

The amplitude (2.61) is the contact term we already mentioned in Section 2.2.1. It is just the track of the disappeared vacuum diagrams. The sum of the amplitudes F and F gives the Feynman a b amplitude for spin 1 particles. As we shall see later on, contact terms are indeed essential in getting 2 fully covariant results (i.e., independent of u). Other examples will be considered in the following sections. To incorporate the contact term from the very beginning, one should not consider the diagram of Fig. 3b, but add to the diagram of Fig. 3a the diagram indicated in Fig. 4, which is obtained from Fig. 3 by deleting the spurion line and marking the internal line by a cross. This contribution can be also derived from a counter term added to H*/5 in order to get H*/5, as it was done for spin 1 in u 2 Ref. [38]. Associating the crossed line to a fermion carrying the momentum l, we assign to this line the factor !uL h(u ) l)/(2u ) l). For the diagram of Fig. 4 this rule gives Eq. (2.61) back.

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Fig. 4. Contact interaction contributing to the exchange of a spin 1 particle in the s channel. 2

We can thus formulate the rules of the covariant light-front graph technique for the case of spin 1 2 particles. 1. Transform the Feynman graph of a given order to the set of the light-front graphs in the same way as described in point 1 of the rules for scalar particles. Without making change in the orientation, replace the ordinary lines corresponding to antifermions by double lines, so that the number of fermions minus the number of antifermions is conserved. Both initial fermions and antifermions are shown by lines incoming to a diagram, and final fermions and antifermions by outgoing lines. 2. Consider the diagrams with internal fermion and antifermion lines labelled at their extremities by two successive numbers (i.e., the ends of the lines are directly connected by the spurion line, see, e.g., Fig. 3). In addition to all the diagrams we already have, we create from the latter diagrams another set of diagrams, by deleting those spurion lines, which connect the ends of the fermion (and antifermion) lines. Put on the corresponding fermion and antifermion lines a cross (see, e.g., Fig. 4). Draw any diagram with incoming lines at the left and outgoing lines at the right. 3. The analytical expression for the amplitude is written from the left to the right. The factors in this expression are written in the order where they are met when one goes through a diagram, starting at the right, from an outgoing fermion line, and continuing in the direction opposite to the orientation of the fermion lines. For an incoming antifermion line, one starts at the left, and continues in the direction of orientation of the (double) antifermion lines. 4. To each internal continuous line with four-momentum k, associate the propagator (kK #m)h(u ) k)d(k2!m2) for a fermion, and the propagator (m!kK )h(u ) k)d(k2!m2) for an antifermion, the factor !uL h(u ) k)/(2u ) k) for each crossed fermion line, the factor uL h(u ) k)/(2u ) k) for each crossed antifermion line, and the factor 1/(q !ie) for each internal j dashed line with four-momentum uq . j 5. Associate with each vertex the factor g», where » depends on the type of coupling (1, ic ,c , etc.) 5 k in the Hamiltonian H"!gtM »t2. All the four-momenta at each vertex, including the spurion momenta, satisfy the conservation law, i.e., the sum of incoming momenta is equal to the sum of outgoing momenta (incoming and outgoing momenta always correspond to the incoming and outgoing lines). 6. With any outgoing fermion [antifermion] line with momentum p, associate the spinor uN (p) [v(p)] and with any incoming fermion [antifermion] line associate the spinor u(p)[vN (p)]. 7. Integrate (with d4k/(2p)3) over those four-momenta of the internal particles which remain unfixed after taking into account the conservation laws, and over all q for the spurion lines from j !R to R.

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8. Repeat the procedure described in 1—7 for all n! possible numberings of the vertices. One should also take into account the standard sign factors appearing from permutation of fermion fields. In the case of a diagram containing boson lines connected to fermion loops, one should go through the outgoing boson line, and then, having reached the fermion line, pass it as indicated in the rules, i.e., in the direction opposite to the orientation of the fermion line and along the orientation of the antifermion line. Both lines in the loop, single and double, are oriented in the same direction, but one makes a cycle when passing through them. We emphasize that the fermion and antifermion are distinguished by the type of the continuous lines (single or double). Both the fermion and antifermion lines are directed from the smaller to the larger number (in the time ordered graph they both propagate from left to right, i.e., in the direction of time increase in time ordered graph). In contrast to the standard versions of the Feynman rules, any incoming line, both for antifermion and fermion, corresponds to the initial state, and outgoing lines — to the final state. The fermion and antifermion lines are followed in opposite directions (opposite to the orientation for a fermion line and along the orientation for an antifermion line). The fermion and antifermion propagators differ from each other. In many practical applications, the contact term (or so called instantaneous interaction) can be conveniently incorporated for spin 1 particles by replacing the spin part of the propagator (m$kK ) 2 (for those lines where the contact term contributes at all) by [m$(kK !uL q)] (however, the delta-function in the propagator d(k2!m2) still depends on the argument (k2!m2)). Together with the fact that the contact term originates from the lines which ends are labelled by two successive numbers, this rule coincides with the rule given in Refs. [39,40]: the contact terms modify only the propagators corresponding to the “lines extending over a single time interval”. Two successive numbers just determine a single time interval. The above replacement to incorporate systematically the contact term can always be made in perturbation theory. However, it cannot be made in more complicated cases. In particular, the graph for the wave function, for instance, indicated in Fig. 10 below, does not contain any external crossed lines. Hence, the contribution of the contact terms to the deuteron electromagnetic form factors or electrodisintegration amplitude, in the impulse approximation for example, cannot be incorporated by the above substitution in the propagators, since it would create an object with an external crossed line. The contact terms in the deuteron electromagnetic form factors and electrodisintegration will be properly taken into account in Section 7.2.1. If a diagram contains two fermions (but not fermion and antifermion) turning into a boson, like in the case of the deuteron form factor (the vertex NNd), one should pass through one fermion line from the right to the left, and then start again from the same final vertex and pass through the second line. Both lines are followed in the direction opposite to their orientation. Such example is given below in Section 6.3.3. 2.2.4. Spin 1 and coupling with derivatives For particles with spin 1 the rules are similar to the case of particles with spin 1. Like in Feynman 2 rules, external lines for spin 1 particles are associated with polarization vectors. However, in the case of spin 1 particles, there is no difference between propagators of particle and antiparticle. The

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propagator has the form

A

B

kk D (k, m)" !g # k l h(u ) k)d(k2!m2) . kl kl m2

(2.62)

The contact term is given by k u #k u uu l k! k l (k2!m2) . D "! k l kl 2(u ) k)m2 4(u ) k)2m2

(2.63)

Like in the case of spin 1, the contact term contributes only in the lines extending over a single time 2 interval. It can be incorporated simply by replacing the spin part of the corresponding propagators (g !k k /m2) by [g !(k!uq) (k!uq) /m2]. kl k l kl k l In the case of massless vector boson (the photon for instance) the form of the propagator, as usual, depends on the gauge. For a general gauge, the momentum dependence of the gauge term in the propagator induces also a corresponding contact term. We do not investigate here the general case and give the photon propagator for the Feynman gauge: D (k, m"0)"!g h(u ) k)d(k2) . (2.64) kl kl There is no contact term in this gauge. It is sometimes more convenient to use in this formulation of LFD the light-cone gauge defined by u ) A"0. The spin-1 propagator for a massless particle in this gauge is [40]:

A

B

u k #u k l k h(u ) k)d(k2!m2) D (k, m)" !g # k l kl kl u)k

(2.65)

and the corresponding contact term has the form: D "!u u /(u ) k)2 . (2.66) kl k l The contact term (2.66) can be again incorporated by the replacement k P k!uq in the spin part of Eq. (2.65). Consider finally the coupling with a derivative, e.g., when the interaction Lagrangian contains a term like ­ u(x). Since all the lines are oriented, any line is either incoming in the vertex, or k outgoing from it. Let this line be associated with the four-momentum k. The derivative results in the multiplication of a vertex by the extra factor ik for an outgoing line, and by the factor !ik for k k an incoming line. The contact term contributes to the lines involving the derivative of a field and having the same topology, as in the case of spins 1 and 1. That means that the ends of these lines 2 should be connected by the spurion line. Like in the case of spin 1 propagator, the contact term can be incorporated by the replacement k P k !u q, where uq is the spurion momentum, and this k k k spurion line should connect the ends of the particle line, corresponding to the field with derivative coupling. 2.2.5. Ultraviolet and infrared behavior A peculiarity of the covariant light-front amplitudes is that they have no any ultraviolet divergences for finite values of all the spurion four-momenta. All the ultraviolet divergences in light-front diagrams appear after integrations over q with infinite limits [35]. Indeed, the j

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energy—momentum conservation (including the spurion four-momentum) is valid at any vertex. Since all the four-momenta are on the corresponding mass shells, we have at each vertex a real physical process as far as the kinematics is concerned. For finite initial particle energies and for finite incoming spurion energy, the energies of the particles in the intermediate states are thus also finite. Hence, the integrations over the particle momenta for fixed spurion momenta are constrained by a kinematically allowed finite domain. It is the same reason that provides finite imaginary part of a Feynman diagram found by replacing the Feynman propagators 1/(k2!m2#ie) by the delta-functions !ipd(k2!m2). In both cases the internal particle lines are associated with the delta-functions. The only source of the ultraviolet divergences in the light-front amplitudes is the infinite intermediate spurion energies, i.e., infinite q . This is the reason why divergences may appear at the j upper limit of integration over q . Since q are scalar quantities, one can introduce an invariant j j cutoff in terms of these variables. This way of regularizing the divergent diagrams is another advantage of the covariant formulation of LFD. We illustrate this property of the light-front amplitudes in Section 2.3.3 for the example of the self-energy. For massless particles, light-front amplitudes may have infrared divergences, like in the case of Feynman diagrams. Another peculiarity of LFD is the appearance of “zero modes”. For constituents of zero mass, for instance, the state vector may contain components with u ) k"0 for non-zero four-momentum k. In the standard approach, this corresponds to the finite light-front energy k~"k2 /k for both M ` k "0 and k2 "0. Zero modes can also appear in theories with spontaneously broken symmetry. ` M They make the equivalence between LFD and the instant form of quantization in which nontrivial vacuum structures (condensates) appear [9,41,42]. The detailed discussion of the above problems is beyond the scope of the present review. 2.3. Simple examples 2.3.1. Exchange in t-channel Consider the diagrams shown in Fig. 5. It corresponds to the exchange of a scalar particle of mass k between two scalar particles, in the t-channel. These diagrams determine, in the ladder approximation, the kernel of the equation for the calculation of the light-front wave function. The external spurion lines indicate that the amplitude is off-energy shell. According to the light-front graph technique for spinless particles, the amplitude has the form:

P

dq 1 K"g2 h(u ) (k !k@ ))d((k !k@ #uq !uq)2!k2) 1 1 1 1 1 q !ie 1 dq 1 #g2 h(u ) (k@ !k ))d((k@ !k #uq !uq@)2!k2) 1 1 1 1 1 q !ie 1 g2h(u ) (k !k@ )) 1 1 " k2!(k !k@ )2#2qu ) (k !k@ ) !ie 1 1 1 1

P

g2h(u ) (k@ !k )) 1 1 # . k2!(k@ !k )2#2q@ u ) (k@ !k )!ie 1 1 1 1

(2.67)

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Fig. 5. Exchange by a particle in t-channel.

The two items in Eq. (2.67) correspond to the two diagrams of Fig. 5. They cannot be non-zero simultaneously. On the energy shell, i.e., for both q"q@"0, the expression for the kernel is identical to the Feynman amplitude: g2 K(q"q@"0)" . k2!(k !k@ )2!ie 1 1

(2.68)

Note that the off-shell amplitude (2.67) depends on u. This agrees with the fact that it is related by Eqs. (2.49) and (2.50) to the S-matrix on a finite light-front plane in the interaction region. It depends therefore on this plane. In the case of a scalar amplitude, this dependence is given in terms of extra scalar variables (in addition to the Mandelstam variables s and t). These variables are the scalar products of u with the four-momenta, as seen from Eq. (2.67). Hence, the amplitude has extra singularities as a function of these new variables, which can be found similarly to singularities of the Feynman amplitudes [43]. On the energy shell, corresponding to q"q@"0 and to the light-front plane shifted to infinity, out of the interaction region, the dependence of the amplitude on u disappears. In more complicated cases, when a Feynman diagram corresponds to the sum a few light-front diagrams (like in the case of the box diagrams considered in Appendix B), the amplitude for a particular light-front diagram may depend on u even on the energy shell. This dependence disappears in the sum of all amplitudes at a given order. In this case the singularities of different amplitudes cancel each other in the sum. The off-energy shell amplitude corresponds to a diagram with external spurion lines. This term — off-energy shell — is borrowed from the old fashioned perturbation theory. As mentioned above, the latter is obtained, if one introduces in Eq. (2.44), instead of u, the four-vector j with j"(1, 0, 0, 0). The difference of initial and final four-momenta differs from zero by jq. Due to k"0 the sums of initial and final spatial components of momenta is still equal to each other, but for q O 0 initial and final energies are not equal to each other (their difference is equal to q). This just characterizes the ”off-energy shell” amplitude, which can be a part of a bigger diagram. In the light-front amplitude for arbitrary u, all the four-vector components do not satisfy the conservation law. Their difference is proportional to uq. Hence the term ‘‘off-energy shell’’ may be inappropriate for the light-front amplitude, since not only energies, but also three-momenta are not conserved. We emphasize again that the four-momenta are always on the mass shells, whereas the Feynman amplitude depends on the four-momenta which may be off-mass shell, but the conservation law is always valid.

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2.3.2. Compton scattering The Compton scattering amplitude from an electron (through s-channel) is indicated in Fig. 6a, and, for the contact term, in Fig. 6b. According to the above rules of graph technique, we associate with Fig. 6a the following expression:

P

dq M "g2 uN @(p@)eL @*(m#pL )eL u(p)h(u ) p )d(p2!m2) d(4)(p#k#uq!p ) d4p a 1 1 1 1 1 q!ie "g2uN @(p@)eL @*

m#pL #kK uL eL u(p)#g2uN @(p@)eL @* eL u(p) , m2!(p#k)2 2u ) (p#k)

(2.69)

where eL "c ej(k) and similarly for eL @*. The polarization vector of the photon is denoted by ej(k). k k k The order of factors in Eq. (2.69) corresponds to following the fermion line from the right to the left, in the direction opposite to its orientation. The amplitude for the contact term contribution, indicated in Fig. 6b, is uL M "!g2uN @(p@)eL @* eL u(p) . b 2u ) (p#k)

(2.70)

We omit here the theta-function h(u ) (p#k)), which is always 1. The expressions (2.69) and (2.70) reproduce the amplitudes (2.60) and (2.61). In the sum, M #M , the omega-dependent items cancel each other. a b Now consider the crossed amplitude, indicated in Fig. 7 and Fig. 8. The amplitude corresponding to Fig. 7a has the form:

P

dq M "g2 uN @(p@)eL (m#pL )eL @*u(p)h(u ) p )d(p2!m2) d(4)(p!k@#uq!p ) d4p a 1 1 1 1 1 q!ie "g2uN @(p@)eL

m#pL !kK @ eL {*u(p)h(u ) (p!k@)) m2!(p!k@)2

#g2uN @(p@)eL

uL eL @*u(p)h(u ) (p!k@)) . 2u ) (p!k@)

(2.71)

Fig. 6. (a) The direct diagram for electron Compton scattering with an electron in the intermediate state. (b) The contact term corresponding to the diagram (a).

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Fig. 7. (a) The cross diagram for electron Compton scattering with an electron in the intermediate state. (b) The contact term corresponding to the diagram (a).

Fig. 8. (a) The cross diagram for electron Compton scattering with a positron in the intermediate state. (b) The contact term corresponding to the diagram (a).

The contact term contribution in Fig. 7b can be written as follows: uL M "!g2uN @(p@)eL eL @*u(p)h(u ) (p!k@)) . b 2u ) (p!k@)

(2.72)

It has the form of the second term in Eq. (2.71) and cancel it in the sum M #M . a b The amplitude of Fig. 8a differs from Fig. 7a by the time ordering of the vertices. If we draw the vertices in the order of increasing numbers from the left to the right, this amplitude appears as a time ordered diagram. It contains the creation of a e`e~-pair by the photon, i.e., includes

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a positron in the intermediate state, shown by the double line, and its subsequent annihilation. The corresponding amplitude has the form

P

dq M "g2 uN @(p@)eL (m!pL )eL @*u(p)h(u ) p )d(p2!m2) d(4)(k@!p#uq!p )d4p a 1 1 1 1 1 q!ie "g2uN @(p@)eL

m#pL !kK @ eL {*u(p)h(u ) (k@!p)) m2!(p!k@)2

! g2uN @(p@)eL

uL eL {*u(p)h(u ) (k@!p)) , 2u ) (k@!p)

(2.73)

and for the contact term indicated in Fig. 8b: uL M "g2uN @(p@)eL eL {*u(p)h(u ) (k@!p)) . b 2u ) (k@!p)

(2.74)

The first item in Eq. (2.71) differs from zero in the region u ) (p!k@)'0, whereas in Eq. (2.73) this occurs in the region u ) (p!k@)(0. The sum of these two contributions reproduce the Feynman amplitude in the u-channel, in the whole kinematical range. In order to show how to deal with antifermion states, consider now the Compton scattering amplitude from an antifermion. The s-channel diagram is similar to Fig. 6 with the single lines replaced by the double ones. The corresponding amplitude has the form: m!(pL #kK ) * uL M "g2vN (p)eL eL @ v@(p@)!g2vN (p)eL eL @*v@(p@) , a m2!(p#k)2 2u ) (p#k)

(2.75)

while the contact term gives uL M "g2vN (p)eL eL @ *v@(p@) . b 2u ) (p#k)

(2.76)

The order of factors in Eqs. (2.75) and (2.76) corresponds to following the antifermion line from the left to the right, in the direction along its orientation. Note that in Eq. (2.76) the sign of the contact term originating from the antifermion line is opposite to the fermion contact term, Eq. (2.70). The corresponding u-dependent contribution in Eq. (2.75) has also an opposite sign and thus disappears in the sum M #M . a b The amplitude for the cross diagram with an antifermion intermediate state is also similar to Fig. 7 and is given by

P

dq M "g2 vN (p)eL @ *(m!pL )eL v@(p@)h(u ) p )d(p2!m2) d(4)(p!k@#uq!p ) d4p a 1 1 1 1 1 q!ie "g2vN (p)eL @*

m!(pL !kK @) eL v@(p@)h(u ) (p!k@)) m2!(p!k@)2

! g2vN (p)eL @*

uL eL v@(p@)h(u ) (p!k@)) . 2u ) (p!k@)

(2.77)

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The corresponding contact term is uL M "g2vN (p)eL @* eL v@(p@)h(u ) (p!k@)) . b 2u ) (p!k@)

(2.78)

The kinematics of the Compton scattering from fermion and antifermion, i.e., the expression of the intermediate momentum p through the external momenta, as given in Eq. (2.77), is here identical. 1 Besides the spinors u and v, the difference is in the signs of the spin parts of the propagators and of the contacts terms. Finally we give the amplitude corresponding to the scattering on an antifermion with a fermion intermediate state (same as Fig. 7, but with single lines replaced by the double ones and vice versa):

P

dq d(4)(k@!p#uq!p ) d4p M "g2 vN (p)eL @*(m#pL )eL v@(p@)h(u ) p )d(p2!m2) 1 1 1 1 1 a q!ie "g2vN (p)eL @*

m!(pL !kK @) eL v@(p@)h(u ) (k@!p)) m2!(p!k@)2 uL eL v@(p@)h(u ) (k@!p)) 2u ) (k@!p)

(2.79)

uL eL v@(p@)h(u ) (k@!p)) . M "!g2vN (p)eL @* b 2u ) (k@!p)

(2.80)

# g2vN (p)eL @* and the contact term:

The difference between Eqs. (2.77) and (2.78) and Eqs. (2.79) and (2.80) is in the sign of the argument of the corresponding theta functions. The sum of both contributions gives the Feynman amplitude for Compton scattering on an antifermion. As mentioned above, the contact term can be incorporated everywhere by the following replacement in the spin matrix of the propagator: pL P pL !uL q. 1 1 2.3.3. Self-energy contributions Another simple example is the self-energy diagram shown in Fig. 9. The corresponding amplitude (equal to the self-energy up to a factor) has the form:

P

d4k dq 1 , R(p@)"g2 h(u ) k)d(k2!m2)h(u ) (p #uq !k))d((p #uq !k)2!m2) 1 1 1 1 (2p)3 q !ie 1 (2.81) with p "p@!uq@. 1 Let q"p #uq . The integral over d4k is thus reduced to the well-known calculation of the 1 1 imaginary part of the Feynman amplitude, when all the propagators are replaced by the deltafunctions:

P

p d(k2!m2)d((q!k)2!m2) d4k" Jq2!4m2 . 2Jq2

(2.82)

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Fig. 9. Self-energy loop.

Inserted in Eq. (2.81), it gives:

P

g2 = Jp2!4m2#q@ dq@ 1 1 1 . (2.83) R(p )" 1 16p2 q@ !ie 2 2 Jp2 #q@ 4m ~p1 1 1 1 The logarithmic divergence is at the upper limit of the integration over q@ . This is a particular 1 manifestation of a general property of the light-front amplitudes, discussed above in Section 2.2.5. One can introduce the invariant cutoff in terms of q@ . In this way, after renormalization, the 1 standard expression for the self-energy amplitude is obtained.

3. The two-body wave function Since the wave functions of a composite system are the Fock components of the state vector, their transformation properties under four-dimensional rotations are determined by the corresponding properties of the state vector investigated in the previous section. This allows one to construct their angular momentum explicitly. The angular condition, necessary to construct the angular momentum operator, is here of particular relevance. We pay also particular attention to the normalization of the wave function. The comparison with the BS wave function is made, and a simple example (Wick—Cutkosky model) is developed. 3.1. General properties of the wave function The wave functions we consider are the Fock components of the state vector defined on the light-front plane u ) x"p. This means that they are coefficients in an expansion of the state vector /Jj(p) with respect to the basis of free fields:

P

Dp, jT ,/Jj(p)"(2p)3@2 UJj1 1 2 2(k , k , p, uq)as1(k )as2(k )D0T jpjp 1 2 p 1 p 2 u d3k d3k 1 2 ]d(4)(k #k !p!uq) exp(iqp)2(u ) p)dq 1 2 (2p)3@2J2e 1 (2p)3@2J2e 2 k k

P

#(2p)3@2 UJj (k , k , k , p, uq)as1(k )as2(k )as3(k )D0T j1p1j2p2j3p3 1 2 3 p 1 p 2 p 3

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]d(4)(k #k #k !p!uq) exp(iqp)2(u ) p) dq 1 2 2 d3k d3k d3k 1 2 3 ] #2 . (3.1) (2p)3@2J2e 1 (2p)3@2J2e 2 (2p)3@2J2e 3 k k k Here j in /Jj(p) is the projection of the total angular momentum of the system on the z-axis in the rest frame, where p"0 and p , p , p are the spin projections of the particles 1—3 in the 1 2 3 corresponding rest systems. We emphasize in Eq. (3.1) the presence of the factor d(4)(k #2#k !p!uq)2(u ) p) dq. 1 n Formally it can be obtained from the relation Eq. (2.20): exp(iPK 0 ) a)/ (p)"exp(ip ) a)/ (p#u ) a) u u

(3.2)

with

P

PK 0" + as(k)a (k)k d3k . k p p k p Indeed, after substituting Eq. (3.1) in Eq. (3.2), the action of the operator exp(iPK 0 ) a) on the n-body sector gives the factor exp[i(k #2#k ) ) a] in the integrand of the l.h.s. of Eq. (3.2), whereas the 1 n factor exp[i(p#uq) ) a] appears in the r.h.s. The delta function in Eq. (3.1) ensures the equality of these factors and, hence, the relation Eq. (3.2). In the particular case where u"(1, 0, 0,!1,) the delta-function d(4)(k #k !p!uq) 1 2 gives, after integration over q, the standard conservation laws for the (o,#)-components of the momenta, but does not constrain the minus-components. From Eq. (3.1) one can see that the wave function depends on the orientation of the light front, from its argument uq. This important property of any Fock component is very natural. As explained in the previous section, any off-energy shell amplitude is related to the S-matrix defined on a light-front plane in the interaction region and therefore depends on its orientation (see Eq. (2.49)). The bound state wave function is always an off-shell object (q O 0 due to binding energy). Therefore, it also depends on the orientation of the light-front plane. This property is not a peculiarity of the covariant formulation of LFD. The covariance allows however to parametrize this dependence explicitly. We will investigate below this dependence for a few systems. 3.1.1. Transformation properties We derive in this section the transformation properties of the relativistic wave functions under four-dimensional rotations. To do this, we write Eq. (2.9) in a more explicit form: /Jj(p) P /@Jj(gp)"º 0(g)/Jj(p) , (3.3) u gu J u where º 0(g) is given for infinitesimal transformations by Eq. (2.10) and g is a Lorentz transformaJ tion and/or rotation. Here j is the projection of the angular momentum operator on the z-axis in the system at rest, p"0. After transformation, the new state /@ does not correspond to a definite projection of J. We then expand /@ with respect to the states /: /@Jj(gp)"+ D(J) MR(g,p)N/Jj{(gp) . gu j{j gu j{

(3.4)

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Here D(J) MR(g, p)N is the matrix of the rotational group and R(g, p) is the rotation operator: j{j R(g, p)"¸~1(gp)g¸(p) , (3.5) where ¸(p) is the Lorentz transformation, corresponding to the velocity €"p/p , i.e., for example, 0 ¸(K)(m, 0, 0, 0)"(mK /JK2, mK/JK2) . 0 The Euler angles that determine the rotation R(g, p) can be expressed in terms of the momentum p and the parameters of the transformation g. The explicit expression of the Euler angles in terms of g and p will not be needed. To obtain the transformation properties of the state vector, we first substitute /@ from Eq. (3.4) in Eq. (3.3), and represent / in the form of the expansion Eq. (3.1). Since asD0T in Eq. (3.1) is the state p vector of a free particle with spin j, the operator as transforms in accordance with the law p º 0(g)as(k)º~1 (g)"+ D(j) MR(g, k)Nas (gk) . (3.6) J p J0 p{p p{ p{ Comparing the left- and right-hand sides of the resulting equation, we thus obtain for the two-body wave function: UJj (gk , gk , gp, guq)" + D(J)*MR(g, p)ND(j11) 1MR(g, k )ND(j22) 2MR(g, k )N j1p1j2p2 1 2 p p{ 1 p p{ 2 jj{ j{p{1p{2 ]UJj{ (k , k , p, uq) (3.7) j1p{1j2p{2 1 2 and similarly for any n-body Fock component. Here j, p , p are the projections of the spins on the 1 2 z-axis in the rest frame of each of the particles. We emphasize that the transformation (3.7) is a purely kinematical one. As mentioned above, in the usual formulation of LFD, the transformation of the system of reference changes the position of the light-front plane. Indeed, under the transformation x P x@"gx, the state vector is transformed from the plane t#z"0 in a system A to the plane t@#z@"0 in a system A@. Introducing u(0)"(1, 0, 0,!1) we indeed obtain different planes u(0) ) x"0 and u(0) ) x@"(g~1u(0)) ) x"0. In the covariant formulation of LFD, the state vector remains to be defined on one and the same plane u ) x"u@ ) x@"0 in both systems A and A@. This property ensures the kinematical transformation law (3.7). The dependence on the surface is then given by the dynamical dependence of the wave function on u. 3.1.2. Parametrization in the spinless case We will mainly concentrate on the two-body wave function. Generalization to the n-body case is straightforward. Due to the conservation law k #k "p#uq , (3.8) 1 2 the light-front wave function can be shown graphically like a two-body scattering amplitude as indicated in Fig. 10. The broken line corresponds to the fictitious spurion. We emphasize again that although we assign a momentum uq to the spurion, there is no any fictitious particle in the physical state vector. The basis in Eq. (3.1) contains the particle states only. Due to this analogy, the decomposition of the wave function in independent spin structures and their parametrization is

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Fig. 10. Graphical representation of the two-body wave function on the light front. The broken line corresponds to the spurion (see text).

analogous to the expansion of a two-body amplitude in terms of invariant amplitudes. We will use this analogy below. Let us consider a system of two spinless particles in a state with zero angular momentum. According to Eq. (3.7), its wave function t"U 1 2 is scalar and, hence, depends on invariant j /j /J/0 variables only. We can first use the Mandelstam variables: s"(k #k )2"(p#uq)2, t"(p!k )2, u"(p!k )2 , 1 2 1 2

(3.9)

with s#t#u"M2#2m2 .

(3.10)

The wave function depends therefore on two independent scalar variables: t"t(s, t). It will be convenient in the following to introduce another pair of variables (k, n). We denote by k the relativistic momentum which corresponds, in the c.m.-system where k #k "0, to the usual 1 2 relative momentum between the two particles. Note that this choice of variable does not assume, however, that we restrict ourselves to this particular reference frame. We denote by n the unit vector in the direction of x in this system. Note that due to the conservation law (3.8), the total momentum p of the system in this reference frame is not zero. In terms of these variables, the wave function takes a form close to the non-relativistic case. Making the appropriate Lorentz transformation, we get

C

P k )P 1 k"¸~1(P)k "k ! k ! 1 1 JP2 10 JP2#P

D

,

(3.11)

0

n"¸~1(P)x/D¸~1(P)xD"JP2¸~1(P)x/u ) p ,

(3.12)

where P"p#uq ,

(3.13)

and ¸~1(P) is the Lorentz boost. The unit vector n is reminiscent of the unit vector p/DpD which appears in the infinite momentum frame.

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From these definitions, it follows that under a rotation and a Lorentz transformation g of the four-vectors from which k and n are formed, the vectors k and n undergo only rotations: k@"R(g, P) k , n@"R(g, P) n ,

(3.14)

where R is the rotation operator (3.5). Therefore k 2 and n ) k are invariants and can be expressed in terms of s and t. For the wave function with zero angular momentum we thus obtain [4]: t"t(k, n),t(k2, n ) k) .

