Physics Reports vol.331

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DIFFRACTION IN DEEP INELASTIC SCATTERING

A. HEBECKER Institut f u( r Theoretische Physik der Universita( t Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

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Di!raction in deep inelastic scattering A. Hebecker Institut fu( r Theoretische Physik der Universita( t Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany Received December 1999; editor: R. Petronzio Contents 1. Introduction 1.1. Preface 1.2. Models for di!raction 1.3. The semiclassical method and its relation to other approaches 2. Basic concepts and phenomena 2.1. Kinematics 2.2. Fundamental observations 2.3. Di!ractive structure function 3. Semiclassical approach 3.1. Eikonal formulae for high-energy scattering 3.2. Production of qq pairs 3.3. Higher Fock states 3.4. Field averaging 4. From soft pomeron to di!ractive parton distributions 4.1. Soft pomeron 4.2. Pomeron structure function 4.3. Di!ractive parton distributions 4.4. Target rest frame point of view 5. Two gluon exchange 5.1. Elastic meson production

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5.2. Factorization 5.3. Charm and high-p jets , 5.4. Inclusive di!raction 6. Models for the colour "eld of the proton 6.1. Small colour dipole 6.2. Large hadron 6.3. Stochastic vacuum 7. Recent experimental results 7.1. Di!ractive structure function 7.2. Final states 7.3. Meson production 8. Conclusions Acknowledgements Appendix A. Derivation of Eikonal formulae Appendix B. Spinor matrix elements Appendix C. Derivation of di!ractive quark and gluon distribution C.1. Di!ractive quark distribution C.2. Di!ractive gluon distribution Appendix D. Inclusive parton distributions References

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Abstract Di!erent theoretical methods used for the description of di!ractive processes in small-x deep inelastic scattering are reviewed. The semiclassical approach, where a partonic #uctuation of the incoming virtual photon scatters o! a superposition of target colour "elds, is used to explain the basic physical e!ects. In this approach, di!raction occurs if the emerging partonic state is in a colour singlet, thus fragmenting

E-mail address: [email protected] (A. Hebecker) 0370-1573/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 0 5 - 3

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independently of the target. Other approaches, such as the idea of the pomeron structure function and two gluon exchange calculations, are also discussed in some detail. Particular attention is paid to the close relation between the semiclassical approach and the method of di!ractive parton distributions, which is linked to the relation between the target rest frame and the Breit frame point of view. While the main focus is on di!ractive structure functions, basic issues in the di!ractive production of mesons and of other less inclusive "nal states are also discussed. Models of the proton colour "eld, which can be converted into predictions for di!ractive cross-sections using the semiclassical approach, are presented. The concluding overview of recent experimental results is very brief and mainly serves to illustrate implications of the theoretical methods presented.  2000 Elsevier Science B.V. All rights reserved. PACS: 12.38.!t; 13.60.Hb; 12.38.Lg Keywords: Deep inelastic scattering; Di!raction; Semiclassical

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1. Introduction 1.1. Preface The term di!raction is derived from optics, where it describes the de#ection of a beam of light and its decomposition into components with di!erent frequencies. In high-energy physics it was originally used for small-angle elastic scattering of hadrons. If one of the hadrons, say the projectile, is transformed into a set of two or more "nal state particles, the process is called di!ractive dissociation or inelastic di!raction. Good and Walker have pointed out that a particularly intuitive physical picture of such processes emerges if the projectile is described as a superposition of di!erent components which scatter elastically o! the target [1]. Since the corresponding elastic amplitude is di!erent for each component, the outgoing beam will contain a new superposition of these components and therefore, in general, new physical states. These are the dissociation products of the projectile. Even at very high energy, the above processes are generically soft, i.e., the momentum transfer is small and the dissociation products have small p . Therefore, no immediate relation to pertur, bative QCD is apparent. By contrast, di!ractive jet production, observed at the CERN Spp S collider in proton}antiproton collisions [2], involves a hard scale. Although one of the hadrons escapes essentially unscathed, a high-p jet pair, which is necessarily associated with a high virtuality in the , intermediate states, is produced in the central rapidity range. The cross-section of the process is parametrically unsuppressed relative to non-di!ractive jet production. This seems to contradict a namK ve partonic picture since the colour neutrality of the projectile is destroyed if one parton is removed to participate in the hard scattering. The interplay of soft and hard physics necessary to explain the e!ect provides one of the main motivations for the study of these &hard di!ractive' processes. The present review is focussed on di!raction in deep inelastic scattering (DIS), which is another example of a hard di!ractive process. This process became experimentally viable with the advent of the electron}proton collider HERA, where DIS at very small values of the Bjorken variable x can be studied. In the small-x or high-energy region, a signi"cant fraction of the observed DIS events have a large rapidity gap between the photon and the proton fragmentation region [3,4]. In contrast to the standard DIS process cHpPX, the relevant reaction reads cHpPX>, where X is a high-mass hadronic state and > is the elastically scattered proton or a low-mass excitation of it. Again, these events are incompatible with the namK ve picture of a partonic target and corresponding simple ideas about the colour #ow. NamK vely, the parton struck by the virtual photon destroys the colour neutrality of the proton, a colour string forms between struck quark and proton remnant, and hadronic activity is expected throughout the detector. Nevertheless, the observed di!ractive cross-section is not power suppressed at high virtualities Q with respect to standard DIS. The main theoretical interest is centred around the interplay of soft and hard physics represented by the elastic or almost elastic scattering of the proton and the scattering of the highly virtual photon respectively. Di!ractive DIS is much simpler than hard di!raction in hadronic reactions since only one non-perturbative object is involved. In a large fraction of the events, the momentum transfer to the proton is very small. Therefore, one can hope to gain a better understanding of the bound state dynamics of the proton by studying di!ractive DIS. Inclusive reactions of the virtual

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photon with hadronic targets are well-studied theoretically and constrained by a large amount of DIS data. It is a challenge to utilize this knowledge for the investigation of the non-perturbative dynamics of di!raction and, thereby, of the proton structure. The paper is organized as follows. In the remainder of the Introduction, di!erent approaches to di!ractive DIS are put into historical and physical perspective. The semiclassical model, which is particularly close to the interests of the author, is emphasized. Section 2 is concerned with the fundamental observations, the basic concepts required for their understanding, and the necessary kinematic considerations. The semiclassical approach, which is used as the starting point for the discussion of other models, is introduced in Section 3. In Section 4, the concepts of soft pomeron, triple pomeron vertex and pomeron structure function are discussed. Di!ractive parton distributions, which are the more fundamental objects from the point of view of perturbative QCD, are introduced. The semiclassical approach is put in relation to the method of di!ractive parton distributions, and explicit formulae for these quantities are derived. In Section 5, two gluon exchange calculations are discussed. The emphasis is on their rigorous validity in speci"c kinematic situations and on their partial correspondence to the semiclassical approach. Section 6 introduces three models for the colour "eld of the proton, which can be converted into predictions for di!ractive cross-sections using the methods of Sections 3 and 4. A discussion of recent experimental results and their description by theoretical models is given in Section 7, followed by the Conclusions. 1.2. Models for diwraction The claim that di!raction in DIS should be a leading twist e!ect, and the understanding of the fundamental mechanism underlying such processes can be traced back to the famous paper of Bjorken and Kogut [5]. Their argument is based on a qualitative picture of DIS in the target rest frame, where the incoming virtual photon can be considered as a superposition of partonic states. The large virtuality Q sets the scale, so that states with low-p partons, i.e., aligned con"gurations, , are suppressed in the photon wave function. However, in contrast to high-p con"gurations, these , aligned states have a large hadronic interaction cross-section with the proton. Therefore, their contribution to the DIS cross-section is expected to be of leading twist. Naturally, part of this leading twist contribution is di!ractive since the above low-p con"gurations represent transverse, ly extended, hadron-like objects, which have a large elastic cross-section with the proton. Note that the basic technical methods date back even further, namely, to the calculation of k>k\ pair electroproduction o! an external electromagnetic "eld by Bjorken et al. [6]. They derive the transition amplitude of the incoming virtual photon into a k>k\ pair, which corresponds to the qq wave function of the virtual photon employed by Nikolaev and Zakharov in their seminal work on di!raction at HERA [7]. The above physical picture, commonly known as the aligned jet model of di!ractive and non-di!ractive DIS, can naturally be extended to account for a soft energy growth of both processes. (Note that we are dealing with the underlying energy dependence of the soft scattering process, not with the additional x-dependence induced by Altarelli}Parisi evolution, that is added on top of it.) Since a large sample of hadronic cross-sections can be consistently parametrized using the concept of the Donnachie}Landsho! or soft pomeron [8], it is only natural to assume that this concept can also be used to describe the energy dependence of the interaction of the aligned jet

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component of the small-x virtual photon and the target. This gives rise to the desired non-trivial energy dependence although, as will be discussed in more detail later on, this energy dependence is not su$ciently steep in the di!ractive case. Note, however, that the above soft energy dependence may also be logarithmic, which has the advantage of explicit consistency with unitarity at arbitrarily high energies. A more direct way of applying the concept of the soft pomeron to the phenomenon of hard di!raction was suggested by Ingelman and Schlein in the context of di!ractive jet production in hadronic collisions [9]. Their idea of a partonic structure of the pomeron, which can be tested in hard processes, applies to the case of di!ractive DIS as well [10]. Essentially, one assumes that the pomeron can, like a real hadron, be characterized by a parton distribution. This distribution factorizes from the pomeron trajectory and the pomeron}proton vertex, which are both obtained from the analysis of purely soft hadronic reactions. The above non-trivial assumptions are often referred to as &Regge hypothesis' or &Regge factorization'. The Ingelman}Schlein approach described above is based on the intuitive picture of a pomeron #ux associated with the proton beam and on the conventional partonic description of the pomeron photon collision. In the limit where not only the total proton}photon centre-of-mass energy but also the energy of the pomeron}photon collision becomes large, the concept of the triple pomeron vertex can be applied [11]. At HERA, this limit corresponds to rapidity gap events with very large di!ractive masses. The concept of fracture functions of Veneziano and Trentadue [12] or, more speci"cally, the di!ractive parton distributions of Berera and Soper [13] provide a framework for the study of di!ractive DIS that is "rmly rooted in perturbative QCD. Loosely speaking, di!ractive parton distributions describe the probability of "nding, in a fast moving proton, a parton with a certain momentum fraction x, under the additional requirement that the proton remains intact losing only a certain fraction of its momentum. This idea is closely related to the concept of a partonic pomeron described above, but it gives up the Regge hypothesis, thus being less predictive. This weaker factorization assumption is sometimes referred to as &di!ractive factorization', as opposed to the stronger Regge factorization assumption. An essential feature of di!ractive DIS is the colour singlet exchange in the t channel, necessary to preserve the colour neutrality of the target. None of the above approaches address this requirement explicitly within the framework of QCD. Instead, the colour singlet exchange is postulated by assuming elastic scattering in the aligned jet model, by using the concept of the pomeron, or by de"ning di!ractive parton distributions. If one insists on using the fundamental degrees of freedom of QCD, the simplest possibility of realizing colour singlet exchange at high energy is the exchange of two gluons [14]. One could say that two gluons form the simplest model for the pomeron. The fundamental problem with this approach is the applicability of perturbative QCD to the description of di!ractive DIS. As will be discussed in more detail below, the hard scale Q of the photon does not necessarily justify a perturbative description of the t channel exchange with the target. Di!ractive processes where the t channel colour singlet exchange is governed by a hard scale include the electroproduction of heavy vector mesons [15], electroproduction of light vector mesons in the case of longitudinal polarization [16] or at large t [17], and virtual Compton scattering [18}20]. In the leading logarithmic approximation, the relevant two-gluon form factor of the proton can be related to the inclusive gluon distribution [15]. Accordingly, a very steep

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energy dependence of the cross-section, which is now proportional to the square of the gluon distribution, is expected from the known steep behaviour of small-x structure functions. The origin of this steep rise itself may be attributed to a combination of the x-dependence of the input distributions and their Altarelli}Parisi evolution [21], or to the BFKL resummation of large logarithms of x [22]. To go beyond leading logarithmic accuracy, the non-zero momentum transferred to the proton has to be taken into account. This requires the use of &non-forward' or &o!-diagonal' parton distributions (see [18,23] and Refs. therein), which were discussed in [19,20] within the present context. Although their scale dependence is predicted by well-known evolution equations, only limited information about the relevant input distributions is available. In particular, the simple proportionality to the square of the conventional gluon distribution is lost. The perturbative calculations of meson electroproduction discussed above were put on a "rmer theoretical basis by the factorization proof of [24], where the conditions required for the applicability of perturbation theory are discussed in detail. As shown explicitly in the simple model calculation of [25], QCD gauge invariance ensures that the non-perturbative meson formation process takes place after the scattering o! the hadronic target. The situation becomes even more complicated if two gluon exchange calculations are applied to more general di!ractive "nal states. Examples are the exclusive electroproduction of heavy quark pairs [26}29] or high-p jets [30}33]. A straightforward perturbative analysis shows that the , virtual photon side of the process is dominated by small transverse distances, establishing two gluon exchange as the dominant mechanism. However, it is not obvious whether the very de"nition &exclusive', meaning exclusive on a partonic level, can be extended to all orders in perturbation theory. As will become clear later on, additional soft partons in the "nal state may destroy the hardness provided by the large p or the heavy quark mass. , From the point of view of the full di!ractive cross-section, perturbative two gluon exchange corresponds to a higher twist e!ect. However, it can be argued that it dominates the kinematic region where the mass M of the di!ractively produced "nal state X is relatively small, M;Q [34]. More generally, higher twist e!ects in di!raction, their calculability and possibilities for their experimental observation are discussed in [34}36]. Eventually it is possible to attempt the description of the full cross-section of di!ractive DIS within the framework of two gluon exchange. In such a general setting, perturbation theory cannot be rigorously justi"ed and two gluons are basically used as a model for the soft colour singlet exchange in the t channel. Nevertheless, it is interesting to see to what extent concepts and formulae emerging from perturbation theory can describe a wider range of phenomena. In particular, much attention has been devoted to the energy dependence of di!ractive cross-sections both in the framework of conventional BFKL summation of gluon ladders [37,38] and within the colour dipole approach to small-x resummation [30,39]. Two gluon exchange calculations are emphasized in the recent review [40]. An apparently quite di!erent approach emerged with the idea that soft colour interactions might be responsible for the large di!ractive cross-section at HERA. The starting point is BuchmuK ller's observation of the striking similarity between x and Q dependence of di!ractive and inclusive DIS at small x [41]. It is then natural to assume that the same hard partonic processes underlie both cros-sections and that di!erences in the "nal states are the result of non-perturbative soft colour exchange [42,43].

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In Ref. [42], boson}gluon fusion was proposed as the dominant partonic process, and di!raction was claimed to be the result of soft colour neutralization of the produced qq pair. A simple statistical assumption about the occurrence of this colour neutralization lead to a surprisingly good "t to the data. Note that similar ideas concerning the rotation of quarks in colour space had previously been discussed by Nachtmann and Reiter [44] in connection with QCD vacuum e!ects on hadron}hadron scattering. A closely related approach was introduced in [43], where the assumption of soft colour neutralization was implemented in a Monte Carlo event generator based on perturbation theory. Normally, the partonic cascades underlying the Monte Carlo determine the colour of all partons produced and, with a certain model dependence, the hadronic "nal state. The introduction of an ad hoc probability for partons within the cascade to exchange colour in a non-perturbative way leads to a signi"cant increase of "nal states with rapidity gaps, resulting in a good description of the observed di!ractive events. Both of the above models share a fundamental theoretical problem: the soft colour exchange is introduced in an ad hoc manner, independently of kinematic con"guration or space}time distances of the partons involved. If all relevant distances are large, our ignorance of the true mechanism of non-perturbative interactions justi"es the namK ve assumption of random colour exchange. However, in many other situations, e.g., if a small colour dipole is involved, colour exchange is suppressed in a perturbatively calculable way. This is known as colour transparency. The above simple soft colour models do not properly account for this e!ect. One possibility of incorporating the knowledge of the perturbative aspects of QCD while keeping the essential idea of soft colour exchange as the source of leading twist di!raction is the semiclassical treatment. 1.3. The semiclassical method and its relation to other approaches The semiclassical approach to di!ractive electroproduction evolved from the attempt to justify the phenomenologically successful idea of soft colour exchange [42,43] within the framework of QCD. The starting point is the proton rest frame picture of DIS advocated long ago in [5] and developed since by many authors (see in particular [7]). To describe di!raction within this framework, the scattering of energetic partons in the photon wave function o! the hadronic target needs to be understood. While the scattering of small transverse size con"gurations is calculable perturbatively, i.e., via two gluon exchange, di!erent models may be employed for larger size con"gurations. A rather general framework is provided by the concept of the dipole cross-section p(o) utilized in the analysis of [7]. It is helpful to begin by recalling the discussion of high-energy electroproduction of k>k\ pairs o! atomic targets given by Bjorken et al. in [6]. There, the closely related QED problem was solved by treating the high-energy scattering of k>k\ pairs o! a given electromagnetic "eld in the eikonal approximation. An analogous QCD process was considered by Collins et al. in [45], where the production of heavy quark pairs in an external colour "eld was calculated. In Ref. [46], Nachtmann developed the idea of eikonalized interactions with soft colour "eld con"gurations as a method for the treatment of elastic high-energy scattering of hadrons. The semiclassical approach to di!raction, introduced in [47], combines the concepts and methods outlined above. The proton is modelled by a soft colour "eld, and the interactions of the fast partons in the photon wave function with this "eld are treated in the eikonal approximation.

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As a result, the non-perturbative proton structure is encoded in a combination of non-Abelian phase factors associated with the partons. In the simplest case of a qq #uctuation, two such phase factors are combined in a Wegner}Wilson loop, which carries all the information about the target. Since the outgoing qq pair can be either in a colour singlet or in a colour octet state, both di!ractive and non-di!ractive events are naturally expected in this approach. Thus, both types of events are described within the same framework. The colour state of the produced pair is the result of the fundamentally non-perturbative interaction with the proton colour "eld, very much in the spirit of the soft colour neutralization of [42]. The essential di!erence between the semiclassical calculation and the soft colour proposal is the recognition that the possibility of soft colour exchange is intimately related to parton level kinematics. The semiclassical approach allows for a consistent treatment of the scattering of both small and large transverse size con"gurations. The requirement of colour neutrality in the "nal state of di!ractive events is explicitly shown to suppress the former ones. This con"rms the familiar qualitative arguments on which the aligned jet model is based [5]. A further essential step is the inclusion of higher Fock states in the photon wave function. In the framework of two gluon exchange, corresponding calculations for the qq g state were performed in [30]. The semiclassical analysis of the qq g state [48] demonstrates that, to obtain leading twist di!raction, at least one of the three partons has to have small p and has to carry a small fraction of , the photon's longitudinal momentum. This is a natural generalization of the aligned jet model, which was previously discussed on a qualitative level in [49]. A fundamental prediction derived from the semiclassical treatment of the qq g state is the leading twist nature of di!ractive electroproduction of heavy quark pairs [50] and of high-p jets [51]. , These large cross-sections arise from qq g con"gurations where the gluon is relatively soft and has small p , so that the t channel colour singlet exchange remains soft in spite of the additional hard , scale provided by the qq pair. As mentioned previously, considerable work has been done attempting to employ the additional hardness provided by "nal states with jets or heavy quarks in order to probe mechanisms of perturbative colour singlet exchange [26}33]. In fact, the semiclassical approach is also well suited to follow this line of thinking. As demonstrated in [51], the features of di!ractive jet production from qq and qq g #uctuations of the photon, both treated consistently in the semiclassical framework, are very di!erent. In the qq case, a small-size colour dipole tests the small distance structure of the proton colour "eld. It can be shown within the semiclassical framework that the relevant operator is the same as in inclusive DIS, thus reproducing the well-known relation with the square of the gluon distribution [15]. In the qq g case, the additional soft gluon allows for a non-perturbative colour neutralization mechanism even though the "nal state contains two high-p jets. , The semiclassical treatment of leading twist di!raction is equivalent to a treatment based on di!ractive parton distributions [52]. To be more precise, the results of [48] lead to the conclusion that a leading twist contribution to di!raction can arise only from virtual photon #uctuations that include at least one soft parton. Starting from this premise, the di!ractive production of "nal states with one soft parton and a number of high-p partons was considered in [52]. It was shown that , the cross-section can be written as a convolution of a hard partonic cross-section and a di!ractive parton distribution. The latter is given explicitly in terms of an average over the target colour "eld con"gurations underlying the semiclassical calculation.

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From this point of view, the semiclassical prediction of the dominance of qq g "nal states and soft colour neutralization in high-p jet and heavy quark production appears natural. It corresponds , simply to the dominant partonic process, boson}gluon fusion, with the gluon taken from the di!ractive gluon distribution. The additional gluon in the "nal state is, from the point of view of parton distributions, merely a consequence of the preserved colour neutrality of the target. From the point of view of the semiclassical calculation, it is the scattering product of the original gluon from the photon wave function that is necessary for the softness of the colour singlet exchange. Even without an explicit model for the proton colour "eld, the semiclassical approach provides an intuitive overall picture of inclusive and di!ractive small-x DIS and predicts a number of qualitative features of the "nal state. However, the ultimate goal is to develop the understanding of non-perturbative colour dynamics that is needed to calculate the required "eld averages. A very successful description of a number of hadronic reactions was obtained in the framework of the stochastic vacuum. Di!erent e!ects of the non-trivial structure of the QCD vacuum in high-energy hadron}hadron scattering were considered in [44]. Field strength correlators in a theory with vacuum condensate were employed in [53,54] to describe the pomeron. The model of the stochastic vacuum was introduced by Dosch and Simonov [55] in the context of gluon "eld correlators in the Euclidean theory. Based on ideas of [53,54] and, in particular, on Nachtmann's suggestion to describe high-energy hadron}hadron scattering in an eikonal approach [46], the stochastic vacuum model was adapted to Minkowski space and applied to high-energy processes by Dosch and KraK mer [56]. In the present context of di!ractive and non-di!ractive electroproduction, the work on inclusive DIS [57], C- and P-odd contributions to di!raction [58], and exclusive vector meson production [59}61] is particularly relevant. Results on di!ractive structure functions have recently been reported [62]. In the special case of a very large hadronic target, the colour "eld averages required in the semiclassical approach can be calculated without specifying the details of the non-perturbative colour dynamics involved. McLerran and Venugopalan observed that the large target size, realized, e.g., in an extremely heavy nucleus, introduces a new hard scale into the process of DIS [63]. From the target rest frame point of view, this means that the typical transverse size of the partonic #uctuations of the virtual photon remains perturbative [64]. Thus, the perturbative treatment of the photon wave function in the semiclassical calculation is justi"ed. Note that the small size of the partonic #uctuations of the photon does not imply a complete reduction to perturbation theory. The long distance which the partonic #uctuation travels in the target compensates for its small transverse size, thus requiring the eikonalization of gluon exchange. For a large target it is natural to introduce the additional assumption that the gluonic "elds encountered by the partonic probe in distant regions of the target are not correlated. In this situation, inclusive and di!ractive DIS cross-sections become completely calculable. A corresponding analysis of inclusive and di!ractive parton distributions was performed in [65]. Starting from the above large target model, expressions for the inclusive quark distribution, which had previously been discussed in a similar framework in [66], and for the inclusive gluon distribution were obtained. Di!ractive quark and gluon distributions were calculated, within the same model for the colour "eld averaging, on the basis of formulae from [52]. The resulting structure functions, obtained from the above input distributions by Altarelli}Parisi evolution, provide a satisfactory description of the experimental data [65].

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A number of further approaches to di!ractive DIS were proposed by several authors. For example, it was argued in [67] that a super-critical pomeron with colour-charge parity C "!1 A plays an essential role. As a result, the main features of the namK ve boson}gluon fusion model of [42] are reproduced while its problems, which were outlined above, are avoided. In a di!erent approach, a geometric picture of di!raction, based on the idea of colourless gluon clusters, was advertised (see, e.g., [68]). There can be no doubt that far more material relevant to the present subject exists which, because of the author's bias or unawareness, is not appropriately re#ected in this introduction or the following sections.

2. Basic concepts and phenomena In this section, the kinematics of di!ractive electroproduction at small x is explained in some detail, and the notation conventionally used for the description of this phenomenon is introduced. The main experimental observations are discussed, and the concept of the di!ractive structure function, which is widely used in analyses of inclusive di!raction, is explained. This standard material may be skipped by readers familiar with the main HERA results on di!ractive DIS. 2.1. Kinematics To begin, recall the conventional variables for the description of DIS. An electron with momentum k collides with a proton with momentum P. In neutral current processes, a photon with momentum q and virtuality q"!Q is exchanged, and the outgoing electron has momentum k"k!q. The exchange of Z bosons can be neglected for the Q values relevant in this review. In inclusive DIS, no questions are asked about the hadronic "nal state X , which is only known to 5 have an invariant mass square ="(P#q). The Bjorken variable x"Q/(Q#=) characterizes, in the namK ve parton model, the momentum fraction of the incoming proton carried by the quark that is struck by the virtual photon. If x;1, which is the relevant region in the present context, Q is much smaller than the photon energy in the target rest frame. In this sense, the photon is almost real even though Q
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Fig. 1. Di!ractive electroproduction. The full hadronic "nal state with invariant mass = contains the elastically scattered proton and the di!ractive state with invariant mass M.

total invariant mass of the outgoing proton and the di!ractive state X is given by the standard + DIS variable =. The following parallel description of inclusive and di!ractive DIS suggests itself. In the former, virtual photon and proton collide to form a hadronic state X with mass =. The 5 process can be characterized by the virtuality Q and the scaling variable x"Q/(Q#=), the momentum fraction of the struck quark in the namK ve parton model. In the latter, a colour neutral cluster of partons is stripped o! the proton. The virtual photon forms, together with this cluster, a hadronic state X with mass M. The process can be character+ ized by Q, as above, and by a new scaling variable b"Q/(Q#M), the momentum fraction of this cluster carried by the struck quark. Since di!raction is a subprocess of inclusive DIS, the struck quark from the colour neutral cluster also carries a fraction x of the proton momentum. Therefore, the ratio m"x/b characterizes the momentum fraction that the proton loses to the colour neutral exchange typical of an elastic reaction. This exchanged colour neutral cluster loses a momentum fraction b to the struck quark that absorbs the virtual photon. As expected, the product x"bm is the fraction of the original proton's momentum carried by this struck quark. Since the name pomeron is frequently applied to whichever exchange with vacuum quantum numbers dominates the high-energy limit, many authors use the notation x/ "m, thus implying that the proton loses a momentum fraction m to the exchanged pomeron. Therefore, x, Q and b or, alternatively, x, Q and m are the main kinematic variables characterizing di!ractive DIS. A further variable, t"(P!P), is necessary if the transverse momenta of the outgoing proton and the state X relative to the cHP axis are measured. Since the proton is a soft + hadronic state, the value of "t" is small in most events. The small momentum transferred by the proton also implies that M;=. To see this in more detail, introduce light-cone co-ordinates q "q $q and q "(q , q ). It is !   ,   convenient to work in a frame where the transverse momenta of the incoming particles vanish, q "P "0. Let D be the momentum transferred by the proton, D"P!P, and m"P"P , , N the proton mass squared. For forward scattering, P "0, the relation , t"D"D D "!mm (2.1) > \ N holds. Since m"(Q#M)/(Q#=), this means that small M implies small "t" and vice versa. Note, however, that the value of "t" is larger for non-forward processes, where t"D D !D . > \ ,

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So far, di!ractive events have been characterized as those DIS events which contain an elastically scattered proton in their hadronic "nal state. An even more striking feature is the large gap of hadronic activity seen in the detector between the scattered proton and the di!ractive state X . It + will now be demonstrated that this feature, responsible for the alternative name &rapidity gap events', is a direct consequence of the relevant kinematics. Recall the de"nition of the rapidity y of a particle with momentum k, 1 k #k 1 k  . (2.2) y" ln > " ln  2 k !k 2 k \   This is a convenient quantity for the description of high-energy collisions along the z-axis. Massless particles moving along this axis have rapidity !R or #R, while all other particles are characterized by some "nite intermediate value of y (for a detailed discussion of the role of rapidity in the description of di!ractive kinematics see, e.g., [69]). In the centre-of-mass frame of the cHp collision, with the z-axis pointing in the proton beam direction, the rapidity of the incoming proton is given by y "ln (P /m ). At small m, the rapidity N > N of the scattered proton is approximately the same. This is to be compared with the highest rapidity y of any of the particles in the di!ractive state X . Since the total plus component of the

 + 4-momentum of X is given by (m!x)P , and the pion, with mass m , is the lightest hadron, none + > L of the particles in X can have a rapidity above y "ln((m!x)P /m ). Thus, a rapidity gap of +

 > L size *y"ln(m /(m!x)m ) exists between the outgoing proton and the state X . For typical values L N + of m&10\ (cf. the more detailed discussion of the experimental setting below) the size of this gap can be considerable. Note, however, that the term &rapidity gap events' was coined to describe the appearance of di!ractive events in the HERA frame, i.e., a frame de"ned by the electron}proton collision axis. The rapidity in this frame is, in general, di!erent from the photon}proton frame rapidity discussed above. Nevertheless, the existence of a gap surrounding the outgoing proton in the cHp frame clearly implies the existence of a similar gap in the ep frame. The exact size of the ep-frame rapidity gap follows from the speci"c event kinematics. The main conclusion so far is the kinematic separation of outgoing proton and di!ractive state X in di!ractive events with small m. + The appearance of a typical event in the ZEUS detector is shown in Fig. 2, where FCAL, BCAL, and RCAL are the forward, central (barrel) and rear calorimeters. The absence of a signi"cant energy deposit in the forward region is the most striking feature of this DIS event. In the namK ve parton model of DIS, a large forward energy deposit is expected due to the fragmentation of the proton remnant, which is left after a quark has been knocked out by the virtual photon. In addition, the whole rapidity range between proton remnant and current jet is expected to "ll with the hadronization products of the colour string that develops because of the colour charge carried by the struck quark. Thus, the event in Fig. 2 shows a clear deviation from typical DIS, even though the outgoing proton left through the beam pipe and remains undetected. Hence, the name rapidity gap events is also common for what was called di!ractive electroproduction in the previous paragraphs. The de"nition of di!raction used above is narrower than necessary. Without losing any of the qualitative results, the requirement of a "nal state proton P can be replaced by the requirement of a low-mass hadronic state >, well separated from the di!ractive state X . In this case, the + argument connecting elastically scattered proton and rapidity gap has to be reversed: the existence

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Fig. 2. Di!ractive event in the ZEUS detector ("gure from [3]).

of a gap between X and > becomes the distinctive feature of di!raction and, under certain + kinematic conditions, the interpretation of > as an excitation of the incoming proton, which is now almost elastically scattered, follows. However, this wider de"nition of di!raction has the disadvantage of introducing a further degree of freedom, namely, the mass of the proton excitation. If one insists on using only the previous three parameters, x, Q and m, the de"nition of a di!ractive event becomes ambiguous. More details are found in the discussion of the experimental results below and in the relevant experimental papers. 2.2. Fundamental observations Rapidity gaps are expected even if in all DIS events a quark is knocked out of the proton leaving a coloured remnant. The reason for this is the statistical distribution of the produced hadrons, which results in a small yet "nite probability for "nal states with little activity in any speci"ed detector region. However, the observations described below are clearly inconsistent with this explanation of rapidity gap events. The "rst analysis of rapidity gap events at HERA was performed by the ZEUS collaboration [3]. More than 5% of DIS events were found to possess a rapidity gap. The large excess of the event numbers compared to namK ve parton model expectations was soon con"rmed by an H1 measurement [4]. The analyses are based on the pseudo-rapidity g"!ln tan(h/2), where h is the angle of an outgoing particle relative to the beam axis. Pseudo-rapidity and rapidity are identical for massless particles; the di!erence between these two quantities is immaterial for the qualitative discussion below.

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Fig. 3. Distribution of g , the maximum rapidity of a calorimeter cluster in an event, measured at HERA ("gure

 from [3]).

In the ZEUS analysis, a rapidity g was de"ned as the maximum rapidity of a calorimeter

 cluster in an event. A cluster was de"ned as an isolated set of adjacent cells with summed energy higher than 400 MeV. The measured g distribution is shown in Fig. 3. (Note that the smallest

 detector angle corresponds to g "4.3; larger values are an artifact of the clustering algorithm.)

 To appreciate the striking qualitative signal of di!raction at HERA, the measured g distribu  tion has to be compared with namK ve expectations based on a purely partonic picture of the proton. This is best done using a parton-model-based Monte Carlo event generator. The corresponding g distribution, which is also shown in Fig. 3, is strongly suppressed at small g . This



 qualitative behaviour is expected since the Monte Carlo (for more details see [3] and Refs. therein) starts from a partonic proton, calculates the hard process and the perturbative evolution of the QCD cascade, and "nally models the hadronization using the Lund string model (see, e.g., [70]). According to the Lund model, the colour string, which connects all "nal state partons and the coloured proton remnant, breaks up via qq pair creation, thus producing the observed mesons. The rapidities of these particles follow a Poisson distribution, resulting in an exponential suppression of large gaps. It should be clear from the above discussion that this result is rather general and does not depend on the details of the Monte Carlo. QCD radiation tends to "ll the rapidity range between the initially struck quark and the coloured proton remnant with partons. A colour string connecting these partons is formed, and it is highly unlikely that a large gap emerges in the "nal state after the break-up of this string. However, the data shows a very di!erent behaviour. The expected exponential decrease of the event number with g is observed only above g K1.5; below this value a large plateau is seen.



 Thus, the namK ve partonic description of DIS misses an essential qualitative feature of the data, namely, the existence of non-suppressed large rapidity gap events. To give a more speci"c discussion of the di!ractive event sample, it is necessary to de"ne which events are to be called di!ractive or rapidity gap events. It is clear from Fig. 3 that, on a qualitative level, this can be achieved by an g cut separating the events of the plateau. The resulting

 qualitative features, observed both by the ZEUS [3] and H1 collaborations [4], are the following.

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There exists a large rapidity interval where the g distribution is #at. For high cHp energies =,

 the ratio of di!ractive events to all DIS events is approximately independent of =. The Q dependence of this ratio is also weak, suggesting a leading-twist contribution of di!raction to DIS. Furthermore, the di!ractive mass spectrum is consistent with a 1/M distribution. A number of additional remarks are in order. Note "rst that the observation of a #at g distri  bution and of a 1/M spectrum are interdependent as long as masses and transverse momenta of "nal state particles are much smaller than M. To see this, observe that the plus component of the most forward particle momentum and the minus component of the most backward particle momentum are largely responsible for the total invariant mass of the di!ractive "nal state. This gives rise to the relation dM/M"d ln M&dg , which is equivalent to the desired result.

 Furthermore, it has already been noted in [4] that a signi"cant contribution from exclusive vector meson production, e.g., the process cHpPop, is present in the rapidity gap event sample. A more detailed discussion of corresponding cross-sections, which have by now been measured, and of relevant theoretical considerations is given in Sections 5 and 7. The discussion of other, more speci"c features of the di!ractive "nal state, such as the presence of charmed mesons or high-p , jets, is also postponed. 2.3. Diwractive structure function The di!ractive structure function, introduced in [71] and "rst measured by the H1 collaboration [72], is a powerful concept for the analysis of data on di!ractive DIS, which is now widely used by experimentalists and theoreticians. Recall the relevant formulae for inclusive DIS (see, e.g., [73]). The cross-section for the process epPeX can be calculated if the hadronic tensor, 1 (2.3) = (P, q)" 1P" jR (0)"X21X" j (0)"P2(2p)d(q#P!p ) , J I 6 IJ 4p 6 is known. Here j is the electromagnetic current, and the sum is over all hadronic "nal states X. Because of current conservation, q ) ="= ) q"0, the tensor can be decomposed according to









q q 1 = (P, q)" g ! I J = (x, Q)# P # q IJ  I 2x I IJ q






1 P # q = (x, Q) . J 2x J 

(2.4)

The data is conveniently analysed in terms of the two structure functions F (x, Q)"(P ) q) = (x, Q) , (2.5)   F (x, Q)"(P ) q) = (x, Q)!2x= (x, Q) . (2.6) *   Introducing the ratio R"F /(F !F ), the electron}proton cross-section can be written as *  * dp 4pa y CNC6 "  1!y# F (x, Q) , (2.7) dx dQ xQ 2[1#R(x, Q)] 





where y"Q/sx, and s is the electron}proton centre-of-mass energy squared. In the namK ve parton model or at leading order in a in QCD, the longitudinal structure function F (x, Q) vanishes, and Q *

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R"0. Since R corresponds to the ratio of longitudinal and transverse virtual photon crosssections, p /p , it is always positive, and the corrections associated with a non-zero R are small at * 2 low values of y. In the simplest de"nition of di!raction, the inclusive "nal state X is replaced by the state X P, + which consists of a di!ractively produced hadronic state with mass M and the scattered proton. This introduces the two additional kinematic variables m and t. However, no additional independent 4-vector is introduced as long as the measurement is inclusive with respect to the azimuthal angle of the scattered proton. Therefore, the decomposition in Eq. (2.4) remains valid, and the two di!ractive structure functions F"(x, Q, m, t) can be de"ned. The di!ractive cross-section reads *





4pa y dp  1!y# CNCN6+ " F"(x, Q, m, t) , xQ 2[1#R"(x, Q, m, t)]  dx dQ dm dt

(2.8)

where R""F"/(F"!F"). In view of the limited precision of the data, the dominance of the *  * small-y region, and the theoretical expectation of the smallness of F", the corrections associated * with a non-zero value of R" are neglected in the following. A more inclusive and experimentally more easily accessible quantity can be de"ned by performing the t integration,



F"(x, Q, m)" dt F"(x, Q, m, t) .  

(2.9)

The results of the "rst measurement of this structure function, performed by the H1 collaboration, are shown in Fig. 4. (The underlying cross-section includes events with small-mass excitations of the proton in the "nal state.) Far more precise measurements, since performed by both the H1 and ZEUS collaborations, are discussed in Section 7 together with di!erent theoretical predictions. The main qualitative features of di!ractive electroproduction, already discussed in the previous section, become particularly apparent if the functional form of F" is considered. The b and  Q dependence of F" is relatively #at. This corresponds to the observations discussed earlier that  di!raction is a leading twist e!ect and that the mass distribution is consistent with a 1/M 6 spectrum. The success of a 1/mL "t, with n a number close to 1, re#ects the approximate energy independence of the di!ractive cross-section. More speci"cally, however, and in view of more precise recent measurements, it can be stated that the "tted exponent is above 1, so that a slight energy growth of the di!ractive cross-section is observed. This will be discussed in more detail later on. Note "nally that the formal de"nition of F" as an integral of F" [72] is not easy to   implement since the outgoing proton or proton excitation is usually not tagged. Therefore, most measurements rely on di!erent kinematic cuts, in particular an g cut, and on models of the

 non-di!ractive DIS background. A somewhat di!erent de"nition of F", based on the subtraction  of &conventional DIS' in the M distribution, was introduced in the ZEUS analysis of [74]. From 6 a theoretical perspective, the direct measurement of F" by tagging the outgoing proton appears  most desirable. Recently, such a measurement has been presented by the ZEUS collaboration [75] although the statistics are, at present, far worse than in the best available direct analyses of F" [76,77]. 

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Fig. 4. First measurement of the di!ractive structure function F"(x, Q, m). The "t is based on a factorizable  m dependence of the form m\  ("gure from [72]).

3. Semiclassical approach In this section, the main physical idea and the technical methods of the semiclassical approach to di!raction are introduced. Although historically the semiclassical approach is not the "rst relevant model, it is, in the author's opinion, well suited as a starting point for the present review. On the one hand, it is su$ciently simple to be explained on a technical level within the limited space available. On the other hand, it allows for a clear demonstration of the interplay between the relevant kinematics and the fundamental QCD degrees of freedom. The underlying idea is very simple. From the proton rest frame point of view, the very energetic virtual photon develops a partonic #uctuation long before the target. The interaction with the target is modelled as the scattering o! a superposition of soft target colour "elds, which, in the high-energy limit, can be calculated in the eikonal approximation. Di!raction occurs if this partonic system is quasi-elastically scattered o! the proton. This means, in particular, that both the target and the partonic #uctuation remain in a colour singlet state.

A. Hebecker / Physics Reports 331 (2000) 1}115

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3.1. Eikonal formulae for high-energy scattering The amplitude for an energetic parton to scatter o! a given colour "eld con"guration is a fundamental building block in the semiclassical approach. This amplitude is the subject of the present section. The two other basic ingredients for the di!ractive cross-section, i.e., the amplitudes for the photon to #uctuate into di!erent partonic states and the integration procedure over all colour "eld con"gurations of the target proton are discussed in the remainder of this section. The essential assumptions are the softness and localization of the colour "eld and the very large energy of the scattered parton. Localization means that, in a suitable gauge, the colour "eld potential A (x) vanishes outside a region of size &1/K, where K is a typical hadronic scale. Later I on, it will be assumed that typical colour "eld con"gurations of the proton ful"l this condition. Softness means that the Fourier decomposition of A (x) is dominated by frequencies much smaller I than the energy of the scattered parton. The assumption that this holds for all "elds contributing to the proton state is a non-trivial one. It will be discussed in more detail at the end of this section. The relevant physical situation is depicted in Fig. 5, where the blob symbolizes the target colour "eld con"guration. Consider "rst the case of a scalar quark that is minimally coupled to the gauge "eld via the Lagrangian L "(D U)H(DIU)   I with the covariant derivative

(3.1)

D "R #igA . (3.2) I I I In the high-energy limit, where the plus component of the quark momentum becomes large, the amplitude of Fig. 5 then reads i2pd(k !k )¹"2pd(k !k )2k [;I (k !k )!(2p)d(k !k )] . (3.3)      , , , , It is normalized as is conventional for scattering processes o! a "xed target (see, e.g., [78]). The expression in square brackets is the Fourier transform of the impact parameter space amplitude, ;(x )!1, where , ig  (3.4) A (x , x )dx ;(x )"P exp ! \ > , > , 2 \ is the non-Abelian eikonal factor. The unit matrix 13S;(N ), with N the number of colours, A A subtracts the "eld independent part, and the path ordering operator P sets the "eld at smallest x to the rightmost position. > This formula or, more precisely, its analogue in the more realistic case of a spinor quark was derived by many authors. In the Abelian case, the high-energy amplitude was calculated in [6] in






Fig. 5. Scattering of a quark o! the target colour "eld.

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Fig. 6. Typical diagrammatic contribution to the eikonal amplitude, Eq. (3.3). Attachments of gluon lines with crosses correspond to vertices at which the classical external "eld appears.

the framework of light-cone quantization. This result was taken over to QCD in [45]. A derivation in covariant gauge, based on the solution of the equation of motion for a particle in the colour background "eld, was given in [46]. In [48], the amplitude for the scattering of a fast gluon o! a soft colour "eld was derived by similar methods. For completeness, a derivation of the amplitude in Eq. (3.3), based on the summation of diagrams of the type shown in Fig. 6, is given in Appendix A of the present review. For the purpose of this section, it is su$cient to explain the main elements of Eq. (3.3) in a physical way, without giving the technical details of the derivation. To begin with, it is intuitively clear that the eikonal factor ;(x ) appears since the fast parton, , travelling in x direction and passing through the target at transverse position x , is rotated in > , colour space by the "eld A (x) that it encounters on its way. I Furthermore, the above amplitude is given for the situation in which the parton is localized at x K0. This co-ordinate of the fast parton does not change during the scattering process. Thus, \ A is always evaluated at x K0, and the x dependence is not shown explicitly. I \ \ The energy d-function in Eq. (3.3) is an approximate one. It appears because the energy of the parton cannot be signi"cantly changed by the soft colour "eld. Finally, due to the explicit factor k , the amplitude grows linearly with the parton energy, as is  expected for a high-energy process with t-channel exchange of vector particles, in this case, of gluons. The amplitude of Eq. (3.3) is easily generalized to the case of a spinor quark, where the new spin degrees of freedom are characterized by the indices s and s (conventions of Appendix B). In the eikonal approximation, which is valid in the high-energy limit, helicity #ip contributions are suppressed by a power of the quark energy (see Appendix A for more details). Thus, the generalization of Eq. (3.3) reads i2pd(k !k )¹ "2pd(k !k )2k [;I (k !k )!(2p)d(k !k )]d . (3.5)   QQY    , , , , QQY Similarly, the amplitude for the scattering of a very energetic gluon o! a soft colour "eld is readily obtained from the basic formula, Eq. (3.3). Note that, although the fast gluon and the gluons of the target colour "eld are the same fundamental degrees of freedom of QCD, the semiclassical approximation is still meaningful since an energy cut can be used to de"ne the two di!erent types of "elds. The polarization of the fast gluon is conserved in the scattering process. The main di!erence to the quark case arises from the adjoint representation of the gluon, which determines the representation of the eikonal factor. Thus, the amplitude corresponding to Fig. 7 reads (3.6) i2pd(k !k )¹ "2pd(k !k )2k [;I A(k !k )!(2p)d(k !k )]d , , , , , HHY   HHY    where j, j are the polarization indices and ;I A is the Fourier transform of ;A(x ), the adjoint , representation of the matrix ;(x ) de"ned in Eq. (3.4). ,

A. Hebecker / Physics Reports 331 (2000) 1}115

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Fig. 7. Scattering of a gluon o! the target colour "eld.

Fig. 8. Electroproduction of a qq pair o! the target colour "eld.

Clearly, the above eikonal amplitudes have no physical meaning on their own since free incoming quarks or gluons cannot be realized. However, they serve as the basic building blocks for the high-energy scattering of colour neutral objects discussed in the next section. 3.2. Production of qq pairs In this section, the eikonal approximation is used for the calculation of the amplitude for qq pair production o! a given target colour "eld [48]. Both di!ractive and inclusive cross-sections are obtained from the same calculation, di!raction being de"ned by the requirement of colour neutrality of the produced pair. The qualitative results of this section are una!ected by the procedure of integrating over all proton colour "eld con"gurations, which is discussed in Section 3.4. The process is illustrated in Fig. 8. The corresponding ¹ matrix element has three contributions, (3.7) ¹"¹  #¹ #¹  , O O OO where ¹  corresponds to both the quark and antiquark interacting with the "eld, while ¹ and O OO ¹  correspond to only one of the partons interacting with the "eld. O Let < (p, p) and < (k, k) be the e!ective vertices for an energetic quark and antiquark interO O acting with a soft gluonic "eld. For quarks with charge e, the "rst contribution to the amplitude of Eq. (3.7) reads



i i dk u (p)< (p, p) e. (q) < (k, k)v (k) , i2pd(k #k !q )¹  "ie QY O PY    OO p. !m !k. !m O (2p)

(3.8)

where q"p#k by momentum conservation, e(q) is the polarization vector of the incoming photon, and r, s label the spins of the outgoing quarks. At small x, the quark and antiquark have large momenta in the proton rest frame. Hence, the propagators in Eq. (3.8) can be treated in a high-energy approximation. In a co-ordinate system where the photon momentum is directed along the z-axis, the large components are p and k . It is > > convenient to introduce, for each vector k, a vector kM whose minus component satis"es the mass shell condition, kM "(k #m)/k , \ , >

(3.9)

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while the other components are identical to those of k. The propagators in Eq. (3.8) can be rewritten according to the identities 1/(k. #m)" v (kM )v (kM )/(k!m)#c /2k , P P > > P

(3.10)

1/(p. !m)" u (p )u (p )/(p!m)#c /2p . (3.11) Q Q > > Q To obtain the "rst term in a high-energy expansion of the scattering amplitude ¹  , the terms OO proportional to c in Eqs. (3.10) and (3.11) can be dropped. > After inserting Eqs. (3.10) and (3.11) into Eq. (3.8), the relations u (p)< (p, p)u (p)"2pd(p !p )2p [;I (p !p )!(2p)d(p !p )]d , (3.12) QY O Q    , , , , QQY v (k)< (k, k)v (k)"2pd(k !k )2k [;I R(k !k )!(2p)d(k !k )]d , (3.13) P O PY    , , , , PPY which correspond to Eq. (3.5) and its antiquark analogue, can be applied. Writing the loop integration as dk"(1/2)dk dk dk and using the approximation d(l )K2d(l ) for the energy > \ ,  > d-functions, the k integration becomes trivial. The k integral is done by closing the integration > \ contour in the upper or lower half of the complex k plane. The result reads \ ie a(1!a) q dk u (p )e. (q)v (kM ) ¹  "! , N#k QY PY OO 4p > , ;[;I (p !p )!(2p)d(p !p )][;I R(k !k )!(2p)d(k !k )] (3.14) , , , , , , , , where



p "(1!a)q , k "aq , N"a(1!a)Q#m . (3.15) > > > > Thus, a and 1!a characterize the fractions of the photon momentum carried by the two quarks, while (N#k ) measures the o!-shellness of the partonic #uctuation before it hits the target. In the , following, the quark mass is set to m"0. The above expression for ¹  contains terms proportional to ;;R, ;, and ;R, as well as OO a constant term. The amplitudes ¹ and ¹  , which contain terms proportional to ; and ;R and O O a constant term, are derived by the same methods. Calculating the full amplitude according to Eq. (3.7), the terms proportional to ; and ;R cancel. Thus, the colour "eld dependence of ¹ is given by the expression [;I (p !p );I R(k !k )!(2p)d(p !p )d(k !k )] . , , , , , , , , Introducing the fundamental function

(3.16)

(3.17) = , (y )";(x );R(x #y )!1 , , , , V , which encodes all the information about the external "eld, the complete amplitude can eventually be given in the form





ie a(1!a) ¹"! q dk u (p )e. (q)v (kM ) e\ D, V, = I , (k !k ) , , N#k QY PY V , , 4p > , V ,

(3.18)

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where = I , is the Fourier transform of = , (y ) with respect to y , and D "k #p is the total V V , , , , , transverse momentum of the "nal qq state. From the above amplitude, the transverse and longitudinal virtual photon cross-sections are calculated in a straightforward manner using the explicit formulae for u (p )e. (q)v (kM ) given in QY PY Appendix B. Summing over all qq colour combinations, as appropriate for the inclusive DIS cross-section, the following result is obtained:

   



dp 2eQ = I (k !k )  * " , , (3.19) (a(1!a)) dk V, , , N#k da dk (2p) V, , , dp e k = I (k !k )  2 " , . (3.20) (a#(1!a)) dk , V, , , da dk N#k 2(2p) V, , , Note that only a single integration over transverse coordinates appears. This is a consequence of the d-function induced by the phase space integration over D , applied to exp[!iD x ] from , , , Eq. (3.18) and to the corresponding exponential from the complex conjugate amplitude. The contraction of the colour indices of the two = matrices is implicit. Consider the longitudinal cross-section in more detail. The integrand can be expanded around k "k . Shifting the integration variable k to l "k !k , the Taylor expansion of the , , , , , , denominator in powers of l yields , 1 1 2l k , , #2 . " ! (3.21) N#(k #l ) N#k (N#k) , , , , From the de"nition of the colour matrix = , (y ) in Eq. (3.17), it is clear that V ,

 



dl = I (!l )"(2p)= , (0)"0 , , V, , V

(3.22)

dl l = I (!l )"i(2p)R = , (0) . , , V, , , V

(3.23)

Using rotational invariance, i.e., k k Pd k, the result G H  GH , dp 4ea(1!a) Nk * " , "R = , (0)" , V da dk (2p)(N#k ) , V , , is obtained. It evaluates to the total longitudinal cross-section



e p " * 6pQ

(3.24)



"R = , (0)" . (3.25) , V V, The transverse contribution can be evaluated in a similar way. In the perturbative region, where a(1!a)


e[a#(1!a)] [N#k] dp , 2 " "R = , (0)" . (3.26) , V 2(2p)(N#k ) da dk V, , , While, in the longitudinal case, the a and k integrations were readily performed, a divergence is , encountered in the transverse case, Eq. (3.26). The k integration gives rise to a factor 1/a(1!a), so ,

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that the a integral diverges logarithmically at aP0 and at aP1. Thus, the region of small a(1!a), where the expansion around k "k does not work, is important for the total cross-section , , p , which therefore cannot be obtained by integrating Eq. (3.26). Instead, the endpoint region, 2 where the large distance structure of = , (y ) is important, can be separated by a cut-o! k V , (K;k;Q). The complete leading twist result for the transverse cross-section, where contributions suppressed by powers of K/k or k/Q have been dropped, reads






Q e ln !1 p "" 2 k 6pQ



V,

"R = , (0)" , V

  



e I/ k = I (k !k )  , # da dk  dk , V, , . (3.27) , , (2p) N#k  V, , The resulting physical picture can be summarized as follows. For longitudinal photon polarization, the produced qq pair has small transverse size and shares the photon momentum approximately equally. Only the small distance structure of the target colour "eld, characterized by the quantity "R = , (0)", is tested. For transverse photon polarization, an additional leading twist , V contribution comes from the region where a or 1!a is small and k&K. In this region, the qq , pair penetrating the target has large transverse size, and the large distance structure of the target colour "eld, characterized by the function = , (y ) at large y , is tested. This physical picture, , V , known as the aligned jet model, was introduced in [5] on a qualitative level and was used more recently for a quantitative discussion of small-x DIS in [7]. It is now straightforward to derive the cross-sections for the production of colour singlet qq pairs corresponding, within the present approach, to di!ractive processes. Note that the cross-sections in Eqs. (3.19) and (3.20) can be interpreted as linear functionals of tr(= , (y )=R, (y )), where the V , V , trace appears because of the summation over all colours of the produced qq pair in the "nal state. Introducing a colour singlet projector into the underlying amplitude corresponds to the substitution 1 (3.28) tr(= , (y )=R, (y ))P tr = , (y )tr =R, (y ) V , V , V , V , N A in Eqs. (3.19) and (3.20). This change of the colour structure has crucial consequences for the subsequent calculations. Firstly, the longitudinal cross-section, given by Eqs. (3.24) and (3.25), vanishes at leading twist since the derivative R = , (0) is in the Lie-algebra of S;(N ), and therefore tr R = , (0)"0. A , V , V Secondly, for the same reason the ln Q term in the transverse cross-section, given by Eq. (3.27), disappears. The whole cross-section is dominated by the endpoints of the a integration, i.e., the aligned jet region, and therefore determined by the large distance structure of the target colour "eld. At leading order in 1/Q, the di!ractive cross-sections read p""0 , *

(3.29)

   



 e k tr = I , (k !k )  , V , dk , p"" da dk  . (3.30) , , 2 (2p)N N#k V, A  , The cut-o! of the a integration, k/Q, has been dropped since, due to the colour singlet projection, the integration is automatically dominated by the soft endpoint.

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In summary, the leading-twist cross-section for small-x DIS receives contributions from both small- and large-size qq pairs, the latter corresponding to aligned jet con"gurations. The requirement of colour neutrality in the "nal state suppresses the small-size contributions. Thus, leading twist di!raction is dominated by the production of pairs with large transverse size testing the non-perturbative large-distance structure of the target colour "eld. 3.3. Higher Fock states Given the leading order results for qq pair production of the last section, it is natural to ask about the importance of radiative corrections. A systematic procedure for calculating to all orders in perturbation theory does not yet exist in the semiclassical framework. However, as will be seen in the following chapters, the summation of leading logarithms in Q is understood. Here, the particularly important case of the di!ractive production of a quark}antiquark}gluon system (see Fig. 9) is discussed in some detail [48]. The purpose of this discussion is to establish, as one of the essential features of di!ractive DIS, the necessary presence of a wee parton in the wave function of the incoming virtual photon. Consider the sum of diagram Fig. 9a and its analogue with the gluon radiated from the antiquark. If the quark propagators with momenta p and l are rewritten according to Eqs. (3.10) and (3.11) and the c terms are dropped, the corresponding amplitude can be given in the form > dk dl ¹ ¹ . (3.31) i2pd(p #k #l !q )¹ " QYQPYPHYH QPH      (2p) (2p) QPH Here ¹ is the amplitude for the scattering of the qq g system o! the target colour "eld, and ¹ is the remainder of the diagram, describing the #uctuation of the virtual photon into the partonic state. In the following calculation, the k integration will be performed in such a way that the \ gluon propagator goes on shell. Anticipating this procedure, the sum over the intermediate gluon polarizations j is restricted to the two physical polarizations, de"ned with respect to the on-shell vector kM . According to Eqs. (3.5) and (3.6), the contribution to ¹ where quark, antiquark and gluon interact with the "eld reads

 

(¹ ) "i2pd(p !p )2p [;I (p !p )!(2p)d(p !p )]d OOE QYQPYPHYH    , , , , QYQ ;i2pd(l !l )2l [;I R(l !l )!(2p)d(l !l )]d    , , , , PYP ;i2pd(k !k )2k [;I A(k !k )!(2p)d(k !k )]d .    , , , , HYH

(3.32) (3.33) (3.34)

Fig. 9. Diagrams for the process cHPqq g. Two similar diagrams with the gluon radiated from the produced antiquark have to be added.

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As explained at the beginning of Section 3.2, contributions where not all of the partons interact with the "eld have to be added. The photon-qq g transition amplitude ¹, with all colour indices suppressed, reads





i i ieg Q u (p ) e. (kM ) e. (q)#e. (q) e. (kM ) v (lM ) . ¹ " H p. #k. !m P QPH pkl Q !l. !k. !m H

(3.35)

The colour structure of ¹ is given by the combination of the ; matrices in ¹, the indices of  which are partially contracted according to the perturbative amplitude ¹. This colour structure is characterized by the colour tensor



(= I )? " ?@

V, W, X,

e V, N, \N, >W, I, \I, >X, J, \J,  (=(x , y , z ))? , , , , ?@

(3.36)

where (=(x , y , z ))? "(;A(y ))?@(;(x )¹@;R(z )) !¹? , (3.37) , , , ?@ , , , ?@ ?@ and ¹? are the conventional S;(N ) generators with adjoint (a"1,2, N!1) and fundamental ?@ A A (a, b"1,2, N ) indices. The notation = is chosen to stress the similarity with the qq case, where A the external "eld is tested by the function =,=, de"ned in Eq. (3.17). In analogy to this equation, the last term in Eq. (3.37) subtracts the unphysical contribution where none of the partons is scattered by the external "eld. One can think of x , y and z as the transverse positions , , , at which quark, gluon and antiquark penetrate the proton "eld, picking up corresponding non-Abelian eikonal factors. The indices a, b and a correspond to the colours of the produced quark, antiquark and gluon. To obtain the complete result, the amplitude ¹ , which is the sum of the diagram in Fig. 9b and 
 its analogue with the gluon radiated from the antiquark, has to be added. The calculation is similar to the case of the amplitude ¹ . Since the gluon is radiated after the quark pair passes the target  "eld, the colour structure is determined by the same function = that appeared in the qq production amplitude of the last section. The diagrams in Fig. 9, with summation over all possible colours of the qq g "nal state, represent an a correction to inclusive qq pair production according to Fig. 8. The factor a is accompanied Q Q by a factor ln Q. Higher Fock states lead to additional factors a ln Q, and an all-orders Q summation can be performed using standard renormalization group techniques, i.e., Altarelli} Parisi evolution. The implementation of this summation in the semiclassical framework is explained in more detail at the end of Section 4.4 and in Appendix D. Hence, the inclusive case is not further discussed in this section. For the di!ractive case, the following situation can be anticipated on the basis of the experience of the last section. There are three qualitatively di!erent kinematic con"gurations. First, if all three transverse momenta, p , k and l are soft, the large distance structure of the target "eld is tested, , , , and a leading twist contribution to the DIS cross-section results. This contribution will not be calculated here (cf., however, the discussion in Section 4.4). Second, if all three momenta are large, p , k , l 
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Fig. 10. Space}time picture in the case of fast, high-p quark and antiquark, passing the proton at small transverse , separation, with a relatively soft gluon further away.

Fig. 11. Space}time picture in the case of fast, high-p quark and gluon, passing the proton at small transverse , separation, with a relatively soft antiquark further away.

di!ractive contribution arises. The two possible situations, depicted in Figs. 10 and 11, are discussed quantitatively below. The results of this calculation justify the physical picture outlined above. Consider "rst the case of high-p quark and antiquark, i.e., p , l


De"ning D,p#k#l!p!k!l to be the total momentum transferred from the proton, = I  can be given in the form



e\ V, D,

(= I )? " ?@



e W, I, \I, >X, J, \J,  =(x , x #y , x #z )? , , , , , , ?@

(3.40) V, W, X, where l is given by Eq. (3.38). The l integration gives a d-function of the variable z , thus , , , resulting in the "nal formula





dl , (= I )? " ?@ (2p)

e\ V, D,



e W, I, \I, =(x , x #y , x )? . , , , , ?@

(3.41) V, W, This expression shows that in the kinematic situation with two high-p quark jets and a relatively , soft gluon the leading twist contribution is not a!ected by the transverse separation of the quarks. It is the transverse separation between quark-pair and gluon which tests large distances in the proton "eld and which can lead to non-perturbative e!ects. The colour singlet projection of the colour tensor = reads



S(=)"

2 (=)? ¹? . ?@ @? N!1 A

(3.42)

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Using the identity (;A)?@"2 tr[;\¹?;¹@] ,

(3.43)

where ;3S;(N ), the contribution relevant for di!raction, i.e., the production of a colour singlet A qq g-system, takes the form





dl , S(=)" (2p)

e\ V, D,

V,

1 (2(N!1) A

tr[= I A, (k !k )] , V , ,

(3.44)

where =A, (y )";A(x );AR(x #y )!1 . (3.45) , , , V , This is analogous to the quark pair production of the previous section (cf. Eq. (3.17)). However, now the two lines probing the "eld at positions x and x #y correspond to matrices in the , , , adjoint representation. An intuitive explanation of this result is that the two high-p quarks are , close together and are rotated in colour space like a vector in the octet representation (cf. Fig. 10). To make this last statement more precise, recall that an upper bound for the Io!e-time of the #uctuation with two high-p quarks is given by q /p . This means that the distance between the ,  , point where the virtual photon splits into the qq -pair and the proton cannot be larger than q /p .  , As long as the pair shares the longitudinal momentum of the photon approximately equally, i.e., a(1!a)"O(1), the opening angle is &p /q . Hence, when quark and antiquark hit the proton, ,  their transverse distance is &1/"p ";1/K. , The diagram in Fig. 9b and its analogue with the gluon radiated from the antiquark do not contribute to the di!ractive cross-section for high-p qq pair production. This is obvious since the , colour "eld is probed only by the qq system, the small transverse size of which prevents colour singlet exchange. Thus, the full cross-section follows from Eq. (3.31), the simpli"ed colour tensor of Eq. (3.44), and the formulae of Appendix B, which are used for the evaluation of the qq c and qq g vertices in Eq. (3.35). The results for longitudinal and transverse photon polarization read aQp a a dp ,  Q * " f (aQK , k ) , , da dp da dk 2p(N!1) [a(1!a)]QK   A , , a[a#(1!a)][p#N] a a dp ,  Q 2 " f (aQK , k ) ,  , [a(1!a)]QK  da dp da dk 16p(N!1) , , A

(3.46) (3.47)

with

 


f (aQK , k )"  ,





dk I A, (k !k )  2kG kH tr = V , , dGH# , , , , (2p) aQK #k aQK  ,

V, p , , QK "Q# a(1!a)

N"a(1!a)Q .