(3.15)

It is seen from Eq. (3.15) that the relativistic light-front wave function depends not only on the relative momentum k but on another variable — the unit vector n. Finally, we introduce the third set of variables in which the wave function can be parametrized, in analogy to the equal-time wave function in the infinite momentum frame. We define the variables: x"u ) k /u ) p , R "k !xp , (3.16) 1 1 1 and represent the spatial part of R as R"R #Ro, where R is parallel to x and R is orthogonal , , M to x. Since R ) u"R u !R ) x"0 by definition of R, it follows that R "DR D, and, hence, 0 0 @ 0 , R2 "!R2 is invariant. Therefore, the two scalars on which the wave function should depend on M can be R2 and x: M t"t(R2 , x) . (3.17) M Using the definitions of the variables R2 and x, we can readily relate them to k2 and n ) k: M (3.18) R2 "k2!(n ) k)2, x"1(1!(n ) k/e )) . k M 2 The inverse relations are

C

D A B

R2 #m2 1@2 1 R2 #m2 !m2, n ) k" M !x . k2" M x(1!x) 2 4x(1!x)

(3.19)

The variables introduced above can be easily generalized to the case of different masses and an arbitrary number of particles [5]. The corresponding variables q , n are still constructed according i to Eqs. (3.11) and (3.12) and the variables R , x according to Eq. (3.16), i.e., x "u ) k /u ) p and iM i i i R "k !x p. i i i 3.1.3. Normalization The state vector is normalized by Eq. (2.26): Sp@, j@Dp, jT "2p d(3)(p!p @) dj{j . (3.20) u u 0 The Fock components are normalized so as to provide the condition Eq. (3.20). Substituting the state vector Eq. (3.1) in the left-hand side of Eq. (3.20), we get u

Sp@, j@Dp, jT "2p d(3)(p!p@) Nj{j , u 0

(3.21)

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with Nj{j"+ Nj{j , dj{j , (3.22) n n where Nj{j is the contribution to the normalization integral from the n-body Fock component. We n represent it in terms of all three sets of variables introduced above:

P

Nj{j"(2p)3 n

+

* UJj{ UJj d(4)(k #2#k !p!uq) 1 n j1p12jnpn j1p12jnpn

p12pn n d3k i 2(u ) p) dq ]< (2p)32e i k i/1

P P

A BA B A BA B

n n n d3q * i UJj{ UJj 2 d(3) + q 2 + e i < 2 1p1 jnpn j1p1 jnpn j i q (2p)32e i 2 q p1 p n i/1 i/1 i/1 n d2R dx n n * Mi i . "(2p)3 + UJj{ UJj 2 d(2) + R d + x !1 2 < (3.23) 2 1p1 jnpn j1p1 jnpn j Mi i (2p)32x 2 i p1 p n i/1 i/1 i/1 For the state with zero total angular momentum the normalization condition has the form: "(2p)3

+

+ N "1 . (3.24) n n In this case, the two-body contribution to the normalization integral reads, for constituents of equal masses:

P P

1 d3k d3k 2 2(u ) p) dq N " t2(k , k , p, uq)d(4)(k #k !p!uq) 1 2 (2p)3 1 2 1 2 2e 1 2e 2 k k 1 d3k 1 d2R dx M " t2(k, n) " t2(R , x) , (3.25) M 2x(1!x) (2p)3 e (2p)3 k where t"UJj (k , k , p, uq)D and we imply in Eq. (3.25) summation over the spin indices of j1p1j2p2 1 2 J,j/0 the constituents. To obtain Eq. (3.25) we used the fact that the first integral is the two-body phase volume:

P

P

P

d3k d3k k dX 2 2(u ) p) dq" (2) 1 k 2(u ) p) dq , (2)d(4)(k #k !p!uq) 1 2 2e 1 2e 2 8e k k k and then use the equality M2#2(u ) p)q"(k #k )2"4e2, which gives: 2(u ) p) dq"8k dk. The 1 2 k change of variables in Eq. (3.23) is made similarly. For the state with J"0 the integral (Eq. (3.25)) and any N do not depend on u. n For the wave function of a system with total angular momentum J"1, integral (Eq. (3.23)) does 2 not depend on u as well, since it is impossible to construct any u-dependent terms. At first glance, one could construct the following u-dependent structure: uN @(p)uL u(p)/u ) p (uN @ and u may correspond

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to different spin projections). However, since p uN @(p)c u(p)" k uN @(p)u(p) , k m

(3.26)

we get: uN j{uL uj/u ) p"uN j{uj/m"2djj{, i.e., this structure does not depend on u. Hence, any Nj{j is n proportional to dj{j. For the sum of them we get Nj{j"Adj{j, and the wave function for a system with J"1 is normalized by the condition A"1. As an example, we give for J"1 the normaliz2 2 ation integral for a n-body system:

P

n d3k i 2(u ) p) dq , + DU(1@2)p (3.27) 2 D2d(4)(k #2#k !p!uq) < n p1 p 1 n (2p)32e 2 ki p, p1 pn i/1 which equals to 1 in the case where the system consists of n particles only. We emphasize that condition (Eq. (3.20)), independent of the light-front plane, is a dynamical property of the state vector provided by the Poincare´ group. When J is different from 0, 1, the 2 integral Nj{j depends in general on u, though this dependence disappears in the sum (Eq. (3.22)) n calculated with all Fock components. For an exact state vector one should still write: Nj{j"Adj{j, and the normalization condition is reduced to A"1, like in the spin 1 case. However, for spins 2 higher than 1 the u-dependence of Nj{j does not disappear automatically if the state vector is 2 approximated by a limited number of Fock components, i.e., if a finite sum is retained in Eq. (3.22). Consider, for example, a system with J"1 (the deuteron for instance). Extracting from the Fock components the polarization vector e(j)(p) of spin 1 system, we represent Nj{j in the form: k n 1 N "(2p)3 n 2

Nj{j"e*(j{)(p)Ikle(j)(p) . n k n l

(3.28)

The general structure of the tensor Ikl is the following: n Ikl"!A gkl#B pkpl#C (pkul#pluk)#D (pkul!pluk) n n n n n

A

B

ukul g #E # kl , n (u ) p)2 3M2

(3.29)

where A , B , C , D , E are constants, M is the mass of the composite system. For example, n n n n n Eq. (3.28) has the form in the rest system:

A

B

1 E Nj{j"A dj{j# n nj{nj! dj{j , n n 3 M2

(3.30)

where nj is a unit vector in the direction of x. After integration of Eq. (3.30) over n, according to Eq. (2.38), the irreducible structure (nj{nj!dj{j/3) gives zero, and we get the relation A "N . n n Using Eq. (3.24), we get +A "+N "1. As it was mentioned above in Section 2.1.3, the same n n condition has to be valid without integration over the directions of u. This means that the exact Poincare´ covariant state vector should provide zero values for the sum of the constants C , D , E n n n (+C "+D "+E "0), although each of them, or a partial sum of them may not be. While it is not n n n required to do so, the u-dependent structure in Eq. (3.30) disappears after averaging over n (or x).

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The general normalization condition A"1 thus is, with Eq. (3.29),

A

B

1 pp 1 A" d Nj{j"!Ikl g ! k l "1 , 3 kl M2 3 j{j

(3.31)

where Ikl"+ Ikl . (3.32) n n We emphasize that normalization Eq. (3.31), according to Eqs. (3.22) and (3.23), contains the sum over the n-body Fock components for all n. It cannot be generally fulfilled for a two-body component, if the contribution of other components is not negligible. The explicit expression for A in terms of the deuteron wave function is found in Section 5.1.2. 2 3.1.4. New representation One can see from Eq. (3.7) that the relativistic wave function, in contrast to the non-relativistic one, is transformed in each index by different rotation matrices. It is therefore convenient to use a representation in which the wave function is transformed in each index by one and the same rotation operator R(g, P), rotating, according to Eq. (3.14), the variables k and n. We define the wave function in this new representation as follows: WJj (k , k , p, uq), + D(J)*MR(¸~1(P), p)ND(j11) @1MR(¸~1(P), k )N jj{ j1p1j2p2 1 2 pp 1 j{,p@1,p@2 (k , k , p, uq) , (3.33) ] D(j22) @2MR(¸~1(P), k )NUJj{ pp 2 j1p@1j2p@2 1 2 where, e.g., R(¸~1(P), p) is given by Eq. (3.5) with g"¸~1(P). Under the transformation g, the operator R(¸~1(P), k ) is factorized as follows [44]: 1 R(¸~1(gP), gk )"R(g, P)R(¸~1(P), k )R~1(g, k ) . (3.34) 1 1 1 Using also the property DMR R N"DMR NDMR N, we find that wave function (3.33) is indeed 1 2 1 2 transformed like Eq. (3.7), in which the arguments of all the D-functions are replaced by R(g, P)"¸~1(gP)g¸(P) .

(3.35)

We thus have Wj1 2(gk , gk , gp, guq)" + D(J)*MR(g, P)ND(j11) 1MR(g, P)N jj{ 1 2 p p{ pp j{,p{1p{2 (3.36) ]D(j22) 2MR(g, P)NWj{1 2(k , k , p, uq) . p p{ p{ p{ 1 2 Eq. (3.36) together with Eq. (3.14) shows that in this new representation and in the variables k, n the relativistic wave function transforms exactly as a non-relativistic wave function under a rotation R. This strongly simplifies the spin structure of the relativistic wave function, making it as close as possible to the non-relativistic one. The only difference is the dependence of the wave function on

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the extra variable n. We shall illustrate this dependence in the case of the Wick—Cutkosky model in this section, and in the case of the deuteron in Section 5. In the particular case of spin 1, the matrix D(1@2) coming in the transformation (3.33), takes the 2 pp{ form [45]: (k #m)(P #JP2)!r ) P r ) k 10 0 1 D1@2MR(¸~1(P), k )N" . 1 [2(k #m)(P #JP2)(k P !k ) P#mJP2)]1@2 10 0 10 0 1

(3.37)

The transformation to representation (3.33) is similar to the Melosh transformation [46]. We will come back to this point in Section 3.5. 3.2. Equation for the wave function The equation for the wave function is obtained from the equation for the vertex part shown graphically in Fig. 11. It is the analog, for a bound state, of the Lippmann—Schwinger equation for a scattering state. Let us first explain its derivation for the case of spinless particles. In accordance with the rules given in Section 2.2.2, we associate with the diagram of Fig. 11 the following analytical expression:

P

C(k , k , p, uq)" C(k@ , k@ , p, uq@)h(u ) k@ )d(k{2!m2)h(u ) k@ )d(k{2!m2) 1 2 1 2 1 1 2 2 dq@ d4k@ 2 . (3.38) ]d(4)(k@ #k@ !p!uq@) d4k@ K(k@ , k@ , uq@; k , k , uq) 1 2 1 1 2 1 2 q@!ie (2p)3 The factor d(4)(2) d4k@ is kept for convenience. The kernel K is an irreducible block which is 1 calculated directly by the graph technique once the underlying dynamics is known. We should then express the vertex C through the two-body wave function. This can be done by comparing, for example, two ways of calculating the amplitude for the breakup of a bound state by some perturbation: (1) by means of the graph technique (the result contains C); (2) by calculating the matrix element of the perturbation operator between the bound state and the free states of n particles (the result contains U). We thus get:4 C(k , k , p, uq) , U(k , k , p, uq)" 1 2 1 2 s!M2

(3.39)

4 The coefficient in relation Eq. (3.39) differs from previous references (see Ref. [5]) due to the different normalization coefficient in Eq. (3.1) and the different normalization of the amplitudes given by the rules of the graph technique.

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Fig. 11. Equation for the two-body wave function.

where s"(k #k )2"(p#uq)2. The corresponding relation for the n-body case has the 1 2 same form. In any practical calculation of the amplitude, we associate C with the vertex shown in Fig. 10 and then express C in terms of / by Eq. (3.39). Taking into account that s!M2"2(u ) p)q, and keeping all the spin indices, the equation for the wave function Uj1 2 has the pp form [47]: [(k #k )2!M2]Uj1 2(k , k , p, uq) 1 2 pp 1 2 m2 + Uj1 2(k@ , k@ , p, uq@) "! p{ p{ 1 2 2p3 p{1p{2 ]»p{11p{22 (k@ , k@ , uq@; k , k , uq)d(4)(k@ #k@ !p!uq@) 1 2 1 2 pp 1 2 d3k@ d3k@ 22(u ) p) dq@ , ] 1 (3.40) 2e 1 2e 2 k{ k{ where we have defined »"!K/4m2. This equation can be extended in a similar way for the many-body component of the state vector. For many practical applications, it may be useful to express this equation in terms of the variables k and n, i.e., express the wave function in terms of Wj1 2(k , k , p, uq),Wj1 2(k, n). With pp pp 1 2 transformation (3.33), one thus gets

P

P

m2 d3k@ [4(k2#m2)!M2]Wj1 2(k, n)"! + W{j1 2(k@, n)»p{11p{2 2(k@, k, n, M2) . pp p{ p{ pp 2p3 e k{ p{1p{2 The integration variable k@ in Eq. (3.41) is defined analogously to k in Eq. (3.11):

(3.41)

k@"¸~1(P@)k@ , (3.42) 1 where P@"k@ #k@ . Eq. (3.41) can also be obtained from Eq. (3.40) simply by transcribing it in the 1 2 system of reference where k #k "0. In this system, the rotation operators in Eq. (3.33) are the 1 2 unit operators, the wave function Uj1 2 turns into Wj1 2. However, W{j1 2 in the integrand differs pp pp p{ p{ from Wj1 2. The wave function W@(k@, n) obtains the same form as W(k, n) (however, expressed p{ p{ through the variable k@) in the system where k@ #k@ "0, but not in the system where k #k "0. 1 2 1 2 This fact is marked by a “prime”. The function W{j1 2 is obtained from Uj1 2(k@ , k@ , p, uq@) by simply p{ p{ p{ p{ 1 2

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expressing the arguments k@ , k@ , p, uq@ through k, k@, n. The relation between these variables has the 1 2 form [48]: e2 !e2 (e !e )2 e2 #e2 e2 !e2 k n# k{ k (n ) k@)n , k # k{ k n ) k@ , k@ "k@# k{ k@ " k{ (3.43) 1 10 2e 2e e 2e 2e e k k k{ k k k{ (e !e )2 e2 #e2 e2 !e2 e2 !e2 k (n ) k@)n , k ! k{ k n ) k@ , k n! k{ k@ " k{ (3.44) k@ "!k@# k{ 20 2 2e e 2e 2e e 2e k k{ k k k{ k 4e2!M2 4e2#M2 p"!xq"! k n, p " k , (3.45) 0 4e 4e k k 4e2!M2 4e2 !M2 4e2 !M2 4e2!M2 n, u q" k , xq@" k{ n , u q@" k{ . (3.46) xq" k 0 0 4e 4e 4e 4e k k k k Since formulae Eqs. (3.43), (3.44), (3.45) and (3.46) are the Lorentz transformations between the systems with k #k "0 and k@ #k@ "0, the function W{j1 2(k@, n) can be also obtained from 1 2 1 2 p{ p{ Wj1 2 (k@, n) by a rotation of spins by means of the D-functions: p{ p{ (3.47) W{j"D(j2)sMR(¸~1(P@), k@ )NWjD(j1)MR(¸~1(P@), k@ )N . 2 1 For spin 1 particles, the matrix D(1@2)MR(¸~1(P), k)N is given explicitly in Eq. (3.37). In the simple 2 case of a scalar particle, the equation for the wave function in terms of the variables k, n has the following form:

P

m2 d3k@ (4(k2#m2)!M2)t(k, n)"! t(k@, n)»(k@, k, n, M2) . (3.48) 2p3 e k{ An equation of such a type was also considered in Refs. [32,49—53]. In the non-relativistic limit, Eq. (3.48) turns into the Schro¨dinger equation in momentum space, the kernel » being the non-relativistic potential in momentum space, and the wave function no longer depends on n. We emphasize that the wave function, which is an equal-time wave function on the light front, turns into the ordinary wave function in the non-relativistic limit where cPR. This reflects the fact that in the non-relativistic limit two simultaneous events in one frame are simultaneous in all other frames. In the variables R and x, Eq. (3.48) can be rewritten in the form: M m2 d2R@ dx@ R2 #m2 M M !M2 t(R , x)"! t(R@ , x@)»(R@ , x@; R , x, M2) . (3.49) M M M M 2p3 2x@(1!x@) x(1!x)

A

B

P

In this form, this equation is nothing else than the Weinberg equation [3]. The advantages of the equation for the wave function in the form (3.48) compared with Eq. (3.49) are its similarity to the non-relativistic Schro¨dinger equation in momentum space, and its simplicity in the case of particles with spin. These properties make Eq. (3.48) very convenient for practical calculations. The kernel of Eq. (3.48) depends on the vector variable n. We shall see that this

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dependence, especially the part which depends on M2, as given by Eqs. (3.45) and (3.46) for instance, is associated with the retardation of the interaction. From this point of view, the dependence of the wave function t(k, n) on n is a consequence of retardation. In absence of explicit calculations, a very useful information on the global contribution of many-body Fock components can be obtained by considering the dependence of the interaction on the total mass M of the system. The current normalization procedure is then to insert in the norm operator the derivative of the potential with respect to M2. In the case of a J"0 state made of two spinless particles, the normalization is defined by

P

C

D

1 d3k d3k@ * 4m2 ­»(k@, k, n, M2) t (k@, n) e d(k!k@)! t(k, n)"1 , (3.50) k (2p)3 e e (2p)3 ­M2 k k{ where the second term accounts for the many-body contribution to the norm, + N . The n;2 n extension to states with JO 0 is straightforward. There are many justifications for the introduction of the derivative of the potential » in Eq. (3.50). A general derivation can be done similarly to the normalization condition of the BS function [54]. Apart from the fact that it appears in the description of a system of two coupled components in which one component is explicitly retained, it is suggested by the requirement that two-state vectors describing two infinitesimally close states should be orthogonal. Examination of Eq. (3.48) then shows that such a property is fulfilled by introducing in the scalar product of the two state vectors the derivative of » with respect to M2 as it is done in Eq. (3.50). The extension to the normalization immediately follows from the fact that the orthogonality and the renormalization are defined from the same operator, namely the time component of a conserved current. The consistency of Eq. (3.50) with the general normalization condition (3.22) supposes that the potential » is determined to all orders in the coupling constant entering the interaction. In practice, this is limited to the exchange of one or two bosons, which may be already enough for the applications considered in the following. Let us just mention here that the examination of a few examples (Wick—Cutkosky model, model-perturbative calculation) shows that the n dependence of the integrand in Eq. (3.50) due to the derivative of the potential tends to cancel that of the first term. This is in accordance with the general discussion given above in Section 3.1.3. When the state is described in relativistic quantum mechanics with a fixed number (two) of particles with ­»/­M2"0, the normalization for J"0 is reduced to N "1. 2 3.3. Relation with the Bethe—Salpeter function To find the relation between the light-front wave function and the Bethe—Salpeter function we should start from the integral that restricts the variation of the arguments of the Bethe—Salpeter function to the light-front plane:

P

I " d4x d4x d(u ) x ) d(u ) x ) U(x , x , p) exp(ik ) x #ik ) x ) , 1 2 1 2 1 2 1 1 2 2

(3.51)

where k , k are the on-shell momenta: k2"k2"m2, and U(x , x , p) is the Bethe—Salpeter function 1 2 1 2 1 2 [1], U(x , x , p)"S0D¹(u(x )u(x ))DpT . 1 2 1 2

(3.52)

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Representing the d-functions in Eq. (3.51) by the integral form

P

1 d(u ) x )" exp(!iu ) x a ) da , 1 1 1 1 2p

P

1 d(u ) x )" exp(!iu ) x a ) da , 2 2 2 2 2p

introducing the Fourier transform of the Bethe—Salpeter function U(k, p), U(x , x , p)"(2p)~3@2 exp[!ip ) (x #x )/2]UI (x, p), x"x !x , 1 2 1 2 1 2

P

U(l, p)" UI (x, p) exp(il ) x) d4x ,

(3.53)

where l"(l !l )/2,p"l #l ,l and l are off-mass shell four-vectors, and making the change of 1 2 1 2 1 2 variables a #a "q, (a !a )/2"b we obtain 1 2 2 1

P

I"J2p

`= d(4)(k #k !p!uq) dq 1 2 ~=

P

]

`= U(l "k !uq/2#ub, l "k !uq/2!ub) db . 1 1 2 2 ~=

(3.54)

On the other hand, integral (3.51) can be expressed in terms of the two-body light-front wave function. We assume that the light-front plane is the limit of a space-like plane, therefore the operators u(x ) and u(x ) commute, and, hence, the symbol of the ¹ product in Eq. (3.52) can be 1 2 omitted. In the considered representation, the Heisenberg operators u(x) in Eq. (3.52) are identical on the light front u ) x"0 to the Schro¨dinger operators (just as in the ordinary formulation of field theory the Heisenberg and Schro¨dinger operators are identical for t"0). The Schro¨dinger operator u(x) (for the spinless case for simplicity), which for u ) x"0 is the free field operator, is given by Eq. (2.51):

P

1 d3k u(x)" [a(k) exp(!ik ) x)#as(k) exp(ik ) x)] . (2p)3@2 J2e k

(3.55)

We represent the state vector DpT,/(p) in Eq. (3.52) in the form of expansion (3.1). Since the vacuum state on the light front is always “bare”, the creation operator, applied to the vacuum state S0D gives zero, and in the operators u(x) the part containing the annihilation operators only survives. This cuts out the two-body Fock component in the state vector. We thus obtain (2p)3@2(u ) p) I" 2(u ) k )(u ) k ) 1 2

P

`= t(k , k , p, uq)d(4)(k #k !p!uq) dq . 1 2 1 2 ~=

(3.56)

Comparing Eqs. (3.54) and (3.56), we find

P

(u ) k )(u ) k ) `= 1 2 U(k#bu, p) db , t(k , k , p, uq)" 1 2 p(u ) p) ~=

(3.57)

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where the argument p in Eq. (3.57) is expressed through the on-shell momenta k , k as 1 2 p"k #k !uq, in contrast to off-mass shell relation p"l #l . For the Bethe—Salpeter func1 2 1 2 tion U(l , l ) parametrized in terms of the momenta l , l formula Eq. (3.57) reads: 1 2 1 2 (u ) k )(u ) k ) `= 1 2 t(k , k , p, uq)" U(l "k !uq/2#ub, l "k !uq/2!ub) db . 1 2 1 1 2 2 p(u ) p) ~= (3.58)

P

In ordinary LFD, Eqs. (3.57) and (3.58) correspond to the integration over dk . This equation ~ makes the link between the Bethe—Salpeter function U and the wave function t defined on the light front specified by u. It should be noticed however that Eq. (3.58) is not necessarily an exact solution of Eq. (3.48), since, as a rule, different approximations are made for the Bethe—Salpeter kernel and for the light-front one. In the ladder approximation, for example, the Bethe—Salpeter amplitude contains the box diagram, including the time-ordered diagram with two exchanged particles in the intermediate state, as indicated in Fig. 41 in Appendix B.3. This contribution is absent in the light-front ladder kernel. The quasipotential type equations for the light-front wave function derived by restricting arguments of the Bethe—Salpeter amplitude to the light-front plane z#t"0 and corresponding electromagnetic form factors were studied in Refs. [55,56]. 3.4. Application to the Wick—Cutkosky model 3.4.1. Solution in the covariant formulation of LFD As a simple, but nevertheless instructive, example, we shall derive in this section the light-front wave function of a system consisting of two scalar particles with mass m interacting through the exchange of a massless scalar particle calculated in the ladder approximation. This is the so-called Wick—Cutkosky model. The diagrams that determine the kernel are shown in Fig. 5. The kernel is given by Eq. (2.67) with k"0. Going over from the kernel K to the potential »"!K/(4m2), introducing the constant a"g2/(16pm2), and expressing Eq. (2.67) by means of the relations (3.43)—(3.46) in terms of k, k@, n, we obtain [57]: »"!4pa/K2 ,

(3.59)

where

A

BK

K

(e !e )2 n ) k@ n ) k 1 k # e2 #e2! M2 K2"(k@!k)2!(n ) k@)(n ) k) k{ ! . k{ k 2 e e e e k{ k k{ k

(3.60)

For k, k@;m, Eq. (3.59) turns into the Coulomb potential in momentum space 4pa . »(k@, k)K! (k@!k)2

(3.61)

For a;1,De D"DM!2mD"ma2/4;m, the wave function is concentrated in the non-relativistic b region of momenta. The non-relativistic wave function of the ground state in the Coulomb

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potential has the form: 8Jpmi5@2 t(k)" , (k2#i2)2

(3.62)

where i"JmDe D"ma/2 (it is normalized, however, according to Eq. (3.25) with e + m). The k b integral over d3k@ in Eq. (3.48) is concentrated in the region k@ + i. Therefore, at k
P

m»(0, k, n, M2) t(k, n)"! t(k@) d3k@ . (2p)3(k2#i2)

(3.63)

Note, that in Eq. (3.63) :t(k@) d3k@"(2p)3t(r"0). The wave function (3.62) is normalized to 1. Substituting in the r.h.s. of Eq. (3.63) the expressions (3.59) and (3.60) for » and Eq. (3.62) for t, we obtain 8Jpmi5@2 . (3.64) t(k, n)" Dn ) kD (k2#i2)2 1# e k This relativistic wave function of the ground state with zero total angular momentum is a good approximation of a more exact one in the range k'i. Corrections of order a log(a) should be considered in the range k(i (see Ref. [58]). Although the kernel, Eqs. (3.59) and (3.60), contains the modulus Dn ) k@/e !n ) k/e D, one can show that the solution has no “cusp” at n ) k" 0. This cusp k{ k in Eq. (3.64) appears due to our approximations. One can check on this simple example that it is the retardation of the interaction that is the dynamical reason for the dependence of the wave function on the variable n. The non-relativistic Coulomb expression for kernel (3.61) does not contain retardation and does not depend on n while the relativistic kernel (3.59) contains retardation and depends on n. This leads to the dependence of the wave function on the argument n. It may seem at first sight that the dependence of the wave function on n is not due to the contribution of the many-body sectors. Indeed, the coupling of the two-body sector with the other sectors must contain a coupling constant, whereas the parameter that determines the dependence of the wave function on n is the nucleon mass and does not contain the coupling constant. This argument should be taken with some care however. It can be seen from Eq. (3.48) that the operator k2#m2!M2/4 acting on the wave function is equivalent to the potential acting on this same wave function. It is thus proportional to the coupling constant. Such a result underlies the disappearance of the n dependence of the integrand of Eq. (3.50) previously mentioned. We emphasize that retardation leads to both the n-dependence and the presence of the carriers of the interaction in the intermediate state, which contribute to the many-body sectors. However, these two effects, being important in full measure in a truly relativistic system, can manifest themselves in a different way in weakly bound systems. Neglecting the many-body sectors does not necessarily entails to neglect the n-dependence of the wave function at k+m. The above mentioned a log(a) correction to Eq. (3.64) for instance, which originates from the last n-dependent term in Eq. (3.60), is n independent and has no counterpart from the many-body sector. As we shall see also

A

B

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259

in Section 5, it is necessary to take into account the n-dependence of the wave function even when one restricts to the two-nucleon sector. The wave function of the 2p state can be found analogously. In the representation (3.33), it has the form [57]: 8pi7@2m1@2 1 tj(k, n)" 1 Dn ) kD 2 3 J6 k2# i2 1# 4 e k (2e !M)2 (k2#1i2) k 4 (h(!n ) k)!h(n ) k)) ] k½ (k/k)#½ (n) (n ) k)! 1j 1j 4e M 2m k

A

G

BA C

B

DH

. (3.65)

The wave function corresponding to the angular momentum l"1 contains the spherical function ½ (n). This is an illustration of the fact that the vector n participates in the construction of the total 1j angular momentum on the same ground as the relative momentum k. The dynamical difference between the solution with kEn and k o n is obviously related to the property that some of the components of the angular momentum J, before using the angular condition, depend on the interaction. 3.4.2. Solution in the Bethe—Salpeter approach The exact expression for the Bethe—Salpeter function in the Wick—Cutkosky model is found in the form of the integral representation [59,54] and, for zero angular momentum, reads:

P

i `1 g(z, M) dz U(l, p)"! . (3.66) J4p ~1 (m2!M2/4!l2!zp ) l!ie)3 Substituting Eq. (3.66) in Eq. (3.57), and making the change of variable bPb/u ) p, we find

P

P

ix(1!x) `= `1 g(z, M) dz db t"! , (3.67) 2p3@2 [(k2#i2)(1!zz )#b(z !z)!ie]3 ~= ~1 0 0 where z "1!2x,i2"m2!M2/4. 0 For fixed zOz , the integrand, as a function of b, has one pole of third order, and therefore the 0 integral is zero. For z"z , the integrand does not depend on b and the integrand over b diverges. 0 To find the value of the integrand (3.67), we first calculate the integral over dz from z !e to z #e 0 0 at eP0, retaining only the term proportional to e, and then over db. As a result, one obtains [60] g(1!2x, M) t" . 25Jpx(1!x)(k2#i2)2

(3.68)

The spectral function g(z, M) is determined by a differential equation [59,54] and has no singularity at z"0. The exact wave function (3.68) is therefore analytic at x"1. The approximate explicit 2 solution found in Ref. [59] for g(x, M) has the form: g(z, M)"26pJmi5@2(1!DzD) .

(3.69)

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Inserting Eq. (3.69) in Eq. (3.66) and integrating over z, one can recover the approximate solution of the Bethe—Salpeter equation given by Eq. (7.2). The discontinuity of the spectral function g(z, M) at z"0 is again the result of our approximation, since the solution (3.69) corresponds to an asymptotically small binding energy. Substituting Eq. (3.69) in Eq. (3.68), we reproduce expression (3.64) for the wave function. The discontinuity of Eq. (3.69) at z"0 results in the “cusp”of Eq. (3.64) at n ) k" 0. 3.5. Angular momentum and angular condition The wave function (3.65) is consistent with the general structure of the light-front wave function of a system with total angular momentum equal to 1 (for spinless constituents): tj(k, n)"f ½ (k/k)#f ½ (n) , (3.70) 1 1j 2 1j with f "f (k2, n ) k). The states with higher angular momentum are constructed similarly, using 1,2 1,2 Clebsch—Gordan coefficients. In this case, it is clear that the angular momentum operator in the representation (3.33) has the form: J"!i[k]­/­k]!i[n]­/­n] .

(3.71)

The same expression for J was found in Refs. [32]. It can be shown to be quite general for spinless particles. Acting on the wave function (3.64) for zero angular momentum and on any function depending on the scalars k2 and n ) k, the operator (3.71) gives zero. The generalization for the case of constituents with non-zero spins is evident: J"!i[k]­/­k]!i[n]­/­n]#s #s , (3.72) 1 2 where s are the spin operators of the constituents. 1,2 As we explained in Section 3.1.4, the relative momentum k and the spin operators s , s have 1 2 identical transformation properties. In the standard approach, the angular momentum operator obtains the same form after performing the Melosh transformation [46,29], once the n-dependent term is neglected in Eq. (3.72). In addition, the factor n ) k in f (k2, n ) k) turns into k . At first 1,2 z glance, the dependence of the wave function on k can be interpreted as a violation of rotational z invariance (see Ref. [61]). Indeed, this wave function is not eigenfunction of the operator L2, where L is the usual angular momentum operator: L"!i[k]­/­k] .

(3.73)

The rotations changing the position of the light-front plane t#z"0 are dynamical, the corresponding angular momentum operator contains the interaction, and does not therefore coincide with Eq. (3.73). In the covariant formulation, the action of the dynamical angular momentum operator has to coincide with the action of the operator (3.71). This is provided by the so called angular condition (2.17) derived in Section 2.1. As explained in Section 2.1, the solutions of Eq. (3.48) with non-zero angular momentum are always degenerate if one does not apply the angular condition. This condition eliminates this unphysical degeneracy. This can be seen easily in the example of the wave function (3.70).

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For this aim, it is convenient to represent Eq. (3.70) in the form: w(k, n)"f k#f n . 1 2

(3.74)

By analogy with Eq. (2.39), one can construct the operator: A"(n ) J )2 .