(3.48) (3.49)

At this point, the fundamental assumption of the softness of the gluon can be quantitatively justi"ed. To achieve this, "x a and p and perform the integration over a and k . As in Eq. (3.30), , , these two integrations are dominated by the region a;1 and k&K. Thus, the di!ractive , production of a qq g system with high-p quark and antiquark occurs predominantly in a generalized ,

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aligned jet con"guration, where the gluon is the wee parton in the photon wave function. It is this soft gluon that is responsible for the resulting leading twist cross-section with colour singlet exchange. As already pointed out above, another leading twist contribution to di!raction arises from the kinematic region where the quark or the antiquark is the soft particle and, accordingly, the outgoing antiquark}gluon or quark}gluon system has large transverse momentum. The calculation proceeds along the same lines as in the soft gluon case. The main qualitative di!erence is in the colour structure. In analogy to the discussion leading to Eq. (3.41), the large p of quark , and gluon results in a small transverse separation (cf. Fig. 11), so that e!ectively the high-p , quark}gluon system is colour rotated like a single quark. Accordingly, the "eld is tested by the same function =, built from ; matrices in the fundamental representation, that appeared in the qq production cross-section. Notice also that, in contrast to the soft gluon case, the diagram in Fig. 8b has to be taken into account. The "nal results for the transverse and longitudinal cross-sections read a a (N!1) Q dp * "  Q A f (aQK , k ) , , 2pN a(1!a)QK   da dp da dk A , , dp a a (N!1) QK #Q 2 "  Q A QK !2Q(QK #Q)# f (aQK , k ) , , da dp da dk 8pNp QK  a(1!a)  , , A ,





(3.50) (3.51)

with

 



dk k tr = I , (k !k )  , , V , , . (3.52) (2p) aQK #k V, , These cross-sections include both the soft quark and the soft antiquark regions. The kinematic variables do not correspond to Fig. 9. They are generic in the sense that p and !p are the , , transverse momenta of the two hard jets, a and 1!a are the corresponding momentum fractions, and the soft parton, in this case the quark or the antiquark, is characterized by the transverse momentum k and the longitudinal momentum fraction a. These conventions are chosen to , emphasize the similarity with the soft gluon result of Eqs. (3.46)}(3.49). Again, the a and k integrations are dominated by the region a;1 and k&K. , , f (aQK , k )"  ,

3.4. Field averaging So far, electroproduction o! a "xed &soft' colour "eld has been considered. As a consequence, electroproduction cross-sections approach constant values as xP0. However, a proper treatment of the target requires the integration over all relevant colour "eld con"gurations. Following the discussion of [48], consider "rst the elastic scattering of a quark o! a proton. Although this process is unphysical since quarks are con"ned, it can serve to illustrate the method of calculation. Therefore, in the following, con"nement is ignored, and quarks are treated as asymptotic states. The generalization to the physical case of electroproduction is straightforward and will be discussed subsequently. A point-like quark with initial momentum q scatters o! the proton, which is a relativistic bound state with initial momentum p. Let m be the proton mass, and s and t the usual Mandelstam N variables for a 2P2 process. In the high-energy limit, s
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annihilation of the incoming quark with an antiquark of the proton is negligible. The amplitude is dominated by diagrams with a fermion line going directly from the initial to the "nal quark state. Therefore, the proton can be described by gluon "elds only. It is convenient to formulate our approach in terms of the SchroK dinger representation of quantum "eld theory (cf. [79]). In this representation, the Hilbert space is identi"ed with the linear space of wave functionals U[A], where the argument runs over all smooth (classical) "elds A(x). Given a state that is characterized by a functional U, the probability for the quantum "eld to assume the value A(x) is proportional to "U[A]". In the present context, the target proton is described by the SchroK dinger wave functional U [A] depending on the gluon "eld only. Quarks are . integrated out, yielding a modi"cation of the gluonic action. For a scattering process, the amplitude can be written in the proton rest frame as





2 DAe 1  1q"q2 . (3.53) 1qP"qP2" lim DA DA UH [A ]U [A ] 2 \2 .Y 2 . \2  \2  2 Here the "elds A and A are de"ned on three-dimensional surfaces at constant times !¹ and \2 2 ¹, and A is de"ned in the four-dimensional region bounded by these surfaces. The "eld A has to coincide with A and A at the boundaries, and the action S is de"ned by an integration over \2 2 the domain of A. The amplitude 1q"q2 describes the scattering of a quark by the given external  "eld A. The initial state proton, having well de"ned momentum P, is not well localized in space. However, the dominant "eld con"gurations in the proton wave functional are localized on a scale K&K . The "eld con"gurations A(x, t), which interpolate between initial and "nal proton state, /!" are also localized in space at each time t. Assume that the incoming quark wave packet is localized such that it passes the origin x"0 at time t"0. At this instant the "eld con"guration A(x, t) is centred at





x[A], dx E (x) ) x 

dx E (x) , 

(3.54)

where E (x) is the energy density of the "eld A(x, t) at t"0. The amplitude (3.53) can now be  written as

 



2 DAe 1d(x[A]!x)1q"q2 . 1qP"qP2" lim dx DA DA UH U 2 \2 .Y .  \2 2 Using the transformation properties under translations, 1q"q2 x "e q\qYx1q"q2 , UH [¸x A]U [¸x A]"e P\PYxUH [A]U [A] , *  .Y . .Y . where ¸x A(y),A(y!x) ,

(3.55)

(3.56)

(3.57)

one obtains, 1qP"qP2"2m (2p)d(P #q!P!q) N



+ ,



1q"q2 . 

(3.58)

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Here + , denotes the operation of averaging over all "eld con"gurations contributing to the  proton state which are localized at x"0 at time t"0. It is de"ned by







1 2 W[A], lim DA DA UH U DAe 1d(x[A])W[A] (3.59) 2 \2 .Y . 2m + ,  \2 N 2 for any functional W. The normalization + , 1"1 follows from  1P"P2"2P (2p)d(P !P) . (3.60)  More complicated processes can be treated in complete analogy as long as the proton scatters elastically. In particular, the above arguments apply to the creation of colour singlet quark}antiquark pairs,



1qq "cH2 , (3.61)   where k and k are the sums of the momenta in the initial and "nal states, respectively. The  generalization of this simplest di!ractive process to a process with an additional fast "nal state gluon, cHPqq g (cf. Section 3.3), is straightforward. In contrast to the quark-proton scattering discussed above, here a colour neutral state is scattered o! the proton. Therefore no immediate contradiction with colour con"nement arises. However, it has to be assumed that the hadronization of the produced partonic state takes place after the interaction with the proton, which is described in terms of fast moving partons. The amplitude for the scattering o! a soft external "eld contains an approximate energy d-function, giving rise to the de"nition of a functional F, 1qq P"cHP2"2m (2p)d(k !k ) N 

+ ,

(3.62) 1qq "cH2 "2pd(k#k !q)F[A] ,  O O where q, k , k  are the momenta of the incoming photon and the outgoing quark and antiquark, O O respectively. Since the energy transferred to the proton is small, Eq. (3.61) can now be written as



1qq P"cHP2"2m (2p)d(k !k ) N 

F[A] . (3.63)  When calculating the cross-section from Eq. (3.63), the square of the 4-momentum-conserving d-function translates into one 4-momentum-conserving d-function using Fermi's trick. The spatial part of this d-function disappears after the momentum integration for the "nal state proton. The squared amplitude, integrated over the phase space U of the outgoing proton and normalized to the total space}time volume, reads



+ ,



1 dU"1qq P"cHP2""4pm d(k#k !q) N O O <¹



F[A] + ,



,

(3.64)

 as it should for the scattering o! a superposition of external "elds. From the above discussion, a simple recipe for the calculation of di!ractive processes at high energy follows: The partonic process is calculated in a given external colour "eld, localized at x"0 at time t"0. The weighted average over all colour "elds contributing to the proton state is taken on the

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amplitude level. It is assumed that the typical contributing "eld is smooth on a scale K and is localized in space on the same scale K. Finally, the cross-section is calculated using standard formulae for the scattering o! an external "eld. The above discussion was limited to the case of di!raction and did not address the important question of how inclusive and di!ractive electroproduction processes are interrelated. To arrive at a combined treatment of both processes, additional assumptions concerning the treatment of the target colour "eld have to be introduced. A corresponding discussion, following the analysis of [65], is outlined below. As explained previously, in the semiclassical framework it is natural to expect di!raction to occur whenever the produced qq pair emerges in a colour singlet state. Non-di!ractive events are expected in the colour octet case. A combined treatment clearly forbids the use of the proton wave functional for the description of the "nal state. Thus, instead of the amplitude in Eq. (3.61), one has to consider the corresponding amplitude in a &mixed' representation, 1qq A"cHP2, where the "nal state consists of the outgoing qq pair and a colour "eld con"guration A. This formalism allows for both the creation of colour singlet and non-singlet partonic states. For simplicity, time evolution of the "eld between the actual scattering process and the moment at which the "nal state "eld con"guration A is de"ned is neglected. The squared amplitude, summed over all "elds A and normalized to the total space}time volume, reads





1 DA"1qq A"cHP2""4pm d(k#k !q) DA "U [A ]F[A ]" . N O O  N   <¹

(3.65)

Here the integral over A on the l.h. side replaces the phase space integral for the outgoing proton in Eq. (3.64). The index &loc' symbolizes that, on the r.h. side of Eq. (3.65), the integration is restricted to "elds localized at, say, x"0. This can be achieved using translation covariance of the proton wave functional and of the matrix element 1qq "cH2 , in an argument similar to the one leading to  Eqs. (3.59) and (3.64). When writing = , (y ), it has so far always been assumed that the functional dependence on the V , classical colour "eld con"guration A is implicit, so that one should really read = , (y )[A ]. As   V , can be seen from Eq. (3.65), the full inclusive cross-sections are obtained from the previous formulae by the substitution



tr(= , (y )[A ]=R, (y )[A ])P DA "U [A ]" tr(= , (y )[A ]=R, (y )[A ]) . (3.66) V ,  V ,   .  V ,  V ,  The same applies to the di!ractive cross-sections, which are obtained by introducing a colour singlet projector on the amplitude level (cf. Eq. (3.28)). Decomposing the "eld A in Eq. (3.65) into its Fourier modes AI (k), the path integral can be   written as





" “ dAI (k) , (3.67)  k ; q       where the cut-o! "q" is required to ensure that the basic precondition for the semiclassical treatment, the softness of the target colour "eld with respect to the momenta of the fast particles, is respected.

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This cut-o! induces a non-trivial energy dependence of the squared amplitude in Eq. (3.65) and therefore of both the inclusive and di!ractive cross-sections. At present, no complete derivation of the explicit form of that energy dependence from "rst principles exists. However, a number of interesting related developments, approaching the highenergy limit of QCD from the perspective of colour "elds and eikonalized interactions, can be found, e.g., in [80,81]. On the basis of the above qualitative picture, a soft, non-perturbative energy growth was ascribed to the input parton distributions used in the phenomenological analysis of [65] (see Sections 6.2 and 7.1 for more details). The discussion of the present section is closely related to Nachtmann's original proposal to treat high-energy hadron}hadron scattering in the eikonal approximation [46]. The above viewpoint corresponds to the rest frame of the proton, where the colour "eld encountered by the fast partons of the projectile is naturally considered to be part of the proton state. The viewpoint of [46] corresponds to the centre-of-mass frame of high-energy processes. In hadron}hadron scattering, both incoming particles are characterized by their parton content. These partons then interact via gluon "elds which belong to neither of the two hadrons. So far, this is completely general. However, once the eikonal approximation is used to treat the parton propagators in the colour "eld background, one is forced to rely on modelling the light-cone wave function of the proton. In this case, the method becomes more predictive but less general than the wave functional treatment of the proton target discussed above.

4. From soft pomeron to di4ractive parton distributions In the previous section, small-x DIS was described as the eikonalized interaction of a partonic #uctuation of the virtual photon with a superposition of colour "eld con"gurations of the proton. Di!raction occurs whenever the partonic #uctuation preserves its overall colour neutrality. An essential feature of this approach is its formulation exclusively in terms of the fundamental degrees of freedom of QCD. It is now appropriate to step back and take a look at di!raction from the historical perspective of soft hadronic physics (see [82] for a recent review of hadronic di!raction). It will soon become clear how partonic ideas arise in this framework and in which way they are related to the results of the last section. 4.1. Soft pomeron Before introducing the pomeron, a concept from soft hadronic physics, it is appropriate to give a simple argument why, even at very high photon virtualities, di!ractive DIS is largely a soft process. Such an argument was presented by Bjorken and Kogut in the framework of their aligned jet model [5] (see also [83]). The underlying physical picture is based on vector meson dominance ideas. At high energy or small x, the incoming photon with virtuality q"!Q #uctuates into a hadronic state with mass M, which then collides with the target (see Fig. 12a). The corresponding cross-section for transverse

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Fig. 12. Vector meson dominance inspired picture of inclusive (a) and di!ractive (b) electroproduction.

photon polarization is estimated by dp dP(M) 2& ) p(M) , dM dM

(4.1)

where the probability for the photon to develop a #uctuation with mass M is given by M dM dP(M)& , (M#Q)

(4.2)

and p(M) is the cross-section for this #uctuation to scatter o! the target. The above expression for dP(M) is most easily motivated in the framework of old-fashioned perturbation theory, where the energy denominator of the amplitude is proportional to the o!-shellness of the hadronic #uctuation, Q#M. If this is the only source for a Q dependence, the numerator factor M is necessary to obtain a dimensionless expression. (See the original derivation of Gribov [84] for more details.) Bjorken and Kogut assume that, for large M, the intermediate hadronic state typically contains two jets. In the rest frame of this state, the two jets are back-to-back. Given an isotropic distribution of the direction of the jet axis, there is only a small probability for this axis to be aligned with the cHp collision axis. To make this statement more precise, de"ne aligned con"gurations by the requirement p (K (where K is a soft hadronic scale and p is the transverse , , momentum of the jets relative to the cHp axis). Simple geometry implies that these aligned con"gurations are suppressed by a probability factor &K/M. Furthermore, Bjorken and Kogut assume that the interaction cross-section with the proton is small for con"gurations that are not aligned, i.e., con"gurations with high p . (This e!ect is now known under the name of colour , transparency.) By contrast, aligned con"gurations have a large, hadronic cross-section &1/K. Thus, p(M) can be estimated by the product this cross-section and the above probability factor for the appearance of an aligned con"guration, p(M)&(1/K) ) (K/M)&1/M. The resulting transverse cHp cross-section, 1 dp 2& , dM (M#Q)

(4.3)

can be interpreted as the total high-energy cross-section of target proton and aligned jet #uctuation of the photon, i.e., of two soft hadronic objects. Therefore, a similar elastic cross-section is expected, p"&p (cf. Fig. 12b). The resulting di!ractive structure function, as de"ned in the previous section, 2 2 reads F"(m, b, Q)&b/m . 

(4.4)

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It is interesting that the very simple arguments outlined above result in an expression for F" that captures two important features of the HERA data: the leading-twist nature of di!raction  and the approximate 1/m behaviour. The main problems of the model are the precise energy dependence, which is measured to be somewhat steeper than 1/m, and the limit bP0, where a constant or even rising behaviour of F" is observed.  The above discussion shows, in very simple terms, what was also one of the main qualitative results of the more technical treatment of the last section: di!ractive electroproduction is based on a soft hadronic high-energy cross-section. Such cross-sections are very successfully described within the framework of pomeron exchange [8]. Let us recall the basic underlying concepts (see [85] for a detailed discussion or [86] for a brief introduction). Using analyticity and crossing symmetry, the amplitude ¹ (s, t), depicted in Fig. 13, can be  related to the amplitude ¹   (s, t), where s"t, t"s, and bared numbers denote antiparticles.  The partial wave expansion for this crossed amplitude reads  ¹   (s, t)" (2l#1)a (s)P (cos h) , (4.5)  J J J where h is the centre-of-mass frame scattering angle, which is a function of s, t and the particle masses, and P are Legendre polynomials. Let the two functions a (l, t) with g"#1 and g"!1 J E be the analytic continuations to complex l of the two sequences +a (t), l"0, 2, 4,2, and J +a (t), l"1, 3, 5,2,. In the simplest non-trivial case, the only singularity of a (l, t) is a single J E t-dependent pole at l"a(t). It can then be shown that, in the limit sPR, ¹ (s, t)"b (t)b (t) f (a(t)) (s/s )?R , (4.6)    E  where s is an arbitrary scale factor, b and b are two unknown functions of t, and    f (a(t))"(1#ge\ p?R)/sin pa(t) (4.7) E is the signature factor, depending on the signature g of the relevant Regge trajectory a(t). If a (l, t) E has a more complicated analytic structure, the rightmost singularity in the l plane dominates the behaviour at large s. Within the present context, the essential predictions of the asymptotic expression Eq. (4.6) are the power-like energy dependence s?R and the factorization of the unknown t dependence into the two vertex factors b and b . This last feature, which underlies the graphic representation of   reggeon exchange in Fig. 13, is relevant if the same Regge trajectory governs di!erent scattering processes. Note also that, for positive t"s and integer l, a(t) describes the positions of poles of

Fig. 13. Scattering process 12P34 via reggeon exchange.

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the physical amplitude ¹   (s, t). Such poles are expected whenever an on-shell particle with  appropriate mass m"s and angular momentum l can be created in the collision of 1 and 3 . Indeed, most Regge trajectories pass through known physical states with mass m"t and angular momentum a(t). The Froissart bound [87] on the high-energy growth of total cross-sections, p p 4 ln(s/s ) , (4.8)  m  p where m is the pion mass and s is an unknown scale factor, implies that a(0)41 for all Regge p  trajectories. However, it was observed early on that a very good "t to pp and pp cross-sections, which were measured to rise at high energy, could be obtained by assuming the dominance of a single pole with a(0)'1 [88]. The corresponding trajectory, originally introduced with a(0)"1 [89], is known as the pomeron trajectory (cf. the Pomeranchuk condition of [90]). Donnachie and Landsho! demonstrated that a large set of di!erent hadronic cross-sections can be "tted with an intercept a/ (0)"1.08 [8]. In spite of the power-like growth of Eq. (4.6), the predicted cross-sections are so small that the Froissart bound is not violated below the Planck scale. It is then argued that unitarity is not a serious problem at all realistic energies. However, one should keep in mind that all following analyses based on a pomeron trajectory with a(0)'1 are, strictly speaking, not self-consistent in the framework of Regge theory. In fact, it is likely that the rightmost singularity in the complex l plane is not a single pole but a cut, in which case many of the following results would have to be reconsidered. Given the universal success of the Donnachie}Landsho! pomeron, it is natural to apply the same trajectory to the quasi-elastic amplitude of di!ractive electroproduction. Thus, the diagram in Fig. 12b is interpreted in terms of pomeron exchange (cf. Fig. 14). In this situation, the soft energy dependence of the pomeron and the measured t dependence of the proton}proton}pomeron vertex are naturally combined with the aligned jet model prediction of Eq. (4.3). Ignoring the vertex factor from the upper part of Fig. 13, one can write (4.9) dp /dt dM&b (t) s?/ R\/(M#Q) . 2 NN The non-trivial energy dependence represents a clear improvement compared to Eqs. (4.3) and (4.4). However, our ignorance of the upper vertex prevents an unambiguous prediction of the M distribution. In fact, the obtained suppression at large M, which corresponds to the region of small b, appears to be the main qualitative problem of the presented model. Furthermore, the treatment of the hard scale Q is rather namK ve in view of the impressive successes of QCD perturbation theory in the description of inclusive structure functions.

Fig. 14. Di!ractive electroproduction via pomeron exchange.

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As is shown in the next section, a better understanding of the M dependence is possible in the framework of Regge theory. The obtained results allow for an interpretation of the upper vertex in Fig. 14 in terms of a cH-pomeron collision and thus for the application of QCD perturbation theory. 4.2. Pomeron structure function It is the purpose of this section to explain the interpretation of Fig. 14 in terms of a cH-pomeron collision, thereby providing the background for the ensuing partonic treatment of the pomeron. Before doing so, it is helpful to recall Mueller's generalization of the optical theorem and the resulting triple-pomeron interpretation of single di!ractive dissociation. The optical theorem relates the two-particle total cross-section to the imaginary part of the forward amplitude. This is illustrated by the "rst equality in Fig. 15. Here (s is the centre-of-mass energy of particles 12, i.e., the mass of the system X, and the discontinuity across the real axis is de"ned by disc A(s)"A(s#ie)!A(s!ie) . (4.10) Q The last equality in Fig. 14 illustrates the high-energy limit, which is dominated by pomeron exchange. Mueller's generalization [91] (see also [85]), which is illustrated by the "rst equality in Fig. 16, relates a one-particle inclusive cross-section to a six-particle amplitude. The derivation uses crossing symmetry to reinterpret the outgoing particle 3 as the incoming antiparticle 3 . In very much the same way as for the usual optical theorem, completeness of the sum over X and unitarity are employed to relate the inclusive cross-section to the discontinuity of the amplitude. Note, however, that in Fig. 16 the discontinuity is taken in M, which is the squared mass of the system X and di!erent from the variable s characterizing the original process with incoming particles 1 and 2.

Fig. 15. Graphic representation of the optical theorem for a two-particle total cross-section. In the rightmost diagram, which is relevant in the pomeron-dominated high-energy limit, the cut is represented by a dotted line.

Fig. 16. Graphic representation of Mueller's generalization of the optical theorem. In the rightmost diagram, relevant for s/MPR and MPR, the cut is only through the pomeron coupled to particle 1.

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The rightmost diagram in Fig. 16 is obtained in the double limit s/MPR and MPR [92] (see also [85]). If s<M and M is much larger than any of the other variables, the process 12P3X is dominated by pomeron exchange, and the amplitude receives a factor f (a(t)) (s/M)?/ R. E The appearance of the ratio s/M is a non-trivial result of the relevant kinematics. The remaining M dependence of the amplitude is given by the cut that is left after the two pomerons coupled to particles 2 and 3 are factorized. If the high-energy limit of this amplitude, which corresponds to MPR, is again dominated by pomeron exchange, a factor f (a(0)) (M)?/  results. The appearE ance of these three &pomeron propagators' is illustrated by the three zigzag lines on the r.h. side of Fig. 16. The corresponding cut amplitude reads




 







s ?/ R  M ?/  disc G(t) f (a(0)) b (0) , (4.11) disc ¹" b (t) f (a(t))   E E M s  where G(t) is the unknown triple-pomeron vertex. Since the t dependence of G(t) is measured to be weak and since, for the pomeron, g"#1, the approximations G(t)KG(0) and "f "K1 can be used E at small "t". The resulting cross-section for the process 12P3X is given by







dp s ?/ R\ M ?/ \ 1 "G(0)b (0)" . (4.12) " "b (t)"  dt dM 16pMs  M s   This expression for the cross-section suggests the following interpretation [93]. Incoming particle 2 radiates a pomeron carrying a fraction m"M/s of its momentum. Then, this pomeron collides with particle 1 producing the di!ractive state X with mass M. According to this interpretation, Eq. (4.12) can be rewritten as dp "f/ (m, t)p / (M) ,  dt dm

(4.13)

where 1 "b (t)"m\?/ R f/ (m, t)" 16ps   is the pomeron #ux factor, characterizing the probability for the transition 2P3/, and

(4.14)

(4.15) p / (M)""G(0)b (0)"(M/s )?/ \    is the total cross-section for the collision of the pomeron and particle 1. Since the pomeron is not a real particle, the decomposition of Eq. (4.12) into pomeron #ux and total cross-section is ambiguous. The de"nitions given here correspond to interpreting b (t) and f (a(t)) (s/M)?/ in  E Eq. (4.11) as pomeron-particle-particle vertex and pomeron propagator, respectively. The rest of the diagram corresponds to the pomeron-particle scattering amplitude, which is used to de"ne the pomeron-particle cross-section. In their seminal paper on di!ractive jet production in hadron}hadron scattering, Ingelman and Schlein exploited this picture to predict the rate of high-p jets within the di!ractive "nal state , X [9]. They suggested that, according to the parton model, hard processes in the collision of particle 1 and pomeron can be described in terms of a convolution of two parton distributions and a hard partonic cross-section. The idea of introducing a parton distribution for the pomeron

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Fig. 17. Di!ractive DIS via the pomeron structure function.

proved to be very successful phenomenologically and was widely used in subsequent analyses of hard di!ractive processes. Donnachie and Landsho! applied this idea to the case of di!ractive DIS [10,11], where particle 1 is a highly virtual photon and can therefore be treated in perturbation theory. In this approach, HERA di!raction at the level of a few per cent of the total DIS cross-section was predicted as early as 1987 [10]. In contrast to Ingelman and Schlein, who focussed on the gluon distribution of the pomeron, Donnachie and Landsho! introduced a quark distribution, which can be directly probed by the virtual photon. The resulting physical picture of di!raction is shown in Fig. 17. In complete analogy to Eq. (4.13), the di!ractive cross-section can be written as dp 2 "f/ (m, t) pAH/ , 2 dt dm

(4.16)

where, according to the conventional parton model, the photon}pomeron cross-section is given in H/ terms of the quark distribution q/ (b, Q) of the pomeron, pA "(pe/Q) 2 bq/ (b, Q). Here b"x/m 2 is the fraction of the pomeron momentum carried by the struck quark, and the factor 2 is introduced to account for the antiquark contribution. The resulting di!ractive structure function reads (4.17) F"(x, Q, m, t)"f/ (m, t) 2bq/ (b, Q) ,  and a non-trivial Q dependence is naturally expected on the basis of the Altarelli}Parisi evolution of the quark distribution. The above normalization of pomeron #ux and pomeron-particle cross-section is consistent with [85]. However, other normalizations are also frequently used (compare the conventions of [9}11,93,94]). 4.3. Diwractive parton distributions Loosely speaking, the analysis of di!raction in terms of di!ractive parton distributions is equivalent to the analysis in terms of the partonic pomeron of Ingelman and Schlein, &minus its Regge content' [95]. Di!ractive parton distributions provide a direct, perturbative-QCD-based approach to the hard part of the process, without introducing any less well established nonperturbative concepts. The basic theoretical ideas are due to Trentadue and Veneziano, who proposed to parametrize semi-inclusive hard processes in terms of &fracture functions' [12], and to Berera and Soper, who

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de"ned similar quantities for the case of hard di!raction [13] and coined the term &di!ractive parton distributions'. The following discussion is limited to the latter, more specialized framework. In short, di!ractive parton distributions are conditional probabilities. A di!ractive parton distribution df "(y, m, t)/dm dt describes the probability of "nding, in a fast moving proton, a parton G i with momentum fraction y, under the additional requirement that the proton remains intact while being scattered with invariant momentum transfer t and losing a small fraction m of its longitudinal momentum. Thus, the corresponding cHp cross-section can be written as [96]








K df "(y, m, t) dp(x, Q, m, t)AHNNY6 " dy p( (x, Q, y)AHG G , (4.18) dm dt dm dt V G H where p( (x, Q, y)A G is the total cross-section for the scattering of a virtual photon characterized by x and Q and a parton of type i carrying a fraction y of the proton momentum. The above factorization formula holds in the limit QPR with x, m and t "xed. At leading order and in the case of transverse photon polarization, only the quark distribution contributes. For one quark #avour with one unit of electric charge, the well-known partonic cross-section reads pe p( (x, Q, y)AHO" d(1!y/x) , 2 Q

(4.19)

giving rise to the di!ractive cross-section dp(x, Q, m, t)AHNNY6 2pe x df "(x, m, t) O " , dm dt dm dt Q

(4.20)

where the factor 2 is introduced to account for the antiquark contribution. The main di!erence to fracture functions is the requirement of a speci"c t transferred to the "nal state hadron. Since fracture functions are de"ned to include the t integration, a non-negligible contribution to the relevant cross-section arises from processes where the outgoing hadron, in this case the proton, belongs to the current fragmentation region. This contribution complicates the Q dependence of fracture functions, but not of di!ractive parton distributions. Even if the t integration is performed, the above problem does not arise in di!ractive kinematics, where m is small and the production of a very energetic forward proton in the fragmentation of the current is strongly suppressed. It is now obvious that Eqs. (4.16) and (4.17) of the last section form a special case of the present more general framework. There, Regge phenomenology and the assumption of a partonic pomeron lead to the speci"c form d f "(x, m, t) "f/ (m, t) bq/ (b) x O dm dt

(4.21)

for the di!ractive quark distribution. However, the factorizing dependence on b"x/m and the expression for the pomeron #ux factor f/ (m, t) given in the last section have not been derived in the framework of QCD.

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The operator de"nition of di!ractive parton distributions can be obtained as follows [96]. Consider "rst a complex scalar "eld (x), which can be written as



dk , (4.22)

(x)" dkI [a( (k)e\ IV#bK R(k)e IV], dkI " (2p)2k  where a( and bK are the annihilation operators of particles and antiparticles, respectively. In this "eld theory, the number of particles with momentum in the interval dk is measured by the operator dNK "dkI a( R(k)a( (k) .

(4.23)

The conventional inclusive distribution f (y) of partons of the "eld is given by the number of particles in a momentum interval dk "P dy, normalized to the size of the interval dy. Note that, > > in this section, the co-ordinate system is such that the proton and photon momenta have large plus and minus components, respectively. This is customary for the discussion of DIS in the Breit frame and contrasts with the remainder of this review, where the proton rest frame is emphasized and, correspondingly, the photon momentum is de"ned to have a large plus component. The particle number is measured in the state of a hadron with momentum P, and integration over all transverse momenta k is assumed. Thus, the explicit formula reads , dNK "P2 , (4.24) (2p) 2P d(P !P) f (y) dy" dk 1P" ,  dk , where the prefactor (2p) 2P d(P !P) is a result of the normalization of the hadronic state "P2,  1P"P2"(2p) 2P d(P !P) . (4.25)  It can then be shown that, in terms of the fundamental "eld , the distribution of scalar partons reads





yP f (y)" > dx e\ W.> V\ 1P" R(0, x , 0 ) (0, 0, 0 )"P2 . \ \ , , 4p

(4.26)

In the case of a gauge theory, this de"nition has to be supplemented with a link operator






i V\ A (0, y , 0 )dy , (4.27) ; \ "P exp ! > \ , \ V  2  connecting the two scalar "eld operators. Before generalizing this de"nition of the parton distribution to the di!ractive case, it is convenient to rewrite it in the form



yP#

(0, 0, 0 )"P2 . (4.28) dx e\ W.> V\  1P" R(0, x , 0 ); \ "X2 1X"; f (y)"  , \ \ , V  4p 6 In the di!ractive case, the operators describing the creation and annihilation of the parton are the same. However, the proton is required to appear in the "nal state carrying momentum P. Thus, the

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above de"nition is changed to



d f "(y, m, t) yP O " > dx e\ W.> V\  \ dm dt 64p ; 1P" R(0, x , 0 ); \ "P, X2 1P, X";

(0, 0, 0 )"P2 , (4.29) \ , V   , 6 where 1P, X" denotes the outgoing state, the only restriction on which is the presence of a scattered proton with momentum P. The above formulae for inclusive and di!ractive distributions of scalar partons are generalized to the case of spinor quarks by the substitution (see, e.g., [97]) 1 tM (0, x , 0 )c t(0, 0, 0 ) ,

R(0, x , 0 ) (0, 0, 0 )P \ , > , \ , , 2yP > and to the case of gluons by the substitution

(4.30)

1

R(0, x , 0 ) (0, 0, 0 )P FR(0, x , 0 )>IF(0, 0, 0 )> . (4.31) \ , , \ , ,I (yP ) > The operator expressions appearing in the de"nitions of both the inclusive and di!ractive parton distributions are ultraviolet divergent. They are conveniently renormalized with the MS prescription, which introduces the scale k as a further argument. The distribution functions then read f (x, k) and d f "(x, m, t, k)/dm dt. Accordingly, Eq. (4.18) has to be read in the MS scheme, with a k dependence appearing both in the parton distributions and in the partonic cross-sections. The claim that Eq. (4.18) holds to all orders implies that these k dependences cancel, as is well known in the case of conventional parton distributions. Since the partonic cross-sections are the same in both cases, the di!ractive distributions obey the usual Altarelli}Parisi evolution equations



d f "(x, m, t, k) K dy d f "(y, m, t, k) d G " P (x/y) H . (4.32) dm dt y GH dm dt d(ln k) H V with the ordinary splitting functions P (x/y). Clearly, this is equivalent to the assertion that, in the GH operator de"nition of Eq. (4.29), the ultraviolet divergences are independent of the "nal state proton P. If this is the case, the Altarelli}Parisi evolution of the distribution functions follows from the operator de"nitions exactly as in the inclusive case of Eq. (4.28). Thus, for the analysis of di!ractive DIS, it is essential to gain con"dence in the validity of the factorization formula Eq. (4.18). Berera and Soper "rst pointed out [96] that such a factorization proof could be designed along the lines of related results for other QCD rocesses [98] (see [97] for a review). Using Mueller's method of cut vertices [99], Grazzini et al. [100] proved, in the framework of a simple scalar model, that the above factorization property holds for &extended fracture functions'. These objects di!er from fracture functions in that they are di!erential in the momentum transfer t. They are thus equivalent to the di!ractive parton distributions discussed here.

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Fig. 18. Leading regions in inclusive (a) and di!ractive (b) DIS (cf. [101]).

Collins [101] showed that factorization holds in full QCD by generalizing the essential step of dealing with soft gluon interactions, which lies at the heart of many previous factorization proofs [97], to the case of di!ractive DIS. A brief summary of the essential arguments is presented below. To begin with, recall the main ideas of the factorization proof for inclusive DIS. It is based on the dominance of contributions from so-called leading regions, shown in Fig. 18a. Here the hard subgraph is denoted by H, the subgraphs with momenta collinear to P and to the produced jets are denoted by A and J 2J respectively, and the soft subgraph is denoted by S. The analysis is  L performed in the Breit frame, where lines in H have typical virtuality Q, lines in A and J 2J have  L small virtualities but may have large longitudinal momentum components of order Q, and all components of momenta in S are small compared to Q. The rationale behind the discussion in terms of leading regions is the realization that the momentum integrations are dominated by pinch singularities, i.e., poles of the propagators that cannot be avoided by deforming the integration contours. In the limit of large Q, such singularities are associated with the leading regions shown in Fig. 18. By power counting, only one hard line may connect the A and H subgraphs, while arbitrarily many soft (dashed) lines may connect the soft subgraph with other parts of the diagram. An essential part of the factorization proof is the demonstration that the soft subgraph can be factored out. More speci"cally, it has to be shown that the soft lines are not important for the subgraphs H and J 2J , so that a perturbative hard cross-section with free partons in the "nal  L state can be used. This is achieved using Ward identities, which, however, can only be applied in the region where all components of the soft momenta are small and of comparable size. Di$culties arising in the region where one component of a soft momentum is much smaller than the other components can be solved by appropriately deforming the integration contour. To see this in more detail, consider the particularly simple diagram of Fig. 18, where there is only one current jet, and a single soft gluon connects the corresponding jet subgraph with subgraph A. Let the soft gluon with momentum l couple to outgoing particles with momenta k and k in the (  subgraphs J and A, respectively. The particle propagators i (k !l)!m#ie (

and

i (k #l)!m#ie 

(4.33)

attached to the gluon vertices produce poles in the complex l and l planes that lie above and > \ below the real axis, respectively. This is also true for further l dependent propagators in J and

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Fig. 19. Leading order process in DIS with additional exchange of a soft gluon between the target and current jet subgraphs (cf. [101]).

A since, due to completeness, "nal state interactions can be disregarded in both subgraphs. Thus, regions where either l or l are too small can always be avoided by deforming the integration > \ contour. This takes us back to the genuinely soft region where all components of l are small and of comparable size, and where Ward identities can be used to factor out the soft subgraph. It is precisely this part of the factorization proof that is a!ected in the case of di!raction. The requirement of a "nal state proton leads to the presence of "nal state interactions in subgraph A of Fig. 19 (as shown in Fig. 18b) that cannot be neglected. Since both initial and "nal state interactions appear in A, poles on both sides of the l plane exist, and the contour can no longer be deformed to \ avoid the dangerous region of too small l . However, Collins was able to demonstrate [101] that \ deformations of the l contour can instead be used to show that this dangerous region does not > produce a non-factorizing contribution. The argument is based on a detailed analysis of the pole structure in subgraphs J 2J (or the single jet subgraph J in the simple example above). It leads to  L the conclusion that, exactly as in the inclusive case, the soft subgraph can be factored out. The result is a convolution of the calculable hard part with di!ractive parton distributions. Note that di!ractive factorization does not hold in the case of hadron}hadron collisions. A general argument for this was given in [102], and the e!ect was also found in the model calculation of [13]. The reason is that, in contrast to the DIS case, the use of completeness in the "nal state cannot be avoided in the factorization proof for hadron}hadron collisions. This completeness is lost if a "nal state proton with a given momentum is required, and a breakdown of the factorization theorem results. 4.4. Target rest frame point of view In this section, the connection between the target rest frame point of view, used in the semiclassical approach, and the Breit frame point of view, relevant for the two previous sections, is established. In particular, the consistency of the semiclassical approach with the concept of di!ractive parton distributions and with the factorization formulae of the last section are demonstrated. Even before the all-orders factorization proofs of [100,101], it was suggested that di!ractive factorization could be understood in the semiclassical picture in the proton rest frame [103]. This was then explicitly shown in the leading order analysis of [52], on which the present section is largely based. Calculating the cross-section with the methods of Section 3, a result is obtained that can be written as a convolution of a partonic cross-section and a di!ractive parton distribution.

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Within the semiclassical model, this di!ractive parton distribution is explicitly given in terms of integrals of non-Abelian eikonal factors in the background "eld. To be more speci"c, it is explicitly shown that the amplitude contains two fundamental parts: the usual hard scattering amplitude of a partonic process, and the amplitude for soft eikonal interactions with the external colour "eld. The latter part is determined by the scattering of one of the partons from the photon wave function. This parton has to have small transverse momentum and has to carry a relatively small fraction of the longitudinal photon momentum. In a frame where the proton is fast, this parton can be interpreted as a parton from the di!ractive structure function. The special role played by the soft parton in the photon wave function was also discussed in [30,33,104,105] in the framework of two gluon exchange. However, the present approach has the two following advantages: "rstly, by identifying the hard part as a standard photon}parton scattering cross-section, the necessity for non-covariant photon wave function calculations is removed. Secondly, once it is established that the main contribution comes from the soft region, non-perturbative e!ects are expected to become important. The eikonal approximation provides a simple, self-consistent model for this non-perturbative region. The explicit calculation closely follows the calculation of Section 3. It is convenient to start with the particularly simple case of scalar partons. The kinematic situation is shown symbolically in Fig. 20. The process is split into two parts, the hard amplitude for the transition of the photon into a virtual partonic state and the scattering of this state o! the external "eld. To keep the amplitude for the "rst part (denoted by H) hard, the transverse momenta p ( j"1,2, n) are required to be large, i.e., &Q. The momentum k is small, i.e., &K, and the H, , corresponding parton carries only a small fraction (&K/Q) of the longitudinal photon momentum in the proton rest frame. While the hardness condition for particles 1 through n is introduced &by hand', simply to make the process tractable, the softness of the last particle follows automatically from the requirement of leading twist di!raction. The cross-section for the scattering o! a soft external "eld reads 1 "¹"2pd(q !q ) dXL> where q"k# p . dp"   H 2q 

(4.34)

All momenta are given in the proton rest frame, ¹ is the amplitude corresponding to Fig. 20, and dXL> is the usual phase space element for n#1 particles. According to Eq. (3.3), each of the particles interacts with the external "eld via the e!ective vertex <(p, p)"2pd(p !p )2p [;I (p !p )!(2p)d(p !p )] .    , , , ,

(4.35)

Fig. 20. Hard di!ractive process in the proton rest frame. The soft parton with momentum k is responsible for the leading twist behaviour of the cross-section.

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The amplitude ¹ can now be built from the hard amplitude ¹ (symbolized by H in Fig. 20) & together with the e!ective vertices de"ned by Eq. (4.35) and the appropriate propagators. It is convenient to consider "rst the amplitude ¹, which is de"ned like ¹ but without the transversespace d-function terms of Eq. (4.35),






dp i H 2pd(p !p )2p ;I (p !p ) i2pd(q !q )¹" ¹ “ & H H H H, H,   (2p) p H H i ; 2pd(k !k )2k ;I (k !k ) .    , , k






 (4.36)

In this equation, colour indices have been suppressed. Notice also that some of the produced partons are antiparticles. The corresponding matrices ; have to be replaced by ;R. To keep the notation simple, this is not shown explicitly. The integrations over the light-cone components p can be performed using the appropriate H> energy d-functions. After that, the p integrations are performed by picking up the poles of the H\ propagators 1/p . The result is H dp 2k H,  ;I (k !k ) . (4.37) ¹" ¹ “ ;I (p !p ) & H, H, (2p) k , , H Here poles associated with the p dependence of ¹ have been disregarded since their contribuH\ & tion is cancelled by diagrams where part of the hard interaction occurs after the scattering o! the external "eld. Next, a change of integration variables is performed,








dp Pdk . (4.38) L, , Since the external "eld is assumed to be soft, it can only transfer transverse momenta of order K, i.e., p Kp for all j. In general, the amplitude ¹ will be dominated by the hard momenta of H, H, & order Q. Therefore, it can be assumed that ¹ is constant if the momenta p vary on a scale K. In & H, this approximation, the integrations over p ( j"1,2, n!1) can be performed in Eq. (4.37), H, resulting in d-functions in impact parameter space. These manipulations give the result



dk 2k , ¹ (2p) k &






“ ;(x ) ;(x #y )e\ V, D, \ W, I, \I,  , (4.39) , , , V, W, H where D is the total momentum transferred from the proton to the di!ractive system and, in particular, D "k # p . It is intuitively clear that the relative proximity of the high-p H, , , , partons in impact parameter space leads to the corresponding eikonal factors being evaluated at the same position x . , Now, the colour structure of the amplitude will be considered in more detail. Spelling out all the colour indices and introducing explicitly the colour singlet projector P, the relevant part of the amplitude reads: ¹"






? 2?L ;(x #y )@YP  2 L . ¹ "¹? 2?L @ “ ;(x ) & , , , @ ? ? @Y   2  L ? ? H

(4.40)

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Using the fact that ¹ is an invariant tensor in colour space, and introducing the function & = de"ned in Eq. (3.17), the following formula is obtained: (4.41) ¹ "¹ !¹? 2?L @P  2 L "¹? 2?L @=@YP  2 L . & ? ?@ & @ ? ? @Y     Here the "rst equality states the explicit relation between ¹ and the true amplitude ¹, where the trivial contribution of zeroth order in A has been subtracted. This subtraction corresponds to the unit matrix on the r.h. side of Eq. (3.17). For colour covariance reasons ¹? 2?L @P  2 L "const.;d@ . @Y & ? ? @Y Since the photon is colour neutral, the following equality holds:

(4.42)

¹? 2?L @¹H  2 L ""¹? 2?L @P  2 L """const."N . (4.43) &? ? @ A & & ? ?@ Here the partons are assumed to be in the fundamental representation of the colour group S;(N ). A Combining Eq. (4.41) with Eqs. (4.42) and (4.43), it becomes clear that the colour structure of the hard part decouples from the eikonal factors, 1 (4.44) "¹ "" "tr[=]""¹ " . &   N A The hard part will be interpreted in terms of an incoming small-k parton that collides with the , virtual photon to produce the outgoing partons 1 through n. Therefore, a factor 1/N for initial A state colour averaging is included in the de"nition of "¹ ". & In the expression for the cross-section, the two functions = appear in the combination (4.45) (tr[= , (y )]) (tr[= , (y )])He\ V, \V, D, , V , V , with independent integrations over x , x , y and y . If the external "eld is su$ciently smooth, the , , , , functions = vary only slowly with x and x . Therefore, after integration over x and x , the , , , , expression in Eq. (4.45) produces an approximate d-function in D . Furthermore, it is assumed that , the measurement is su$ciently inclusive, i.e., the hard momenta p , are not resolved on a soft H scale K. This corresponds to a D -integration, which gives an approximate d-function in x !x . , , , Since the expression in Eq. (4.45) will always appear under x , x and D integration, the above , , , considerations justify the substitution (4.46) e\ V, \V, D, P(2p)d(x !x )d(D ) . , , , Combining Eqs. (4.34), (4.39) and (4.44), the following formula for the cross-section results:

  

1 dp" "¹ " & 2q V, 



dk tr[= I , (k !k )]  , V , , (2k )(2p)d( p )d(q !q ) dXL> .  H,   (2p) (N k A (4.47)

Note that the soft momentum k has been neglected in the transverse d-function. , To "nally establish the parton model interpretation of di!raction, the hard partonic crosssection based on "¹ " has to be identi"ed in Eq. (4.47). Consider the process & cH(q)#q(yP)Pq(p )#2#q(p ) , (4.48)  L

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where the photon collides with a parton carrying a fraction y of the proton momentum and produces n high-p "nal state partons. The cross-section is approximately given by , 1 dp( (y)" "¹ "(2p)d(q!k! p ) dXL , (4.49) H 2(s( #Q) & where s( "( p ) and the quantities "¹ " and k are the same as in the previous discussion. Eq. H & (4.49) is not exact for several reasons. On the one hand, "¹ " is de"ned in terms of the unprimed & momenta p , which di!er slightly from p . On the other hand, the vector k is slightly o! shell H H and has, in general, a non-zero transverse component. However, both e!ects correspond to K/Q corrections, where Q stands generically for the hard scales that dominate ¹ . & Using Eq. (4.49), the cross-section of Eq. (4.47) can now be rewritten as

 

 



s( #Q dk tr[= I , (k !k )]  dk , V , , (2k ) dp( (y) . (4.50)  2pq (2p) (2p)2k k (N V,   A The light-cone component k is given by !k "yP "ym . Note that the minus sign in this \ \ \ N formula comes from the interpretation of the parton with momentum k as an incoming particle in Eq. (4.49). This is, in fact, the crucial point of the whole calculation: due to the o!-shellness of k, the corresponding parton can be interpreted as an incoming particle in both the process of Eq. (4.48) and in the soft scattering process o! the external "eld. The latter process, where an almost on-shell parton with momentum k scatters softly o! the external "eld changing its momentum to the on-shell vector k, is most easily described in the proton rest frame. By contrast, the natural frame for the hard part of the diagram is the Breit frame or a similar frame. In such a frame, the k component is large and negative, so that the above parton can be interpreted as an almost \ on-shell particle with momentum !k, colliding head-on with the virtual photon. Substituting the variables y and m for k and k , the cross-section, Eq. (4.50), takes the form \  K d f "(y, m) dp Q " dy p( (y) , (4.51) dm dm V where the di!ractive parton distribution for scalars is dp" dk



\







d f "(y, m) b 1 Q " dm m 1!b



 



dk (k) dk tr[= I , (k !k )]  , , , V , , . (4.52) (2p)N (2p) kb#k (1!b) A V, , , This result is in complete agreement with the concepts described in the last section. In contrast to the formulae presented there, the above parton distribution is inclusive in t. However, as will be seen in an example calculation in Section 6.1, parton distributions di!erential in t can also be obtained in this formalism. For an external colour "eld that is smooth on a soft scale K and con"ned to a region of approximate size 1/K, the function tr[= , (y )] is also smooth and vanishes at y "0 together with V , , its "rst derivative. From this it can be derived that the k and k integrations in Eq. (4.52) are , , dominated by the soft scale. This justi"es, a posteriori, the softness assumption for one of the partons used in the derivation. The qualitative result is that the eikonal scattering of this soft parton o! the proton "eld determines the di!ractive parton distribution. The hard part of the photon evolution can be

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explicitly separated and expressed in terms of a standard cross-section for photon}parton collisions. This partonic cross-section, which appears explicitly on the r.h. side of Eq. (4.51), is infrared divergent at higher orders. In terms of the diagram Fig. 20, these divergences correspond to kinematic con"gurations where more than one of the partons of the photon wave function have small p . For example, the qq g #uctuation of the photon with all three partons soft contributes to , di!raction at leading twist (cf. Section 3.3). As in inclusive DIS, the infrared divergences of the partonic cross-sections are cancelled by the ultraviolet divergences of parton distributions at higher order. In Eq. (4.52), no such divergence (and therefore no explicit factorization scale dependence) is present. However, a higher-order calculation of this quantity in the semiclassical framework will reveal the divergence and introduce an explicit dependence on the factorization scale. Having worked out the kinematics in the simple scalar case, it is straightforward to extend the calculation to realistic quarks and gluons. The introduction of spinor or vector partons does not a!ect the calculations leading to the generic expression in Eq. (4.51). However, those parts of the calculation responsible for the speci"c form of Eq. (4.52) have to be changed if the soft parton is a spinor or vector particle. Referring the reader to Appendix C for details, only the "nal formulae for di!ractive quark and gluon distributions in the semiclassical approach are presented below. They read



 



dk k tr[= I , (k !k )]  2 dk (k) d f "(y, m) , , , , V , , O , " (2p) kb#k (1!b) m (2p)N dm A V, , , for the case of a realistic spinor quark, and




b 1 d f "(y, m) E " m 1!b dm

 

dk (k) , , (2p)(N!1) , V A

(4.53)



dk tr[= I A, (k !k )]tGH  V , , , , (2p) k b#k (1!b) , ,

(4.54)

with




2kG kH 1!b , (4.55) tGH"dGH# , , b k , for the case of a gluon. It was checked explicitly that these distribution functions together with the appropriate partonic cross-sections reproduce the results of Sections 3.2 and 3.3. Henceforth, the simpli"ed notation dq(y, m)/dm and dg(y, m)/dm will be used for the di!ractive quark and gluon distribution calculated above. The application of these di!ractive quark and gluon distributions to di!ractive DIS is summarized in an intuitive way in Fig. 21. On the l.h. side, the two lowest-order processes, qq and qq g state production, are shown from the target rest frame point of view. On the r.h. side, the same two processes are shown from the Breit frame point of view, which is conventionally used to discuss the partonic interpretation of DIS. The essence of the calculations presented in the present section is the identi"cation of these two viewpoints, as a result of which explicit formulae for the di!ractive parton distributions, expressed in terms of the target colour "eld, are obtained. Note that it is incorrect to interpret the r.h. side of Fig. 21 as a two-step process, where a colour-neutral cluster is "rst emitted by the proton and then probed by the virtual photon. If this was the case, the two-gluon or two-quark cluster relevant in this calculation would necessarily lead

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Fig. 21. Di!ractive DIS in the proton rest frame (left) and the Breit frame (right); asymmetric quark #uctuations correspond to di!ractive quark scattering, asymmetric gluon #uctuations to di!ractive boson}gluon fusion.

Fig. 22. Inclusive DIS in the proton rest frame (left) and the Breit frame (right); asymmetric #uctuations correspond to quark scattering (a), symmetric #uctuations to boson}gluon fusion (b).

to parton distributions symmetric in b and 1!b. A counter example to this is provided by the model distributions derived in Section 6.2. Thus, the idea of a &pre-formed' colour neutral cluster, as it is usually associated with the pomeron structure function, is not supported by the present calculation. At this point, it is appropriate to add a brief discussion of the target rest frame vs. Breit frame interpretations of inclusive DIS [65]. The leading order semiclassical calculation of inclusive DIS, which amounts essentially to inclusive qq pair production o! an external "eld, was given in Section 3.2. It was shown that, in contrast to the di!ractive case, both asymmetric and symmetric qq con"gurations contribute to the leading twist cross-section. As explained in more detail in Appendix D, these con"gurations correspond, in the parton model, to leading order quark scattering, testing the inclusive quark distribution, and boson}gluon fusion, testing the inclusive gluon distribution (cf. Fig 22). The symmetric con"gurations have a small transverse size and test directly the one-gluon component of the target colour "eld. This is the reason why, in the

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semiclassical framework, the leading order calculation is already sensitive to the gluon distribution of the target. The inclusive gluon distribution plays a very special role. In contrast to both the inclusive quark distribution and the di!ractive quark and gluon distributions, it is only sensitive to the short distance structure of the proton "eld, and it is enhanced by an explicit factor 1/a (see Appendix D). Q As a result, the dominance of the inclusive over the di!ractive DIS cross-section, which is of fundamental importance for the successful phenomenological analysis of [65] (cf. Section 7.1), emerges.

5. Two gluon exchange So far, inclusive di!raction, as parametrized, e.g., by the di!ractive structure function F", was at  the centre of interest of this review. It was argued that, for inclusive processes, the underlying colour singlet exchange is soft, and two corresponding approaches, the semiclassical framework and the pomeron picture, were described in some detail. In perturbative QCD, the simplest possibility of realizing colour singlet exchange is via two t channel gluons. In fact, the colour singlet exchange in certain more exclusive di!ractive processes is, with varying degree of rigour, argued to be governed by a hard scale. In such cases, two gluon exchange dominates. Some of these processes are discussed in the present section. Finally, attempts to approach the whole di!ractive cross-section in two-gluon models are described. 5.1. Elastic meson production Elastic meson electroproduction is the "rst di!ractive process that was claimed to be calculable in perturbative QCD [15,16]. It has since been considered by many authors, and a fair degree of understanding has been achieved as far as the perturbative calculability and the factorization of the relevant non-perturbative parton distributions and meson wave functions are concerned. To begin with, consider the electroproduction of a heavy qq bound state o! a given classical colour "eld. This calculation represents an alternative derivation Ryskin's celebrated result [15] for elastic J/t production. The relevant amplitude is shown in Fig. 23. In the non-relativistic limit, the two outgoing quarks are on-shell, and each carries half of the J/t momentum. Thus, the two quark propagators with momenta p"k"q/2 and the J/t vertex are replaced with the projection operator g e. (k. #m). ( ( Here g"3C( m /64pa , (5.1) ( CC (  C( is the electronic decay width of the J/t particle, m "2m is its mass, and e its polarization CC ( ( vector [106]. Using the notation and calculational technique of Section 3.2, the amplitude of Fig. 23 can now be written as







dk i i tr e. (k. #m)< (p, p) e. (q) < (k, k) , i2pd(q !q )¹  "ie g A ( (2p) ( O   OO p. !m A !k. !m O

(5.2)

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Fig. 23. Leading order amplitude for the elastic production of a J/W particle o! an external colour "eld.

where e "(2/3)e is the electric charge of the charm quark. The Dirac structure is simpli"ed A employing the identities g g (5.3) !g e. (k. #m)" ( (k. !m)e. (k. #m)" ( v (k)v (k) e. u (p)u (p) ( P P ( Q Q ( ( 2m 2m P Q as well as Eqs. (3.10) and (3.11). The further calculation proceeds along the lines of Section 3.2 using, in particular, the quark scattering vertices Eqs. (3.12) and (3.13). Adding the two contributions where only the quark or only the antiquark is scattered to Eq. (5.2), the full amplitude ¹"¹  #¹ #¹  takes the form O O OO ie g a(1!a) ¹"! A ( q dk [u (p )e. (q)v (kM )] [v (kM )e. (q)u (p )] , N#k QY A PY PY ( QY 8pm > , PYQY ;[;I (p !p );I R(k !k )!(2p)d(p !p )d(k !k )] . (5.4) , , , , , , , , Note that this is very similar to Eq. (3.14) with the ; matrix structure replaced according to Eq. (3.16). The main di!erence is that the produced quarks are projected onto the J/t state. Since both Q and m are considered to be hard scales while ; and ;R are governed by the soft hadronic scale K, the integrand in Eq. (5.4) can be expanded in powers of the soft momentum k . , The leading power of the amplitude is given by the "rst non-vanishing term. In the case of forward production, p "k "0, the dependence on the external colour "eld takes the form , ,





dk k tr[;I (p !p );I R(k !k )!(2p)d(p !p )d(k !k )] , , , , , , , , , ,



" dk k , ,

 

V, W,

tr[;(x ) ;R(y )!1] e I, W, \V,  , ,

. (5.5) tr = , (y )" , V , W  V, Now, the crucial observation is that precisely the same dependence on the external "eld is present in the amplitude for forward Compton scattering shown in Fig. 24. In the case of longitudinal photon polarization, the transverse size of the qq pair is always small, and the target "eld enters only via the second derivative of = that appears in Eq. (5.5). Thus, comparing with the parton model result for longitudinal photon scattering, this derivative can be identi"ed in terms of the gluon distribution of the target proton [51], "!(2p)R, W

!R, W



V

,

tr = , (y )" , "2pa xg(x) . Q V , W 

(5.6)

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Fig. 24. The Compton scattering amplitude within the semiclassical approach.

Using this relation and Eq. (5.4), the amplitudes for the forward production of transversely and longitudinally polarized J/t mesons by transversely and longitudinally polarized virtual photons are obtained. To go from amplitudes for scattering o! an external "eld to usual covariant amplitudes, a factor 2m has to be introduced. Under the additional assumption Q<m, the N ( covariant amplitudes for longitudinal and transverse polarization read m s , ¹ " (¹ , ¹ "!i64pa g e (xg(x)) 2 * Q ( Q * 3Q

(5.8)

where (s is the centre-of-mass energy of the cHp collision. It is not surprising that the gluon distribution of Eq. (5.6), calculated according to Fig. 24, shows no scaling violations and only the trivial Bremsstrahlungs energy dependence &1/x. The reason for this is the softness assumptions of the semiclassical calculation. Firstly, the eikonal approximation implies that all longitudinal modes of the external "eld are much softer than the photon energy. Secondly, the reduction of the "eld dependence to a transverse derivative is only justi"ed if the scales governing the quark loop, i.e., Q in the case of Fig. 24 and Q and m in the case of Fig. 22, are harder than the transverse structure of =. These two approximations, evidently valid for a given soft "eld, are also justi"ed for a dynamical target governed by QCD as long as only leading logarithmic accuracy in both 1/x and Q is required. Thus, a non-trivial dependence on 1/x and Q can be reintroduced into Eq. (5.8) via the measured gluon distribution, keeping in mind that the result is only valid at double-leading-log accuracy. Note that this is precisely what was claimed in the original two-gluon exchange calculation of [15]. Note also that, in distinction from this presentation, the calculation of [15] obtains the gluon distribution by coupling the t channel gluons to the quarks of the target and identifying the logarithmic integral over the transverse gluon momentum as the logarithm accompanying the usual quark-to-gluon splitting function. The "rst essential extension of the above fundamental result is related to the treatment of the bound state produced. Brodsky et al. [16] showed that, at least for longitudinal photon polarization, a perturbative calculation is still possible in the case of light, non-perturbative bound states like the o meson. The calculation is based on the concept of the light-cone wave function of this meson. Referring the reader to [107] for a detailed review, a brief description of the main ideas involved is given below (cf. [108]). For this purpose, consider the generic diagram for the production of a light meson in a hard QCD process given in Fig. 25.  Note that this is also consistent with the result for longitudinal photon scattering given in Eq. (3.25) since





tr = , (y )" , " tr[R , = , (0) R , =R, (0)] , V , W  W V W V V, V, which follows from unitarity of the ; matrices. !R, W

(5.7)

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Fig. 25. Generic diagram for meson production in a hard process.

Assume that, as shown in this "gure, all diagrams can be cut across two quark-lines, the constituent quarks of the meson, in such a way as to separate the hard process H from the soft meson formation vertex V, which is de"ned to include the propagators. The amplitude can be written as





¹" dk ¹ (k)<(k)" &







q  dz ¹ (z) > dk dk <(k)" dz ¹ (z) (z) , \ , & & 2  

(5.9)

where z"k /q , and the last equality is simply the de"nition of the light-cone wave function of > > the meson. The two crucial observations leading to the "rst of these equalities are the approximate k and k independence of ¹ and the restriction of the z integration to the interval from 0 to 1. \ , & The "rst is the result of the hard scale that dominates ¹ , the second follows from the analytic & structure of <. In QCD, the k integration implicit in u usually has an UV divergence due to gluon , exchange between the quarks. Therefore, one should really read " (z, k), where the cut-o! k is of the order of the hard scale that governs ¹ . At higher orders in a , the hard amplitude & Q ¹ develops a matching IR cut-o! dependence. & A more rigorous de"nition of the light-cone wave function, required, in particular, for the discussion of higher-order corrections, can be given in the operator language. Without going into further detail, note that the above wave function satis"es the relation



1meson(q)"uR(y)u(!y)"02"



dz e \XOYW (z) ,

(5.10)



where y"0 and u is the "eld operator, for simplicity a scalar, corresponding to the particle content of the meson. The case of the o meson is more complicated because it is a vector particle built from spin-(1/2) constituents. The matrix element relevant for the exclusive electroproduction of a longitudinally polarized vector meson reads



1o(q)"tM (y)cIt(!y)"02"





dz e \XOYW[qI(e ) y)f (z)#eIf (z)#yI(e ) y)f (z)] , ? @ A

(5.11)

where e is the polarization vector of the meson and f are leading twist qq distribution functions ?@A (see [109] for more details on these quantities and related issues). After the convolution with the appropriate hard production amplitude, only the combination i R

(z)"f (z)# f (z) M @ 2 Rz ?

(5.12)

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survives. Under the simplifying assumption that the o is built from one #avour of quarks with one unit of electric charge (which does not a!ect the "nal result if normalized by the decay width) the longitudinal amplitude reads [16,110]



s 

(z) . (5.13) ¹ "!i4pa e (xg(x)) dz M * Q 3Qm z(1!z) M  The underlying calculation of the hard scattering process is analogous to the case of the J/t discussed above. Although the shape of the light-cone wave function is not known, its M normalization can be related to the electronic width CM of the o via the relations CC  CM "8pa f /3m . (5.14) dz (z)"(2 m f , MM CC  M M M  If the non-relativistic approximation used in the J/t case was to be applied here, a wave function

proportional to d(z!1/2) would result. In this case, Eqs. (5.13) and (5.14) would give an M unambiguous prediction for the o production amplitude in terms of its electronic width, precisely as for the J/t amplitude. Note that, in the transverse case, the hard amplitude generates a z dependence which is even more singular at the endpoints than the factor 1/z(1!z) of Eq. (5.13). This can be rephrased by saying that the qq wave function of the transverse photon is enhanced relative to the qq wave function of the longitudinal photon by a factor of 1/z in the limit zP0 (cf. Eqs. (3.19) and (3.20), where the squares of the photon wave functions enter and a corresponds to z). While the expected fall-o! of the o meson wave function at the endpoints is su$cient to render Eq. (5.13) "nite, this is probably not true in the transverse case. Thus, perturbative calculability in the sense of the J/t and the longitudinal o amplitudes cannot be claimed. Since the gluon distribution is de"ned by the imaginary part of the forward Compton scattering amplitude, the above result for longitudinal vector meson production, ¹ &ixg(x), is, strictly * speaking, only the imaginary part of the full amplitude. Assuming that the energy dependence is given by xg(x)&(1/x)?, where the intercept a belongs to an even signature trajectory, the real part follows from the signature factor of Eq. (4.7). If a!1 is small, the ratio of real and imaginary part is approximately (p/2)(a!1), and the real part correction can be introduced into Eq. (5.13) via the substitution [16,111]



p R(xg(x)) . ixg(x)Pixg(x)# 2 R ln(1/x)

(5.15)

The discussion of vector meson production given above was limited to the double-leading-log approximation as far as the colour singlet exchange in the t channel is concerned. To go beyond this approximation, the concept of &non-forward' or &o!-diagonal' parton distributions, introduced some time ago (see [18,23] and Refs. therein) and discussed by Ji [19] and Radyushkin [20] in the present context, has to be used. Recent reviews of these quantities and their evolution, which interpolates between the Altarelli}Parisi and the Brodsky}Lepage evolution equations, can be found in [112]. Recall "rst that the semiclassical viewpoint of Figs. 23 and 24 is equivalent to two gluon exchange as long as the transverse size of the energetic qq state is small. So far, the recoil of the

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Fig. 26. Elastic J/t production (three further diagrams with the gluons connected in di!erent ways have to be added). The two gluon lines and the incoming and outgoing proton are labelled by their respective fractions of the plus component of PM ,(P#P)/2.

target in longitudinal direction has been neglected. However, such a recoil is evidently required by the kinematics. For what follows, it is convenient to use a frame where q is the large component of \ the photon momentum. In Fig. 26, the exchanged gluons and the incoming and outgoing proton with momenta P and P are labelled by their respective fractions of the plus component of PM ,(P#P)/2. If D is the momentum transferred by the proton, mPM "D /2. The variable y is an > > integration variable in the gluon loop. The lower part of the diagram in Fig. 26 is a generalization of the conventional gluon distribution (cf. Eqs. (4.26) and (4.31)). It can be described by the non-forward gluon distribution



1 dx e\ W.M > V\ 1P"FR(0, x , 0 )>IF(0, 0, 0 )>"P2 . H (y, m, t)" \ \ , ,I E 4pyPM >

(5.16)

Recently, it has been shown [113] that, as in the case of conventional parton distributions [114], no time ordering of the operators in Eq. (5.16) is required. The description of elastic meson production in terms of non-forward parton distributions is superior to the double-leading-log approach of [15,16] since a corrections to the hard amplitude, Q meson wave function and parton distribution function can, at least in principle, be systematically calculated. However, the direct relation to the measured conventional gluon distribution is lost. A new non-perturbative quantity, the non-forward gluon distribution, is introduced, which has to be measured and the evolution of which has to be tested } a very complicated problem given the uncertainties of the experiment and of the meson wave functions involved. Over the recent years, the theory of non-forward parton distributions has developed into an active research "eld in its own rights, a detailed account of which is beyond the scope of this paper (see, however, [112] for recent reviews). Important issues include the investigation of di!erent models for non-forward distribution functions [115], helicity-#ip distributions [116], the further study of non-forward evolution equations [117], and possibilities of predicting the non-forward from the forward distribution functions [118]. The latter suggestion relies on the observation that, at su$ciently high Q, the non-trivial m dependence (cf. Fig. 26 and Eq. (5.16)) is largely determined by the Q evolution. Furthermore, following the basic results of [15,16], a number of interesting phenomenological analyses of meson production have appeared. The analyses of [119] and [120] focus, among other issues, on the e!ects of the meson wave functions. More details of these approaches will be given in Section 7.3, when experimental results are discussed. The e!ects of Fermi motion and quark o!-shellness have recently also been discussed in [121]. Other interesting topics discussed in the literature include the form of the energy dependence [122], shadowing e!ects [123], Sudakov suppression [124], e!ects of polarization [125], the t slope [120,126], and the intrinsic transverse

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momentum of the hadron [127]. Vector meson production at large momentum transfer ("t"
 

¹"





dz dy H(y, x/2) ¹ (Q, y/x, z, k) (z, k) , & 4

Fig. 27. Leading regions in elastic meson production (cf. [24]).

(5.17)

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where ¹ is the hard scattering amplitude, is the light-cone wave function of the vector meson & produced, and H is the non-forward parton distribution of the proton. It could, for example, be the non-forward gluon distribution H of the last section. The variable x"x is the usual DIS E

Bjorken variable. Eq. (5.17) implies that the soft subgraph in Fig. 27 is irrelevant, and only two parton lines, i.e., the minimal number required by the exchanged quantum numbers, connect the hard subgraph H with both A and B. According to [24], the essential steps of the proof include the demonstration that the leading regions are indeed given by Fig. 27, the derivation of a power counting formula showing that H is connected to A and B by the minimal number of lines, and the demonstration of factorization of the soft subgraph using Ward identities. An important complication compared to conventional factorization proofs results from the fact that "nal state interactions are important for both the outgoing proton and the produced meson. Thus, the soft gluons of S have to be factorized from both of these subgraphs at the same time. Furthermore, the so-called endpoint regions for the two quark lines connecting H and B have to be analyzed in detail. Endpoint regions are those kinematic domains where one of the two quarks carries almost all of the meson's longitudinal momentum. They are thus endpoints of the z integration in Eq. (5.17). An essential result of [24] is the suppression of these endpoint regions to all orders of perturbation theory in the case of longitudinal photon polarization. In the case of transverse polarization, it was shown that the amplitude is suppressed by a power of Q relative to the longitudinal case. However, the endpoints are not suppressed relative to the intermediate z region, and therefore the factorization formula Eq. (5.17) cannot be established in the transverse case. A discussion of helicity and transversity parton distributions, measurable via the polarization of the produced meson, is also contained in [24]. This will not be reproduced here. More recently, factorization proofs similar to [24] have been given for the process of deeply virtual Compton scattering [130], where the situation is simpler since no soft meson wave function has to be factorized. Note also that the factorization proofs of [24,130], which use Breit frame kinematics, do not rely on the limit of small x. However, it is instructive to see in a particularly simple situation how factorization works speci"cally at small x, from the point of view of the target rest frame commonly used for the description of small-x processes [25]. For this purpose, consider a very energetic scalar photon that scatters o! a hadronic target producing a scalar meson built from two scalar quarks (see Fig. 28). The quarks are coupled to the photon and the meson by point-like scalar vertices ie and ij, where e and j have dimension of mass. The coupling of the gluons to the scalar quarks is given by !ig (r #r ), where r and r are the momenta of the directed quark lines, and g is the strong gauge I I coupling.

Fig. 28. The leading amplitude for a point-like meson vertex.

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Under quite general conditions [25], the gluon momenta satisfy the relations l , l ;q , > > > l , l ;P and l&l&!l . Then the lower bubble in Fig. 28 e!ectively has the structure \ \ \ , FIJ(l, l, P)Kd(P l ) F(l ) PIPJ , (5.18) \ > , which is de"ned to include both gluon propagators and all colour factors. A similar expression was found by Cheng and Wu [131] in a tree model for the lower bubble. Assume that F restricts the gluon momentum to be soft, l ;Q. In the high-energy limit, it , su$ces to calculate



M"



dl dl ¹IJF K ¹ F , IJ (2p) 4(2p) >> \ \

(5.19)

where ¹IJ"¹IJ(l, l, q)"¹IJ#¹IJ#¹IJ (5.20) ? @ A is the sum of the upper parts of the diagrams in Fig. 28. Note that, because of the symmetry of F with respect to the two gluon lines, the amplitude IJ ¹IJ of Eq. (5.20) is used instead of the properly symmetrized upper amplitude ¹IJ (l, l, q)"[¹IJ(l, l, q)#¹JI(!l,!l, q)] . (5.21)   The two exchanged gluons together form a colour singlet and so the symmetrized amplitude ¹IJ satis"es the same Ward identity as for two photons,  ¹IJ (l, l, q)l l "0 . (5.22)  I J Writing this equation in light-cone components and setting l "l , as appropriate for forward , , production, it follows that, for the relevant small values of l , l , l and l , \ \ > > ¹ &l (5.23)  >> , in the limit l P0. Here the fact that the tensor ¹IJ , which is built from l, l and q, has no large ,  minus components has been used. The l integration makes this equation hold also for the \ original, unsymmetrized amplitude,



dl ¹ &l . \ >> ,

(5.24)

This is the crucial feature of the two-gluon amplitude that will simplify the calculation and lead to the factorizing result below. Consider "rst the contribution from diagram (a) of Fig. 28 to the l integral of ¹ , which is \ >> required in Eq. (5.19),





dk z(1!z) ij dl ¹ "!4egq . (5.25) \ ?>> > (2p) N#(k #l ) k(q!k) , , Here N"z(1!z)Q, z"k /q and the condition l "0, enforced by the d-function in > > > Eq. (5.18), has been anticipated.

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Now  dl ¹ and  dl ¹ each carry no l dependence. So, to ensure the validity of \ @>> \ A>> , Eq. (5.24), the sum of the three diagrams must be





ij dk dl ¹ "4egq z(1!z)N , \ >> > (2p) k(q!k)

(5.26)

where





1 1 l , ! & . (5.27) N#k N#(k #l ) (N#k ) , , , , Note the 1/Q behaviour obtained after a cancellation of 1/Q contributions from the individual diagrams. This cancellation, which is closely related to the well-known e!ect of colour transparency [132], has been discussed in [54] in the framework of vector meson electroproduction. Introduce the k dependent light-cone wave function of the meson , ij iq d(k !zq ) . (5.28)

(z, k )"! > dk dk \ > (2p)k(q!k) > > , 2 N"



The "nal result following from Eqs. (5.19) and (5.26) is a convolution of the production amplitude of two on-shell quarks and the light-cone wave function:




 

z(1!z) dz d k

(z, k ) . (5.29) , (N#k ) , , This corresponds to the O(l ) term in the Taylor expansion of the contribution from Fig. 28a, given , in Eq. (5.25). At leading order, factorization of the meson wave function was trivial since the point-like quark}quark}meson vertex <(k, (q!k))"ij was necessarily located to the right of the all other interactions. To see how factorization comes about in the simplest non-trivial situation, consider the vector meson vertex M"iegs

dl , l F(l ) , 2(2p) ,



<(k, (q!k))"

dk ijj , (2p) k(q!k)(k!k)

(5.30)

which corresponds to the triangle on the r.h. side of Fig. 29. Here, the dashed line denotes a colourless scalar coupled to the scalar quarks with coupling strength j. The diagram of Fig. 29 by itself gives no consistent description of meson production since it lacks gauge invariance. This problem is not cured by just adding the two diagrams 28(b) and (c) with the blob replaced by the vertex <. It is necessary to include all the diagrams shown in Fig. 30.

Fig. 29. Diagram for meson production with the vertex modelled by scalar particle exchange.

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Fig. 30. The remaining diagrams contributing to meson production within the above simple model for the meson wave function.

The same gauge invariance arguments that lead to Eq. (5.24) apply to the sum of all the diagrams in Figs. 29 and 30. Therefore, the complete result for ¹ , which is now de"ned by the sum of the >> upper parts of all these diagrams, can be obtained by extracting the l term at leading order in the , energy and Q. Such a term, with a power behaviour &l /Q, is obtained from the diagram in , Fig. 29 (replace ij in Eq. (5.25) with the vertex < of Eq. (5.30)) by expanding around l "0. It can , be demonstrated that none of the other diagrams gives rise to such a leading-order l contribution , (see [25] for more details). The complete answer is given by the l term from the Taylor expansion of Eq. (5.25). The , amplitude M is precisely the one of Eqs. (5.29) and (5.28), with ij substituted by < of Eq. (5.30). The correctness of this simple factorizing result has also been checked by explicitly calculating all diagrams of Fig. 30. The above simple model calculation can be summarized as follows. The complete result contains leading contributions from diagrams that cannot be factorized into quark-pair production and meson formation. However, the answer to the calculation can be anticipated by looking only at one particular factorizing diagram. The reason for this simpli"cation is gauge invariance. In the dominant region, where the transverse momentum l of the two t-channel gluons is small, gauge , invariance requires the complete quark part of the amplitude to be proportional to l . The leading , l dependence comes exclusively from one diagram. Thus, the complete answer can be obtained , from this particular diagram, which has the property of factorizing explicitly if the two quark lines are cut. The resulting amplitude can be written in a factorized form. 5.3. Charm and high-p jets , In the two previous sections, the exclusive production of vector mesons was described as an example of a di!ractive process with hard colour singlet exchange. As a di!erent possibility of keeping the colour singlet exchange in di!raction hard, the di!ractive production of heavy quarks

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[26}29] and of high-p jets [30}33] was considered by many authors. However, as will become , clear from the discussion below, both processes can be associated with either soft or hard colour singlet exchange, and it is necessary to distinguish the two mechanisms carefully [50,51]. The semiclassical approach provides a very convenient framework for this analysis. One might expect the hard scale, provided by the transverse momentum of the jets, to ensure the applicability of perturbation theory. Indeed, the production of "nal states containing only two high-p jets can be described by perturbative two-gluon exchange [133,134]. This process has been , studied in detail by several groups and higher-order corrections have already partially been considered [30}33]. Below, the two simplest con"gurations, qq and qq g, are discussed following [51]. In both cases, di!ractive processes are obtained by projecting onto the colour singlet con"guration of the "nal state partons. Although the discussion focusses on high-p jets, all qualitative results carry over to , the case of di!ractive charm production [50]. Technically, it is not important whether the hard scale in the di!ractive "nal state is p or m. One has simply to replace the high-p qq jets with cc , A , jets, whose transverse momentum will automatically be &m . However, there is a clear phenomA enological di!erence. On the one hand, m is "xed and not very large, while the hard scale p can, A , at least in principle, be arbitrarily high. On the other hand, charm production is simpler to analyse since it does not require the identi"cation of jets. Consider the production of a di!ractive qq "nal state. Using the results of Section 3.2, for the transversely polarized photon one easily "nds



 



ea p tr = I (p !p )  dp , , 2 " O O  (a#(1!a)) dp , . , 2N (2p) N#p dt da dp A V, , , R

(5.31)

There are two essential di!erences compared to Eq. (3.20): "rstly, the colour trace is taken at the amplitude level to ensure colour neutrality of the qq state; secondly, two independent x integra, tions are applied to the two factors = and =R. This is the result of Eq. (5.31) being di!erential in t at t"0, in contrast to Eq. (3.20), where the t integration has been performed. The cross-section for large transverse momenta is calculated by expanding the integrand around p "p , as exercised in Section 3.2 for inclusive electroproduction in the longitudinal case. Note , , however that, due to the colour singlet condition, the "rst two terms (cf. Eqs. (3.22) and (3.23)) do not contribute, so that the leading contribution comes from the third term, which is proportional to the second derivative of = at the origin (compare the discussion of di!raction in Section 3.2). Even higher terms of the Taylor series give rise to contributions suppressed by powers of p, thus , demonstrating the dominance of the short distance behaviour of tr = , (y ). The leading order V , result reads






 



R  p   dp ea , 2 R, tr = , (0) . " O O  (a#(1!a)) W V Rp N#p  dt da dp 384p V, , , , R

(5.32)

As the derivation illustrates, this cross-section describes the interaction of a small qq pair with the proton. Hence, it is perturbative or hard. According to Eq. (5.6), the cross-section Eq. (5.32) is proportional to the square of the gluon distribution. In order to obtain the t integrated

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cross-section, one has to multiply Eq. (5.32) by the constant


 
 

C"

dp dt dt

dp dt

&K , (5.33) R! where K is a typical hadronic scale. The resulting cross-section integrated down to the transverse momentum p yields a contribution to the di!ractive structure function F" which is suppressed ,  by K/p . , As discussed in Section 3.3, a leading twist di!ractive cross-section for jets with p &Q requires , at least three partons in the "nal state, one of which has to have low transverse momentum. It can be written as a convolution of ordinary partonic cross-sections with di!ractive parton distributions. In the case of high-p quark jets there is an additional wee gluon. The partonic process is , then boson}gluon fusion, and the cross-section 

K dp( A EOO(y, p ) dg(y, m) dp , 2 " dy 2 (5.34) dp dm dm dp  , V , involves the di!ractive gluon distribution of Eq. (4.54). In addition to boson}gluon fusion, the QCD Compton process can also produce high-p jets. In , this case either the quark or the antiquark is the wee parton. The corresponding cross-section



H



dp K dp( (y, p )AHOEO dq(y, m) 2 " dy 2 , , (5.35) dm dp  dp  dm , V , involves the di!ractive quark distribution of Eq. (4.53). An analogous relation holds in the antiquark case. The cross-sections of Eqs. (5.34) and (5.35) for di!ractive boson}gluon fusion and di!ractive Compton scattering can be evaluated along the lines described in [50]. In the leading-ln(1/x) approximation, one obtains for the longitudinal and transverse boson}gluon fusion cross-sections: R ea a [a(1!a)]Qp dp , ln(1/x)hA , * " O O  Q 2p (N#p) da dp  , , R ea a (a#(1!a))(p #N) dp , 2 " O O  Q ln(1/x)hA , 16p (N#p ) da dp  , , "tr =A, (y )" V , . hA " y W , V, , Similarly, one "nds for the QCD-Compton cross-sections:



dp Q 16R ea a * " O O  Q hF , da dp [a(1!a)]QK  27n , dp 4R ea a QK #Q 2 " O O  Q QK !2Q(QK #Q)# hF , da dp 27pQK p  a(1!a) , , p , , QK "Q# a(1!a)





(5.36) (5.37) (5.38)

(5.39) (5.40) (5.41)

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where the constant hF is de"ned analogously to Eq. (5.38) but with the ; matrices in the fundamental representation. Comparing Eqs. (5.36) and (5.37) with Eqs. (5.39) and (5.40), it is apparent that the con"gurations with a wee gluon are enhanced by ln(1/x) at small x relative to those with a wee quark or antiquark. The origin of this enhancement can be understood as follows. Eqs. (5.34) and (5.35) provide cross-sections di!erential in m, p and y or, equivalently, in m, p and a, since , , y"x[Q#p /a(1!a)]/Q. The above p -spectra are obtained after performing the m-integra, , tion from y to some "xed m ;1. The integral is dominated by m




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Fig. 31. The fraction of di!ractive events with p above p for Q of 10 GeV and 100 GeV (lower and upper curve in , , each pair).

distinction between qq "nal states, where M"M, and qq g "nal states, where M(M. In H H practice, however, this requires the contribution of the wee gluon to the di!ractive mass, which is responsible for the di!erence between M and M, to be su$ciently large. To quantify the H expectation within the semiclassical approach, consider the transverse photon contribution to the di!erential di!ractive cross-section dp/dM dM. H In the case of a qq "nal state this cross-section can be obtained directly from Eq. (5.43), dp 16MQ(1!i 2 "R ea ap[mG(m)]Cd(M!M) . (5.45) O O  Q H 3(M#Q)i dM dM H Here the d-function setting M"M is only precise up to hadronization e!ects, which are H expected to be of the order of the hadronic scale. The dependence on the transverse momentum cuto! enters via the variable i"4p /M. , H By contrast, the mass distribution for di!ractive processes with three-particle "nal states is not peaked at M"M. Concentrating, as before, on the transverse photon polarization and on the H contribution from the di!ractive gluon distribution, the following formula can be derived from Eq. (5.34) dg(y, m) Q#M dp H [2 Arctanh(1!i!(1!i] . 2 "2pR ea a y (5.46) O O  Q dm (Q#M) dM dM H H Explicit results can be obtained by using model calculations for the di!ractive gluon distribution (cf. Section 6) or utilizing a simple parametrization (cf. the numerical predictions of [50,51].) Note that the ln(1/x)-enhancement present in Eq. (5.37) can be recovered if the M integration is performed in Eq. (5.46). The origin of this enhancement is the integration measure dM/M, which appears since y(dg(y, m)/dm)&1/M for M su$ciently large. The main contribution to the total cross-section comes from the region where M is signi"cantly larger than M. Thus it is clear, even H

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without detailed calculations, that the distribution of Eq. (5.46) di!ers qualitatively from Eq. (5.45). With su$cient statistics, a determination of the relative weight of soft colour singlet exchange, relevant in the qq g case, and hard colour singlet exchange, relevant in the qq case, should be feasible. The above calculations can be summarized as follows. The di!ractive production of a qq "nal state with high p or with charm quarks proceeds via hard colour singlet exchange. The description , in the semiclassical framework reproduces the two-gluon exchange calculations. By contrast, high-p jets or charm in qq g "nal states are predominantly produced via boson}gluon fusion. The , colour neutralization mechanism is soft, and the cross-section is proportional to the di!ractive gluon distribution. The boson}gluon fusion mechanism is distinguished by a much harder p spectrum and a di!ractive mass that is, on average, much larger than the invariant mass of the , two-jet system. The energy dependence of the process could be very di!erent for the above two mechanisms. For hard colour singlet exchange, a steep rise is expected from the known small-x behaviour of the gluon distribution. For soft colour singlet exchange, the m dependence is expected to be less steep. In Ref. [27], the importance of higher-order a corrections to di!ractive charm production was Q estimated in the framework of two-gluon exchange. A sizeable enhancement of the cross-section was found. Clearly, the cc g "nal state is part of these corrections. However, the above discussion shows that this "nal state is dominated by the region where the gluon is soft. In this case, the cc g contribution is not perturbatively calculable and cannot be considered an a correction to the hard Q cc process. Thus, the systematic calculability of corrections to hard cc or jet production is an interesting open problem, which may be related to the problem of de"ning &exclusive' jet production at higher orders. 5.4. Inclusive diwraction There are a number of attempts by several authors to approach the bulk of the di!ractive DIS data, as described by the structure function F", from the perspective of two gluon exchange in the  t channel. The degree to which perturbation theory is taken seriously varies signi"cantly in the di!erent investigations discussed below. In what can be possibly called the most modest approach, two gluons with an appropriate form-factor-like coupling to the proton are used as a simple model for colour singlet exchange, even if this exchange is believed to be non-perturbative. In the context of the di!ractive electroproduction cross-section at HERA, two gluon exchange calculations were performed in [7], where the importance of the soft region was pointed out, and the result was parametrized in terms of an e!ective two gluon form factor of the proton. Furthermore, the two gluon calculation was used to describe both di!ractive and inclusive cross-sections in terms of the qq component of the light-cone wave function of the virtual photon and of p(o), the cross-section for a qq pair to interact with the hadronic target [7]. If the qq component dominates, which is, however, a non-trivial assumption, this approach allows one to link the di!ractive cross-section at t"0 to the inclusive DIS cross-section via the optical theorem. In Ref. [31], the di!ractive structure function was discussed on the basis of the exchange of two gluons with non-perturbative propagators in the sense of the Landsho!}Nachtmann model [53].

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Further analyses extend the two gluon exchange calculations to include the qq g component of the incoming photon [33,104,134]. Going one step further in the direction of perturbation theory, the known relation between the qq cross-section p(o) and the inclusive gluon distribution [135], p p(o)" a [xg(x)]o#O(o) , 3 Q

(5.47)

may be employed for the calculation of di!ractive cross-sections. This is similar to what was discussed in Sections 5.1 and 5.3 in the case of meson production and high-p jet or charm , production, respectively. However, such a relation to the gluon distribution, employed, e.g., in [30,136], is problematic since the di!ractive structure function is dominated by large transverse sizes of the qq #uctuation of the photon, where Eq. (5.47) is not valid. It was emphasized in [137] that small, perturbative values of o become more important if the analysis is restricted to small di!ractive masses. In spite of the evident problems with the applicability of perturbation theory to the di!ractive structure function, it is still interesting to take the perturbative approach even further, calculating the BFKL leading-log corrections [22] to the two gluon exchange amplitude. A possible formal justi"cation of such a treatment can be obtained by considering di!ractive DIS o! a heavy quark}antiquark state, a so-called onium, which provides a perturbative scale in addition to the Q of the virtual photon. The focus is clearly on the energy dependence of the cross-section, which so far cannot be quantitatively described by non-perturbative QCD based methods. A detailed discussion of the BFKL technique of summing leading logarithms in the high-energy limit of perturbative QCD amplitudes is beyond the scope of the present review. The essential qualitative result, relevant for the following brief overview, concerns the scattering of two small colour dipoles at very high centre-of-mass energy (s. It states that all corrections of the form a ln s Q to the above process, which proceeds via two gluon exchange at leading order, can be summed. This results in a power-like growth of the amplitude with s. One may think of the energy logarithms as being associated with gluonic ladders, although the ladder topology does not exhaust all relevant diagrams. More recently, a colour dipole picture of the BFKL amplitude has been developed [30,138}140]. In this picture, each of the colliding colour dipoles radiates gluons, thus creating new colour dipoles with smaller energy, which are the source of further gluon radiation. Eventually, two dipoles, one from each of the two colliding cascades, interact via simple two gluon exchange. The equivalence with the original BFKL technique has been established for the most fundamental, but not for all relevant applications. In particular, the role of the large N limit, which is used in addition to the A leading logarithmic approximation [138,139], is not yet fully understood. For illustration, the particularly simple formula for the total cross-section of two onia with radius R and mass m is given (see, e.g., [139]). It reads (s/m)?/ \ , (5.48) p(s)"16pRa Q ((7/2)a N f(3) ln(s/m) Q A where a/ "1#(4N a /p) ln 2 is the intercept of the BFKL pomeron. The reader is referred to the A Q original papers [22] and to the modern introductory text [86] for further details.

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The most straightforward application of the above perturbative techniques to di!raction at HERA relies on postulating the exchange of a BFKL pomeron between target proton and partonic photon #uctuation. For forward di!raction, the conventional BFKL amplitude has to be introduced between, say, the qq #uctuation of the photon and a quark of the proton. More generally, the BFKL amplitude at non-zero t is needed. Corresponding formulae appear as a bye-product in the more general investigations of [37,38]. In the framework of the colour dipole approach, they are discussed in [141,142]. Clearly, the method provides the desired rise of the cross-section with energy or, equivalently, of F" with 1/m. However, with the simple asymptotic BFKL result, the rise  appears to be far too strong (cf. Section 7.1 more details). It is a further, even more challenging theoretical problem to consider the double limit =<M
Fig. 32. Triple pomeron vertex in di!ractive electroproduction of large masses.

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[145], the use of BLM scale setting improves the situation dramatically. Note also the improved small-x equation of [146], which incorporates the next-to-leading order BFKL kernel as well as renormalization group constraints on the relevant collinear limits, thereby improving the stability of resummed results.

6. Models for the colour 5eld of the proton In this section, three di!erent models for the procedure of averaging over the target colour "eld con"gurations are described. Using the formulae of Section 4.4, these models give rise to predictions for the di!ractive quark and gluon distribution of the target hadron and thus for di!ractive electroproduction cross-sections. 6.1. Small colour dipole A particularly simple model based entirely on perturbation theory has recently been suggested by Hautmann et al. [147]. The authors study di!raction as quasi-elastic scattering o! a special target photon that couples to only one #avour of very massive (M


tr = , tr =R, V V



,

(6.1)

V where F (with i"q, g) is a linear functional depending on tr = , (y ) tr =R, (y ), interpreted as G V , V , a function of y and y . To be di!erential in t, one simply writes , , df" 1 G " F tr = , tr =R, e O, V, \V,  , (6.2) V V dm dt 4p G  V, V, with q "!t. The "eld responsible for tr = is created by a small colour dipole which, in turn, is , created by the special photon that models the target. At leading order in perturbation theory, the colour "eld of a static quark is analogous to its electrostatic Coulomb "eld. The "eld of a quark travelling on the light cone in x direction at transverse position 0 has therefore the following line \ , integral along the x direction, > dk e I, V, ig ,) ! A dx "!ig . (6.3) (2p) k 2 \ > , ,

 







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It is exactly this type of line integral that appears in the exponents of the non-Abelian phase factors ; and ;R that form = (cf. Eq. (3.4)). A straightforward calculation shows that the function tr = produced by a dipole consisting of a quark at o and an antiquark at 0 reads , , (1!e\ I, M, )(1!e\ I, M, ) g(N!1)¹ A 0 tr = , (y )"! V , (2p)k k 2 I, I, , ,   ;(1!e I, W, )(1!e I, W, )e I, >I, V, , (6.4)





where ¹ "1/2 and ¹ "N have to be used for the fundamental and adjoint representation, $  A respectively. The "nal formulae for the di!ractive parton distributions of the target are obtained after integrating over the transverse sizes of the colour dipoles with a weight given by the qq wave functions of the incoming and outgoing target photon. They read





df" 1 G " dz do dz do F , , 4p G dm dt

 

V, V

 ,



tr = , tr =R, e O, V, \V,  V V

1 ; [tH(z, o , p , e )t (z, o , 0 , e )][tH(z, o , p , e )t (z, o , 0 , e )] , (6.5) A , , , A , , , A , , , A , , , 2 CCY where tr = , (y ) is produced by the "eld of a quark at o and an antiquark at 0 , and tr = , (y ) is , , V , V , produced by the "eld of a quark at o and an antiquark at 0 , as detailed in Eq. (6.4). , , The wave function t (z, o , 0 , e ) characterizes the amplitude for the #uctuation of the incomA , , , ing target photon with polarization e and transverse momentum 0 into a qq pair with momentum , fractions z and 1!z and transverse separation o . Similarly, the wave function tH(z, o , p , e ) , A , , , characterizes the amplitude for the recombination of this qq pair into a photon with polarization e and transverse momentum p "!q . The summation over the helicities of the intermediate , , quark states, which are conserved by the high-energy gluonic interaction, is implicit. The required product of photon wave functions can be calculated following the lines of [48] and using the matrix elements of Appendix B. It reads explicitly tH(z, o , p , e ) t (z, o , 0 , e ) A , , , A , , , N ee " A O tr UR(z, k , M, e )U(z, k , M, e )e M, I, \I, >XN,  , , , , , 2(2p) , , I I where the notation of [147],



(6.6)

1 U(z, k , M, e )" [(1!z) e ) p k ) p!z k ) p e ) p#iM e ) p] , (6.7) , , , , , , , (k #M) , has been used, M is the quark mass, and p are the "rst two Pauli matrices. Note that for p "0,  , the average of the diagonal elements (e "e ) in Eq. (6.6) reproduces the well-known formula for , , the square of the photon wave function [7]. Inserting Eq. (6.6) into Eq. (6.5) and introducing explicitly the required functionals F speci"ed by G Eqs. (4.53) and (4.54), the formulae of [147] for di!ractive quark and gluon distribution are exactly reproduced. For the lengthy "nal expressions the reader is referred to the original paper, where a number of plots, based on the numerical evaluation of these formulae, is also given. Qualitatively,

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the behaviour can be summarized as follows. The quark distribution b(d f "/dm dt) falls o! like b as O bP0 and approaches a constant value, which is small compared to intermediate b points, as bP1. The gluon distribution b(d f "/dm dt) approaches a sizeable constant as bP0 and a small constant E as bP1. For t"0, the behaviour at bP1 changes } quark and gluon distributions vanish approximately as (1!b) and (1!b). The overall normalization of the gluon distribution is found to be much larger than that of the quark distribution. Even though a real proton is very di!erent from a small colour dipole, it would certainly be interesting to perform a phenomenological analysis on the basis of the above model. 6.2. Large hadron In this section, the colour "eld averaging procedure is described for the case of a very large hadronic target, where a quantitative treatment becomes possible under minimal additional assumptions. The following discussion is based on [64,65], where the large hadron model was developed and applied to a combined analysis of both di!ractive and inclusive structure functions (see also Section 4.4 and Appendix D of this review). McLerran and Venugopalan observed that the large size of a hadronic target, realized, e.g., in an extremely heavy nucleus, introduces a new hard scale into the process of DIS [63]. From the target rest frame point of view, this means that the typical transverse size of partonic #uctuations of the virtual photon remains small [64], thus justifying the perturbative treatment of the photon wave function in the semiclassical calculation. The basic arguments underlying this important result are best explained in the simple case of a longitudinally polarized photon coupled to scalar quarks with one unit of electric charge. As far as the Q-behaviour of the total cHp cross-section is concerned, this is analogous to the standard partonic process where a transverse photon couples to spinor quarks [148]. In analogy to [7], the longitudinal cross-section can be written as



p " do p(o)= (o) , , * *

(6.8)

with the square of the wave function of the virtual photon given by



3a = (o)"  da NK (No) . *  4p

(6.9)

Here o""o " is the transverse size of the qq pair, a is the longitudinal momentum fraction of the , photon carried by the quark, N"a(1!a)Q, and K is a modi"ed Bessel function. Note that, in  contrast to [7], = is de"ned to include the integration over a. * Within the semiclassical approach, the dipole cross-section p(o) is given by



2 p(o)" dx tr[1!;(x );R(x #o )] , , , , , 3

(6.10)

but it is convenient to formulate the following arguments in terms of the more general quantity p(o).

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Fig. 33. Qualitative behaviour of the function p(o).

The functional form of p(o) is shown qualitatively in Fig. 33. For conventional hadrons of size &1/K (where K&K ), its typical features are the quadratic rise at small o (p(o)&o with /!" a proportionality constant O(1)) and the saturation at p(o)&1/K, which occurs at o&1/K. Consider now the idealized case of a very large target of size g/K with g<1 (g&A for a nucleus). It is easy to see that at small o the functional behaviour is given by p(o)&go while saturation has to occur at p(o)&g/K for geometrical reasons. It follows that the change from quadratic rise to constant behaviour takes place at o&1/(gK, i.e., at smaller o than for conventional targets. From the above behaviour of p(o), the dominance of small transverse distances in the convolution integral of Eq. (6.8) will now be derived. For this purpose, a better understanding of the function = (o) is necessary. Recalling that K (x)&ln(1/x) for x;1 while being exponentially *  suppressed for x<1, it is easy to see that = (o)&Q ln(1/oQ) for o;1/Q and = (o)&1/oQ * * for o<1/Q. Here numerical constants and non-leading terms have been suppressed. Under the assumption K;gK;Q, the integral in Eq. (6.8) can now be estimated by decomposing it into three regions with qualitatively di!erent behaviour of the functions = (o) and * p(o),




p "p' #p'' #p'''" * * * *



 

/ EK  # # p do p(o)= (o) . *  / EK

(6.11)

Of the three contributions

  

/ g p' & , do goQ ln(1/oQ)& * Q  1 g EK do go & ln(Q/gK) , p'' & * oQ Q / g 1 g  p'''& do & , * K oQ Q K E the second one dominates, giving the total cross-section g p & ln(Q/gK) . * Q

(6.12)

(6.13)

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It is crucial that the third integral is dominated by contributions from its lower limit. Therefore, the overall result is not sensitive to values of o that are larger than 1/(gK. Phrased di!erently, for su$ciently large targets the transverse size of the qq component of the photon wave function stays perturbative. This result can be carried over to the realistic case of a transverse photon and spinor quarks, which is obtained simply by substituting K (oN) with 2[a#(1!a)]K (oN) in Eq. (6.9). From   the above derivation, one can also expect the result to hold for the transverse size of the qq g component of the photon wave function, both in the case of inclusive and di!ractive scattering. Note that this does not imply a complete reduction to perturbation theory since the long distance which the partonic #uctuation travels in the target compensates for its small transverse size, thus requiring the eikonalization of gluon exchange. Within this framework, it is natural to introduce the additional assumption that the gluonic "elds encountered by the partonic probe in distant regions of the target are not correlated (cf. [81] and the somewhat simpli"ed discussion in [64]). Thus, one arrives at the situation depicted in Fig. 34, where a colour dipole passes a large number of regions, each one of size &1/K, with mutually uncorrelated colour "elds A ,2, A .  L Consider the fundamental quantity = , (y ) [A] =R, (y ) [A] which, after specifying the V , IJ V , GH required representation and appropriately contracting the colour indices ijkl, enters the formulae for inclusive and di!ractive parton distributions (cf. Sections 4.4 and Appendix D). According to Eq. (3.17), this quantity is the sum of four terms, the most complicated of which involves four ; matrices, +; , [A] ;R, , [A], +; , , [A] ;R, [A], V >W GH V >W V IJ V "+; , [A ]2; , [A ];R, , [A ]2;R, , [A ], V L V  V >W  V >W L GH ;+; , , [A ]2; , , [A ];R, [A ]2;R, [A ], . V >W L V >W  V  V L IJ

(6.14)

The crucial assumption that the "elds in regions 1,2, n are uncorrelated is implemented by writing the integral over all "eld con"gurations as

   2

"





,

(6.15)

L

i.e., as a product of independent integrals. Here the appropriate weighting provided by the target wave functional is implicit in the symbol  . 

Fig. 34. Colour dipole travelling through a large hadronic target.

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Under the integration speci"ed by Eq. (6.15), the ; matrices on the r.h. side of Eq. (6.14) can be rearranged to give the result





+; , [A] ;R, , [A], +; , , [A] ;R, [A], V V >W GH V >W V IJ

 

2 +; , [A ];R, , [A ]2; , [A ];R, , [A ], V  V >W  V L V >W L GH  L (6.16) ;+; , , [A ];R, [A ]2; , , [A ];R, [A ], . L V L V >W  V  IJ V >W To see this, observe that the A integration acts on the integrand +; , [A ];R, , [A ],  V >W  GYHY  V ;+; , , [A ] ;R, [A ], transforming it into an invariant colour tensor with the indices ijkl. V >W  V  IYJY The neighbouring matrices ; , [A ] and ;R, [A ] can now be commuted through this tensor  V  V structure in such a way that the expression +; , [A ] ;R, , [A ],   +; , , [A ] ;R, [A ],    V >W  GH V >W  V  IJ V emerges. Subsequently, the A integration transforms this expression into an invariant tensor with  indices ijkl. Repeating this argument, one eventually arrives at the structure displayed on the r.h. side of Eq. (6.16). To evaluate Eq. (6.16) further, observe that it represents a contraction of n identical tensors "



+; , [A ] ;R, , [A ], +; , , [A ] ;R, [A ], , (6.17) V K V >W K GH V >W K V K IJ K where the index m refers to any one of the regions 1,2, n into which the target is subdivided. At this point, the smallness of the transverse separations y and y , enforced by the large size of the target, , , is used. In fact, for a target of geometrical size &n/K (where n<1), the relevant transverse distances are bounded by y&y&1/nK. Assuming that size and x dependence of typical "eld con"gurations A are characterized by , K the scale K, it follows that the products ; , ;R, , and ; , , ;R, are close to unit matrices for all V >W V V V >W relevant y and y . Therefore, it is justi"ed to write , , (6.18) ; , [A ];R, , [A ]"exp+i¹?f ?(x , y )[A ], , K V >W K , , K V where ¹? are the conventional group generators and f ? are functions of x and y and functionals , , of A . Eq. (6.18) and its y analogue are expanded around y "y "0 (which corresponds to K , , , f ?(x ,0)"0) and inserted into Eq. (6.17). At leading non-trivial order, the result reads , F "d d (1!cC (y#y))#c(y y )¹? ¹? , (6.19) GHIJ GH IJ  0 , , GH IJ where C is the Casimir number of the relevant representation (C "C ) and the constant c is 0 0 $ de"ned by F " GHIJ



f ?(x , y ) f @(x , y )"cd?@(y y )#O(yy) . (6.20) , , , , , ,  Note that the absence of terms linear in f ? and the simple structure on the r.h. side of Eq. (6.20) are enforced by colour covariance and transverse space covariance. The absence of an explicit x dependence is a consequence of the homogeneity that is assumed to hold over the large , transverse size of the target. Neglecting boundary e!ects, the x integration is accounted for by ,

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multiplying the "nal result with a parameter X&n/K that characterizes the geometrical crosssection of the target. Substituting the n tensors F on the r.h. side of Eq. (6.16) by the expression given in Eq. (6.19) GHIJ and contracting the colour indices as appropriate for the inclusive and di!ractive case respectively, one obtains, in the large-N limit, A 1 L +; , ;R, , , +; , , ;R, , "d 1! cC (y !y ) , (6.21) V V >W GH V >W V HG 0 , 2 0 ,  1 L +; , ;R, , , +; , , ;R, , "d 1! cC (y #y) , (6.22) V V >W GG V >W V HH 0 , 2 0 ,  where d is the dimension of the representation. 0 Since n is assumed to be large and the typical values of y and y do not exceed 1/nK, the formula (1!x/n)LKexp[!x] can be applied to the r.h. sides of Eqs. (6.21) and (6.22). Furthermore, contributions proportional to +; , ;R, , , d , d +; , , ;R, , and d d have to be GH IJ V V >W GH IJ GH V >W V IJ added to obtain the complete expression for = , (y ) =R, (y ) . The corresponding calculations V , GH V , IJ are straightforward and the result reads

 

 

  V

,



 

tr(= , (y )=R, (y ))"Xd [1!e\?0 W!e\?0 WY#e\?0 W, \W, ] , 0 V , V ,

(6.23)

tr = , (y )tr =R, (y )"Xd [1!e\?0 W][1!e\?0 WY] , V , V , 0

(6.24)

V,  where a "ncC /2 plays the role of a saturation scale. 0 0 The above calculation, performed at large N and for the case of a large target subdivided into A many uncorrelated regions, has no immediate application to realistic experiments. However, it provides a set of non-perturbative inclusive and di!ractive parton distributions which are highly constrained with respect to each other. For the purpose of a phenomenological analysis, it is convenient to consider X and a,ncN /4 as new fundamental parameters, giving rise to the A following formulae for the basic hadronic quantities required in Section 4.4 and Appendix D,

  

tr(=F, (y )=F,R(y ))"XN [1!e\?W!e\?WY#e\?W, \W, ] , A V , V ,

(6.25) V,  1 tr =F, (y )tr =F,R(y )"XN [1!e\?W][1!e\?WY] , (6.26) V , V , A N , A V  1 tr =A, (y )tr =A,R(y )"XN[1!e\?W][1!e\?WY] . (6.27) V , V , A N , A V  Here the indices F and A stand for the fundamental and adjoint representation. A similar, Glauber type y dependence has been recently used in the analyses of [149,150]. Note that according to Eqs. (6.25)}(6.27) the di!ractive structure function is not suppressed by a colour factor relative to the inclusive structure function, as originally suggested in [42].

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Using Eq. (6.26) in the generic semiclassical formula Eq. (4.53) for the di!ractive quark distribution, the explicit result dq(b, m) aXN (1!b) A " h (b) , O dm 2pm

(6.28)

is obtained. Here h (b) is an integral over two Feynman-type parameters, O (b#x (b#x



h (b)"4 O








 (1!b#((b#x)) (1!b#((b#x)) dx dx x x  (x#x)(b#(1!b) # (b#x (b#x





,

(6.29)

which can be evaluated analytically at b"0 and b"1 yielding h (0)"1/2 and h (1)"3p/8!2. O O Analogously, Eqs. (6.27) and (4.54) give rise to an explicit formula for the di!ractive gluon distribution. It reads dg(b, m) aXN(1!b) A " h (b) , E dm 2pmb

(6.30)

where h (b) is given by the two-dimensional integral E 1!b#3(1#x)b 1!b#3(1#x)b  (1#x)(1!b#(1#x)b) (1#x)(1!b#(1#x)b) h (b)"2 dx dx E x x  (x#x)b#(1!b) # 1#x 1#x














.

(6.31)

This integral is easily evaluated for b"0 and b"1 yielding h (0)"4 ln 2 and h (1)" E E 45p/32!17/2. For general b, h (b) and h (b) can be evaluated numerically. The results can be O E inferred from the solid curves in Fig. 35, where the distribution dR/dm"6dq/dm is displayed to account for the 3 generations of light quarks and antiquarks. The total normalization, the value of m, and the Q evolution given in Fig. 35 are not relevant for the present section and will be discussed in the context of the phenomenological analysis of Section 7.1. The di!ractive distributions displayed in Fig. 35 are multiplied by b and thus re#ect the distribution of momentum carried by the partons. The quark distribution is peaked around

Fig. 35. Di!ractive quark and gluon distributions in the large hadron model at the initial scale Q and after Q evolution. 

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bK0.65, thus being harder than the distribution b(1!b) suggested in [10]. It vanishes like b for bP0 and like (1!b) at large b; the gluon distribution b dg/dm, on the other hand, approaches a constant for bP0 and falls o! like (1!b) at large b. This asymptotic behaviour in the small- and large-b region is in agreement with the results obtained in the perturbative approach of [147] at t"0. In spite of the (1!b) behaviour, gluons remain important even at large b, simply due to the large total normalization of this distribution. Very recently, a closely related discussion of di!ractive and inclusive structure functions has been given in [151]. The authors focus on the process where the target hadron remains intact, discussed in the "rst part of Section 3.4, and its relation to inclusive DIS. Technically, the results obtained are very similar to those of the large hadron model as described above [65]. It is also emphasized in [151] that the results can be generalized to the case of conventional hadrons if the assumption of a Gaussian distribution of colour sources of [63] is correct. 6.3. Stochastic vacuum A further fundamentally non-perturbative approach to high-energy hadronic processes, which has recently been applied to di!ractive structure functions by Ramirez [62], is based on the model of the stochastic vacuum. The model was originally developed by Dosch and Simonov in Euclidean "eld theory [55]. A detailed description of the stochastic vacuum approach to high-energy scattering, introduced originally in [56] and closely related to the eikonal approach of [46], can be found in [152] (see also [153] for a comprehensive review). The method was "rst applied to di!ractive DIS in [59], where vector meson production processes were considered. Here, only a brief description of the main underlying ideas will be given. The fundamental assumption underlying the model of the stochastic vacuum of [55] is that of a convergent cumulant expansion for the vacuum expectation value of path ordered products of "eld operators. To calculate the average 122 of the path ordered exponential




R



 



Q QL\  R ds 2 ds OK (s )2OK (s ) , OK (s) ds " ds  L  L     L  the so-called path ordered cumulants ((2)), de"ned by P exp

(6.32)

112"((1)) , 11, 22"((1, 2))#((1))((2)) , 11, 2, 32"((1, 2, 3))#((1))((2, 3))#((1, 2))((3))#((1))((2))((3)) , 11, 2, 3, 42"((1, 2, 3, 4))#2#((1))((2))((3))((4)) ,

(6.33)

are introduced. An expansion in the above cumulants can be applied to the Wegner}Wilson loop in a non-Abelian gauge theory. The supposition that the cumulants are decreasing su$ciently fast with increasing distance between the operators leads to the area law in the purely gluonic case. Neglecting all cumulants higher than quadratic in the "elds amounts to the assumption of a Gaussian process, where all higher correlators can be obtained from the two-point Green's function. All of the above is naturally formulated in Euclidean space. The two-point correlator or, more precisely, its analytic continuation to Minkowski space, is the fundamental object in applications to high-energy scattering.

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In the treatment of di!ractive DIS, the stochastic vacuum approach describes the hadron as well as the virtual photon in terms of fast partons moving in opposite directions. The colour "eld facilitating the interaction is soft in the centre-of-mass frame of the cHp collision (cf. the general discussion of soft hadronic processes in [46]). In this situation, the partons from both sides interact with the "eld in an eikonalized way, and the actual model of the stochastic vacuum is used to evaluate the correlation function of the resulting oppositely directed light-like Wegner}Wilson lines. In the simplest case, where the photon #uctuates into a qq pair and the hadron is modelled as a quark}diquark system, the interaction amplitude of two Wegner}Wilson loops has to be calculated (see Fig. 36). Introducing the notation

 

="P exp !ig

.1



A (z) dzI I

(6.34)

for the path ordered integral around the loop RS, which is extended to light-like in"nity in both directions, this amplitude can be written as





1 1 . tr(= (b/2, R )!1) tr(= (!b/2, R )!1) J(b, R , R )"       N N  A A

(6.35)

The notation of this section follows the original papers reviewed (see, e.g., [59,152]), which means that the function de"ned in Eq. (6.34) di!ers by a unit matrix from the closely related function = used previously in the context of the semiclassical approach. The brackets 122 denote the  vacuum expectation value or, in the functional language, the integration over all colour "eld con"gurations. The line integrals in Eq. (6.35) are transformed into surface integrals with the help of the non-Abelian Stokes theorem



P exp

.1



!igA (z) dz "P exp I I 1



1



!igF (z, u) dRIJ(z) . IJ

(6.36)

Fig. 36. Wegner}Wilson loops formed by the paths of quarks and antiquarks inside two dipoles. The impact parameter b connects the centres of the two loops, while R and R point from the quark to the antiquark line of each dipole. All   three are vectors in the transverse plane of the collision ("gure from [57]).

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Here F (z, u) are the "eld strength tensors F (z), parallel-transported to a common reference IJ IJ point u. The operator P denotes an appropriate surface ordering of these matrix valued tensors 1 (see [152] and Refs. therein for further details). The surface S that is chosen for each of the two Wegner}Wilson loops is the upper side of the pyramid with the loop as base and the origin of the co-ordinate system as apex (see Fig. 35). The reference point u is also chosen to be at the origin. The interaction amplitude for two Wegner}Wilson loops, in its explicit form of Eq. (6.35) or expressed through surface integrals according to Eq. (6.36), cannot be evaluated by direct application of the cumulant expansion methods discussed above. The problem is that those methods are adopted for one path ordered integral, in contrast to the two path (or surface) ordered integrals required for J(b, R , R ). In Ref. [56] it was suggested to apply the Gaussian factorization   hypothesis directly to products of the "elds F? (z, u), IJ 1F(1)2F(2n)2"

1F(i )F( j )221F(i )F( j )2 , (6.37)   L L    where the single argument of F stands for Lorentz and colour indices and space-time co-ordinates. Note, however, that this factorization assumption is not equivalent to the original approach of [55] (cf. the discussion in [153,154]). Eq. (6.35) can now be evaluated if the fundamental correlator 1gFA (x, u)FB (y, u)2 (6.38) IJ NM  is known. Under the further assumption that this correlator depends neither on the path used for transporting the "eld to u nor on the position of the reference point u itself, the most general form reads dAB1gFF2 +i(g g !g g )D(z/a) 1gFA (x, u)FB (y, u)2 " IM JN IN JM IJ MN  12(N!1) A 1 #(1!i) [R (z g !z g )#R (z g !z g )]D (z/a), , N JM J N IM M IN  2 I M JN

(6.39)

where D and D are, a priori, two independent functions, and 1gFF2 is the gluon condensate.  Note that the model of the stochastic vacuum is formulated in Euclidean "eld theory and the analytic continuation to Minkowski space is non-trivial. The intricacies of this process and, in particular, the constraints it imposes on the shape of the functions D and D will not be discussed  here. For completeness, the correlation functions used, e.g., in [59] are given:



k 27p dk i e\ IX? . D (z/a)"D(z/a)"  (2p) (k!(3p/8)) 4

(6.40)

Detailed information about the parameters entering Eq. (6.39) and the functional form of D and D is available from low energy hadronic physics as well as from lattice calculations (see, e.g.,  the recent results of [155]). Given these low-energy parameters, a large number of soft hadronic high-energy scattering processes is successfully described by the model of the stochastic vacuum. For the present review, it is su$cient to state that, within the present model, the behaviour of the correlator in Eq. (6.39) is quantitatively known.

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Thus, the method for evaluating the fundamental dipole amplitude of Eq. (6.35) is now established. The amplitude vanishes for small dipoles, as expected from colour transparency, and grows linearly for large dipoles, as suggested by the geometric picture of string}string scattering. The calculation of di!ractive processes is straightforward as soon as a speci"c quark}diquark wave function of the hadronic target is chosen. The procedure of folding the dipole amplitude with the virtual photon and the hadron wave functions introduces a certain model dependence on the hadronic side. An essential di!erence between the present framework and the semiclassical approach discussed above is the frame of reference where the model for the hadronic target is formulated. From the perspective of the virtual photon, the situations are similar: the qq #uctuation scatters o! a soft colour "eld. However, in the semiclassical approach (cf., e.g., Section 6.2), this "eld is modelled on the basis of ideas about hadron colour "elds in the rest frame of the hadron. By contrast, the stochastic vacuum model uses an intermediate frame, where both projectile and target are fast (e.g., the centre-of-mass frame), as the natural frame for the colour "eld. The target hadron is described in terms of quarks interacting with this "eld in an eikonalized way, and the "eld, as seen by the projectile, is a result of the distribution of these partons and of the gluon dynamics in the stochastic vacuum model. As an interesting consequence of the above discussion, the conditions required for the eikonal approximation to work are di!erent in the two approaches. While, in the semiclassical model, the partons from the photon wave function have to be fast in the target rest frame, the stochastic vacuum approach requires them to be fast in, say, the centre-of-mass frame. The latter is a far more stringent condition, which can be numerically important in phenomenological applications [61,62]. The reader is referred to Sections 7.1 and 7.3 for a brief discussion of recent applications of the presented model to di!ractive electroproduction data.

7. Recent experimental results In this section, a brief discussion of experimental results in di!ractive electroproduction, focussing in particular on the di!ractive structure function, on speci"c features of the di!ractively produced "nal state, and on exclusive meson production is given. The discussion is aimed at the demonstration of interesting features of di!erent theoretical models and calculational approaches in the light of the available data (see also [156] for recent reviews). From an experimentalist's perspective, what follows can only be considered a very brief and incomplete overview. 7.1. Diwractive structure function As already explained in Section 2, the observation that di!ractive DIS is of leading twist and the ensuing measurement of di!ractive structure functions lie at the heart of increased recent interest in the "eld. While the basic experimental facts were stated at the beginning of this review, the present section supplies some additional details concerning F" and compares observations with theoretical  ideas.

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To begin with, a brief discussion of the most recent and most precise data on F", produced by the  H1 and ZEUS collaborations [76,77], is given. As mentioned previously, the most precise analyses have to be based on measurements where the scattered proton is not tagged. In Ref. [76], this problem is handled by starting from a cross-section dp CNC67 , (7.1) dx dQ dm dt dM 7 where, by de"nition, the two clusters X and > are separated by the largest rapidity gap in the hadronic "nal state of the event. This de"nition allows for the possibility that the proton breaks up into a "nal state > with mass square P"M (cf. Fig. 1). A structure function F" is then de"ned, 7  as in Section 2.3, after integration over t and M . The event selection of [76] is based on the 7 requirement that X is fully contained in the main detector while > passes unobserved into the beam pipe. This implies that M :1.6 GeV and "t":1 GeV, thus approximately specifying the relevant 7 integration region for M and t. A rapidity cut ensures that X is well separated from >. 7 The most recent ZEUS analysis [77] is based on the so-called M method, introduced in [74]. 6 Similar to what was discussed above, it is assumed that X is contained in the detector while > escapes down the beam pipe. Thus, M is simply de"ned to be the full hadronic mass contained 6 in the main detector components. In contrast to the H1 analysis, the di!ractive cross-section, integrated over all t and M , is de"ned by subtracting from this full cross-section the non7 di!ractive contribution. The subtraction is based on the ansatz dN "D#c exp(b ln M ) (7.2) 6 d ln M 6 for the event distribution in the region of not too large masses. Here D is the di!ractive contribution and the second term is the non-di!ractive contribution. The exponential ln M 6 dependence is expected as a result of the Poisson distribution of particles emitted between current and target jet regions in non-di!ractive DIS. This leads to an exponential distribution of rapidity gaps between the detector limit and the most forward detected particles and therefore to the above M dependence. The di!ractive contribution is not taken from the "t result for D but is determined 6 by subtracting from the observed number of events the non-di!ractive contribution that corresponds to the "t values of b and c. For more details concerning the data analysis and the experimental results themselves the reader is referred to the original papers [76,77] and to the "gures in the remainder of this section, where theoretical models are compared with measured values of F". It can be seen from the direct  comparison of H1 and ZEUS data found in [77] that the two methods give similar results, although the H1 values for F" have a tendency towards a faster Q rise for any given b.  The M method of ZEUS has the advantage of subtracting events which happen to have 6 a rapidity gap in spite of non-singlet colour exchange. It also allows for di!ractive events with relatively forward particles, which are excluded by the rapidity cut of H1 [157]. However, the subtraction of the non-di!ractive background also introduces new uncertainties, in particular the dependence on the region in M where the "t according to Eq. (7.2) is performed. 6 Note also that a measurement of F", based on the use of the leading proton spectrometer of  ZEUS, was reported in [75]. The results for F", obtained by explicit t integration, are probably  the cleanest ones from both the theoretical and experimental perspectives. They agree with the

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latest F" measurements discussed above, but at the moment they are, unfortunately, less precise  because of limited statistics. The observed t dependence of F" can be parametrized as e@R, with  b"7.2$1.1(stat.)> (syst.) GeV\. \  7.1.1. Leading twist analysis As mentioned previously, the leading twist behaviour of small-x di!raction is one of its most striking features. For this reason, and in keeping with the general perspective of this review, it is convenient to start the discussion with calculations, such as the semiclassical approach, that focus on the leading twist nature of the process. Detailed numerical predictions exhibiting this behaviour were "rst made in [7] based on simple aligned jet model calculations with soft two gluon exchange. Further theoretical and numerical developments by these and other authors [30,134,141], focussing, in particular, on high-mass di!raction and higher twist e!ects, are discussed below. Data analyses based on the idea of parton distributions of the pomeron [9] and their Q evolution were performed early on by many authors [158,159]. The most recent H1 publication [76] includes such a partonic analysis as well. Here, a detailed description of the approach of [65] will be given, which is based on di!ractive parton distributions in the sense of Berera and Soper [13], calculated in the semiclassical approach with a speci"c model for the averaging over all target colour "elds (cf. Section 6.2). As can be seen from Figs. 37}39, a satisfactory description of F" is achieved in this framework.  In the presented approach, the di!ractive distributions of Section 6.2 and the inclusive distributions of Appendix D are used as non-perturbative input at some small scale Q . They are evolved to  higher Q using the leading-order DGLAP equations [21]. The non-perturbative parameters of the model as well as the scale Q are then determined from a combined analysis of experimental data  on inclusive and di!ractive structure functions. At "rst sight, the semiclassical description of parton distribution functions always predicts an energy dependence corresponding to a classical bremsstrahlung spectrum: q(x), g(x)&1/x. However, one expects, in a more complete treatment, a non-trivial energy dependence to be induced since the "eld averaging procedure encompasses more and more modes of the proton "eld with increasing energy of the probe (cf. the discussion at the end of Section 3.4). This energy dependence is parametrized in the form of a soft, logarithmic growth of the normalization of di!ractive and inclusive parton distributions with the collision energy &1/x, consistent with the unitarity bound. As a result, the additional parameter ¸ is introduced into the formulae of both Section 6.2 and Appendix D, where the overall normalization factor X has to be replaced according to XPX(¸!ln x) .

(7.3)

The following expressions for the di!ractive parton distributions are obtained: dq(b, m, Q ) aXN (1!b)(¸!ln m) A  " h (b) , O 2pm dm

(7.4)

dg(b, m, Q ) aXN(1!b)(¸!ln m)  " A h (b) , E dm 2pbm

(7.5)

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Fig. 37. The structure function F"(m, b, Q) computed in the semiclassical approach with H1 data from [76]. The open  data points correspond to M44 GeV and are not included in the "t.

where the functions h (b) are de"ned in Section 6.2. Corresponding expressions for the inclusive OE input distributions of quarks and gluons are given in Appendix D. Thus, the input distributions depend on a, X, ¸, and on the scale Q at which these distributions  are used as a boundary condition for the leading-order DGLAP evolution. At this order, the measured structure function F coincides with the transverse structure function. In de"ning   Note that the two variables a and X cannot be combined into one since the inclusive quark distribution depends on them in a more complicated way.

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Fig. 38. The structure function F"(m, b, Q) computed in the semiclassical approach with ZEUS data from [77]. The  open data points correspond to M44 GeV and are not included in the "t.

structure functions and parton distributions, all three light quark #avours are assumed to yield the same contribution, such that the singlet quark distribution is simply six times the quark distribution de"ned above. Valence quarks are absent in the semiclassical approach. Charm quarks are treated entirely as massive quarks in the "xed #avour number scheme [160]. Thus, n "3 in the D DGLAP splitting functions, and only gluon and singlet quark distributions are evolved. The structure functions F and F" are then given by the singlet quark distribution and   a massive charm quark contribution due to boson}gluon fusion. Explicit formulae can, for example, be found in [159]. For the numerical studies, the values K D "144 MeV *-L  (a (M )"0.118), m "1.5 GeV, m "4.5 GeV are used, and the massive charm quark contribuQ 8 A @ tion is evaluated for a renormalization and factorization scale k "2m . A A The resulting structure functions can be compared with HERA data on the inclusive structure function F (x, Q) [161] and on the di!ractive structure function F"(m, b, Q) [76,77]. These data   sets from the H1 and ZEUS experiments are used to determine the unknown parameters of the model. The following selection criteria are applied to the data: x40.01 and m40.01 are needed to justify the semiclassical description of the proton; with Q being a "t parameter, a su$ciently large  minimum Q"2 GeV is required to avoid that the data selection is in#uenced by the current value of Q ; "nally M'4 GeV is required in the di!ractive case to justify the leading-twist  analysis. The optimum set of model parameters is determined from a minimization of the total s (based on statistical errors only) of the selected data. As a result, Q "1.23 GeV, ¸"8.16, X"(712 MeV)\, a"(74.5 MeV) . (7.6)  All parameters are given with a precision which allows reproduction of the plots, but which is inappropriate with respect to the crudeness of the model. The distributions obtained with these

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Fig. 39. Dependence of the di!ractive structure function F" on b and Q, compared to data from H1 (left) and ZEUS  (right) [76,77]. Open data points correspond to M44 GeV. The charm content of the structure function is indicated as a dashed line.