(3.75)

Due to Eq. (3.71), this operator can be represented in the form: A"(n ) L)2. Any n-dependence of the kernel does not violate conservation of the projection of the operator L on the direction n. Hence, the operator A is conserved. As a scalar, it commutes with the angular momentum operator J: [J, A]"0. Therefore, any solution of Eq. (3.48) with definite angular momentum is characterized also by the eigenvalues of the operator A: At "a t . a a

(3.76)

For J"1, there are two eigenstates with a"0 and a"1 respectively: w "n(n ) k)h (k2, n ) k) , (3.77) 0 0 w "[k!n(n ) k)]h (k2, n ) k) , (3.78) 1 1 where h are arbitrary scalar functions. In the case where h do not depend on n ) k, the states 0,1 0,1 w are eigenstates of the usual angular momentum operator squared: L 2w "1(1#1)w . 0,1 0,1 0,1 Rewritten in terms of the spherical functions, they still have no definite projection of the usual angular momentum on the z-axis. Thus, w is proportional to + a ½ (k/k) (with a "½* (n)) 1m 0 m m 1m m and, hence, is a superposition of states with different projections m. In the non-relativistic limit h "h , and the degeneracy of the states (3.77) and (3.78) is nothing but the usual degeneracy 0 1 relative to projections of the angular momentum. In the practical case of the approximate two-body equation (3.48), the states (3.77) and (3.78) are not degenerate but correspond to two different energies. Both of them are non-physical. It is difficult to project explicitly the general angular condition (2.18) on the two-body sector. However, model solution (3.65), found from the known Bethe—Salpeter function, does not contain any ambiguity, and hence, satisfies the angular condition. It has the form of Eq. (3.74), i.e., is a combination of two states with a"0 and a"1 (neglecting at this level of approximation the energy splitting). This example illustrates the result, proved in Section 2.1, that the angular condition allows to construct the physical solution from unphysical degenerate solutions. Another example is given in Ref. [32]. The angular condition does not eliminate the n-dependence of the wave function. The angular condition in the ladder approximation was found in Ref. [33].

4. The nucleon–nucleon potential The understanding of the NN interaction should ultimately rely on QCD. However, the non-perturbative character of this theory in the low-energy range makes this goal quite far away. For many purposes, using effective degrees of freedom such as mesons in the t-channel or nucleons

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and baryon resonances in the s-channel in order to describe this interaction will still be relevant, provided it is used in a given domain of energy and momentum. In the present context, this approach is natural because (i) the meson exchange picture provides a reasonably good description of the NN interaction and (ii) it allows one to consider specific relativistic corrections. The effects due to a deeper understanding of hadronic physics will have to be considered, involving in the simplest case off-shell effects, the effective character of some meson exchanges or of hadronic form factors. 4.1. Mesonic degrees of freedom in nuclei The relevance of mesonic degrees of freedom in describing the NN interaction stems from the study of peripheral partial waves phase shifts at low energy. These ones are dominated by the exchange of the pion, due to both its low mass and the low-energy domain. In related area, dealing for instance with electromagnetic properties of light nuclei, the relevance of the pion degree of freedom relies on meson exchange current contributions to the deuteron electrodisintegration near threshold [62], where it allows to achieve agreement with experiment at momentum transfers around Q2&0.5 (Gev/c)2[63]. As this contribution is not required by a pure phenomenological description of the NN interaction, it is a clear evidence for the role of the pion in the description of nuclear forces. In both cases (NN interaction and deuteron electrodisintegration cross-section), the role of the pion is related to the peculiar nature of this particle within QCD. The pion is the Goldstone boson associated to the spontaneous breaking of chiral symmetry which provides its low mass. The p-exchange is now a common component of all the models of the NN interaction. Beyond single pion exchange, one expects contributions due to 2, 3,2 pion exchanges. They have shorter and shorter ranges, 1 fm or less, and it becomes more and more difficult to identify their contribution to the NN interaction as the number of exchanged pions increases. At this point, models differ from each other. Some models [64], following the philosophy of the Reid Soft Core (RSC) potential [65], simply parametrize this part in terms of Yukawa potentials, but without precise relation to meson exchanges. As a support for such an approach, one may argue that nucleons have a radius of the order of 0.8 fm (as far as this radius can be identified to the proton charge radius) and consequently, below a distance of 1.5 fm, nucleons began to overlap, making the description of NN interaction in this range in terms of exchange of mesons quite doubtful. A second approach relies on the use of dispersion relations [66] to calculate the irreducible 2p-exchange contribution to the NN scattering amplitudes. The interaction potential is obtained by removing the 2p-exchange contribution resulting from the iteration of one p-exchange which is automatically generated by the dynamical equation used to calculate the two-body wave function. This allows one to extend the potential to distances as low as 0.8 fm. Notice that the full 2p-exchange scattering amplitude has covariance properties that are lost by the separate pexchange and 2p-exchange contribution to the NN potential. The short range part is largely phenomenological, with the consequence that the Lorentz structure of the NN amplitude may not be correctly determined, quite similarly to the previous approach. An important point to be noticed is that this approach relies on physical pN scattering amplitudes with the consequence that no form factor is introduced at the pNN(pND2) vertices. Off-shell effects, which have a shorter range, are incorporated in the phenomenological part. On the other hand, the pN scattering amplitude

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satisfies properties expected from chiral symmetry, although a pseudoscalar pNN coupling is used. The large contribution from the excitations of NNM pairs (Z-diagrams) to the pN scattering amplitude expected in this case is cancelled here by a contribution involving mainly the D resonance intermediate state (calculated via dispersion relations). Finally, although it does not incorporate explicit o-exchange contribution, this approach can account for it, through the 2p interaction in a P-state in the t-channel. A third approach is based on a field-theoretical description of the meson—nucleon interaction, including vertex form factors [7,67,68]. In the simplest approximation, the NN interaction results from a sum of contributions involving single meson exchanges: p for the long-range part, “p” giving attraction at intermediate distances (1 fm) and accounting for the exchange of 2p in a S-state, o accounting for the exchange of 2p in a P-state, and u for the short-range repulsive part. Other well known mesons such as g, a , d may also be considered, but are quantitatively less important. 1 While some phenomenology is involved in the vertex form factors or through adjustments of coupling constants, this approach has the great advantage to provide a parametrization of the NN interaction in terms of explicit exchanges of mesons, which are expected to be important, if not dominant, in some channels. The fit to NN scattering is quite satisfactory and there is no real indication at present that such an approach breaks down, even at short distances where it could fail. Because it keeps the full Lorentz structure attached to the different exchanged mesons, this approach is quite appropriate to the present purpose. The above approach can be improved by considering two-meson exchanges. This is especially important for a better account of the “p” exchange contribution, whose effective character is well known. As in the dispersion relations approach, the iterated one meson (pion in practice) exchange has to be removed from the full 2p-exchange contribution to the NN amplitude. The above contribution also involves D resonances (and other higher resonances) in intermediate states in a quite similar way as Van der Waals forces in atomic physics are generated. The non-local character attached to such contributions is likely to be accounted for only in an approximate way by single meson exchanges (as “p” and o mesons). 4.2. The non-relativistic NN potential The non-relativistic NN potential is to be used with the Schro¨dinger equation. In momentum space, which is more appropriate for an extension to relativistic calculations, it reads:

P

k2 d3k@ t(k)# SkD»Dk@Tt(k@)"Et(k) , m (2p)3

(4.1)

where k represents one half of the relative momentum of the two nucleons, k"1(k !k ), and 2 1 2 similarly for k@. In r-space, where many models were developed, the corresponding equation would be

P

e2 ! t(r)# d3r@ SrD»Dr@Tt(r@)"Et(r) . m

(4.2)

Typically, in a meson exchange theory, SkD»Dk@T takes the following form in the simplest case: g2 »(k, k@),SkD»Dk@TJ , k2#(k!k@)2

(4.3)

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while in r-space, one would get exp(!kr) »(r, r@),SrD»Dr@TJg2d(3)(r!r@) . 4pr

(4.4)

where r represents the relative distance between the two nucleons, r"r !r . The same interac1 2 tion that is non-local in momentum space, Eq. (4.3), is local in r-space. This explains why the last one is often preferred in practical calculations. 4.2.1. Spin structure of the non-relativistic NN potential Quite generally, the NN non-relativistic Galilean invariant potential contains 5 terms with different spin structure (assuming that strong interactions conserve isospin). It may be written in momentum space as: i »(k, k@)"» (k, k@)#» (k, k@)r ) r #» (k, k@) (r #r ) ) [k]k@] C SS 1 2 SO 2 2 1 #» (k, k@)M(k!k@)2r ) r !3(k!k@) ) r (k!k@) ) r N T 1 2 1 2 #» 2(k, k@)r ) [k]k@]r ) [k]k@] , (4.5) SO 1 2 where » (k, k@),» (k, k@),» (k, k@),» (k, k@) and » 2(k, k@) stand, respectively, for central, spin—spin, C SS SO T SO spin—orbit, tensor and quadratic spin—orbit forces. They are predominantly scalar functions of (k!k@)2 (with further slight dependence on k2 and k{2) and contain isospin independent and dependent terms: »(k, k@)"»0(k, k@)#»1(k, k@)s ) s . (4.6) 1 2 Other structures, which have an off-energy shell character, are discarded on the basis that their effect can be re-absorbed in the shorter-range part of the interaction model, mostly phenomenological. Such terms have been considered in Ref. [69]. In r-space, and assuming a dependence of »(k, k@) on (k!k@)2 only, the above interaction takes the following form:

G

r #r 2)l »(r, r@)"d(3)(r!r@) » (r)#» (r)r ) r #» (r) 1 C SS 1 2 SO 2

A

» (r) T

B

A

BH

3r ) r r ) r r ) l r ) l#r ) l r ) l 1 2 !r ) r #» (r) 1 2 2 1 #2 2 1 2 SO r2 2

,

(4.7)

where the points indicate the existence of other terms involving » 2, but usually neglected. The SO potentials » (r) and » (r) can have a linear k2 dependent term [68,70], which, by using an C SS appropriate transformation [70], can be dealt with in coordinate space without too much difficulty. In momentum space, such terms can be kept to any order without further difficulty. These terms have quite different origins. Some simply come from the mathematical structure of Dirac spinors. Other ones come from the dependence on the s-variable arising from two-meson exchange (Js is the energy of the system in its c.m. system). This s-dependence can be turned into a k2 dependence if one assumes sK4(k2#m2). This procedure amounts to neglect off-shell effects that can be incorporated phenomenologically through a fit to experimental data of the short-range part of the NN interaction.

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Typically, the effects under consideration are of order k2/m2 and look like kinematical relativistic corrections. Knowing their origin is, however, not sufficient to fix them in the NN potential. The equation, which the NN potential has to be associated with, has to be precised in order to have an unambiguous prescription. Thus, for the Schro¨dinger equation (or equivalent ones such as the Lippmann—Schwinger equation for the ¹-matrix for instance), the appropriate definition of »(k, k@) is

S

S

m uN p1(k)O up{1(k@)uN p2(!k)O up{2 (!k@) m 1 2 »p{22p{11(k, k@)" , (4.8) pp e 4m2(k2#(k!k@)2) e k k{ where the spinors are normalized as uN u"2m, hence the factor 4m2 in the denominator in Eq. (4.8), and O represents the vertex describing the interaction of nucleons with the mesons of interest. The factors Jm/e and Jm/e , that are neither 1, as in the non-relativistic limit nor m/e and m/e as k k{ k k{ one may expect from the normalization factors relative to the two nucleons in the initial and final states, are required to satisfy a unitarity condition [66]. Furthermore, their omission in earlier calculations of the NN potential was responsible for a large energy dependence of the 2p-exchange contribution (obtained by removing the iterated one pion exchange from the full 2p-exchange contribution to the NN scattering amplitude). The presence of the factors Jm/e and Jm/e in k k{ Eq. (4.8) is important in comparing relativistic calculations to non-relativistic ones. They already include some sizeable relativistic kinematical corrections. Anticipating a comparison with the light-front equation for the NN system, as given by Eq. (3.48) for instance, we can rewrite the Schro¨dinger equation Eq. (4.1) with Eq. (4.8) for the potential and E,k2/m, as follows: 0

AS B

P

AS

B

e d3k@ uN (k)O u(k@) uN (!k)O u(!k@) e kt(k) "!m2 1 2 k{t(k@) . (4.9) m (2p)3e@ 4m2(k2#(k!k@)2) m k Quite generally, the on-shell NN amplitude, from which the NN potential is derived, can be expressed as a sum of independent invariants. Their number, five, is in relation with the number of independent terms in the non-relativistic limit given by Eq. (4.5). Their choice is not unique however. They may be built from the different tensors 1(S),c (PS),c (»),c c (A) and p (¹). Other 5 k k 5 kl choices, in closer relation to the 2p-exchange contribution can also be made [66]. They differ by off-shell effects, which may be accounted for in a phenomenological way by the fit of shorter-range contributions to experimental data, as already mentioned. (k2!k2) 0

4.2.2. Physical inputs The NN potential due to the exchange of single mesons which we consider here can be calculated from the following Lagrangian densities describing the meson—nucleon couplings.5 (i) Pseudoscalar mesons (n, g) (PS coupling): L*/5"i g tM c t/(14) ; 5

(4.10)

5 In the present paper we use the definition of coupling constants corresponding to g2 /4p+14, in contrast to Refs. pNN [48,71,72], where the definition with g2 +14 has been used. pNN

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(ii) Scalar mesons (p, d): L*/5"g tM t/(4) ;

(4.11)

(iii) Vector mesons (o, u): L*/5"tM [gck/(7)#( f/4m)pkl(­ /(7)!­ /(7))]t . k k l l k

(4.12)

For simplicity, the isospin degrees of freedom have been omitted. Other types of coupling may be considered, such as the pseudo-vector one for the n (or g) meson for instance: (iv) Pseudoscalar mesons (p, g) (PV coupling): g tM ckc t­ /(14) . L*/5"! 5 k 2m

(4.13)

For a transition involving on-mass-shell nucleons described by positive energy spinors, couplings (i) and (iv) are equivalent. They strongly differ off-mass shell as it is well known. The first one can give rise to larger effects due to the so-called Z-diagram it leads to, while the second one does not. The choice of the coupling is deeply connected with the way in which chiral symmetry is chosen to be realized. As far as nucleons described by positive energy spinors are retained in the description of the NN system, the potential is unique. As will be seen later in Section 4.3.1, different potentials can be obtained on the light front depending on the choice of the particular coupling used at the meson—nucleon vertex (like for instance the PS or PV pNN couplings). Using the standard approximation made in the meson propagator consisting in taking k !k@ "0, neglecting there0 0 fore retardation effects, one gets in the NN c.m.s.:

SG

»(k, k@)"

[uN (k)c u(k@)] [uN (!k)c u(!k@)] m 5 1 5 2 s )s g2 1 2 4m2(k2#(k!k@)2) e pNN p k

! g2 pNN

#

[uN (k)u(k@)] [uN (!k)u(!k@)] 1 2 4m2(k2#(k!k@)2) p

C

D

g2 f uNN uN (k) (g#f )c ! (k #k@ ) u(k@) k 2m 1 1k 4m2(k2 #(k!k@)2) u 1

C

D

f ] uN (!k) (g#f )ck! (k #k@ )k u(!k@) 2 2m 2 2

HS

m , e k{

(4.14)

where k , k (k@ , k@ ) are the initial (final) on-shell nucleon momenta. Only a few significant ex1 2 1 2 changes have been displayed in Eq. (4.14). Other ones, g, d or o can easily be obtained by removing

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or inserting the appropriate isospin dependence. Hadronic form factors may also be introduced by giving to the meson—nucleon coupling constants a dependence on the meson four-momentum. 4.2.3. Choice of the parametrization There are various single meson exchange models in the literature. They are not all equivalent, with the consequence that they should be used within a defined scheme. One of the most ambitious model is the field theory motivated Bonn model (Bonn-E) [7]. Its energy dependence, which has a relationship with retardation effects included here in the kernel of the interaction, has prompted the authors to derive energy-independent models (now denoted QA and RA) that can be used more easily. The emphasis on reproducing the deuteron D state properties of the original model has biased this derivation. The model is unable to account for the observable mixing angle e beyond 1 100 MeV. These drawbacks have been corrected in later versions (QB, QC, RB) [67] by relaxing the constraints on the deuteron D state properties that are known to be model dependent [73]. Other models differ by the values of the coupling constants, including the associated form factors. This is expected for short-range contributions which have an effective character anyway, but this also occurs for the pNN coupling constant, which may be sensitive to isospin breaking effects [68]. The last one is important for a fine description of the NN interaction but unimportant in the present review. The difference with Bonn QB and QC models [67], which motivation was shortly explained above, involves changes in both the pNN coupling and those determining short-range contributions. The difference between momentum and coordinate space models has conceptually an other origin. Working in one or the other should indeed be equivalent. In practice however, this is true as far as the linearly k2-dependent terms, which are tractable in r-space models, represent an accurate description of the full momentum dependence of the NN interaction model. Recent developments, dealing with the well known p-exchange, show that off-shell effects involving fourth-order terms in the nucleon momentum are relevant for describing the NN interaction in coordinate space [69]. Their neglect explains a large part of the differences in the bare predictions obtained from the Bonn QB [67] and Paris [70] models, which reproduce the same NN scattering data otherwise. In particular, the deuteron D-state probability of coordinate space models is expected to be systematically between 0.5% and 1% higher than in momentum space models. Other differences may be mentioned. The genuine term, r ) [k]k@] r ) [k]k@], in Eq. (4.5) is 1 2 approximately replaced by the term r ) l r ) l#r ) l r ) l in coordinate space models. This can 1 2 2 1 introduce some bias in fitting the potential parameters to NN scattering data. The full 2p-exchange contribution considered in the Paris model does not seem to bring significant improvement upon a single meson exchange model, probably because these ones contain enough parameters (coupling constants together with vertex form factors) to accurately account for these NN data. This is indirectly confirmed by the Argonne model [64] which also does well with these data although it does not incorporate k2 dependent terms as in the Bonn R, Paris or Nijmegen models and, on the other hand, fit the deuteron quadrupole moment to the observed one without any regards to the contribution of meson exchange currents. Altogether, it seems that NN scattering data put strong constraints on the NN interaction models and within a given scheme do not allow for much freedom. Differences are closely related to the scheme under consideration and, most often, involve terms of order (k/m)2, that are of relativistic origin. For some part, the present models may be unitary equivalent, implying the existence of

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corrections pertinent to the model under consideration for any calculation of the physical observables. The choice of some particular model depends on a prejudice [69], models that keep track of these theoretically motivated (k/m)2 terms should deserve a special attention however. 4.3. Meson exchange interaction on the light front As already mentioned, the approach to the non-relativistic NN interaction is semi-phenomenological. This means that the meson—nucleon field theory is used to obtain a form of the NN interaction kernel (OBEP, as a rule), and the parameters of this kernel are found from fitting the experimental data in the framework of the Schro¨dinger equation. A similar approach can be based on the relativistic quantum mechanics in the front form. Besides the Hamiltonian, one should introduce the interaction in all the generators of the Poincare´ group, which change the orientation of the light-front plane [2]. This approach has been developed in a number of studies (see, in particular, Refs. [26,27,74—80]). We do not describe them here in details (see Refs. [28,29] for a review). One should mention that in this approach the full program of fitting the phase shifts and the relativistic kernel has not yet been realized. The kernel in these references is considered as purely phenomenological, i.e., it is not derived from light-front field theory, even in the framework of OBEP. It is assumed that it does not depend on the orientation of the light-front plane. However, if one uses the field theory to construct the off-energy shell light-front kernel, this one always gets a dependence on the light front through the four-vector u or its spatial part n (see examples in Section 4.3.1). It seems reasonable therefore to use this kernel as an input in the light-front version of relativistic quantum mechanics. This cannot be done by a simple replacement of the n-independent kernel in the above versions of the relativistic quantum mechanics by the n-dependent one. This would destroy the commutation relations found for the n-independent mass operator. In order to restore the covariance, one should find a solution of the Poincare´ group algebra with this new n-dependent mass operator. Due to covariance, the exact on-shell amplitudes in this approach will not depend on u (with the accuracy of numerical calculations). It is quite probable that in both phenomenological light-front approaches (with the u-independent and dependent kernels), in spite of the completely different behavior of the off-shell amplitudes, the phase shifts and the binding energies can be equally fitted, at least in the two-body problem. This implies that there exists an unitary transformation between corresponding sets of the Poincare´ generators. However, this does not imply their practical equivalence, since this unitary operator has to be dynamical and highly complicated, since it has to change the off-shell behavior of all the partial amplitudes. It hardly can be useful in practice. It seems therefore reasonable to develop a phenomenological scheme inspired from field theory and to start with the OBEP kernel in the light-front relativistic approach. This way seems rather promising, since it reconciles the main features of the field-theoretical kernel on the light front with relativistic covariance. It has not yet been developed in its full form. 4.3.1. OBEP on the light front The contribution due to single meson exchange, which is shown in Fig. 5, can be obtained from the meson—nucleon interaction described by the Lagrangians given by Eqs. (4.10), (4.11), (4.12) and (4.13). The analytical expression for the pion exchange in the pseudo-scalar coupling, for instance,

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269

is a straightforward generalization of Eq. (2.67). It has the form

G

g2 h(u ) (k@ !k ))F2 1 1 »p{22p{11" pNNs ) s pp 4m2 1 2 k2!(k@ !k )2#2q@u ) (k@ !k )!ie 1 1 1 1

H

h(u ) (k !k@ ))F2 1 1 # k2!(k !k@ )2#2qu ) (k !k@ )!ie 1 1 1 1 ][uN p2(k )c up{2(k@ )] [uN p1(k )c up{1(k@ )] , 2 5 2 1 5 1

(4.15)

where F is a phenomenological form factor, assumed to depend on the same argument as the denominator of the kernel. Here and below it is introduced “by hand”. The g exchange contribution can be obtained from Eq. (4.15) by removing the isospin factor and replacing g2 by g2 . The contributions due to p- and d-exchanges are obtained from the pNN gNN previous ones by removing the c matrices and changing the overall sign. As for the vector meson 5 exchange, it contains a peculiarity because of the contact terms due to the vector meson propagator and the coupling with derivative. We therefore explain this point in more details. The starting expression for the vector meson exchange contribution, obtained by applying the rules of the graph technique to the diagram of Fig. 5, has the form:

P C C

D D C

f K" uN (k ) gca# pa{a(i)(k!uq ) u(k@ ) 1 1 a{ 1 2m

D

(k!uq ) (k!uq ) f 1a 1 b uN (k ) gcb# pb{b(!i)(k!uq ) u(k@ ) ] !g # ab 2 1 b{ 2 k2 2m dq 1 ]d[(k@ !k #uq !uq@)2!k2]h(u ) (k@ !k )) 1 1 1 1 1 q !ie 1

P C C

D D C

f # uN (k ) gca# pa{a(!i)(k!uq ) u(k@ ) 1 1 a{ 1 2m

D

(k!uq ) (k!uq ) f 1a 1 b uN (k ) gcb# pb{b(i)(k!uq ) u(k@ ) ] !g # ab 2 1 b{ 2 k2 2m dq 1 , ]d[(k !k@ #uq !uq)2!k2]h(u ) (k !k@ )) 1 1 1 1 1 q !ie 1

(4.16)

where k is the meson momentum. The factors (i) and (!i) at (k!uq ) are associated with outgoing 1 and incoming lines, according to Section 2.2.4. The factor uq , everywhere in the difference 1 (k!uq ) in the expression (4.16), incorporates the contact terms for both the vector meson 1 exchange and for the coupling with derivative.

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After integrating over q and introducing the form factor F, we get for the potential »"!K/4m2:

G

F2 f »p{22p{11" uN p1(k ) (g#f )c ! (k #k@ ) 1 a 2m 1 1a pp 4m2(K2#k2)

H

if # p ua{[qh(u ) (k !k@ ))!q@h(u ) (k@ !k ))] up{1(k@ ) 1 1 1 1 1 2m a{a

G

f ]uN p2(k ) (g#f )ca! (k #k@ )a 2 2 2m 2

H

if # pb{au [qh(u ) (k@ !k ))!q@h(u ) (k !k@ ))] up{2(k@ ) b{ 1 1 1 1 2 2m g2 ! [uN p1(k )uL up{1(k@ )][uN p2(k )uL up{2(k@ )]qq@ , 1 1 2 2 4m2k2(K2#k2)

(4.17)

where K2 is given by Eq. (3.60). The contribution of the isovector exchange (o-meson) is obtained by multiplying Eq. (4.17) by the factor s ) s . The terms in Eq. (4.17) proportional to q, q@ and qq@ were omitted in Ref. [48]. 1 2 For reasons already explained, the pseudovector pNN coupling is sometimes preferred to the pseudoscalar one. To illustrate the treatment of terms corresponding to derivative couplings of the meson field, we just give the result here:

G

g2 h(u ) (k@ !k ))F2 1 1 »p{22p{1 1" pNN pp 4m2 k2!(k@ !k )2#2q@u ) (k@ !k )!ie 1 1 1 1 uL c q uL c q@ ] uN p2(k ) c ! 5 up{2(k@ ) uN p1(k ) c # 5 up{1(k@ ) 2 5 2 1 5 1 2m 2m

C

A

B

DC

A

B

D

h(u ) (k !k@ ))F2 1 1 # k2!(k !k@ )2#2qu ) (k !k@ )!ie 1 1 1 1 uL c q@ uL c q ] uN p2(k ) c # 5 up{2(k@ ) uN p1(k ) c ! 5 up{1(k@ ) 2 5 2 1 5 1 2m 2m

C

A

B

DC

A

B

DH

s )s . 1 2

(4.18)

The comparison with the expression obtained with the pseudo-scalar pNN coupling evidences differences which depend on the four-vector u and vanish in the limit of on-energy shell nucleons k (q"q@"0). It is also interesting to compare the interaction on the light front, Eq. (4.15), with the Bonn-Q model (4.14), which contains some relativity. Differences involve again terms depending on the four-vector u , which vanish on-energy shell. Further differences, due to the time component of k the four-momentum of the meson in its propagator (retardation effects), vanish on energy shell. The difference in the normalization factors Jm/e and Jm/e@ , discussed in Section 4.2.1, is related to k k the particular equation for the wave function.

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4.3.2. Beyond OBEP on the light front The interaction due to single meson exchanges can produce contributions depending on the arbitrary four vector u. While this dependence vanishes for the Born amplitude (which is calculated for on-shell particles), there is no guarantee that this property holds to all orders. This implies an asymptotic scattering behavior, and therefore a scattering amplitude which depends on u. A quite similar problem arises in other approaches based on a description referring to the plane determined by p"jx with the particular choice j"(1, 0, 0, 0), but here the difficulty is more striking as results may depend on the arbitrary orientation of the three-vector x(n) while physical ones should not. For example, the relativistic wave function originating from the p-exchange, calculated in a perturbative way in the first 1/m order, reads [72]:

P

m g2 d3k@ (k2!k@2) [r ] r ] ) [(k!k@) ] n] 1 2 dt(1)p(k, n)"! pNNs ) s t(0)(k@) . (4.19) 8m2 1 2k2#i2 (2p)3 m k2#(k!k@)2 The quantity i2 is positive in the case of a bound state such as the deuteron, and it is negative for a scattering state. The defect of the wave function (4.19) is that it has a n-dependent residue at the pole at k"ii, that can contribute to an observable amplitude. A similar defect in the scattering state wave function appears in the dependence of asymptotical states on n, i.e., in the on-energy-shell amplitude, in contradiction with the results of Section 2.2. Knowing that the sum of the contributions of all irreducible diagrams to all orders should reproduce results obtained from Feynman diagrams, which are independent of u, one can guess that higher-order terms in the interaction are essential for removing the above unpleasant feature. In this section, we explore their contributions and the role they may play. As will be shown below, the defect of the wave function (4.19) is corrected by incorporating the contact term in the NN kernel [81]. To give some insight about the relevance of these extra terms in the interaction, we will concentrate on a selected one, where the contact term discussed for the spin 1 particle (Eq. (2.60)) is 2 operating (see Fig. 12). This will be done by retaining the relevant lowest 1/m order terms, and for contributions in relation with the single p-exchange. At this order, only p- and u-exchanges contribute, so in Fig. 12 one wavy line corresponds to p-exchange and the other wavy line corresponds to p and u. The corresponding contribution to the kernel has the form: d»p,p`u(k, kA, n),d»p,p`u#d»p,p`u 1 2 g2 d3k@ [(k!k@) ] n] "! pNNs ) s [r ] r ] »p`u(k@!kA) 2 1 2 1 8m2 (2p)3 k2#(k!k@)2 [(k@!kA) ] n] !»p`u(k!k@) , (4.20) k2#(k@!kA)2 where »p`u represents the dominant contribution to the interaction at zeroth order in 1/m [81]. The insertion of d»p,p`u in an equation such as Eq. (3.41) provides an extra, n-dependent, contribution to the wave function. In a perturbative calculation it is given by:

P C D

dtp,p`u(k, n),dtp,p`u(k, n)#dtp,p`u(k, n) 1 2 m d3kA "! [d»p,p`u(k, kA, n)#d»p,p`u(k, kA, n)]t(0)(kA) , 1 2 k2#i2 (2p)3

P

where t(0)(kA) is a zeroth-order non-relativistic wave function.

(4.21)

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Fig. 12. Irreducible diagram with contact term.

One should also take into account all the iterations of the above contributions with the non-relativistic potential »p`u, since these iterations remain in the first 1/m order. They are represented as

P

m d3k@ »p`u(k!k@)[dt(1) p(k@, n)#dtp,p`u(k@, n) dtp,p`u(k, n)"! 1 3 k2#i2 (2p)3 #dtp,p`u(k@, n)#dtp,p`u(k@, n)] , (4.22) 2 3 where the integrand incorporates the function dtp,p`u itself. Therefore Eq. (4.22) is an in3 homogeneous equation for dtp,p`u. 3 It turns out (see below) that dtp,p`u#dtp,p`u"0. Hence, the full correction to the wave 2 3 function reads dtp(k, n)"dt(1) p#dtp,p`u . 1

(4.23)

It can be written as

P

g2 m d3k@ [(k!k@) ] n] dtp(k, n)"! pNNs ) s [r ] r ] 1 2 1 2 8m2 k2#i2 (2p)3 k2#(k!k@)2

C

]

P A

B

D

1 k2 k@2 t (k@)! d3kA d(3)(k@!kA)# »p`u(k@!kA) t (kA) . 0 (2p)3 m 0 m

(4.24)

Applying Eq. (3.41) at the lowest order considered here allows to replace the integral in the last bracket by !(i2/m)t (k@). It is then easily seen that this term adds to the first one in the bracket to 0 cancel the front factor m/(k2#i2) at the r.h.s. of Eq. (4.24). One thus gets

P

g2 d3k@ [r ] r ] ) [(k!k@) ] n] 1 2 dtp(k, n)"! pNNs ) s t (k@) . 1 2 0 8m2 (2p)3 k2#(k!k@)2

(4.25)

The pole in Eq. (4.19) at k"ii (k"k in continuous spectrum case) has now disappeared. This 0 result is important since it guarantees that the observables, such as asymptotic normalization or phase shifts, do not depend on the arbitrary direction n, though the wave function (4.25) depends on it.