"tted parameters yield a good qualitative description of all data on inclusive and di!ractive DIS at small x, as illustrated in Figs. 46, 37 and 38. The starting scale Q is in the region where one would  expect the transition between perturbative and non-perturbative dynamics to take place; the two other dimensionful parameters X¸ and a are both of the order of typical hadronic scales. The approach fails to reproduce the data on F" for low M. This might indicate the  importance of higher twist contributions in this region (see, e.g., [36]). It is interesting to note that a breakdown of the leading twist description is also observed for inclusive structure functions [162], where it occurs for similar invariant hadronic masses, namely =:4 GeV. The perturbative evolution of inclusive and di!ractive structure functions is driven by the gluon distribution, which is considerably larger than the singlet quark distribution in both cases. With the parameters obtained above, it turns out that the inclusive gluon distribution is about twice as large as the singlet quark distribution. By contrast, the relative magnitude and the b dependence of the di!ractive distributions are completely independent of the model parameters. Moreover, their absolute normalization is, up to the slowly varying factor 1/a (Q ), closely tied to the normalization Q  of the inclusive gluon distribution.

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In spite of the (1!b) behaviour, gluons remain important even at large b, simply due to the large total normalization of this distribution. As a result, the quark distribution does not change with increasing Q for bK0.5 and is only slowly decreasing for larger values of b. The dependence of the di!ractive structure function on b and Q is illustrated in Fig. 39, where the predictions are compared with experimental data [76,77] at "xed m"0.003 (H1) and m"0.0042 (ZEUS). The underlying di!ractive parton distributions dg/dm and dR/dm"6 dq/dm at input scale Q and their Q evolution are shown in Fig. 35.  It is an essential feature of the above semiclassical analysis that the rise of F (x, Q) and of  F"(m, b, Q) at small x and m has the same, non-perturbative origin in the energy dependence of the  average over soft "eld con"gurations in the proton. With increasing Q, the x dependence is enhanced by perturbative evolution in the case of the inclusive structure function while, in the di!ractive case, the m dependence remains unchanged. 7.1.2. Importance of higher twist To illustrate the importance of higher twist contributions, consider the parametrization of Bartels et al. [36] (cf. Figs. 40 and 41). In this analysis, four di!erent contributions to the di!ractive structure function F" are considered. These are F2  and F2  } the leading twist contributions of OOE  OO qq and qq g components of the transverse photon wave function } supplemented by *F*  and *F2  OO OO } the higher twist contributions of the qq component of longitudinal and transverse photon wave function. As has already been explained in Section 3.2, there is no leading twist contribution to di!raction from the qq component of the longitudinal photon. Nevertheless, this component is important because of its dominance in the region bP1. The approach of [36] is to write down generic expressions for the above four contributions to the structure function. These expressions incorporate the qualitative knowledge about b, m and Q dependence of each contribution. This knowledge is based on the two gluon exchange calculations of [33,104] and is largely in agreement with what can be learned from the semiclassical treatment presented in this review. For example, the leading twist contributions are expected to have a softer 1/m behaviour than the higher twist contributions, which are expected to behave as (mg(m)) (cf. Section 5.3). Furthermore, the longitudinal qq component is the only one known not to vanish at t"0 and bP1. The relative weights of the four contributions, which are, in principle, determined by photon wave function and structure of the target, are allowed to vary in the "t of [36]. The results of the two independent "ts to ZEUS and H1 data (cf. Figs. 40 and 41) are in good agreement with physical expectations. Note especially the dominance of *F*  at bP1 and of F2  at bP0. OOE OO Note also the steep m dependence of *F*  , the need of which is visible, in particular, in the top OO right-hand bins of Fig. 41. Thus, the data clearly has room for the higher twist contributions introduced in [36]. It would certainly be desirable to include such contributions in the leading twist analysis of [65], with its consistent treatment of the Q evolution. An obvious problem to be solved before such a development can be realized is the m dependence of leading and higher twist terms. As an interesting feature of the data, the authors of [36] notice that the H1 results can be described by two di!erent sets of parameters: one in which F2  is enhanced at large b (correspondOOE ing to Fig. 41) and a second one which is more similar to the ZEUS "t. The solution with enhanced

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Fig. 40. Fit to ZEUS data for F"(m, b, Q) by Bartels et al. [36]. Solid line: total result; dashed line: F2  ; dotted line: F2  ;  OO OOE dashed-dotted line: *F*  (data from [163], see also [77]). OO

qq g contribution can be interpreted as the analogue, within the framework of this parametrization, of the singular gluon proposal of H1 [76]. As can be seen from the QCD analysis in [76], the H1 data exhibit some preference for a gluon distribution that is large for high b, such that for low values of Q most of the pomeron's

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Fig. 41. Fit to H1 data for F"(m, b, Q) by Bartels et al. [36]. Solid line: total result; dashed line: F2  ; dotted line: F2  ;  OO OOE dashed-dotted line: *F*  (data from [76]). OO

momentum is carried by a single gluon. This singular gluon proposal, which has some similarity with the namK ve boson}gluon fusion model of [42], is hard to justify theoretically. In particular, the validity of the partonic leading twist picture at b&1 and not too large Q is questionable. Given the successful analysis of [65] and the alternative, less singular

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parametrizations in [36,76], a singular di!ractive gluon distribution is not unavoidable with the present data. Note that the QCD analysis of [76] has the important advantage of explicitly dealing with the reggeon contribution, which, although suppressed in the high-energy limit, represents an important correction in the upper region of the m range. (Recall, however, that this may be di!erent in the ZEUS measurement, where at least a part of the reggeon contribution is subtracted.) A reggeon contribution is also included in the recent, more detailed analysis of H1 data [164] in the framework of the parametrization of [36]. For a discussion of reggeon exchange in di!ractive DIS the reader is referred, e.g., to [165]. A further interesting analysis, based on the model of the stochastic vacuum (cf. Section 6.3) and including both leading and higher twist contributions, was reported by Ramirez [62]. As an example, results for F" and for the transverse contribution F" at Q"12 GeV and x"0.0075  2 are shown in Fig. 42. The presented calculation is limited to the qq #uctuation of the photon and does not include Altarelli}Parisi evolution. Therefore, the author limits the discussion of data to the high-b region, where gluon radiation e!ects are not yet dominant. The model as used in previous analyses tends to overshoot the data. Modi"cations proposed by Rueter (see [61,166]) subtract integration regions where one of the quarks is too slow for the eikonal approach to work, and where the transverse size of the pair is too small for the non-perturbative model to be applicable. The modi"ed model gives a good description of the data (cf. Fig. 42). It is particularly interesting to see the absolute magnitude of the longitudinal structure function at large b as a prediction of an explicit calculation. Furthermore, the model shows a non-vanishing contribution from the qq component at bP1. This is an improvement compared to the treatment of [65] which is related to the better description of the dynamics of the outgoing proton at non-zero t in the calculation of [62].

Fig. 42. Di!ractive structure functions at Q"12 GeV and x"0.0075 in the model of the stochastic vacuum, calculated by Ramirez [62], compared to H1 data from [76]: mF"(m, b, Q) and mF"(m, b, Q) in the modi"ed model  2 (long-dashed and solid lines respectively); mF"(m, b, Q) in the model without modi"cations (short-dashed line); "gure  from [62].

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Note that the above approach allows, in principle, for the inclusion of explicit qq g contributions of the photon wave function as well as for the all orders resummation of ln Q terms, in a way similar to [65]. It would be very interesting to see the results of such an extended analysis. The analysis of Bertini et al. [35] is devoted to the in#uence of higher twist terms on the Q dependence of di!ractive DIS. It is found that, similar to what is known about the longitudinal qq contribution, the higher twist terms in the transverse cross-section are short distance dominated and are therefore perturbatively calculable. These contributions a!ect the Q dependence at bP1 and should be subtracted from the cross-section before a conventional DGLAP analysis of the large b region is performed. The data analyses of [167,168] are based on the dipole picture of the BFKL pomeron discussed brie#y in Section 5.4. The di!ractive structure function is described as a sum of two components: the elastic component, where the qq #uctuation of the virtual photon scatters o! the proton [142], and the inelastic component, where, based on the original qq #uctuation, a multi-gluon state develops, giving rise to high-mass di!raction [39]. While the "rst contribution is modelled by BFKL pomeron exchange in the t channel, the second one involves the triple-pomeron coupling and a cut gluon ladder representing the "nal state. In Ref. [167], a combined analysis of both inclusive and di!ractive structure functions was performed. The inclusive structure function was "tted on the basis of a colour dipole BFKL calculation, which, however, contains a number of free parameters and a smaller pomeron intercept than namK ve perturbation theory would suggest. Introducing further parameters for the normalization of the elastic and inelastic components discussed above, qualitative agreement with both the di!ractive and inclusive DIS data was achieved. A further phenomenological investigation of di!raction on the basis of colour dipole BFKL was performed in [168], where a good 7 parameter "t to the di!ractive structure function was presented. It is certainly a challenging problem to gain a better understanding of the dynamics, thus reducing the number of required parameters. In particular, the applicability of perturbative methods needs further investigation. Recently, an analysis of di!raction has appeared [150] that is based on the previous inclusive DIS "t [149] with a Glauber-type model for the dipole cross-section p(o) (cf. Eq. (6.10)). Motivated by the idea of saturation (see [169]), an energy or x dependence of p(o) is introduced according to the formula






o p(x, o)"p 1!exp !  4R (x) 



,

(7.7)

where the function R (x) vanishes as xP0. The authors "nd a similar energy dependence of the  di!ractive and inclusive cross-section. Two new parameters required for the analysis of di!raction are the di!ractive slope and the "xed coupling constant a . The very successful description of F" is Q  based on the qq and qq g #uctuations of the virtual photon. It includes higher twist terms but no Altarelli}Parisi evolution. In the di!erent approaches discussed above, separate predictions for transverse and longitudinal photon cross-sections can be derived. Corresponding measurements, which are, however, very di$cult, would be immensely important in gaining a better understanding of the underlying colour singlet exchange. An interesting possibility of obtaining the required polarization information is

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the measurement and analysis of the distribution in the azimuthal angle, which, at non-zero t, characterizes the relative position of the leptonic and hadronic plane of a di!ractive event [170]. 7.2. Final states A further tool for studying the mechanism of di!raction is provided by the details of the hadronic "nal states. In this section, the focus is on events with a di!ractive mass of the order of Q or larger. Di!ractive production of single particles forms the subject of the next section. The "rst topic is the production of jets with high transverse momentum relative to the cHp collision axis within the di!ractively produced "nal state. Such di!ractive dijet events were reported soon after the beginning of the investigation of di!raction at HERA [171]. Given the kinematic constraints and the available number of events, a direct analysis of jet cross-sections in di!ractive DIS proved di$cult. Therefore, event shape observables such as thrust were "rst considered by both collaborations [172]. One of the main results was the alignment of the di!ractive state along the cHp axis and, furthermore, the presence of signi"cant transverse momentum components. If this is to be interpreted in terms of di!ractive parton distributions, a large gluonic component seems to be required. A direct analysis of di!ractive dijet events has recently been reported by the H1 collaboration [173]. In the following, this analysis is discussed in more detail since, for such measurements, a qualitative comparison with di!erent theoretical models is simpler and less a!ected by hadronization e!ects. As an illustration, one of the distributions published in [173] is reproduced in Fig. 43.

Fig. 43. Di!erential cross-section in transverse jet momentum for events where the di!ractively produced hadronic state contains two jets. Photon virtualities vary in the range 7.5(Q(80 GeV. The data are compared to predictions of RAPGAP Monte Carlo models with leading-order pomeron and reggeon parton distributions: quarks only ("t 1), &#at' gluon dominated ("t 2) and &peaked' gluon dominated ("t 3). The "gure is from [173], where further details can be found.

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As explained in Section 5.3, jets in the di!ractive "nal state can be the result of either boson}gluon fusion, with the gluon coming from the di!ractive gluon distribution of the proton, or of so-called exclusive dijet production, where the jets are associated with the pure qq component of the photon wave function. The former case, viewed in the target rest frame, is associated with the qq g component of the photon wave function, where the gluon has small p and the colour singlet , exchange in the t channel is soft. In Fig. 43, the data points are compared to predictions based on the boson}gluon fusion scenario. They are well described if the di!ractive parton distributions are dominated by the gluon, a situation realized, e.g., in the semiclassical calculation based on the colour "eld model of Section 6.2. It would certainly be interesting to substitute the di!ractive parton distributions underlying the Monte Carlo results of Fig. 43 (see, e.g., [174] for a recent review of Monte Carlo generators used for HERA di!raction) with the model distributions of Section 6. However, given the precision of the data, which is at present consistent with the two very di!erent gluon distributions of "t 2 and "t 3, it appears unlikely that any gluon dominated model can be ruled out. The most important qualitative conclusion to be drawn is that the data favours boson}gluon fusion rather than exclusive dijet production. As discussed in Section 5.3, the exclusive process has a much steeper p distribution, which would be re#ected in the quality of the "t in Fig. 43. This , conclusion is made even clearer by the distribution of the variable z  / , which characterizes the fraction of the proton's momentum loss absorbed by the jets. In exclusive jet production, z  / "1, which, even taking into account the smearing caused by hadronization, is in sharp contrast to the data. Thus, the observations suggest that only a small fraction of the observed dijet events are related to exclusive jet production with hard colour singlet exchange. It would be important to try to quantify this statement. A further interesting aspect of the di!ractive "nal state is the presence of open charm. So far, no published results by either the H1 or ZEUS collaboration are available on this subject. However, preliminary data have recently been presented by both collaborations [175,176]. Cross-section measurements for the production of DH! mesons in di!ractive events are reported. It is found in [176] that both = and Q dependences of di!ractively produced open charm as well as the fraction of charmed events are consistent with the observations made in inclusive DIS. This is in agreement with the semiclassical scenario of [50] or, equivalently, with a boson}gluon fusion scenario based on gluon dominated di!ractive parton distributions. It should "nally be mentioned that a number of more inclusive di!ractive "nal state analyses focussing, in particular, on charged particle distributions [177], transverse energy #ow [178] and the multiplicity structure [179] were reported by both collaborations. Although a detailed discussion of the obtained results cannot be given here, it is important to note that all observations seem to be consistent with the physical picture of a partonic pomeron. Clearly, this is also in accord with the more general concept of di!ractive parton distributions. As expected, particle distributions and energy #ow support the by now well-established picture of colour singlet exchange with the target. In agreement with the jet oriented analyses, a signi"cant gluonic component is required by the data. It was emphasized by H1 (see [177,179]) that a relatively hard di!ractive gluon distribution is favoured. As already mentioned in the Introduction, an approach describing di!ractive DIS by the assumption of soft colour interactions in the "nal state was proposed by Edin et al. in [43]. The idea of soft colour exchange is similar to the namK ve boson}gluon fusion model of [42] (see also [180]

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for further developments), but the Monte-Carlo based implementation is quite di!erent. After the hard scattering, the physical state is given by a number of quarks and gluons } those created in the scattering process and those from the proton remnant. It is assumed that, at this point, colour exchange can take place between each pair of colour charges, the probability of which is described by a certain phenomenological parameter R. This changes the colour topology and leads, in certain cases, to colour singlet subsystems, giving rise to rapidity gaps in the "nal state. The number of gap events initially increases with R, but saturates or even decreases at larger values of this parameter since further colour exchanges can destroy neutral clusters previously created. A more detailed description of the soft colour interaction model can be found in [181], where the consistency of the approach with the large transverse energy #ow in the proton hemisphere, a feature of HERA data that appears to be &orthogonal' to the rapidity gaps, is demonstrated. In a more recent development [182], the colour exchange is described on the level of colour strings rather than on the partonic level discussed above. Furthermore, e>e\ data is included in the analysis. In conclusion it can be said that an impressive agreement with many features of hadronic "nal states in small-x DIS has been achieved within the framework of soft colour interactions. It is an interesting question in how far the perturbatively known suppression of interactions with small colour-singlet objects, i.e., colour transparency, is consistent with this approach. 7.3. Meson production Exclusive meson production in DIS is, both theoretically and experimentally, a very interesting and active area that certainly deserves more space than is devoted to it in the present review. The main theme of the previous sections was the semiclassical approach, the applicability of which to meson production processes is limited by the lack of a genuine prediction for the energy dependence of the amplitude. Although, as discussed in Section 5.1, Ryskin's double-leading-log result is reproduced, two essential problems in going beyond it, the dynamics of the energy growth and details of the meson wave function, have not been addressed. Nevertheless, this review would be incomplete without at least a brief discussion of the most important new experimental results and their implications for theory. Recent analyses of exclusive vector meson electroproduction have been published by ZEUS for o and J/t mesons [183] and by H1 for o mesons [184]. New preliminary results on exclusive J/t production were also reported by H1 [185], with previous measurements published in [186]. Agreement is observed between the most recent results of the two experiments [184]. For illustration, the ZEUS results for a particularly interesting quantity, the forward longitudinal o meson cross-section, are shown in Fig. 44. As discussed in Sections 5.1 and 5.2, this quantity can, at least in principle, be calculated in perturbation theory. If the asymptotic form of the meson wave function (z)&z(1!z) is used, the amplitude of Eq. (5.13) gives rise to the cross-section M formula [16,110] dp dt



12pCM m a(Q)[xg(x, Q)] CC M Q " . a Q R 

(7.8)

The main qualitative predictions of this formula are in good agreement with experimental results. Taking into account the anomalous dimension of the gluon distribution, the observed

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Fig. 44. Forward longitudinal o electroproduction cross-section dp/dt" as measured by the ZEUS collaboration R [183]. Shaded areas indicate normalization uncertainties in addition to the error bars shown. The curves are based on calculations by Martin et al. [129] (solid line), by Frankfurt et al. [119] (dashed}dotted line) and on colour dipole model calculations of Nemchik et al. [120] (dashed line). The "gure is from [183], where further details can be found.

Q dependence of roughly Q\ is consistent with Eq. (7.8). The data indicate that the energy dependence becomes more pronounced with growing Q, as expected from the square of the gluon distribution. However, taking the above formula at face value and using, e.g., the MRS(A) gluon distribution [187], the absolute normalization of the cross-section comes out too high. Several improvements on the result of [16] were discussed in [119]. In particular, it was found that the Fermi motion of the quarks in the vector meson leads to a suppression factor in the cross-section. Furthermore, in comparing the recent ZEUS data with the perturbative calculation, the gluon distribution is evaluated at a scale Q (Q. The new scale is determined from the  average transverse size of the qq pair, which turns out to be larger in the o production process than, say, in longitudinal electroproduction. Such a rescaling is certainly legitimate in view of the leading logarithmic nature of the approach. It leads to a further reduction of the cross-section. While both suppression e!ects together produce results that are too small, either one of them gives a reasonable "t (see Fig. 43 and [183]). Note also the claim [188] that, at least in the J/t case, Fermi motion e!ects might be considerably smaller than expected. The approach of [129], brie#y mentioned in Section 5.1, also gives a reasonable "t to the o production data. Both the results of FKS [119] and MRT [129] shown in Fig. 43 are based on

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the ZEUS 94 NLO gluon distribution, which, in the relevant parameter region, is not too di!erent from, e.g., MRS(A) [187]. Note that all calculations which derive their total normalization from the identi"cation of the two gluon exchange amplitude with the gluon distribution are a!ected by the large experimental uncertainties of the latter. The results of [120], also displayed in Fig. 44, are based on calculations that emphasize the non-perturbative aspects of the process. The interaction of the qq pair with the target is described in terms of the colour dipole cross-section p(o, l), where l is the photon energy. The amplitude for the production of a qq pair of transverse size o is convoluted with a meson wave function with non-trivial o dependence. In the relevant region, where Q is not yet very large, these nonperturbative e!ects and the contributions from p(o, l) at relatively large o are important. It is proposed that the energy and Q dependence of the measured cross-section will allow for the &scanning' of the dipole cross-section p(o, l) as a function of o (cf. [189]) and provide evidence for BFKL dynamics. The violation of s channel helicity conservation is found to be small. However, evidence for a helicity #ip amplitude on the level of 8$3% of the non-#ip amplitude was reported [184] (see also the very recent analysis of [190]). A particularly interesting feature of the data is the relatively small ratio R"p /p of longitudinal and transverse cross-sections, which reaches the value RK3 * 2 at QK20 GeV [184]. This is in sharp contrast to a namK ve extrapolation of the perturbative result of Eq. (5.8), leading one to expect R&Q/m. Even though the transverse cross-section is not M perturbatively calculable in the case of light vector mesons, one would still expect a linear growth of R with Q, which is not favoured by the data [183,184]. In the stochastic vacuum approach of [59}61], the supposition that, at presently accessible values of Q, the perturbative regime is not yet reached is taken even more seriously. The analysis is based on a fundamentally non-perturbative model of the interaction with the target hadron (cf. Section 6.3) and is thus ideally suited for studying the transition region to photoproduction, as well as the transverse cross-section, where QCD factorization theorems fail. Clearly, the approach depends on the model for the o meson wave function, but certain observables, such as the ratio of longitudinal and transverse cross-sections or elastic slopes, represent relatively robust predictions. In particular, the ratio of longitudinal and transverse o meson production cross-sections is well described [60]. In its original form, the model of the stochastic vacuum predicts constant high-energy crosssections. This limits the applicability of the model in the HERA regime, where the energy growth of meson electroproduction cross-sections cannot be neglected. Recently, the model has been extended [61] by introducing a phenomenological energy dependence based on two di!erent pomerons coupled to small and large dipoles, very much in the spirit of [191]. Furthermore, for small dipole con"gurations, a perturbative two gluon exchange contribution was added. This allowed for a good description of both photo- and electroproduction of vector mesons and of F (x, Q) in the HERA regime.  The presently available data can also be described in the generalized vector dominance approach of [192]. In contrast to the calculations discussed so far, this approach predicts both longitudinal and transverse vector meson production cross-sections to fall like &1/Q at asymptotically large virtualities. Note also the recent perturbative model calculation of [121], which "nds an asymptotic behaviour of R"p /p that is qualitatively di!erent from the expected linear Q growth. The * 2

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calculations of [121,129,192] reproduce the experimentally observed #at behaviour of R at high Q. The most striking feature of the measured J/t production cross-sections is the strong energy growth &(=) . Its approximate Q independence up to Q&13 GeV suggests that, in the present data, the charm quark mass rather than the photon virtuality play the dominant role in making the colour singlet exchange hard. The #avour-symmetric ratio p(J/t)/p(o)"8/9 is not yet reached at Q"13 GeV. Fitting the t dependence of o and J/t electroproduction cross-sections with the function e@R, values of b&8 GeV\ and b&5 GeV\, respectively, were obtained [183]. The exclusive electroproduction of excitations of both the o and the J/t meson have also received much theoretical interest. However, experimental data on this subject is only beginning to emerge [193].

8. Conclusions In the above review, di!erent methods employed in the treatment of di!ractive electroproduction processes were put into context. The semiclassical approach, which was chosen as the starting point of the discussion, is well suited to develop a basic understanding of the underlying physical e!ects. In this approach, non-di!ractive and di!ractive DIS are treated along parallel lines. Di!raction occurs if, after scattering o! the target, the partonic #uctuation of the virtual photon emerges in a colour singlet state. Altarelli}Parisi evolution of both the di!ractive and inclusive structure function is related to the presence of higher Fock states of the photon. These #uctuations have a small transverse size and do not a!ect the soft colour exchange with the target. The application of the semiclassical approach to experimental results is particularly simple if the approach is used to derive both di!ractive and inclusive parton distributions at some small input scale. In this case, the analysis of all higher-Q data proceeds with standard perturbative methods. Di!erent models for the underlying colour "elds can be compared to di!ractive and inclusive structure function data in a very direct way. If the scattering amplitude of the partonic #uctuation of the photon and the target grows with energy, then, in the semiclassical framework, this growth enters both the di!ractive and inclusive input distributions in precisely the same way. Such an energy growth is expected to be generated by the process of averaging over all relevant colour "eld con"gurations. However, no explicit non-perturbative calculation exists so that, at present, the energy dependence remains one of the most challenging aspects of the method. The idea of the pomeron structure function combines methods of Regge theory with the partonic description of hard scattering processes in QCD. In the treatment of di!ractive electroproduction, both the semiclassical approach and the pomeron structure function are equivalent as far as the hard part of the process, i.e., the Q evolution, is concerned. In this respect, they represent di!erent realizations of the more general and less predictive concept of di!ractive parton distributions. However, as far as the colour singlet exchange mechanism is concerned, the pomeron structure function idea is very di!erent in that it assumes the existence of a &pre-formed' pomeron state, o! which the virtual photon scatters. By contrast, in the semiclassical approach the di!ractive

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character of an event is only determined after the photon #uctuation has passed the target. No pre-formed colour singlet object is required. From this point of view, the semiclassical description of di!raction shows similarity to the idea of probabilistic soft colour exchange added on top of standard Monte Carlo models. However, in contrast to this method, the semiclassical calculation keeps track of the relevant transverse distances in the parton cascade, ascribing soft colour exchange only to large size con"gurations. In an apparently quite di!erent approach, the t channel colour singlet exchange, characteristic of di!ractive processes, is realized by the exchange of two t channel gluons. This is justi"ed for certain speci"c "nal states, like longitudinally polarized vector mesons or two-jet systems, where the colour singlet exchange is governed by a hard scale. From the semiclassical perspective, this e!ect manifests itself in the small transverse size of the photon #uctuation at the moment of passing the external colour "eld. Thus, only the small distance structure of the "eld is tested, which corresponds to the two gluon exchange amplitude discussed above. The fundamental advantage of two gluon exchange calculations is the understanding of the energy dependence which, one may hope, can be achieved in this framework by the resummation of large logarithms. However, leading order resummation predicts a far too steep energy growth, and the recently obtained next-to-leading correction is very large, thus calling the whole method into question. It is also possible that the mechanism responsible for the energy dependence in soft processes is fundamentally di!erent from the above perturbative ideas. Nevertheless, leading logarithmic two gluon calculations were successful in relating certain di!ractive cross-sections to the conventional gluon distribution, the energy growth of which is measured. Going beyond leading logarithmic accuracy, the non-forward gluon distribution was introduced. This is a promising tool for the further study of di!ractive processes with hard colour singlet exchange. On the experimental side, much attention has been devoted to the di!ractive structure function, which represents one of the most interesting new observables in small-x DIS. Leading twist analyses, based on the concept of di!ractive parton distributions, are successful in describing the bulk of the data. It has been demonstrated that a simple large hadron model for the target colour "elds allows for the derivation of a consistent set of di!ractive and inclusive parton distributions in the semiclassical framework. Other colour "eld models, such as a perturbative dipole "eld and the model of the stochastic vacuum, promise a successful data analysis as well. It can be hoped that future work with the data will expose the qualitative di!erences among the available models, thus advancing our understanding of the proton bound state and non-perturbative QCD dynamics. The available data shows the importance of higher twist contributions in the small-mass region of di!ractive structure functions. A consistent theoretical description of both leading and higher twist e!ects has yet to be developed. Furthermore, the energy dependence of the di!ractive structure function and, even more importantly, its relation to the energy dependence of the inclusive structure function are interesting unsolved problems. While the namK ve summation of logarithms appears to be disfavoured, no new standard framework has emerged. The proposal of an identical, non-perturbative energy dependence of both di!ractive and inclusive cross-sections at some small virtuality Q is phenomenologically successful. This energy growth is re#ected in the  m dependence of the di!ractive structure function, which is not altered by the perturbative Q evolution.

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The detailed investigation of di!ractive "nal states provides a further broad "eld where di!erent approaches can be tested. Here, considerable improvements on present data are expected in the near future. So far, the analyses performed emphasize the dominance of the soft colour exchange mechanism and the importance of the di!ractive gluon distribution. Di!ractive meson production is, on a qualitative level, well described by perturbative QCD wherever the dominance of the hard scale is established. However, large uncertainties remain, and non-perturbative contributions are important even at relatively high values of Q. In particular, the ratio of longitudinal and transverse o meson production cross-sections is far smaller than suggested by simple perturbative estimates. A better understanding of the dynamics of meson formation appears to be required. In summary, it is certainly fair to say that di!ractive electroproduction at small x proved a very rich and interesting "eld. Its investigation over the recent years has lead, in many di!erent ways, to the improvement of our understanding of QCD dynamics on the interface of perturbative and non-perturbative physics. However, many important problems, such as the energy dependence and the systematic treatment of higher twist contributions, remain at present unsolved.

Acknowledgements I am deeply indebted to W. BuchmuK ller, who introduced me to the subject of di!raction at HERA, for our very interesting and fruitful collaboration and for his continuing support of my work over the recent years. I am also very grateful to S.J. Brodsky, T. Gehrmann, P.V. Landsho!, M.F. McDermott and H. Weigert, with whom I had the privilege to work on several of the topics described in this review and from whose knowledge I have greatly bene"ted. Furthermore, I would like to thank J. Bartels, M. Beneke, E.R. Berger, J.C. Collins, M. Diehl, J.R. Forshaw, L. Frankfurt, G. Ingelman, B. Kopeliovich, L. McLerran, A.H. Mueller, D.E. Soper, M. Strikman, T. Teubner, B.R. Webber and M. WuK stho! for their many valuable comments and discussions. Finally, I am especially grateful to H.G. Dosch and O. Nachtmann for their encouragement, for numerous interesting discussions, and for their comments on the "nal version of the manuscript. Appendix A. Derivation of Eikonal formulae A particularly simple way to derive the eikonal formulae of Section 3.1 is based on the direct summation of all Feynman diagrams of the type shown in Fig. 6 in the high-energy limit. In the case of scalar quarks, the one gluon exchange contribution to the S-matrix element reads



S (p, p)"!ig(p #p ) dx AI(x)e VNY\N .  I I

(A.1)

Working in a covariant gauge and in the target rest frame, all components of A are expected to be of the same order of magnitude. If, in the high-energy limit, p and p are the large momentum > > components, the approximation (p #p )AIK(p #p )A /2 can be made. Assuming that the I I > > \ x dependence of A is soft, one can write



dx e V\ N> \N> A (x , x , x )K4pd(p !p )A (x , x ) , \ \ > \ , > > \ > ,

(A.2)

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where the x dependence of A has been suppressed on the r.h. side since it is irrelevant for the \ process. More precisely, one has to think of the incoming and outgoing particles as wave packets, localized, say, at x K0, which is the x position where A has to be evaluated. Eq. (A.1) takes the \ \ form








ig S (p, p)"2pd(p !p )2p dx e\ V, N, \N,  ! dx A (x , x ) , , > \ > ,  > > > 2

(A.3)

which, within the approximation k d(k )Kk d(k ), is precisely the "rst term of the expansion of > >   Eq. (3.3) in powers of A. The n gluon exchange contribution to the S-matrix element can be written as

    

dpG\ ig ie VGNG\NG\ L (pG #pG\)A (xG) S (p, p)" “ ! dxG > \ L (2p) > (pG\)#ie 2 G ig ; ! dx(p#p)A (x)e VN\N , > > \ 2



 (A.4)

where xG is the space}time variable at vertex i, the momentum of the quark line between vertex i and vertex i#1 is denoted by pG, and the initial and "nal momenta are p"p and p"pL. With Eq. (A.2), all p integrations in Eq. (A.4) become trivial, and all momentum plus components > become identical with p. Integrations over the momentum minus components are performed > using the approximate relation



e\ N\ W> \X>  2pi dp "!h(y !z ) . (A.5) \ p p !p #ie > > p > \ , > The resulting step functions translate into path ordering of the matrix valued "elds A along the plus direction,





1 dx2dxL h(x!x)2h(xL!xL\)+2," dx2dxL P+2, . > > > > > > > > n!

(A.6)

Now that all dependence on the intermediate transverse momenta p2pL\ originating in the , , propagators has disappeared, the corresponding integrations are straightforward. They produce d-functions in transverse co-ordinate space,



dp e N, W, \X, "(2p)d(y !z ) . , , ,

(A.7)

Making use of Eqs. (A.5)}(A.7), the expression for the nth-order contribution to the S-matrix element "nally takes the form S (p, p)"2pd(p !p )2p L > > >








ig 1 L dx e\ V, N, \N,  P ! dx A (x , x ) . , > \ > , 2 n!

(A.8)

Since this is precisely the nth term of the expansion of Eq. (3.3) in powers of A, the derivation of the eikonal formula in the scalar case is now complete.

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To obtain the eikonal formula for spinor quarks, Eq. (3.5), write down the analogue of Eq. (A.4) using the appropriate expressions for quark propagator and quark}quark}gluon vertex. Applying the identity of Eq. (3.11) to each of the quark propagators, neglecting the c term, which is > suppressed in the high-energy limit, and making use of the relation u (k)(!igA. )u (k)K!igk A d , (A.9) QY Q > \ QQY valid if k Kk are the big components, the exact structure of Eq. (A.4) is recovered. The only > > di!erence is an additional overall factor d , corresponding to the conservation of the quark QQY helicity. The further calculation, leading to the analogue of Eq. (A.8), is unchanged, and the eikonal formula follows. The further generalization to the gluon case, Eq. (3.6), proceeds along the same lines. Begin by writing down the analogue of Eq. (A.4), i.e., the S-matrix element for Fig. 6 with a gluon line instead of the quark line. The expression contains conventional three-gluon vertices and gluon propagators in Feynman gauge. Contributions with four-gluon vertices are suppressed in the high-energy limit. The colour structure is best treated by introducing matrices AA "A?(¹?A )@A"!iA?f ?@A ,

(A.10)

where f ?@A are the usual structure constants appearing in the three-gluon vertex. Now, the product of matrices A in Eq. (A.4) is, in the gluonic case, simply replaced by an identical product of adjoint representation matrices AA . Next, the gIJ tensor of each gluon propagator !igIJ/k is decomposed according to the identity






 mIkJ kImJ mImJ # ! k , (A.11) gIJ" eI eJ # G G (mk) (mk) (mk) G where m is a light-like vector with a non-zero minus component, m"(0, 2, 0 ), and the polarization , vectors are de"ned by ek"em"0 and e"!1. An explicit choice is given by






2(k e ) , G, , e , (A.12) G, k > where the transverse basis e "(1, 0) and e "(0, 1) has been used. If, in the high-energy limit, , , gIJ appears between two three-gluon vertices, the last three terms of Eq. (A.11) can be neglected. Note that, for the second and third term, this is a non-trivial statement since the vector k in the numerator could, in principle, compensate for the suppression by the k in the denominator. > However, this is prevented by the gauge invariance of the three gluon vertex. Thus, the analogue of Eq. (A.4) is written down with three-gluon vertices and propagators proportional to e " 0, G

 gIJK eI eJ . (A.13) G G G Now, each of the three-gluon vertices < (!k, k, k!k) (where all momenta are incoming and IJN colour indices are suppressed) appears between two transverse polarization vectors and simpli"es according to eI (k)< (!k, k, k!k)ANA eJ (k)K!igk AA d , G > \ GGY GY IJN

(A.14)

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where AA is the external "eld. This is similar to what was found in the quark case in Eq. (A.9), with the di!erence that helicity conservation is replaced by polarization conservation. Therefore, as before, the structure of Eq. (A.4) is recovered, but with the external "eld in the adjoint representation and with an additional polarization conserving d-function. The further calculation, leading to the analogue of Eq. (A.8), is unchanged, and the eikonal formula in the gluonic case follows.

Appendix B. Spinor matrix elements In this appendix, the spinor matrix elements of the type u (k)e. (q)u (k), required, e.g., for the QY Q calculation the transition from virtual photon to qq pair, are listed. Using light-cone components for vectors, a"(a , a , a ), and orienting the photon momentum > \ , along the positive z-axis, q"(q ,!Q/q , 0 ), the longitudinal and transverse polarization > > , vectors can be de"ned as e "(q /Q, Q/q , 0 ), e "(0, 0, e ($)) , * > > , ! ,

(B.1)

where e ($)"(1,$i)/(2. , The Dirac representation of c matrices and the conventions of [78] for Dirac spinors are used. Introducing the matrix




e"



0

1

!1 0

.

(B.2)

and the two-component spinors s


 1



"

0

, s


 0

\

"

1

,

(B.3)

the two independent Dirac spinors, with s"$1/2, for negative and positive frequency solutions can be explicitly written as rk s es Q u (k)"(k #m , v (k)"!(k #m k #m Q , rk Q  Q  s es Q k #m Q  where r is the vector formed by the three Pauli matrices. The result for longitudinal photon polarization can be obtained from the relation






u (k)c u (k)"d 2(k k , QY  Q QYQ  






(B.4)

(B.5)

which holds at leading order in the high-energy expansion. From this, the matrix element u (k)e. (q)u (k) is obtained using the gauge invariance of the qq -photon vertex, u (k)q. u (k)"0. QY * Q QY Q Since the quark mass and transverse momentum do not enter the leading order relation, Eq. (B.5), they may be neglected so that the relation v (k)"!u (k) holds. Thus, the complete result takes Q \Q

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the form u (k)e u (k)"v (k)e v (k)"!v (k)e u (k)"!u (k)e u (k)"d 2Q(a(1!a) , QY * Q QY * Q \QY * Q \QY * Q QYQ where a"k /q and e "e (q).   * * Neglecting terms suppressed in the high-energy limit, the relevant matrix elements for transverse photon polarization read




K(k !k )/0   , k k /k k QYQ ,>  ,\ 




0/m(k !k )   , k k /k k QYQ ,>  ,\ 

2 k,> k /k,\ k u (k)e. (q)u (k)"! QY >\ Q k k 0/m(k !k )     2 k,> k /k,\ k v (k)e. (q)v (k)"! QY >\ Q k k m(k !k )/0    



(B.7)



(B.8)











2 !m(k #k )/0 k,> k /k,\ k u (k)e. (q)v (k)"# , QY >\ Q k k k k /k k 0/m(k #k ) QYQ   ,>  ,\    2 0/!m(k #k ) k,> k /k,\ k v (k)e. (q)u (k)"# . QY >  \ Q k k k k /k k m(k #k )/0 QYQ   ,>  ,\   

(B.9)

(B.10)

Here the notation k "k $ik is used, and the four entries (11), (12), (21) and (22) of the ,!   two-by-two matrices on the r.h. sides of the above equations correspond to the combinations (ss)"(##), (#!), (!#) and (!!). For each of these four entries, the expression         before and after the oblique stroke &/' corresponds to positive and negative photon polarization, respectively. For example, u

>

(k)e.



2 (q)u (k)"! m(k !k ) . > \   k k  

(B.11)

The calculation of the transition from virtual photon to qq -gluon con"guration requires, in addition to the qq -photon vertex, the knowledge of the qq -gluon vertex (see Section 3.3). The corresponding spinor matrix elements are easily obtained from Eqs. (B.7)}(B.10) if the con"guration is rotated in such a way that the gluon momentum is parallel to the z-axis. In the high-energy limit, such a rotation corresponds to the substitutions k Pk !q (k /q ), k Pk !q (k /q ) , , , , > > , , , > >

(B.12)

where q is now interpreted as the gluon momentum. Note that similar matrix elements are commonly used in light-cone perturbation theory (see, e.g., Tables II and III of [194]).

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Appendix C. Derivation of di4ractive quark and gluon distribution The explicit formulae for di!ractive quark and gluon distributions, Eqs. (4.53) and (4.54), can be derived by appropriately adapting the scalar calculation of Section 4.4 to the case of spinor or vector particles. The most economic procedure is to "rst identify the piece of the old squared amplitude "¹" that depends explicitly on the spin of the soft parton. This piece, which is essentially just the squared scattering amplitude of the soft parton and the external "eld, is symbolically separated in Fig. 45. The two independent integration variables for the intermediate momentum of the soft parton are denoted by k and kI in the amplitude ¹ and its complex conjugate ¹H, respectively. Note also that k "kI and k "kI at leading order. > > \ \ It is straightforward to write down the factor B that corresponds to the box in Fig. 45 for the Q scalar case. Since the o!-shell denominators k and kI  as well as the two eikonal factors and energy d-functions are present for all spins of the soft parton, they are not included into the de"nition of B. All that remains are the explicit factors 2k Kk from the e!ective vertex, Eq. (4.35). Therefore, the  > result for the scalar case reads simply B "k . (C.1) Q > The next step is the calculation of the corresponding expressions B and B , given by the box in O E Fig. 45, in the case where the produced soft parton is a quark or a gluon. Introducing factors B /B O Q and B /B into Eq. (4.52) will give the required di!ractive parton distributions (d f "/dm) and E Q O (d f "/dm). E C.1. Diwractive quark distribution Observe "rst, that the analogue of Eq. (4.35) for spinors is simply c < (p, p)"2pd(p !p ) > [;I (p !p )!(2p)d(p !p )] . , , , , O   2

(C.2)

The Dirac structure of < follows from the fact that, in the high-energy limit, only the light-cone O component A of the gluon "eld contributes. The normalization is consistent with Eqs. (3.12) \ and (3.13).

Fig. 45. Symbolic representation of the square of the amplitude for a hard di!ractive process. The box separates the contributions associated with the soft parton and responsible for the di!erences between di!ractive distributions for scalars, spinors and vector particles.

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Consider now the Dirac propagator with momentum k. The hard part ¹ requires the & interpretation of the quark line as an incoming parton, which collides head on with the photon. Therefore, it is convenient to switch to the Breit frame and to de"ne a corresponding on-shell momentum l, given by l "!k , l "!k and l "l /l . The propagator can now be written \ \ , , > , \ as u (l)u (l) c 1 "! Q Q Q ! \ . (C.3) k 2l k. \ This is technically similar to Section 3.2, where an on-shell momentum was de"ned by adjusting the minus component of k. In the present treatment, however, the plus component of l"!k is adjusted, so that a partonic interpretation in the Breit frame becomes possible. It has now to be shown that the second term on the r.h. side of Eq. (C.3) can be neglected, since it is suppressed in the Breit frame by the hard momentum l "yP . This can be intuitively understood by observing \ \ that this term represents a correction for the small o!-shellness of the quark, which is neither important for ¹ , nor for the soft high-energy scattering o! the external "eld. & To see this more explicitly, write the relevant part of the amplitude in the form






c u (l)u (l) 1 Q Q Q # \ ¹ ,!(C #C ) . (C.4) u (k)c ¹ "!u (k)c > &   > k. & 2l k \ It will now be shown that the c term C is suppressed with respect to C . \   The spinor matrix element in C can be estimated using the relation  4(p p ) (C.5) u (p)c u (p)" , , , Q > Q (p p Q \ \ valid in the limit where the minus components of p and p become large (cf. Table II of [194]). In the soft region, k&k &k&K, and !k &k &Q are the large components in the Breit frame. , , \ \ Thus, [u (k)c u(l)]&K/Q, and the "rst term on the r.h. side of Eq. (C.4) is estimated to be > 1 (C.6) C & [u (l)¹ ] , &  Q where factors O(1) have been suppressed. Introducing a vector a"(1, 0, 0 ), so that c /2"a. , the c term can be written as , \ \ 1 (C.7) C " [u (k)c u (a)] [u (a)¹ ] . Q &  > Q l \ Q Since [u (k)c u(a)] vanishes for k "k/k P0, in the Breit frame, where k is large, the estimate > > , \ \ [u (k)c u(a)]&"k "/(k &K/(Q can be made. With l &Q, one obtains \ > , \ K [u (a)¹ ] K C & & , & (C.8) Q [u (l)¹ ] C &  (Q where it has been assumed that no speci"c cancellation makes u (l)¹ small, i.e., & [u (l)¹ ]/[u (a)¹ ]&(Q. Eq. (C.8) establishes the required suppression of the c term in Eq. (C.3). & & \

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To proceed with the evaluation of B note that, when the soft part is separated in Fig. 45, the Q spinor u (l)Kv (l) from Eq. (C.3) has to be considered a part of the hard amplitude ¹ . Therefore Q \Q & the analogue of Eq. (C.1) reads c c (C.9) B " u (k) > u (l)u (lI ) > u (k) , Q Q Q QY 2 QY 2 QQY where lI is de"ned analogous to l, but using the momentum kI instead of k. The spin summation decouples from the hard part if the measurement is su$ciently inclusive. The above expression can be evaluated further to give 1 2(l lI ) ,, , B " k u (lI )c u (l)"k Q 2 > Q > Q > (l lI Q \\

(C.10)

where the last equality again uses Eq. (C.5). Simple kinematics leads to the result




2(k kI ) m!y , , . (C.11) y k , Comparing this with Eq. (C.1), the di!ractive quark distribution, Eq. (4.53), is straightforwardly obtained from the scalar case, Eq. (4.52). Note that the virtual fermion line corresponds to a right-moving quark with momentum k in the proton rest frame and to a left-moving antiquark with momentum l in the Breit frame. Therefore, the above result has, in fact, to be interpreted as a di!ractive antiquark distribution. The di!ractive quark distribution is identical. B "k Q >

C.2. Diwractive gluon distribution To obtain the di!ractive gluon distribution, the procedure of the last section has to be repeated for the case of an outgoing soft gluon with momentum k in Fig. 45. Calculating the contribution separated by the box will give the required quantity B , in analogy to Eqs. (C.1) and (C.11). E It will prove convenient to introduce two light-like vectors m and n, such that the only non-zero component of m is m "2 in the proton rest frame, and the only non-zero component of n is \ n "2 in the Breit frame. Since Breit frame and proton rest frame are connected by a boost along > the z-axis with boost factor c"Q/(m x), the product of these vectors is (mn)"2c. N Furthermore, two sets of physical polarization vectors, e and e (with i"1, 2) are de"ned by G G the conditions ek"ek"0, e"e"!1, and em"en"0. An explicit choice, written in lightcone co-ordinates, is











2(k e ) 2(k e ) , G, , e , G, , 0, e and e " , (C.12) G, G G, k k > \ where the transverse basis e "(1, 0) and e "(0, 1) has been used. Note that the above , , equations hold in the proton rest frame, in the Breit frame, and in any other frame derived by a boost along the z-axis. e " 0, G

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These de"nitions give rise to the two following representations for the metric tensor:







mIkJ kImJ mImJ # ! k (C.13) gIJ" eI eJ # G G (mk) (mk) (mk) G nIkJ kInJ nInJ " eI eJ # # ! k . (C.14) G G (nk) (nk) (nk) G The amplitude for the process in Fig. 45, with the lowest parton being a gluon in Feynman gauge, is proportional to



A"eI(k)< (k, k) ¹J "eI(k)< (k, k) gJMg ¹N , (C.15) E IJ & E IJ MN & where




mk km mm A"e< ee# # ! (k) E (mk) (mk) (mk)



nk kn nn ee# # ! (k) ¹ , & (nk) (nk) (nk)

(C.16)

where the appropriate contractions of vector indices are understood. Several estimates involving products of < and ¹ with speci"c polarization vectors will be required. E & Note that both e and n are O(1) in the Breit frame. For appropriate polarization, the amplitude (e¹ ) involves no particular cancellation, i.e., it has its leading (formal) power behaviour in the & dominant scale Q. Therefore, (n¹ ) is not enhanced with respect to (e¹ ), & & (n¹ )/(e¹ )&O(1) . (C.17) & & By analogy, it can be argued that (e< m) is not enhanced with respect to (e< e): since both e and E E m are O(1) in the proton rest frame and (e< e) has the leading power behaviour for appropriate E polarizations e and e, the following estimate holds: (e< m)/(e< e)&O(1) . (C.18) E E Gauge invariance requires (k¹ ) to vanish if k"0. Since the amplitude ¹ is dominated by & & hard momenta O(Q), and k&K;Q, this leads to the estimate (k¹ )/(e¹ )&k/Q . (C.19) & & Analogously, from (e< k)"0 at k"0, the suppression of this quantity at small virtualities k can E be derived, (e< k)/(e< e)&k/k . (C.20) E E > For this estimate it is also important that none of the soft scales involved in < , like k or the gauge E , "eld A , can appear in the denominator to compensate for the dimension of k. I

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All the vector products nk, mk, ne, me, mn, and ee can be calculated explicitly. Using the relations in Eqs. (C.17)}(C.20), it is now straightforward to show that (e< e)(ee)(e¹ ) is indeed the leading E & term in Eq. (C.16). The other terms are suppressed by powers of Q or k . > Thus, the leading contribution to "A", with appropriate polarization summation understood, reads "A"" [(e (k)< e )(e e )(e ¹ )][(e (k)< e )(e e )(e ¹ )]H , (C.21) J E G G H H & J E GY GY HY HY & GHGYHYJ where the arguments k and kI of the polarization vectors in the "rst and second square bracket, respectively, have been suppressed. In the high-energy limit, the scattering of a transverse gluon o! an external "eld is completely analogous to the scattering of a scalar or a spinor, (C.22) e (p)< (p, p)e (p)"2pd(p !p ) 2p d ;I A(p !p ) , , , JY E J    JYJ the only di!erence being the non-Abelian eikonal factor, which is now in the adjoint representation. In analogy to the spinor case, the polarization sum decouples from the hard part for su$ciently inclusive measurements, so that the squared amplitude is proportional to (C.23) "A"""¹ "(e (k)< e (k))(e (k)< e (kI ))H (e (k)e (k)) (e (kI )e (kI )) . G H G H & J E J J E J GH Note that there is no summation over the index l. Recall the de"nition of B, the soft part of the amplitude square, given at the beginning of the last section and illustrated in Fig. 45. The corresponding expression in the case of a soft gluon can now be read o! from Eqs. (C.22) and (C.23): B "k (e (k)e (k))(e (kI )e (kI )) . E > G H G H GH This is further evaluated using the explicit formulae in Eq. (C.12) and the identity

(C.24)

e? e@ "d?@ (a, b3+1, 2,) . G, G, G Comparing the resulting expression,

(C.25)











2kG kH 1!b 2kI G kI H 1!b B "k dGH# , , dGH# , , , (C.26) E > b b k  k , , to Eq. (C.1), the di!ractive gluon distribution of Eq. (4.54) is obtained. Note that the factor N appearing in the denominator of Eq. (4.52) has been replaced by the A dimension of the adjoint representation, N!1. A Appendix D. Inclusive parton distributions This appendix is devoted to the calculation of inclusive parton distributions and inclusive structure functions in the semiclassical framework. Inclusive DIS was discussed in Section 3.2,

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where the cross-section for qq production o! a colour "eld was derived, at the end of Section 4.4, where the parton model interpretation of this cross-section was outlined, and in Section 7.1, where it was part of the semiclassical analysis of di!ractive and inclusive structure function data. Here, the above scattered information is collected, and a more coherent account, including further technical details, is given (cf. [65]). Recall the leading twist cross-section p for qq pair production by a transversely polarized 2 photon obtained in Section 3.2 (cf. Eq. (3.27)). The corresponding transverse structure function reads






   



 

Q 4 ln !1 F (x, Q)" 2 k 3(2p)

tr(R , = , (0)R , =R, (0)) W V W V

V, I k = I (k !k )  2 , . (D.1) dk , V, , # dN dk  , , N#k (2p)  V, , To map this calculation onto the conventional parton model framework, identify the result as F (x, Q)"2xq(x, Q). The corresponding quark distribution reads 2 2 Q xq(x, Q)" ln !1 tr(R , = , (0)R , =R, (0)) W V W V 3(2p) k V, I 2 # N dN K (yN) tr(= , (y )=R, (y )) . (D.2)  V , V , (2p) , ,  W V Here the modi"ed Bessel function K has been introduced so that, in both terms, the functions  = appear in co-ordinate space. This makes it particularly clear that the "rst term is only sensitive to the short distance behaviour, while the second term depends on the non-perturbative longdistance structure of the colour "eld. Note that the sum of both terms is independent of k. The corresponding gluon distribution at small x is most easily calculated as










1 3p RF (x, Q) " tr(R , = , (0)R , =R, (0)) . (D.3) xg(x, Q)" ) 2 W V W V R ln Q 2pa , a Q V Q Eqs. (D.2) and (D.3) can serve as the starting point for a conventional partonic analysis of inclusive DIS. To gain more physical insight into the correspondence of the semiclassical and the parton model approach, return to the starting point, Eq. (D.1). It is instructive to view F as a sum of two terms: 2 F  , the contribution of asymmetric con"gurations where quark or antiquark are slow, a(k/Q 2 or 1!a(k/Q (Fig. 22a), and F , the contribution of symmetric con"gurations where both 2 quark and antiquark are fast, a, 1!a'k/Q (Fig. 22b). In a frame where the proton is fast, say, the Breit frame, the asymmetric and symmetric contribution to F correspond to photon-quark 2 scattering and photon}gluon fusion respectively. The symmetric part is dominated by small qq pairs, i.e., by the short distance contribution to the Wilson-loop trace,



V

,



1 tr(= , (y )=R, (y ))" y V , V , 2

V

,

tr(R , = , (0)R , =R, (0))#O(y) . W V W V

(D.4)

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The corresponding contribution to the structure function is related to the "rst term on the r.h. side of Eq. (D.2), which generates the gluon distribution, Eq. (D.3). It can also be written as








e  Q F (0, Q)" O tr(R , =F, (0)R , =F,R(0)) , (D.5) dz P (z) ln !1 V 2 W W V OE 2p k V,  where P (z) is the conventional gluon}quark splitting function. OE The other splitting functions appear if a corrections to F and F , associated with higher Fock Q 2 * states of the virtual photon, are considered in the semiclassical approach. For example, the qq g parton con"guration involves, in the case where one of the quarks carries a small fraction of the photon momentum, a ln Q term associated with P (z). OO The splitting function P (z) is most easily derived by considering an incoming virtual scalar EE which couples directly to the gluonic action term F FIJ. Such a current was used previously in IJ [195] to study small-x saturation e!ects. The relevant lowest-order Fock state consists of two gluons. As reported in [65], it can be checked explicitly that the semiclassical calculation of the corresponding high-energy scattering process yields the usual gluon}gluon splitting function. Recently, a next-to-leading order calculation of the semiclassical gluon distribution, which is based on the above methods, has become available [196]. Since the semiclassical approach exactly reproduces the well-known DGLAP splitting functions, the large logarithms ln(Q/k) can be resummed in the conventional way, by means of the renormalization group. To this end, the parton distributions q(x, Q) and g(x, Q) are evaluated using DGLAP evolution equations, with the input distributions q(x, Q ) and g(x, Q ) given by   Eqs. (D.2) and (D.3). Here Q is some small scale where logarithmic corrections are not yet  important. The parton model description of the structure function at leading order includes only photon}quark scattering. The leading logarithmic term from the photon}gluon fusion process appears now as part of the resummed quark distribution. The large hadron model of Section 6.2 provides an expression for the basic function  , tr(= , (y )=R, (y )). Inserting the explicit formula of Eq. (6.25) into Eqs. (D.2) and (D.3), the V , V , V following compact expressions for the inclusive parton distributions at a low scale Q are obtained  aXN Q A ln  !0.6424 , xq(x, Q )" (D.6)  3p a






2aXN A . (D.7) xg(x, Q )"  pa (Q ) Q  As discussed in Section 7.1, the analysis of [65] introduces a soft energy dependence into these input distributions by ascribing a logarithmic growth to the total normalization, XPX(¸!ln x) .

(D.8)

Note also the observation of [197] that the small-x structure function is well described by a simple ln(1/x) with an additional ln Q enhancement, similar to the e!ects of Altarelli}Parisi evolution. The DGLAP evolution of the above inclusive parton distributions and corresponding di!ractive distributions given in Section 7.1 provides predictions for both the inclusive and di!ractive structure functions. In the numerical analysis, the inclusive distributions are multiplied with (1!x)

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Fig. 46. The inclusive structure function F (x, Q) at small x computed in the semiclassical approach with data from  [161]. The data with Q"1.5 GeV are not included in the "t.

to ensure vanishing of the distributions in the limit xP1, which is required for the numerical stability of the DGLAP evolution. A combined "t to small-x HERA data gives a good description of the experimental results. While plots of F" are given in Figs. 37 and 38, here, corresponding results for the inclusive structure  function are presented (see Fig. 46). Since the underlying model is only valid in the small-x region, data points above x"0.01 are not considered. To appreciate the quality of the "ts, recall that, within the large hadron model, the di!ractive (Eqs. (7.4) and (7.5)) and inclusive (Eqs. (D.6) and (D.7)) parton distributions are highly constrained with respect to each other.

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SELF-FOCUSING AND TRANSVERSE INSTABILITIES OF SOLITARY WAVES

Yuri S. KIVSHAR , Dmitry E. PELINOVSKY
Optical Sciences Centre, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

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Self-focusing and transverse instabilities of solitary waves Yuri S. Kivshar , Dmitry E. Pelinovsky
Optical Sciences Centre, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia
Department of Mathematics, University of Toronto, Toronto, Ont. M5S 3G3, Canada Received April 1999; editor: A.A. Maradudin Contents 1. Introduction 2. Physical models 2.1. Basic soliton equations 2.2. Suppression of wave collapse: physical mechanisms 3. Criteria for soliton self-focusing 3.1. Geometric optics approach 3.2. Linear eigenvalue problems 3.3. Analogy with modulational instability 4. Equations for soliton parameters 4.1. Direct asymptotic expansions 4.2. Averaged Lagrangian method 4.3. Gas dynamics equations 4.4. Reduction to the NLS equation 4.5. Higher-order perturbation theory 4.6. Di!erent scenarios of soliton self-focusing 4.7. Exact solutions for the soliton self-focusing 5. Long-scale approximation 5.1. Basic asymptotic equations 5.2. Derivation 5.3. Analysis 5.4. Extensions

120 122 122 125 129 129 131 134 135 136 139 141 143 144 146 147 148 148 149 153 158

6. Short-scale approximation 6.1. Basic asymptotic equations 6.2. Derivation 7. Some other models 7.1. NLS-type models 7.2. KdV-type models 7.3. Kinks in the Cahn}Hilliard equation 8. Experimental observations 8.1. Self-focusing and bright solitons 8.2. Dark solitons 9. Concluding remarks 9.1. Random or periodic #uctuations 9.2. Self-focusing of coupled waves 9.3. Transverse vs. longitudinal instabilities 9.4. Transverse instabilities of nonlinear guided waves 9.5. Higher-order localized modes 9.6. Ring-like models and ring solitons 9.7. Instabilities in higher dimensions Acknowledgements References

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Abstract We give an overview of the basic physical concepts and analytical methods for investigating the symmetrybreaking instabilities of solitary waves. We discuss self-focusing of spatial optical solitons in di!ractive nonlinear media due to either transverse (one more unbounded spatial dimension) or modulational (induced by temporal wave dispersion) instabilities, in the framework of the cubic nonlinear SchroK dinger (NLS) equation and its generalizations. Both linear and nonlinear regimes of the instability-induced soliton 0370-1573/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 1 0 6 - 4

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dynamics are analyzed for bright (self-focusing media) and dark (self-defocusing media) solitary waves. For a defocusing Kerr medium, the results of the small-amplitude limit are compared with the theory of the transverse instabilities of the Korteweg}de Vries solitons developed in the framework of the exactly integrable Kadomtsev}Petviashvili equation. We give also a comprehensive summary of di!erent physical problems involving the analysis of the transverse and modulational instabilities of solitary waves including the soliton self-focusing in the discrete NLS equation, the models of parametric wave mixing, the Davey} Stewartson equation, the Zakharov}Kuznetsov and Shrira equations, instabilities of higher-order and ring-like spatially localized modes, the kink stability in the dissipative Cahn}Hilliard equation, etc. Experimental observations of the soliton self-focusing and transverse instabilities for bright and dark solitons in nonlinear optics are brie#y summarized as well.  2000 Elsevier Science B.V. All rights reserved. PACS: 42.65.!k

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1. Introduction Wave instabilities are probably the most remarkable physical phenomena that may occur in nonlinear systems (see, e.g., Infeld and Rowlands (1990) and references therein). Modulational instability and breakup of a continuous-wave (c.w.) "eld of large intensity was "rst predicted and analyzed in the context of waves in #uids (Benjamin and Feir, 1967; Zakharov, 1968). The similar e!ect is self-focusing of light in optical media with a nonlinear response (Askar'yan, 1962; Chiao et al., 1964; Talanov, 1964; Kelley, 1965; Ostrovsky, 1966; Bespalov and Talanov, 1966) which is responsible for the appearance of hot spots and associated optical damage in media irradiated by high power laser pulses. One of the important physical processes associated with the development of modulational instability is the generation of a train of spatially (beams) or temporary (pulses) localized waves (Yuen and Ferguson, 1978), the e!ect observed experimentally in di!erent physical systems, e.g. in the #uid dynamics (Benjamin and Feir, 1967; Yuen and Lake, 1975; Melville, 1982; Su, 1982), nonlinear beam propagation (Campillo et al., 1973, 1974; Iturbe-Castillo et al., 1995), electrical transmission lines (Mizumura and Noguchi, 1975; MarquieH et al., 1994, 1995), optical "bers (Tai et al., 1986), etc. One of the fundamental models describing the nonlinearity-induced modulational instability is the generalized nonlinear SchroK dinger (NLS) equation, written in one spatial dimension as follows: iu #u #(r#1)"u"Pu"0 , (1.1) R VV where r is the power of nonlinearity. In the case r"1, this model describes the propagation of an electric "eld envelope in an optical waveguide, the famous model known to be integrable by means of the inverse scattering transform (see, e.g., Ablowitz and Segur, 1981, and references therein). Modulational instability can be viewed as the simplest case of the so-called symmetry-breaking instability, when a solution of a nonlinear system of a certain dimension (e.g., an uniform c.w. background) is subjected to a broader class of perturbations. Another example is the instability of low-dimensional solitary waves to perturbations involving higher dimensions. This is a typical case of the so-called transverse instability of plane solitary waves, "rst discussed for long-wave solitons of the Korteveg}de Vries (KdV) equation (Kadomtsev and Petviashvili, 1970) and envelope solitons of the NLS equation (Zakharov, 1967; Zakharov and Rubenchik, 1973), and then investigated by di!erent methods for a variety of nonlinear soliton-bearing models (see, e.g., Yajima, 1974; Washimi, 1974; Schmidt, 1975; Spatschek et al., 1975; Katyshev and Makhankov, 1976; Laedke and Spatschek, 1978; Andersen et al., 1979a,b; Ablowitz and Segur, 1979, 1980; Akhmediev et al., 1992; Soto-Crespo et al., 1991, 1992; Kuznetsov and Rasmussen, 1995) with demonstrations in numerical simulations (e.g., Degtyarev et al., 1975; Pereira et al., 1977; Infeld and Rowlands, 1980). Why this type of soliton instabilities becomes important and deserves a special attention and discussion these days? First of all, the recent progress in developing and employing nonlinear optical materials, led to several important discoveries of self-focusing of light, self-trapped beam propagation, and spatial optical solitary waves in di!erent nonlinear materials, including photorefractive (Shih et al., 1995, 1996), quadratic (Torruellas et al., 1995), and saturable non-Kerr (Tikhonenko et al., 1995, 1996b) media. It is expected that the development of novel band-gap materials based on quadratic or cubic nonlinearities will eventually lead to the experimental

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observation and manipulation of the so-called light bullets, self-focused states of light localized in both space and time, the building blocks of the future all-optical photonics devices. Secondly, di!erent types of symmetry-breaking instabilities have been recently described theoretically and observed experimentally in nonlinear optics. This includes the observation of a breakup of bright soliton stripes in a bulk photorefractive medium due to the transverse modulational instability and the formation of a sequence of two-dimensional self-trapped beams (Mamaev et al., 1996a,b), the "rst observation of the generation of pairs of optical vortex solitons due to the transverse instability of dark-soliton stripes (Tikhonenko et al., 1996a; Mamaev et al., 1996c); the theory and the "rst experimental demonstration of spatial modulational instability in quadratically nonlinear (or s) optical media (Fuerst et al., 1997a,b; DeRossi et al., 1997a,b); a decay of ring-shape optical beams with nonzero angular momentum into higher-dimensional solitary waves, observed experimentally in a rubidium vapour (a saturable defocusing medium) (Tikhonenko et al., 1995), a quadratic nonlinear crystal (Petrov et al., 1998), and investigated theoretically as a general solitonic e!ect (Firth and Skryabin, 1997). It is also worth mentioning the pulse compression and all-optical switching in waveguide arrays induced by modulational instability of nonlinear localized modes in discrete systems (e.g., Aceves et al., 1994a,b, 1995). These results, applied to nonlinear systems of a di!erent nature, call for a systematic overview of the basic ideas and analytical methods of the soliton stability theory previously discussed only for particular examples or techniques (Makhankov, 1978; Kuznetsov et al., 1986; Trubnikov and Zhdanov, 1987; Rypdal and Rasmussen, 1989; Infeld and Rowlands, 1990; Kamchatnov, 1997). The main purpose of this survey is twofold. First of all, we give an overview of the basic physical concepts and analytical methods in the theory of the symmetry-breaking instabilities of solitary waves. We analyze in detail linear and nonlinear regimes of self-focusing of planar bright and dark solitons described by two conventional models, (i) the elliptic and hyperbolic versions of the cubic NLS equation, and (ii) the Kadomtsev}Petviashvili (KP) equation, the well-known two-dimensional generalization of the KdV equation. Second, we discuss all possible scenarios of the instability-induced soliton dynamics and give a summary of the results on the transverse soliton instabilities for di!erent types of physical models and more general types of nonlinearity. In particular, we show that the self-focusing instability leads to either the formation of localized waves stable in higher dimensions with a singular, nonsingular or decaying amplitude evolution, or long-lived periodic oscillations between planar and modulated quasi-plane soliton states. For illustration of the basic asymptotic methods, we compare our results with those for modulational instability of a c.w. background within the generalized NLS equation. The paper is organized as follows. In Section 2 we introduce our basic models and discuss their physical applications and generalizations, including a brief overview of physical mechanisms for suppressing wave collapse in the NLS model. Then, in Section 3 we discuss the general criteria for the transverse self-focusing of solitary waves in dispersive and di!ractive nonlinear media. The asymptotic analysis of this phenomenon is presented in Section 4, where we derive the modulation equations for the parameters of slowly modulated self-focusing solitons. Modi"cations of the asymptotic approach for small-amplitude (long-scale and short-scale) expansions are discussed in Sections 5 and 6, respectively. Section 7 gives a brief summary of some generalizations of the NLSand KdV-type models, and it presents also some other types of nonlinear models, including discrete lattices, the Zakharov}Kuznetsov and Shrira equations, the Davey}Stewartson equation, the models of parametric wave mixing in di!ractive media with a quadratic optical response, the

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Cahn}Hilliard equation, etc. In Section 8 we discuss the experimental observations of self-focusing and soliton instabilities in nonlinear optics, and Section 9 gives a list of some closely related or unsolved problems which might be useful for the future studies.

2. Physical models Throughout this paper, we distinguish two di!erent phenomena: self-focusing collapse of a spatially localized wave and self-focusing (or transverse) symmetry-breaking instability of a planar solitary wave. Collapse (or blowup) occurs when the amplitude of an unstable solitary wave localized in all dimensions grows to in"nity in a "nite time. As a matter of fact, the wave collapse is a particular scenario of the instability-induced evolution of a solitary wave under the action of perturbations of the same dimension. Transverse instability is an instability of a solitary wave that is localized in one (longitudinal) dimension and nonlocalized but perturbed in other (transverse) dimensions. The latter phenomenon is more generic, it may occur in the systems where the blowup instability is suppressed or it does not happen at all. There are known several di!erent scenarios of the long-term dynamics of the soliton transverse instability. Only in the systems where both blowup and transverse instability coexist, we expect that a perturbation along the soliton front may break a plane soliton into a chain of localized modes, each undergoing a further transformation into a collapsing mode. Here we discuss the basic soliton-bearing models which we consider below for analyzing the soliton transverse instability (Section 2.1), and also some physically important generalizations which are required for the suppression of the blowup instability whenever it may occur (Section 2.2). 2.1. Basic soliton equations The fundamental model to analyze the soliton transverse self-focusing is the (2#1)-dimensional (i.e. two spatial and one temporal variables) NLS equation which can be written as follows: it #t #p t #2p "t"t"0 , (2.1) R VV  WW  where p "$1 de"nes the type of the cubic nonlinearity, i.e. focusing (at p "#1) or defocusing   (at p "!1), and p "$1 de"nes the type of the wave dispersion/di!raction.   The most well-known applications of Eq. (2.1) are in nonlinear optics (see, e.g., a number of books, Shen, 1984; Boyd, 1992; Newell and Moloney, 1992; Agrawal, 1995). In particular, in the theory of spatial optical solitons (e.g., Boardman and Xie, 1993; Kivshar, 1998a), this model always appears with p "#1 (di!raction), and it can be derived for the beam propagation in a bulk  medium from Maxwell's equations in the so-called paraxial approximation, taking into account two transverse and one longitudinal spatial dimensions. The same equation appears for the case of temporal modulations of the (1#1)-dimensional solitons in a waveguide with the so-called anomalous dispersion, i.e., u(k)'0, where u"u(k) is the frequency of a wave which is a function of its wave number. For this latter case, the variable t stands for the propagation coordinate, and the variable y plays a role of the retarded time in the reference frame moving with the group velocity. For p "#1, the model (2.1) is usually referred to as the elliptic cubic NLS equation. 

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For the so-called normal dispersion, i.e., for u(k)(0, the NLS equation appears with p "!1,  and it is usually called the hyperbolic cubic NLS equation. In this latter case, the NLS equation (2.1) is less studied in the literature. However, it also provides a fundamental model for describing the nonlinear dynamics of a broad class of waves, e.g., deep-water gravitational waves (Zakharov, 1968; Martin et al., 1980; Yuen and Lake, 1986), lower-hybrid (Litvak et al., 1979) and cyclotron (Myra and Lin, 1980) waves in magnetized plasmas, etc. In nonlinear optics, Eq. (2.1) with p "!1  describes the spatio-temporal dynamics of an optical beam in a nonlinear medium with positive Kerr e!ect which undergoes self-focusing in space and self-modulation in time (see, e.g., Chernev and Petrov, 1992a). In the self-focusing medium (p "#1), the c.w. solution of Eq. (2.1) of the form  t(t)"o exp(2iot), is modulationally unstable to spatial periodic perturbations &e GV with the modulation wave numbers i selected in a "nite band, 0(i(i . As a result of the development

 of such an instability, we expect the generation of a train of localized waves (beams or pulses), usually called bright NLS solitons. An individual planar bright soliton is described by the yindependent localized solution of the NLS equation (2.1), that we write here in a general form, t(x, t)"U (x!2vt!s; u)e TV\TR>SR>F , @ where, for the case of the cubic nonlinearity,

(2.2)

U (x; u)"(u sech((ux) . @ In Eq. (2.2), the parameters 2v and u stand for the soliton velocity and frequency (or its propagation constant), which represent the translational and oscillatory degrees of freedom of a bright NLS soliton, respectively. The constant parameters s and h are the soliton initial position and phase, respectively. In many problems of the soliton dynamics, the envelope bright soliton (2.2) is analyzed at rest; its translational degree of freedom is not excited and it can be eliminated by the standard Galilei transformation. As was "rst shown by Zakharov and Rubenchik (1973), a plane bright soliton (2.2) is unstable to higher-dimensional perturbations both in the elliptic and hyperbolic cases. This phenomenon is called the soliton transverse instability, for the spatial case, or modulational instability, for the temporal case. In the elliptic case (p "#1), the unstable transverse perturbations &e NW have wave numbers  within the "nite interval 0(p(p , where p "(3u. It was shown numerically (Degtyarev et al.,   1975) that the self-focused beams undergo collapse in a "nite time (or at "nite propagation distance). To avoid a catastrophic collapse in realistic physical models, an e!ective nonlinearity saturation should be taking into account, and it leads to a "nite wave amplitude (Litvak et al., 1991) (see also Section 2.2 below). In the hyperbolic case (p "!1), the unstable transverse perturbations also have a "nite-interval  wave numbers, i.e., 0(p(p , but the derivation of the cuto! value p is a complicated problem   which led to contradictory conclusions (Kuznetsov et al., 1986; Rypdal and Rasmussen, 1989). Recently, a simple result was obtained by an asymptotic analysis, p "(u (Pelinovsky and Sulem,  1999). Numerical simulations (Pereira et al., 1978) revealed a breakup of a planar soliton into localized modes which move apart and spread out due to the action of the wave dispersion. In addition, travelling instabilities of a bright soliton was also discovered numerically for transverse

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wave numbers p (p(p (Martin et al., 1980) but this issue remained an open problem until   recently (Pelinovsky and Sulem, 1999). In a defocusing medium (p "!1), the c.w. solution is modulationally stable in the elliptic  problem (p "#1) and localized waves can exist on a nonvanishing background as dips of lower  intensity, usually called dark solitons, t(x, t)"U (x!2vt!s; v)e\ MR\F ,  where

(2.3)

U (x; v)"k tanh(kx)#iv, k"(o!v .  Here the boundary conditions for the function t(x, t) are speci"ed by the amplitude o of the c.w. background, i.e., "t"Po as xPR, and the parameter v de"nes the soliton velocity, "v"4o. As was "rst shown by Kuznetsov and Turitsyn (1988), a dark soliton (2.3) is unstable to transverse perturbations with the wave numbers, 0("p"(p (v), where  (2.4) p (v)"[!(o#v)#2(v!vo#o ] .  Recent numerical (McDonald et al., 1993; Law and Swartzlander, 1993), analytical (Pelinovsky et al., 1995), and experimental (Tikhonenko et al., 1996a) results revealed that the instability leads to the generation of a train of vortex solitons with alternative polarities, as a possible scenario of the self-focusing process in a defocusing medium. In addition, the self-focusing of dark solitons may also display long-lived intermediate oscillations between a quasi-planar soliton and a train of vortex solitons (Pelinovsky et al., 1995). Although the transverse instability of dark solitons has been considered also for the hyperbolic NLS equation (Rypdal and Rasmussen, 1989), we notice that p "!1 the c.w. background is  modulationally unstable for both signs of p . This implies that the wave background is likely to be  destroyed by growing perturbations and experimental observations of dark solitons and their self-focusing dynamics becomes overshadowed by the background instability. Therefore, we omit the case of dark solitons in the hyperbolic NLS equation. In the small-amplitude asymptotic limit, the transverse self-focusing of a plane dark soliton is described by a rather universal model known as the KP equation (Kadomtsev and Petviashvili, 1970). The analytical approximation resulting in the KP equation corresponds to small-amplitude long-wave modulations of the c.w. background within the asymptotic expansion: (2.5) t"[o!eu(X#2o¹, >, q)#O(e)] exp+!2iot#ieR(X#2o¹, >, q), ,  where X"ex, ¹"et, >"ey, q"et, and e;1. As follows from Eq. (2.1) for p "!1 and  p "#1, the function u"R satis"es the KP equation with the positive dispersion,  6 (4u #12uu #u ) "4u , (2.6) R V VVV V WW where we have used the conventional notations (x, y, t) for the stretched variables (X, >, q) and also put o"1. A plane KdV soliton is described by a steady-state solution of Eq. (2.6): u";(x!vt!s; v) ,

(2.7)

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where v is the soliton velocity, s is the soliton initial coordinate, and the soliton shape is described by a sech function, ;(x; v)"v sech((vx) . The KP equation (2.6) has been studied in many papers devoted to the soliton self-focusing, because it commonly appears in di!erent physical applications, and it is integrable by means of the inverse scattering transform (see, e.g., Ablowitz and Segur, 1981). In particular, we would like to mention a number of analytical (Murakami and Tajiri, 1992; Pelinovsky and Stepanyants, 1993) and numerical (Infeld et al., 1994, 1995) results which showed that, under a periodic transverse perturbation, a plane KdV soliton transforms into a chain of two-dimensional KP solitons. The NLS equation (2.1) and the KP equation (2.6) are two basic models we have selected in this paper to demonstrate di!erent asymptotic approaches in the theory of the symmetry-breaking soliton instabilities. These models are generic, and many of the techniques we discuss here can be further employed for the analysis of more complicated nonlinear systems. 2.2. Suppression of wave collapse: physical mechanisms Physical models discussed above are drastically simpli"ed. Therefore, they do not take into account all possible processes which may occur in realistic physical systems. This is a general feature of any theoretical model, that holds as a good approximation provided it does not display some exotic properties or singular dynamics. However, there exist several physical models where smooth and localized initial conditions develop singularities in "nite time (or at "nite distances). For example, the power-law KdV equation u #uNu #u "0 describes the development of R V VVV a singularity from localized initial conditions for p'4; the NLS equation it #  t#"t"Pt"0 R " has collapsing solutions for Dr52, where D stands for the space dimension. There exist some other examples, including the modi"ed KP equation, the Boussinesq equation, the model for dispersionless three-wave interaction, the continuum limit of the Toda lattice with a transverse degree of freedom, the subcritical Ginzburg}Landau equation, etc. (see, e.g., Zakharov, 1991; Turitsyn, 1993b; BergeH , 1998a; Sulem and Sulem, 1999). In the framework of the elliptic NLS equation, a spatially localized wave can develop a singularity in a "nite time provided its total power exceeds a certain critical (threshold) value (e.g., Rypdal and Rasmussen, 1989). However, from the physical point of view, it is clear that such a catastrophic self-focusing wave collapse cannot proceed inde"nitely. Moreover, the critical collapse that corresponds to the marginal condition Dr"2, should be sensitive to small structural perturbations of the NLS equation. There are known many di!erent physical mechanisms that can suppress or even completely eliminate collapse. Therefore, the models possessing the collapse dynamics are valid for describing only the initial and intermediate regimes of the beam/pulse self-focusing, and they should be modi"ed for a later stage of the instability-induced dynamics. Below, we give a summary of some physical mechanisms which appear mostly in applications to nonlinear optics problems (see also Goldman, 1984; BergeH , 1998a). It is worth mentioning that the modulation theory developed by Fibich and Papanicolaou (1998, 1999) may allow to analyze the e!ects of a rather general class of perturbations on critical self-focusing.

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2.2.1. Nonlinearity saturation One of the well-known ways of stabilizing the blowup instability of optical beams above the self-focusing power threshold comes from the fact that in practice the refractive index of an optical medium saturates at high powers of optical beams. Several di!erent models of the saturable nonlinearity were used to demonstrate the existence of stable self-trapped beams in two and three dimensions, including exponential saturation (e.g., Wilcox and Wilcox, 1975; Kaw et al., 1975; Vidal and Johnston, 1996), two-level type model (e.g., Marburger and Dawes, 1968; Gustafson et al., 1968), cubic}quintic nolinearity with a defocusing contribution of the next-order quintic nonlinear response (e.g., Zakharov et al., 1971; Wright et al., 1995; Josserand and Rica, 1997), the so-called threshold nonlinearity (e.g., Snyder et al., 1991), logarithmic nonlinearity (Bialynicki-Birula and Mycielski, 1979; Snyder and Mitchell, 1997a). A similar mechanism is produced by the so-called cavitation e!ect due to repulsion of the ions from the region of the energy concentration in an ultra-intense laser beam (Komashko et al., 1995). The physical mechanism behind the stabilizing action of the nonlinearity saturation is very simple: When a beam increases its amplitude the e!ective action of nonlinearity decreases therefore preventing the further self-focusing and collapse. In most of the cases, this stabilizing mechanism is associated with the existence of stable multi-dimensional solitary waves. 2.2.2. Nonparaxial and vector focusing In applications to nonlinear optics, the NLS equation appears as a result of several approximations. The most important one is the so-called paraxial approximation which allows to derive the NLS equation for the beam dynamics from the scalar wave (or Helmholtz) equation. Presenting the "eld in the form E(r , z)"t(r , z) e IX, this approximation means that the resulting scalar equation , , for t in an isotropic medium, it # t#"t"t#et "0 , X XX

(2.8)

can be considered in neglection of the last term with the small parameter e which is de"ned by a ratio of the initial beam radius to the di!raction length. Eq. (2.8) is ill-posed as a Cauchy problem and, therefore, its direct numerical integration can meet some problems (see, e.g., Sheppard and Haelterman, 1998). Beginning with Feit and Fleck (1988), it was argued that no singularity forms if beam nonparaxiality is included. This result was further supported by asymptotic analysis of Fibich (1996) who demonstrated for the equation above that the critical self-focusing is indeed arrested for any small e (see also Fibich and Papanicolaou, 1998, 1999). More accurate models of nonparaxial optical self-focusing should include vectorial ewects of the coupling between TE and TM components and backscattering, both the e!ects lead to additional power losses. Vectorial coupling becomes signi"cant when a beam focuses to a peak which has the width comparable with the wavelength. As a result, the weak-guidance approximation, based on the assumption that the term ( E) can be dropped from the Maxwell wave equations, becomes invalid. Vectorial e!ects (see, e.g., Milsted and Contrell, 1996, and references therein) support the existence of stable self-trapped beams with the coupled transverse and longitudinal "elds of the same order (e.g., Eleonskii et al., 1973), and, therefore, the e!ective coupling of TE and TM components can also arrest collapse (Chi and Guo, 1995).

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2.2.3. Dissipation and diwusion Dissipation of the beam energy during its self-focusing is one of the important physical mechanisms that should prevent its collapse. The e!ect of a linear damping &!ilt in the NLS equation has been analyzed numerically (Goldman et al., 1980), by means of the collective coordinates (Rasmussen et al., 1994), and by the moment method (PeH rez-GarciD a et al., 1995). In general, the linear damping increases the length of self-focusing collapse but it does not remove it completely. For a given input power P, there exists a critical damping l (P) such that for l(l (P)   the beam collapses completely even in the presence of the energy dissipation. Another important mechanism of the energy absorption is the thermal self-action of light due to heat diwusion which can be described by the coupling between the NLS equation and the heat di!usion equation (e.g., Litvak et al., 1975; Bertolotti et al., 1997, and references therein). The e!ect of the heat di!usion is more dramatic, and the beam amplitude remains "nite even for the powers larger then the critical power of self-focusing. Similar e!ects are produced by a nonlocal response of the medium (see, e.g., Suter and Blasberg, 1993) and by a dissipative interaction of the focused beam with stimulated Brillouin scattering which results in the self-focusing suppression and modi"cation of the beam structure (Rubenchik et al., 1995). 2.2.4. Nonlocal interaction The e!ective saturation e!ect in the soliton self-focusing is also produced by nonlocality of the nonlinear interaction, which can be described by the nonlocal NLS equation,



it #  t#t <("r!r")"t(r)" dr"0 , R "

(2.9)

where <("x") is the potential of the interaction between the particles. The standard NLS model follows from Eq. (2.9) under the assumption of the delta-like point interaction. As was demonstrated by Turitsyn (1985) for a few important physical types of the potential <("x"), Eq. (2.9) does not possess collapsing solutions, so that collapse can be indeed arrested by higher-order derivative terms produced by the nonlocal interaction. The physical mechanism of the stabilizing action of nonlocality is even more clear in the critical limit of highly nonlocal media when Eq. (2.9) becomes a linear equation for an e!ective quantum oscillator (Snyder and Mitchell, 1997b). 2.2.5. Higher-order dispersion and discreteness The similar stabilizing action of nonlocal nonlinearity is produced solely by a nonlocal linear dispersion (Gaididei et al., 1996). In the simplest case, the problem can be reduced, by keeping only the lowest-order term, to the NLS equation with the fourth-order dispersion, c it #  t#"t"Pt# t"0 , R " 2

(2.10)

with small cO0. The e!ect produced by this higher-order dispersion depends on the sign of c, so that for c'0 it leads to resonant radiation of linear waves and defocusing of a localized wave. For

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c(0, Eq. (2.10) can possess stable localized solutions of the form t(r, t)" (r) e SR, provided




(Dr!2)P(2ru

dP , du

where P""t" d"r is the total beam power (Karpman, 1996), the result not yet veri"ed numerically. Now, it becomes clear that the discreteness of physical models may also lead to the arrest of collapse and/or stable localized states (e.g., Bang et al., 1994; Aceves et al., 1995; Christiansen et al., 1996a,b) because it generates a higher-order dispersion term of a proper sign. Indeed, if we use the perturbation theory to approximate the lattice discreteness by a continuum model taking some higher-order terms in the corresponding Taylor expansion (a small parameter is a ratio of the lattice spacing to the characteristic length of the wave), the resulting forth-order dispersion produces a stabilizing e!ect on the localized waves. In the opposite case of a strong discreteness, the size of the self-focused beam is limited by the lattice spacing, so that no singularity can develop in principle (see, e.g., Laedke et al., 1994). 2.2.6. Normal dispersion As was suggested by Strickland and Corkum (1994), the normal group-velocity dispersion increases the peak power of two-dimensional self-focusing of short light pulses, therefore it should also provide a limiting (saturating) mechanism of the wave collapse. Numerical simulations (e.g., Chernev and Petrov, 1992a,b; Rothenberg, 1992a,b; Luther et al., 1994a,b,c; Ryan and Agrawal, 1995) demonstrated that the normal dispersion tends to ease self-focusing by spreading the pulse along the propagation direction, and therefore increases the threshold for the two-dimensional collapse through a temporal pulse-splitting process (see discussions and references in BergeH and Rasmussen, 1996a,b; BergeH , 1998a). Normal dispersion removes the critical self-similarity of the wave self-focusing as is seen through the analysis of modulation equations (Luther et al., 1994c; Fibich et al., 1995) leading to a non-self-similar one peak splitting and, as is observed in numerical simulations, a fragmentation process "rst de"ned as fractal collapse by Zharova et al. (1986). In general, the virial-type arguments and self-similar analysis (BergeH et al., 1996, 1998) suggest that a pulse does not develop a spatial singularity being splitted into small-size beams (&cells') which are sequentially formed and "nally spread out with a power lower than the critical power of self-focusing. The similar behaviour is basically preserved for a general power-law nonlinearity &"t"Pt, so that it can be shown (BergeH et al., 1996c) that in the critical case D"2/r no collapse can occur and the longitudinal extension of the solution never reaches zero for any D. Recently, an extensive numerical simulations of the di!erent regimes of the pulse propagation and self-focusing in Argon have been carried out by Mlejnek et al. (1998) on the basis of the model of the hyperbolic NLS equation coupled to a rate equation describing plasma generation. By varying the pressure p as a control parameter, they observed numerically a number of di!erent scenarios, from a low pressure regime (p(1 atm) to full blowup arrested by normal dispersion at high pressures (p'100 atm). In all the examples discussed above, we have seen that even small additional terms in the NLS equation may have a large e!ect on the global dynamics of self-focusing. However, in many numerical studies it was shown that there is very little di!erence between self-focusing described by

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the cubic NLS equation and the extended models during the xrst focusing cycle until the collapse is arrested. For that reason, the cubic NLS equation still serves as the fundamental model equation for self-focusing in the prefocal region even though it neglects higher-order e!ects.

3. Criteria for soliton self-focusing Fundamental physical mechanisms leading to the transverse self-focusing of solitary waves in di!erent types of dispersive and di!ractive nonlinear systems are similar to the mechanisms governing self-focusing and modulational instabilities of small-amplitude quasi-harmonic wave packets (see, e.g., Kadomtsev, 1976; Rabinovich and Trubetskov, 1984). The instability occurs when transverse modulations along the front of a planar solitary wave decrease a local value of the soliton energy. There exist two basic analytical methods to study the soliton self-focusing, the geometric optics approach and linear stability analysis. Here we brie#y overview both these approaches (Sections 3.1 and 3.2) and show a link between the methods for the analysis of transverse instabilities of planar solitons and modulational instability of c.w. modes in dispersive nonlinear systems (Section 3.3). 3.1. Geometric optics approach The geometric optics approach in the theory of linear and nonlinear waves is based on the assumption that a transversely modulated plane wave remains locally close to its steady-state pro"le, so that each individual segment of the wave evolves along an individual ray [see, e.g., the book by Anile et al. (1993) for many di!erent applications of this method]. In the dynamics of quasi-plane solitons, this assumption is valid provided the soliton remains stable against longitudinal perturbations preserving its symmetry. The geometric optics method for the soliton transverse self-focusing was "rst developed by Ostrovsky and Shrira (1976) who considered one-parametric solitons propagating in an isotropic medium (see also Shrira, 1980). To present the basic results of this method, we introduce an orthogonal system of coordinates (a, b) and consider the evolution of the local soliton velocity v, the width of a way tube D, and the phase angle h that the ray makes with the x-axis, see Fig. 1(a). Geometrically, these three parameters are connected through the following kinematic relations, 1 Rv Rs 1 RD Rs # "0, ! "0 . Rb v Ra Ra D Rb

(3.1)

This system is closed with the help of the energy conservation law for a ray tube, *=(v)"const ,

(3.2)

where ="=(v) is the energy density. System (3.1),(3.2) is elliptic provided dv/dD'0, and then it predicts a transverse instability of a plane soliton. Since the ray width D is typically inversely proportional to the soliton amplitude, the ray equations de"ne a universal criterion for the self-focusing instability of one-parametric solitons in an isotropic medium: transverse self-focusing of a plane soliton occurs when the soliton velocity decreases with an increase of its amplitude. In this case, the corresponding rays form a converging cylindrical wave, as shown schematically in Fig. 1(b).

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Fig. 1. (a) Coordinate system for the application of the geometric optics approach. (b) Schematic of the soliton self-focusing and the instability mechanism described by the geometric optics.

This simple physical mechanism explains the transverse instability of the dark NLS solitons (Kuznetsov and Turitsyn, 1988), for which the dependence of the velocity v on the amplitude a is de"ned by the relation, v"(o!a [see Eq. (2.3)]. Similarly, this criterion is valid for the KdV solitons if one considers the solitons within the original isotropic problem in a medium with a positive dispersion (Kadomtsev and Petviashvili, 1970). It is straightforward to generalize Eqs. (3.1) and (3.2) of the geometric optics approach to the case of the solitons modulated with respect to their frequency rather than velocity. As a result, the same criterion of the soliton self-focusing is valid in the case when the soliton frequency u decreases with the soliton amplitude a. This is valid for the bright NLS solitons that have the frequency u in an e!ective gap of the linear spectrum band, u"u !ca, where u is the cuto! frequency of the   linear band, and c is positive. Indeed, the bright solitons are known to be unstable with respect to transverse perturbations (Zakharov and Rubenchik, 1973). The geometric optics method is not applicable directly to certain nonlinear models, e.g. those described by Eqs. (2.1) and (2.6), where the dependences u"u(a) and v"v(a) have already been renormalized. In addition, Eqs. (3.1) and (3.2) are limited by the wave propagation in an isotropic medium and by the case of one-parametric solitons. For instance, the hyperbolic NLS equation is beyond the applicability of the ray method because (i) this model is inherently anisotropic, and (ii) the symmetry-breaking instability of bright solitons leads to a self-consistent coupled dynamics of both translational and oscillatory degrees of freedom of the bright NLS soliton. An alternative and mathematically more accurate method is based on the derivation of the Whitham modulation equations for the soliton parameters (Whitham, 1974). The Whitham equations describe both longitudinal and transverse modulations of a quasi-plane nonlinear wave, and they can be derived for both periodic and localized waves (see, e.g., examples of the application of this method in the book by Infeld and Rowlands (1990)). If a resulting system of the Whitham equations is elliptic, the solitary waves are unstable and their evolution leads to the self-focusing dynamics. In a particular case of transverse modulations, the Whitham modulation theory reduces to an averaged Lagrangian method which we discuss below in Section 4.2.

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3.2. Linear eigenvalue problems Another basic approach for analyzing the self-focusing soliton instabilities is based on the analysis of a linear eigenvalue problem that is obtained by linearizing the equations of motion near the exact solitary wave solution. Because a steady-state solution for a solitary wave depends only on a few parameters (soliton coordinates and phases), one can separate variables and reduce the analysis of the corresponding linear problem to the analysis of an eigenvalue spectrum of a certain linear operator. For the bright NLS solitons, linear perturbations are considered in the form, dt"t(x, y, t)!U (x; u) e SR"[u(x)#iw(x)] e SR>CR> NW , @ where p is the transverse wave number, C is an eigenvalue that determines the instability growth rate, U (x; u) is given by Eq. (2.2), and the real functions u(x) and w(x) satisfy the linear eigenvalue @ problem following from Eq. (2.1) at p "#1,  (L K #p p)u"!Cw, (L K #p p)w"Cu , (3.3)     where LK "!R#u!2U, L K "L K !4U .  V
  @ For the dark NLS solitons, the linear eigenvalue problem follows from Eq. (2.1) at p "!1 and  p "#1 after substituting,  dt"t(x, y, t)!U (m; v) e\ MR" (m) e\ MR>CR> NW ,  where m"x!2vt, U (m; v) is given by Eq. (2.3), and (m) satis"es the linear eigenvalue equation,  LK #p "iC , (3.4) where LK "!R#2ivR !2(o!2"U ")#2U (H) . K K   For the KdV solitons, the linear eigenvalue problem follows from the KP equation (2.6) after the substitution, du"u(x, y, t)!;(m, v)"=(m) eCR> NW , where m"x!vt, ;(m; v) is given by Eq. (2.7), and ="=(m) satis"es the problem (L K =) !4p="4C= , KK K

(3.5)

with LK "!R#4v!12; . K Self-focusing instability of solitary waves occurs if the corresponding linear eigenvalue problem possesses at least one localized eigenmode with an eigenvalue C"C(p) having a positive real part for p'0. Additionally, if a soliton is stable against longitudinal perturbations preserving its symmetry, there exist no unstable eigenvalues at p"0, i.e. C(0)"0.

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In most of the cases, no exact solutions can be found for linear eigenvalue problems of this type. However, there exist several analytical methods to approximate the unstable eigenvalue C"C(p) and to derive the instability threshold condition. These methods are based on the asymptotic expansions for either small wave numbers (pP0) or "nite wave numbers near a critical (cut-o!) value (pPp ). We de"ne these asymptotic approximations as long-scale and short-scale expan sions, respectively. The long-scale asymptotic method is based on the Taylor expansions of the instability eigenmode for small values of the wave number p. Such a technique is sometimes called the p-expansion method, and it may often detect the existence of self-focusing soliton instability since the instability typically occurs for large spatial wavelengths (i.e. small p) of transverse modulations. The method is originated from the "rst papers (Kadomtsev and Petviashvili, 1970; Zakharov and Rubenchik, 1973), and it is commonly used nowadays (see, e.g., Infeld and Rowlands, 1990). Using the discrete-spectrum modes of linear eigenvalue problem at p"0 and C"0 usually known in an explicit analytical form (the so-called neutral modes), one can construct the Taylor expansion of the eigenfunctions and eigenvalues of the corresponding linear problem and derive the "rst terms of the asymptotic expansions. These "rst terms de"ne the spatial symmetry of perturbations leading to the transverse instability, and also provide an approximate expression for the instability growth rate C(p) for small p. Below, we present the neutral modes which generate the corresponding long-scale expansions, and the asymptotic approximation for C(p) derived for the basic soliton equations. (i) Bright NLS solitons in the elliptic problem (Zakharov and Rubenchik, 1973; Kuznetsov et al., 1986)






p 4 p#O(p) ; u"0, w"U (x; u), C(p)"4up! 1# @ 3 3

(3.6)

(ii) Bright NLS solitons in the hyperbolic problem (Zakharov and Rubenchik, 1973; Kuznetsov et al., 1986)






4 p 4 !1 p#O(p) ; u"[U (x; u)] , w"0, C(p)" up! @ V 9 3 3

(3.7)

(iii) KdV solitons (Shrira and Pesenson, 1983; Pesenson, 1991) 4 4 pC#O(p) ; ="; (m; v), C(p)" vp! K 3 3(v

(3.8)

(iv) Dark NLS solitons (Kuznetsov and Turitsyn, 1988)

"[U (m; v)] , B K

4 2(o#v) C(p)" (o!v)p! pC#O(p) . 3 3o(o!v

(3.9)

We would like to point out that numerical coe$cients in Eqs. (3.6) and (3.7) are slightly di!erent from those which can be found in the original papers (Zakharov and Rubenchik, 1973; Kuznetsov et al., 1986) because of misprints. The short-scale asymptotic method is based on the Taylor expansion with respect to another small parameter (p !p), where p is a critical (or cuto!) value of p where C vanishes, i.e. C(p )"0. This   

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method uses the expressions for the eigenmodes of the linear eigenvalue problems at p"p and  C"0. The eigenmodes can be often found by a direct analysis of the linearized equations. Below, we present the critical eigenmodes, eigenvalues, as well as the asymptotic approximations for the growth rate C(p) in the limit pPp .  (i) Bright NLS solitons in the elliptic problem (Janssen and Rasmussen, 1983; Kuznetsov et al., 1983) u"sech((ux), w"0, p "(3u ,  24up  (p !p)#O((p !p)) ; C(p)"  (n!6) 

(3.10)

(ii) Bright NLS solitons in the hyperbolic problem (Pelinovsky, 2000) u"0, w"tanh((ux), p "(u ,  16up  (p!p#O(p !p) . C(p)"   3n

(3.11)

We notice that in this case the critical eigenfunction is delocalized. (iii) KdV solitons (Gorshkov and Pelinovsky, 1995a, 1995b) (3 ="sech((vm)!2 sech((vm), p " v,  2 4 C(p)" p (p !p)#O((p !p)) ;  3  

(3.12)

(iv) Dark NLS solitons (Pelinovsky et al., 1995) 3v sinh(km) (p#3k)

"! #i  , 2k cosh(km) 4k cosh(km) p "[!(o#v)#2(v!vo#o] ,  p C(p)"  (p !p)#O((p !p)) .  b(k) 

(3.13)

In the latter case, the dependence b"b(k) was found numerically [see Fig. 2(b) in Pelinovsky et al., 1995] within the domain 1.884b/k43.00, the lower limit is for k"1 (v"0) and the upper limit is for kP0 (vPo). A complete spectrum of unstable eigenvalues can be found only in special cases, e.g. if the linear eigenvalue problem is integrable. For example, this is the case of the transverse instabilities of the KdV solitons because the KP equation (2.6) is known to be integrable by means of the inverse scattering transform (see, e.g., Ablowitz and Segur, 1981). Using the inverse scattering technique, Zakharov (1975) constructed the spectrum of the transverse instability of the KdV solitons (see also Alexander et al., 1997). Indeed, one can verify that the eigenfunction,






2  R p . =" [eGK sech((vm)], i" v! Rm (3

(3.14)

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solves Eq. (3.5) for






2 2  C(p)" p v! p . (3 (3

(3.15)

The expansion of Eq. (3.15) in the limits pP0 (iP(v) and pP(3v/2 (iP0) coincide with the results of the asymptotic analysis given by Eqs. (3.8) and (3.12). 3.3. Analogy with modulational instability Modulational instability of a monochromatic nonlinear wave occurs via the four-wave interaction between the wave of the main frequency and generated wave satellites which form a certain resonant con"guration (see Infeld and Rowlands, 1990, and references therein). A possibility of such a resonant four-wave mixing follows from the structure of the dispersion relation of linear waves, the dependence of the wave frequency u(k) on the wave number k. The linear dispersion relation for the NLS equation (1.1) is u(k)"k, and such a frequency dependence allows resonant four-wave interactions according to the following frequency mixing rule: 2u(2k)"u(k#dk)#u(k!dk) ,

(3.16)

where dk"(3k. Such a resonant wave-mixing mechanism explains the appearence of modulational instability of a fundamental wave with respect to the parametric excitations of two wave satellites. The instability follows from the analysis of the linearized version of the NLS equation (1.1). To show this, we present the linearized solution in the form, u"[o#(e #ie )e NV>CR#(eH#ieH)e\ NV>CHR]e IV\ IR> P>MPR , (3.17) S T S T where e and e are coupled through the relation, by the expression S T e /e "(C#2ikp)/p . (3.18) T S Then, the instability growth rate C"C(p) is de"ned by a solution of a quadratic equation, (C#2ikp)"2r(r#1)oPp!p .

(3.19)

A simple analysis of Eq. (3.19) indicates that the c.w. solution, u"oe P>MPR, is unstable against modulations with the wave numbers p selected within the band 0(p(p , where  p "[2r(r#1)oP] .  Nonlinear dynamics of modulational instability of a c.w. solution can be also analyzed in the limit of long-wave modulations, i.e. when both p and C are small. To do so, we use the well-known presentation of the complex "eld in the #uid-dynamics form (see, e.g., Spiegel, 1980), u"[q(X, ¹)]e F62C ,

(3.20)

where X"ex, ¹"et, and e;1. Then, the NLS equation reduces to a system of two coupled equations for real h and q, q #2(qh ) "0 , 2 66 q q h #h !(r#1)qP!e 66 ! 6 "0 , 2 6 2q 4q






(3.21)

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Coupled equations (3.21) possess a c.w. solution, q"q and h"(r#1)qP t#h , where q and     h are constants. Linear expansion around the c.w. solution produces the characteristic equation  (3.19) with q "o. In the limit eP0, system (3.21) transforms into the gas-dynamics equations  (Trubnikov and Zhdanov, 1987) which describes a dispersionless limit of the NLS equation (Kamchatnov, 1997). Equivalently, the gas dynamics equations coincide with the geometric optics approximation for the wave propagation in nonlinear di!ractive media, where the "rst equation gives the energy balance, while the second equation is a kinematic relation for the wave phase. The modulational instability of a c.w. background can be then studied within the coupled equations (3.21) in the limit of small but "nite values of e, by applying the Whitham modulation theory (see, for a review, Kamchatnov, 1997). However, the applicability of the Whitham theory is limited by integrable systems because it is based on exact periodic solutions. It is a purpose of our survey to show that the modulation equations which describe the development of the soliton self-focusing instability are similar to those of the #uid dynamics form (3.21) of the NLS equation. Moreover, these modulation equations can be investigated systematically by means of the regular asymptotic expansion methods. The latter methods use only a balance between the amplitude and the spatial scale of the soliton modulations and do not rely upon the integrability properties of the modulation equations. Thus, we expect that the methods described here are more general than the Whitham theory, and they can be applied to much broader class of problems related to the symmetry-breaking instabilities of nonlinear waves.

4. Equations for soliton parameters The analytical approaches described above are based on two di!erent approximations imposed on the soliton transverse modulations. The geometric optics approach describes the evolution of strongly nonlinear long-wavelength modulations in the dispersion less limit, whereas the linear eigenvalue problem allows to describe the evolution of perturbations of all scales but within a small amplitude (linear) approximation. Depending on a type of balance between linear and nonlinear e!ects in the soliton dynamics, di!erent analytical methods have been employed in the literature for deriving the modulation equations describing nonlinear regimes of the soliton self-focusing. In this section, we review some of those methods and also discuss their applicability limits. The basic assumptions we use here are the following: (i) a planar soliton is stable against the symmetry-preserving perturbations, and (ii) the instability occurs when the wavelength of the soliton transverse modulations is much larger than the soliton width whereas the amplitude of the soliton modulations may be not small. The modulation equations are derived below by means of direct asymptotic expansions (Section 4.1) or, equivalently, by an averaged Lagrangian method (Section 4.2). These equations generate, in the leading order, ill-posed (elliptic) equations for the dynamics of an equivalent (unstable) gas which displays the development of singularities in "nite time (Section 4.3). The gas-dynamics equations represent the dispersionless limit of the NLS equation written in the hydrodynamical form (3.21) (see Section 4.4). We can improve the applicability of the asymptotic equations by extending the perturbation theory into higher orders to include dispersive and/or dissipative e!ects (see Section 4.5). The bounded scenario of the transverse self-focusing can be then described within the extended models (Section 4.6). For

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integrable evolution equations, the soliton self-focusing can be studied in the framework of exact analytical solutions (Section 4.7). 4.1. Direct asymptotic expansions We consider the NLS equation (2.1) and the KP equation (2.6) in the asymptotic limit of long transverse modulations, i.e. we write the "elds as t"t(x, >, t) and u"u(x, >, t), where >"ey in the stretched coordinate and e;1. Then, those equations transform to the perturbed NLS and KdV equations, respectively, it #t #2p "t"t"ep t , (4.1) R VV   77 (4u #12uu #u ) "4eu . (4.2) R V VVV V 77 The soliton dynamics induced by such e!ective perturbations can be analyzed in the framework of a regular soliton perturbation theory (Kivshar and Malomed, 1989; Pelinovsky and Grimshaw, 1997). This theory implies that a planar soliton evolves adiabatically under the action of long-scale transverse perturbations, i.e. it is locally close to the pro"le of a planar solitary wave solution but with the parameters varying slowly with > and also depending on slow time ¹"et. Then, the soliton perturbation theory allows us to derive, in a systematic way, a system of modulation equations for the soliton parameters. We show below how to derive those equations for several important physical cases. 4.1.1. Bright NLS solitons in the elliptic problem The transverse self-focusing of a bright soliton in the elliptic problem is generated by the phase and frequency modulations [see Eq. (3.6)]. Therefore, we neglect the parameters v and s in Eq. (2.2) and assume the following asymptotic series:






 (4.3) t(x, y, t)" U (x; u)#e (x, t; >, ¹)# eL (x, t; >, ¹) e FC , @  L L where U (x; u) is the soliton pro"le (2.2), the parameters are some functions of slow variables, i.e. @ u"u(>, ¹) and h"h(>, ¹), and (x, t; >, ¹) is a "rst-order correction to the soliton shape.  Substituting Eq. (4.3) into Eq. (4.1) for p "#1 and p "#1 and neglecting the terms for   L n52, we "nd the linear equation for the function "u #iw ,    RU RU @u # @ u , w #LK u "H ,!e\(h #h !u)U #e R    2 7 @ Ru 77 Ru 7 (4.4) RU @ !u #L K "H , (u #2u h )#h U , R   7 7 77 @ Ru 2 where L K and L K are de"ned after Eq. (3.3). It is clearly seen from Eq. (4.4) that the asymptotic   balance occurs when the phase factor (h #h !u) becomes of order of O(e) and the terms of 2 7 order of O(e) should be moved from Eq. (4.4) into the next-order equation for . One of the typical  approximation is however to keep all terms into the same equation (4.4) and proceed with a solution. A solution can be found by expanding u and w through a complete set of the   continuous spectrum eigenfunctions (Kaup, 1990). The discrete and associated eigenfunctions are






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to be removed from the expansion by means of the orthogonality conditions,







 RU @ H dx"0 . (4.5) Ru  \ \ It can be shown that the "rst condition gives a power balance for a transversely modulated bright soliton, where the power is de"ned as N" "t" dx, and the second equation is a condition \ the value of the soliton energy N (u), i.e. that the correction term does not change  @ N"N (u)#O(e). The latter condition can be omitted in the soliton perturbation theory in one @ dimension (Pelinovsky and Grimshaw, 1997) because it just renormalizes the energy conservation equation into the next asymptotic order. However, the second equation (4.5) has its own meaning for the soliton dynamics in two dimensions as the kinematic relations (3.1) within the geometric optics approach. Evaluating integrals in Eqs. (4.5), we derive a system of modulation equations for the parameters of a bright NLS soliton, U H dx"0, @ 

u #2u h #4uh "0 , 2 7 7 77 u u h #h !u#ek 3 7 !4 77 "0 , 2 7 u u where







(4.6)





1 p k" 1# . 12 12 Eqs. (4.6), without the terms of order of O(e) represent the energy conservation law and the eikonal equation of the geometric optics method. The additional terms of the order of O(e) describe the e!ect of dispersion on the soliton self-focusing. 4.1.2. Bright NLS solitons in the hyperbolic problem In the hyperbolic problem, the self-focusing instability of a bright NLS soliton is induced via the coordinate and velocity modulations [see Eq. (3.7)] coupled to the phase and frequency modulations. As a result, a variation of all four parameters of a bright NLS soliton (2.2) should be taken into account in the asymptotic expansion,






 (4.7) t(x, y, t)" (m; u)#e (m, t;>, ¹)# eL (m, t;>, ¹) e TK> FC ,   L L where "(1!4s U (m; u), m"x!2s/e, and all parameters (v, s, u, h) are assumed to be some  7 @ functions of slow variables > and ¹. The amplitude factor (1!4s appears due to a curvature of 7 the soliton front induced by the coordinate modulations. We investigate here only the leading order of the modulation equations. To "nd these equations, we derive from Eq. (4.1) at p "#1  and p "!1 the system of equations for the function "u #iw ,     w #(1!4s )LK u "!(v !2v h #4vv s )m #2s #4s

, R 7   2 7 7 7 7  77 K 7 K7 !u #(1!4s )L K "(2vs #4v s !h ) # #4v s m

R 7  77 7 7 77  2 7 7 K !2h #4vs . (4.8) 7 7 7 7

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Solvability of this system is determined by four orthogonality conditions for the neutral and associated modes of the discrete spectrum. As a result of applying those conditions, we obtain the following system of modulation equations: 4 (1!12s ) 7 us "0 , v !2v h #2u s #4vv s # 2 7 7 7 7 7 7 3 (1!4s ) 77 7 s !v(1!4s )!2s h "0 , 2 7 7 7 (4.9) (1#4s ) 7 u(2vs !h )"0 , u !2u h ##4vu s !8uv s #4 2 7 7 7 7 7 7 77 77 (1!4s ) 7 h !h !(u#v)(1!4s )"0 . 2 7 7 At v"s"0, this system reduces to the modulation equations (4.6) after changing R PiR and 7 7 neglecting the terms of order of O(e). On the other hand, we are not able to decouple the evolution of the parameters (v, s) from that of the parameters (u, h), and therefore the self-focusing dynamics of a bright NLS soliton in the hyperbolic problem involves a coupling between all soliton degrees of freedom. 4.1.3. KdV solitons Transverse self-focusing of a KdV soliton is an e!ect of the coordinate and velocity modulations [see Eq. (3.8)]. Therefore, we are looking for the asymptotic expansion of Eq. (4.2) in the form,  u";(m; v)#eu (m, t; >, ¹)# eLu (m, t; >, ¹) , (4.10)  L L where m"x!s/e, ;(m; v) is the soliton pro"le (2.7), and the soliton parameters depend on slow variables, i.e. v"v(>, ¹) and s"s(>, ¹). The "rst-order correction u to the soliton shape  ; satis"es the linear equation, R; 4u !(L K u ) "H ,4e\(s #s !v); !4(v #2s v ) !4s ; , 77 R K  2 7 K 2 7 7 Rv where L K is given by Eq. (3.5), and the terms of order of O(e) are neglected. Solutions to this equation can be presented through the continuous-spectrum eigenfunctions subject to the orthogonality conditions for discrete and associate eigenmodes,












 R; K (4.11) H dm!4va dm"0 .  Rv \ \  Here a"a(>, ¹) is an integration constant with respect to the variable m. The "rst condition reproduces a balance equation for the momentum of a transversely modulated KdV soliton, i.e. P" u dx, while the second one is equivalent to the condition that the "rst-order correction term u\does not change the value of the soliton momentum P (v), i.e. P"P (v)#O(e). Using    these conditions, we derive a system of modulation equations for the parameters of a KdV soliton, ;H dm"0, 

v #2v s #vs "0 , 2 7 7  77 (4.12) s #s !v!ea"0 . 2 7 System (4.12) is not closed because the parameter a is not speci"ed so far. The parameter a is associated with a radiation "eld generated by a soliton. As a consequence, the term of order of O(e)

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included into Eqs. (4.12) describes dissipative e!ects in the soliton dynamics induced by radiation. The dispersive e!ects, similar to those of Eqs. (4.6), have the order of O(e), and therefore they are negligible in comparison with dissipative e!ects. In the limit eP0, system (4.12) reproduces the momentum conservation equation and the eikonal equation of the geometric optics approach. 4.1.4. Dark NLS solitons The scenario of the self-focusing dynamics of dark solitons is similar to that of the KdV solitons. The corresponding analysis can be developed for Eq. (4.1) for p "!1 and p "#1. We   reproduce here only main steps of the corresponding asymptotic analysis. First, we look for a solution for a perturbed soliton in the form






 (4.13) t(x, y, t)" U (g; v)#e (g, t; >, ¹)# eL (g, t; >, ¹) e\ MR> F , B  L L where U (g; v) is given by Eq. (2.3), g is a streched coordinate that describes a curved soliton front, B x!2s/e g" , (1#4s 7 and the parameter h is associated with the radiation emitted by the dark soliton. The "rst-order correction satis"es the linear inhomogeneous equation,  s 4is s 2 !v U #iU !h U ! 7 72 gU LK "!2ie\  BE B2 2 B (1#4s ) BE (1#4s 7 7 2(1!4s ) 4s 16s s 7 s U ! 7 7 77 gU ! (U #ih U ) , (4.14) # 7 BE (1#4s ) BEE (1#4s ) 77 BE (1#4s B7E 7 7 7 where the operator L K is de"ned above, see Eq. (3.4). As the result, the slowly varying parameters of a dark soliton, v"v(>, ¹) and s"s(>, ¹), satisfy the modulation equations following from Eq. (4.14),






4 (o!v)s 4vv s 77 ! 7 7 "0 , v # 2 3 (1#4s ) (1#4s (4.15) 7 7 s 2 v! "0 . (1#4s 7 In the limit vP!o#v , sP!o¹#s , where "v ", "s ";1 and o"1, these equations coincide   with Eqs. (4.12) for a KdV soliton, provided the terms of order of O(e) are neglected. 4.2. Averaged Lagrangian method An alternative and rather simple method for deriving the modulation equations in the theory of the soliton transverse self-focusing is to apply the average Lagrangian method also known as a variational approach. Such a method is a particular case of the general Whitham modulation theory (Whitham, 1974) when the soliton modulations are assumed to be transverse with respect to a planar soliton. The average Lagrangian method is based on a variational problem equivalent to

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the nonlinear evolution equation, dS"0, where S is the action expressed through the Lagrangian ¸, S"R dt ¸ dx dy. A trial function is usually chosen in the form of a steady-state soliton but  \ with the parameters slowly varying with respect to the transverse coordinates and time. Then, a variation of the soliton parameters within the averaged variational problem reproduces the leading order of the modulation equations for the soliton self-focusing (see Makhankov, 1978; Trubnikov and Zhdanov, 1987). However, we show below that the straightforward application of the averaged Lagrangian method may meet some di$culties associated with the appearance of diverging integrals responsible for the radiation emitted by solitary waves. To describe those e!ects, a proper trial function should include a nonlocalized radiation component. 4.2.1. Bright NLS solitons in the elliptic problem Lagrangian for the NLS equation (2.1) is de"ned as (for p "p "#1),   i ¸" (tHt !ttH)!"t "!"t "#"t" . R R V W 2

(4.16)

As a trial function, we take a bright NLS soliton, t"U (x; u)e FC, with the slowly varying @ parameters u"u(>, ¹) and h"h(>, ¹). Integrating ¸ with respect to x, we "nd the averaged Lagrangian in terms of the soliton parameters u and h,



1 u 1  ¸ dx"!(u(h #h )# u!ek 7 , (4.17) 1¸2" 2 7 3 u 2 \ where k is the same as in Eqs. (4.6). The variation of 1¸2 with respect to h yields the energy conservation law for the system (4.6), whereas the variation with respect to u generates an eikonal equation. The soliton modulation equations were "rst obtained with the help of the averaged Lagrangian method by Makhankov (1978). 4.2.2. KdV solitons Lagrangian for the KP equation (2.6) is de"ned as ¸"w w !w !w#w , (4.18) R V  VV W V where u"w . As a trial function, we consider a KdV soliton, u";(x!s/e; v), where s"s(>, ¹) V and v"v(>, ¹) are slowly varying parameters, so that w"(v tanh[(v(x!s/e)]. Integrating ¸ with respect to x, we "nd the averaged Lagrangian in the form,



1¸2"





4 v v 4 * ¸ dx"! v(s #s )# v#el 7 !e 7 lim dx , 2 7 5 v 4v 3 \ * \*

(4.19)

where






1 n l" 1! . 6 6 The last term in Eq. (4.19) is diverging, and it is of order of O(e). The appearance of such a secular term indicates the necessity to include a nonlocalized correction into a trial function for applying the averaged Lagrangian method. Thus, the self-focusing dynamics of a KdV soliton is

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accompanied by a strong interaction between the soliton and the radiation it induces. This fact was overlooked by Katyshev and Makhankov (1976; see also Makhankov, 1978) where the modulation equations for a KdV soliton were analyzed neglecting the diverging term of the averaged Lagrangian (4.19). As a result, the modulation equations led to wrong predictions, see discussions in Laedke and Spatschek (1979). As a matter of fact, the averaged Lagrangian (4.19) for a KdV soliton is only applicable in neglecting the terms of order of O(e) when this method reproduces the equations of the geometric optics approach. In some other (rather exotic) problems when the self-focusing dynamics of a long-wave soliton is not associated with the generation of a radiation "eld up to the order of O(e), the averaged Lagrangian method is as much e!ective as in the case of a bright soliton of the NLS equation. For example, this is the case of the soliton self-focusing described by the Benjamin}Ono equation (see Pelinovsky and Shrira, 1995). 4.3. Gas dynamics equations A consistent asymptotic analysis requires neglecting the terms of order of O(e) or O(e) in the soliton modulation equations. Such an approximation corresponds to the dispersion-less limit of nonlinear evolution equations (Kamchatnov, 1997) when the results in the leading order of the asymptotic expansions coincides with the gas-dynamics equations. However, the self-focusing phenomenon is equivalent to the dynamics of an unstable gaseous medium (Trubnikov and Zhdanov, 1987) rather than to a standard evolution of a quasi-linear system described by characteristics (Whitham, 1974). In particular, the initial-value problem is elliptic (i.e., ill-posed) so that the characteristic velocities are complex. As a result, the Riemann method cannot be applied in such an unstable case. Instead, an analytical technique based on the so-called hodograph transformation allows us to construct exact analytical solutions to the gas-dynamics equations (Trubnikov and Zhdanov, 1987). These exact solutions describe the formation of singularities in the dispersion-less elliptic initial-value problem. Within the original problem, a singularity of the gas dynamics equations resembles an initial stage of a growth of a self-focusing spike along the front of a planar soliton. Following Trubnikov and Zhdanov (1987), we reproduce here the analytical solutions for the gas-dynamics equations (4.6) and (4.12) in the limit eP0 that describes the self-focusing dynamics of a planar soliton under the action of a periodic transverse modulation. 4.3.1. Bright NLS solitons in the elliptic problem The exact analytical solution to Eqs. (4.6) in the limit eP0 is obtained after the transformation u"r and h "z, where 7 sin g sinh m , z"z(m, g)"! , (4.20) r"r(m, g)" (cosh m!cos g) (cosh m!cos g) and the variables m and g are implicitly related to ¹ and > as follows: m(cos g!cosh m) m sin g C¹" , p>"g# . sinh m sinh m

(4.21)

Here C is the instability growth rate in the dispersionless limit, i.e. C"2p [see Eq. (3.6)]. The parameters (u, h) of a bright soliton reproduce, for large negative time, a planar bright soliton with

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the boundary condition uP1 as ¹P!R (see Trubnikov and Zhdanov, 1987). The analytical solution u"u(>, ¹) expressed through Eq. (4.21) is shown in Fig. 2(a).It has the form of a bubble which becomes tighter and higher as ¹P0. 4.3.2. KdV solitons Exact analytical solution to Eq. (4.12) in the limit eP0 is obtained by means of the transformation v"r and s "(3z, where r"r(m, g) and z"z(m, g) are the same as in Eq. (4.20) while m and 7 g are de"ned this time by the expressions (m coth m!1)(cosh m!cos g) , C¹"! sinh m sin g 3(m coth m!1)(cosh m!cos g) !m , p>"g# sinh m sinh m






(4.22)

where C"(2/(3p) [see Eq. (3.8)]. The analytical solution v"v(>, ¹) is shown in Fig. 2(b). It displays the development of a singularity similar to Fig. 2(a). Thus, the gas-dynamics equations exhibit the singular solutions for the transverse soliton self-focusing. The principal further problem is to predict whether the singularities of the gasdynamics equations still persist within the original evolution problem, or they are removed by taking into account a weak dispersion or dissipation. For example, the development of modulational instability described by the cubic NLS equation in one dimension displays a bounded scenario (`recurrencea) (Yuen and Lake, 1986). Also, the characteristic features of the soliton self-focusing might be di!erent for the nonlinear regime of the soliton dynamics. The

Fig. 2. (a) The analytical solution (4.21) describing the transverse self-focusing of a bright NLS soliton within the gas dynamics equations: 1!¹"!1.6, 2!¹"!1.0, and 3!¹"!0.4. (b) The analytical solution (4.22) describing the transverse self-focusing of a KdV soliton for the same times as above.