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In r-space Eq. (4.25) takes a simpler form:

CA

B D

g2 exp(!kr) dtp(r, n)"i pNNs ) s [r ] r ] ) e ] n t (r) , 1 2 1 2 r 0 8m2 r

(4.26)

which evidences a radial part in close correspondence with that appearing in the pair term contribution to meson exchange currents or also in relativistic components in other approaches. To calculate the contribution dtp,p`u one has to substitute for dt(1) p#dtp,p`u in the integrand 3 1 in Eq. (4.22) the wave function (4.25). The result of this substitution coincides with !dtp,p`u. 2 Hence, Eq. (4.22) turns into dtp,p`u(k, n)#dtp,p`u(k, n) 2 3

P

m d3k@ "! »p`u(k!k@)[dtp,p`u(k@, n)#dtp,p`u(k@, n)] . 2 3 k2#i2 (2p)3

(4.27)

This is the homogeneous equation relative to dtp,p`u#dtp,p`u. However it does not determine 2 3 any eigenvalue, since the energy E in i2"!mE is already fixed by the equation for the complete wave function. Therefore its solution is zero: dtp,p`u(k, n)#dtp,p`u(k, n)"0 . 2 3

(4.28)

The significance of this result may be obtained from a comparison with other relativistic approaches [12]. In these approaches, a term like Eq. (4.26) is associated with a pair of nucleons with positive and negative energies. Their interaction due to p and u exchanges produces a correction proportional to »p!»u (instead of »p#»u as in Eq. (4.22)). The second term in Eq. (4.20) has therefore the effect to remove the contribution that should be absent in any case. In this section, we have considered some of the second order n-dependent contributions to the irreducible interaction ». This has been done to lowest 1/m order. These contributions which are the first ones of an infinite series seem to play an important role to make the theory consistent, accounting in particular for the n independence of the asymptotic wave function, which are related to physical properties. They also show that solving the equation of motion without using the appropriate interaction may lead to wrong results. This consideration makes the relationship with other relativistic approaches closer. The present study is, to a large extent, exploratory. It suggests that the interaction should fulfill some constraints whose determination may be helpful for further work.

5. Applications to the nucleon–nucleon system As a direct application of the formalism developed above, we consider in this section the two nucleon system. Since its non-relativistic phenomenology has been developed over more than twenty years, it allows a sensible discussion of the new features brought about by relativity.

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5.1. The deuteron wave function 5.1.1. Spin structure The construction of the deuteron wave function is a nice illustration of how to construct a state with non-zero angular momentum in the covariant formulation of LFD. For this nucleus, Jp"1` (pseudovector particle), the decomposition of the wave function Uj2 1 in independent spin strucpp tures has the general form [48]: Uj2 1(k , k , p, uq)"Jmej(p)uN p2(k )/ º uN p1(k ) , pp 1 2 k 2 k c 1

(5.1)

with 1 u (k !k ) uL (k !k ) 2k 2 k#u c #u k #u 1 / "u 1 2m k 3u ) p 4 2mu ) p k 1 2m2 i mu uL k , #u c e (k #k ) (k !k ) u #u 52m2u ) p 5 kloc 1 2l 1 2o c 6(u ) p)2

(5.2)

and m is the nucleon mass. The light-front deuteron wave function is determined by six invariant functions u , depending on two scalar variables, e.g., s"(k #k )2 and t"(p!k )2. The extra 1~6 1 2 1 spin structures in front of u are constructed by means of the four-vector u. Other possible 3~6 structures (e.g., ejuN uL c º uN , etc.) are expressed through the six structures given by Eq. (5.2). The k 2 k # 1 wave function (5.1) is kinematically transformed under rotations and Lorentz transformations g of its arguments according to Eq. (3.7). The transformation to the new representation (3.33) has the form: Wj2 1(k , k , p, uq)" + D(1)*MR(¸~1(P), p)ND(1@2) MR(¸~1(P), k )N pp 1 2 1 jj{ p1p@1 j{,p@1,p@2 ]D(1@2) MR(¸~1(P), k )NUj{@2 @1(k , k , p, uq) , p2p@2 2 pp 1 2

(5.3)

where, e.g., R(¸~1(P), p) is given by Eq. (3.5), in which g"¸~1(P), and P is given by Eq. (3.13). The matrix D(1@2) in Eq. (5.3) is defined by Eq. (3.37). pp{ The transformation law of the wave function (5.3) is given by Eq. (3.36). In this new representation given by Eq. (5.3), the rotation operator is the same for all the spin indices and coincides with the operator rotating the variables k, n in Eq. (3.14). In the variables k and n, the problem of constructing the wave function is the same as in the non-relativistic case: we have to construct the general form of a pseudovector from the variables k, n, the Pauli matrices and two-component spinors. Since p ,p "$1, the wave function is 1 2 2 a 2 ] 2-matrix. In this respect, the only difference from the non-relativistic case is the presence of an additional vector n that increases the number of independent structures. The decomposition of the wave function (5.3) takes therefore the form [47]: Wj2 1(k, n)"Jmws2tj(k, n)p ws1 , p y p pp

(5.4)

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with

A

B

1 3k(k ) r) 1 1 r#f !r #f (3n(n ) r)!r) w(k, n)"f 22 32 1 J2 k2

S

#f

1 3 i (3k(n ) r)#3n(k ) r)!2(k ) n)r)#f [k ] n] 5 2k 4 2k

#f

J3 [[k ] n] ] r] , 6 2k

(5.5)

where w is the two-component nucleon spinor normalized to wsw"1. The relation between tj and w is the same as the relation between the spherical function ½j (n) and n. The scalar functions 1 f depend on the scalars k2 and n ) k or k,DkD and z"n ) k/k. For an isospin zero state, the Pauli 1~6 principle results in f (k, z)"f (k,!z), f (k, z)"!f (k,!z) . (5.6) 1,2,3,5 1,2,3,5 4,6 4,6 The reduction of the number of variables can be summarized as follows. The initial wave function (3.7) depends on 4 four-vectors, i.e., on 16 variables. Four mass-shell constraints reduce this number to 12. The four-dimensional conservation law (3.8) reduces it to 8. Lorentz-invariance eliminates the dependence on the velocity € of the system of reference, i.e., reduces the number of variables to 5. The wave function (5.5) just depends on the 5 independent components of two vectors k, n(n2"1). Because the rotational invariance eliminates their dependence on the three Euler angles, the scalar functions f depend only on the two variables (k and z or R2 and x). i M In the region k;m, the functions f which are of relativistic origin become negligible, f do 3~6 1,2 not depend anymore on z and turn into the S- and D-waves: f +u , f +!u . From the general 1 S 2 D decomposition (5.5), one recovers the usual non-relativistic wave function:

C

D

1 1 3k(k ) r) w (k )"u (k) r!u (k) !r . NR S J2 D 2 k2

(5.7)

Representation (5.2) is, however, convenient in the calculation of physical observables, like electromagnetic form factor for instance. The decomposition (5.5) is more adapted for comparison with the non-relativistic limit and will be chosen in this section to present numerical results. The relations between the functions u and f can be found by comparing Eqs. (5.1) and (5.2) with 1~6 1~6 Eqs. (5.4) and (5.5) in the reference system where k #k "0. Indeed, both representations 1 2 coincide in this system, since all the D-matrices in Eq. (5.3) turn into unit matrices. These relations are given in Appendix C. The number of components of the wave function is a priori not the same in different relativistic approaches. For example, the Bethe—Salpeter amplitude of the deuteron is determined by eight components [21] (see Section 5.2), whereas the Gross wave function [13] contains four components. It does not follow, however, that these components do not lead to physical observable consequences. In a given framework, they are unambiguously related to measured cross sections. The convenience of one approach or another depends on the problem under consideration. The relation between the Bethe—Salpeter and light-front components of the deuteron functions is given in Section 5.2. We emphasize that within the LFD the number of components is the same both in

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the non-covariant approach and in the covariant formulation. In the non-covariant approach, these components manifest themselves as six independent matrix elements of the wave function tm1 2 with m"0,$1, p "$1 instead of two matrix elements in the non-relativistic case. The pp 1,2 2 representation of the wave function (5.5) in terms of the well defined spin structures and corresponding components f is however much more transparent than in terms of those matrix 1~6 elements. Note that when a deuteron state at rest, and described from two positive energy spinors for its constituents, is seen from a different inertial frame, it automatically acquires four extra components depending on the velocity of this frame. This also holds for the 1S scattering state (one extra 0 component in this case). The equality of the total number of six components thus obtained with the number of components in the light-front approach is not a coincidence, since it is related in both cases to the appearance of an extra vector. The extra components in the light-front approach just represent the dynamical counterpart arising from the boost of the system at rest to some velocity €, and the vector n is the counterpart of the direction € when v P c. 5.1.2. Two-body contribution to the deuteron normalization For the contribution of the two-body Fock component, the normalization integral (3.23) is rewritten as:

P

1 d3k d3k 22(u ) p) dq . 1 Nj{j" + U*2Jj{2 1 1UJj d(4)(k #k !p!uq) 2 2 1 1 2 j j 1 2 p j p p j p (2p)3 2e 1 2e 2 k k p1p2 Substituting wave function (3.1) into integral (5.8), we get

(5.8)

Nj{j"e*j{(p)Iklej(p) k l 2 where

P

m d3k d3k 1 22(u ) p) dq , Ikl" ¹rM/k(kK #m)/l(kK !m)Nd(4)(k #k !p!uq) 2 1 1 2 (2p)3 2e 1 2e 2 k k

(5.9)

and / is given by Eq. (5.2). Substituting in Eq. (3.31) the integral given by Eq. (5.9) with wave function (5.2), one can obtain the normalization condition for the deuteron wave function in terms of the components u . 1~6 Expressing the functions u through f by the formulae (C.3)—(C.8) given in Appendix C we can 1~6 1~6 represent it in terms of the f functions. It is however simpler to do this directly in terms of f . 1~6 1~6 For this goal, consider integral (5.8) in the reference system where k #k "0. The deuteron wave 1 2 function is given by Eq. (5.5). We then get for Nj{j, in the basis where j@, j"x, y, z: 2

P

m d3k Nij" ¹rMts(k, n)t (k, n)N . j i 2 (2p)3 e k

(5.10)

The general structure of integral (5.10) is given by Eq. (3.30) and A is found from Eq. (3.31): 2 A "1d Nij . 2 3 ij 2

(5.11)

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Substituting in Eq. (5.10) the wave function (5.5), we get

P

P

m =k2 dk 1 I dz , A " 2 (2p)2 e 0 k ~1

(5.12)

where I"f 2#f 2#f 2#f 2 (3#z2)#f 2(1!z2)#f 2(1!z2) 1 2 3 4 5 6 !f f (1!3z2)#4f f z#4f f z . 2 3 2 4 3 4

(5.13)

In the case where the kernel of the interaction, », is independent of M, one has A "1. Keeping the 2 dominating contribution in the region k;m, where the functions f are negligible and 3~6 f "u , f "!u , we recover the non-relativistic normalization condition for the deuteron wave 1 S 2 D function:

P

1 (u2#u2 )k2 dk"1 . D 2p2 S 5.1.3. Equation for the wave function Keeping all the spin indices, the equation for the wave function Uj1 2 has the form already pp indicated in Eq. (3.40). In terms of the variables k and n, this equation can be rewritten as:

P

d3k@ m2 [4(k2#m2)!M2]w 1 2(k, n)"! + [w@(k@, n)p ] 1 2»p{11p{A22 (k@, k, n, M)(p ) A1 1 pp y p{ y p p{ p p p e 2p3 k{

(5.14)

According to our discussion of Section 3.2, the function w@ 1 2 is obtained from Uj1 2(k@ , k@ , p, uq@) pp p{ p{ 1 2 simply by expressing the arguments k@ , k@ , p, uq@ through k, k@, n, as given by Eqs. (3.43), (3.44), (3.45) 1 2 and (3.46). Expressing the function w through Uj1 2 with the help of the D-matrices, in Eq. (5.3), the pp wave function w@(k@, n) can then be obtained from w(k@, n) by a rotation of spins according to: w@"D(1@2)sMR(¸~1(P@), k@ )NwD(1@2)MR(¸~1(P@), k@ )N , 2 1

(5.15)

where the matrix D(1@2)MR(¸~1(P), k)N is defined by Eq. (3.37). 5.2. Connection with the Bethe—Salpeter amplitude The Bethe—Salpeter function of the deuteron U(p, k , k ) is a 4 ] 4 matrix depending on the 1 2 two off-shell nucleon momenta k and k and the on-shell deuteron momentum p"k #k 1 2 1 2 with p2"M2. Both nucleons in the vertex d P np are final, whereas, as a convention, the index a in the matrix U corresponds to the final nucleon and the index b corresponds to the initial ab nucleon. It is therefore convenient to separate from UBS at the right the charge conjugation matrix º: # U"tBSº #

(5.16)

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In order to construct the Bethe—Salpeter function tBS, we will start with the “initial” expression amK #b(k ) m)/m and then multiply it by kK and kK from right and left respectively and from both 1 2 sides. Here m,ej(p) is the deuteron polarization vector with p ) m"0, and k"(k !k )/2. We thus 1 2 obtain

C

D C D

D

C

(k ) m) (k ) m) kK kK (k ) m) 1# 2 a mK #b tBS" a mK #b # a mK #b 1 1 m 2 2 m m 3 3 m m

C

D

(k ) m) kK kK 1. # 2 a mK #b 4 4 m m m

(5.17)

The functions a , b depend on the scalars k ) p and k2. One can see from this equation that the i i deuteron Bethe—Salpeter function, tBS, contains eight spin structures. This fact agrees with another classification [17,82,83] using the partial waves 2S`1¸o, where the index o corresponds to the so J called o-spin, which operates on the positive- and negative-energy states exactly in the same manner as usual spin operates on the usual spin states. In these terms, the Bethe—Salpeter function of the deuteron is determined by the following eight coupled waves [17]: 3S`, 1

3D`, 3S~, 3D~, 1P% , 3P0 , 1 1 1 1 1

1P0 , 1

3P% . 1

(5.18)

The indices “#” and “!” mean “up” (##) and “down” (!!) o-states of both nucleons, the indices e (even) and o (odd) mean the triplet and singlet o-spin states. From the Pauli principle it follows that the function U satisfies the symmetry relation: U (k , k , p)"U (k , k , p). Hence, ab 1 2 ba 2 1 tBS(1, 2)"!º [tBS(2, 1)]5º . To provide the symmetry relation in a simplest way, we transcribe # # Eq. (5.17) in the form: (k ) m) (mK kK !kK mK #2mmK ) (k ) m) 1 2 tBS(k , k , p)"g #g mK #g #g (kK !kK #2m) 1 2 1 m 2 3 4 2 m m2 1

C

D

(k ) m) (kK #m) (k ) m) (mK kK #kK mK ) (kK k !m) 1 2 # 2 1 g mK #g #g (kK #kK ) . #g 6 7 8 2 5 m m m2 1 m m

(5.19)

The symmetry implies: g (k ) p, p2)"g (!k ) p, p2) for i"1!4, 6, 7 and g (k ) p, p2)" i i 5,8 !g (!k ) p, p2). 5,8 The relation between the Bethe—Salpeter and the light-front wave function is given by Eq. (3.58). We show below that Eq. (3.58) together with the deuteron Bethe—Salpeter function (5.19), coincides with the deuteron light-front wave function (5.2). We substitute Eq. (5.19) in Eq. (3.58). The momenta k , k in Eq. (5.19) are off the mass shell. 1 2 However, after the following substitution in Eq. (3.58): k P l "k !uq/2#ub, 1 1 1

k P l "k !uq/2!ub 2 2 2

we have to put k , k on mass shell. Due to that, we may use the Dirac equation 1 2 uN (k )(kK !m)"(kK #m)º uN (k )"0. After applying the Dirac equation, the function tBS in the 2 2 1 # 1

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279

integrand of Eq. (3.58) reads: 1 tBS(k !uq/2#bu, k !uq/2!bu, p) P g [(k ) m)#(u ) m)b]#g mK 1 2 1m 2

C

D

1 1 1 #g 2(u ) m)b! (mK uL !uL mK )q #g [(k ) m)#(u ) m)b]2buL 3m 4 2 m2 2(u ) m) 1 1 uL (b2!q2/4)!g [(k ) m)#(u ) m)b]uL q . (5.20) # g [(mK uL !uL mK )b!(u ) m)q]!g 6 m2 8m 5m The function g does not contribute. One can see that the structures at g in Eq. (5.20) 7 1,2,4,6,8 coincide with the corresponding structures at u — in Eq. (5.2). The expression (mK uL !uL mK ) at 1 4,6 g and g in Eq. (5.20) can be transformed to the form c e m p k u at u in Eq. (5.2). For this 3 5 5 kloc k l o c 5 aim we decompose kK (mK uL !uL mK )kK in the complete set of 4 ] 4 matrices: 2 1 M,kK (mK uL !uL mK )kK "A#Bc #Ckc #Dkic c #Eklip . 2 1 5 k k 5 kl

(5.21)

The coefficients are given by: A"1¹r(M), B"1¹r(c M), 2E "1¹r(ip M) and C "D "0. 4 4 5 kl 4 kl k k Sandwiching Eq. (5.21) between the spinors uN (k ) and º uN (k ), and using again the Dirac equation, 2 # 1 we find

C

4 uN (mK uL !uL mK )º uN " uN ic e m k k u !(k ) m)(u ) p)!mK m(u ) p) 2 # 1 s 2 5 kloc k 1l 2o c

D

uL (u ) m) # 1(s!M2)(x!1)(u ) m)#1(s!M2)m º uN , 2 2 2 # 1 (u ) p)

(5.22)

where, as usual, s"(k #k )2"4(k2#m2). This structure contributes to u and to other struc1 2 5 tures. By this way, we reproduce exactly the six structures in the light-front deuteron wave function (5.20). Comparing their coefficients, one can easily find the expressions for u in terms of the integral i of g over b. The arguments of the functions g "g (k2, p ) k) are replaced in the integrand by i i i k2P!k2#2m2(x!1)b@ , 2 p ) kP!1(x!1)(s!M2)#m2b@ , 2 2 where we introduce the dimensionless variable b@"b(u ) p)/m2, and, hence, depend on k2, n ) k"!2e (x!1). This provides the explicit link between Bethe—Salpeter and light-front k 2 wave functions. 5.3. Two nucleon wave function in the J p"0` scattering state This relativistic scattering state wave function has been calculated in Ref. [71]. Before coming to the relativistic case we remind some definitions for the non-relativistic continuous spectrum wave function in coordinate and momentum spaces.

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5.3.1. Non-relativistic wave function We start with the known S-wave continuous spectrum wave function in coordinate space, t (r), p having the usual asymptotical behavior: sin(pr) e*pr sin(pr#d) t (r)D +tA(r)" #f "e*d , p r?= p pr r pr

(5.23)

where f"e*dsin d/p is the scattering amplitude. The corresponding energy is E "p2/m. p The Fourier transform6 t (k) of t (r) is expressed through the off-shell scattering amplitude p p f (k, p):

P

2p2 4pf (k, p) t (k)" t (r) exp(!ik ) r)d3r" d(k!p)# , p p p2 k2!p2!ie

(5.24)

where the S-wave off-shell amplitude f (k, p) satisfies the Lippmann—Schwinger equation. The variable k is the argument of the Fourier transform, whereas p is related to the eigenvalue E . p Because of the asymptotical behavior (5.23), valid for any short-range interaction, the momentum space wave function (5.24) has a singular term at k"p and a quickly decreasing off-energy shell dynamical part f (k, p). It is convenient to extract the global phase factor from the wave function: t (r)"e*d(p)u (r) p p and from the amplitude:

(5.25)

f (k, p)"e*d(p)g(k, p) .

(5.26)

The function u (r) is then real and has the asymptotical form: p sin(pr#d) . u (r)D +uA(r)" p r?= p pr The function g(k, p) is real with g(p, p)"sin d/p. The Fourier transform of uA thus gives: p 2p2 4pf uA(k)" uA(r) exp(!ik ) r) d3r" d(k!p)# , p p p2 k2!p2!ie

P

(5.27)

(5.28)

where f"f (p, p) is the on-shell scattering amplitude given above. Subtracting Eq. (5.28) from Eq. (5.24), one finds:

P

sin d = sin(kr) #(k2!p2) (u (r)!uA(r)) r dr . g(k, p)" p p p k 0

(5.29)

6 We use the same notation both for the wave function t (r) in coordinate space and for its Fourier transform t (k). p p The difference is indicated by the arguments.

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By relations (5.24) and (5.26), g(k, p) determines the continuous spectrum wave function in momentum space. The difference, u (r)!uA(r), decreases faster for r greater than the interaction p p range (exponentially for the Yukawa-type potentials) and the integral (5.29) can be easily computed. 5.3.2. Spin structure of the relativistic wave function in LFD This wave function is the relativistic light-front generalization of the non-relativistic np wave function in the 1S state. The partial scattering amplitude (scattering through Jp"0` state) can be 0 factorized as follows: Fa11a22"e*dA 1 2ss1 2(1S ) , pp aa 0 pp

(5.30)

where s(1S ) is the final 1S -state spin function on energy shell normalized to 1: 0 0 s

1 (1S )" uN a1(p )c º uN a2(p ) , 0 1 5 # 2 aa 2J2m 1 2

(5.31)

In the non-relativistic limit in the center-of-mass system of the final nucleons, s(1S ) turns into its 0 non-relativistic counterpart: i s(1S )+ ws p ws "C00 1 ws (1)ws2(2) , 0 (1@2)p (1@2)p2 p1 p J2 1 y 2

(5.32)

where w represents the two-component nucleon spinor (see Appendix A), and C00 1 is the p (1@2)p (1@2)p2 Clebsh—Gordan coefficient. The amplitude A does not depend on the spin projections of the final nucleons a and a . We 1 2 have separated the phase factor e*d, where d is the 1S phase shift. We expand the off-energy shell 0 amplitude A for forming the 0`-state in the most general invariant amplitudes [71]:

C

D

4m2 2muL A"uN c º uN a #uN ! c º uN a , 1 5 # 2 1 1 u)p 5 # 2 2 s

(5.33)

where s"(k #k )2, uN "uN p1,2(k ) and a are scalar functions. In the representation (5.3) 1 2 1,2 1,2 1,2 * (without the matrix D1 ), the amplitude (5.33) can be decomposed as follows [71]

A

B

1 ir ) [k ] n] g p ws2 , G 1 2" ws1 g # 2 y p pp k J2 p 1

(5.34)

where the scalar functions g depend on k and z"n ) k/k. This procedure is completely analogous 1,2 to the one used above for the deuteron wave function. The relativistic amplitude (5.34) contains not only the 1S state with spin zero (the first item), but also a state with spin 1 (the second item). 0 Therefore, it can be misleading to use the notation 1S . We refer here to the Jp"0` state. From the 0 Pauli principle it follows: g

(k, z)"g (k,!z) . 1,2 1,2

(5.35)

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The relations between the invariant amplitudes in the representations (5.33) and (5.34) are the following: J2 a " g , 1 p 1

J2e2 kg . a " 2 pmk 2

(5.36)

5.3.3. Equation for the wave function The inhomogeneous equation for the relativistic light-front amplitude has a form similar to that determining the bound state wave function (3.40):

P

1 F(p , p ; k@ , k@ , uq@) F(p , p ; k , k , uq)"K(p , p ;k , k , uq)# 1 2 1 2 1 2 1 2 1 2 1 2 (2p)3 ]K(k@ , k@ , uq@;k , k , uq)d(4)(k@ #k@ !uq@!k !k #uq) 1 2 1 2 1 2 1 2 dq@ d3k@ d3k@ 1 2. (5.37) ] (q@!ie) 2e @1 2e @2 k k The delta-function, together with the integration on one of the momenta, is here kept for symmetry. Substituting here amplitude (5.30) and using the variables k, n in the system of reference where P"0 we find

A

B

P

A» d3k@ ir ) [k ] n] m2 m2 g # g "! e~*ds»! . 1 2 k 4pe 32p2e (k@2!p2!ie) e J2 k{ k k 1

(5.38)

with »"!K/4m2. The products s» and A» contain summation over spin indices. 5.4. Numerical results In order to have a first estimate of the relativistic corrections to the deuteron wave function, one can calculate the components f —f perturbatively, starting from the non-relativistic solution 1 6 u and u : S D d3k@ !m2 + (w@ p ) @1 2»p{11p{2A2 (k@, k, n, M)(p ) A2 2 , (5.39) w 1 2(k, n)" NR y p p{ p p ypp e pp 2p3(4(k2#m2)!M2) k{ p{1p{2 where w@ is given by Eq. (5.15) with the replacement of w by w defined by Eq. (5.7). ApproximaNR NR tion (5.39) is based on the fact that due to the small deuteron binding energy, the deuteron wave function is concentrated at non-relativistic values of k@ and the integral in the r.h.s. of Eq. (5.14) is dominated by the same domain. From Eq. (5.39) one can find f — (see Ref. [48] for more details). 16 The comparison of Eq. (5.39) with the equation to be used with the Bonn-Q models (as written in Eq. (4.9) for instance) indicates that they are identical in the limit where n dependent terms and boost effects under the integral are omitted. In this limit, the light-front wave function should be equal to Je /m tB0//-Q(k). This one could therefore provide an approximate zeroth order in solving k Eq. (5.39).

P

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283

The same approximate solution can be found for the 1S scattering state [71]. In this case, one 0 replaces A in the integrand of Eq. (5.38) by J2 A" uN @ c º uN @ g (k@, p) p 1 5 # 2 1

(5.40)

with g (k@, p)"(e /m)g(k@, p) and g(k@, p) given in Eq. (5.29). 1 k{ We use the one-boson-exchange interaction with the parameters of the Bonn-QA model [7]. This one was used in earlier calculations [48,71] and in view of the exploratory character of the present work, we have not used a more realistic model (Section 4.3.2). The non-relativistic deuteron wave function was also taken from Ref. [7]. For the 1S scattering state, we use the Paris potential 0 [70] and the wave function was taken from [84]. Results for the deuteron components [48] are displayed in Figs. 13—15. All the wave functions are normalized by JA , as given by Eq. (5.12). In Figs. 13 and 14 the functions f (k, z) and f (k, z) 2 1 2 in (MeV/c)~3@2 are shown for a fixed value of z"0.6 in the region 04k41.5 GeV/c. They are compared to their non-relativistic counterparts (dashed lines). All the functions are shown together in Fig. 14 as a function of k. The wave functions g [71] for the 1S scattering state, for E "1.5 MeV (p"37.52 MeV/c), in 1,2 0 p (MeV/c)~1, are shown in Fig. 16. In view of the results given in Refs. [48,71], one can make the following remarks. 1. In the region k5 0.4—0.5 GeV/c, the components f of the deuteron wave function, and g of the 5 2 1S scattering state are comparable to the non-relativistic components. They are dominant 0

Fig. 13. The function f , in (MeV/c)~3@2, in absolute value, as a function of k at a fixed value of z"n ) k/k"0.6. f is 1 1 positive at the origin and changes sign twice. The dashed line is the non-relativistic S wave function calculated with the Bonn potential, and the solid line is the relativitic result calculated in perturbation theory (see text). Fig. 14. The same as in Fig. 13, but for the function f . 2

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Fig. 15. The relativistic components f —f , in (MeV/c)~3@2, as a function of k, at z"0.6. 1 6 Fig. 16. The relativistic wave functions g of the 1S scattering state, in (MeV/c)~1, for E "1.5 MeV, at z"0.6 (solid 1,2 0 p line). The dashed line is the non-relativistic 1S scattering state wave function. 0

2.

3.

4. 5. 6.

above k"0.5 GeV/c. These components are calculated here with the pseudo-scalar representation at the pNN vertex. We shall come back to this point in the next section. The functions f ,!f and g are close to the input non-relativistic wave functions in the region 1 2 1 0(k(0.5 GeV/c. They are however strongly modified for larger momenta. This indicates that relativistic corrections are important in this region. Some of the differences at low k, for f for 2 instance, may disappear once the low energy NN scattering data or the asymptotic normalization, A and A , are required to be produced. D S Calculating the phase shift by means of the relativistic function, g (p, p)"sin d/p, we get d"48° 1 in comparison with the value 61° found from the non-relativistic input Paris wave function. The deviation of 13° indicates that the relativistic effects on the low-energy phase shift are sizeable and advocate for a new fit of the parameters in the relativistic kernel. This deviation shows the actual uncertainties of the relativistic kernels. The relative contribution of the relativistic components f — to the normalization integral is of 36 the order of 1%. At k of the order of 1 GeV/c all the components of the wave function strongly depend on z. This is in contrast with the components f , f and g in the non-relativistic, small k, region. 1 2 1 The n dependence of the wave function is a property of the wave function related to the off-energy shell amplitude. At k"p the scattering amplitude should not depend on n, otherwise

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it would violate rotational invariance. In practice this dependence could appear due to approximations. It was found in Ref. [71] that, at k"p+37.5 MeV/c,g is negligible compared 2 to g and the z-dependence of g is of the order of 10~3. These facts indicate that in this respect 1 1 these calculations are self-consistent. The existence of the relativistic extra components f and g which dominate already at moderate 5 2 momenta may seem surprising given the success of the non-relativistic phenomenology in the two nucleon systems. This fact is indeed related to the dominance of meson exchange currents in the deuteron electrodisintegration amplitude. It shows that the calculation of the light-front wave function omitting its n-dependence cannot be considered as realistic. We shall come back to this point in Section 7.3.1, where the physical significance of these components will be investigated in leading order in a 1/m expansion.

6. The electromagnetic amplitude The physical amplitude and electromagnetic form factors should not depend on a particular choice of orientation of the light-front plane. This is however only the case in a given order of perturbation theory or in an exact calculation. In any approximate calculation, the light-front electromagnetic amplitude depends on the orientation of the light front, i.e., on the four-vector u. This dependence is non-physical for any on-shell amplitude. We will show in this section, for systems with spins 0, 1, 1 and transitions between them, how to extract the physical form factors 2 from the u-dependent electromagnetic vertex. The covariant formulation of LFD will prove to be very powerful in resolving the ambiguities arising from the light-front orientation. 6.1. Factorization of the electromagnetic amplitude The form factors are experimentally extracted from the scattering amplitude of a charged particle (usually the electron) from a bound system. One of the contribution to the amplitude is shown graphically in Fig. 17. For simplicity, we consider a two-body system to start with, but all our results are obviously applicable to many-body systems. In the Feynman approach, the amplitude of this process is factorized in the product of the electromagnetic vertex of the projectile, the propagator 1/q2 of the virtual photon and the electromagnetic vertex of the bound system. This latter is then expressed through the physical form factors. The complete set of light-front diagrams should, of course, be factorized and reproduces this structure. However, the separate amplitude represented in Fig. 17 does not in general factorize in this formalism. Indeed, the expression for this amplitude, corresponding to the photon propagator and the adjoining spurion lines has the form:

P

dq dq 2 3 . M" (2)h(u ) (p@!p))d((p@!p#uq !uq )2) 3 2 q !ie q !ie 2 3

(6.1)

The dots represent the parts of the amplitude which contains the matrix element of the current J . o They will be discussed in the next section.