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aforementioned problems can be solved only within the modulation equations modi"ed by dispersive or dissipative corrections. 4.4. Reduction to the NLS equation The gas-dynamics equations are similar to the hydrodynamics form (3.21) of the NLS equation. Therefore, some predictions about the long-term dynamics of the soliton transverse instability may be extracted by deriving the e!ective power p of a nonlinear term in the gas-dynamics equations. In some exceptional cases, further remarkable reductions to the NLS equation (1.1) may also take place. 4.4.1. Bright NLS solitons in the elliptic problem It was "rst realized by Degtyarev et al. (1975) that the truncated modulation equations (4.6) are nothing but the hydrodynamics form (3.21) of the quintic NLS equation (1.1) for r"2. Indeed, the corresponding transformation is given by




u(x, t)"




u  ih exp , 3 4(ke

where the new variables are > x" , 4(ke

¹ t" . 4(ke

Such a reduction leads to several important consequences. First of all, a steady-state periodic solution to Eq. (1.1) resembles a steady-state transversely modulated soliton in two dimensions, while the soliton solution to Eq. (1.1) approximates a two-dimensional soliton in Eq. (2.1). Second, it is well-known that the steady-state solutions are weakly unstable for the quintic NLS equation and possess singularities associated with the so-called critical collapse (Rasmussen and Rypdal, 1986). The latter feature enables us to predict that the self-focusing of a planar bright soliton within the elliptic NLS equation leads to the formation of two-dimensional localized modes along the soliton front which then collapse according to the critical NLS equation. A self-similar form of each individual localized mode is described by an asymptotic solution of Eq. (1.1) at r"2 (Fibich and Papanicolaou, 1998; Pelinovsky, 1998). We express this solution in terms of the parameters of a bright NLS soliton,






(X > , 2ke



2 X(¹) d¹ , (4.23)  where the scaling law for the singularity formation modi"ed by the radiation-induced factor is the following: u&3X sech

h&

log"log(¹ !¹)"  X(¹)P as ¹P¹  (¹ !¹)  (see also Fraiman, 1985; Malkin, 1990). It is well-known that the two-dimensional elliptic NLS equation (2.1) for p "p "#1 also possesses the critical collapse dynamics. Thus, the reduced  

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modulation equations (4.6) preserve this principal property of the model (2.1). We exploit this analogy further in Section 7.1.1, where some generalizations of the NLS equation are considered. 4.4.2. KdV solitons Exact reduction of the modulation equations (4.12) to a NLS model cannot be veri"ed because these equations have dissipative rather than dispersive perturbative terms. However, in the leading order, the modulation equations (4.12) coincide with the dispersionless limit (eP0) of the NLS equation (1.1) for r"2/3. The corresponding transformation is given by the relation




u(x, t)"




3  ijs exp v , 5 e

where x"j>/e, t"j¹/e, and j is arbitrary. As is well-known, the NLS equation (1.1) with r"2/3 is subcritical (Rasmussen and Rypdal, 1986), i.e. it does not display any singularity dynamics or any instability of steady-state localized solution. Therefore, we can expect that the self-focusing of a KdV soliton is suppressed by the nonlinear and dissipative e!ects and displays a bounded scenario of the soliton evolution. However, this hypothesis can only be checked within the extended modulation equations. 4.5. Higher-order perturbation theory Here we extend the asymptotic analysis to include higher-order approximations for the cases of the bright NLS and KdV solitons. To do so, we construct a self-consistent solution to the linear equation of the "rst-order perturbation theory. This perturbation term determines distortions of the soliton shape due to the self-focusing dynamics as well as the structure of radiation emitted outside the soliton core. Depending on the type of the radiation emitted, the modulation equations include either dispersive or dissipative terms. We show that a balance between the dispersive and dissipative e!ects depends on properties of a nonlinear system under consideration. 4.5.1. Bright NLS solitons in the elliptic problem For a systematic asymptotic procedure we set u #2u h #4uh "eu , 2 7 7 77 2 (4.24) h #h !u"ehM . 2 7 2 Then, neglecting in Eq. (4.4) the terms of order of O(e), we "nd the "rst-order perturbation term induced due to the soliton adiabatic dynamics, "u (x; >, ¹)#iw (x; >, ¹), where u "0 and     (4.25) w "h xU (x; u) . @   77 Generally speaking, a solution of an initial-value problem associated with Eq. (4.4) is decomposed through the wave packets of the continuous spectrum which may evolve also at the fast time scale t. Here we have supposed that the induced wave packets are self-consistent with the adiabatic self-focusing dynamics of a solitary wave. The "rst-order perturbation (4.25) to the soliton shape (2.2) represents a quadratically growing chirp of a complex phase of t [see Eq. (4.3)]. The phase chirp is responsible for radiation emitted by a bright NLS soliton but such a radiation is

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exponentially small in e (Pelinovsky, 1998). Therefore, this e!ect is negligible for the dynamics of a self-focusing bright soliton. Then, one can proceed to higher orders where the linear equations for

"u (x; >, ¹) and "iw (x; >, ¹) are de"ned as follows:     !L K u "hM U #w #2h w #h w !U !2U w ,   2
2 7 7 77  @77 @  (4.26) RU @ #hM w !u !2h u !h u !w !4U w u !2w , !L K w "!u 2  2 7 7 77  77 @      2 Ru where L K and L K are the same as in Eqs. (3.3). Using the orthogonality conditions (4.5) and Eqs.   (4.24), we extend the modulation equations for the soliton self-focusing to include the second-order e!ects. After a simple algebra, the generalized modulation equations for the soliton parameters can be written in the form, u h u h h 77 # 77 77 "0 , u #2u h #4uh #e k h ! 7 777 #k 2 7 7 77  7777  u 2u 2u (4.27) u (16k#k ) k  u !  h "0 , h #h !u#e 3k 7 ! 2 7 77 4u 77 4u u












where k is the same as in Eqs. (4.6), k "(p/6), and k "(7p/120). The O(e) terms describe   completely the dispersive e!ects of the soliton dynamics while the dissipative e!ects are negligible (exponentially small) in terms of e. The modulation equations (4.27) represent an extended version of the modulation equations (4.6) of the "rst-order asymptotic theory. 4.5.2. KdV solitons We apply the leading-order system (4.12) and "nd the explicit form of the "rst-order perturbation term u "u (m; >, ¹), where   R R; s [m tanh((vm)]#a 1!2 . (4.28) u " 77  6(v Rm Rv






It follows from Eq. (4.28) that the perturbation term u is not localized at in"nity, i.e. u Pu! as   mP$R. This indicates the generation of a radiation tail similar to that which appears in the instability-induced dynamics of a KdV soliton (Pelinovsky and Grimshaw, 1996). Moreover, the second-order perturbation term u also grows at in"nity, i.e. u &u!m as mP$R. Proceeding to   6 the second-order approximation, we "nd the linear equation for u "u (m; >, ¹),   K R; #u #2s u #s u !au ! ; dm#3u u , (4.29) (L K u ) "4 vb#v 2 Rv 2 7 7 77  K 77  K K  where L K is given after Eq. (3.5), b"b(>, ¹) is another integration constant, and v is the extension 2 of v to the next order. Then, applying the orthogonality condition (4.11) to Eq. (4.29), we extend 2 the modulation equations for a KdV soliton to the order of O(e),








v #2v s #vs #e(2vb!as )"0 ,  77 2 7 7  77 (4.30) s #s !v!ea"0 . 2 7 Parameters a and b de"ne the pro"le of the asymptotic series (4.10) extended outside the soliton core, u"eu!(X, >, ¹), where X"ex. The boundary conditions for the radiation "elds u!

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calculated at the soliton position, i.e. at X"s(>, ¹), are given by the matching conditions, s u!" "a$ 77 , 6Q 3(v




1 1 1 1 v 1 s # v ! 7 u!" "b# (a #2s a #s a)G 6 6Q 2 7 7 77 77 77 v 9 6 4v v



(4.31) .

Outside the soliton core, the function u!(X, >, ¹) satis"es the reduced wave equation in two dimensions, u! "u! . (4.32) 26 77 The boundary conditions (4.31) supplemented by system (4.30) determine the radiation "elds u! propagating according to the wave equations (4.32). A proper solution of this (radiation) problem closes the system (4.30) by additional relations between the parameters a and b and the soliton parameters s and v. We solve this problem in a small-amplitude approximation (see Section 5.2.3). In Eqs. (4.30), the terms of order of O(e) describe the radiation-induced dissipative e!ects in the soliton dynamics while the dispersive e!ects [of order of O(e)] are beyond this approximation. 4.6. Diwerent scenarios of soliton self-focusing As has been discussed above, the extended modulation equations include either dispersive or dissipative e!ects in the e!ective elliptic-type asymptotic problem. The solution of those equation in the form of a (unstable) c.w. mode always corresponds to an unperturbed planar soliton. For example, the modulation equations (4.6) and (4.9) have the c.w. solution, u"u and  h"u ¹#h , where u and h are constants; this solution corresponds to a planar bright NLS     soliton. On the other hand, Eqs. (4.12) and (4.15) have the c.w. solution v"v and s"v ¹#s    corresponding to unperturbed KdV and dark solitons. Although the gas-dynamics equations always lead to a formation of singularities for elliptically unstable problems, the higher-order dispersive or dissipative e!ects may suppress an exponential growth of modulations. Depending on the type of higher-order e!ects, we can distinguish, in general, four diwerent types (or scenarios) of the instability-driven soliton self-focusing dynamics: E wave collapse or formation of a chain of two-dimensional localized singularities along the front of a planar soliton; E monotonic transition from a planar soliton to a periodic chain of two-dimensional solitons; E breakup of a planar soliton into localized states which gradually decay due to the action of the wave dispersion; E quasi-recurrence, i.e. a periodic growth and damping of transverse modulations along the front of a planar soliton. Unfortunately, the extended modulation equations that describe all such types of the solitons self-focusing dynamics, can be usually solved only numerically. As an exception, some exact solutions for the soliton self-focusing can be found in an explicit analytical form, if an original nonlinear equation is integrable by the inverse scattering transform method (see Section 4.7). Nevertheless, the extended modulations equations for the soliton parameters can be analyzed

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e!ectively by employing small-amplitude asymptotic expansions summarized below in Sections 5 and 6. Applying those methods, we are able to demonstrate that the bright NLS solitons of the elliptic NLS equation display the "rst scenario of the soliton self-focusing, the KdV solitons decay according to the second scenario, and the bright NLS solitons in the hyperbolic NLS equation follow the third scenario of the soliton self-focusing. Dark solitons, depending on their initial parameters, can display either second or fourth scenarios of the soliton instability-induced decay. 4.7. Exact solutions for the soliton self-focusing Some nonlinear evolution equations describing the soliton self-focusing in two or more dimensions are solvable by means of the inverse scattering transform method (e.g., Ablowitz and Segur, 1981). These so-called integrable models possess a rich functional structure of explicit solutions that allow to describe, in some particular cases, both linear and nonlinear regimes of the soliton self-focusing (see, e.g., Zakharov, 1975; Kuznetsov et al., 1984; Pelinovsky and Stepanyants, 1993; Allen and Rowlands, 1997). Here we present exact solutions for the soliton self-focusing obtained in the framework of the KP equation (2.6). Analytical solution of the KP equation can be expressed in a bilinear form, u(x, y, t)"R log q , V

(4.33)

where the function q"q(x, y, t) possesses a determinant representation (Pelinovsky and Stepanyants, 1993). We discuss here only a scalar reduction of this solution de"ned by the following expression for q:



q"1#

V

\

" " dx ,

(4.34)

where the function " (x, y, t) satis"es the system of linear equations, 2 (3

i # "0, # "0 . W VV R VVV

In a particular case when &exp[(vx#i((3v/2)y!(vt], the solution given by Eqs. (4.33) and (4.34) reproduces a single KdV soliton (2.7). To "nd more general solutions, we select (x, y, t) in the form of two exponentials,











i(3v i(3 y!(vt #(2i)a exp ix# iy!it ,

"(2(v) exp (vx# 2 2

(4.35)

where v, i, and a are real parameters. Then, the corresponding exact solution of the KP equation has the form (Zakharov, 1975) (i(v) q(x, y, t)"1#e(TV\TR#4a eG>(TV\TR>CR cos(py)#aeGV\TR>CR , (i#(v)

(4.36)

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where C and p are de"ned as (3 (v!i), C"i(v!i) . p" 2 In the limit aP0, a Taylor expansion of Eqs. (4.33) and (4.36) leads to the exact solution (3.14) of the linear eigenvalue problem (3.5) obtained for the KP equation. Exact solution (4.36) for "nite values of the parameter a describes a monotonic transition (&splitting') of a planar soliton moving initially with the velocity v into a chain of two-dimensional KP solitons, the so-called lumps, propagating with the velocity v "v#(vi#i, and a complimentary small-amplitude planar  KdV soliton moving with a small velocity v "i.  Numerical simulations (Infeld et al., 1994, 1995) completely con"rmed this result as the basic physical mechanism of a decay of a plane KdV soliton in two dimensions. However, in three dimensions the two-dimensional lump solitons are known to be unstable and, moreover, the instability of three-dimensional azimuthally symmetric solitons were found to lead to gradual collapse (Kuznetsov et al., 1983; Kuznetsov and Musher, 1986). Recent numerical simulations (Senatorski and Infeld, 1998) con"rmed the scenario of the graduate collapse and display steepening, narrowing, and slow disintegration of two-dimensional solitons in higher dimensions.

5. Long-scale approximation Long-scale small-amplitude asymptotic approximation is based on the assumption that the amplitude of the long-scale transverse perturbation remains small compared to the amplitude of a planar (unperturbed) soliton. This approximation simpli"es the extended modulation equations derived in Section 4 and it allows to reduce them to a number of universal asymptotic equations (Section 5.1). The main objective for this reduction is to improve the predictions of the gas dynamics equations by describing a balance between weak nonlinear (e.g. quadratic or cubic) and linear (e.g. dispersive or dissipative) terms of the extended modulation equations (Section 5.2). Then, the asymptotic equations can be employed for constructing analytical solutions describing the development of the transverse instability in the case of periodic modulations of the soliton front (Section 5.3). Extensions of these equations can also be derived for the problems, where some of the coe$cients vanish in the small-amplitude limit (Section 5.4). 5.1. Basic asymptotic equations Two universal asymptotic equations appear within the long-scale small-amplitude approximation depending on a type of a balance between the dispersive and dissipative e!ects. If the dispersive e!ects are dominant over the dissipative ones, the small-amplitude approximation reduces the extended modulation equations to the elliptic Boussinesq equation (see, e.g., Pelinovsky and Shrira, 1995), H #aH #e(bH #c H H #c H H )#O(e)"0 , 22 77 7777  7 72  2 77

(5.1)

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where the coe$cients a and b are determined from a linear approximation and they are positive within the instability band (see Section 3.2), and the coe$cients c and c are responsible for the   e!ect of quadratic nonlinearity. In the opposite case, when the dissipative terms are dominating, the small-amplitude approximation results in the elliptic Shrira}Pesenson equation (see Shrira and Pesenson, 1983), S #aS #e(!bS #c S S #c S S )#O(e)"0 , 22 77 277  7 72  2 77

(5.2)

where the coe$cients a, b, c , and c have the same meaning as in Eq. (5.1). The hyperbolic   analogue of the governing equation (5.2) was "rst derived by Shrira and Pesenson (1983) (see also Pesenson, 1991) for describing the weakly nonlinear dispersive waves propagating along a transversely stable planar soliton. The reduction to a linear problem can be made by substituting H, S&eCR> N7, where the dependence C"C(p) has the form, C"ap!ebp ,

(5.3)

for Eqs. (5.1), and C"ap!ebpC ,

(5.4)

for Eq. (5.2), respectively. For some problems, the coe$cients c and c in Eqs. (5.1) and (5.2) may   vanish. For example, this occurs for bright solitons in the hyperbolic NLS equation and for dark solitons in the limit of zero velocities (the so-called &black solitons'). Then, the asymptotic equations should include higher-order (namely, cubic) nonlinear terms (see Section 5.4). 5.2. Derivation Equations for the long-scale modulations have constant-background solutions corresponding to an unperturbed planar soliton. Then, the purpose of the small-amplitude asymptotic expansion technique is to derive a nontrivial evolution equation governing the evolution of small-amplitude modulations of the background solution. Such evolution equations are valid for certain time intervals when the amplitude of the perturbation mode becomes comparable with the background amplitude. Since the instability leads to a growth of the amplitude, the applicability of the small-amplitude evolution equations is generally limited by a certain time interval. Nevertheless, we show that the results obtained in the small-amplitude approximation give a surprising good agreement with the results that can be obtained by some other methods for analyzing modulational and self-focusing instabilities. 5.2.1. Modulational instability We start with the hydrodynamical form (3.21) of the NLS equation (1.1) and set the asymptotic scaling, h"(r#1)qP ¹#eH(X, ¹) . 

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Then, the kinematic equation of system (3.21) de"nes the asymptotic expansion for q(X, ¹), e q"q # H  r(r#1)qP\ 2  r!1 1 1 H ! H ! H #O(e) . (5.5) # e 6 2 2r(r#1)qP\ 2r(r#1)qP\ 266 r(r#1)qP\    This expansion describes small-amplitude long-scale perturbations propagating along the constant background q"q . It follows from the "rst equation of system (3.21) that the function  H"H(X, ¹) satis"es the elliptic Boussinesq equation (5.1) (written for the variables X and ¹) with the coe$cients





a"2r(r#1)qP , b"1, c "4, c "2r . (5.6)    Linear dispersion relation (5.3) with these coe$cients corresponds to the linear relation (3.19) for k"0 and q "o.  5.2.2. Bright NLS solitons in the elliptic problem We impose the asymptotic expansion for the extended modulation equations (4.27), h"u ¹#eH(>, ¹) , 

(5.7) (16k#k )  H #O(e) . u"u #eH #e H ! 7 277  2 4u  This expansion describes small-amplitude long-scale disturbances at a planar soliton background with the propagation constant u"u . It follows from Eqs. (4.27) and (5.7) that the function  H(>, ¹) satis"es asymptotically the elliptic Boussinesq equation (5.1) with the coe$cients











p 4 , c "4, a"4u , b" 1#   3 3

c "4 . 

(5.8)

The linear part (5.3) reproduces the dispersion relation (3.6). The instability region is limited by a critical wave number p"p "(3u . Expansion (3.6) however gives an approximation for   p beyond the applicability of this expansion, i.e. ep "3(u /(3#p+0.836(u . Neverthe    less, the existence of the instability band cut o! is described by the dispersive term of the Boussinesq equation (5.1), and therefore it provides an improved version of the elliptic gas dynamics equations, where the instability band is not bounded. We mention that the same small-amplitude approximation can be applied to the truncated version of the modulation equations (4.6) and it also results in the elliptic Boussinesq equation (5.1) but with the numerical constant b replaced by 16k. 5.2.3. KdV solitons We start with the extended modulation equations (4.30) for a KdV soliton and the associated radiation problem (4.31) and (4.32) and impose the asymptotic expansion, s"v ¹#eS(>, ¹),  a"eA(>, ¹), b"eB(>, ¹), u!"e;!(X, >, ¹), and v"v #eS #e(S !A)#O(e). For  2 7

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small-amplitude expansions, system (4.30) reduces to a single equation,






4v 4 S #  S #e 4S S # S S !A #2v B #O(e) . 22 7 72 3 2 77 2  3 77

(5.9)

In order to connect the parameters A and B with the parameter S, we solve the problem for the radiation "eld. In the reference frame propagating with the soliton velocity v"v wave equation  (4.32) is rewritten as (5.10) ;! "  "v ;! "  #;! "  .  66 6T 2 77 6T 2 26 6T 2 Now we connect the >-derivatives with the ¹-derivatives using the leading order of modulation equation (5.9), i.e.



3 ;! "  "! . ;! 77 6T 2 4v 22 6T 2  Then, two characteristic velocities for the radiative "elds can be found from Eq. (5.10), i.e. ;!";!(X!v ¹!j ¹), where j "2v and j "!(2v /3). Thus, we come to the con ! >  \  clusion that the radiation "eld ;> in front of a KdV soliton propagates with the velocity equal to a triple soliton velocity in a laboratory reference frame, while the radiation "eld ;\ behind the soliton moves with the third of the soliton velocity. This explicit solution of the wave emission problem results in the di!erential relations for the boundary values (4.31), (5.11) ;!"  "!j ;!"  . ! 6 6T 2 2 6T 2 Substituting Eq. (4.31) taken into the small-amplitude approximation into Eqs. (5.11), we de"ne the parameters A and B in terms of S, 1 1 A" S , B"! S . (5.12) 3(v 77 2(v 772   It follows from Eqs. (4.31) and (5.12) that the radiation "eld behind the KdV soliton is not excited along the characteristics j , i.e. ;\,0, while the radiation "eld in front of the KdV soliton is \ generated, and it can be determined from the boundary condition, 2 S . (5.13) ;>"  " 6T 2 3(v 77  We notice that the radiation induced due to the long-scale soliton self-focusing di!ers from a typical adiabatic evolution of a KdV soliton, when the radiation is emitted behind the KdV soliton (Pelinovsky and Grimshaw, 1996). The relations (5.12) enable us to close the asymptotic equation (5.9) for the function S(>, ¹) and reduce it to the elliptic Shrira}Pesenson equation (5.2) with the coe$cients 4v 4 4 a"  , b" , c "4, c " . (5.14)   3 3 3(v  The linear part (5.4) reproduces the result of the linear stability analysis (3.8). If we use the approximation C+2(v p/(3 to simplify the last term in Eq. (3.8), then this expansion coincides 

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with the exact result (3.15). The instability band is limited by the transverse wave numbers, p(p ,  where ep "(3v /2.   5.2.4. Dark NLS solitons In small-amplitude limit, a dark NLS soliton transforms into a KdV soliton. As a result, the elliptic Shrira}Pesenson equation (5.2) can be derived for a dark soliton of a "nite amplitude. In order to derive this equation, we use the asymptotic representation (4.13) together with the small-amplitude expansion, s"v ¹#eS(>, ¹), h"eH(>, ¹), and v"v #eS !2ev S #   2  7 O(e). The "rst-order correction (l; >, ¹) can be found from Eq. (4.14) within the small amplitude approximation,











RU RU H km km B ! 2 tanh(km)# !2S

"Q imU ! B #v B  Ro 7 cosh(km)  cosh(km) Rv 4k






2 RU 1 m k!v  i tanh(km) , # S ! v m B ! v # 77 3  Rv 3  cosh(km) 3kv  where m"x!2v t!2S(>, ¹), k"(o!v ), and Q"Q(>, ¹) is a parameter associated with   radiation. The analysis shows (see Pelinovsky et al., 1995a, for details) that S"S(>, ¹) satis"es the extended evolution equation,





8v 1 4 S # kS #e !8v S S !  S S #v H ! Q #O(e) , 77  7 72  77 2 2 22 3 3 2 77

(5.15)

where H and Q are related to S through a solution of the radiation problem. We notice that the "rst linear and quadratic terms of Eq. (5.15) can also be obtained from Eq. (4.15) within the smallamplitude approximation. The radiation "eld, tPt!(X, >, ¹) as mP$R and X"ex&O(1), can be found in the asymptotic form (2.5), where the boundary conditions for u! and R! are de"ned at the soliton X"2v ¹, 




1 2(k!v) 1 u!" S "! v Q! H $ 6T 2  2 77 2 3k o 2v "QG S . R!" 6 6T 2 3k 77



, (5.16)

Outside the soliton, the radiation "eld u!"u!(X, >, ¹), where X"ex, satis"es the scalar wave equations, u! !4o(u! #u! )"0 . 22 66 77

(5.17)

Using the analysis similar to the case of KdV solitons, we "nd from Eq. (5.17) that the radiation "elds are generated along the characteristic directions u!"u!(X!2v ¹, >), where ! $2o(o!v )#3ov   , v " ! 4o!v 

(5.18)

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Together with Eqs. (5.16), this relation allows us to "nd the parameters H and Q as follows: v 2(o#k) H"!  S , Q" S . 77 ok 2 3ok

(5.19)

As a result, we derive from Eq. (5.15) the elliptic Shrira}Pesenson equation (5.2) with the coe$cients, 2(o#v ) 8 4  , c "!8v , c "! v . (5.20) a" (o!v ), b"     3  3 3o(o!v  The linear part (5.4) corresponds to the relation (3.9). In addition, we "nd from Eqs. (5.16) and (5.19) the boundary conditions for the radiation "elds, 2[!ov $(2v !o)]   S . (5.21) " u!" 77 6T 2 3o(o!v  Thus, in contrast to the evolution of an unstable KdV soliton, the radiation "elds are excited in both directions. In the limit v P!o#v and SPSI , results coincide with those for a KdV     soliton at o"1, so that the "eld u\ vanishes. 5.3. Analysis Asymptotic reduction either to the elliptic Boussinesq equation (5.1) or to the Shrira}Pesenson equation (5.2) helps us to construct explicit asymptotic solutions for describing the transverse soliton self-focusing. We mention that the Boussinesq equation (5.1) can be transformed, within the same asymptotic approximation, to the integrable Boussinesq equation, where the exact solutions can be found by regular methods (see Pelinovsky and Shrira, 1995, and references therein). However, there exists a universal method that allows to construct these asymptotic solutions. This method does not use the properties of integrability but, instead, reduce the elliptic equations to two partial (`one-wavea) equations by means of the Laplace (complex) coordinates. We apply this method to both Eqs. (5.1) and (5.2) as well as to their extentions. 5.3.1. The Boussinesq equation The Laplace coordinates are de"ned by the relations, z"¹#ia\>, z "¹!ia\> .

(5.22)

Then, the function H"H(>, ¹) can be expanded into an asymptotic series, H"B(z, q)#BM (z , q)#eH (z, z , q)#O(e) , (5.23)  where q"e¹ and BM is a complex conjugate to B. The straightforward asymptotic analysis reduces Eq. (5.1) to the KdV equation for the function B(z, q), b (c #c )  B B # B "0 . B !  X XX 2a XXXX XO 2a

(5.24)

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The complex conjugated function BM satis"es the same equation (5.24), while the correction term H can be explicitly found as the following:  (c !c )  (BBM #B BM ) . H "  (5.25)  X X 4a The KdV equation (5.24) has a family of soliton solutions (Ablowitz and Segur, 1981). In particular, a one-soliton solution for B(z, q) reproduces, through Eq. (5.23), an asymptotic solution describing the transverse soliton self-focusing under the action of a periodic perturbation, 12bp sinh(C¹) H"! , (5.26) (a(c #c ) [cosh(C¹)!cos(p>)]   where C is the linear instability growth rate (5.3). In application to the bright NLS solitons in the elliptic problem, the asymptotic solution (5.26) describes the appearance of a periodic chain of large-amplitude (singular) localized modes along the planar soliton front. The corresponding analytical solution u"u(>, ¹) is expressed through Eqs. (5.7) and (5.26) and it is shown in Fig. 3(a).The variable t(x, y, t) [see Eq. (4.3)] for the elliptic NLS equation is displayed in Fig. 3(b). Thus, the asymptotic solution is singular as ¹P0 at the center of a self-focusing region. This singular structure di!ers quantatively from that predicted by the elliptic gas dynamics equations [see Eq. (4.21) or Fig. 2(a)] because the latter equations are limited by a shorter time scale. The formation of collapsing two-dimensional modes along the front of a planar bright soliton was numerically con"rmed for the elliptic NLS equation (Degtyarev et al., 1975; Litvak et al., 1991). 5.3.2. The Shrira}Pesenson equation We introduce the same Laplace coordinates z and z as in Eq. (5.22) and expand S"S(>, ¹) into the asymptotic series, S"B(z, q)#BM (z , q)#eS (z, z , q)#O(e) , (5.27)  where q"e¹ and BM is a complex conjugate to B. The straightforward asymptotic analysis reduces Eq. (5.2) to the Burgers equation for the function B(z, q), b (c #c )  B B # B "0 . B !  X XX 2a XXX XO 2a

(5.28)

The complex conjugate function BM satis"es the same equation (5.28), while the correction term S can be found through B as follows:  (c !c )  (BBM #B BM ) . S "  (5.29)  X X 4a We mention that the reduction of the KP equation (2.6) to the Burgers equation (5.28) was "rst presented by Shrira and Pesenson (1983) for describing the propagation of "nite-amplitude modulations along the transversely stable planar KdV soliton in a medium with a negative dispersion (see also Pesenson, 1991). The corresponding KP equation di!ers from Eq. (2.6) by the negative sign in front of the y-derivative term. In the latter problem, the small-amplitude asymptotic approach allows to describe a shock wave propagating along the stable soliton. This

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Fig. 3. (a) The analytical solution (5.26) describing the transverse self-focusing of a bright NLS soliton in the elliptic problem within the long-scale analysis: e"0.2 and 1!¹"!0.9, 2!¹"!0.6, and 3!¹"!0.3. (b) The corresponding pro"le of "t(x, y)" at ¹"!0.2.

phenomenon is typical for nonlinear dynamics of stable solitons in the negative-dispersion medium (Zakharov, 1986; Anders, 1995) and it leads to the formation of resonant soliton triads (see also Infeld and Rowlands, 1990, and references therein). In the problems of the transverse soliton self-focusing, the Burgers equation is written in terms of the complex Laplace coordinates z. However, one can still proceed with constructing an asymptotic solution for the soliton self-focusing under the action of a periodic perturbation. To do this, we "nd a one-soliton solution of Eq. (5.28) for B(z, q) which reproduces, with the help of Eq. (5.27), an asymptotic solution of Eq. (5.2), 2b +C¹#log[cosh(C¹)!cos(p>)], , (5.30) S"! (c #c )   where C is the instability growth rate (5.4). In application to the KdV solitons, the asymptotic solution (5.30) describes a chain of large-amplitude (nonsingular) localized mode growing along the front of a planar soliton. The asymptotic solution for v"v(>, ¹) follows from Eqs. (5.27) and (5.30) and it is shown in Fig. 4(a). In addition, we "nd from Eq. (5.13) the radiation "eld ;>";>(X!3v , >) generated in front of the self-focusing soliton,  1!cos(p>) cosh[k(X!3v ¹)]  (5.31) ;>"k [cosh(k(X!3v ¹))!cos(p>)]  where k"p/(3. The corresponding function u [see Eq. (4.10)] satisfying the KP equation (2.6) is presented in Fig. 4(b).

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Fig. 4. The analytical solution (5.30) describing the transverse self-focusing of a KdV soliton within the long-scale analysis: (a) e"0.2 and 1!¹"!0.5, 2!¹"!0.125, 3!¹"0.125, and 4!¹"0.5. (b) The corresponding pro"le of u(x, y) at ¹"0.75.

It is remarkable that the spatial structure defened by Eq. (5.31) is identical to a chain of two-dimensional solitons in the asymptotic limit of a long period of transverse modulation. Still a singularity is presented at the centers of the self-focusing, this feature is intrinstical for the long-wave small-amplitude expansions. In spite of the presence of singularities, the asymptotic solution shown in Fig. 4(b) clearly corresponds to a decay of a planar soliton with velocity v"v  into a chain of two-dimensional solitons propagating with the velocity v "3v and a residuent   planar soliton moving with the velocity v "v !eC. This asymptotic solution agrees, in the limit   pP0 (iP(v), with exact solution (4.36). Thus, in the long-scale asymptotic approximation, we have improved drastically the predictions of the gas dynamics equations [see Eq. (4.22) and Fig. 2(b)]. In application to the dark NLS solitons, the asymptotic solution (5.30) describes the self-focusing instability of a "nite-amplitude dark soliton. If the dark soliton has the velocity v within the  interval !o(v (!o, then the radiation "elds u! describe chains of two-dimensional dark   solitons. The corresponding solution for the function t"t(x, y, t) [see Eq. (4.13)] is shown in Fig. 5(a). The scenario of the splitting of a dark soliton into a residuent dark soliton with generation of two chains of two-dimensional dark solitons agree with numerical simulations of the NLS equation (Pelinovsky et al., 1995). For v "!o the radiation "eld u> vanishes because the generation of this "eld implies v "0 >   [see Eq. (5.18)]. For !o(v (0 the "eld u> acquires an `oppositea polarity to the "eld u\ [see   Eq. (5.21)]. This corresponds to a n-phase shift between two-dimensional dark solitons generated in the opposite directions of the radiation "elds. When a dark soliton transforms into a stationary black soliton, i.e. v tends to zero, the coe$cients c and c for the quadratic nonlinear terms in    Eq. (5.2) vanish. In this case, the self-focusing dynamics becomes symmetric in space [see Figs. 5(b) and (c)], and it is governed by an e!ective equation with cubic nonlinear terms (see Section 5.4).

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Fig. 5. (a) The level lines of "t(x, y)" for the transverse self-focusing of a dark NLS soliton within the long-scale analysis: e"0.1, v "!0.75 and ¹"1.5. (b)}(c) The level lines of "t(x, y)" for the transverse self-focusing of a black NLS soliton  within the long-scale analysis: e"0.1, v "0., and (b) ¹"!0.2 and (c) ¹"0.2. 

5.3.3. Necessary criteria for collapse Here we discuss the necessary condition for collapse, that can be obtained in the framework of the extended modulation equations. It follows from Eqs. (5.23) and (5.25) for the particular solution (5.26) that the correction H has the same sign of in"nite singularity as the leading term  H "B#BM provided the following condition helds:  dc"(c !c )/(c #c )(0 . (5.32)     In the opposite case, i.e. when dc'0, the correction H has the opposite sign of a singularity and,  being taking together with the main term, it can prevent the singularity to be developed in the latter

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case. Other words, summation of the (singular) asymptotic series (5.23) may result in a regular solution, while no regular solution is expected for dc(0. We conclude therefore that, in the extended modulation equations, collapse takes place if the small-amplitude asymptotic expansion satisfy dc(0. Similar conclusion follows also for the Shrira}Pesenson equation by analyzing Eqs. (5.27) and (5.29). In application to the modulational instability of the NLS equation (1.1), we "nd from Eq. (5.32) that the collapse is supported by the power nonlinearity with r'2. In the opposite case, i.e. when r(2, the bounded scenarios of the soliton self-focusing are likely to happen. This conclusion agrees with the conventional classi"cation of the NLS equations into supercritical and subcritical cases (Rasmussen and Rypdal, 1986). For instance, the modulational instability in the integrable NLS equation (r"1) results in long-lived periodic oscillations discussed in Section 4.6 (Yajima, 1983; Ma, 1984). In application to the KdV and dark NLS solitons, we "nd from Eqs. (5.14) and (5.20) that the condition dc'0 is satis"ed. As a result, the instability-induced dynamics of those solitons displays a bounded scenario of the soliton self-focusing, i.e. a monotonic transition to a chain of twodimensional solitons. Finally, the bright NLS solitons in the elliptic problem satisfy the condition dc"0, i.e. c "c (see   Eq. (5.8)). This is a marginal case between unbounded and bounded soliton dynamics, and it still possesses critical collapse (Rasmussen and Rypdal, 1986). 5.4. Extensions In a number of nonlinear problems, the small-amplitude evolution equations (5.1) and (5.2) turn out to be inconsistent because either linear or nonlinear coe$cients vanish. In these special cases, (5.1) and (5.2) should include higher-order corrections. Here we discuss two such cases which include dark solitons of near-zero velocity and bright solitons in the hyperbolic NLS equation. 5.4.1. &Almost Black' NLS solitons When v approaches zero, the coe$cients c and c vanish and one should reconsider    the small-amplitude asymptotic expansion to include the higher-order (cubic) nonlinear terms into the asymptotic balance with the linear (dissipative) term in Eq. (5.2). This can be done by a simple scaling s"(eS(>, ¹) and the extention of v following from Eqs. (4.15), v"eS !2eS S #O(e). Then, it can be shown that the function S"S(>, ¹) satis"es the 2 2 7 modixed elliptic Shrira}Pesenson equation,






2 4 16o 4o S #e ! S !8S S S ! S S ! S S #O(e)"0 . S # 2 7 27 3 2 77 22 3 277 3 7 77 3 77

(5.33)

A standard reduction based on the Laplace coordinates, z and z given by Eq. (5.22) and the asymptotic expansion S"B(z, q)#BM (z , q)#eS (z, z , q)#O(e), where q"e¹, leads to the modi xed Burgers equation for the function B, b (d !(d #d )a)   BB # B "0 . B #  X XX 2a XXX XO 2a

(5.34)

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Then, the asymptotic solution can be obtained in the form,




S"



3b(a  log "F" , [d !(d #d )a]p   

(5.35)

where F"eC2> N7#(1#eC2> N7) , and C is the linear instability growth rate (5.4). This solution describes the transverse instability of a black soliton under the action of a periodic perturbation. The soliton instability develops symmetrically in space, and it results in a break-up of the planar soliton at the places of location of two-dimensional vortices, the process is accompanied by small radiation propagating with the velocities v!"$o [see Figs. 5(b) and (c)]. This scenario agrees with the results of numerical  simulations of the vortex generation in the defocusing NLS equation (McDonald et al., 1993; Law and Swartzlander, 1993; Pelinovsky et al., 1995). 5.4.2. Bright NLS solitons in the hyperbolic problem The system of modulation equations (4.9) derived for the soliton self-focusing in the framework of the hyperbolic NLS equation can be reduced, in the small-amplitude approximation, to a system of coupled equations. The scaling s"eS(>, ¹) and h"u ¹#eH(>, ¹) results in the asymptotic  expansions for v and u, v"eS #e(4S S !2S H )#O(e) , 2 2 7 7 7 u"u #e(H #4u S !S )#O(e) .  2  7 2 Then, the coupled system for S"S(>, ¹) and H"H(>, ¹) follows from Eqs. (4.9),






4 4 p 16 4 S # u S #e !1 S # u S S ! S S 22 3  77 7777  7 77 9 3 3 3 2 77 4 # 8S S S !4S H # S H #O(e)"0 , 2 7 27 72 7 3 77 2



(5.36)

H !4u H #8u S S !2S S #O(e)"0 . 22  77  2 77 2 22 Here we have included the linear dispersion term from Eq. (3.7), which can be obtained by means of a direct asymptotic analysis. The important property of the coupled equations (5.37) is that the reduction to the Laplace coordinates fails since the nonlinear (cubic) term vanishes. Moreover, we can show that any power nonlinearity cannot stabilize the growth of linear perturbations because the nonlinearity vanishes. Indeed, the system of modulation equations (4.9) possess an explicit (complex-valued) solution, i(u  ((1!u!1) , S "u(>!c ¹), H "! 7  7 (3 u u 2i(u   , u" , v" (1!4u) (3 (1!4u

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Fig. 6. The level lines of "t(x, y)" for the transverse self-focusing of a bright NLS soliton in the hyperbolic problem within the long-scale analysis.

where c and u are constants and u"u(>). The complex-valued parameter c depends usually on   u and this provides an e!ective nonlinearity in the evolution equations of a `single-wavea approximation. Here c"c "const and the nonlinearity vanishes identically for any power. As  a result, the development of self-focusing instability of a planar soliton cannot be stabilized in the hyperbolic NLS equation and this leads eventually to a break-up of the soliton. The corresponding solution is shown in Fig. 6. This scenario corresponds to numerical simulations reported earlier by Pereira et al. (1978; see Fig. 4). Since two-dimensional solitons do not exist in the hyperbolic NLS equation, each individual part of a planar soliton spreads out and gradually decays due to dispersion.

6. Short-scale approximation Asymptotic equations obtained in the long-scale small-amplitude approximation discussed in Section 5 still possess singularities within the validity of the asymptotic expansion technique. This occurs even in the problems where the exponential growth of the soliton amplitude due to the transverse self-focusing is bounded by nonlinear e!ects leading to the formation of a chain of non-singular two-dimensional solitons. The reason for such singularities of the asymptotic expansions can be explained by the fact that, within the long-scale asymptotic approximation, twodimensional solitons formed due to the development of the transverse instability have large amplitudes compared to the amplitude of the initially unstable planar soliton. Therefore, although the long-scale expansion method is usually very simple for the asymptotic analysis, the applicability of the asymptotic results is limited by the temporal and spatial domains where such asymptotic singularities appear. The short-scale approximation described in this section is based on the asymptotic analysis near the cuto! of the instability band. Under the action of short-scale transverse modulations, twodimensional solitons formed in result of the development of instability have the amplitudes compared with the amplitude of an initially unstable planar soliton. If the soliton self-focusing is bounded by nonlinear e!ects (no collapse), the asymptotic technique is free of singularities and it allows to describe, in a self-consistent manner, the nonlinear regime of the soliton self-focusing. A disadvantage of this approach is that, in most of the cases, the cuto! wave number of the instability band and the corresponding linear eigenmode cannot be found analytically, so that the

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whole asymptotic scheme may be developed only formally with the subsequent numerical calculations of the parameters and eigenfunctions. Here we discuss the basic asymptotic equation of the short-scale approximation (Section 6.1) and also present a derivation of this equation for the important examples (Section 6.2). 6.1. Basic asymptotic equations From the physical point of view, the long-scale soliton transverse instability corresponds to either phase or coordinate modulations which destabilize the propagation of a planar soliton. In contrast, the short-scale instability is associated with the instability of the amplitude modulations along the front of a planar soliton. Such an origin of the instability is supported by the existence and bifurcations of the transversely periodic solitary-wave structures that occur near the cuto! wave number (Laedke et al., 1986). Therefore, although the existence of the cuto! of the instability band can be predicted within the long-scale asymptotic equations (5.1) and (5.2), the rigorous asymptotic analysis should be developed on the basis of direct asymptotic expansions of nonlinear evolution equations near p"p .  We assume that a perturbation applied to a planar soliton is nearly periodic along the transverse coordinate, i.e. it can be written in the form, du(x, y, t)&[a(>, ¹)e N W#aH(>, ¹)e\ N W] ;(x) ,

(6.1)

where p is the critical transverse wave number for the instability cuto!, ;(x) is the corresponding  eigenmode of the linear eigenvalue problem. A slowly varying complex amplitude a"a(>, ¹) depends on a slow time ¹"et and the strechted transverse coordinate >"ey [notice that a di!erent streched coordinate, >"ey, has been used in the expansions of Sections 4 and 5]. Perturbation du in the form (6.1) describes, in the leading order, the amplitude modulations along the front of a planar soliton. The main target of the asymptotic analysis is to derive a governing equation for the amplitude a(>, ¹). As a matter of fact, the universal equation that appears in the short-scale small-amplitude asymptotic expansions is the unstable NLS equation (Wadati et al., 1991), !ip a #ba #c"a"a"0 , (6.2)  7 22 where the coe$cient b is determined from the linear analysis, and the coe$cient c describes the nonlinear e!ects. According to the linear analysis, the instability appears for longer transverse perturbations, i.e. for p(p , this condition speci"es b to be positive. Indeed, for linear perturba tions of the form a&eC2> DN 7, we obtain C"!b\p D , (6.3)  N where C is the instability growth rate (C'0) while D is the deviation of the transverse wave N number p from p , i.e. p"p #eD so that D (0 in the instability domain. Within the unstable   N N NLS equation (6.2), the instability domain is not bounded, and therefore the initial-value problem is ill-posed. However, this model still can provide accurate results for the transverse self-focusing under the action of periodic or multi-periodic perturbations (see Janssen and Rasmussen, 1983; Gorshkov and Pelinovsky, 1995a).

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We should also mention that an extension of the underlying model (6.2) to include higher-order terms of the asymptotic expansions may bound the instability band making the initial-value problem to be well-posed, similar to the case of the extended long-scale modulation equations described below in Sections 4 and 5. However, for the self-consistent predictions of the instabilityinduced soliton dynamics, it is usually su$cient to analyze only the unstable NLS equation (6.2) or its analogues. The coe$cient c in Eq. (6.2) determines whether the nonlinear dynamics of the transverse instability remains bounded or not. If c(0, Eq. (6.2) describes an unbounded scenario of the nonlinear dynamics, i.e. collapse may occur as a result of the self-focusing process. If c'0, long-lived bounded oscillations (&quasi-recurrence') is the most typical scenario for, at least, intermediate regimes of the nonlinear dynamics. At last, the case c"0 is special, and the unstable NLS equation (6.2) should be modi"ed by higher-order nonlinear terms. For example, this situation occurs in integrable models where the transverse self-focusing displays a monotonic (resonant) transition from a planar soliton to a transversely modulated soliton-like structure. Radiative e!ects, which are usually beyond the leading-order approximation described by the unstable NLS equation (6.2), should be also taken into account for those special (resonant) cases. 6.2. Derivation Since the asymptotic analysis is straightforward, we present here only the main steps of the derivation and the "nal results obtained by the short-scale expansion technique referring to the original papers where more details can be found. We consider the basic soliton equations and construct the asymptotic solutions for di!erent scenarios of the instability-induced evolution of a planar soliton under the action of periodic short-scale transverse modulations. 6.2.1. Bright NLS solitons in the elliptic problem We start from the following asymptotic expansion for the NLS equation (2.1) for p "p "#1   [cf. Eqs. (4.3) and (5.7)],





 (6.4) t" U (x; u)#e (x, y; >, ¹)# eL (x, y; >, ¹) e R>CF , @  L L where u"1#eh #O(e), U (x; u) is given by Eq. (2.2), and the "rst-order perturbation 2 @

(x, y;>, ¹) is selected according to the structure of the linear eigenmode (3.10), 

"[a(>, ¹)e N W#aH(>, ¹)e\ N W] sech x , 

(6.5)

where the cuto! wave number is p "(3. Such a perturbation describes nearly periodic amplitude  modulations along a planar bright soliton. Self-consistent phase modulations are introduced through the slowly varying function h"h(>, ¹). The short-scale asymptotic analysis was considered by Janssen and Rasmussen (1983) where the "rst terms of the asymptotic series (6.4) were found for a particular case of periodic modulations, a"A(¹)e DN 7,

""a"u (x)#[ia e N W#iaH e\ N W]w (x)#[ae N W#aHe\ N W]u (x) ,   2 2  

(6.6)

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where u "!4 sech x#2 sechx, while the real functions w (x) and u (x) can be expressed    through the Legendre functions (see Appendix B in the original paper by Janssen and Rasmussen, 1983). Furthermore, the parameter h(>, ¹) is related to the amplitude a(>, ¹) through the relation h "8"a" , 2 and the modulation amplitude a(>, ¹) satis"es the unstable NLS equation (6.2) with the coe$cients,











1 p 108 27 b" !1 , c"! # I , 4 6 2 35 where I is given by Eq. (B6) in the paper by Janssen and Rasmussen (1983). It was found numerically that I+0.164 and therefore c+!5.300. The unstable NLS equation (6.2) describes an unbounded scenario of a singularity formation for the amplitude of a modulated bright NLS soliton. Fig. 7(a) shows the corresponding phase plane of the dynamical system (6.2) for the periodic perturbation a"A(¹)e DN 7. A particular analytical solution for a separatrix trajectory represents a nonlinear regime of the soliton self-focusing,




A(¹)"

2b  C cosech(C¹) , "c"

(6.7)

where C is de"ned by Eq. (6.3). This asymptotic solution, rewritten for the original "eld t with the help of Eqs. (6.4) and (6.7), is presented in Fig. 7(b). The unbounded dynamics of the soliton self-focusing is associated with the development of a two-dimensional collapse known to occur for the elliptic NLS equation (Janssen and Rasmussen, 1983). 6.2.2. KdV solitons We analyze the short-scale transverse modulations of the KdV soliton in the framework of the KP equation (2.6). We start from the following asymptotic expansion [cf. Eqs. (4.10) and (5.9)],  u";(m; v)#eu (m, y; >, ¹)# eLu (m, y; >, ¹) ,  L L

(6.8)

Fig. 7. (a) The phase plane corresponding to Eq. (6.2) for a bright NLS soliton under a periodic transverse perturbation. (b) The pro"le of "t(x, y)" within the short-scale analysis with p"1, e"1, and ¹"!0.8.

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where m"x!t!es, v"1#es #O(e), ;(m; v) is the pro"le of a KdV soliton (2.7), and the 2 "rst-order perturbation u (m, y; >, ¹) is speci"ed by the linear eigenmode,  u "[a(>, ¹)e N W#aH(>, ¹)e\ N W](sech m!2 sech m) , 

(6.9)

where p "(3/2. Perturbation of this form describes nearly periodic amplitude modulations  along a planar KdV soliton. The amplitude modulation induces also the coordinate modulation described by the parameter s"s(>, ¹). The second-order term of the asymptotic expansions was found explicitly by Gorshkov and Pelinovsky (1995a), and it can be written in the following form: (6.10) u ""a"w (m)#[a e N W#aH e\ N W]w (m)#[ae N W#aHe\ N W]w (m) , 2     2 where w (m)"3m tanh m sech m!7 sech m#6 sech m, w (m)"(R/Rm)(m sech m), w (m)"!2 sech m    #3 sech m. The coordinate s(>, ¹) is related to the amplitude a(y, ¹) as s ""a", and the 2 amplitude a(>, ¹) satis"es Eq. (6.2) with b"3/4 and c"0. Thus, the asymptotic expansion done in the framework of the integrable KP equation corresponds to a critical soliton self-focusing, and therefore it should include higher-order nonlinear terms. Such a modi"cation of the asymptotic procedure was developed by Gorshkov and Pelinovsky (1995a, 1995b) who derived the unstable Eckhaus equation (see, e.g., Calogero and Eckhaus, 1987) for the amplitude a"a(>, ¹) rescaled to the order of (e, 4 ! ip a #a #"a"a#2a("a") "0 . 22 2 3  7

(6.11)

The last term in Eq. (6.11) describes an e!ective dissipation due to the emission of radiation behind the KdV soliton. Within the asymptotic analysis (see Gorshkov and Pelinovsky, 1995a), the generation of a radiation "eld behind the soliton, u"eu\, is de"ned by the boundary condition at the soliton front, u\" "2("a") . (6.12) 62 2 In the framework of the short-scale approximation, the generation of the radiation "eld in front of the KdV soliton does not occur. Model (6.11), derived by means of an extended short-scale asymptotic analysis, predicts a monotonic transition of an unstable planar KdV soliton to a steady-state transversely modulated soliton structure. Fig. 8(a) presents the phase plane of the dynamical system (6.11) for the periodic modulations in the form a"A(¹)e DN 7. A particular solution for the separatrix trajectory can be found analytically,



A(¹)"



C C  , e 2 sech(C¹) 2

(6.13)

where C is de"ned by Eq. (6.3) for b"3/4. A monotonic transition from a planar soliton to a transversely modulated structure described by the solution (6.13) is accompanied by the generation of a radiation "eld u\ behind the solitary wave, u\"C sech(CX) .

(6.14)

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Fig. 8. (a) The phase plane corresponding to Eq. (6.2) for a KdV soliton under a periodic transverse perturbation. (b) The pro"le of u(x, y) within the short-scale analysis with p"0.7, e"1, and ¹"7.5.

Radiation "eld (6.14) has a structure of a planar KdV soliton (2.7) of smaller amplitude and velocity. Thus, within the short-scale asymptotic analysis, the analytical solution describes a splitting of an initially unstable planar KdV soliton moving with the velocity v "1 into a chain of  two-dimensional KP solitons moving with the velocity v "1#eC and a complimentary planar  KdV soliton moving with a small velocity v "eC remaining behind the modulated soliton  chain. The asymptotic solution for the "eld u [see Eqs. (6.8) and (6.14)] is shown in Fig. 8(b). The asymptotic solution (6.13) and(6.14) agrees with the exact solution (4.36) calculated in the limit pPp (iP0).  6.2.3. Dark NLS solitons The analysis of short-scale self-focusing of dark solitons is similar to that for KdV solitons. In particular, the short-scale asymptotic expansion has the form [cf. Eq. (4.13)],





 (6.15) t" U (m; v)#e (m, y; >, ¹)# eL (m, y; >, ¹) e\ MR ,   L L where m"x!2vt!es, U (m; v) is de"ned by the pro"le of a dark soliton (2.3), and the "rst-order  perturbation (m, y; >, ¹) is given by  3v sinh(km) (p#3k)

"[a(>, ¹)e N W#aH(>, ¹)e\ N W] ! #i  , (6.16)  2k cosh(km) 4k cosh(km)





where k"(o!v and p "[!(o#v)#2(v!vo#o]. The instability dynamics is  described again by the e!ective NLS equation (6.2) with the coe$cients calculated numerically (Pelinovsky et al., 1995). It was found that the coe$cient c"c(k) is positive and cP0 as kP0 (the limit of a KdV soliton). The radiation "elds u! propagate to the right and to the left with the sound speed $2o. Their pro"les are generated by certain boundary conditions (see Pelinovsky et al., 1995) similar to Eq. (6.12). Thus, the short-scale self-focusing of a dark soliton displays the bounded

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Fig. 9. (a) The phase plane corresponding to Eq. (6.2) for a black NLS soliton under a periodic transverse perturbation. (b) The level lines of "t(x, y)" within the short-scale analysis with p"0.5, e"1, and ¹"1.5.

scenario of long-lived oscillations of a modulated dark soliton. Fig. 9(a) displays the corresponding phase plane of the dynamical system (6.2) for the periodic perturbation a"A(¹)e DN 7. A particular analytical solution for the separatrix trajectory is given by the explicit result,




A(¹)"

2b  C sech(C¹) , c

(6.17)

where C is the same as in Eq. (6.3). The asymptotic solution for the "eld t [see Eqs. (6.15) and (6.17)] is presented in Fig. 9(b). The analytical asymptotic results agree with numerical simulations of the defocusing NLS equation which revealed formation of a train of vortex pairs from a planar soliton (McDonald et al., 1993; Pelinovsky et al., 1995) and long-lived periodic oscillations (Pelinovsky et al., 1995). As kP0, one can modify the asymptotic analysis and derive the mixed unstable NLS and Eckhaus equation that describes a transformation of a small-amplitude planar soliton into a chain of two-dimensional solitons accompanied by some intermediate oscillations (see Pelinovsky et al., 1995).

7. Some other models In the previous sections, we have presented an overview of di!erent approaches for analyzing the soliton self-focusing phenomena on the basis of a few fundamental nonlinear models. Di!erent varieties of soliton-bearing models still appear in physical problems of di!erent physical context, and they bring many novel modi"cations of the classical methods. Here we review a few more examples, which we classify into several groups, namely the NLS-type models (Section 7.1), the KdV-type models (Section 7.2), and the kink-type models (Section 7.3).

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7.1. NLS-type models There exist several di!erent modi"cations of the NLS equation (2.1). Some of them have been already mentioned in Section 2.2 in the discussions of realistic physical models employed to describe the suppression of wave collapse. Here we analyze more examples of this type. 7.1.1. Power-law nonlinearities A balance between power-law nonlinearity and wave di!raction can be described by the generalized NLS equation in D dimensions, it #  t#(r#1)"t"Pt"0 , (7.1) R " where  is the Laplace operator in D dimensions, and r is the power of nonlinearity. It is " well-known that the D-dimensional soliton of the model (7.1) is stable provided D(2/r, and unstable otherwise. In the latter case Eq. (7.1) displays collapse of localized solitons (Rasmussen and Rypdal, 1986). The c.w. background is modulationally unstable within the generalized NLS equation (7.1). Being related to the analysis of the soliton stability, the modulational instability of the c.w. background develops into singularities for the case D'2/r while, in the opposite case, it results in long-lived periodic oscillations between the c.w. and modulated states. Stable Ddimensional solitons can be formed at later stages of the instability-induced dynamics. The case r"2/D corresponds to a weak instability of a soliton which results in a critical collapse of slowly growing perturbations (see, e.g., Rasmussen and Rypdal, 1986). Here we discuss how these well-known properties of the generalized NLS equation (7.1) are related to the symmetry-breaking instabilities and the transverse self-focusing of planar solitons. It was shown in Section 4.4 that the modulation equations for transverse perturbations of a one-dimensional soliton of Eq. (2.1) reduce to the one-dimensional NLS equation (1.1) for r"2. Here we generalize this result and show that the modulation equations for the transverse perturbations of a d-dimensional (d(D) soliton of Eq. (7.1) with the parameters D and r reduce to the same NLS equation (7.1) but with the di!erent parameters, DK and r( , where 2r . DK "D!d, r( " 2!rd

(7.2)

The generalized NLS equation (7.1) follows from the Lagrange function, i ¸" (tHt !ttH)!" t"!" t"#"t"P> , R R , , 2

(7.3)

where the gradient vector includes d dimensions parallel to a planar soliton, while the vector ,

includes DK "D!d dimensions transverse to the soliton. The planar soliton solution can be , written in the form, t"uP f (X )e FC , (7.4) , where X "(ux , h/e"ut, and the function f satis"es the equation for a steady-state normalized , , envelope,  f!f#(r#1) f P>"0. The transverse modulation of a planar soliton can be , described by Eq. (7.4) with the varying soliton parameters, u"u(X , ¹) and h"h(X , ¹), where , ,

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X "e x , ¹"et, and e;1. Integrating the function (7.3) with respect to x , we "nd the following , , , averaged Lagrangian (see also Kuznetsov et al., 1986; Trubnikov and Zhdanov, 1987),  ¸ dx , 1¸2" \  f  dX , \ in the form

(7.5)

2!rd uP>\PBP 1¸2"!u\PBP[h #( h)]# 2 , 2(r#1)!rd !eku\P\PBP( u) , , where  ( f#X ) f ) dX , , , '0 . k" \ P 4 f  dX , \ For r"1 and d"1 this expression reduces to Eq. (4.17). The modulation equations can be obtained from Eq. (7.5) by varying 1¸2 with respect to the parameters u and h, (u\PBP) #2 (u\PBP h)"0 , 2 , , 4r  u 2(1!2r)!rd ( u) , ! , "0 , h #( h)!u!ek 2 , 2!rd u u 2!rd





(7.6)

The new function tK (g, q)"cu\PBPe FH satis"es Eq. (7.1) written for the variables g"X /j and , q"¹/j with the parameters DK and r( given by Eqs. (7.2). The constants c and j are de"ned as



c"



2!rd \PBP 4r (ke . , j" 2(r#1)!rd (2!rd)

For r(2/d, the modulation equations (7.6) are elliptic, and this property indicates immediately the existence of a self-focusing instability of planar (d-dimensional) solitons. The scenario of the development of the self-focusing instability depends on the ratio between DK and 2/r( , according to the scheme discussed above. It follows from Eq. (7.2) that the critical NLS equation, with respect to the modulational instability of a c.w. background, i.e. D"2/r, remains the critical NLS equation with respect to the self-focusing instability of a planar soliton, i.e. DK "2/r( . In addition, the subcritical (D(2/r) and the supercritical (D'2/r) cases of Eq. (7.1) transform to the corresponding cases of Eqs. (7.6). This implies that the long-term instability-induced dynamics of a modulated soliton should display a collapse scenario, for DK 52/r( , and a quasi-recurrence scenario, for DK (2/r( . Thus, we come to the conclusion that there exists one-to-one correspondence between modulational instability of continuous waves and self-focusing instability of solitons in nonlinear dispersive/di!ractive media. For r52/d, the modulation equations (7.6) are hyperbolic, i.e. they fail to predict the selffocusing instability of a planar (d-dimensional) soliton. This is explained by the appearance of longitudinal instabilities of planar solitons within the generalized NLS equation (7.1). If a planar soliton is unstable against the symmetry-preserving (longitudinal) perturbations, none of the methods discussed in this survey can help to describe the instability (see, e.g., discussions in Pelinovsky and Grimshaw, 1997).

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7.1.2. Parametric quadratic solitons For three decades optical self-trapped beams (or spatial solitons) con"ned in the transverse plane were commonly believed to be a prerogative of media with cubic-like nonlinearities or their generalizations. However, about 25 yr ago, Karamzin and Sukhorukov (1974) predicted a possibility to achieve beam the self-trapping in an optical medium with a quadratic nonlinear response in the form of mutually interacting and coupled beams of the fundamental and second-harmonic "elds, the so-called parametric quadratic solitons. The "eld of quadratic solitons has acquired importance only recently (e.g., Buryak and Kivshar, 1994, 1995; Torner et al., 1996; He et al., 1996; see also the review papers by Stegeman et al., 1996, and Kivshar, 1998b) being also actively stimulated by experiment. Since the parametric solitons are, strictly speaking, solitary waves, i.e. localized solutions of nonintegrable equations, a crucial issue is their stability, including the symmetry-breaking instabilities. To investigate the symmetry-breaking instabilities of quadratic solitons, we consider a nearly phase-matched interaction of two beams in a quadratic medium governed by two coupled equations for the normalized "eld envelopes u , at the fundamental frequency u, and u , at the   second-harmonic frequency 2u (see, e.g., De Rossi et al., 1997a,b) c 1 iu #  u !  u #u uH"0 ,   X 2 ,  2 RR 1 c u iu #  u !  u #dku #  "0 , X 2p ,   2 RR 2

(7.7)

where  "R#R, is the transverse Laplacian, z is the propagation distance, the normalized time , V W t is in the reference frame travelling at the common group velocity, dk,Dkz "(k !2k )k r is B     the phase-mismatch, and p,k /k . Here k and k are the wave numbers at the corresponding     frequencies and r is the characteristic beam width.  Similar to the NLS solitons, two-wave quadratic solitons of a plane geometry display the symmetry-breaking instabilities in a planar geometry (De Rossi et al., 1997a,b; Skryabin and Firth, 1998b) or in higher dimensions (Skryabin and Firth, 1998b). Importantly, it was proven for di!erent cases (Kanashov and Rubenchik, 1981; BergeH et al., 1995; Turitsyn, 1995) that the equations for quadratic solitons possess no collapse dynamics for localized solutions. Therefore, as was shown in the recent studies, the development of the instability leads either to the formation of a train of higher-dimensional solitons (De Rossi et al., 1997b), similar to the elliptic NLS equation, or to the complete disintegration and radiative decay of a plane soliton (De Rossi et al., 1997a), similar to the case of the hyperbolic NLS equation. Because the model (7.7) is not integrable, explicit results can be obtained when the soliton pro"le is known in a closed analytical form (see, e.g., Buryak and Kivshar, 1995). The full analysis of the linear eigenvalue problem was performed numerically for both the cases (DeRossi et al., 1995a,b). Once established that the plane solitons are unstable, a crucial issue is their long-range evolutions. For the transverse instability, whenever the eigenfunction pro"les follow those of the bell-shaped soliton, the dynamics of the instability process should show no signi"cant changes along the trapping dimension and remain essentially one dimensional. However, in the model (7.7) the problem of long-range evolutions of plane solitons is complicated by a large number of e!ective frequency modes (at least two carriers and two pairs of sidebands). De Rossi et al., 1995a,b; see also Baboiu and Stegeman, 1998) investigated a nonlinear stage of the soliton symmetry-breaking

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instability by integrating numerically Eqs. (7.7) with slightly modulated initial front corresponded to one of the instability eigenmodes. A typical result for the focusing case (De Rossi et al., 1997b) is that a stripe breaks up into a periodical sequence of spots forming a lattice of trapped waves, which are naturally expected to be the (2#1)-dimensional solitons, the so-called &neck instability' (see Fig. 10, left column). In the problem of temporal or modulational instability, the temporal break-up of guided modes in waveguides leads to spatio-temporal trapping for the case of the anomalous dispersion with no qualitative changes in comparison with the transverse instability. Conversely, in the normal dispersion regime no spatial analogy exists. The unstable modes are anti-symmetric and they lead to spatio-temporal wave breaking with characteristic snake-like shapes (De Rossi et al., 1997a), followed by the radiative decay of the soliton (see Fig. 10, right column). Thus, parametric solitons undergo symmetry-breaking instabilities in the way quite similar to the cases discussed above in the framework of the hyperbolic and elliptic NLS equations with cubic nonlinearity. The resulting collapse-free soliton dynamics can be compared with the e!ect produced by a saturable nonlinearity in the NLS models, in particular, the transverse break-up of plane solitons leads to the soliton buncing (via the neck instability), and oscillations around a lattice of stable higher-dimensional solitons. Recently, Skryabin and Firth (1998c) analyzed also the symmetry-breaking instabilities in a more general case of nondegenerate three-wave mixing which describes a phase matched interaction between three waves satisfying the resonant condition, u "u #u , so that the model (7.7)   

Fig. 10. Self-focusing modulational instability of two-wave parametric quadratic solitons in the cases of (a) normal and (b) anomalous dispersion. In the latter case, the stripe decays into a periodic train of solitary waves stable in higher dimension.