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Fig. 17. Particular amplitude corresponding to the scattering of an electron from a bound system.

For definiteness, we suppose that u ) (p@!p)'0. Integrating Eq. (6.1) over dq by means of the 3 d-function, we get, with q , p@!p:

P

dq 2 . (6.2) [2(u ) q)q !q2!ie](q !ie) 2 2 This amplitude is not a product of the 1/q2 term with the form factors. However, as seen from Eq. (6.2), the factorization can be obtained under the special condition M" (2)

u ) (p@!p)"u ) q"0 ,

(6.3)

implied as the limit u ) q P #0. Under this condition, amplitude (6.2) obtains the form M & f (q)/q2. The integration over dq in Eq. (6.2) is absorbed by the form factors. The correspond2 ing electromagnetic vertex can then be represented by Fig. 18 with the off-mass-shell photon momentum q2 O 0. In the standard formulation of LFD with u"(1, 0, 0, !1), condition (6.3) turns into the usual condition q "0. This condition can only be achieved for q240. ` 6.2. Extracting the physical form factors The problem of extracting the physical form factors from the elementary electromagnetic amplitude in the LFD is rather general and appears for a system with any spin. As it was discussed in Section 2, the sum of all the on-energy-shell amplitudes in a given order of perturbation theory does not depend on the four-vector u, whereas any given amplitude can be u-dependent. This is so also for the on-energy shell electromagnetic amplitudes. The amplitude corresponding to a separate diagram can indeed depend on u. The diagram of Fig. 17 for instance is not the only one contributing to the form factor. Other contributions are shown in Fig. 19.

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287

Fig. 18. Electromagnetic vertex of a bound system.

Fig. 19. Amplitude which cannot be expressed through the bound state wave function.

The matrix element of the exact current, J "Sp@DJ (0)DpT, can be represented as the sum of all o o possible contributions: J "J(1)#J(2)#2#J(n)#2 , (6.4) o o o o where the superscripts is a short notation for indicating the various levels of approximation. In the case of the deuteron form factors for instance, the terms in Eq. (6.4) correspond to the impulse approximation, meson exchange current contribution, etc. Any contribution in Eq. (6.4) consists of the sum of two parts: a u independent one denoted by F(n), and a u dependent one denoted by o B(n)(u): o (6.5) J(n)"F(n)#B(n)(u) . o o o The explicit form of Eq. (6.5) for systems with spin 0, 1 and 1 is given below in Section 6.3. 2 Since the matrix element of the exact current does not depend on u, the u-dependent parts cancel each other: B(1)(u)#B(2)(u)#2#B(n)(u)#2"0 , o o o

(6.6)

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and, hence, the physical matrix element is determined by the sum of the u-independent parts only: J "F(1)#F(2)#2#F(n)#2 . (6.7) o o o o It follows that the problem of restoring the independence of any approximate physical amplitude on u is reduced to the separation, and omission, of the “pure” u-dependent terms. We emphasize that this definition of the form factors, after separating out u-dependent contributions, still leads to approximate form factors. However, our procedure guarantees that the form factors do not receive any spurious, non-physical contributions. Since these spurious, u-dependent, contributions can be as large as the physical ones, in particular at high momentum transfer, our procedure is the most adequate when one wants to draw any conclusion on a particular model, or to compare various relativistic approaches. Moreover, it leads to an unambiguous determination of the form factors, independent of the orientation of the light front or of the choice of the component of the current which is used to calculate the amplitude. This is not the case in the usual formulation on the light-front plane t#z"0, as it is already known for the deuteron form factors [85,86]. In the case of spin 0 and 1 systems, the criterion of irreducibility for the u-dependent contribution is rather evident (though with a subtle point discussed in Section 6.3.3). It can be read directly from the decomposition of the current: the terms proportional to u do not contribute to the physical form factors. In the case of spin 1 particle, the unambiguous separation of the purely 2 u-dependent part requires some care and is discussed in Section 6.3.2. In a more general language, the extraction of the physical form factors can be explained as follows. The physical electromagnetic amplitude J , Sp@DJ (0)DpT does not depend on the hypero o surface where the state vector DpT is defined and, hence, does not contain any term proportional to u , provided the current operator J (0) is the full operator. This current operator has to contain the o o interaction. Indeed, the commutation relation between J (0) and the four-dimensional angular o momentum JK is similar to Eq. (2.33), since it is universal for any four-vector: ab (1/i)[J (0), JK ]"g J (0)!g J (0) . (6.8) o ab ob a oa b Since JK in Eq. (6.8) contains the interaction, the current J (0) has to contain the interaction too. ab o The consistency of the transformation properties of the current and of the state vector ensures the independence of the matrix element of the current J (0) on the quantization plane orientation. o In practical calculations however, this consistency is violated twice: (i) the diagram of Fig. 18 corresponds to the free current and (ii) the state vector is approximated by its two-body component. Due to these reasons, the vertex J obtains spurious contributions proportional to u . They o o are, however, completely explicited in our covariant formulation, while they are hidden in the usual formulation of the LFD with t#z"0. 6.3. Light-front electromagnetic vertex The calculation of any amplitude with composite systems in the initial and final states is done according to the rules of the graph technique discussed in Section 2.2, with the coupling constants at the vertices replaced by the vertex functions C and Cs introduced in Section 3.2 for the final and initial systems, respectively. We detail in this section how to perform the calculations for spin 0, 1 2 and 1 bound states.

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6.3.1. Spin 0 system Let us consider the structure of the vertex of Fig. 18 for a system of spin 0. It has the general form: (p#p@)2 JI "(p#p@) F(Q2)#u B (Q2) , o o o 2u ) p 1

(6.9)

in which the factor (p#p@)2/(2u ) p) is separated for convenience. The tilde on top of the electromagnetic amplitude JI here and below indicates that this amplitude is an approximate one o and contains u-dependent non-physical contributions, as compared to the exact physical amplitude with no tilde. As usual, we have Q2"!q2 , !(p@!p)2. The invariant functions F and B could depend in principle on u ) p and u ) p@. However, the four-vector u is defined up to an 1 arbitrary number, and, hence, the theory is invariant relatively to the replacement u P au, where a is a number. The form factors F and B can therefore depend only on the ratio u ) p@/u ) p. But 1 since u ) q"0, we have u ) p@/u ) p"1, and the functions F and B (and similar functions for the 1 case of spin 1 and 1) depend on Q2 only. 2 The general physical electromagnetic amplitude of a spinless system is given by J , Sp@DJ (0)DpT"(p#p@) F(Q2) . (6.10) o o o The main difference of the amplitude (6.9) with respect to Eq. (6.10) is the presence of an additional contribution, proportional to u . Note that expression (6.9) satisfies the conservation of the current o JI qo"0 since u ) q"0. This conservation is automatically fulfilled in this particular case due to o the simplicity of structure (6.9), but it cannot be imposed a priori. We will see below that for a system with spin 1, some u-dependent terms violate indeed current conservation. In the simple case where the constituents are also spinless (Wick—Cutkosky model), the amplitude indicated in Fig. 18 reads:

P

JI " [(p#uq !k ) #(p@#uq !k ) ]h(u ) (p!k ))d((p#uq !k )2!m2) o 2 1o 1 1o 1 2 1 ]h(u ) (p@!k ))d((p#uq !k )2!m2)h(u ) k )d(k2!m2) 1 1 1 1 1 dq dq d4k 1 2 1, ]CC@ (6.11) (q !ie) (q !ie) (2p)3 1 2 where the vertex functions C, C@ depend on the momenta in the initial and final vertices. Integrating by means of the d-functions over dq dq dk , and expressing C through t by means of Eq. (3.39), 1 2 0 we get

P

d3k 1 (p#p@#uq #uq !2k ) 1, 1 2 1 ot@t h(u ) (p!k )) JI " (6.12) o (2p)3 1 2e (1!u ) k /u ) p)2 1 k 1 where q "[m2!(p!k )2]/(2u ) (p!k )) and q "[m2!(p@!k )2]/(2u ) (p@!k )). 1 1 1 2 1 1 From Eq. (6.12) one clearly sees that J contains terms proportional to u and, hence, has the o o structure of Eq. (6.9). However, to avoid any misunderstanding, we emphasize that the contribution uq #uq is not the only source of u-dependence. Even in the absence of this term and in the 1 2

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case where the wave function t does not depend on n, the term proportional to u survives because of o momentum conservation at the electromagnetic vertex. In the spinless case, the prescription to extract the physical contribution to the form factor, F(Q2), from the u-dependent amplitude JI can be obtained immediately by multiplying both sides o of Eq. (6.9) by u . We thus get o JI ) u F(Q2)" . (6.13) 2u ) p With Eqs. (6.12) and (6.13) we obtain

P

1 d2R dx M F(Q2)" t(R2 , x)t((R !xD)2, x) . M M (2p)3 2x(1!x)

(6.14)

We have represented here, and in the following, the four-momentum transfer q by q"(q , D, q ) 0 , with D ) x"0 and q is parallel to x. Since u ) q"0, we have Q2"!q2"D2. , In the usual light-front formulation, with u"(1, 0, 0,!1), Eq. (6.13) corresponds to expressing the form factor through the JI component. This is well known, and Eq. (6.14) has been found in ` Ref. [87]. However, this procedure cannot be extended to the calculation of physical form factors of systems with total spin 1 and 1, as we shall see in the next sections. 2 One can similarly calculate for comparison the non-physical form factor B : 1 2 (R2 #m2!M2(1!x)2!xR ) D#Q2/2) M M B (Q2)" 1 (4M2#Q2) (2p)3

P

d2R dx M ] t(R2 , x)t((R !xD)2, x) . M M 2x(1!x)3

(6.15)

Here and below we use M for the total mass of the system, m being used for the mass of its constituents. For a system with small binding energy De D;m, the wave function is mainly concentrated at " x & 1/2, R2 & mDe D. For such a system, the function B at Dq2D;m2 has smallness &De D/m in M " 1 " comparison with F or &Q2/M2, if Q2/M2'De D/m. However, at Q25m2 the function B becomes " 1 comparable with F. We will illustrate this in Section 7.1 in the Wick—Cutkosky model. 6.3.2. Spin 1 system 2 The electromagnetic physical amplitude for a spin 1 fermion is determined by the two standard 2 form factors: iF (Q2) uN @p qlu,uN @C u . J "F (Q2)uN @c u# 2 ol o o 1 o 2M

(6.16)

Here uN @"uN p{(p@), u"up(p) are the spin 1 fermion bispinors and M is the fermion mass. The charge 2 (G ) and magnetic (G ) form factors are expressed through F and F as follows: E M 1 2 Q2 G "F ! F , G "F #F . (6.17) E 1 4M2 2 M 1 2

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The covariant light-front spin 1 electromagnetic vertex was firstly considered in Ref. [85] and 2 investigated in detail in Ref. [88] (see also Ref. [89]). Like in the case of spin 0, the dependence of the corresponding amplitude on the extra four-vector u increases the number of independent terms in the decomposition of the amplitude. Besides the two structures entering in Eq. (6.16), one can construct the following three structures: 1 1 1 uN @uL u P , uN @u u , uN @uL u u , o o o u )p (u ) p)2 u )p

(6.18)

where P"p#p@, uL "u ck. k Other structures, like non-gauge-invariant ones: uN uq , uN uL uq , uN @p u ul and also o o ol iuN @c u e Pkqluc (not independent from the previous three), contradict ¹-invariance. 5 oklc At first glance, it seems that all the structures in Eq. (6.18) exhibit no u-independent parts, since, like in the previous section, they are proportional to u/u ) p in the first and second degree. This is indeed true for the last two structures which are explicitly proportional to u /u ) p. o This is, however, not the case for the first structure uN @uL uP /u ) p which is directly proportional to o P , as one of the physical amplitudes (after Gordon decomposition). It could in principle contain o a part which has to contribute to the physical form factors and therefore cannot be simply omitted together with the full structure. This becomes evident in the particular situation where p@"p. In this case, due to the identity uN (p)c u(p)"uN up /M, we get uN uL uP /u ) p"uN uP /M, i.e., a non-zero k k o o u-independent contribution only. It indicates that this part exists also when p@ O p. In order to construct the appropriate structure, we demand that it is orthogonal to the physical contribution given in Eq. (6.16). Since there are only two physical contributions, any other contribution proportional to P and orthogonal to them is completely independent and therefore o non-physical. Thus, eliminating this structure from the vertex, we shall not loose any physical contribution to the form factors. For the two structures uN @ºku and uN @»lu, we define here the orthogonality by the following condition: ¹r[(pL @#M)ºo(pL #M)»M ]"0 (6.19) o with »M "c »sc . This definition is motivated by the procedure, detailed in the next subsection, to o 0 o 0 calculate the form factors. The only alternative to the structure uN @uL u/u ) p is the term uN @(uL /u ) p!a)u with arbitrary a, since all the possible scalar structures sandwiched with the spinors are proportional either to uN @uL u or to uN @u. The value of a is then determined by the orthogonality conditions. This gives: a"4M/P2"1/(1#g)M where g"Q2/4M2 .

(6.20)

Indeed, for this value of a, the following two orthogonality conditions are simultaneously satisfied:

C C

A

¹r (pL @#M)co(pL #M)

BD

4M uL ! P2 u )p

A

P "0 , o

BD

4M uL ! ¹r (pL @#M)ipolq (pL #M) l P2 u )p

P "0 . o

(6.21)

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The reason why a unique value of a satisfies two conditions is a consequence of the fact that the expression (uL /u ) p!1/((1#g)M)), being substituted in Eq. (6.21), is proportional to eL , where e is the following four-vector:

A

B

JP2 u)p e" u!2P , P2 2u ) p

(6.22)

orthogonal to all available four momenta: e ) p"e ) p@"e ) q"0 .

(6.23)

For convenience, we shall normalize e by the condition e2"!1. Note also that the term uN @(uL /u ) p!1/((1#g)M))u is proportional to uN @eL u. The two last structures in Eq. (6.18) are not orthogonal to c and ip ql. However, they are o ol proportional to u , and, hence, do not contribute to the u-independent structures associated with o F , F and vice versa. In this respect, the situation does not differ from the spin 0 and 1 cases. We 1 2 emphasize that in this way the structures proportional to u /u ) p and P are separated by different o o formal criterions. Another way to separate the physical and non-physical structures is the following. One can simply subtract the two physical structures with indefinite coefficients from uN @uL u P /(u ) p) and o then multiplying at the left by u@ and at the right by uN and taking sum over polarizations get a 4 ] 4-matrix M "(pL @#M)º (pL #M). This matrix can be decomposed in the full set of o o 4 ] 4-matrices, similar to Eq. (5.21): M "A #B c #C ck#D ic ck#E ipkl . o o o 5 ok ok 5 okl Now the matrix M can be analyzed in terms of the tensors A , B , etc. We require them to be o o ok irreducible or proportional to u. The tensor A turns out to be proportional to P , and therefore is o o set to zero. The tensor B equals to zero. The trace of C is set to zero in order to construct an o ok irreducible tensor. The antisymmetric part of it is proportional to u. These two conditions fix the coefficients, and we obtain the final structure º which exactly coincides with o [uL /u ) p!1/((1#g)M)]P . With these coefficients, the tensor D turned out to be proportional o ok to u, and the tensor E is irreducible and proportional to u for the antisymmetric part. In this okl analysis, all the three non-physical structures in Eq. (6.18) are treated on an equal footing, since the tensors obtained after decomposition of the last two are proportional to u and therefore are o non-physical. It should also be mentioned that in the Breit system, where P"0, the term uN @eL u obtains a particularly transparent form. In this system, the spatial part of e is the unit vector directed along x: e"x/u "n, whereas e "0. Therefore, the term uN @eL u becomes proportional to n: 0 0 uN @eL u"!uN @(p@)n ) cu(p) , (6.24) with n orthogonal to both p and p@"!p. In an arbitrary system of reference one can find in uN @eL u an u-independent part. Omitting this full structure, we inevitably omit also this u-independent part. Its form depends on the criterions we used above (orthogonality and irreducibility). These criterions, strictly speaking, are however a matter of convention. They are imposed as natural and reasonable, but are not mathematically derived from some more general principles.

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The above consideration shows that we achieve (with the structure uN @eL uP ) optimal separation of o non-physical and physical contributions. Collecting the various possible contributions, we obtain the following expression for the spin 1 2 light-front electromagnetic amplitude7 [88]: JI "uN @CI u , o o

A

B

1 M M2 uL iF ! P #B u #B uL u . CI "F c # 2 p ql#B 1 u ) p (1#g)M o 2u ) p o 3(u ) p)2 o o 1 o 2M ol

(6.25)

The electromagnetic vertex (6.25) is gauge invariant since JI qo"0 (with u ) q"0). As mentioned o above, the possible non-gauge-invariant terms are forbidden by ¹-invariance. 6.3.2.1. The physical form factors. We can now express the physical form factors F and 1 F through the full vertex function CI . We multiply JI by [uN p{(p@)coup(p)]*, [uN p{(p@)ipolq /(2M)up(p)]*, l 2 o o etc. and sum over polarizations. We thus obtain the following quantities: c "¹r[O co] , c "¹r[O ipolq ]/(2M) , c "¹r[O (uL /u ) p!1/(1#g)M)]Po , 1 o 2 o l 3 o c "¹r[O ]uoM/u ) p , c "¹r[O uL ]uoM2/(u ) p)2 , (6.26) 4 o 5 o where O "(pL @#M)CI (pL #M)/(4M2) . (6.27) o o with the inhomogeneous part By this way, we get a linear system of five equations for F , F , B 1 2 1~3 determined by c . The way of calculating c through traces with the matrix O just corres1~5 1~5 o ponds to the operation used in the definition of the orthogonality condition (6.19). The values of c can be calculated in specific models for the structure of the composite system, as shown in the 1~5 next section. Solving the above-mentioned system of equations relative to F and F , we find 1 2 1 F " [(c #4c !2c )(1#g)#2(c #c )!2c (1#g)2] , (6.28) 1 4(1#g)2 3 4 1 1 2 5 1 F " [(c #4c !2c )(1#g)#2(c #c )!2(c #c )(1#g)2] . 2 4g(1#g)2 3 4 1 1 2 5 4

(6.29)

In spite of g in the denominator in Eq. (6.29) and below, there is no singularity at Q2"0. From Eq. (6.17), we can now easily obtain G and G : E M G "c /2 , (6.30) E 4 1 [(c #2c !2c )(1#g)#2(c #c )!2c (1#g)2] , (6.31) G " 4 1 1 2 5 M 4g(1#g) 3

7 A similar expression for CI was given in Ref. [85]. In this reference however, as well as in Ref. [89], the structure o uL P /u ) p was used instead of the structure proportional to B in Eq. (6.25). In this case, the form factor G (0) does not o 1 E coincide with the normalization integral.

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with c given in Eq. (6.26). These formulae are quite general in the light-front approach. They are 1~5 applicable to any model of the spin 1 fermion structure. 2 6.3.2.2. Comparison with other approaches. In the usual formulation of LFD on the plane t#z"0, the form factors of spin 1 systems are found from the plus-component of the current (see, 2 e.g., Refs. [89—92]), i.e., with our notation, from the contraction of JI in Eq. (6.25), with u . ¹his o o contraction eliminates the contributions of B , but does not eliminate the term with B . The form 2,3 1 factors F@ and F@ deduced in this way are thus given by 1 2 iF u)p JI ) u"uN @ F uL # 2 p uoql#2B uL ! u 1 1 2M ol (1#g)M

C C

D

A

iF@ ,uN @ F@ c # 2 p ql u uo . 1 o 2M ol

BD

(6.32)

where 2gB 2 1 , F@ "F # F@ "F # B . 1 1 1#g 2 2 1#g 1

(6.33)

Eq. (6.32) shows that the structure of JI ) u (or JI ) indeed coincides with the standard representa` tion of the nucleon electromagnetic vertex, Eq. (6.16). However, F@ and F@ in Eq. (6.32) are not the 1 2 physical form factors, but their superposition (6.33) with the non-physical contribution B . 1 The expression for B can be found from the above-mentioned system of equations, leading to 1 1 B "! [(c #4c !2c )(1#g)#2(c #c )!4c (1#g)2] . (6.34) 1 4 1 1 2 5 8g(1#g) 3 Together with Eq. (6.33), one has for F@ and F@ : 1 2 F@ "c /2 , F@ "(c !c )/(2g) . (6.35) 1 5 2 5 4 From Eqs. (6.35) and (6.17) one can find the corresponding G@ and G@ : E M G@ "G , (6.36) E E G@ "G #2B "[c (1#g)!c ]/(2g) . (6.37) M M 1 5 4 The difference between the two approaches — the first one separating out all the u-dependent terms to define the physical form factors, and the second based on the component JI of the electromag` netic current and incorporating non-physical item with B — is in the magnetic form factor. This 1 difference is of relativistic origin and has to disappear for form factors of extremely non-relativistic systems (i.e. in the limit where the average internal momenta go to zero) at finite Q2. It also disappears in an exact calculation leading to all B "0. We shall estimate this difference in a simple i quark model for the nucleon structure in the next section. 6.3.2.3. Off-energy shell effects in electromagnetic form factors. Though all the four-momenta are on the mass shell, this does not mean that there is no off-shell effects in the form factors at all. If, for

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295

Fig. 20. Off-energy shell form factor.

example, the nucleon form factors are used in the calculation of the deuteron form factors, they are part of a larger diagram and contain external spurion lines. They are off-energy shell. Such a form factor is shown in Fig. 20. In the standard formulation of LFD this corresponds to the fact that the minus-component of the nucleon momenta and the photon momentum are not related by the conservation law. Hence, the off-energy shell form factor can depend on the minus-component of the momenta, in addition to its Q2-dependence. In the covariant formulation of LFD the graph for the off-energy shell form factor contains external spurion lines. The off-shell effects are parametrized in terms of the corresponding spurion four-momenta. Besides Q2, the off-shell form factor F depends therefore on two variables [43]: F"F(Q2, s, s@)

where s"(p!uq)2, s@"(p@!uq@)2 .

(6.38)

Since u ) q"0, no other independent scalar products can be constructed. The spin structure of the off-energy shell electromagnetic vertex differs from Eq. (6.25) by three extra items:

A

B

iF 1 M M2 uL C0&&"F c # 2 p ql#B ! P #B u #B uL u o 1 o 2M ol 1 u ) p (1#g)M o 2u ) p o 3(u ) p)2 o #B

(s@!s) (s@!s) i(s@!s) q #B uL q #B p ul , 4 M3 o 5M2(u ) p) o 62M(u ) p) ol

(6.39)

where all the form factors depend on Q2, s, s@ and are symmetric relative to the permutation s Q s@. The last three spin structures have been already mentioned after Eq. (6.18). Their opposite ¹-parity is corrected in Eq. (6.39) by the factor (s@!s), changing the sign due to permutation of initial and final momenta. We emphasize that, in contrast to Eq. (6.25), the u-dependent items in Eq. (6.39) appear not because of approximation, but due to the off-energy shell continuation. Of course, in an approximate calculation of the off-energy shell form factors these item appear due to both reasons. The form factors B in an exact electromagnetic amplitude turn into zero on the energy shell 1~3 q"q@"0 P s"s@"M2. The off-shell electromagnetic vertex of a spin-zero particle differs from Eq. (6.9) by the extra term proportional to (s@!s)q /M2. o The extra form factors and the dynamical dependence of all the form factors on the off-shell variables s, s@ are usually unknown and are neglected in almost all calculations.

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6.3.3. Spin 1 system The electromagnetic form factors of a system with spin 1 (the deuteron, for instance) are defined by the following decomposition [93] of the electromagnetic vertex:

G C

D

H

qkql Sj@DJ DjT"e*j{(p@) P F (q2)gkl#F (q2) #G (q2)(gkql!gl qk) ej(p) o k o 1 2 2M2 1 o o l (6.40) ,e*j{(p@)¹klej(p) . k o l Here ej(p) is the deuteron polarization vector, p and p@ are the initial and final deuteron momenta, k j and j@ are the corresponding helicities, P"p#p@ and q"p@!p. Charge (F ), magnetic (F ) and C M quadrupole (F ) form factors are expressed through F , F and G as follows: Q 1 2 1 2g F "!F ! [F #G !F (1#g)] , (6.41) C 1 1 1 2 3 F "G , (6.42) M 1 F "!F !G #F (1#g) , (6.43) Q 1 1 2 where g"Q2/4M2. In the absence of polarization, the cross section of electron—deuteron scattering has the form:

A B

dp dp " (6.44) (A(q2)#tan2 1h B(q2)) , 2 dX dX 0 where h is the electron scattering angle in the laboratory system, and (dp/dX) is the Mott cross 0 section:

A B

dp a2 cos2(h/2) . " dX 4E2 sin4(h/2) 0 The functions A(q2), B(q2) are expressed through the form factors as follows:

(6.45)

A(q2)"F2(q2)#8g2F2 (q2)#2gF2 (q2) , (6.46) C 9 Q 3 M B(q2)"4g(1#g)F2 (q2) . (6.47) 3 M Unpolarized experiments are able to separate A and B, but do not allow one to extract all three form factors. To do this, one should measure a polarization observable. Due to its spin one, the deuteron can have, besides vector polarization, quadrupole polarization. The corresponding observable is ¹ given by: 20 ¹ Dq2D [A#B tan2 1 h] 20 2 1 8 8 1 1 "! gF F # g2F2 # g 1#2(1#g)tan2 h F2 . (6.48) C Q Q M 9 3 2 3 J2

C

A

B D

The deuteron electromagnetic properties are reviewed in Ref. [94].

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Decomposition (6.40) is valid for the full electromagnetic current operator. Considering the one-body current J0, as indicated in Fig. 18, we get the following analytical expression: o * Sj@DJ0DjT"e j{(p@)JI klej(p) , o k o l m h(u ) k )h(u ) k )h(u ) k@ ) d3k 1, 1 2 2 JI kl" ¹r[/{k(kK @ #m)C (kK #m)/l(kK !m)] (6.49) o 2 o 2 1 (2p)3 2e 1 (1!u ) k /u ) p)2 k 1 where C is the electromagnetic vertex of the nucleon given by Eq. (6.16) (with M replaced by m). o The matrix /k is given by Eq. (5.2), while the matrix /@l is obtained from / by the replacement l k Pk@ . This expression is derived using the graph technique presented in Section 2. It is similar to 2 2 the calculation indicated in Section 6.3.1 for spinless particles. The difference with Eqs. (6.11) and (6.12) is due to the non-zero spins. We explain below, how Eq. (6.49) is derived. According to the remark in the end of Section 2.2.3, we start in the graph 18 from the outgoing deuteron line (i.e., from the vertex NN P d@), follow the line in the direction opposite to the orientation of any nucleon line and reach its end, i.e., the vertex d P NN. Then we start again from the vertex NN P d, and follow another nucleon line, also in the direction opposite to its orientation. We thus obtain:

P

e*j{(p@)M[c /@kº c ]s(kK @ #m)C (kK #m)/lº N ej(p) MkK #mN . (6.50) k 0 # 0 2 o 2 # ba l 1 ba The first matrix element M 2N in Eq. (6.50) corresponds to the upper nucleon line in the diagram, ba the second one to the lower line. We attach the deuteron wave function to the upper nucleon line. The first row contains the factor [c /@kº c ]s which originates from the complex conjugated final 0 # 0 wave function of the deuteron: [ej{(p@)uN (k@ )/{kº uN (k )]*"e*j{(p@)u(k )[c /@kº c ]su(k@ ) . k 2 # 1 k 1 0 # 0 2 We keep in Eq. (6.50) the matrix indices a and b explicitly. Since both nucleon lines are equivalent, the order of indices b, a is the same. This means that one of the factors in Eq. (6.50) (we take the second one) is the transposed matrix. With the wave function (5.2) we get: [c /@kº c ]s"!º /@k 0 # 0 # and, hence, obtain the factor !º (kK #m)5º "(kK !m) , # 1 # 1 that gives the trace in Eq. (6.49). Like in the case of a spinless system, the light-front vertex (6.49) depends on the four-vector u. Its expansion in terms of form factors contains therefore extra terms. This expansion has the general form [85]: 1 * e j{(p@)JI klej(p) where JI kl"¹kl#Bkl(u) , Sj@DJI DjT" o l o o o o 2u ) p k

(6.51)

where ¹kl has the structure given by Eq. (6.40) and is determined by the physical form factors, o F ,F and G . We recall that tilde is used to distinguish approximate amplitudes and matrix 1 2 1

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elements, containing u-dependent non-physical contributions, from the u-independent exact amplitudes. The tensor Bkl contains the u-dependent terms: o M2 qkql ukul qkul!qluk Bkl" u B gkl#B #B M2 #B o 2 M2 3 (u ) p)2 4 2u ) p 2(u ) p) o 1

C

D

ukul qkul!qluk gkul#gl uk o #B P M2 #B P #B M2 o 5 o (u ) p)2 6 o 2u ) p 7 u )p #B q 8 o

qkul#qluk . 2u ) p

(6.52)

Here P"p#p@ and B ,2, B are invariant functions. The tensor Bkl as well as ¹kl is symmetrical 1 8 o o with respect to the simultaneous permutations p H p@ (q P !q) and k H l. The structures at B and B in Eq. (6.52) do not satisfy the conservation of current, that is qoJI j{j O 0. 7 8 o The total number of independent spin structures can be calculated without their explicit construction. Owing to the u-dependence, the helicity indices in the vertex are not constrained by the equality j"j #j (j is the photon helicity, j , j are the hadron helicities). Therefore we have 1 2 1 2 3 ] 3 ] 3"27 initial matrix elements. P-invariance reduces this number down to 14. ¹-invariance (or C-invariance in annihilation channel) eliminates in addition 4 structures and finally 10 independent structures remain. This calculation automatically takes into account the conservation of the electromagnetic current which would lead to a relation between B and B . However, 7 8 u-dependent terms in Eq. (6.52) do not necessarily satisfy this condition. The functions B and 7 B are therefore independent and we obtain 11 structures (or invariant form factors): three of them 8 are in ¹lk and the remaining eight ones are in Blk. p p From the point of view of the usual light-front approach with t#z"0, the dependence of the vertex (6.51) on the orientation of the light front means lack of relativistic covariance. Hence, formula (6.40), based on this covariance, is not any longer valid. Therefore, the number of independent matrix elements in Eq. (6.51) is not reduced to 3, but is larger. The increase of the number of matrix elements just corresponds to the increase of the number of independent spin structures in Eq. (6.52). In the formulation of the light-front with t#z"0, the contribution of these structures is not separated out from the physical form factors. In the covariant formulation, the dependence on the light front is parametrized explicitly by Eqs. (6.51) and (6.52). However, like in the spin 1 case, and in the absence of explicit evaluation of the u-dependent 2 terms, one can see some ambiguity in the procedure: what has to be considered as a non-physical contribution? Consider for example the term proportional to B : P e*j{(p@)u u ej(p)/(u ) p)2. Mul5 o k k l l tiplying it by e@, e* and summing over j@, j, we get:

A

B

uu u u p@ u up p@ p k l! k l ! k l # k l . (6.53) P + ej{(p@)e*j{(p@) k{ l{ ej (p)e*j(p)"P o k k{ l{ l o (u ) p)2 (u ) p)M2 (u ) p)M2 (u ) p)2 M4 j{,j It contains, apart from u-dependent contributions, the term p@ p /M4. This one cannot be removed k l from Eq. (6.53) and be included in the physical part, since it is required by the transversality of the tensor Eq. (6.53). The orthogonality condition, similar to the spin-1 case (obtained by the generaliz2 ation of Eq. (6.19)), results in the charge form factor F which does not coincide at q"0 with the C

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normalization integral, in contrast to the spin-1 case. We therefore do not impose this condition in 2 the spin-1 case, and define the non-physical structures as proportional to the first and higher degrees of u, including the terms where u is contracted with the polarization vector. After that, the physical form factors can be unambiguously separated from the spurious u-dependent contributions. This allows one to obtain the contribution to the physical form factors in terms of the electromagnetic amplitude in an explicit form by a generalization of formula (Eq. (6.13)). This approach is confirmed by comparison with the Bethe—Salpeter solution in the Wick—Cutkosky model (see Section 7.1.2). 6.3.3.1. Contributions to physical form factors. The explicit expressions for the form factors are obtained by solving Eq. (6.51) relative to F , F and G . The procedure is described in Ref. [85]. 1 2 1 The result is given below:

C

D

uo qq P u #P u uu l k#P2 k l F "JI kl g ! k l! k l , 1 o 2u ) p kl q2 2u ) p 4(u ) p)2

(6.54)

C

D

uo qq P u #P u uu q u !q u F l k#M2 k l ! k l l k , 2 "!JI kl g !2 k l! k l o 2(u ) p)q2 kl q2 2u ) p (u ) p)2 2u ) p 2M2

(6.55)

G

1 goq !goq gou #gou l k# k l l k G " JI kl 2 k l 1 4 o q2 u)p

C

D

uo q u !q u q P !q P uu P u #P u l k# k l l k#P2 k l ! k l l k # !P2 k l u)p 2(u ) p)q2 q2 2(u ) p)2 2(u ) p) #Po

C

D

H

uu q u #q u q u !q u l k . k l l k! k l !qo k l (u ) p)2 (u ) p)q2 (u ) p)q2

(6.56)

These expressions determine the electromagnetic form factors. In spite of the fact that u enters the r.h.s. of Eq. (6.54)—Eq. (6.56), these expressions do not depend on u. The form factor F , Eq. (6.54) which coincides at Q2"0 with F (0), does not coincide with the 1 C normalization integral (3.31). This is due to the fact that the normalization is obtained on the basis of the decomposition (3.29), where the non-physical structure proportional to E, contains gkl besides ukul. This is required by the irreducibility of this structure and is compatible with the angular condition. In the decomposition of the electromagnetic vertex, we have not separated out this term from the terms proportional to ukul, as we explained above. We believe that the concordance of F (0) with the normalization integral is beyond the accuracy of the current C calculations which omits all the Fock components except for the two-body one. In practical calculations of the deuteron elastic form factors, the difference is very small. 6.3.3.2. Comparison with other approaches. The deuteron electromagnetic form factors were already calculated in LFD in Refs. [95—101]. More generally, spin-1 form factors were considered in

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Refs. [85,86,102—104]. Using the exact decomposition (6.40), one can express the matrix elements Sj@DJ DjT , J in terms of the form factors: ` j{j J "!F #gF , (6.57) 11 1 2 J "!gF , (6.58) 1~1 2 J "!J2g(F !gF #G /2) , (6.59) 1 2 1 10 J "!(1!2g)F !2g2F #2gG . (6.60) 00 1 2 1 The four matrix elements in Eqs. (6.57), (6.58), (6.59) and (6.60) can be expressed through three form factors and, hence, are not independent from each other. They satisfy the relation [98]: (6.61) (1#2g)J #J !2J2gJ !J "0 , 10 00 11 1~1 which can be checked by direct substitution. This condition is called “angular condition” (do not confuse with the angular condition for the state vector discussed in Section 2.1.3). In principle, with the exact matrix elements J one can calculate any triplet of matrix elements j{j and find the form factors. It is not so however for the approximate ones JI . The following triplets j{j of the matrix elements were used in the literature: JI , JI , JI and JI , JI , JI in Ref. [98] 11 1~1 00 11 1~1 10 (the so called solutions A and B, respectively); JI , JI , JI in Ref. [102]. For example, taking the 1~1 10 00 triplet JI , JI , JI and solving the system of equations (6.57), (6.58) and (6.60) (with J P JI ) 11 1~1 00 j{j j{j relative to F , F , G , one finds 1 2 1 FGK~A"!JI !JI , 1 11 1~1 /g , (6.62) FGK~A"!JI 2 1~1 1 GGK~A"! [(1!2g)JI #JI !JI ] . 11 1~1 00 2g In Refs. [95,100] the form factors were expressed through different combinations of four matrix elements. The matrix elements in the r.h.s. of Eq. (6.62) calculated in LFD are, however, not given by the decomposition (6.40), but should be calculated according to Eqs. (6.51) and (6.52). The matrix elements of Eq. (6.51) have the form: JI "!F #gF , 1~1 1 2 JI "!gF , 1,~1 2 JI "!J2g(F !gF #G /2)#Jg/2B , (6.63) 1,0 1 2 1 6 JI "!(1!2g)F !2g2F #2gG !2gB #B #B . 0,0 1 2 1 6 5 7 The matrix elements JI , JI have the same form as J , J , whereas JI , JI differ from 11 1~1 11 1~1 10 00 J , J by the items containing the non-physical form factors B , B , B . Other non-physical form 10 00 5 6 7 factors B and B do not contribute to these matrix elements. 1~4 8

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The matrix elements JI do not satisfy condition (6.61). Substituting JI in Eq. (6.61) instead of j{j j{j J , we get j{j D , (1#2g)JI #JI !2J2gJI !JI "!(B #B ) . (6.64) 11 1~1 10 00 5 7 This function was calculated for the deuteron in Ref. [98] and for o-meson in Refs. [90,103]. Note that D in Eq. (6.64) should not be confused with the perpendicular component of the momentum transfer q. Substitution in Eq. (6.62) of the matrix elements JI from Eq. (6.63) gives j{j FGK~A"F , FGK~A"F , 1 1 2 2 1 (6.65) GGK~A"G !B ! D . 1 1 6 2g This solution gives F , F without any non-physical contribution (though still approximate), but 1 2 GGK~A contains the non-physical form factor B and, in addition, the function D"!(B #B ), 1 6 5 7 Eq. (6.64), responsible for the violation of condition (6.61). The different solutions were compared numerically in Ref. [103] (for the case of o-meson form factor) and analytically in Ref. [104]. The results are summarized in Table 1. Calculations by analytical formulae from this table exactly coincide with the numerical calculations [103]. From the Table 1, one can see that all the methods give superpositions of the physical form factors with the non-physical ones. Most of them contain the non-physical contribution D and differ from each other by its coefficient. This is so not only for the methods considered above, but also for any other method based on the JI component. If D"0, the form factors do not depend on ` the prescription, as it was pointed out in [90,103]. But G still contains B . 1 6 Note that the terms Bkl contribute to the matrix elements of the J(0) current component if at least o ` one of the projections j, j@ equals to zero. In Ref. [97] it was proposed to avoid using the zero Table 1 The set of form factors F , F , G calculated in six versions discussed in Section 6.3.3 1 2 1 No.

Version

Form factor F

Form factor F

Form factor G

1 2

GK-B [98] GK-A [98]

F 1 F 1

F 2 F 2

3

BH [102]

F

4

D

1 F # D 1 1#2g F #D 1

G !B 1 6 1 G !B ! D 1 6 2g 2 G !B ! D 1 6 1#2g G !B 1 6

5

CCKP [95]

6

FFS [100]

1

1 F # D 1 2(1#g) F 1

2

2

1 F # D 2 g F 2 1 F ! D 2 2(1#g)2

1

1 G !B ! D 1 6 1#g The same as for CCKP

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projections of j, j@, but consider the matrix elements with j, j@"$1 of the current components J(0) and J(0)"MJ(0), J(0)N. Taking into account that u "0 for u"(1, 0, 0,!1), one gets ` M x y ` * * e j{(p@)Blkej(p)"0, e j{(p@)Blkej(p)"0 for j, j@"$1 . l ` k l M k We see that this method indeed gives the solution for the form factors, in which the non-physical contributions vanish entirely. In both methods the complete separation of the physical form factors from non-physical ones is realized at the expense of incorporating, besides JI , other components of the current. A similar ` situation takes place with the form factors of a spin 1 system [88]. These other components can 2 receive contributions not contained in JI (for example from the instantaneous part of the fermion ` propagators, which do not contribute to JI due to c2 "0). The instantaneous contributions ` ` (contact terms) have been already taken into account in the deuteron electrodisintegration [72]. As we shall see in the following section, these contributions have a very important physical interpretation in terms of meson exchange currents in the non-relativistic approach. They can therefore be interpreted also as a two-body current. Considering the JI -components only, these ` contributions would be lost. This also shows that the consideration of other components of the current, beside JI , is inevitable if one wants to keep all physical contributions, and only these, to ` the form factors. 6.3.4. The transitions 1`—0` and 1~—0~ The pseudo-vector—scalar transition 1`—0` corresponds to the deuteron electrodisintegration amplitude near threshold (a few MeV in the center-of-mass system of the np pair). In this case, the final state is dominated by the singlet S-wave and the amplitude does not depend on the direction of the relative np-momentum. The amplitude of the vector—pseudoscalar transition 1~—0~, corresponds, for example, to the o!p transition, and has the same structure. For definiteness we will discuss here the deuteron electrodisintegration. In the impulse approximation, LFD was applied to the deuteron electrodisintegration amplitude in Ref. [105] and, in its explicitly covariant form, in Ref. [106]. The corresponding amplitude can be written as: M

4pa " uN (k@ )cou(k )ej(p)Fks(1S ) , 2?3 e e k o 0 q2

(6.66)

where F describes the vertex d#c* P 1S . The function s(1S ) is the spin function of the final ko 0 0 nucleons in the 1S -state we discussed in Section 5.3. The vertex F can be represented as follows: 0 ko 1 F " e qlPcA , (6.67) ko 4m2 oklc where P"p#p@, p@"p #p and A is a dimensionless scalar function: A"A(Q2, l). Tensor 1 2 (6.67) is the only one which has the appropriate properties expected from covariance, transversality and parity conservation to describe the vertex dc* P np(1S ). Let us emphasize that vertex (6.67) 0 automatically satisfies the transversality condition F qo"0. In this particular case, the Lorentz ko invariance leads to gauge invariance for the amplitude. This is a consequence of the particular

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1S final state. The cross-section of inclusive electron—nucleus scattering is usually represented in 0 terms of the structure fuctions ¼ and ¼ (see, e.g., Ref. [107]): 1 2

A BA

AB B

dp h dp " ¼ #2tan2 ¼ . 2 1 dX 2 dX dE@ e 0

(6.68)

The structure functions ¼ and ¼ can be expressed through the unique scalar amplitude 2 1 A [106]: p*Q2 ¼ " DAD2 , 2 3 ) 26p2m4

(6.69)

¼ "(1#l2/Q2)¼ . 1 2

(6.70)

Here p* is the c.m.-momentum of the final nucleons. All the dynamical information about the process is therefore contained in the transition form factor A. We stress that this result follows only from the assumption that the final 1S state dominates and is valid independently of the presence or 0 absence of final state interaction, meson exchange currents and isobar configurations. Their contribution only influences the amplitude A. The particular amplitude for the vertex dc* P np(1S ) in any approximate calculation, denoted 0 by FI , does not coincide, however, with F in Eq. (6.67). The tensor FI depends in that case on u. ko ko ko Following the arguments presented in Section 6.2, we can decompose FI on the general invariant ko amplitudes. It has the form: 1 FI " e qlpcA#e qlucB #e plucB ko oklc 1 oklc 2 2m2 oklc 1 #(» q #» q )B #(» u #» u )B # (» p #» p )B , k o o k 3 k o o k 4 2m2u ) p k o o k 5

(6.71)

where » "e ua qb pc. Decomposition (6.71) contains the symmetrical structures in front of the k kabc functions B . Corresponding antisymmetric terms (like » q !» q ) are not independent and 3,4,5 k o o k can be expressed through the first three contributions. We emphasize that FI depends on u even kl for the components of the wave function which do not depend on u. In the latter case, u enters through the rules of the graph technique. Similarly to the case of the deuteron form factors, decomposition (6.71) enables us to separate the u-independent parts from non-physical udependent ones so that one can unambiguously extract the physical form factors from the initial tensor FI . From Eq. (6.71), we immediately find ko m2 A"! ekolcq u FI . l c ko Q2 (u ) p)

(6.72)

This simple formula is the analogue of formulae (6.54)—(6.56) for the deuteron form factors. 6.3.4.1. Comparison with other approaches. The transition form factor A given by Eq. (6.72) cannot be reduced to the o"# component of the amplitude FI . For example, in the reference system ko

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where q "0, and q is parallel to the x-axis, Eq. (6.72) is reduced to 0 !m2q x(FI !FI ) . (6.73) A" y` `y Q2p ` This formula includes FI with o"y. `y The contribution of the plus-component is given, as usual, by the contraction FI uo. If FI would ko ko coincide with F in Eq. (6.67), i.e., would not contain any non-physical contributions, we indeed ko could find the form factor from the plus-component by the equation: 2m2 eaklcu q p (F uo) . A"! a l c ko Q2(u ) p)2 However, applying this equation to FI , we get ko 2m2 A@"! eaklcu q p (FI uo)"A!B . a l c ko 5 Q2(u ) p)2

(6.74)

(6.75)

Like in the cases considered above, we get a superposition of the physical form factor A and non-physical one B . Rewritten in terms of components, contraction (6.75) obtains the form: 5 !2m2q xFI A@" . (6.76) y` Q2p ` The difference with Eq. (6.72) is reduced to the replacement of FI in Eq. (6.76) by the antisymmety` ric combination (FI !FI )/2 in Eq. (6.73). This seems very natural since the physical amplitude y` `y (6.67) is antisymmetric relative to permutation of the indices o, k, and, hence, any symmetrical contribution is a spurious one and has to be excluded (independent of the fact that it depends on u or not). This is another reason to separate it. We emphasize that the difference between the amplitudes A and A@ is of relativistic origin and disappears in the non-relativistic limit. 6.3.5. The transitions 0~—1` and 0`—1~ The pseudo-scalar—pseudo-vector transition 0~—1` corresponds, for example, to p!A and to 1 K!K transitions. The scalar—vector transition amplitude 0`—1~ has the same form. 1 The general transition amplitude reads: (6.77) SPS(p@)DJ DP»(p, j)T"[F ((p ) q)gk!p qk)#F (q qk!q2gk)]ej(p) . o 1 o o 2 o o k It is determined by two form factors F and F . 1 2 In an approximate calculation, this amplitude should incorporate the u-dependent contributions. These are given by SPS(p@)DJI DP»(p, j)T"GI kej(p) , o o k where GI "F ((p ) q)g !p q )#F (q q !q2g ) ok 1 ok o k 2 o k ok # A@p (p@ !q (q ) p@)/q2)#Cp@ q #C@p q k o o k o k o # B q2u p@ /u ) p#B p@ u #B@ p u #B u q #B u u . 1 k o 2 k o 2 k o 3 k o 4 k o

(6.78)

(6.79)

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The terms A@, C@, B@ do not contribute to Eq. (6.4), since pkej(p)"0. They appear however before 2 k contraction with ej(p). The term Cp@ q appears, since an approximate amplitude is not gauge k k o invariant. It should be also separated. Its appearance has nothing to do with the u-dependence or LFD. In the Feynman approach its separation is equivalent to the construction, by hand, of a gauge invariant amplitude by the following subtraction: G P G !(G qo{)q /q2. ok ok o{k o One can construct other gauge non-invariant terms, proportional to p@ p ,q p and q p@ , but they k o k o k o are expressed through the structures included in Eq. (6.79). The term » » with » "e uaqbpc is k o k kabc also expressed through them. Solving Eq. (6.79) relative to F , F one gets 1 2 GI qkuo ukuo F " ok ! #(m2 !m2 !q2) , (6.80) 1 14 17 q2 (u ) p) 2(u ) p)2

C G

D

GI ukuo qouk F " ok [(m2 !m2 )2!q2(2m2 !q2)] #(m2 !m2 #q2) 2 (q2)2 14 17 14 17 14 2(u ) p)2 2(u ) p)

H

uoqk ukpo#uop@k # (m2 !m2 ) #q2 #qkqo!q2gko . 17 14 (u ) p) (u ) p)

(6.81)

Here m and m are the masses of pseudo-scalar and pseudo-vector mesons. 14 17 6.3.5.1. Comparison with other approaches. In Ref. [108] the form factors are found from the component o"#. Contracting Eq. (6.77) with u , we get o SPS(p@)DJI DP»(p, j)Tuo"[F ((p ) q)uk!(u ) p)qk)!F q2uk]ej(p) . (6.82) o 1 2 k Contracting Eq. (6.78) with u , we reproduce Eq. (6.82) with the new form factors F@ , F@ given by o 1 2 F@ "F , F@ "F !B . 1 1 2 2 1 Hence, the JI ` component provides F without any spurious contributions, while the form factor 1 F@ is determined by: 2 uk GI uo F@ "F !B " ok m2 !pk . (6.83) 2 2 1 q2(u ) p) 17u ) p

C

D

It includes the spurious contribution B . 1 7. Electromagnetic observables The formalism and methods presented above are applied to different physical systems and processes: pion, nucleon and deuteron electromagnetic form factors, o!p and ed!enp transitions. With standard assumptions and approximations, we easily reproduce results given in the literature. Some of these results, however, like the asymptotic behavior of the pion form factor and meson-exchange current contributions in ed!enp amplitude, obtain a new and clear physical interpretation in our approach, since they directly reflect the presence of extra relativistic components in the corresponding wave functions.

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7.1. Electromagnetic form factors in the Wick—Cutkosky model The Wick—Cutkosky model [59] allows to get the explicit form of both the Bethe—Salpeter functions and the light-front wave functions for states with angular momentum J"0 and J"1. This gives us the opportunity to compare the electromagnetic form factors calculated by two methods: by the standard one, with the use of the Feynman rules and through the Bethe—Salpeter amplitude, and by means of the light-front graph technique through the light-front wave function. This comparison is especially non-trivial in the J"1 case, where the physical form factors should be calculated after separating out the non-physical (u-dependent) amplitude from the physical one. 7.1.1. Spin 0 For a spin-0 system consisting of spinless constituents, the form factor is expressed through the light-front wave function by Eq. (6.14). Its expression through the Bethe—Salpeter function U(k, p) corresponding to the Feynman graph of Fig. 21 reads:

P

d4k (p#p@) F(t)"i (p#p@!2k) U(1p!k, p)U(1p@!k, p@)(m2!k2) . o o 2 2 (2p)4

(7.1)

The Bethe—Salpeter function is given by Eq. (3.66). Substituting here the spectral function (3.69), one can find the explicit form of the Bethe—Salpeter amplitude: U(k, p)"!ic[(m2!1 M2!k2) (m2!(1 p#k)2!i0) (m2!(1 p!k)2!i0)]~1 , 2 2 2

(7.2)

where c"25Jpmi5 with i"JmDe D"ma/2. " The analytical (in the asymptotical region) and numerical comparison of the form factors calculated through the light-front wave function, Eq. (6.14), and through the Bethe—Salpeter amplitude, Eq. (7.1), was carried out in Ref. [85]. The form factors calculated by these two approaches are compared in Table 2.

Fig. 21. Feynman electromagnetic vertex of a two-body system.

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307

Table 2 Electromagnetic form factor of a system with spin J"0 in the Wick—Cutkosky model for a"0.08 F(t) DtD/m2

BS

LF

NR

0 0.01 1 25 100 225

1.00E#00 5.14E!01 6.02E!04 1.01E!06 6.40E!08 1.28E!08

9.67E!01 4.98E!01 5.75E!04 9.76E!07 6.20E!08 1.24E!08

1.00E#00 5.17E!01 6.23E!05 1.05E!06 6.55E!08 1.29E!08

Note: Columns BS, LF and NR represent the results of the calculation using the Bethe—Salpeter amplitude Eq. (7.2), the light-front wave function Eq. (3.64) and non-relativistic wave function Eq. (3.62), respectively.

Both approaches give the same asymptotical behavior of the form factors at DtD<m2: F(t) +

C

A BD

a DtD 16a4m4 1# log 2p m2 t2

,

(7.3)

where a"g2/(16pm2), g is the coupling constant in the Wick—Cutkosky model. In contrast to Ref. [85] the LF form factor in Table 2 has not been renormalized at t"0. In the light-front approach, the form factor at low t, as well as the normalization, is smaller than the Bethe—Salpeter one by 4a/3p. For the normalization, the discrepancy has its origin in the contribution of the derivative of potential to the norm operator, as indicated in Eq. (3.50). For the form factor, this corresponds to omitted contributions indicated in Fig. 22 (the so-called recoil contributions). If corrections to the form factor and to the norm (which coincide with each other at Q2"0) are included, then the renormalization of the light-front result, as done in Ref. [85], is not required. At high t, the results obtained in LFD and in the Bethe—Salpeter approach are very similar. Non-relativistic calculation is also close to them, when (a/2p)log(DtD/m2);1. This is a peculiarity of the Wick—Cutkosky model. In the asymptotical region where (a/2p)log(DtD/m2)'1 LFD and Bethe—Salpeter form factors are still close to each other, while they differ from non-relativistic one. The asymptotics of the non-physical form factor B , which appears in the decomposition (6.9) 1 and is given by Eq. (6.15), has the form: B (t) + 32a4m4/t2 . (7.4) 1 In the intermediate region, B exceeds F: B (t)/F(t) + 2 while it is negligible in the non-relativistic 1 1 region. According to [85], B becomes comparable with the physical form factor F(t) at DtD & m2. 1 Therefore its separation is important. In the spin-0 case it is separated automatically, if one proceeds from the o"# component of the current. However, this is not so for the spin-1 case. 7.1.2. Spin 1 For a system with angular momentum J"1 made out from two spinless constituents, the light-front matrix elements JI kl of the current are obtained from the spin-0 case Eq. (6.12) with o

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Fig. 22. Amplitudes corresponding to admixtures of the three-body sector in the state vector. These amplitudes correspond to retardation effects.

the replacement of the spin-0 wave functions t@, t by the spin-1 wave functions t@ , t . In fourk l dimensional notations, t has the form: k uk (k !k )k 2 #u . (7.5) Uj(k , k , p, uq)"e(j)(p)tk, tk"u 1 2 1 2 k 1 u )p 2 The explicit form of the components u in Eq. (7.5) can be found from the comparison of Eq. (7.5) 1,2 with Eq. (3.65) in the reference frame where k #k "0 . (7.6) 1 2 Substituting JI lk in Eqs. (6.54), (6.55) and (6.56), in order to separate the non-physical contributions, o we find three form factors F , F , G . 1 2 1 The form factor G calculated in the two approaches in Ref. [86] is shown by the solid line in 1 Fig. 23. The two calculations give the curves indistinguishable from each other. In order to show the importance of a proper separation of the non-physical contribution we show also in Fig. 23 the form factor corresponding to the solution A from Ref. [98] (dashed line). It is given by Eq. (6.65) and contains the non-physical contributions D and B . It considerably differs from the correct form 6 factor. According to Ref. [86], this difference in the model is mainly due to D, whereas the B contribution is negligible. 6

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309

Fig. 23. The form factor G in the Wick—Cutkosky model. The solid line represents the calculation with the Be1 the—Salpeter function (7.6) and by Eq. (6.56) with the light-front wave function (3.65). Dashed line illustrates the form factor incorporating the non-physical contribution by Eq. (6.65) (solution A from Ref. [98]).

In the asymptotical region, these Bethe—Salpeter and light-front form factors are given by the following analytical expressions [85]:

C

A BD A BD

a6m6 a DtD F +! 1# log 1 DtD3 4p m2

C

a DtD 48a6m8 1# log F +! 2 4p m2 t4 a6m6 . G + 1 DtD3

,

(7.7)

,

(7.8)

(7.9)

We emphasize that, generally speaking, using two-body Bethe—Salpeter function does not correspond to two-body Fock component, but incorporates implicitly many-body sectors of the state vector. 7.2. Applications to the quark model We shall investigate in this section a few simple examples on the application to the quark model of the formalism developed in this review. We shall discuss in some details the pion wave function and form factors, and give some results for other observables.

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7.2.1. The pion form factor 7.2.1.1. Pion wave function. The spin structure of the pion wave function as a system consisting of quark and antiquark in the Jp"0~ state is identical to the np wave function in Jp"0` state, Eq. (5.33). The negative parity is generated automatically by opposite internal parities of the quark and antiquark and therefore does not change the spin structure. The pion wave function has thus the form:

C

D

1 1 uL t" uN (k ) A #A c v(k )"uN (k )Ov(k ) , 1m 2u ) p 5 1 2 1 J2 2

(7.10)

where O"(A /m#A uL /u ) p)c , and m is the quark mass. For simplicity, we do not take into 1 2 5 account isospin. The representation of this wave function in terms of the variables k and n is almost identical to Eq. (5.34):

A

B

ir ) [n ] k] 1 g w , t" ws g # 2 1 2 1 k J2

(7.11)

with the following relations between the invariant functions:

A

B

m e m g # g , A " kg . A "! 1 2 2 1 k k 2 2e k The normalization condition is a particular case of Eq. (3.23):

P

N " + t* 1 2t 1 2D , jj jj 2 j1j2 where

(7.13)

1 1 d3k 1 d2R dx d3k 1 " M D" " . (2p)3 (1!x)2e 1 (2p)3 e (2p)3 2x(1!x) k k Substituting in Eq. (7.13) the wave function represented by Eqs. (7.10) and (7.11), get:

PC

N " 2

(7.12)

D P

(m2#R2 ) M A2#4A A #4x(1!x)A2 D" [g2#(1!z2)g2]D , 1 2 2 1 2 m2x(1!x) 1

(7.14)

(7.15)

where z"n ) k/k. Obviously, in the case where the interaction potential between quarks is independent of M, as often assumed in phenomenological approaches, N should be equal to one. 2 7.2.1.2. Pion form factor. The diagrammatical representation of the pion electromagnetic current in the impulse approximation almost coincides with Fig. 18. For the interaction of a virtual photon with a quark, the horizontal line has to be replaced by the double line of antiquark. This current has the form:

P

1 JI cq" ¹r[!O@(kK @ #m)j (kK #m)O(m!kK )] D, o 2 o 2 1 (1!x)

(7.16)

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311

where O depends on the initial momenta, and O@ depends on the final momenta. The quark current j is taken as: o if j "f c # 2 p ql . (7.17) o 1 o 2m ol In this equation f and f are the quark form factors. The minus sign in ¹r[!O@2] is from the 1 2 fermion loop. We remind that both lines in the loop, single and double, corresponding to the quark and antiquark, are in the same direction. But since, according to the rules of the graph technique, they are followed in the same and opposite orientations respectively, one makes a loop when going through them. The pion form factor for the cq interaction is given by:

P

1 Fcq(Q2)" ¹r[!O@(kK @ #m)(u ) j)(kK #m)O(m!kK )] D. 2 2 1 2u ) p(1!x)

(7.18)

The pion form factor for the cqN interaction is found similarly:

P

1 D. FcqN (Q2)" ¹r[!O(m!kK )(u ) j)(m!kK @ )O@(m#kK )] 2 2 1 2u ) p(1!x)

(7.19)

The momenta k , k@ in Eq. (7.16) are the quark momenta, whereas they are the antiquark momenta 2 2 in Eq. (7.19). After a trivial generalization for the case of different masses of the quark and anti-quark, these formulas can be applied to the K-meson form factors. The final form factor is expressed through the momentum transfer D (with Q2"D2) and the integration variables R , x with the use of the M invariants indicated in Appendix D. Let us for a moment put A "0. We thus have: 2 D2 A A@ 1 1 D. (7.20) Fcq(Q2)"2 (m2#R !xR ) D) f ! xf M M 1 2 2 x(1!x)m2

PC

D

With A expressed through g by Eq. (7.12) the form factor Eq. (7.20) exactly coincides with the 1 1 expressions given in Ref. [109]. We emphasize that the calculation of Eq. (7.20), and more generally of any form factor or transition amplitude is, due to the covariance of our approach, a simple routine for analytical computer calculations. When A is different from zero, but for pointlike quarks, i.e., with f "1, f "0 in the current 2 1 2 (7.17), calculation of Eq. (7.18) gives

PC

Fcq(Q2)"

D

m2#R2 !xR ) D M M A A@ #2(A A@ #A@ A )#4x(1!x)A A@ D , 1 1 1 2 1 2 2 2 x(1!x)m2

(7.21)

where A@ "A (R !xD, x). We shall use this expression in the following to evaluate the 1,2 1,2 M asymptotical behavior of the pion form factor. Note that for the interaction of the virtual photon with the antiquark, in Eq. (7.19), we get FcqN "!Fcq. Hence, the full form factor of a system consisting of identical qqN is zero:

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Fcq(t)#FcqN (t)"0. We get opposite sign of FcqN automatically, without introducing the negative charge of the antiquark. This negative charge is automatically taken into account in this formalism. It can be obtained from the antiquark contribution, multiplying it by a sign factor depending on the process under consideration. For the o!p transition, the quark and antiquark contributions have the same sign. The transition amplitude FI can be obtained in this case from the p elastic amplitude J by replacing in ko o Eq. (7.16) the initial pion wave function O by the wave function of the vector meson / . The form of k this wave function coincides with the deuteron wave function (5.1). 7.2.1.3. Asymptotical behavior of the pion form factor. It is well known that QCD provides a 1/Q2 asymptotical behavior for the pion form factor [40]. As it was shown above, the pion wave function contains in general two components. The first one leads to the usual non-relativistic wave function, while the second one is of purely relativistic origin and u-dependent. We shall show in this section that the asymptotical 1/Q2 behavior of the pion form factor is entirely determined by this last u-dependent component of the pion wave function [110]. This piece is essential in order to explain the difference with the form factor of the J"0 state in the Wick—Cutkosky model, which asymptotical behavior is 1/Q4. The asymptotical behavior of the form factor can be related to the asymptotical behavior of the wave function. We shall calculate the latter in a qqN model with a one-gluon exchange kernel. The equation for the wave function corresponding to the diagram indicated in Fig. 11 has the form:

P

1 1 t(k , k , p, uq)"! d(4)(p#uq@!k@ !k@ )2(u ) p) dq@ 1 2 1 2 s!M2 (2p)3 ] uN (k )c (kK @ #m)h(u ) k@ )d(k@2!m2) d4k@ 2 k 2 2 2 2 ] O@(m!kK @ )h(u ) k@ )d(k@2!m2) d4k@ ckv(k ) 1 1 1 1 1 ] K(k@ , k@ , uq@; k , k , uq) . 1 2 1 2

(7.22)

Here M is the pion mass. We use the gluon propagator in the Feynman gauge and K is the scalar part of the gluon propagator (after separation of !g ). It coincides with the scalar t-channel kl exchange amplitude, given by Eq. (2.67) (where one should put k"0). The matrix O@ is the “inner” part of the pion wave function as defined in Eq. (7.10). Prime means that it depends on the momenta in the loop. For simplicity, we omit here any color degrees of freedom which are irrelevant for our consideration. To find the asymptotical behavior of the wave function, we shall use the iterative procedure already explained in Section 5 to find the deuteron wave function. We suppose that O@ is concentrated in a finite region of the quark momenta and the integral (7.22) is dominated by this domain. These momenta are negligible relative to the asymptotical ones we are interested in. This means that we can evaluate all the factors except for O@ and the delta-functions at zero relative quark momentum. This corresponds to R @ "0 and x@ "x@ "1. In that case, the quark fourM 1 2 2 momenta are equal to each other: k@ "k@ "(p#uq )/2 , 1 2 0

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313

where q "(4m2!M2)/(2u ) p). We thus get the following replacement in Eq. (7.22): 0 1 d(4)(p#uq@!k@ !k@ )2(u ) p) dq@ 1 2 (2p)3

P

] h(u ) k@ )d(k@2!m2) d4k@ O@h(u ) k@ )d(k@2!m2) d4k@ N O , (7.23) 2 2 2 1 1 1 0 where O is the integrated value of the wave function. 0 We substitute in Eq. (7.22) the one-component wave function obtained from Eq. (7.10) with A "0. Together with replacement (7.23), this gives in Eq. (7.22) O@"A c /m, where A is 2 0 5 0 a constant. The second component A of the wave function is then automatically generated by 2 Eq. (7.22). We thus find for the pion wave function: 2A 0 uN (k )(pL #uL q !4m)c v(k )K . t"! 0 5 1 s!M2 2

(7.24)

Substituting here p from the relation p#uq"k #k and using the Dirac equation, we finally 1 2 obtain 2A 0 uN (k )(uL (q!q )#2m)c v(k )K . t" 0 5 1 s!M2 2

(7.25)

The pion wave function (7.25) has the form of Eq. (7.10) with the following two components: 4m2 A K, A + A K A " 2 0 1 s!M2 0

(7.26)

with s"(R2 #m2)/x(1!x). The kernel K given in Eq. (2.67) can be rewritten in terms of the M variables R , x. For x 4 x@, it reads: M x@ x 2 x x@ 2 K(R@ , x@; R , x, M2)"g2 k2# 1! m2# R ! R@ M M x x@ x M x@ M

C

A B A

# (x@!x)

A BD

m2#R{2 M !M2 x@(1!x@)

B

~1 ,

(7.27)

with k"0. For x 5 x@K(R@ , x@; R , x, M2) is obtained from Eq. (7.27) by the replacement: M M x % x@, R % R@ . As indicated above, one should then put R@ "0, x@"1. M M M 2 We will see below that this wave function enters in the form factor at x"1. Then s + 4R2 . As 2 M follows from Eq. (7.27), the kernel K at R@ "0, x@"1, x"1 obtains the simple form: K"g2/R2 . M 2 2 M The components of the wave function which dominate in the asymptotical region are then given by A "g2m2A /R4 , A "g2A /R2 . (7.28) 1 0 M 2 0 M In this region, the component A decreases more slowly than A . 2 1 The form factor is now easily calculated through the components A , A according to Eq. (7.21). 1 2 The integral for the form factor is dominated by the region where one of the two wave functions (initial or final ones) takes its non-relativistic value, whereas the other wave function corresponds

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to the asymptotical one. This is equivalent to the standard calculation discussed in the literature in the usual formulation of LFD [40]. Hence, similarly to Eq. (7.22), we can replace A@ , A@ in 1 2 Eq. (7.21) by A@ "A d(2)(R@ )d(x!1/2) , A@ "0 . 1 0 M 2 and take the asymptotical values (7.28) for A and A . We thus find: 1 2 A A F(Q2)" 0(2A #A ) + 0A . 1 2 p3 p3 2 The substitution of A from Eq. (7.28) in Eq. (7.29) with R "D/2 and D2"Q2 gives 2 M 4g2A2 1 0 F(Q2) + . p3 Q2

(7.29)

(7.30)

From Eqs. (7.29) and (7.30) we see that the asymptotic behavior J1/Q2 of the pion form factor is indeed determined by the extra component A of the pion wave function, related to the 2 u-dependent spin structure. This originates from the slower decrease of A J 1/R2 in comparison 2 M to A J 1/R4 (see Eq. (7.28)). The enhancement of A is determined by the factor 1 M 2 q(u ) p)"(s!M2)/2+2R2 in Eq. (7.25). This extra factor originates from the part uq in the M conservation law p#uq"k #k used to derive Eq. (7.25) and therefore accompanies namely the 1 2 u-dependent spin structure. 7.2.2. The nucleon form factors As we have already seen in the case of the two-body systems, it is convenient to parametrize the covariant light-front wave function in terms of the following three sets of variables: (i) The four-momenta k of each constituent. In these variables, the wave function of any i three-quark system has the following form: t"t(k , k , k , p, uq; p , p , p , p) , (7.31) 1 2 3 1 2 3 where p is the nucleon spin projection and p are the quark spin indices, with the following 1,2,3 conservation law [5]: k #k #k "p#uq . (7.32) 1 2 3 (ii) The three-vector variables q which are identical, in the c.m.s., to the constituent momenta. i They are constructed similarly to the two-body case, Eq. (3.11): q "¸~1(P)k , (7.33) i i where P"p#uq"k #k #k and ¸~1(P) is the Lorentz boost operator defined in Eq. (3.11). 1 2 3 The sum of q gives zero, as it should be for the relative momenta: q #q #q "0. i 1 2 3 (iii) The four-vectors: R "k !x p, where x "u ) k /u ) p. They can be represented as i i i i i R "(R , R , R ), where R ) x"0, R is parallel to x. We thus have the following relations: i i0 iM i, iM i, R2 "!R2 , iM i R #R #R "0 , 1M 2M 3M x #x #x "1 . 1 2 3

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315

We do not pretend here to a calculation of the form factors in a realistic model for the nucleon. In order to illustrate our approach and estimate the effect of eliminating the unphysical B contribu1 tion from the physical form factors, we consider a simple model in which two neutral “quarks” labelled 2 and 3 are coupled to zero total angular momentum, so that the spin of the nucleon is determined by the charged “quark” labelled 1. We put the masses of all the quarks equal to each other. The corresponding wave function reads: t(k , k , k , p, uq; p, p , p , p )"[uN p2(k )c º uN p3(k )][uN p1(k )up(p)]t (k , k , k , p, uq) , (7.34) 1 2 3 1 2 3 2 5 # 3 1 0 1 2 3 where º "c c is the charge conjugation matrix. The spinor structure of Eq. (7.34) coincides with # 2 0 one of the invariants given in Ref. [111]. In order to show the method more clearly, we do not symmetrize the wave function Eq. (7.34) and do not introduce isospin. So, we have only one (charged) nucleon — “proton”. In Eq. (7.34) t is a scalar function. 0 We consider the charged quark 1 as a structureless particle coupled to the electromagnetic current: j "uN (k@ )c u(k ). The electromagnetic nucleon vertex Eq. (6.25) is shown in Fig. 24 and o 1 o 1 has the form:

P

c JI "uN @CI u" [uN (p@)(kK @ #m) o (kK #m)u(p)]¹r[(kK #m)(kK #m)]t t@ D(R , x ) . (7.35) o o 1 2 3 0 0 i i x 1 1 Here t and t@ depend on the initial and final momenta, respectively. Details of the calculation can 0 0 be found in Ref. [88]. At zero momentum transfer, the difference G@ (0)!G (0) obtains the simple form: M M G@ (0)!G (0)"2B (0)"!Sk 2T/3m2 . M M 1 Here Sk2T is the average quark momentum. The difference between G@ and G is of course of M M relativistic origin. We indicate in Fig. 25 the ratio R"G /G@ of the physical form factors G obtained with our M M M formalism and the form factor G@ which would be obtained with the usual procedure (using the M J component of the current), as discussed in Section 3.3. Both form factors are calculated with ` a simple harmonic oscillator radial wave function [88]. We recall that the electric form factor GI is E the same in both approaches.

Fig. 24. Nucleon electromagnetic amplitude in the three-quark model.

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Fig. 25. The ratio G /G@ as a function of Q2 in (GeV/c)2 [88]. M M

The difference between G and G@ is of relativistic origin. It is already sizable at Q2"0 for the M M magnetic moment (enhancement of about 15%). The two form factors largely differ at high momentum transfer, where corrections of the order of 50% are to be expected. The physical form factor G , i.e., the form factor free from any ambiguity from the position of the light front, is always M larger than G@ in this model. According to Fig. 25, the magnetic radius is larger for G than M M for G@ . M 7.3. Application to the nucleon—nucleon systems 7.3.1. Light-front dynamics and meson-exchange currents 7.3.1.1. Non-relativistic phenomenology. The qualitative, and to a large extent also quantitative, understanding of electromagnetic observables in the few-body systems in terms of meson-exchange currents is now well established [6]. This is particularly transparent for isovector magnetic transitions like the deuteron electrodisintegration cross-section near threshold. These two-body currents appear in that case already at order 1/m, while they are of higher order for charge and isoscalar magnetic transitions. Last but not least, these currents are also required by the low-energy soft-pion theorems. In order to extend these non-relativistic analyses to a wide range of momenta, let say for Q2'M2, a relativistic framework is necessary. In order to be fully consistent, the relativistic analyses have to give, as a limiting case, the well-known results in terms of meson-exchange currents. While genuine two-body currents like the mesonic contribution indicated in Fig. 26 has to be considered explicitly in both LFD and non-relativistic formalisms, some of them may appear differently depending on the underlying framework. Physical amplitudes should, however, be the same.

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317

Fig. 26. Two-body meson exchange current: the mesonic current. Fig. 27. Two-body meson exchange current: the pair term.

This is the case for the so-called pair term which, in the non-relativistic approach, is indicated on Fig. 27 (we assume here that the pNN vertex is of the pseudo-scalar type, we shall come back to this point at the end of this section). This contribution cannot survive on the light front because of the absence of vacuum fluctuation diagrams, together with the particular constraint u ) q"0. A careful analysis of the deuteron electrodisintegration amplitude near threshold in LFD is therefore of particular interest. 7.3.1.2. Relativistic impulse approximation. The vertex d#c* P np(1S ) is represented graphically 0 in Fig. 28. It corresponds to the following amplitude: FI "c ko

P

¹r[c (kK @ #m)CV(kK #m)/ (kK !m)] d3k 1, 5 2 o 2 k 1 u & e (1!u ) k /u ) p)2 k1 1

(7.36)

where c"i2~11@2p~3Jm, / is the deuteron wave function Eq. (5.2), u is the final state wave k & function and C is the nucleon electromagnetic vertex Eq. (6.16). The superscript V refers to the o isovector part of the form factors which is the only one which contributes here. In the plane wave approximation, we simply have FI "iJ2m ¹r[c (kK #m)CV(kK #m)/ (kK !m)] . (7.37) ko 5 & o 2 k 1 The expression of the electromagnetic current used here involves the Pauli and Dirac form factors F and F . They differ off-energy-shell from the Sachs form factors used in Ref. [106]. The 1 2 present one for its F part is consistent with the minimal substitution in the Dirac equation. Sachs 1 form factors may be used, but extra terms have then to be introduced in the current to get the exact result.

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Fig. 28. The electromagnetic vertex c*d P np(1S ). 0

The amplitude A, determining the cross section Eq. (6.68) is found by Eq. (6.72). In the plane wave approximation and for zero c.m.-energy in the final state, the amplitude A calculated with the deuteron wave function Eq. (5.2) has the form: m5@223@2 (2GV u !GVu ) A" M 2 E 5 e k

(7.38)

with k"D/2. We introduce in Eq. (7.38) the charge and magnetic nucleon form factors according to Eq. (6.17). The deuteron components u do not contribute to Eq. (7.38). Expressing u , 1,3,4,6 2 u through f , f , f by Eqs. (C.4) and (C.7) (and neglecting the small contributions from f and f ) 5 1 2 5 3 4 we get

G

H

1 J3m f (D/2) , A"2m3@2 GV [u (D/2)# u (D/2)]!GV M S D E D 5 J2

(7.39)

where D"JQ2. Neglecting f in Eq. (7.39), we recover the usual expression in the plane wave 5 approximation. 7.3.1.3. Meson exchange currents and 1/m expansion. In order to make the connection with the non-relativistic phenomenology, it is instructive to calculate the f component in leading 5 order in a 1/m expansion. Following the iterative procedure used in Refs. [48,71], and recalled in Section 5.4, f component has the following form in momentum space and in 1/m 5 order [72]:

P

d3k@ 3g2 J2 (k2!k@2) f [k ] n] /k"! pNN 5 i (k2#i2) k2#(k!k@)2 8J3p2m2 ]

C

A

B D

1 3k@k@ i j!d u (k@) [(k!k@)]n] . u (k@)d ! S ij ij D j 2 k @2 J2 1

(7.40)

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In the limit where the momentum k@ is restricted by the integration domain to low momenta, one can take (k2!k@2)/(k2!i2) K 1. We shall come back to this approximation later on. The coordinate space integral for the f component has the form: 5

P

C

D

3g2 1 =exp(!kr) pNN f (k)"! (kr#1) j (kr) u(r)# w(r) dr , 5 1 r 8J3pm2 0 J2

(7.41)

where u and w are the S and D state deuteron wave functions in coordinate space. The extra component g of the final state wave function can be calculated simply from isospin arguments, 2 leading to the replacement !3g2 P !(3#1)g2 "!4g2 in the amplitude. pNN pNN pNN The amplitude A has thus the following form, in coordinate space:

G PC A B P C

A"4p1@2m3@2 GV M

=

A BD DA BH

rD rD 1 u(r)j ! w(r)j 0 2 2 2 J2 0

r dr

G Vg2 1 =exp(!kr) rD # E pNN (kr#1) u(r)# w(r) j dr . 1 2 mD r J2 0

(7.42)

The structure of the second term is that of a two-body contribution of order g2 /4p. It is similar pNN to the one given by the pair term in the non-relativistic MEC analyses [6,62,112,113]. However, its strength is smaller by a factor of 2. 7.3.1.4. Contribution from the contact term. The one-to-one correspondence with the non-relativistic phenomenology in terms of MEC is indeed recovered if one collects all the contributions of order g2 /4p to the light-front amplitude. The missing contribution originates in this partipNN cular example from the contact term which appears once we have singled out the one-pion exchange mechanism in generating the f (g ) component of the deuteron (1S ) wave function. It is 5 2 0 indicated in Fig. 29. There are four diagrams corresponding to the contact terms. Two of them, Fig. 29a and Fig. 29b, correspond to the contact interaction at the left of the photon line. Two other diagrams, Fig. 29c and Fig. 29d, correspond to the right contact term with the two possible time orderings for the lower vertex. Each diagram can be calculated in terms of the kinematical invariants given in Section D.2 of Appendix D. In leading order in 1/m, the contact term contribution is exactly equal to the contribution from the f (g ) components, and therefore the total 5 2 amplitude of the deuteron electrodisintegration near threshold is strictly equivalent in LFD and in the non-relativistic phenomenology including meson-exchange currents (more precisely the pair term) [72]. This analysis can even be extended to the whole domain of momenta in Eq. (7.40), i.e., without neglecting the factor (k2!k@2)/(k2#i2) in the integrand. This term gives in fact a wrong asymptotic behavior of the wave function. As shown above in Section 4.3.2 and in Ref. [81], incorporation of the complete contact interaction which arises with the other components of the kernel (NN potential) like p and u exchange and of a higher-order iteration, gives a correction which turns the factor (k2!k@2)/(k2#i2) into 1. For the deuteron electrodisintegration, the dominant component u of the deuteron wave 5 function does not contribute to the amplitude corresponding to the contact term. For the elastic

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Fig. 29. Contact terms contribution to the electromagnetic interactions with the deuteron.

ed scattering with representation Eq. (6.16) for the nucleon electromagnetic vertex C , the contact o terms contribute to the deuteron electromagnetic form factor G only and do not contribute to 1 F and F . 1 2 7.3.1.5. The relevance of p exchange. The 1/m analysis of the relativistic contributions to the deuteron electrodisintegration amplitude is based on the p-exchange interaction with the pseudoscalar pNN vertex. It is also well known that, in leading 1/m order, the pair contribution associated with the pseudo-vector pNN coupling is zero. After minimal substitution in the pNN vertex, generating a genuine NNpc current, the deuteron electrodisintegration amplitude is however equivalent to the one given in the pseudo-scalar representation. In LFD, the equivalence between the two representations is realized in the following way. In the pseudo-vector representation with p-exchange only, the f component in the deuteron wave function is strictly zero in leading order. 5 This is due to the off-shell condition at the pNN vertex. The contribution from the light-front contact term is also of higher 1/m order for this representation. On the other hand, the current originating from the direct coupling of the photon to the pseudo-vector pNN vertex has its analog in LFD, providing the equivalence between relativistic and non-relativistic formulations in leading 1/m order. The accuracy of the 1/m approximation and the contribution of the pion exchange relative to all mesons are shown in Fig. 30. The solid curve in Fig. 30 is the ratio of f for the pion exchange only, 5 calculated in the 1/m approximation by Eq. (7.40), to the f , also for the pion exchange only, but 5

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Fig. 30. The solid curve is the ratio of f ’s in 1/m approximation and beyond, both for the pion exchange only. 5 The dashed curve is the ratio of f for pion only to f incorporating all the meson exchanges, both beyond 1/m 5 5 approximation.

calculated beyond the 1/m approximation. It shows the accuracy of the 1/m approximation, which is very good in the non-relativistic region of k, but changes the result by a large factor for higher k. The dashed curve is the ratio of f for pion only to f incorporating all the meson exchanges, both 5 5 beyond the 1/m approximation. The pion exchange contributes about 60% in the region k+0.4 GeV/c, where f dominates. 5 The cross section of deuteron electrodisintegration is shown in Fig. 31. As it was explained above, there is equivalence, in 1/m order, between the relativistic impulse approximation (including all components of the wave functions, together with the contact interaction), and non-relativistic calculations including the dominant contribution to meson-exchange current (the so-called pair term). The comparison between meson exchange contributions (pair term) and the light-front approach is shown in Fig. 31. The dotted line is the non-relativistic impulse approximation without meson exchange currents. The dashed line is the sum of the non-relativistic impulse approximation and of the contribution from the pair term for p-exchange only. The dot-dash line is the relativistic impulse approximation (with f and g ) together with the contribution from the contact term. The 5 2 relativistic components f , g and the contact term are calculated with p-exchange only and with 5 2

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Fig. 31. The cross section dp/dX dE@ of deuteron electrodisintegration cross-section in cm2/MeV/sr for h "155°. e e The dotted curve is the non-relativistic impulse approximation with the Bonn deuteron wave function [7] and with the Paris scattering state wave function [84]. The dashed curve includes the dominant contribution to meson exchange currents (the standard pair term), calculated with p-exchange only. The dot-dash curve is the relativistic impulse approximation with the light-front deuteron wave function ( f , f , f ) and the scattering state one (g , g ) and 1 2 5 1 2 includes the contact term. The extra components f , g and the contact term have been calculated with pion exchange 5 2 only. It does not include the product f ]g and extra components of the wave functions in the calculation of the con5 2 tact term. The solid curve differs from the dot-dashed one by the contribution of all mesons both in f , g and in the 5 2 contact term.

the pseudo-scalar pNN vertex, according to the iterative procedure indicated in Section 5.4. To be consistent with this iterative procedure, we omit in the amplitude terms of higher order than g2 . The solid curve differs from the dot-dashed one by the contribution of all mesons both in pNN f , g and in the contact term. For the nucleon electromagnetic form factors the dipole fit [114] 5 2 was used. The deviation between the dotted and dashed lines shows the well known influence of mesonexchange currents due to the pion pair term. The deviation between the dashed and dot-dash lines indicates that the influence of relativistic effects to both the deuteron and the scattering state wave function, and to the electromagnetic operator (contact term) are important. The comparison

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between dash-dotted and solid curves shows that the contribution of other mesons, in addition to the pion, considerably influence the cross-section at large Q2. The relativistic calculations in Fig. 31 are higher than non-relativistics ones. This difference is related to a deviation of relativistic scattering state wave function g from non-relativistic one (see 1 Fig. 16). As seen in Fig. 16, the non-relativistic scattering state wave function in momentum space, in contrast to the non-relativistic one, changes sign. On the other hand, one can expect that the recoil-type diagram, indicated on Fig. 22, partially cancels the difference between both calculations. These calculations should therefore be considered as an estimate of relativistic effects and are not compared with experimental data. The difference between the non-relativistic impulse approximation (dotted curve) and the relativistic impulse approximation, calculated with the same wave functions (neglecting f and g ), 5 2 is negligible. This was already noticed in Refs. [105,106].

Fig. 32. The structure function A(Q2) for the deuteron. The dashed line is non-relativistic impulse approximation calculated with the Bonn wave function. The dot-dash curve is the relativistic impulse approximation with relativistic deuteron components f and f . The solid curve is the same as the dot-dash one, but including the f component (in first 1 2 5 degree only). The dotted curve incorporates, in addition, the contact term. The nucleon form factor were taken in the dipole form [114].

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7.3.2. The elastic deuteron form factors The deuteron form factors can be calculated according to Eqs. (6.54), (6.55) and (6.56) which express them through the tensor JI kl. This tensor is expressed through the deuteron wave function o by means of the rules of the graph technique given in Section 2. In the impulse approximation this expression is given by Eq. (6.49). It includes the full deuteron wave function /k, with its six components. We shall take into account below the dominant ones, f , f and f . The contribution of 1 2 5 the contact term to the tensor JI lk is obtained by calculating the amplitude corresponding to the o graph of Fig. 29. All relevant kinematical invariants are given in Appendix D. Like in the case of the electrodisintegration cross-section, the component f of the deuteron wave 5 function was calculated by an iterative procedure, keeping the first degree of the relativistic kernel (of the order of g2). Relativistic contributions of order g4, like terms in f 2, have been removed from 5 the calculation of the form factor. The influence of relativistic effects on the structure function A(Q2) is shown in Fig. 32. The dashed curve corresponds to the non-relativistic impulse approximation [115,116] with the S- and D-waves of the Bonn-QA wave function [7]. The dot-dash line is calculated in the light-front

Fig. 33. Same as Fig. 32 but for the B(Q2) form factor of the deuteron.

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formalism with relativistic deuteron components f and f (solid lines in Figs. 13 and 14), but 1 2 without f . The solid curve is similar to the dot-dash one, but includes the f component (in first 5 5 degree only). The dotted curve incorporates, in addition, the contact term (where the contributions of order g2 are taken into account only). Fig. 33 gives the B(Q2) deuteron form factor. One can see that the function B(Q2) is more sensitive to the different approximations and contributions than A(Q2). The comparison between the dot-dash and solid curves shows that the influence of the extra component f of the deuteron wave function is 5 important. The minimum of B(Q2) in this figure is the consequence of the f component. 5 Fig. 34 shows the influence of different approximations to ¹ . The effect of incorporating the 20 component f is qualitatively similar to that obtained when adding the contribution of the pair 5 current in non-relativistic calculations [117]. Fig. 35 illustrates the influence of B(Q2) of the higher-order terms than g2. Their importance indicates the need to carry out the calculations beyond the iterative procedure. The function A(Q2) is less sensitive to this contribution.

Fig. 34. The deuteron tensor polarization ¹ at h"70° calculated by relativistic formulas for form factors. The 20 designations of the curves are the same as in Figs. 32 and 33. Fig. 35. The structure function B(Q2). The solid curve is the same as in Fig. 33. The dashed curve incorporates the quadratic terms in f . 5

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The results presented in this section should be considered only as indicative. They can be improved in many respects. An exact solution of the equation for the deuteron and scattering state wave functions is indeed feasible. It implies the redefinition of the parameters of the NN kernel. However, we believe a realistic estimate of relativistic corrections to the deuteron can be performed along the lines presented in this section. It necessitates a careful choice of the NN potential to generate the kernel (energy-dependent OBE potential for instance), and a good starting point for the non-relativistic wave functions (with kinematical relativistic corrections indicated in Section 4.2). This may imply also the use of a PV pNN coupling to be consistent with chiral symmetry constraints in leading order. This necessitates also a careful treatment of second-order contributions, as discussed in Section 4.3.2. Finally, the calculation of electromagnetic observables should include retardation effects as indicated in Fig. 22.

8. Concluding remarks We have reviewed the general properties of the covariant formulation of LFD, and its application to relativistic few-body systems. We have first detailed the general transformation properties of the state vector, both kinematical and dynamical, and emphasized the most important features brought about by the covariant formulation of the state vector on the plane u ) x"0. The explicit covariance provides similar advantages over the standard formulation, as the Feynman graph technique in comparison to old fashioned perturbation theory. We have presented a covariant light-front graph technique to calculate physical amplitudes. The covariance enables us also to construct bound and scattering states of definite angular momentum, separating kinematical and dynamical aspects of this problem. The angular momentum of a system is a property of the wave function relative to transformation of the coordinate system and therefore has in our formalism a purely kinematical nature. Dynamics is involved in the composition of this angular momentum from the spins of its constituents. We illustrate our formalism with the most simple and adequate system, the two nucleon system, and construct explicitly the deuteron wave function and the 1S scattering state. In both cases, the 0 dominant relativistic components of the wave function originating from its dependence on the light front position (the variable n) has been found. We also extend our formalism to the p and nucleon wave function in terms of valence quarks and antiquarks. This formulation is essential in order to extract the physical amplitude for all electromagnetic transitions. This is mandatory in all practical calculations in order to avoid non-physical contributions coming from the definite choice of u. We show how this applies to the electromagnetic form factors of spin 0, 1 and 1 systems, as well as for the deuteron electrodisintegration. We show also in 2 this last example how one can recover easily, in a 1/m expansion, the well known results obtained in the non-relativistic phenomenology in terms of meson-exchange currents (the so-called pair term). The dominant n-dependent components of the wave functions are essential in order to reproduce these results. This shows that the calculations which omit this dependence cannot be considered as realistic. These developments have been made particularly simple, and intuitive, by the three-dimensional nature of the formalism, and the absence of vacuum fluctuations. This enables a transparent link with the non-relativistic framework developed over the last 20 yr in the microscopic structure of few-nucleon systems.

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We expect many new developments in the near future along the lines presented in this review, both in nuclear and particle physics. One of them concerns the particular way to implement the independence of various results on the light-front orientation. Some restricted examples (determination of the NN-interaction, normalization of the wave function, etc.) have been considered in this review to show how this can be done, but more extensive work is required. Another aspect, which is related, concerns the construction of the generators satisfying, in the truncated Fock space, the commutation relation of the Poincare´ group algebra, that also provides independence of observables on the light-front orientation. This feature of LFD — the dependence of approximate on-shell amplitudes on u, which has to be excluded from observables — has, however, a positive aspect. It gives a quantitative measure of the incompleteness of a given approximation. The model calculations in other approaches where this feature is absent are often still incomplete, but this does not manifest itself explicitly. In nuclear physics, the exact calculation of the two-body wave function is worth to be done. This, however, necessitates to reparametrize the NN potential starting from the same relativistic formalism. This is essential in order to have quantitative predictions for electromagnetic observables in the few GeV range. In particle physics, the structure of the pion and nucleon can be investigated easily in this framework, and electromagnetic form factors calculated in the highmomentum region. Application to virtual compton scattering can also be considered, as well as the extension to axial transitions. One other particularly interesting application is the structure of heavy quarkonium, like the J/t, in order to investigate the importance of relativistic dynamical corrections, i.e., the importance of higher Fock states in the J/t wave function. Last but not least, the application of the covariant formulation of LFD to field theory should be investigated. Acknowledgements One of the authors (V.A.K.) is sincerely grateful for the warm hospitality of the theory group at the Institut des Sciences Nucle´aires, Universite´ Joseph Fourier, in Grenoble and of Laboratoire de Physique Corpusculaire, Universite´ Blaise Pascal, in Clermont-Ferrand, where part of this work was performed. Appendix A. Notations We use the standard covariant/contravariant notation, with implicit summation over repeated indices, according to

xk"(t, x ),

x "g xl, k kl

A

1

0

0

0 !1 0 g " kl 0 0 !1 0

0

0

The Dirac matrices satisfy the anticommutation relation: c c #c c "2g . k l l k kl

0 0 0 !1

B

.

(A.1)

(A.2)

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The standard representation for the Dirac matrices and spinors is used:

A

c " 0

B

1

0

0

!1

A

c"

,

0

r

!r 0

B

A

, c5"!ic0c1c2c3"

0

!1

!1

0

B

,

(A.3)

where r are the usual Pauli matrices. Note the minus sign for the definition of c . 5 The tensor p is: kl p "(i/2)(c c !c c ) . kl k l l k

(A.4)

Throughout the review the following convention is taken: kK "k ck . k

(A.5)

The Dirac spinors up(k) and vp(k) for spin 1 fermion and antifermion respectively satisfy the 2 following Dirac equations: (kK !m)up(k)"uN p(k)(kK !m)"0 ,

(kK #m)vp(k)"vN p(k)(kK #m)"0 ,

(A.6)

with the normalization: uN p(k)up{(k)"2md "!vN p(k)vp{(k) . pp{

(A.7)

They are given by up(k)"Je #m k

A B

1 wp, r)k (e #m) k

r·k (e #m) wp . vp(k)"Je #m k k 1

A B

(A.8)

where wp are the two-component spinors and e "Jk2#m2. The summation over spin indices in k that case is + up(k)uN p(k)"(kK #m) , a b ab p

+ vp(k)vN p(k)"(kK !m) . a b ab p

(A.9)

The charge conjugation matrix is defined as: º "c c . # 2 0 We use the following properties: p rt"!rp ,º ct "!c º , where the superscript t means y y # k k # transposition. The polarization vector e(j)(p) of a spin 1 particle of mass m satisfies the following general k relations: pke(j)(p)"0 , k

e*j{(p)ej(p)"!d , k k j{j

pp + ej(p)e*j(p)"!g # l k . l k lk m2 j

(A.10)

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Appendix B. Relation to other techniques B.1. Relation to the Weinberg rules As we already mentioned, the approach considered in this report is a generalization of the usual LFD obtained by the replacement of the particular value of the light-front vector u"(1, 0, 0,!1) by the most general one u"(u , x) with the only constraint u2"0. Hence, the graph technique 0 developed in Section 2.2 is the direct generalization of the Weinberg rules [3]. These rules have been obtained from old fashioned perturbation theory in the infinite momentum frame. We shall show below for the case of a spinless particle how to transform the amplitudes calculated in the covariant light-front graph technique to the Weinberg amplitudes. This explicit transformation was found in Ref. [43]. Any diagram of the covariant light-front graph technique can be represented as a sequence of intermediate states, each of them containing, besides the particles, one spurion line and only one. An example of such contribution is indicated in Fig. 36. The expression for the corresponding amplitude has the form:

P

M" M M d(4)(+ p !+k #uq)d(4)(+p@!+k #uq) 1 2 i i i i dq ]
(B.1)

If a particle passes through a vertex without interaction (like in Fig. 18 the particle in the lower part of the diagram propagates without interaction in the upper vertex), the intermediate states can still be factorized. This is ensured by the following representation of the propagator:

P

(2) h(u ) k) d (k2!m2) d4k

P

" (2)(2u ) k )h(u ) k )d(k2!m2) d4k d(4)(k #uq!k ) dq h(u ) k )d(k2!m2) d4k , 1 1 1 1 1 2 2 2 2 (B.2)

which reduces the amplitude with the non-interacting particle to the form of Eq. (B.1).