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is a particular case. These authors found a new branch of unstable eigenvalues corresponding to a two-parameter phase symmetry of the three-wave solitons which appears due to the second invariant of the equations associated with the conservation of the power unbalance of the di!erent frequencies "elds. This new branch was shown to have a dramatic e!ect on the soliton instability in the case of the normal dispersion. Indeed, this new branch of the symmetry preserving unstable eigenvalues corresponds to a neck instability which, under certain conditions, may become dominant even in a normal dispersion case when the corresponding NLS model displays only the snake-like soliton instabilities. The similar results have been very recently obtained for two incoherently coupled NLS equations (Skryabin and Firth, 1999). 7.1.3. The Davey}Stewartson equation There are known several physical situations when a resonant coupling occurs between high- and low-frequency waves. In one-dimensional systems, this e!ect is described by the so-called Zakharov model which, with the same accuracy, reduces to an e!ective NLS equation. For multi-dimensional case, such a simple reduction is no longer valid, leading to a new model described by the Davey}Stewartson (DS) equation. As one of the possible examples of the corresponding physical system, we mention the evolution of weakly nonlinear gravity-capillary waves at a free surface where a fundamental wave is coupled to an induced mean "eld (Ablowitz and Segur, 1981; Craig et al., 1997). If the #uid is deep, the governing model reduces either to the hyperbolic or to the elliptic NLS equation. In the opposite limit of a shallow #uid, the equations can be reduced to the DS equation, it #pt #t #2t(n!"t")"0 , R VV WW (7.8) n !pn !2("t") "0 . VV WW VV Here t is the amplitude of a wave packet, n is a self-consistent mean #ow, and p"$1. The case p"#1 occurs for weak capillary e!ects, and it is referred to as the DS-I equation. The case p"!1 occurs for pure gravity waves and it is referred to as the DS-II equation. Both the cases are known to be integrable by means of the inverse scattering technique. Soliton solutions (bright solitons) are stable with respect to transverse perturbations for p"!1 and unstable for p"#1 (Ablowitz and Segur, 1979). Exact solutions describing the development of the soliton self-focusing instability were constructed for the DS-I equation by Pelinovsky (1994), who considered both dark and bright solitons of Eqs. (7.8) for p"#1. However, it can be shown that the nonvanishing c.w. background is also unstable in this model, therefore, dark solitons do not survive at wave background due to the background instability. Here we reproduce the exact solutions for bright solitons of Eq. (7.8) at p"#1. The DS-I equation has an explicit bilinear representation, q> t" , "t""(R !R) log q, n"2R log q . V W V q

(7.9)

The bilinear functions q(x, y, t) and q>(x, y, t) can be speci"ed in the following particular representation (Pelinovsky, 1994): q>"2 sH,



q"1#

V

\

" " dx



V

\

"s" dx ,

(7.10)

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where the functions and s satisfy the system of linear equations,

# "0, s !s "0, i # "0, !is #s "0 . V W V W R VV R VV We consider the plane-wave solution to Eqs. (7.11),

(7.11)

"c exp[p(x!y)#2ipt]#c exp[i(x!y)#2iit] ,   (7.12) s"c exp[p(x#y)!2ipt]#c exp[i(x#y)!2iit] ,   where i"p!ik, c "(2p), c "(2p)a, c "(2p)b, and a, b, and k are real parameters.    Taking, for simplicity, the case b"0, we can write the following exact solution to Eq. (7.8), 4peNV> NR(1#e\ IV\W>IR>CR) , t(x, y, t)" 2p 2p C C C e\ IV\W>IR> R# e IV\W>IR> R#e R 1#eNV 1# 2p!ik 2p#ik






(7.13)

where C"4pk. If C'0, this solution describes, for tP!R, a planar soliton perturbed by a transverse periodic perturbation that exponentially grows in time. The instability is induced by asymmetric (translational) eigenfunctions in the long-scale limit (kPR), as in the case of bright solitons of the hyperbolic NLS equation. The instability domain is not bounded from above, i.e. CPR as kPR. This unusual feature of the soliton self-focusing is explained by the pro"le of n(x, y, t). It is clear from Eqs. (7.9), (7.10), and (7.12) that the self-focusing of a bright soliton is driven by a transverse periodic modulation of the self-consistent mean #ow n nonlocalized in the direction of the soliton as xP#R. This perturbation in the mean #ow pumps the fundamental wave and induces the soliton transverse instability. At a nonlinear stage of the instability development, the growth of transverse perturbations is stabilized, and it alternates with damping. As a result, the bright soliton returns to its unperturbed planar shape but it acquires a drift velocity, v"!4k. In the framework of this exact solution, the energy of a soliton is not changed and radiation is not generated. In the symmetric case, a"b, the exact solution for the soliton self-focusing in the DS-I equation is presented in Figs. 11(a)}(c). 7.1.4. Discrete NLS equations All the models analyzed above describe nonlinear waves and instabilities in the continuous systems. However, in the solid state physics, continuous models often appear as a limiting case of more general, discrete physical models where the lattice spacing is a fundamental physical parameter. Discreteness introduces a number of new features in the system dynamics, in particular, it modi"es the conditions for modulational instability of plane waves (Kivshar and Peyrard, 1992). A simplest model to demonstrate some basic features introduced by discreteness is a model of an array of optical "bers (or planar optical waveguides) coupled by a weak overlapping of the guided wave "elds excited in each core of the waveguide array (Christodoulides and Joseph, 1988; Kivshar, 1993; Aceves et al., 1994b), iR t #Rt #K(t #t !2t )#2"t "t "0 , (7.14) R L V L L> L\ L L L where the variable t stands for an envelope of the average electric "eld in the "ber with the number L n, and the variables t and x have a reverse meaning in the "ber optics, t is the coordinate along the "ber in the reference frame moving with the group velocity and x is the retarded time. If we neglect

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Fig. 11. (a)}(c) Self-focusing instability in the DS-I system described by the exact analytical solutions.

the coupling, i.e. K"0, Eq. (7.14) becomes a standard (1#1)-dimensional NLS equation, and therefore the coupling introduces the second dimension described by the discrete variable n. We follow Aceves et al. (1994b) and study transverse stability of the plane solution t (x, t)"U (x)e SR, where U (x)"(u sech((ux), to the perturbation in the discrete lattice varyL Q Q ing with n. To do so, we write as usual t (x, t)"[U (x)#dt (x, t)]e SR, where this time the small L Q L perturbation is selected in the form, dt (x, t)"dt(x, t) cos(Qn). Linear eigenvalue problem for dt is L standard, and it is known to possess unstable eigenmodes for the interval "Q"(Q , where Q is   de"ned di!erently for the discrete problem, 4K sin(Q /2)"3u , (7.15)  which, as expected, transforms into the well-known result of the continuum limit for Q;1.

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Aceves et al. (1994b) analyzed numerically the scenario of a decay of a plane soliton in the case of the transverse instability in such a partially discretized lattice. In a sharp contrast with the (2#1) dimensional NLS equation displaying the collapsing dynamics, in the discrete model Eq. (7.14) localized states are formed after the early stage of the instability development and exponential growth of the amplitude. Such localized modes are the analog of multi-dimensional solitons localized in both time (continuous variable) and space (discrete variable) (see e.g., Aceves et al., 1994a; and also Pouget et al., 1993, for the similar localized modes in a discrete two-dimensional lattice). The collapse mechanism can be associated with the initial stage of the evolution, whereas any singular dynamics is suppressed by discreteness (Aceves et al., 1995). Thus, the lattice discreteness in the transverse dimension prevents collapse and it allows to develop a sequence of spatially localized states as a result of the transverse instability of a plane soliton acting, in some sense, as an e!ective nonlinearity saturation. Some further results in the analysis of transverse instabilities of solitons in discrete lattices and waveguide arrays were obtained by Darmanyan et al. (1997), who considered transverse instability of envelope solitons in the model (7.14) but with an arbitrary wave number of the carrier wave. Unlike the case of the continuous NLS equation where the instability growth rate does not depend on the carrier wave number, this is not so even for modulational instability in discrete lattices where the carrier wave modi"es the e!ective dispersion and it can lead to stabilization of plane waves (Kivshar and Peyrard, 1992). Considering the stability of the NLS solitons in the model (7.14) with a moving carrier wave, i.e. t (x, t)"U (x!vt)e SR\OL\GV , L Q Darmanyan et al. (1997) obtained the following result for the cut-o! wave number Q of the  instability band [cf. Eq. (7.15)] 4K cos q sin(Q /2)"3u , (7.16)  which is valid for cos q'0 and therefore it generalizes the result (7.15). Similarly, for cos q'0 the plane soliton forms a train of stable localized modes of higher dimensions (the soliton bunching e!ect), as in the case of the elliptic NLS equation with a saturable nonlinearity. For the case cos q(0 the analysis is more involved, and e!ectively this case coresponds to the hyperbolic NLS equation, and it can be analyzed by an asymptotic technique. In that case, the maximum growth rate is reached near the edge of the Brillouin zone, and asymmetric instability leads to a bending of the initially plane soliton. Thus, the carrier wave number q allows to vary the e!ective dispersion in the lattice and to transform the scenario of the soliton instability from the elliptic case to the hyperbolic one. Recently, Relke (1998) extended this kind of analysis to the case of a waveguide array with a periodically varying coupling constant in the array, i.e. for KPK . In a contrast to continuous L models, in the mixed continuous-discrete NLS model (7.14) the asymptotic analysis of the soliton stability can be performed in the approximation of weak coupling but for arbitrary values of the perturbation wave numbers. For a periodic variation of the coupling constant, K "K#*K cos[p(n#)], Relke (1998) revealed the existence of two types of unstable eigenL  modes, optical and acoustic ones. This analogy comes from the fact that a discrete array of waveguides coupled by a periodically varying constant K form an e!ective diatomic lattice with L the spectrum consisting of two branches separated by a gap (see e.g., Kivshar and Flytzanis, 1992).

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The existence of two branches of unstable eigenvalues open a new mechanism of the instability scenario when at the initial stage of the instability-driven evolution the energy gets redistributed between adjacent solitons in the array, and a kind of doubling of a spacial period occurs (Relke, 1998). 7.2. KdV-type models The integrable KP equation (2.6) is an example of a higher-dimensional model that describes long-wave transverse modulations of a KdV soliton. Here we discuss a few more examples of this type. 7.2.1. The Zakharov}Kuznetsov equation The Zakharov}Kuznetsov (ZK) equation is another, alternative version of a nonlinear model describing two-dimensional modulations of a KdV soliton. It was "rst derived for describing the evolution of the ion density in strongly magnetized ion-acoustic plasmas (Zakharov and Kuznetsov, 1974). If a magnetic "eld is directed along the axis x, the ZK equation in renormalized variables takes the form, u #uu # u "0 . (7.17) R V V The ZK equation appears as a generalization of the KdV equation to two spatial dimensions but, unlike the KP equation, it is not integrable by the inverse scattering transform method (see, e.g., Shivamoggi et al., 1993). Instability of a plane KdV soliton to transverse dimensions, in the framework of the model (7.17), was extensively investigated analytically and numerically (e.g., Laedke and Spatschek, 1982; Laedke et al., 1986; Infeld and Frycz, 1987; Frycz and Infeld, 1989a, 1989b; Allen and Rowlands, 1993, 1995; Infeld, 1985; Infeld et al., 1995; Bettinson and Rowlands, 1998a,b). To analyze the soliton instabilities, Laedke et al. (1986; see also Laedke and Spatschek, 1982) developed a modi"cation of the short-scale small-amplitude asymptotic analysis. Instead of the unstable NLS equation (6.2), for the ZK equation (7.17) they derived a "rst-order evolution equation, !ip a #ba #c"a"a"0 , (7.18)  7 2 where the coe$cients b and c are positive. In contrast to the case of the unstable NLS equation, this short-scale asymptotic model describes a monotonic transition from a planar soliton to a periodic train of two-dimensional solitons generated along the soliton front (Laedke et al., 1986). Such a scenario of the soliton decay was con"rmed numerically by Frycz and Infeld (1989b) and Frycz et al. (1992). Later, Allen and Rowlands (1993) suggested an extension of the multi-scale perturbation approach and also obtained the maximum growth rate for all k in the form of a two-point PadeH approximant (see also the case of obliquely propagating plane solitons discussed by Allen and Rowlands (1995)). 7.2.2. The Shrira and Benjamin}Ono equations Large-amplitude localized structures were experimentally observed in boundary layers generated by subsurface shear #ows (Kachanov et al., 1993). An analytical description of this phenomenon was developed by Shrira (1989) who derived a two-dimensional model generalizing the

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well-known Benjamin}Ono (BO) equation. If a shear #ow is directed along the x-axis, the Shrira model is described by the equation, u #uu #Q+u ,"0 , R V V where Q+u, is the Cauchy}Hadamard integral transform,

(7.19)



> u(x, y, t) dx dy . [(x!x)#(y!y)] \ Linear transverse instability of (1#1) dimensional solitons of the model (7.19) was predicted and analyzed by Pelinovsky and Stepanyants (1994) (see also D'yachenko and Kuznetsov, 1994). Numerical simulations indicated the formation of singular two-dimensional structures in Eq. (7.19) (D'yachenko and Kuznetsov, 1995). This conclusion was con"rmed by Pelinovsky and Shrira (1995) who constructed approximate analytical solutions describing a singular localized mode. Eq. (7.19) can be simpli"ed under the assumption "u ";"u ", and then it takes the form of a twoWW VV dimensional BO equation, 1 Q+u," 2p

u #uu #H+u ,#H+u ,"0 , WW R V VV  where H+u, is the Hilbert integral transform,

(7.20)



1  u(x, y, t) dx H+u," . p x!x \ Transverse modulations of the BO soliton, 2v u(x)" , [1#v(x!s)] can be studied by the averaged Largangian method outlined in Section 4.2, which generates the modulation equations for the soliton parameters, v #(vs ) "0 , 77 2  1 v v s !v# e vs # 7 ! 77 "0 . 2 7 v 2 v




(7.21)



A small-amplitude limit of these equations corresponds to the elliptic Boussinesq equation for which the asymptotic solutions for the transverse soliton self-focusing was obtained by Pelinovsky and Shrira (1995). Furthermore, there exists an exact transformation for the modulation equations (7.21),




 







¹  ¹    =(g), g" >, ¹ !¹ ¹ !¹   3 ¹ !¹  1 E z dz s" ¹ 1!  ! , 2  ¹ =(z) 3¹    v"

 



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where p , =(g)" 1!(1!p cos(pg) p is the transverse wave number for a periodic modulation, ¹ is the time of blowup of the  asymptotic solution. This solution resembles the formation of singular localized modes along the front of a planar soliton due to the development of a transverse instability. Basically the same phenomenon was observed numerically by D'yachenko and Kuznetsov (1995). 7.3. Kinks in the Cahn}Hilliard equation A wide class of the stability problems can be formulated for solitary wave solutions of dissipative models. In such models, the most common localized solution is a kink (or an interface solution) that connects two equilibrium states of the system. Generally speaking, the asymptotic methods described above for conservative models are well applicable for the stability analysis of localized solutions of dissipative models. To give a speci"c example and also point out some new features of dissipative models, here we mention some results for the transverse stability of the (1#1) dimensional kink solution in the Cahn}Hilliard equation (Cahn and Hilliard, 1958) with general nonlinearity,






dF ! u , u "  R du

(7.22)

where F(u) is some general nonlinear free energy. Eq. (7.22) admits a stationary (1#1) dimensional kink solutions u (x) connecting two equilibrium states u"u and u"u , i.e. it is assumed that I   F(u )"F(u )"0, F(u )"F(u )"0, u 4u (x)4u .      I  Eq. (7.22) was derived from classical thermodynamic considerations of the interdi!usion of two components A and B, to describe the phase transition induced rapidly decreasing the temperature from some ¹ '¹ to some ¹ (¹ (for an overview of the physics, see Novick-Cohen and Segel,     1984). In a particular case, the free energy F(u) can be taken in the form, (7.23) F(u)"(1!u) ,  and then the Cahn}Hilliard equation becomes, u " (u!u! u), and its kink solution can be R found in an explicit form, u (x)"tanh(x/(2). I To analyze stability of the (1#1) dimensional kinks u (x) to small perpendicular perturbations I of the wave number k, we follow Bettinson and Rowlands (1996a), and write a solution of Eq. (7.22) in the form, u(r, t)"u (x)#edu(x)e IW W>IX XeAR , I where u (x) is a kink solution of the equation (u )"2F(u ), and du(x) stands for a perturbation I I I amplitude. The linear eigenvalue problem for du(x) can be analyzed by means of the asymptotic expansions for small and large k,(k#k. For small k, the asymptotic result for the growth rate W X

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c(k) is (Bettinson and Rowlands, 1996a) c(k)"!c k!c k ,  

(7.24)

where the positive coe$cients c and c are de"ned by the speci"c form of F(u), e.g. for the potential   (7.23) they are c "(2/3 and c "11/18. In the lowest order, the result (7.24) agrees with that   obtained by a variational method (Shinozaki and Oono, 1993). In the case of large k, the asymptotic expansion in k\ yields: c/k"!1!c /k#2, where,  e.g., c +0.303 for the potential (7.23). Bettinson and Rowlands (1996a) suggested a general PadeH  approximation to describe the full dispersion relation c(k) for all k, which provides a reasonably good agreement (within 1.3% of the exact results) with numerical results and particular cases. This analysis shows that the kink solution of the dissipative model (7.22) is stable to transverse perturbations for all k. Eq. (7.22) admits also both cylindrical and spherically symmetric stationary kink solutions. As has been shown by Bettinson and Rowlands (1996b) for the case of large radius R, the cylindrically symmetric kink solution is stable to perturbations involving angular variation, but is unstable to a general perturbation. In contrast, the spherically symmetric kink solution is stable for all small perturbations. This suggests that the unstable cylindrically symmetric solution may decay into spherically symmetric states similar to the cases discussed above for some conservative models. It is interesting to note that a discrete version of the Cahn}Hilliard equation, which can also have some physical applications, shows new features for the kink stability (see Bettinson and Rowlands, 1998a,b). Analytical kink solution of a discrete model, u "tanh b tanh(nb#s), can be L found for some special form of F(u), so that the asymptotic analysis for small and large k can be developed, similar to the continuum case. As a result of that analysis, a kink is stable to transverse perturbation in the discrete model as well, however, the growth rate vanishes at the edges of the Brollouin zone, k"2pp, p is integer.

8. Experimental observations 8.1. Self-focusing and bright solitons Theoretical predictions of self-focusing of light in an optical medium with nonlinear refractive index (Askar'yan, 1962; Chiao et al., 1964; Talanov, 1964) were followed by experimental evidence of this phenomenon in di!erent optical materials, e.g. glasses, Raman-active liquids, gas vapors, etc. In particular, Pilipetskii and Rustamov (1965) reported the generation of one-, two- and three"laments due to self-focusing of a laser beam in di!erent organic liquids. Later, Garmire et al. (1966) reported a direct observation of the evolution of beam trapping in CS in the simplest  cylindrical mode. They found that the threshold, trapping length, nonlinearity-induced increase in the refractive index in the trapped region, and beam pro"le are consistent with theoretical predictions, and the steady-state input beam of circular symmetry asymptotically collapses to a bright "lament as small as 50 lm. As a matter of fact, this was one of the "rst experimental manifestations of the phenomenon which we now call spatial optical soliton.

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Because ruby-laser beams used in the experiments have intensities far above threshold for self-trapping in CS (25$5 kW), Garmire et al. (1966) also observed the formation of rings around  the self-focused spots and the development of many "laments from an apparently homogeneous beam about 1 mm in diameter and considerably above the threshold power. The former e!ect can be associated with the existence of a set of higher-order circularly symmetric steady-state modes (Yankauskas, 1966; Haus, 1966), whereas the latter e!ect is a direct manifestation of the transverse beam instability, spatial modulational instability of a broad beam. Steady-state self-focusing and self-trapping was observed for several other media, including potassium vapor (e.g., Grischkowsky, 1970), sodium vapor (Bjorkholm and Ashkin, 1974), etc. Additionally, detailed studies of a spatial breakup of a broad optical beam due to self-focusing was reported by Campillo et al. (1973, 1974) who used a 50-cm cell of CS to study self-focusing and  observed that radially symmetric ring patterns created by circular apertures breakup into focal spots having azimuthal symmetry and regular spacing. This kind of e!ect can be associated with the transverse modulational instability of quasi-plane bright rings created by the input beam, and the number of the bright spots and critical powers are in a good qualitative agreement with the simple theory of transverse instabilities, as was discussed in detail later by Campillo et al. (1974). A number of similar experiments were performed later for di!erent types of nonlinear media, including arti"cial Kerr media made from liquid suspensions of submicrometer particles (e.g., Ashkin et al., 1982), where the smallest-diameter self-trapped "laments (&2 lm) were observed. Similar experiments were recently done for vortex rings, i.e. bright rings with a nonzero angular momentum created by passing the laser beam through a di!racting phase mask and then propagating it in a nonlinear medium (a 20-cm cell with rubidium vapor) (Tikhonenko et al., 1995, 1996b) and also for a quadratically nonlinear medium (KTP crystal) (Petrov et al., 1998). An angular momentum introduced in the input beam, strongly a!ected the dynamics of bright spots (in fact, spatial solitons) created by the transverse instability of the rings, so that they can attract and repel each other, or even fuse together. Formation of a variety of di!erent patterns of spots (bright spatial solitons) was investigated by Grantham et al. (1991) in a sodium vapor. They varied the input beam power from 30 to 460 mW and observed spatial bifurcation sequences due to spatial instabilities seeded by intentionally introduced aberrations. They used the structure of the instability gain curve for an inputwavefront-encoding feedback to accelerate particular unstable wave vectors, and observed complicated spatial bifurcations as a function of intensity or detuning, with `2complexity and beauty rivaling that of a kaleidoscopea (Grantham et al., 1991). The analysis of self-focusing based on the spatial (2#1) dimensional NLS equation and associated with the spatial instabilities, bifurcations, and formation of spatial solitons is valid for both c.w. beams and long pulses. In contrast, short pulses undergoing self-focusing do not collapse to wavelength dimensions. A number of experimental results (e.g., Strickland and Corkum, 1991) demonstrated the resistance of short pulses (&50 fs) to self-focusing. In spite of the fact that these process can be modelled by the hyperbolic NLS equation with normal group-velocity dispersion, experimental results (Strickland and Corkum, 1994) indicate that spectral dispersion and other non-slowly-varying are also important to explain di!erent behavior of short pulses. A detailed experimental investigation of the self-focusing dynamics of a femtosecond pulse in a normally dispersive (glass) medium was recently reported by Ranka et al. (1996) who observed one of the main e!ects predicted by the theory based on the hyperbolic NLS equation, i.e. the

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splitting of a short pulse (85}90 fs) into two pulses for the power above the threshold value, P'P +3 MW, and even additional splittings, for higher powers (P+4.8 MW). Moreover,  Ranka et al. (1996) noticed that above the threshold power for pulse splitting, the pulse spectrum undergoes signi"cant broadening which eventually develops, at higher powers, into supercontinuum generation (SCG) or while-light generation "rst observed by Alfano and Shapiro (1970). Such a spectrum broadening con"rms a hypothesis that SCG is a result of the nonlinear dynamics of self-focusing in which the temporal and spatial degrees of freedom are coupled. However, the corresponding model describing both these phenomena, i.e. the pulse splitting and SCG, should be not based on the slowly varying envelope approximation. More recently, Diddams et al. (1998) reported the similar e!ects for the propagation of intense fs pulses in fused silica. Frequencyresolved optical gating was used to characterize the pulse splitting into subpulses which were found to be not generally symmetric, in accordance with the theoretical predictions based on a threedimensional NLS equation that includes the Raman e!ect, linear and nonlinear shock terms, and third-order dispersion (Zozulya et al., 1998). Theoretical prediction and a number of experimental observations of beam self-trapping in photorefractive media allow to observe spatial solitons in crystals at relatively low input powers. The "rst observation of two-dimensional spatial solitons was reported by Shih et al. (1995; see also Shih et al., 1996) who used an electric "eld of 5.8 kV/cm applied to a crystal of strontium barium niobate (SBN) to create an e!ective self-focusing nonlinearity and trap an optical beam into a "lament as small as 9.6 lm at micro-Watt power levels. Experimental observation of breakup of a quasi-plane bright spatial soliton into a sequence of higher dimensional solitons due to the transverse (&neck'-type) instability was observed by Mamaev et al. (1996c). In the experiments, Mamaev et al. (1996c) used a 10 mW beam from a He}Ne laser (j"0.6328 lm) to create a highly asymmetric elliptical beam (15 lm;2 mm) with a controlled waist. The beam was directed into a photorefractive crystal (10;9 mm) of SBN:60, lightly doped with 0.002% by weight Ce. A variable DC voltage was applied along the crystal c( -axis to take advantage of the largest component of the electro-optic tensor of SBN, and to vary an e!ective self-focusing nonlinearity. Fig. 12 shows the near-"eld distributions of the input (a) and outputs [(b) to (f )] beam for di!erent values of the applied voltage, i.e. di!erent values of the nonlinearity. First of all, without the applied voltage, the output beam spreads due to di!raction [Fig. 12(b)]. As the nonlinearity increases, the beam starts to self-focus [Fig. 12(c)] forming a self-trapped channel of light [Fig. 12(d)]. A further increase of nonlinearity leads to the modulational instability, and the self-trapped stripe beam breaks up into a periodic sequence of "laments [Figs. 12(e) and (f )]. No arti"cial seeding was added to the input beam, and it was argued that the instability developed from the natural level of noise present on the beam and /or in the crystal (Mamaev et al., 1996c). As a matter of fact, the transverse instability of a plane spatial soliton in photorefractive media is more complicated phenomenon than that in a cubic Kerr medium, due to the applied electric "eld which makes the problem anisotropic. The equations describing these e!ects can be written as a system of normalized equations for an optical beam envelope, B, and the normalized electrostatic potential induced by the beam, , as follows: iB #  B# B"0 , V X  ,

 # ln(1#"B") "+ln(1#"B"), , , , , V

(8.1)

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Fig. 12. Experimental observation of a plane-soliton decay in a photorefractive medium: (a) input beam, and (b)}(f ) output beams for di!erent values of the applied DC voltage, <"0, 600, 1000, 1560, and 2200 V, respectively (Mamaev et al., 1996c).

where acts on components x and y perpendicular to the direction of propagation of the beam z (see, e.g., Mamaev et al., 1996c). Numerical simulations were carried out by Mamaev et al. (1996c) in order to show that the transverse self-focusing of a plane soliton can occur for Eqs. (8.1), similar to the elliptic NLS equation. First of all, a plane soliton is a stationary solution of Eqs. (8.1), which satisfy an equivalent saturable NLS equation because the second equation of Eq. (8.1) can be integrated to yield, "("B"!B )/(1#"B"). Transverse instability of this stationary solution was analyzed, V  for both focusing and defocusing cases, in the framework of the complete model (8.1) by Infeld and Lenkowska-CzerwinH ska (1997), by means of the asymptotic p-expansions for small wave numbers, and also by using a trial sech-type function for the soliton pro"le. At last but not least, we would like to mention the most recent observations of the soliton instability due to parametric beam self-focusing in quadratic nonlinear media, via the cascaded interaction between the fundamental wave and its second harmonic under the condition of the phase matched second-harmonic generation process (see, e.g., Stegeman et al., 1996, for a comprehensive review of cascaded nonlinearities). The cascading mechanism of self-focusing allows to observe a number of interesting nonlinear e!ects in noncentrosymmetric crystals, similar to those already observed in gases and liquids. In particular, Fuerst et al. (1997a; see also Fuerst et al., 1997b) reported the "rst observation of the transverse modulational instability of a spatial two-wave soliton in a quadratically nonlinear optical medium (see also Section 7.1.2 for discussions of the theoretical results). In the experiments, the transverse instability was demonstrated for a 1 cm long KTP crystal, similar to that earlier used in the experiments on the beam self-focusing (Torruellas et al., 1995). The 35 ps pulses were generated by Nd:YAG laser with 10 Hz repetition at j"1064 nm. A variable elliptical beam was created with an adjustable cylindrical telescope, so that the small dimension of

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Fig. 13. Break-up of an elliptic beam in a quadratic optical medium: (a) input beam, (b) output beam at 48 GW/cm, (c) output beam at 57 GW/cm. Shown is the beam of the fundamental frequency but the similar structures appear for the second harmonic beam (Fuerst et al., 1997a).

the beam was about 20 lm, and the larger dimension was varied to be more than in "ve times larger, at the input face of the crystal. This ratio made reasonable plane soliton since at a 1 cm long crystal the e!ects of di!raction are minimal with such large waists. Figs. 13(a)}(c) present the experimental results (Fuerst et al., 1997a) for the evolution of the input elliptic beam with the ratio of its dimension 12:1, shown in Fig. 13(a). Figs. 13(b) and (c) show the beam of the fundamental frequency at the output of the crystal for intensities of 48 and 57 GW/cm, respectively. The main e!ect observed by Fuerst et al. (1997a) is a breakup of the beam into a sequence of well-de"ned circular spots with the radius 9.5$1.5 lm [see Fig. 13(c)], essentially equal to the value of 10}12 lm obtained for spatial solitons created by 20 lm circular beams (Torruellas et al., 1995). According to the theory, the number of created spatial solitons should depend on the intensity, and indeed the experimental results were found to give a qualitatively good agreement with the results of the simpli"ed theory of modulational instability in quadratic media. 8.2. Dark solitons Most of the experimental demonstrations of spatial and temporal dark solitons have been recently discussed in the paper by Kivshar and Luther-Davies (1998). For the case of spatial dark solitons, the experiments reported the creation of dark soliton stripes in a (2#1)-dimensional geometry. As discussed above, such stripes should be unstable due to transverse modulational instability which leads to stripe breakup and the eventual creation of optical vortex solitons. However, it turns out that this instability was avoided in the early experiments by the use of "nite-sized background beams and weak nonlinearity. By increasing nonlinearity, the transverse instability should be observed even with "nite sized beams. The "rst experiments to verify the existence of this transverse instability, and through it the creation of optical vortex solitons, have been performed by Tikhonenko et al. (1996a) using a continuous wave, Ti:sapphire laser and a nonlinear medium comprised of atomic rubidium vapour. Very similar observations, with less

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evidence of the stripe decay into a sequence of vortex solitons, were performed almost simultaneously by Mamaev et al. (1996a,b) for spatial dark solitons in a photorefractive medium. In the experiments using rubidium vapour (Tikhonenko et al., 1996a), the laser output was a linearly polarized slightly elliptical Gaussian beam with a wavelength tuned close to the rubidium atom resonance line at 780 nm. A p phase jump was imposed across the beam center using a mask and the resulting beam imaged into the nonlinear medium. The rubidium vapor concentration could be increased up to 10 cm\ by changing the cell temperature. Images of the beam at the output of the cell were recorded by a CCD camera and frame capture system. As a matter of fact, a schematic of this experimental arrangement is similar to that used to observe optical vortex solitons (see, e.g., Kivshar et al., 1998, and references therein). The important step in observing the instability was to resonantly enhance the value of nonlinearity of the medium by tuning the laser frequency close (!0.4 to !1.0 GHz) to the rubidium atom D line and the use of the maximum vapor pressure consistent with tolerable absorption. The  power in the beam at the input face of the cell was 240 mW with a 1/e waist of 0.3 mm. A maximum nonlinear refractive index change of order of 10\ was achieved. Fig. 14 shows a series of output intensity pro"les observed experimentally, with increasing cell temperature and with the detuning "xed at approximately 0.85 GHz. For vanishingly small vapor concentration, the beam underwent linear propagation through the medium. With increasing temperature (i.e., increasing nonlinearity), the output beam developed a vertically uniform dark soliton stripe, as shown in Fig. 14(a). Further increase in the temperature led to the growth of a periodic modulation of the uniformity of the stripe. As the temperature was further increased, the breakup of the stripe began, initially appearing as a growing, &snake-type' bending [Fig. 14(b)], then as breaking, with "eld coalescing into dark spots at the in#ection points in the bends [Fig. 14(c)]. At the highest nonlinearity the dark spot assumed close to circular symmetry consistent with the predicted formation of a pair of optical vortex solitons [Fig. 14(d)]. Tikhonenko et al. (1996a) also carried out numerical simulations based on the generalized NLS equation including saturation and dissipative e!ects for comparison with the experimental results. The calculated output intensity distributions showed the same dynamics observed experimentally

Fig. 14. Out beam intensity pro"les demonstrating the instability of a dark soliton stripe as the nonlinearity is increased. The cell temperatures are: (a) 823C (b) 903C, (c) 1123C, and (d) 1253C. The power of the beam at the cell input was 240 mW, corresponding to a maximum intensity +170 W/cm (Tikhonenko et al., 1996a).

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with increasing nonlinearity. The beam defocusing, power depletion and instability growth rates seen in experiments appeared to be well approximated by the simulations. However, the period of the transverse perturbation corresponding to the maximum growth rate appeared to be smaller than that seen in experiment, by a factor of 1.5}2. This discrepancy is most likely due to (i) the physically complicated nonlinear response of rubidium vapour, which was only approximated by the model used in the simulations; (ii) the di$culty in accurately characterizing the initial "eld which was found to sensitively a!ect the simulations. It should be noted that this sensitivity was not observed in the experiments, suggesting that the breakup process may have been partially stabilized by some physical mechanisms not included in the model (e.g., nonlocality in the form of di!usion). Similar experimental results were reported by Mamaev et al. (1996a). In their experiments a biased photorefractive SBN crystal, irradiated with a 10 mW He}Ne laser beam containing a phase step, was used as the nonlinear medium. Numerical simulations using the generalized NLS equation with a saturable nonlinearity demonstrated a similar breakup of the initial stripe into a set of optical vortex solitons as reported by Tikhonenko et al. (1996a). The e!ectively nonlinearity could be varied by increasing the bias voltage on the crystal. When zero voltage was applied, the dark stripe spread due to di!raction as did the background beam. As the applied voltage increased the background beam underwent self-defocusing and a dark-stripe soliton was clearly formed. A further increase in the voltage up to 990 V [the maximum voltage reported was 1410 V (Mamaev et al., 1996a) and 2000 V (Mamaev et al., 1996b)] led to the appearance of the snake-like bending of the dark soliton stripe. However, the "nal state was markedly di!erent from that shown in Fig. 14(d) because it did not present clear evidence of the creation of optical vortex soliton pairs. However, the authors reported the observation of zeroes in the electromagnetic "eld from interferometric measurements of the output beam with the distances between the zeroes being about 40 lm. This measurement indicates that wave-front dislocations similar to single vortices were being formed. The experiments, which reported the "rst observation of the transverse dark-soliton instability, indicate that the scenario of the periodic modulation of a plane soliton is really hard to observe because it is strongly a!ected by radiative losses and dissipative losses in real physical systems. However, a decay of a plane dark soliton into vortex solitons is readily observed being a fundamental physical phenomenon only slightly a!ected by dissipation.

9. Concluding remarks We have discussed above the asymptotic analytical methods for describing linear and nonlinear regimes of the symmetry-breaking instabilities and self-focusing of plane solitons induced by transverse modulations in higher spatial dimensions or self-modulation due to temporal dispersion. The analytical methods are associated with di!erent approximations and approaches, namely E E E E

geometric optics approach, linear stability analysis, gas dynamics approach (Whitham equations), modulation equations for the soliton parameters,

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E long-scale and short-scale asymptotic expansions, E exact solutions for integrable nonlinear equations. Each of those analytical methods is valid in the corresponding asymptotic domain where analytical solutions can be obtained. We have demonstrated the applications of all of those methods to the analysis of the transverse instability of bright and dark envelope solitons in the cubic NLS equation, and also long-wave solitons of the KdV equation. The asymptotic methods allow to describe, in a self-consistent manner, both weak nonlinear and linear e!ects in the soliton self-focusing, and they provide a rather informative and adequate picture of the instability-induced soliton dynamics. We believe that these analytical methods will be useful in the analysis of the soliton self-focusing dynamics in other physically important nonlinear models, and also for some other problems we brie#y summarize below. Indeed, in a context of di!erent physics, there exist a number of problems which are closely connected to the topics discussed above. Some of them still require further investigation, and we believe that the analytical methods and physical concepts discussed above may help to achieve a further progress in this research. Below, we mention just a few such problems. 9.1. Random or periodic yuctuations One of the important physical generalizations of the problems discussed above is the soliton self-focusing and instabilities in the presence of spatial (random or periodic) inhomogeneities or temporal yuctuations of the physical parameters. The e!ect of a periodic spatial modulation of the refractive index in the spatiotemporal pulse evolution was analyzed by a variational aproach (Aceves and De Angelis, 1992), by the virial theorem (Turitsyn, 1993a,b), and by the averaging method (Kivshar and Turitsyn, 1994). More recently, Gaididei and Christiansen (1998) have demonstrated analytically, by means of the virial theorem and the Furutsu}Novikov technique, that delta-correlated random #uctuations &delay' the threshold power for the pulse self-focusing. This is in agreement with the case of periodic variations where, for the averaged equations, the e!ect of renormalized nonlinearity was demonstrated (Kivshar and Turitsyn, 1994), suggesting a shift of the instabilily band. The qualitatively similar e!ects were observed in numerical simulations (see, e.g., Rasmussen et al., 1995; Christiansen et al., 1996a,b). 9.2. Self-focusing of coupled waves Other class of important physical problems is self-focusing of coupled waves. The e!ect of the wave coupling on modulational instability was "rst analyzed by Agrawal (1987; see also Agrawal et al., 1989), in the framework of two coupled NLS equations. Agrawal (1987) pointed out that the cross-phase modulation between two waves leads to a number of interesting new e!ects and, in particular, he predicted that modulational instability of two waves can occur even in the case of normal group-velocity dispersion when a single wave is always stable. For the spatial beams, the e!ects of the wave interaction have been studied for the mutual focusing of coupled waves (Berkhoer and Zakharov, 1970; McKinstrie and Russell, 1988; Agrawal, 1990b; BergeH , 1998b), and also for the transverse modulational instability of both counterpropagating (Vlasov and Sheinina, 1983; Firth and PareH , 1988; Firth et al., 1990) and copropagating (Agrawal, 1990a; Luther and

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McKinstrie, 1990) optical beams, and some of the theoretical predictions were observed experimentally (Grynberg et al., 1988; Gauthier et al., 1990). In general, the instability growth rate of the coupled waves is larger, and a range of unstable wave numbers is broader, than those of either wave alone. Moreover, waves that are modulationally stable by themselves are often unstable in the other's presence, and this is true for both copropagating and counterpropagating waves. The transverse modulational instability of copropagating waves is always convective. An overview of transverse e!ects in other cases, when two or more beams propagate and interact in a nonlinear medium can be found in the paper by Abraham and Firth (1990). A similar kind of transverse instabilities, for both focusing and defocusing media, is expected for two-component (or vector) spatial optical solitons. The simplest problem of this kind is the analysis of the e!ects of vectorial perturbations on scalar solitons (see, e.g., Bondeson, 1979). More general analysis of the transverse instability should include the vectorial nature of solitons, and this is still an open direction of research. Recently, Musslimani et al. [1999; see also Skryabin and Firth (1999) for a more general case of incoherently coupled bright solitons] made the "rst step in this direction and found the long-wave-expansion results for the transverse instability of vector solitons, including a special case of dark-bright soliton pairs. An important class of the problems is associated with the models of interaction between high-frequency and low-frequency "elds, such as the Zakharov equation for the nonlinear Langmuir waves (e.g., Kuznetsov et al., 1986, and references therein), the model for multi-dimensional light bullets supported by nonresonant quadratic nonlinearity (Ablowitz et al., 1997), the IizukaKivshar model for the propagation of gap solitons in quadratically nonlinear gratings (Iizuka and Kivshar, 1999), etc. Some preliminary results (e.g., Kuznetsov et al., 1986; Hadzy ievski and Sx koricH , 1991) are not su$cient to make a general conclusion about the e!ect of the wave coupling of the multi-component soliton self-focusing. At last, Sa!man (1998) suggested another form of the coupled NLS equations describing the interaction of copropagating electromagnetic and matter waves, when the optical self-focusing generates dipole forces on the atoms (the original physical concept was suggested by much earlier Klimontovich and Luzgin, 1979). Numerical studies of the coupled equations revealed Sa!man (1998) the "lamentation due to modulational instability, and the formation of localized waves anticorrelated in the two "elds. Even there exists no analysis of the solitary waves and their stability in two spatial dimensions, the numerical results suggest a typical scenario of the soliton selffocusing with the formation of localized solutions stable in higher dimensions. 9.3. Transverse vs. longitudinal instabilities Several generalized evolution models, such as the saturable NLS equation (Mamaev et al., 1996a,b,c; Tikhonenko et al., 1996a,b) and the model for two- and three-wave mixing in quadratic nonlinear materials (Fuerst et al., 1997a; DeRossi et al., 1997a,b; Skryabin and Firth, 1998b) do not possess properties of the scaling invariance. In such nonscalingly invariant models the longitudinal instabilities of solitary waves are known to occur in a certain range of the soliton parameters. Then, the transverse self-focusing may compete with the longitudinal soliton instabilities, the latter may display rather di!erent types of long-term instability-induced soliton dynamics (Pelinovsky et al., 1996a,b). The analytical studies of such a competition between two instabilites and the soliton

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evolution is still an open problem, although a proper combination of the asymptotic methods developed for the longitudinal instabilities (see, e.g., Pelinovsky and Grimshaw, 1997) with the methods described here are expected to lead to the asymptotic analytical models which will cover the whole class of such complex phenomena. 9.4. Transverse instabilities of nonlinear guided waves An extensive theoretical literature is devoted to nonlinear stationary waves localized at optical interfaces and in layered dielectric media (see, e.g., Maradudin, 1983, and references therein). Such nonlinear guided waves are localized due to the fact that one or more of the dielectric layers shows either a positive or negative nonlinear optical response to an incident electromagnetic wave, and the trapped surface modes may appear above a certain critical power. These are the so-called nonlinear guided waves which can be regarded, in some approximation, as spatial solitons trapped by the interface, and therefore they are expected to be unstable to transverse spatial #uctuations in the dimension parallel to the coupling layers (e.g., Moloney, 1987). As a result of the development of such an instability, localized peaks located within the guided layer or near the interface appear, similar to the decay of plane solitons into two-dimensional solitons (see, e.g., Vysotina et al., 1990). Because a decay of guided waves is a!ected by an attractive potential created by the interface, the instability scenarious, especially in the case of the bending (snake-type) dynamics of the hyperbolic NLS equation, are expected to be di!erent from those for spatial solitons in homogeneous models. However, an analytical study of this type of problems is still absent. 9.5. Higher-order localized modes At last, we would like to mention a variety of problems involving the instability and decay of higher-order localized modes and ring-type soliton structures. Additionally to the radially symmetric (no nodes) localized solution for a self-trapped optical beam, the NLS equation in higher dimensions possesses a discrete number of the so-called higher-order localized modes which consist of bright or dark central spots surrounded by one or more rings. Such higher-order nonlinear localized modes have been "rst suggested as radially symmetric solutions of the (2#1) dimensional cubic NLS equation without (Yankauskas, 1966; Haus, 1966) or with an angular momentum (i.e., &bright vortex solitons'; see Kruglov and Vlasov, 1985; Kruglov et al., 1992), and then the similar solutions were found in higher dimensions and for other nonlinear models, including the saturable NLS equation (e.g., Edmundson, 1997; and reference therein) and the models of two- and three-wave mixing in a quadratic optical medium (Firth and Skryabin, 1997; Torres et al., 1998; Skryabin and Firth, 1998a). Stability of the higher-order localized modes is an important problem which was analyzed by many authors for di!erent models. No universal stability criterion is known for this kind of solutions, but it is clear that the instability of the rings (or &shells', in higher dimensions) is similar to the modulational instability of plane solitons. Indeed, it was shown that all types of higher-order localized modes are unstable with respect to azimuthally dependent perturbations and they decay into "laments, i.e. stable bright solitons of the lowest order. The scenario of the decay depends on the angular momentum of the modes. The modes with a bright central spot have no angular momentum, and the rings break up into "laments that move radially to the initial ring, the similar

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e!ects were observed experimentally for the transverse modulational instability of broad beams (Campillo et al., 1973,1974). The number of the "laments is de"ned by the maximum of the instability growth rate (Feit and Fleck, 1988; Soto-Crespo et al., 1991). The decay of the modes with a dark central spot is a!ected by the angular momentum (Tikhonenko et al., 1995,1996b; Firth and Skryabin, 1997; Petrov et al., 1998). Recently, Skryabin and Firth (1998a) have analyzed stability and dynamics of higher-order localized modes in the problem of two-wave mixing in a di!ractive bulk medium and found another scenario of the decay of these modes. For su$cient negative values of the phase mismatch between the harmonics, rings were found to coalesce with the central peak forming a single oscillating "lament. For positive phase mismatch, the standard break-up into "laments was observed. This coalescence scenario is not observed in the saturable NLS models. For a quadratic nonlineaar material [KTP crystal], the "rst experimental demonstration of the azimuthal selfbreaking of intense beams with a vortex phase dislocation into optical spatial solitons was recently reported by Petrov et al. (1998). Instability of the higher-order localized modes (&light bullets') was investigated numerically in the framework of the three-dimensional saturable NLS equation (Edmundson, 1997). In contrast with the two-dimensional case (e.g., Soto-Crespo et al., 1991), rather than a single structure emerging independent of the weak perturbation, in the three-dimensional case &the shells' of the higher-order modes decay in a miriad of complicated intermediate patterns attributed to a degeneracy in the number of maximally unstable eigenmodes. 9.6. Ring-like models and ring solitons A number of interesting physical e!ects is associated with a ring geometry. Let us bend a plane soliton to form a ring of the length ¸. Then, if the instability band of a plane soliton is characterized by the maximum wave number q , we expect that this instability will be suppressed in the ring

 geometry provided the condition ¸(2p/q (or, for the ring radius R, q R(1) holds. This



 simple observation explains why for the so-called ring solitary waves, the transverse instability is either suppressed or completely eliminated. This is true for many types of ring solitons, including bright (Lomdahl et al., 1980; Afanasjev, 1995), and dark (Kivshar and Yang, 1994) ring solitons of the (2#1) dimensional NLS equation, and the solitons of the so-called cylindrical KdV equation (e.g., Maxon and Viecelli, 1974; Ko and Kuehl, 1979; Stepanyants, 1981). However, the ring solitons do not exist as stationary localized states, they either expand or contract due to an e!ective surface tension introduced by the bending into a ring. If the condition q R(1 is not ful"eld, the ring soliton is

 expected to decay into a number of "laments, similar to the decay of the higher-order localized modes discussed above. Recently, it has been demonstrated numerically (Soljacy icH et al., 1998) that a ring of two-dimensional stable bright solitons in the form of the so-called `necklacea beam possesses an unique robustness due to the stabilizing interaction forces between the solitons in the ring. In particular, a necklace beam expands much slower that a quasi-plane soliton in the ring geometry. 9.7. Instabilities in higher dimensions Many interesting problems related to the soliton self-focusing and symmetry-breaking instabilities appear in higher dimensions. We did not touch this topic at all in the present review paper,

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however, it is worth mentioning, as the most interesting problem of this kind, the transverse instability of a vortex tube subjected to the action of higher dimensions or dispersion. This problem was "rst discussed more than 10 yr ago by Jones et al. (Jones et al., 1986) and the most recent analysis has been presented by Kuznetsov and Rasmussen (1995).

Acknowledgements We are indebted to M.J. Ablowitz, A.V. Buryak, K.A. Gorshkov, A.M. Kamchatnov, V.I. Shrira, Yu.A. Stepanyants, M. Segev and S. Trillo for useful discussions and collaboration. We also thank L. BergeH , G. Fibich, E.A. Kuznetsov, A.A. Nepomnyashchy, J.J. Rasmussen, M. Sa!man, and S. Turitsyn for critical reading of di!erent sections of this manuscript, their valuable comments and additional references. The work of Yuri Kivshar has been partially supported by the Australian Photonics Cooperative Research Centre.

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THERMODYNAMICS AND EXCITATIONS OF THE ONE-DIMENSIONAL HUBBARD MODEL

T. DEGUCHI , F.H.L. ESSLER
, F. GOG HMANN , A. KLUG MPER, V.E. KOREPIN , K. KUSAKABE Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11794-3840, USA
Department of Physics, Theoretical Physics, Oxford University, 1 Keble Road, Oxford OX1 NP, UK Institut f u( r Theoretische Physik, Universita( t zu Ko( ln, Zu( lpicher Str. 77, 50937 Ko( ln, Germany Graduate School of Science and Technology, Niigata University, Ikarashi, Niigata 950-2181, Japan

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

Physics Reports 331(2000) 197}281

Thermodynamics and excitations of the one-dimensional Hubbard model T. Deguchi , F.H.L. Essler
, F. GoK hmann , A. KluK mper, V.E. Korepin *, K. Kusakabe Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11794-3840, USA
Department of Physics, Theoretical Physics, Oxford University, 1 Keble Road, Oxford OX1 NP, UK Institut fu( r Theoretische Physik, Universita( t zu Ko( ln, Zu( lpicher Str. 77, 50937 Ko( ln, Germany Graduate School of Science and Technology, Niigata University, Ikarashi, Niigata 950-2181, Japan Received October 1999; editor: D.K. Campbell Contents 1. Introduction 2. Bethe ansatz for the Hubbard model 2.1. Eigenfunctions and eigenvalues 2.2. SO(4) symmetry 2.3. Discrete Takahashi equations 2.4. Completeness of the Bethe ansatz 3. Lieb}Wu equations for a single down spin (I) } graphical solution 3.1. All charge momenta k real H 3.2. k}K two string 3.3. Summary 4. Lieb}Wu equations for a single down spin (II) } self-consistent solution 4.1. Zeroth order and the discrete Takahashi equations 4.2. First-order corrections 4.3. Summary 5. Lieb}Wu equations for a single down spin (III) } numerical solution 5.1. Numerical method

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5.2. Numerical solution for two electrons 5.3. Numerical solution for three electrons 5.4. Summary 6. Thermodynamics in the Yang}Yang approach and excitation spectrum in the in"nite volume 6.1. Takahashi's thermodynamic equations 6.2. Thermodynamics 6.3. Zero-temperature limit 6.4. Ground state for a less than half-"lled band 6.5. Excitations for a less than half-"lled band 6.6. Excitations in the half-"lled band 6.7. Relation to the root-density formalism 6.8. Summary 7. Thermodynamics in the quantum transfer matrix approach 7.1. The classical counterpart 7.2. Diagonalization of the quantum transfer matrix 7.3. Non-linear integral equations 7.4. Analytical solutions of the integral equations

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* Corresponding author. E-mail addresses: [email protected] (F. GoK hmann), [email protected] (V.E. Korepin).  Present address: Department of Physics, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan. 0370-1573/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 1 0 - 7

T. Deguchi et al. / Physics Reports 331 (2000) 197}281 7.5. Numerical results 7.6. Equivalence of string based thermodynamics with QTM approach 8. Conclusions Acknowledgements Appendix A. Derivation of the wave function A.1. General setting A.2. Two electrons

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A.3. Three electrons A.4. N electrons Appendix B. Inhomogeneous Heisenberg model B.1. Algebraic Bethe ansatz B.2. Explicit expressions for the eigenstates Appendix C. The spectrum of three electrons with one-down spin for ¸"6 References

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Abstract We review fundamental issues arising in the exact solution of the one-dimensional Hubbard model. We perform a careful analysis of the Lieb}Wu equations, paying particular attention to the so-called &string solutions'. Two kinds of string solutions occur: K strings, related to spin degrees of freedom and k}K strings, describing spinless bound states of electrons. Whereas K strings were thoroughly studied in the literature, less is known about k}K strings. We carry out a thorough analytical and numerical analysis of k}K strings. We further review two di!erent approaches to the thermodynamics of the Hubbard model, the Yang}Yang approach and the quantum transfer matrix approach, respectively. The Yang}Yang approach is based on strings, the quantum transfer matrix approach is not. We compare the results of both methods and show that they agree. Finally, we obtain the dispersion curves of all elementary excitations at zero magnetic "eld for the less than half-"lled band by considering the zero-temperature limit of the Yang}Yang approach.  2000 Elsevier Science B.V. All rights reserved. PACS: 71.10.Fd; 71.10.Pm; 71.27.#a Keywords: Hubbard model; Strongly correlated electrons; Thermodynamics; Excitations

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1. Introduction The Hubbard model was introduced as a simple e!ective model for the study of correlation e!ects of d-electrons in transition metals [1,2] (see also [3]). It is believed to provide a qualitative description of the magnetic properties of these materials and of the Mott metal}insulator transition [4]. Despite of its appealing conceptual simplicity rigorous results for the Hubbard model are rare. The dimension of the underlying lattice is a crucial parameter. The most explicit results have been obtained for the one-dimensional case. This is the only case where the spectrum and a complete set of eigenfunctions are known [5]. Rigorous results for the Hubbard model in more than one dimension are so far restricted to rather general theorems, for instance about the spin of the ground state. Two important examples of such kind of theorems are due to Nagaoka [6] and Lieb [7], respectively. Nagaoka's theorem states, that creating a single hole in a half-"lled connected lattice for in"nitely repulsive interaction renders the ground state to be ferromagnetic. Lieb's theorem is valid for arbitrary "nite repulsion. It states that on a bipartite lattice at half-"lling the ground state has spin S"""B"!"A"", where "B" ("A") is the number of sites in the B (A) sublattice. For reviews of rigorous  results about the Hubbard model in more than one dimension the reader is referred to [8,9]. It should be also noted that certain simpli"cations occur in the limit of in"nite lattice dimension [10}12]. In this article we shall concentrate on the one-dimensional Hubbard model. This model has fascinated theoretical physicists for a long time and is becoming progressively more important for experiments, with the synthesis of a growing number of quasi-one-dimensional materials and the re"nement of experimental techniques [13}20]. In one dimension quantum #uctuations play a more important role than in higher dimensions. This leads to new physical phenomena such as spin}charge separation, which cannot be understood within a perturbative approach. There exists a vast literature on the Hubbard model. A good selection of original research articles, covering the most signi"cant results for the period from 1963 to about 1990, is o!ered by the reprint volume [21]. For an overview over more recent results the reader is referred to [22,23]. The reprint volume [24] provides an introduction into the physics of the one-dimensional Hubbard model. It includes a rather exhaustive list of references until about 1992. Most of the literature on the one-dimensional Hubbard model is based on the seminal 1968 paper [5] by Lieb and Wu. In this paper the model was solved by means of the nested Bethe ansatz [25,26], i.e. the problem of diagonalizing the Hamiltonian was reduced to solving a set of coupled non-linear equations known as the Bethe ansatz or Lieb}Wu equations. Lieb and Wu calculated the ground state energy of the system. They showed that the model at half-"lling is an insulator for arbitrary repulsive interaction ;, or, in other words, that the half-"lled model undergoes a Mott transition at a critical strength of interaction ;"0. In 1972 Takahashi proposed a classi"cation of the solutions to the Lieb}Wu equations [27], which is commonly referred to as &Takahashi's string hypothesis' (or simply &string hypothesis'). Analogous classi"cations are used in all models solvable by the Bethe ansatz method (see e.g. [28,29] for the case of the Heisenberg model and [30] for the case of the Anderson model, which bears certain similarities with the Hubbard model). The string hypothesis is the basis of many subsequent publications. In the paper [27], Takahashi used the string hypothesis to obtain a set of non-linear integral equations that determines the thermodynamics of the Hubbard model. By solving these equations in appropriate limiting cases, he was able to calculate the low-temperature speci"c heat [31].

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At the beginning of the 1980s Woynarovich resumed the study of the excitation spectrum of the Hubbard model [32}35] which was started 10 years earlier [36,37]. He gave a detailed analysis of the charge excitations at half-"lling [32] and was the "rst to study gapped, spin-singlet charge excitations at half-"lling. These involve the "rst examples of excitations which in the sequel will be called k}K strings [33]. In his article [32] Woynarovich presented an explicit form of the Bethe ansatz wave function (see Eqs. (2)}(6) below). Since the publication of the reprint volume [24] there have been several interesting developments the most important of which we shall discuss below. Two of the present authors [38,39] showed that the excitation spectrum at half-"lling in the absence of a magnetic "eld is given by the scattering states of only four elementary excitations. Two of these excitations carry charge but no spin, the other two carry spin but no charge. These elementary excitations are called holon, antiholon and spinon with spin-up or -down, respectively. They form the fundamental representation of SO(4). In the same articles [38,39] the S-matrix of the four quasi-particles was obtained. Thus, there is now a complete and satisfactory picture on the level of elementary excitations of spin}charge separation in the one-dimensional Hubbard model at half-"lling. Spin and charge degrees of freedom can be excited separately, but the corresponding quasiparticles do interact, albeit weakly. The interaction is seen in the non-triviality of the S-matrix. Spin-charge separation is one of the most interesting properties of the one-dimensional Hubbard model. Recently, experimental evidence was found for the existence of spin}charge separation in quasi-one-dimensional materials [13,14]. Another interesting recent development is the calculation of the bulk thermodynamic properties of the Hubbard model within the quantum transfer matrix approach [40}44]. In contrast to the traditional approach [27,45,46], the quantum transfer matrix approach leads to a "nite number of non-linear integral equations [44,47,48] that determine the Gibbs free energy. This enables high-precision numerical calculation of thermodynamic quantities such as the charge and spin susceptibilities over the entire range of doping, temperature and magnetic "eld. It further opens the interesting perspective of calculating the correlation length at arbitrary "nite temperature. It was shown in the papers [49}51] how to use a pseudo-particle approach in order to obtain transport properties (optical conductivity) of the one-dimensional Hubbard model. Other quantities of direct relevance for experimental observation are the spectral functions. The problem of their calculation in the case of in"nitely strong repulsion was addressed in [52,53]. There was also progress in the understanding of the algebraic structure of the Hubbard model. Shiroishi and Wadati showed [54], that the R-matrix, which was constructed earlier by Shastry [55,56] and by Olmedilla et al. [57], and which underlies the integrability of the Hubbard model, satis"es the Yang}Baxter equation. Martins and Ramos [58,59] were able to construct a variant of the algebraic Bethe ansatz for the Hubbard model. They obtained the eigenvalue of the transfer matrix of the two-dimensional statistical covering model (see also [60]). This result was later used in the quantum transfer matrix approach to the thermodynamics [44]. Another interesting algebraic result was the discovery of a quantum group symmetry of the Hubbard model on the in"nite line. The Hamiltonian is invariant under the direct sum of two Y(su(2)) Yangians [61]. The relation of these Yangians to Shastry's R-matrix was clari"ed in [62,63], where it was also shown that the eigenstates of the Hubbard Hamiltonian on the in"nite interval, at zero-density transform-like irreducible representations of one of the Yangians. For further articles pushing forward the understanding of the algebraic structure of the Hubbard model see [64}68].

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The purpose of this article is to give a pedagogical introduction to the Bethe ansatz solution of the one-dimensional Hubbard model and at the same time to "ll some gaps in the previous literature. We present a detailed account of the Bethe ansatz solution for periodic boundary conditions and of the thermodynamics of the model. There are two approaches to the thermodynamics. The approach of Takahashi [27,45,46] relies on a string hypothesis for the Hubbard model and is a natural generalization of Yang and Yang's thermodynamic Bethe ansatz for the delta interacting Bose gas [69]. The second approach [44] is built on a lattice path integral formulation of the partition function. We compare both approaches and discuss their speci"c advantages. Special attention is given to an aspect which, although fundamental, was largely ignored in the previous literature, namely k}K strings. k}K strings are spin-singlet bound states of electrons. The Bethe ansatz for the one-dimensional Hubbard model [5] gives the eigenfunctions and eigenvalues of the Hubbard Hamiltonian parametrized by two sets of quantum numbers +k , H and +j ,, which are solutions of the Lieb}Wu equations (see formulae (7) and (8) below). The k and J H j are called charge momenta and spin rapidities, respectively. The Lieb}Wu equations have "nite J and in"nite solutions k , j . They should be considered separately, because they have di!erent H J occupation numbers. Every "nite solution k (or j ) can be occupied only once. If two "nite k (or H J H j ) coincide, the wave function vanishes (see formulae (11) and (12) and below). By contrast, in"nite J k (or j ) can be generally occupied more than once. Later we shall explain that the multiplicities of H J occupation of the in"nite k's and j's are given by the dimensions of the representations of corresponding su(2) symmetry algebras. In order to study this carefully the ®ular' Bethe ansatz was de"ned in [70]. &Regular' means that all k's and j's are "nite. They may be real or complex. Takahashi's string hypothesis [27] is a statement about the structure of the regular solutions of the Lieb}Wu equations in the thermodynamic limit. Except for the real solutions (all k and all H j real) there are solutions involving complex k and j . The complex momenta and rapidities occur J H J in two kinds of con"gurations, which are symmetric with respect to the real axis. These con"gurations are called strings. There are K strings involving only spin rapidities and k}K strings, which involve charge momenta as well. The K strings can be interpreted as bound states of magnons, whereas the k}K strings describe spin-singlet bound states of electrons. The K strings in the Hubbard model are similar to the K strings in the isotropic Heisenberg spin chain, which have been extensively studied in the literature [28,71}73]. Much less attention has been given to the k}K strings, which are peculiar to the Hubbard model. They play an important role at half-"lling, where the k}K two string enters the calculation of the phase shift in the holon}holon scattering [38,39]. Holons are the lowest lying charge excitations of the Hubbard model. At half-"lling they have a gap. Below half-"lling they are gapless, however, whereas all k}K strings lead to gapped excitations in the thermodynamic limit (see Section 6). Hence, k}K strings below half-"lling do not contribute to the low-energy properties of the model. What is their physical signi"cance then? They do contribute to the high-temperature thermodynamic properties of the Hubbard model. This is, in fact, the context in which they were "rst introduced [27]. On the other hand, k}K strings are interesting as a curious kind of excitations, which does not exist in more simple integrable models. According to the string hypothesis the k}K strings approach certain ideal con"gurations as the number of lattice sites becomes large. We call these con"gurations &ideal k}K strings'. They are characterized by m complex j's and 2m complex k's. The j's involved in an ideal k}K string have

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common real part K, and their imaginary parts are (m!1)i;/4,(m!3)i;/4,2,!(m!1)i;/4. In the repulsive case (;'0) the k's are given as k"p!arcsin(K#mi;/4) , k"arcsin(K#(m!2)i;/4) , k"p!k , $ kK\"arcsin(K!(m!2)i;/4) , kK\"p!kK\ , kK"p!arcsin(K!mi;/4) . This means that Re sin(k?)"K. Hence, the sin(k?)'s and the j's form a string in the complex plane. In the thermodynamic limit, when the strings become ideal, the variables describing their width can be eliminated from the Lieb}Wu equations. We call the resulting set of equations discrete Takahashi equations. Some reformulation of the string hypothesis may be necessary before it will be possible to achieve a rigorous mathematical proof. Yet, the string hypothesis has passed many tests, and there is no doubt by now that it describes the physics of the Hubbard model correctly. We summarize our understanding of the issue and present several important tests and consequences of the string hypothesis. We shall mostly concentrate on the Hubbard model below half-"lling, since the case of half-"lling was treated elsewhere [38,39]. Let us outline the plan of this article. In Section 2 we summarize known basic results about the Hubbard model. This section contains a review of the Bethe ansatz solution of Lieb and Wu [5]. We present the wave function in the form given by Woynarovich [32] and discuss its discrete symmetries, namely the symmetries under permutations of electrons and quantum numbers and the particle}hole and spin-reversal symmetries. This leads to the notion of regular Bethe ansatz states. We proceed with explaining the SO(4) symmetry [66,67,74}77], which is characteristic of the Hubbard model. Then we introduce the discrete Takahashi equations. SO(4) symmetry and discrete Takahashi equations are the prerequisites for the proof of completeness of the Bethe ansatz [78], which is reviewed at the end of the section. Section 3 comprises a rigorous analytical study of the Lieb}Wu equations for one down spin. For the sake of pedagogical clarity we mostly focus on the cases of two and three electrons. This is our "rst test of the string hypothesis. The result is positive: k}K strings do exist as solutions of the Lieb}Wu equations. They become ideal in the thermodynamic limit. Furthermore, the counting of solutions implied by the discrete Takahashi equations agrees in this case with the counting obtained directly from the Lieb}Wu equations. Section 4 is devoted to the self-consistent solution of the Lieb}Wu equations for three electrons and one down spin. Self-consistency arguments underly the derivation of the discrete Takahashi equations. In the simple case considered in this section we can be more explicit. We calculate the deviations of the rapidities and momenta (solutions of the Lieb}Wu equations) from their ideal positions (solutions of the discrete Takahashi equations). It turns out that these deviations vanish exponentially in the thermodynamic limit. This is our second positive test of the string hypothesis.