Fig. 36. Amplitude with one intermediate state shown explicitly. Other intermediate states, if any, are absorbed by the amplitudes M and M . 1 2

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From the conservation equation +p #uq"+ k in Eq. (B.1), it follows that i i q"[(+ k )2!( + p )2]/(2u ) + p ) . i i j Integrating over dk we get 0j M M (!2u ) +p )d(4)(+p !+k #uq) dq d3k 1 2 i i i j. M"d(4)(+p !+p@) < (B.3) i i 2p[(+p )2!(+k )2#ie] 2e i i j j In order to transform expression (B.3) to a form analogous to old fashioned perturbation theory in the infinite momentum frame, we introduce the variables:

P

R%95"p !y +p , y "u ) p /(u ) +p ) , i i i j i i j R "k !x +p , x "u ) k /(u ) +p ) , (B.4) i i i j i i j satisfying +y "+ x "1 and R ) u"R%95 ) u"0 with R"(R , R , R ), where R ) x"0 and 0 M , M i i i i R is parallel to x. Taking into account that R2"!R2 , we can write the denominator of Eq. (B.3) , M into the form:

C

D

(R%95)2#m2 R2 #m2 iM i !+ iM i #ie . (B.5) y x i i This form coincides with the one given by the Weinberg rules [3]. The factor d3k /2e in these j j variables turns into d2R dx /2x , the domain of integration over dx being from 0 to 1. The j j j j expression +

(2u ) +p )d(4)(+p !+k #uq) dq i i i turns into 2d(2)(+R )d(+x !1) . iM i By this way, one can transform the expression of any intermediate state. The equivalence between the expression for any amplitude in the covariant formulation of light front dynamics and the one given by the Weinberg rules is thus realized. The vectors R play here the role of the momenta M transverse to the infinite momentum, and the variables x , y are analogous to the fractions of the i i particle momenta with respect to the infinite momentum. Note that the factorization formula Eq. (B.2) is generalized for the propagator of a spin 1 particle 2 as follows:

P

P

u)k 1h(u ) k )d(k2!m2)(kK #m) d4k (2)(kK #m)d(k2!m2) d4k" (2) 1 1 1 1 m

]d(4)(k #uq!k ) dq h(u ) k )d(k2!m2)(kK !uL q#m) d4k . (B.6) 1 2 2 2 2 2 The part !uL q in Eq. (B.6) corresponds to the instantaneous interaction which appears in light front dynamics, since the fermion line with the momentum k now “extends over a single time 2 interval” (see Section 2.2).

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B.2. Relation between the Feynman amplitudes and the Weinberg rules It is also instructive to derive the Weinberg rules directly from the Feynman diagrams. This allows one to avoid some subtle problems of light-front quantization. The direct relation between Feynman and the light-front amplitudes was shown for a few cases in Ref. [118] and investigated in detail in Ref. [119]. The problem of renormalization was investigated in Ref. [120]. We illustrate this relation by the simplest case of the triangle Feynman graph shown in Fig. 21. This example has been considered also in Ref. [121]. Up to a multiplicative factor, the corresponding Feynman amplitude for scalar particles of mass m has the form:

P

Fn"

d4k 1 . (2p)4 (k2!m2#ie)((k!p)2!m2#ie)((k!p@)2!m2#ie)

(B.7)

Introducing the new light front variables: k #k 3, k " 0 ` J2

k !k 3, k " 0 ~ J2

k "(k , k ) , M 1 2

(B.8)

we rewrite Eq. (B.7) in the form

P

P

dk d2k dk 1 ` MMn, Mn" ~ , (B.9) (2p)3 2p/ (k !H ) (k !H )(k !H ) 3 ~ 1 ~ 2 ~ 3 where / "8k (k !p )(k !p@ ) and 3 ` ` ` ` ` k2 #m2!ie H " M 1 2k ` (k !p )2#m2!ie M H "p # M 2 ~ 2(k !p ) ` ` (k !p@ )2#m2!ie M H "p@ # M . (B.10) 3 ~ 2(k !p@ ) ` ` Let p2"p@2"m2 and q2"(p!p@)2(0. For simplicity we put q "0, i.e., p "p@ . At k (0 ` ` ` ` and k 'p all the poles in k of the integrand in Eq. (B.9) are above or below the real axis. ` ` ~ Hence, the integral differs from zero 0(k (p only. Application of the residue theorem gives: ` ` i 1 . (B.11) Mn"! / (H !H )(H !H ) 3 1 2 1 3 Introducing x"k /p , we get ` ` i 1 d2k dx M Fn(t)"! , (B.12) (2p)3 k2 #m2 (k !xD)2#m2 2x(1!x)2 M M !m2 !m2 x(1!x) x(1!x) Fn"

P C

DC

D

where t"q2"!D2. The form Eq. (B.12) reproduces the amplitude given by the Weinberg rules.

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Fig. 37. Box Feynman graph. Fig. 38. Time-ordered box graph.

The next step is the box diagram of Fig. 37. This example is considered in Ref. [119]. The amplitude has the form:

P

d4k Fh" D(k , m)D(k , k)D(k , m)D(k , k) , 1 2 3 4 (2p)4

(B.13)

with D(k, m)"i/(k2!m2#ie) and k "k, k "k!l, k "p#q!k, k "k!p. Particles are 1 2 3 4 supposed to have mass m or k depending on their appearance in the s or t channel. This amplitude can be represented analogous to Eq. (B.9):

P

P

dk k2 dk 1 ` MMh , Mh" ~ , (B.14) (2p)3 2p/ <4 (k !H ) 4 i/1 ~ i where / "16k (k !l )(k !p !q )(k !p ) and H are analogous to Eq. (B.10): 4 ` ` ` ` ` ` ` ` 1~4 k2 #m2!ie H " M , 1 2k ` (k !l )2#k2!i e M H "l # M , (B.15) 2 ~ 2(k !l ) ` ` (k !p !q )2#m2!ie M M , H "p #q # M 3 ~ ~ 2(k !p !q ) ` ` ` (k !p )2#k2!ie M . H "p # M 4 ~ 2 (k !p ) ` ` ¸et p 'l . Then at k (0 and at k 'p #q the imaginary parts Im H have the same ` ` ` ` ` ` 1~4 signs and, hence, the integral for Mh is zero. At 0(k (l (p (p #q there is one pole in ` ` ` ` ` the variable k at the down half plane relative to the real axis and three poles are at upper half ~ plane. The residue reproduces the expression given by the Weinberg rules for the diagram analogous to Fig. 38, as in the case of the triangle graph indicated in Fig. 21. Fh"

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At k 'p 'l , but k (p #q , the amplitude corresponds to a diagram similar to Fig. 38, ` ` ` ` ` ` but up side down. Note that if the poles have imaginary parts of the same signs, the corresponding lines of the time-ordered graph have the same (clockwise or counter clockwise) directions. For the poles with opposite imaginary parts, the directions of lines are opposite. New feature appears at l (k (p , when two pairs of poles are at the opposite sides of the ` ` ` real axis. In this case, any given residue does not reproduce any time-ordered diagram. However, the sum of them can be transformed [119] to a form consisting of two other parts, each of them coinciding with the amplitudes given by the Weinberg rules for the diagrams analogous to those shown below in Figs. 40 and 41. We explain this below in more detail in the case of the covariant formulation of LFD. The general algorithm is developed in Ref. [119]. B.3. Relation between the Feynman amplitudes and the covariant light-front graph technique B.3.1. Spin 0 Like in the case of the Weinberg rules, the Feynman approach and the covariant light-front graph technique give different ways of calculating one and the same S-matrix and, hence, give one and the same on-shell amplitude. However, these amplitudes are represented in different forms. It is therefore instructive to transform explicitly the Feynman amplitudes to the form of covariant light-front amplitudes. We will explain this transformation in the simplest case of spinless particles for the box diagram indicated in Fig. 37. The analytical expression for this diagram is given by Eq. (B.13). For this transformation, we first represent the Feynman propagator in the following form [122]:

P

1 d(4)(uq#k!p) iD(k, m)"! " h(u ) p)d(p2!m2) d4p dq k2!m2#ie q!ie

P

#

d(4)(uq!k!p) h(u ) p)d(p2!m2) d4p dq . q!ie

Substituting Eq. (B.16) in Eq. (B.13), we get

P G

Fh"

K

d4k d(4)(uq #k!p ) 1 1 h(u ) p )d(p2!m2) 1 1 (2p)4 q !ie 1 /(1a)

K H K K H

d(4)(uq !k!p ) 1 1 h(u ) p )d(p2!m2) # d4p dq 1 1 1 1 q !ie 1 /(1b)

G

]

d(4)(uq #k!l!p ) 2 2 h(u ) p )d(p2!k2) 2 2 q !ie 2 /(2a)

d(4)(uq !k#l!p ) 2 2 h(u ) p )d(p2!k2) # d4p dq 2 2 2 2 q !ie 2 /(2b)

(B.16)

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G

]

K

d(4)(uq #k!p!q!p ) 3 3 h(u ) p )d(p2!m2) 3 3 q !ie 3 /(3a)

K H

d(4)(uq !k#p#q!p ) 3 3 h(u ) p )d(p2!m2) d4p dq # 3 3 3 3 q !ie 3 /(3b)

G

]

K

d(4)(uq #k!p!p ) 4 4 h(u ) p )d(p2!k2) 4 4 q !ie 4 /(4a)

K H

d(4)(uq !k#p!p ) 4 4 h(u ) p )d(p2!k2) d4p dq . # 4 4 4 4 q !ie 4 /(4b)

(B.17)

Let us first consider the product of the first items in the four braces in Eq. (B.17). Using schematic notations, we have:

P

P

d4k d(4)(uq #k!p ) d(4)(uq #k!l!p ) 1 1 2 2 (1a)(2a)(3a)(4a)2" (2p)4 q !ie q !ie 1 2 d(4)(uq #k!p!q!p ) d(4)(uq #k!p!p ) 3 3 4 4 (2) dq dq dq dq . ] 1 2 3 4 q !ie q !ie 3 4

(B.18)

The functions h(u ) p ), d(p2!k2) and all other factors in Eq. (B.18) are absorbed in (2). To i i exclude q from the numerator, we make the following change of variables: 1 k#uq "k@ , 1

q !q "q@ , 2 1 2

q !q "q@ , 3 1 3

q !q "q@ , 4 1 4

after which we find

P

P

(2) dq 1 (1a)(2a)(3a)(4a)2" "0 . (q !ie)(q@ #q !ie)(q@ #q !ie)(q@ #q !ie) 1 2 1 3 1 4 1

(B.19)

The numerator (2) in Eq. (B.19) does not depend on q . All the poles of the integrand in Eq. (B.19) 1 relative to q are above the real axis. Therefore integral Eq. (B.19) equals to zero. Similarly we get 1 :(1b)(2b)(3b)(4b)"0. We suppose that all the external momenta are on the mass shells and, for convenience, u ) p'u ) l. Expression (B.17) contains 16 items. However, only three of them differ from zero, namely, :(1a)(2b)(3b)(4b),:(1a)(2a)(3b)(4a) and :(1a)(2a)(3b)(4b). In all other items, except for the ones considered above :(1a)(2a)(3a)(4a) and :(1b)(2b)(3b)(4b), we get the products of the h-functions with mutually incompatible restrictions on the scalar products of u with four-momenta. Let assign to the items (a) the arrow counter-clockwise in the loop and to the items (b) — the clockwise arrow. Put on the external lines with the momenta p and q the arrows incoming to the box and to the lines with momenta l and p#q!l — the outcoming arrows. Then the items which disappear due to the h-functions correspond to the diagrams containing at least one vertex with all incoming or outcoming lines, which are interpreted as vacuum fluctuations. One of these graphs, corresponding to the product :(1b)(2a)(3b)(4b) is shown in Fig. 39. The vertex 2 in Fig. 39 corresponds to the

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Fig. 39. Time-ordered box graph with the vacuum vertex No. 2.

creation of particles from the vacuum. The items :(1a)(2a)(3a)(4a) and :(1b)(2b)(3b)(4b), which are zero, correspond to the loops with all the lines directed clockwise or counter clockwise. Now consider the item

P

(1a)(2b)(3b)(4b)2 "

P

P

P

P

P

d(4)(uq #k!p ) d(4)(uq #l!k!p ) 1 1 2 2 q !ie q !ie 1 2 d(4)(uq #p#q!k!p ) d(4)(uq #p!k!p ) 3 3 4 4 (2) dq . ] (B.20) 1 q !ie q !ie 3 4 Again after replacing k#uq "k@, q #q "q@ we get 1 2,3,4 1 2,3,4 (2) dq 1 (1a)(2b)(3b)(4b)2" , (B.21) (q !ie)(q@ !q !ie)(q@ !q !ie)(q@ !q !ie) 1 2 1 3 1 4 1 where (2) does not depend on q . Calculating the residue at q "ie and deleting the prime over q’s 1 1 we find: (1a)(2b)(3b)(4b)2"i h(u ) k)d(k2!m2) h(u ) (l!k))d((l!k#uq )2!k2) 2

]h(u ) (p#q!k))d((p#q!k#uq )2!m2) h(u ) (p!k))d((p!k#uq )2!k2) 3 4 dq dq d4k dq 3 4 2 . (B.22) ] (q !ie) (q !ie) (q !ie) (2p)3 3 4 2 Expression Eq. (B.22), after separating the factor i (due to its presence in the definition (2.54) of the amplitude), exactly corresponds to the light-front amplitude indicated by the diagram in Fig. 38. The integral :(1a)(2a)(3b)(4a) is calculated analogously (by integrating over the variable q instead of q ) and corresponds to the diagram having the form of the overturned trapezium 3 4 (relative to Fig. 38).

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Fig. 40. Time-ordered box graph. Fig. 41. Time-ordered box graph.

There are two other diagrams in the light-front dynamics shown in Fig. 40 and Fig. 41. We can find both of them in the item :(1a)(2a)(3b)(4b). After similar transformations we get:

P

P

(2) dq 1 (1a)(2a)(3b)(4b)2" , (q !ie)(q #q !ie)(q !q !ie)(q !q !ie) 1 2 1 3 1 4 1

(B.23)

Taking the sum of the two residues at q "#ie and q "!q #ie, we obtain two items: 1 1 2

P

PC

(1a)(2a)(3b)(4b)2"

D

1 1 ! (2) . q q q q (q #q )(q #q ) 2 3 4 2 2 3 2 4

(B.24)

Each of them does not correspond to any diagram. However, Eq. (B.23) indeed gives the sum of the diagrams in Figs. 40 and 41. To transform it to the appropriate form, we shall use the simple formula:

A B

1 1 1 1 " # . ab a#b a b

(B.25)

This formula should be applied recursively to pairs of factors in the integrand having the poles in the integration variables with the imaginary parts of opposite signs. For example, applying Eq. (B.25) to 1/(b b b ), first to 1/b b and then again to 1/b b , we get 2 3 4 2 3 2 4 1 1 1 1 " # # . b b b (b #b )b b (b #b )(b #b )b (b #b )(b #b )b 2 3 4 2 3 3 4 2 3 3 4 4 2 3 2 4 2

(B.26)

We apply Eq. (B.26) to the integrand of Eq. (B.23), i.e., substitute b "q #q !ie, b " 2 2 1 3 q !q !ie and b "q !q !ie. The factors 1/b and b in 1/b b have the poles relative to the 3 1 4 4 1 2 3 2 3 integration variable q with the imaginary parts of opposite signs. The same is true for the factors 1

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b and b in 1/b b . By this way, we transform the integrand of Eq. (B.23) to the form which after 2 4 2 4 integration over dq and standard replacement of variables leads to the expression: 1

P

P

(1a)(2a)(3b)(4b)2"i h(u ) k)d(k2!m2) h(u ) (k!l))d((k!l#uq !uq )2!k2) 2 3 ]h(u ) (p#q!k))d((p#q!k#uq )2!m2) h(u ) (p!k))d((p!k#uq )2!k2) 3 4

P

d4k dq dq dq 2 3 4 #i h(u ) k)d(k2!m2) ] (q !ie)(q !ie)(q !ie) (2p)3 2 3 4 ]h(u ) (k!l))d((k!l#uq !uq )2!k2) 3 4 ]h(u ) (p#q!k))d((p#q!k#uq #uq !uq )2!m2) 2 4 3 d4k dq dq dq 2 3 4 . ]h(u ) (p!k))d((p!k#uq )2!k2) 4 (q !ie)(q !ie)(q !ie) (2p)3 2 3 4

(B.27)

Two items in Eq. (B.27) exactly correspond to the amplitudes for the diagrams of Figs. 40 and 41. The method can be generalized to more complicated diagrams. The general case is considered in Ref. [119] and the renormalization in Ref. [120]. B.3.2. Spins 1/2 and 1 For spin 1, one should use the following representation of the propagator [38]: 2 kK #m iS(k)"! k2!m2#ie

P

"

d(4)(uq#k!p) (pL !uL q#m)h(u ) p)d(p2!m2) d4p dq q!ie

P

!

d(4)(uq!k!p) (pL !uL q!m)h(u ) p)d(p2!m2) d4p dq q!ie

P

"

d(4)(uq#k!p) (pL #m)h(u ) p)d(p2!m2) d4p dq q!ie

P

#

uL d(4)(uq!k!p) (m!pL )h(u ) p)d(p2!m2) d4p dq! . 2(u ) k) q!ie

(B.28)

In the representation given by Eq. (B.28), the propagator contains the spin parts (pL #m) and (m!pL ) with the on-shell four-momentum p2"m2 and the extra contact term !uL /2(u ) k), like in the rules of the graph technique, Section 2.2.3. At u"(1, 0, 0,!1) this contact term coincides with the one given in Refs. [39,123,124].

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For the propagator with spin 1, we have the representation: kk !g # k l kl m2 iD (k)"! kl k2!m2#ie

P

"

A

B

(p!uq) (p!uq) d(4)(uq#k!p) k l h(u ) p)d(p2!m2) d4p dq !g # kl m2 q!ie

P

#

A

d(4)(uq!k!p) (p!uq) (p!uq) k l !g # kl q!ie m2

] h(u ) p)d(p2!m2) d4p dq

P

"

A

B

B (B.29)

pp d(4)(uq#k!p) !g # k l h(u ) p)d(p2!m2) d4p dq kl m2 q!ie

P

#

A

B

d(4) (uq!k!p) pp !g # k l h(u ) p)d(p2!m2) d4p dq kl q!ie m2

uu k u #k u k l (k2!m2) . l k! ! k l 4(u ) k)2m2 2(u ) k)m2

(B.30)

The item in the last line of Eq. (B.30) is the vector contact term. Let now the particle 1 in Fig. 37 have spin 1. Then the contact term for the particle 1 does not 2 contribute to the amplitude indicated in Fig. 38. Indeed, this amplitude was calculated in Eq. (B.21), as the residue at q "ie in the pole of the propagator of particle 1. Replacing this 1 propagator by the contact term, we see that the only pole above the real axis disappears and the amplitude turns to zero. However, the contact terms for the particles 2—4 contribute to the diagram of Fig. 38. According to Eq. (B.28) they can be incorporated by replacing the spin part of the propagators (pL $m) by (pL !uL q$m). These facts agree with the general rule given in Section 2.2: the contact terms have to be associated with these particles which lines are directly connected by a spurion line. This also coincides with the rule given in Ref. [39]: contact terms modify only the propagators corresponding to the “lines extending over a single time interval”. The lines of the particles 2—4 in Fig. 38 extend over single time intervals and are associated with contact terms, while the line for the particle 1 extends over three time intervals and is not associated with any contact term. Another example is given by the diagram of Fig. 40. In this case, the contact term contributes for the exchanged particles (if they have spin) and does not contribute for the particles shown by the horizontal lines. On the contrary, in Fig. 41 the contact term contributes for the horizontal internal lines and does not contribute for the exchanged particles.

Appendix C. Relation between the deuteron components We give here for completeness the relations between the six invariant functions coming in the decomposition of the wave function (5.1), (5.2) and (5.6). These two representations are connected

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by Eq. (5.3) and coincide with each other in the system of reference where P"0. In this system, we have k "!k "k. 1 2 The deuteron polarization vector e(j)(p) in the representation with j"j"x, y, z in the rest k system has the form: e(j)(0)"dj, e(j)(0)"0. In an arbitrary system where pO0, it can be obtained i i 0 by the Lorentz transformation (cf. Eq. (3.11)): e(j)(0) ) p (e(j)(0) ) p)p , e(j)(p)" . e(j)(p)"e(j)(0)# 0 M M(M#p ) 0

(C.1)

In the system of reference where P"0, we have 4e2#M2 p " k , 0 4e k

e2!m2 p"!n k . e k

With these expressions for p we get from Eq. (C.1) (2e !M)2 e(j)(p)"dj# k nn , i i i j 4Me k

4e2!M2 e(j)(p)"! k n . 0 j 4Me k

(C.2)

The polarization vector Eq. (C.2) satisfies the general relations Eq. (A.10). Substituting in Eqs. (5.1) and (5.2) the expressions for the nucleon spinors and Dirac matrices given in Appendix A and the deuteron polarization vector given above, we obtain the structure of the wave function given in Eq. (5.5). Comparing the coefficients of identical spin structures, we can find the expressions of u in terms of f . In the approximation where M + 2m, they have the form: i i m2(2e #m) m2 k u " f # (J2f !f #zf !J3zf ) , 1 2 1 3 4 6 4e k2 4e (e #m) k k k

(C.3)

m u " (J2f !f !f !2zf ) , 1 2 3 4 2 4e k

(C.4)

J2k5 (2e #m)k3 (e2#4me #m2)k k u "! zf ! k zf # k zf 3 3 4e2(e #m) 4e2(e #m)3 1 4e2(e #m)2 2 k k k k k k

A

B

A

B

z2k6 J3m z2k6 3m 1! f # 1# f , # 6me2(e #m)3 4 2me2(e #m)3 6 2k 2k k k k k 3m J3m u "! f # f , 4 4 2k 2k 6 1 u " 5 2

S

3 m2 f , 2 ke 5 k

(e2#4me #m2) k4 k (J2f !f #zf !J3zf )! k f . u " 1 2 4 6 3 6 2me (e #m)2 2me k k k

(C.5)

(C.6)

(C.7)

(C.8)

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Appendix D. Two-body kinematical relations In the calculation of the electromagnetic amplitude we have to express the scalar products of the four vectors u, k , k , k@ , k@ , p, p@, q with each other in terms of the integration variables and 1 2 1 2 the four-momentum transfer squared. The variables and, hence, the kinematical relations for the one-loop diagram of Fig. 18 (impulse approximation) and for the two-loop diagram of Fig. 42 are different. D.1. One-loop diagram The one-loop diagram is shown in Fig. 18. All four-momenta are on the corresponding mass shells: k2"m2, k2"k@ 2"m2, p2"M2, p@2"M2, q2"!D2"!Q2"t, u2"0 . 1 1 2 2 2 We consider the general case of different masses, M being the mass of the initial state, while M is the mass of the final state. We start from the four-vector R already defined in Eq. (3.16): 1 R "k !xp, x"u ) k /u ) p . 1 1 1 Since R ) u"0, it can be represented as R "(R , R , R ) with R ) x"0 and R is parallel to 1 1 0 M , M , x. Similarly we represent q"(q , D, q ) with D ) x"0. With these definitions, we immediately get 0 , (with u ) q"0): u ) k "xu ) p, u ) k "(1!x)u ) p, u ) k@ "(1!x)u ) p, u ) p"u ) p@ . 1 2 2 We get for the various scalar products the following relations. Scalar products with k : 1 k ) k "(s!m2!m2)/2 , 1 2 1 2 k ) k@ "(s@!m2!m2)/2 , 1 2 1 2 k ) p"xM2/2#(1!x)s/2#(m2!m2)/2 , 1 1 2 k ) p@"k ) p#k ) q , 1 1 1 k ) q"!R ) D#xp ) q , 1 M with s and s@ given by

(D.1)

(D.2)

R2 #m2 R2 #m2 1# M 2, s,(k #k )2" M 1 2 x 1!x R@2#m2 R @2#m2 1# M 2. s@,(k@ #k@ )2" M 1 2 x 1!x where R@ "R !xD. M M

(D.3)

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Scalar products with k : 2 k ) k@ "m2#k ) q#(1!x)(s@!M2!s#M2)/2 , 2 2 2 2 k ) p"(1!x)M2/2#xs/2#(m2!m2)/2 , 2 2 1 k ) p@"k ) p#k ) q , 2 2 2 k ) q"R ) D#(1!x)p ) q . 2 M Scalar products with k@ : 2 k@ ) p"k@ ) p@!k@ ) q , 2 2 2 k@ ) p@"(1!x)M2/2#xs@/2#(m2!m2)/2 , 2 2 1 k@ ) q"k ) q!D2 . 2 2 Scalar products with p:

341

(D.4)

(D.5)

p ) p@"(D2#M2#M2)/2 , p ) q"(D2!M2#M2)/2 .

(D.6)

Scalar product with p@: p@ ) q"(M2!M2!D2)/2 .

(D.7)

D.2. Two-loop diagram The two-loop diagrams indicated on Fig. 29 correspond to the contribution of the contact term. One of them is reproduced in Fig. 42, where all the momenta are explicitly indicated. Some of the relations coincide with the ones given above. We repeat them here for completeness. For the scalar products of u we immediately get u ) k "xu ) p, u ) k "(1!x)u ) p, u ) k@ "x@u ) p , (D.8) 1 2 1 u ) k@ "(1!x@) u ) p, u ) p"u ) p@, u ) q"0 . 2 All the following invariants can be transformed to the variables k, k@ and n by the substitution:

A

B

e n)k R "k!(n ) k)n , x" 1 1! ; M e #e e 1 2 1 e@ n ) k@ R@ "k@!(n ) k@)n , x@" 1 1! , M e@ #e@ e@ 1 2 1

A

B

(D.9) (D.10)

taking into account that D ) n"0. We denote in this equation e "Jk2#m2, and similarly for i i quantities with prime. We thus get the following scalar products.

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Fig. 42. Contact term of Fig. 29 a contributing to the deuteron electromagnetic form factors.

Scalar products with k : 1 k ) k "(s!m2!m2)/2 , 1 2 1 2 k ) k@ "!R ) R@ #xDR@ !x@D ) R #x(1!x@)s@/2#x@(1!x)s/2#xx@D2/2 , 1 1 M M M M k ) k@ "k ) p@#x(s@!M2)/2!k ) k@ , k ) p"(1!x)s/2#xM2/2 , 1 2 1 1 1 1 k ) p@"k ) q#k ) p, k ) q"!R ) D#xp ) q. 1 1 1 1 M Scalar products with k : 2 k ) k@ "k@ ) p#x@(s!M2)/2!k ) k@ , 2 1 1 1 1 k ) k@ "k ) p@!k ) k@ #(1!x)(s@!M2)/2 , 2 2 2 2 1 k ) p"xs/2#(1!x)M2/2 , 2 k ) p@"k@ ) q#k ) p , 2 2 2 k ) q"R ) D#(1!x)p ) q . 2 M Scalar products with k@ : 1 k@ ) k@ "(s@!m2!m2)/2 , 1 2 1 2 k@ ) p"k@ ) p@!k@ ) q , 1 1 1 k@ ) p@"(1!x@)s@/2#x@M2/2 , 1 k@ ) q"!R@ ) D#x@p@ ) q . 1 M

(D.11)

(D.12)

(D.13)

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Scalar products with k@ : 2 k@ ) p"k@ ) p@!k@ ) q , 2 2 2 k@ ) p@"x@s@/2#(1!x@)M2/2 , (D.14) 2 @ k@ ) q"R ) D#(1!x@)p@ ) q . 2 M The scalar products with p and p@ are the same as Eqs. (D.6) and (D.7). The kinematics for the final plane wave contribution (neglecting the final state energy) is obtained from the above formulas by the substitution M"2m, R@ "0, x@"1 (or k@"0). The M 2 kinematics for the elastic deuteron form factors is obtained by the replacement MPM.

Appendix E. Three-body kinematical relations The scalar products appearing in the calculation of the amplitude indicated in Fig. 24 are expressed through the integration variables R , x and D. We give them below in the general case iM i of different quark masses. For p ) p@ we have: p ) p@"M2#D2/2 .

(E.1)

In order to find p ) k (i"1, 2, 3), we calculate (k !x p)2"m2!2x p ) k #x2M2"R2"!R2 . i i i i i i i i iM From here we find: R2 #m2 1 i # M2x . p ) k " iM i i 2 2x i Other scalar products are found analogously. We have

(E.2)

p@ ) k "p ) k !R ) D#x D2/2 , (E.3) i i iM i R@2 #m2 1 i # M2x , (E.4) p@ ) k@" iM i i 2 2x i p ) k@"p@ ) k@#R@ ) D#x D2/2 , (E.5) i i iM i k ) k@ "!R ) R@ #x p ) k@ #x p@ ) k !x x p ) p@ for i, j"1, 2, 3 . (E.6) i j i j i j j i i j The scalar product k ) k is obtained from Eq. (E.6) by deleting primes at all variables, the scalar i j product k@ ) k@ is obtained by the following replacement in Eq. (E.6): R PR@, k Pk@, pPp@. The i j i i i i variables x are the same for initial and final states. i Let us now express R@ through D and the integration variables R and x . The expressions are iM iM i not symmetrical relative to the quark 1, which interacts with the photon, and to the spectator quarks 2 and 3. We have R@ "k@ !x p@, R@ "k@ !x p@. With k@ "k , k@ "k we find 2 2 2 3 3 3 2 2 3 3 R@ "R !x q, R@ "R !x q. With R@ "k@ !x p@"R #(1!x )q#u(q@!q), we get 2 2 2 3 3 3 1 1 1 1 1 R@ "R !x D . (E.7) R@ "R #(1!x )D , 1M 1M 1 2,3M 2,3M 2,3

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Note that R@ can be also found from the relation R@ "!R@ !R@ . The arguments 1M 1M 2M 3M q2 (i"1, 2, 3) of the initial wave function are given by i q2"e2i!m2 , i q i where e i is the energy of the particle in the reference system where P"k #k #k "0. It can be 1 2 3 q represented in the invariant form as e i"P ) k /M with P"p#uq, q"(M2!M2)/(2u ) p). That i q gives e i"[p ) k #1x (M2!M2)]/M , i 2 i q where M2 is the invariant mass squared:

(E.8)

3 R2 #m2 i . M2"(k #k #k )2" + iM (E.9) 1 2 3 x i i/1 The arguments q@2 of the final wave function are constructed similarly from the final momenta. i References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

[22] [23] [24] [25] [26] [27]

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