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In Section 5 we complement the analytical considerations of the previous sections with numerical results. Numerical data based on the Lieb}Wu equations are compared with data, which were obtained independently of the Bethe ansatz by direct numerical diagonalization of the Hamiltonian. We "nd perfect agreement between the two numerical methods. The energy levels obtained by the two methods agree within a numerical error of O(10\). Our numerical study con"rms the completeness of the Bethe ansatz and the correctness of the counting of the solutions implied by the string hypothesis. This is our third successful test of the string hypothesis. In Section 6 we review Takahashi's approach to the thermodynamics of the Hubbard model [27]. We show that the dressed energies of all k}K strings (bound states of electrons) follow from Takahashi's integral equations (thermodynamic Bethe ansatz equations) in the zero-temperature limit. We can actually do better. Starting from the thermodynamic Bethe ansatz equations and passing to the zero-temperature limit we obtain a complete classi"cation of all elementary excitations at zero magnetic "eld, below half-"lling. Takahashi derived his equations in order to calculate thermodynamic quantities such as the speci"c heat or charge and spin susceptibilities for the Hubbard model [27,31,79,80]. Nowadays there is an independent method to calculate these quantities, which does not rely on strings. It is called the quantum transfer matrix method [40}42,44,81]. However, the full strength of this approach has been developed only recently in [44] where a "nite set of non-linear integral equations was derived which yields reliable results for half-"lling and "nite doping. In Section 7 we compare the results of both methods and "nd that they agree well. The string hypothesis also passes this signi"cant test. If the string hypothesis would miss one of the elementary excitations, it should be visible in the thermodynamics. Section 8 contains a brief conclusion and a list of interesting open problems. In Appendix A we present a derivation of the Bethe ansatz wave function for the Hubbard model. It can be seen from the derivation that charge momenta and spin rapidities do not have to be real. To every solution of the Lieb}Wu equations there corresponds a well-de"ned periodic wave function. This is particularly the case for the k}K string solutions. In Appendix B we derive the algebraic Bethe ansatz solution of the inhomogeneous isotropic Heisenberg model, which is needed to construct the Bethe ansatz wave function in Appendix A. Appendix C contains the tables of our numerical data for three electrons and one down spin. 2. Bethe ansatz for the Hubbard model 2.1. Eigenfunctions and eigenvalues The Hamiltonian of the one-dimensional Hubbard model on a periodic ¸-site chain may be written as * * (1) H"! (c> c #c> c )#; (n !)(n !) , HN H>N H>N HN Ht  Hs  H Nts H where c> and c are creation and annihilation operators of electrons in Wannier states, and HN HN periodicity is guaranteed by setting c "c . n "c> c is the particle number operator for *>N N HN HN HN electrons of spin p at site j, ; is the coupling constant. The eigenvalue problem for the Hubbard

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Hamiltonian (1) was solved by Lieb and Wu [5] using the nested Bethe ansatz [25]. The Hubbard Hamiltonian conserves the number of electrons N and the number of down spins M. The corresponding SchroK dinger equation can therefore be solved for "xed N and M. Since the Hamiltonian is invariant under particle}hole transformations and under reversal of spins [5], we may set 2M4N4¸. We shall denote the positions and spins of the electrons by x and p , H H respectively. The Bethe ansatz eigenfunctions of the Hubbard Hamiltonian (1) depend on the relative ordering of the x . There are N! possible orderings of the coordinates of N electrons. Any H ordering may be related to a permutation Q of the numbers 1,2, N through the inequality 14x

4x 424x 4¸ . (2) / / /, This inequality divides the con"guration space of N electrons into N! sectors, which can be labeled by the permutations Q. The Bethe ansatz eigenfunctions of the Hubbard Hamiltonian (1) in the sector Q are given as






, . (3) t(x ,2, x ;p ,2,p )" sign(PQ)u (p ,2,p )exp i k x . / /, .H /H  ,  , H .Z1, Here the P-summation extends over all permutations of the numbers 1,2, N. These permutations form the symmetric group S . The function sign(Q) is the sign function on the symmetric group, , which is !1 for odd permutations and #1 for even permutations. The spin-dependent amplitudes u (p ,2,p ) can be found in Woynarovich's paper [32]. They are of the form of the Bethe . / /, ansatz wave functions of an inhomogeneous XXX spin chain, + (4) u (p ,2,p )" A(j ,2,j ) “ F (j ;y ) . . / /, p p+ . pJ J J pZ1+ Here F (j;y) is de"ned as . W\ j!sin k !i;/4 1 .H “ F (j;y)" (5) . j!sin k #i;/4 j!sin k #i;/4 .H .W H and the amplitudes A(j ,2,j ) are given by  + j !j !i;/2 K L A(j ,2,j )" “ . (6)  + j !j K L XKLX+ y in the above equations denotes the position of the jth down spin in the sequence p ,2,p . H / /, The y's are thus &coordinates of down spins on electrons'. Below we shall illustrate the notation through an explicit example. The wave functions (3) are characterized by two sets of quantum numbers +k , and +j ,. These H J quantum numbers may be generally complex. The k and j are called charge momenta and spin H J rapidities, respectively. The charge momenta and spin rapidities satisfy the Lieb}Wu equations + j !sin k !i;/4 H , j"1,2, N , e IH *" “ J j !sin k #i;/4 H J J + j !j !i;/2 , j !sin k !i;/4 H K "“ J , l"1,2, M . “ J j !sin k #i;/4 K j !j #i;/2 H J K H J K$J

(7) (8)

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Derivations of the wave function (3) and the Lieb}Wu equations (7) and (8) are presented in Appendices A and B. The wave functions (3) are joint eigenfunctions of the Hubbard Hamiltonian (1) and the momentum operator with eigenvalues






, , ; (9) E"!2 cos k # (¸!2N), P" k mod 2p . H H 4 H H The &coordinates of down spins' y which enter (4) depend on (p ,2,p ) and on (x ,2, x ). The H  ,  , following example should help to understand the notation. Let ¸"12, N"5, M"2, and let, for example, (x ,2, x )"(7,3,5,1,8), (p ,2,p )"(!!! ). Then x 4x 4x 4x 4x , i.e.          Q"(4,2,3,1,5). It follows that (x ,2, x )"(1,3,5,7,8) and (p ,2,p )"( !!! ). Thus y "1, / / / /  y "5.  Whenever it will be necessary, we shall indicate the dependence of the wave functions (3) on the charge momenta and spin rapidities by subscripts, t"t  2 ,  2 + . Let us consider the I  I _H  H symmetries of the eigenfunctions under permutations, t(x ,2, x ;p ,2,p )"sign(P)t(x ,2, x ;p ,2,p ), P3S , (10) . ., . .,  ,  , , (11) t . 2 .,  2 + "sign(P)t  2 ,  2 + , P3S , I  I _H  H , I  I _H  H (12) t  2 , . 2 .+ "t  2 ,  2 + , P3S . I  I _H  H + I  I _H  H Eq. (10) means that the eigenfunctions respect the Pauli principle. Eqs. (11) and (12) describe their properties with respect to permutations of the quantum numbers. They are totally antisymmetric with respect to interchange of the charge momenta k , and they are totally symmetric with respect H to interchange of the spin rapidities j . Hence, in order to "nd all Bethe ansatz wave functions we J have to solve the Lieb}Wu equations (7) and (8) modulo permutations of the sets +k , and +j ,. The H J k 's have to be mutually distinct, since otherwise the wave function vanishes due to (11). In fact, the H j 's have to be mutually distinct, too. This is called the &Pauli principle for interacting Bosons' J (see [82]). We would like to emphasize that there are no further restrictions on the solutions of (7) and (8). In particular, the spin and charge rapidities do not have to be real. Bethe ansatz states on a "nite lattice of length ¸ that have "nite momenta k and rapidities j , H J a non-negative value of the total spin (N!2M50), and a total number of electrons not larger than the length of the lattice (N4¸) are called regular (cf. [70, p. 562]). There exist two discrete symmetries of the model which can be used to obtain additional eigenstates from the regular ones [5]. The Hamiltonian is invariant under exchange of up and down spins. This symmetry allows for obtaining eigenstates with negative value N!2M of the total spin from eigenstates with positive value of the total spin. This symmetry does not a!ect the number of electrons. Thus, its action on regular states does not lead above half-"lling. States above half-"lling (N'¸) can be obtained by employing the transformation c P(!1)Hc> , HN HN c> P(!1)Hc , p"!, , which leaves Hamiltonian (1) invariant, but maps the empty Fock state "02 HN HN to the completely "lled Fock state "! 2.  For a proper de"nition of the momentum operator see Appendix B of [66].

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2.2. SO(4) symmetry The Hubbard Hamiltonian (1) is invariant under rotations in spin space. The corresponding su(2) Lie algebra is generated by the operators * 1 * * f" c> c , f -" c> c , fX" (n !n ) . Hs Ht Hs Ht Ht Hs 2 H H H [f,f -]"!2fX, [f,fX]"f, [f -,fX]"!f - .

(13)

For lattices of even length ¸ there is another representation of su(2), which commutes with the Hubbard Hamiltonian [74}76]. This representation generates the so-called g-pairing symmetry, * 1 * * g" (!1)Hc c , g -" (!1)Hc> c> , gX" (n #n )!¸ . Hs Ht Ht Hs Hs Ht  2 H H H [g,g- ]"!2gX, [g,gX]"g, [g-,gX]"!g- .

(14)

The generators of both algebras commute with one another. They combine into a representation of su(2)su(2). The g-pairing symmetry connects sectors of the Hilbert space with di!erent numbers of electrons. The operator g-, for instance, creates a local pair of electrons of opposite spin and momentum p. Hence, in order to consider the action of the g-symmetry on eigenstates we write them in second quantized form. * "k ,2, k ;j ,2,j 2" t  2 ,  2 + (x ,2, x ;p ,2,p )c>  2c>, , "02 , ,  , V N V N  ,  + I  I _H  H  2 V  V,  (15) where p "2"p " and p "2"p "!. It is easily seen that  + +> , (fX#gX)"k ,2, k ;j ,2,j 2"(M!¸/2)"k ,2, k ;j ,2,j 2 . (16)  ,  +  ,  + Here M!¸/2 is integer, since ¸ is even. Therefore the symmetry group generated by representations (13) and (14) is SO(4) rather than SU(2);SU(2) [77]. It was shown in [70] that the regular Bethe ansatz states are lowest weight vectors of both su(2) symmetries (13) and (14), f"k ,2, k ;j ,2,j 2"0, g"k ,2, k ;j ,2,j 2"0 . (17)  ,  +  ,  + This is an important theorem. It was the prerequsite for the proof of completeness (see Section D) of the Bethe ansatz for the Hubbard model in [78]. The proof of (17) is direct but lengthy [70]. f and g are applied to states (15), and the Lieb}Wu equations (7) and (8) are used to reduce the resulting expressions to zero. We would like to emphasize that the proof of (17) is not restricted to real solutions of the Lieb}Wu equations. It goes through for all solutions corresponding to regular Bethe ansatz states including the strings. Since the two su(2) symmetries (13), (14) leave the Hubbard Hamiltonian (1) invariant, additional eigenstates which do not belong to the regular Bethe ansatz can be obtained by applying fR and

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gR to regular Bethe ansatz eigenstates. Since fX"k ,2, k ;j ,2,j 2"(M!N/2)"k ,2, k ;j ,2,j 2 , (18)  ,  +  ,  + gX"k ,2, k ;j ,2,j 2"(N!¸)"k ,2, k ;j ,2,j 2 , (19)  ,  +   ,  + a state "k ,2, k ;j ,2,j 2 has spin (N!2M) and g-spin (¸!N). The dimension of the  ,  +   corresponding multiplet is thus given by dim "(N!2M#1)(¸!N#1) . +, The states in this multiplet are of the form

(20)

"k ,2, k ;j ,2,j ;a;b2"(f - )?(g- )@"k ,2, k ;j ,2,j 2 , (21)  ,  +  ,  + where a"0,2, N!2M and b"0,2, ¸!N. Note that states of form (21) can be obtained from regular Bethe ansatz states with NI 5N, M I 5M by formally setting some of the charge momenta and spin rapidities equal to in"nity [29,32,83]. 2.3. Discrete Takahashi equations Let us now formulate Takahashi's string hypothesis [27] more precisely: all regular solutions +k ,, +j , of the Lieb}Wu equations (7) and (8) consist of three di!erent kinds of con"gurations. H J (i) A single real momentum k . H (ii) m j's combining into a K string. This includes the case m"1, which is just a single K . ? (iii) 2m k's and m j's combining into a k}K string. For large lattices (¸<1) and a large number of electrons (N<1), almost all strings are close to ideal, i.e. the imaginary parts of the k's and j's are almost equally spaced. For ideal K strings of length m the rapidities involved are KKH"KK#(m!2j#1)i;/4 . (22) ? ? Here a enumerates the strings of the same length m, and j"1,2, m counts the j's involved in the ath K string of length m. KK is the real center of the string. ? The k's and the j's involved in an ideal k}K string are (for ;'0) k"p!arcsin(K K#mi;/4) , ? ? k"arcsin(K K#(m!2)i;/4) , ? ? k"p!k , ? ? $ kK\"arcsin(K K!(m!2)i;/4) , ? ? kK\"p!kK\ , ? ? kK"p!arcsin(K K!mi;/4) , ? ? K KH"K K#(m!2j#1)i;/4 . ? ?

(23)

(24)

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Again m denotes the &length' of the string, a enumerates strings of length m, and j counts the j's involved in a given string. K K is the real center of the k}K string. The branch of arcsin(x) in (23) is ? "xed as !p/24Re(arcsin(x))4p/2. The string hypothesis assumes that almost all solutions of the Lieb}Wu equations (7) and (8) are approximately given by (22)}(24) with exponentially small corrections of order O(exp(!d¸)), where d is real and positive and depends on the speci"c string under consideration. Using the string hypothesis inside the Lieb}Wu equations (7) and (8) and taking logarithms afterwards, we arrive at the following form of Bethe ansatz equations for strings, which we call discrete Takahashi equations











sin k !KL sin k !K L  +L  +L H ? ! h H ? , (25) k ¸"2p I ! h H H n;/4 n;/4 L ? L ? ,\+Y KL !sin k KL !KK  +K H "2p JL # H ? @ , h ? (26) ? LK n;/4 ;/4 H K @ ¸[arcsin(K L#ni;/4)#arcsin(K L!ni;/4)] ? ? ,\+Y K L!sin k K L!K K  +K H # H ? @ "2p J L# h ? . (27) ? LK n;/4 ;/4 H K @ Here we assumed ¸ to be even. I , JL , and J L are integer or half-odd integer numbers, according to H ? ? the following prescriptions: I is integer (half-odd integer), if (M #M ) is even (odd); the H K K K JL are integer (half-odd integer), if N!M is odd (even); the J L are integer (half-odd integer), if ? L ? ¸!(N!M ) is odd (even). M and M are the numbers of K strings of length n, and k}K strings L L K of length m in a speci"c solution of system (25)}(27). M"  nM , is the total number of j's L L involved in k}K strings. The integer (half-odd integer) numbers in (25)}(27) have ranges





















!¸/2(I 4¸/2 , (28) H  1 (29) "JL "4 N!2M! t M !1 , LK K ? 2 K  1 (30) "J L"4 ¸!N#2M! t M !1 , LK K ? 2 K where t "2 min(m, n)!d . The functions h and H in (25)}(27) are de"ned as h(x)"2 KL KL LK arctan(x), and x x x h #2h #2#2h "n!m" "n!m"#2 n#m!2







H (x)" LK












 #h

2h






x n#m



if nOm ,







x x x x #2h #2#2h #h 2 4 2n!2 2n

if n"m .

(31)

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In terms of the parameters of the ideal strings total energy and momentum (9) are expressed as





,\+Y  +L (32) k ! (2 Re arcsin(K L#ni;/4)!(n#1)p) mod 2p , ? H H L ? ,\+Y  +L ; E"!2 cos(k )#4 Re(1!(K L#ni;/4)# (¸!2N) . (33) H ? 4 H L ? Eqs. (25)}(30) can be used to study all excitations of the Hubbard model in the thermodynamic limit. They are the basis for the derivation of Takahashi's integral equations [27], which determine the thermodynamics of the Hubbard model (see Sections 6 and 7). Applications of (25)}(30) are usually based on the following assumptions. P"

(i) Any set of non-repeating (half-odd) integers I , JL , J L subject to constraints (28)}(30) speci"es H ? ? one and only one solution +k ,, +KL ,, +K L, of Eqs. (25)}(27). H ? ? (ii) The solutions +k ,, +KL ,, +K L, of (25)}(27) speci"ed by a set of non-repeating (half-odd) H ? ? integers I , JL , J L subject to (28)}(30) are in one-to-one correspondence to solutions of the H ? ? Lieb}Wu equations (7) and (8). (iii) For large ¸ and N almost every solution +k ,, +j , of the Lieb}Wu equations (7) and (8) is H J exponentially close to the corresponding solution +k ,, +KL ,, +K L, of the discrete Takahashi H ? ? equations, which means that the strings contained in +k ,, +j , are well approximated by the H J ideal strings determined by +k ,, +KL ,, +K L,. H ? ? 2.4. Completeness of the Bethe ansatz The proof of completeness of the Bethe ansatz given in [78] is based on assumptions (i) and (ii) above. Similar assumptions were proved for other Bethe ansatz solvable models [82]. Note that assumption (ii) does not mean that the classi"cation of the solutions of the Lieb}Wu equations (7) and (8) into strings is actually given by (25)}(30). There may be a redistribution between di!erent kinds of strings. This phenomenon was observed in a number of Bethe ansatz solvable models and was carefully studied by examples [71,73,78] (see also Section 5.2). It turned out that the redistribution did in no case a!ect the total number of solutions of the Bethe ansatz equations. Using (i) and (ii) above, the proof of completeness reduces to a combinatorial problem based on (28)}(30) [78]. From (28) to (30) we read o! the numbers of allowed values of the (half-odd) integers I , JL , J L in a given con"guration +M ,, +M , of strings. These numbers are H ? ? L L (i) ¸ for a free k (not involved in a k}K string), H (ii) N!2M!  t M for a K string of length n, K LK K (iii) ¸!N#2M!  t M for a k}K string of length n. K LK K The total number of ways to select the I , JL , J L (recall that they are assumed to be non-repeating) H ? ? for a given con"guration +M ,, +M , is thus L L  ¸  N!2M! tLK MK “ n(+M ,,+M ,)" K L L N!2M L ML









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  ¸!N#2M! tLK MK ;“ . (34) K M L L Hence, the number of regular Bethe ansatz states for given numbers N of electrons and M of down spins is






n(+M ,,+M ,) , (35) L L +L ,++L , where the summation is over all con"gurations of strings which satisfy the constraints N!2M50 and M"  m(M #M ). Finally, the total number of states (21) in the SO(4) K K K extended Bethe ansatz is n



(M, N)"

+

n (¸)" n (M, N) dim   +, +, " n (M, N)(N!2M#1)(¸!N#1) , (36)  +, where the sum is over all M, N with 042M4N4¸. Sums (35) and (36) were calculated in [78]. It turns out that n (¸)"4* ,  which is the dimension of the Hilbert space of the Hubbard model on an ¸-site chain. Let us list again the essential steps that led to the above proof of completeness:

(37)

(i) Impose periodic boundary conditions. (ii) Take Woynarovich's wave function (3)}(6). (iii) De"ne the Bethe ansatz in the narrow sense of regular Bethe ansatz (see below (12)). This eliminates in"nite k's and j's whose multiplicities are not under control. (iv) Prove the lowest weight theorem (17). Then gluing back solutions with in"nite k's and j's is equivalent to considering multiplets (21). (v) The multiplicities of occupation of in"nite k's and j's are given by dimensions (20) of multiplets (21). (vi) Use Takahashi's integers (28)}(30) for counting of the regular Bethe ansatz states.

3. Lieb}Wu equations for a single down spin (I) } graphical solution In this section we study the Lieb}Wu equations (7) and (8) in the most simple non-trivial case, when there is only one down spin, M"1. For pedagogical clarity some emphasis will be on the most instructive cases N"2 and 3. These are the cases which we also studied numerically (cf. Section 5). Some of the analytical calculations, however, are presented for general N, simply because the general arguments are simple enough and enable treating the cases N"2 and 3 to some extent simultaneously. The Lieb}Wu equations for N"2 and M"1 were studied before in Appendix B of [78]. There the emphasis was on the redistribution phenomenon mentioned in Section 2.4. For ;"0 the

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Hubbard Hamiltonian turns into a free tight binding Hamiltonian, and there is no bound state of electrons (k}K string) left. It is therefore clear that bound states decay as the coupling becomes weaker. In Appendix B of [78] it was shown that each time a k}K string disappears from the spectrum at a certain critical value of the coupling ;'0, a new real solution emerges. Here we take a slightly di!erent point of view. We "x ; and study the solutions for large "nite ¸. It turns out that there is no redistribution phenomenon for the most simple k}K strings consisting of two complex conjugated k's and one real j as ¸PR. These strings always exist for large enough "nite ¸, and their number is in accordance with the counting implied by Takahashi's discrete equations (25)}(30). In this respect, the k}K strings of the Hubbard model are di!erent from the K strings in the XXX spin chain [73]. For M"1 the Lieb}Wu equations (7) and (8) read e IH *"(K!sin k !i;/4)/(K!sin k #i;/4), j"1,2, N , H H , K!sin k !i;/4 H "1 . “ K!sin k #i;/4 H H Eq. (39) can be replaced by the equation for the conservation of momentum, e I >2>I, *"1 .

(38) (39)

(40)

Eqs. (38) and (40) follow from (38) and (39) and vice versa. Let us take the logarithm of (40) and solve (38) for sin k !K. We obtain the equations H (k #2#k ) mod 2p"m2p/¸, m"0,2, ¸!1 , (41)  , sin k !K"(;/4)ctg(k ¸/2), j"1,2, N , (42) H H which are equivalent to (38) and (39) but more convenient for the further discussion. 3.1. All charge momenta k real H The equation sin q!K"(;/4)ctg(q¸/2)

(43)

is easily solved graphically for q as a function of K (see Fig. 1). It has at least ¸ branches of solutions ql (K) belonging to the interval 04q(2p. There is at least one branch with (l!1)2p/¸(q(l2p/¸. Yet, for p/24q43p/2 there may be more than one solution in the interval [(l!1)2p/¸,l2p/¸], if ; or ¸ is too small. We have the following uniqueness condition:




q¸ !1" min R sin q' max R (;/4)ctg O O 2 XOp XOp

" max !(;¸/8) sin\(q¸/2)"!;¸/8 , XOp which is equivalent to ¸'8/; .

(44)

(45)

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We call this condition Takahashi condition. In the following, we will assume the Takahashi condition to hold. Eq. (43) de"nes K as a function of q. We have dK/dq"cos q#(;¸/8)sin\(q¸/2) .

(46)

Then, using (45), dK/dq'0. Hence, all branches ql (K) of solutions of (43) are monotonically increasing, dql (K)/dK'0. We can summarize the properties of the solutions of Eq. (43) in the following lemma. Lemma 3.1. If ¸'8/;, Eq. (43) has exactly ¸ branches of solutions ql (K), which have the properties (i) (l!1)2p/¸4ql (K)4l2p/¸ , (ii) dql (K)/dK'0 , lim ql (K)"(l!1)2p/¸, lim ql (K)"l2p/¸ . (47) K \  Property (iii) can be seen from Fig. 1. Lemma 3.1 is su$cient to characterize and count the real solutions of (38) and (39) (or equivalently (41) and (42)). We are seeking for solutions of (38) and (39) modulo permutations, where all k are mutually distinct. Choose N branches ql (K)(2(ql, (K) of solutions of (43), H and de"ne (iii)

K

Q(K)"ql (K)#2#ql, (K) .

(48)

Then k "qlH (K), j"1,2, N, and K solve (38) and (39), if and only if H Q(K) mod 2p"m2p/¸ .

Fig. 1. Sketch of Eq. (43).

(49)

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Now, using Lemma 3.1, lim Q(K)"(l #2#l !N)2p/¸ ,  , \

K

lim Q(K)"(l #2#l )2p/¸ . (50)  ,  Furthermore, dQ(K)/dK'0. Thus there are precisely N!1 values K , a"1,2, N!1, which ? satisfy (49) for a given choice l (2(l of branches of Eq. (43) (recall that we exclude  , K"$R). We summarize our result in the following K

Lemma 3.2. Let ¸'8/;. Then there are precisely (* )(N!1) solutions with all k real to Eqs. (38) , H and (39). To every choice of N mutually distinct intervals I "[(l !1)2p/¸,l 2p/¸], j"1,2, N, H H H l Ol , there correspond N!1 solutions with k 3I . These solutions are characterized by N!1 H I H H diwerent values of K. Let us show that our result coincides with Takahashi's counting (28)}(30). We have N real k's and one K string of length 1. Since there is no k}K string, there is no JL to specify. M"0, M "1, ?  M "0 for j'1. Thus "J"4(N!t !1)"(N!2), and there are N!1 possible values of J.   H  The number of di!erent sets +I ,2, I , follows from (28) as (* ). This means that Takahashi's  , , counting predicts a total number of (* )(N!1) real solutions, which is in accordance with , Lemma 3.2 as long as the Takahashi condition (45) is satis"ed. In the special cases N"2 and 3 we "nd (*) and 2(*) solutions, respectively.   3.2. k}" two string Let us consider Eq. (42) in the case that two of the k 's are complex conjugated and the others are H real. We may set k "k "q!im, k "k "q#im with real q and real, positive m. It follows \  >  from (42) that sin(q#im)"sin(q)cosh(m)#i cos(q)sinh(m) "K#(;/4)(sin(q¸)!i sinh(m¸))/(cosh(m¸)!cos(q¸)) ,

(51)

or, if we separate real and imaginary part of this equation, sin(q)cosh(m)"K#(;/4)sin(q¸)/(cosh(m¸)!cos(q¸)) ,

(52)

cos(q)sinh(m)"!(;/4)sinh(m¸)/(cosh(m¸)!cos(q¸)) .

(53)

Note that for m"0 Eq. (53) is satis"ed identically and (52) turns into (43). Let us consider the two-electron case N"2 "rst. Then, by Eq. (41), q"mp/¸, m"0,2,2¸!1 .

(54)

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Fig. 2. Sketch of Eq. (56).

Eq. (53) determines m as a function of q"mp/¸, and Eq. (52) determines K. We have q¸"mp. Hence, sin(q¸)"0, cos(q¸)"(!1)K, and (52) and (53) decouple into K"sin(q)cosh(m) ,

(55)

sinh(m)"!(;/4 cos(q))sinh(m¸)/(cosh(m¸)!(!1)K) .

(56)

This decoupling is a peculiarity of the two-particle case, which makes it more simple than the general case. We can discuss Eq. (56) graphically (see Fig. 2). Let us de"ne



th(m¸/2) for m odd , sinh(m¸) " f (m)" cosh(m¸)!(!1)K cth(m¸/2) for m even .

(57)

f (m)'0 for all m'0. Hence, (56) can have solutions for positive m only if p/2(q(3p/2 for ;'0, p/2("q!p"(p for ;(0 .

(58)

For the sake of simplicity let us concentrate on the repulsive case ;'0. It follows from (57) that (56) always has exactly one solution m for every even m which satis"es (58). The condition for K a solution with odd m to exist is that the derivative of the right-hand side of Eq. (56) is larger than the derivative of the left-hand side of Eq. (56) as m approaches zero from the right, i.e. !;¸/8 cos(q)'1 .

(59)

This is satis"ed for all q with p/2(q(3p/2, if and only if ¸'8/; .

(60)

Again we have found the Takahashi condition (45). If the Takahashi condition is satis"ed, then there is one and only one k}K two string solution for every m satisfying (54) and (58), and we can easily count these solutions. Their number as a function of ¸ is di!erent for odd and even ¸, respectively.

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Lemma 3.3. Let ¸'8/;, ;'0, N"2. Eqs. (38) and (39) have k}K two-string solutions only if the momentum q of the center of the string is in the range p/2(q(3p/2. The allowed values of q in that range are quantized as q"mp/¸. For every q"mp/¸ there is one and only one k}K two string. The total number of k}K two strings is ¸ for ¸ odd and ¸!1 for ¸ even. Comparing our result with the prediction of Takahashi's counting (28)}(30) we "nd again agreement. Now there is no free k and no K string, thus no I and no JL . Furthermore, H H ? M "M"1, and M "0 for j'1. Thus "J"4(¸!t !1)"(¸!2), which means that    H  there are ¸!1 possible values of J. This agrees with our lemma since ¸ was assumed to be even in (28)}(30). Note that lim f(m)"1, pointwise for all m'0. This suggests the notion of an ideal string * determined by the equations K"sin(q)cosh(m), sinh(m)"!;/4 cos(q), q"mp/¸ .

(61)

Replacing (55) and (56) by (61) means to replace the curves denoted by &m odd' and &m even' in Fig. 2 by the dashed line. Clearly, solutions of (55), (56) are in one-to-one correspondence to solutions of (61), if the Takahashi condition is satis"ed. We can formulate the following corrolary of Lemma 3.3. Lemma 3.4. Let ¸'8/;, N"2. Let m be a sequence of integers (¸/2(m (3¸/2), such that * * lim m p/¸"q, p/24q43p/2. According to Lemma 3.3 this dexnes a sequence m * of solutions * * K of (56). This sequence has the limit lim m * "!arsinh(;/4 cos(q)) , K * i.e. in the thermodynamic limit all k}K two strings are driven to the ideal string positions.

(62)

Proof. Eq. (62) follows from (56), since the sequence m * is bounded from below by a positive K number. Let us prove the latter statement. Assume the contrary. Then there is a subsequence m *H of K f (m *H )"0. This means (i) m H is odd for m * which goes to zero, and it follows from (56) that lim H L * K m ¸ "0. We conclude that all su$ciently large j, and (ii) lim sinh(m *H ¸ )"lim H K*H H H K H 1 sinh(m *H ) ; ; K " lim "0" lim ! "! , (63) lim ¸ sinh(m *H ¸ ) 8 cos(m H 2p/¸ ) 8 cos(q) H K H * H H H H which is a contradiction. Thus, the lemma is proved. 䊐 Let us now proceed with the case N"3. We have to solve the following system of equations (cf. (41), (42), (52), (53)): 2q#k "m2p/¸, m"0,2,3¸!1 ,  sin(k )!K"(;/4)ctg(k ¸/2) ,  

(64) (65)

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K"sin(q)cosh(m)!(;/4)sin(q¸)/(cosh(m¸)!cos(q¸)) ,

(66)

cos(q)sinh(m)"!(;/4)sinh(m¸)/(cosh(m¸)!cos(q¸)) .

(67)

We choose a branch of solution ql (K) of (65) and insert it into (64). This yields q"mp/¸!ql (K)/2 .

(68)

Using this result, (66) and (67) turn into K"sin(mp/¸!ql (K)/2)cosh(m) ! (;/4)sin(ql (K)¸/2)/(cos(ql (K)¸/2)!(!1)K cosh(m¸)) , cos(mp/¸!ql (K)/2)sinh(m)"!(;/4)sinh(m¸)/(cosh(m¸)!(!1)K cos(ql (K)¸/2)) .

(69) (70)

This is a system of two equations in two unknowns m and K. In contrast to the case N"2, which was considered above, these equations do not decouple. We shall "rst consider the solutions ml (K) of Eq. (70). For this purpose we de"ne K f (m)"sinh(m¸)/(cosh(m¸)!a) , ? ql (K)¸ , a"(!1)K cos 2






; b"! . 4 cos(mp/¸!ql (K)/2)

(71)

With these de"nitions Eq. (70) turns into sinh(m)"bf (m) . (72) ? Note that a is real and "a"41. Hence, f (m)'0 for all positive m, and a necessary condition for (72) ? to have a solution is b'0. This means that p/24mp/¸!ql (K)/243p/2 for ;'0 , p/24"mp/¸!ql (K)/2!p"4p for ;(0 .

(73)

Let us concentrate again on the case ;'0. The inequality for the case ;'0 in (73) holds for all real K, if and only if ¸/24m!l43¸/2!1 (cf. Lemma 3.1). Eq. (72) can be easily solved graphically for "xed a ("a"(1) and b'0 (see Fig. 3). The function f (m) has the following properties: ? (i) a(0: f (m) is monotonically increasing (f  (m)'0) and concave ( f (m)(0), f (0)"0, ? ? ? ? f  (0)"¸/(1!a)5¸/2 and lim f (m)"1. (ii) a'0: f (m) has a single positive maximum m ? K ? ? and a single positive turning point m, m(m, f (0)"0, f  (0)"¸/(1!a)5¸/2 and ? ? lim f (m)"1. These properties are su$cient to conclude that (72) has a unique solution ml (K) K K ? for all real K, if and only if ¸/24m!l43¸/2!1 (recall that ;'0) and the Takahashi condition ¸'8/; is satis"ed. Note that f (m) as a function of a interpolates between the two ? branches of the function f (m), Eq. (57). These two branches are the two dashed lines envelopping the functions f (m) in Fig. 3. ?

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Fig. 3. Sketch of Eq. (72).

Lemmas 3.1 and 3.3 allow us to understand the behavior of the solutions ml (K) of (70) as K KP$R. Lemma 3.1 applied to (70) implies cos((m!l#1)p/¸)sinh(m)"!(;/4)sinh(m¸)/(cosh(m¸)!(!1)K\l>) for KP!R , (74) cos((m!l)p/¸)sinh(m)"!(;/4)sinh(m¸)/(cosh(m¸)!(!1)K\l) for KPR .

(75)

Using Lemma 3.3 we conclude that lim ml (K)"m l , lim ml (K)"m l , (76) K K\ > K K K\ \  where m is the unique solution of Eq. (56). This solution exists (cf. (58) and recall that ;'0), if and K only if ¸/2(m(3¸/2. Hence, the range of validity of (76) is restricted to K

¸/2(m!l(3¸/2!1 . Let us insert ml (K) into (69). Then (69) turns into K K"gl (K) , K where gl (K) is de"ned by K gl (K)"sin(mp/¸!ql (K)/2)cosh(ml (K)) K K ! (;/4)sin(ql (K)¸/2)/(cos(ql (K)¸/2)!(!1)K cosh(ml (K)¸)) . K Using (76) we obtain the asymptotics of gl (K) K lim gl (K)"sin((m!l#1)p/¸)cosh(m l ) , K K\ > \

K

(77)

(78)

(79)

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lim gl (K)"sin((m!l)p/¸)cosh(m l ) . (80) K K\  We see that gl (K) has "nite asymptotics for KP$R. Since gl (K) is continuous in K, we arrive K K at the conclusion that there exists a solution Kl of (78) for every pair (l, m), which satis"es (77). K Hence, we have shown the following: K

Lemma 3.5. Let ¸'8/;, ;'0, N"3. Eqs. (38) and (39) have solutions consisting of one k}K two string and a single real k only if the momentum q of the center of the string is in the range p/2(q(3p/2. For every choice of branch of the real momentum k (cf. Fig. 1), there exist ¸!1 k}K  two strings for odd ¸ and ¸!2 k}K two strings for even ¸. This gives a total number of ¸(¸!1) solutions of this type for ¸ odd and of ¸(¸!2) for ¸ even, respectively. Let us compare with Takahashi's counting (28)}(30). We have a single free k and no K string, H which means that there is one I "I and no JL . It follows from (28) that I may take ¸ di!erent H ? values. Furthermore, M "M"1, and M "0 for j'1. Thus "J"4(¸!3), which leads to  H  ¸!2 possible values of J. Takahashi's counting therefore gives ¸(¸!2) solutions with one k}K string and one real k. This is in aggreement with our lemma. We are now ready to state the following generalization of Lemma 3.4, Lemma 3.6. Let ¸'8/;, N"3. Choose two sequences of integers l and m , such that * * lim (m !l )p/¸"q, p/24q43p/2. This dexnes a sequence of solutions ml* * (K) of Eq. (70), K * * * which has the limit lim ml* * (K)"!arsinh(;/4 cos(q)) (81) K * uniformly in K. Thus all strings corresponding to the sequence ml* * (K) are driven to their ideal K positions. m (K)"0. This can be Proof. There is no K and no subsequence ml*H *H (K), such that lim H l*H K*H K seen in similar way as in the proof of Lemma 3.4. Let us "nally note that our considerations for the case N"3 readily generalize to arbitrary N. For arbitrary N we have to consider N!2 copies of Eq. (65) in the system of Eqs. (64)}(67). We . The further have to replace ql (K) in (68) by Q(K)"ql (K)#2#ql,\ (K) with l (2(l  ,\ properties of Q(K) (monotonicity and asymptotics) then follow from Lemma 3.1, and all considerations go through as in the case N"3. 3.3. Summary In this section we have studied k}K string solutions of the Lieb}Wu equations (7) and (8). We have shown that such solutions exist and that they are driven to certain ideal string positions in the limit of a large lattice. We have further shown that for a large enough lattice of "nite length their number is in accordance with the number of corresponding solutions of the discrete Takahashi equations (25)}(27).

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4. Lieb}Wu equations for a single down spin (II) } self-consistent solution 4.1. Zeroth order and the discrete Takahashi equations In this section we present the self-consistent solution of the Lieb}Wu equations (7) and (8) for the case of three electrons and one k}K string. The Lieb}Wu equations (7) and (8) provide a selfconsistent way of calculation of the deviation of the strings from their ideal positions. We show that every solution of the discrete Takahashi equations gives an approximate solution to the Lieb}Wu equations (7) and (8), and we calculate the leading order corrections. These corrections vanish exponentially fast as the number of lattice sites ¸ becomes large. The Lieb}Wu equations for three electrons and one down spin are (82) e IH *"(K!sin k !i;/4)/(K!sin k #i;/4), j"1,2,3 , H H  K!sin k !i;/4 H “ "1 . (83) K!sin k #i;/4 H H As in Section 3.1 we may replace Eq. (83) by the equation for the conservation of momentum, e I >I >I *"1 .

(84)

Eqs. (82) and (84) follow from (82) and (83) and vice versa. Let us follow the usual self-consistent strategy for obtaining a k}K string solution. As in Section 3.3 we shall use the notation k "k "q!im, k "q#im, m'0, i.e. we assume that k and \  >  k are part of a k}K string. In order to facilitate comparison with the previous literature we shall  also use the abbreviations "Re sin(k )"Re sin(k ) and s"Im sin(k )"!Im sin(k ). We > \ \ > introduce d as a measure of the deviation of the string from its ideal position. Then sin(k )" #is"K#i;/4#d, sin(k )" !is"K!i;/4#dM . \ > Inserting (85) into (82) gives e I\ *"1#i;/2d, e\ I> *"1!i;/2dM .

(85)

(86)

We may consider the "rst equation in (85) as the equation, that de"nes d. The second equation in (85) is not independent. It is the complex conjugated of the "rst one and may be dropped for that reason. Similarly, we may also drop the second equation in (86). Then we are left with six independent equations, (82) for j"3, (84), and the real and imaginary parts of the "rst equations in (85) and (86). Note that k #k "2q. Therefore our six equations are equivalent to   e O*"(K!sin k #i;/4)/(K!sin k !i;/4) , (87)   e I *"(K!sin k !i;/4)/(K!sin k #i;/4) , (88)   sin(k )"K#i;/4#d , (89) \ d"(i;/2)/(e I\ *!1) . (90) Every k}K string solution of the Lieb}Wu equations (82) and (83) gives a solution of Eqs. (87)}(90) (with real q, k , K and real positive m) and vice versa. 

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If ¸ is large and m"Im k "O(1), then d is very small and may be neglected in Eq. (89). Then > (90) decouples from the other equations, which become (91) e O*"(K !sin k#i;/4)/(K !sin k!i;/4) ,      e I *"(K !sin k!i;/4)/(K !sin k#i;/4) , (92)     sin(k)"K #i;/4 . (93) \  If, on the other hand, d is very small in (89), we may neglect it in "rst approximation and solve (93) instead. Now (93) implies that m"O(1) (see below). Then, using (90), we see that d is indeed small for large ¸. This means the assumption d be small for large ¸ is self-consistent. Let us be more precise. We set k"q!im with real q and real, positive m. Separating \ (93) into real and imaginary parts we obtain two equations which relate the three unknowns q, m and K ,  K "sin(q)cosh(m) , (94)  m"!arsinh(;/4 cos(q)) . (95) Let us concentrate on positive coupling ;'0 for simplicity. Since we are assuming that m'0, the range of q is then restricted to, say, p/2(q(3p/2 by Eq. (95). We obtain the uniform estimate m'arsinh(;/4) .

(96)

In order to test, if a solution q, m, k, K of (91)}(93) is a good approximate solution of   (87)}(90), we use it to estimate the modulus of d for large ¸, "d"+(;/2)e\K*((;/2)e\  3* .

(97)

The inequality follows from (96). We conclude that "d" becomes very small for large ¸. For ;"4 and ¸"24, for instance, estimate (97) gives "d"(1.3;10\, whereas the di!erence between two real k 's is of the order of 2p/¸"0.26 (cf. Section 3.2). For large ¸ it becomes impossible to  numerically distinguish between solutions of (87)}(90) and (91)}(93), respectively. If we "x the branch of the arcsin as !p/24arcsin(z)4p/2, it follows from the inequality p/2(q(3p/2 that k"p!arcsin(K #i;/4) . \  Inserting (98) into (91) leads to

(98)

(99) e 0  K > 3*"(K !sin k!i;/4)/(K !sin k#i;/4) .     We thus have eliminated k from the system of Eqs. (91)}(93), and we are left with two independent \ equations (92) and (99). These two equations determine the two real unknowns K and k. Taking   logarithms of (92) and (99) we arrive at Takahashi's discrete equations (25) and (27) for one k}K string and one real k. We have seen that Eqs. (91)}(93) determine Takahashi's ideal strings. Eqs. (87)}(90), on the other hand, are equations for non-ideal strings, which solve the Lieb}Wu equations (82) and (83). Thus, d is a measure for the deviation of the strings from their ideal positions. We have further seen that the assumption that d be small is self-consistent. In particular, every solution of Eqs. (91)}(93),

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which are equivalent to Takahashi's discrete equations, is an approximate solution of Eqs. (87)}(90). The approximation becomes extremely accurate for large ¸. 4.2. First-order corrections Inserting any solution of (91)}(93) into (90) we have d"(i;/2) e\ O*e\K*#O(e\K*) "(;/2) sin(q¸)e\K*#(i;/2) cos(q¸)e\K*#O(e\K*) .

(100)

So the relevant parameter, which controls the deviation of the strings from their ideal positions is e"e\K*. Every solution of (91)}(93) is an approximate solution of (87)}(90). Let us calculate the leading order corrections. We expect them to be proportional to e, q"q#qe#O(e) ,

(101)

m"m#me#O(e) ,

(102)

k "k#ke#O(e) ,    K"K #K e#O(e) .   The e-expansion of d is given in Eq. (100) above. We also introduce

(103) (104)

"sin(q)cosh(m)# e#O(e) ,

(105)

s";/4#se#O(e) ,

(106)

since it is the quantities "Re sin(k ) and s"Im sin(k ) which actually form strings in the \ \ complex plane. The two sets of variables q, m and , s are not independent. Inserting (101) and (102) into the left-hand side of (105) and (106) and comparing leading orders in e we "nd





 s

cos(q)cosh(m) sin(q)sinh(m)

"

sin(q)sinh(m)




!cos(q)cosh(m)

q m

.

(107)

Let us insert (100), (105) and (106) into (89). We obtain to leading order

"K #(;/2)sin(q¸) ,  s"(;/2)cos(q¸) ,

(108) (109)

i.e. we have already found the leading order correction s, Eq. (109). Next, we insert (101), (103) and (104) into (87) and (88) and linearize in e. The resulting equations are 2q"!(;/2¸)(K !cos(k) k)/[(K !sin(k))#;/16] ,      k"(;/2¸)(K !cos(k) k)/[(K !sin(k))#;/16]"!2q .      

(110) (111)

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The equation 2q#k"0 following from (110), (111) is, of course, a consequence of momentum  conservation. Eqs. (107)}(111) are a system of six linear equations for six unknowns, q, m, , s, k and  K . Note that in the derivation of these equations we have used all of Eqs. (87)}(90). Eqs. (108) and  (109) came out of (89), (90) was used to obtain expansion (100) for d in terms of e, and (110), (111) follow from (87) and (88). Eq. (107) is a consequence of the de"nition of and s. Being a set of linear equations, (107)}(111) are readily solved, K "!(;/2)(sin(q¸)#tan(q)th(m)cos(q¸))C(¸)  "!(;/2)(sin(q¸)#tan(q)th(m)cos(q¸))#O(1/¸) , K  k"  (2¸/;)[(K !sin(k))#;/16]#cos(k)    ;(sin(q¸)#tan(q)th(m)cos(q¸)) "! #O(1/¸) , 4¸[(K !sin(k))#;/16]  

"K #(;/2)sin(q¸)"!(;/2)tan(q)th(m)cos(q¸)#O(1/¸) ,  s"(;/2)cos(q¸) .

(112) (113) (114) (115) (116) (117)

The function C(¸) in Eq. (112) which gives the explicit form of the O(1/¸) and O(1/¸) corrections in the remaining equations is





\ cos(q)cosh(m)(1#tan(q)th(m)) C(¸)" 1# (4¸/;)[(K !sin(k))#;/16]#2 cos(k)    "1#O(1/¸) .

(118)

Eqs. (112)}(117) give a complete description of the leading order deviation of a non-ideal string from its ideal position in the presence of one real k. The deviations of q and m, which are determined by q and m, follow from Eqs. (107), (116) and (117). In order to see that C(¸)!1 is indeed of order O(1/¸) on has to use (94) and (95). 4.3. Summary In this section we have presented a self-consistent solution of the Lieb}Wu equations for the case of three electrons and one k}K string. Recall that the existence of these solutions was shown in the previous section. Here we showed that a self-consistent approach naturally leads to Takahashi's discrete equations. We showed that Takahashi's discrete equations provide a highly accurate approximate solution of the Lieb}Wu equations in the limit of a large lattice. We also showed that there is a natural parameter e"e\K* that measures the deviation of solutions of the Lieb}Wu equations from the corresponding solutions of the discrete Takahashi equations. Employing an algebraic perturbation theory we explicitly calculated the leading order deviation in e of a nonideal k}K string from its ideal position in the presence of a real k.

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5. Lieb}Wu equations for a single down spin (III) } numerical solution 5.1. Numerical method In Section 3 the existence of k}K string solutions was analytically proven under the Takahashi condition for the simplest non-trivial cases with one down spin, M"1. The deviation of ideal string solutions given by the discrete Takahashi equations from the corresponding solutions of the Lieb}Wu equations was evaluated analytically in Section 4, and it was shown that the corrections vanish exponentially fast as the lattice size ¸ becomes large. To con"rm these analytical arguments on the existence of k}K strings, we utilize a complementary numerical approach. Although the tractable system size is limited, we can directly obtain k}K strings and verify the completeness of the Bethe ansatz for arbitrary ;. We note that numerical solutions for low-lying particle}hole and K string excitations in small "nite systems can be found in the literature (see e.g. [84]). As far as we know, however, our data are the "rst example of a numerical study that con"rms the completeness of the Bethe ansatz for a small "nite system. We study again the cases N"2 and 3 for both pedagogical clarity and technical simplicity. We shall employ two numerically exact methods, (1) the numerical diagonalization of a real-symmetric matrix using the Householder-QR method (we call it Method 1), (2) a numerical method to solve coupled non-linear equations using the Brent method (we call it Method 2). These techniques themselves are conventional. They allow us to obtain numerically exact solutions for the Hubbard model. Here we use the term &numerically exact' to state that our numerical solutions give exact numbers except for the inevitable rounding error in the computation. Our strategy here is the following: (i) Obtain all energy eigenvalues with "xed N and M as a function of ; using Method 1. This step gives a complete list of energy eigenvalues. (ii) Obtain numerical (real and/or complex) solutions by solving the Lieb}Wu equations with Method 2. (iii) List up all eigenvalues obtained by the two methods and compare them with one another. This step gives a con"rmation of completeness. For our numerical study we used the following form of the Hubbard Hamiltonian: * * H"! (c> c #c> c )#; n n . H>N HN H\N HN Ht Hs H Nts H

(119)

This form is di!erent from (1) by a shift of the chemical potential and by a constant energy shift. For "xed particle number N this leads to a shift of the spectrum by (;/4)(2N!¸). In order to perform a numerical diagonalization we used basis vectors c> 2c>,\+ c> 2c>+ "02 to V W W V represent the Hamiltonian as a matrix in the sector of "xed N and M. The number of di!erent

 We will indicate errors by the di!erence of the left-hand side and the right-side of each equation in (7) and (8), evaluated within our numerical treatment. Then it will become clear that in all the equations which we tested the relative error is negligible and of the order of the rounding error expected for the double-precision calculation.

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con"gurations, x (2(x , y (2(y , in this sector is  ,\+  + ¸ ¸ ; N!M M







and determines the size of the matrix. In order to employ method 2, we rewrote the Lieb}Wu equations into a proper set of real equations, which will be presented in the following subsections. Hereafter in this section we assume that ¸ is an even integer. 5.2. Numerical solution for two electrons 5.2.1. Equations for the k}K string Let us discuss numerical k}K two string solutions for the two electron system with one down spin (N"2, M"1). We shall use the same notation as in Section 3. The variables to be determined are k , k and K, where k is the complex conjugate of k and K is real. We may write > \ > \ k "q#im, k "q!im , (120) > \ where 04q(2p and m'0. Then the total momentum is P"2q"(2mp/¸) mod 2p .

(121)

This equation restricts the admissible values of q. It is easy to obtain K as a function of q and m. We "nd K"sin(q)cosh(m) (cf. Eq. (55)). Thus we are left with a single equation that determines m. For our numerical calculations we wrote it in the form exp(imp#m¸)"(!cos(mp/¸)sinh(m)#;/4)/(!cos(mp/¸)sinh(m)!;/4) .

(122)

This equation is equivalent to (56). Therefore, for positive ;, the allowed values of m are restricted to m"¸/2#1,2,3¸/2!1

(123)

(cf. Eq. (58)). Eq. (122) was already studied in Appendix B of [78]. There it was shown that there is a redistribution phenomenon as ; becomes small. k}K strings corresponding to odd values of m collapse at critical values of ; given by ; "(8/¸)"cos(mp/¸)". K 5.2.2. Numerical solutions for N"2 As a typical example, let us present some numerical k}K two string solutions for N"2 (two electrons). For a k}K two string we show the dependence of energy eigenvalues, imaginary parts of charge momenta and the deviation from the ideal string positions on ;. We put m"16 and ¸"16 (16 sites). Then we have q"p, i.e. k "p#im, k "p!im, K"0 . > \ In the list below the deviation of the string from its ideal position, k  "p$i arsinh(;/4) is ! measured by Im d"sinh(m)!;/4.

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(1) ;"10: Householder E"10.7703296143355 , Bethe Ansatz E"10.7703296143355,

m"1.64723114637774 ,

Im d"1.78994596922166;10\ . (2) ;"1: Householder E"4.13148449882288 , Bethe Ansatz E"4.13148449882289,

m"0.255705305537532 ,

Im d"8.50098694369150;10\ . (3) ;"0.1: Householder E"4.00668762410970 , Bethe Ansatz E"4.00668762410971,

m"5.78176505356310;10\ ,

Im d"3.28498688384022;10\ . (4) ;"0.01: Householder E"4.00062917225929 , Bethe Ansatz E"4.00062917225931,

m"1.77363435624506;10\ ,

Im d"1.52372734873120;10\ . The string with m"16 does exist for any ;'0 and is actually the highest level in the spectrum for N"2. We note that the non-ideal string approaches the ideal k}K string, when ; becomes large, even for such a small system. Yet, in accordance with our expectations, the ideal string does not provide a good approximation for small ; in "nite systems. In Table 1 we show a complete list of eigenstates for the case of one down-spin (M"1, S "0) X and ;"1.5 for a 6-site system (¸"6). Note that the value of ;"1.5 is greater than max ; "1.1547, or, in the language of Section 3, the Takahashi condition is satis"ed. Let us K explain the table. (i) The 36 eigenstates are listed in increasing order with respect to their energy. (ii) S and P denote the spin and momentum of the eigenstate, respectively. (iii) The energy eigenvalues obtained by direct diagonalization of the Hamiltonian (the Householder method) and by the Bethe ansatz method coincide within an error of O(10\). (iv) The last digit for each numerical value has a rounding error. (v) There are 5 k}K string solutions among the 36 eigenstates, which is consistent with the number ¸!1"5 obtained by Takahashi's counting (28)}(30). Let us give more explanations on the table. In fact, it con"rms the completeness of the Bethe ansatz as discussed in Section 2.4. We "rst note that there are 36 eigenstates for the case of two electrons and one down-spin (N"2 and M"1) on a 6-site lattice (¸"6). We recall that the two electrons

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with one up-spin and one down-spin can occupy the same site. The ();()"36   eigenstates in Table 1 can be classi"ed into the following types: (i) Fifteen eigenstates with two real charge momenta k , k and one real spin-rapidity K,   (ii) "ve eigenstates with one k}K two string, (iii) "fteen eigenstates with S"1 and SX"0 belonging to spin triplets, (iv) one g-pairing state. Let us consider case (i). In Table 1 there are 15 states with real charge momenta which have total spin zero (S"0). This agrees with our analytical arguments in Section 3 on the number of real, regular Bethe ansatz solutions (cf. Lemma 3.2 and below), which should be ()"15. Let us also  recall that the I are integer (half-integer) when M #M is even (odd). Thus, for the case H K K K M"1, the I should be half-odd integer, which is in agreement with the results shown in Table 1. H We now consider case (ii). From the analytic discussion in Section 3.3, we should have "ve eigenstates with k}K strings. This is in accordance with our numerical data in Table 1. Recall that we showed in Section 3.3 (below Lemma 3.3) that also Takahashi's counting, using Eqs. (28)}(30), leads to the same number of k}K string solutions. We consider case (iii). There are 15 states with S"1 and S "0. We describe them as triplets in X Table 1. They are obtained by multiplying the spin-lowering operator f? to the regular Bethe states with S"1 and S "1. (For the notation for the SO(4) symmetry see Section 2.2.) The regular Bethe X states with S"1 and S "1 correspond to N"2 and M"0. We recall again that the I 's are X H integer (half-integer) when M #M is even (odd). Thus, the I 's which belong to the SX"0 K K K H states in spin triplets should be integer valued. They should take one of the values !2,!1,0,1,2, or 3, which means that there are ()"15 states according to Takahashi's counting (28)}(30).  There is one g-pairing state. The energy of this state is equal to ;, since the two electrons occupy the same site. Now, let us sum up all the numbers of the di!erent types of eigenstates: 15#5#15#1"36 .

(124)

Thus, we have shown that all the energy eigenstates obtained by Method 1 are con"rmed by Method 2. In particular, we have con"rmed numerically the completeness of the Bethe ansatz. Let us consider the total momentum. Using Eqs. (25)}(27) we can express the total momentum P of the eigenstates with real charge momenta in terms of I and I ,   P"(2p/¸)(I #I ) mod 2p . (125)   This formula is consistent with Table 1. Let us now discuss the ;-dependence of the spectrum. In Fig. 4 we show the spectral #ow from strong coupling to weak coupling, where the Takahashi condition does not hold. In Fig. 4 we show the redistribution phenomenon discussed in Sections 2.4, 3.1 and above. There are "ve k}K two-string solutions in Table 1. The entries 30, 31 and 36 correspond to even m. According to our discussion above, these k}K two strings are stable as ; becomes small. Entry number 36 is the highest energy level in the "gure. Entries number 30 and 31 are degenerate. They correspond to the third highest level at ;"5. The entries number 34 and 35 correspond to odd

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Table 1 Classi"cation of all energy levels for ¸"6, N"2, M"1 and ;"1.5 No.

Energy

S

P/(p/3)

Type of solution

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 28 29 30

!3.82047006625301 !3.00000000000000 !3.00000000000000 !2.60959865138515 !2.60959865138515 !2.00000000000000 !1.86256622153075 !1.86256622153075 !1.53909577258373 !1.00000000000000 !1.00000000000000 !0.594210897273940 !0.594210897273940 0.00000000000000 0.00000000000000 0.00000000000000 0.00000000000000 0.00000000000000 0.00000000000000 0.00000000000000 0.474357244982949 0.474357244982949 1.00000000000000 1.00000000000000 1.43079477929458 1.43079477929458 1.50000000000000 2.00000000000000 2.49213401737360 2.52598233951011

0 1 1 0 0 1 0 0 0 1 1 0 0 0 0 1 1 1 1 1 0 0 1 1 0 0 0 1 0 0

0 1 5 1 5 0 2 4 0 2 4 2 4 3 3 3 3 3 1 5 1 5 2 4 4 2 3 0 0 2

Real I "!0.5,0.5 H Triplet I "0,1 H Triplet I "0,!1 H Real I "!0.5,1.5 H Real I "!1.5,0.5 H Triplet I "1,!1 H Real I "0.5,1.5 H Real I "!0.5,!1.5 H Real I "!1.5,1.5 H Triplet I "0,2 H Triplet I "0,!2 H Real I "!0.5,2.5 H Real I "!2.5,0.5 H Real I "0.5,2.5 H Real I "!2.5,!0.5 H Triplet I "1,2 H Triplet I "!1,!2 H Triplet I "0,3 H Triplet I "2,!1 H Triplet I "!2,1 H Real I "!1.5,2.5 H Real I "!2.5,1.5 H Triplet I "3,!1 H Triplet I "3,1 H Real I "1.5,2.5 H Real I "!2.5,!1.5 H g-pair Triplet I "2,!2 H Real I "!2.5,2.5 H Complex m"8, m"0.710224864788777 Complex m"4, m"0.710224864788777 Triplet I "2,3 H Triplet I "!2,3 H Complex m"7, m"0.313056827256169 Complex m"5, m"0.313056827256169 Complex m"6, m"0.425405934759021

31

2.52598233951011

0

4

32 33 34

3.00000000000000 3.00000000000000 3.63524140640220

1 1 0

5 1 1

35

3.63524140640220

0

5

36

4.36743182146314

0

0

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m"5,7. The corresponding levels are degenerate. At ; "; "1.1547 these k}K two-string   solutions collapse into pairs of real charge momenta. This is indicated by the arrow in Fig. 4. On the other hand, all "ve k}K string solutions do exist as long as the Takahashi condition is satis"ed. 5.3. Numerical solution for three electrons 5.3.1. Equations for the k}K string We shall now consider the set of solutions to the Lieb}Wu equations for N"3. The k}K strings to be searched for have two complex k , a real k and a real K. We express these momenta and H H rapidities by k "q!im, k "q#im, k , K .    The total momentum is 2pm/¸, so that

(126)

k #k #k "2pm/¸, 2p(l!1)/¸(k (2pl/¸ .     We de"ne Re d and Im d by

(127)

sin k "K#i;/4#Re d#i Im d , (128)  sin k "K!i;/4#Re d!i Im d . (129)  We recall that Re d and Im d describe the deviations from the ideal string solutions. Now we derive equations for four variables, k , q,m and K, starting from the Lieb}Wu equations (82) and (83).  Taking logarithms, we obtain a set of equations of the form f "0 (i"1,2,4). Here the f 's are G G de"ned by f "¸k !2 arctan  






K!sin k  !2p(l!) ,  ;/4

(130)

Fig. 4. The spectral #ows for N"2 and M"1 for a 6-site lattice. Solid lines denote the k}K strings. Dashed lines denote the energy of real roots (S"0). Dotted lines denote the triplet states (S"1). The dash}dotted line denotes the energy of the g-pair. At ; "; "1.1547, as indicated by the arrow, two k}K two strings with m"5,7 collapse into real   solutions.

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f "!2¸ arcsin(+((K#Re d#1)#(;/4#Im d)   ! ((K#Re d!1)#(;/4#Im d),)# ¸k !2p(l!1/2!J) ,  f "exp(!2¸m)![(Re d)#(Im d)]/[(Re d)#(;/2#Im d)] ,  Re d Re d #2 arctan f "!2 arctan  Im d ;/2#Im d






! 2 arctan

K!sin k  !pp , ;/4

(131) (132)



(133)

where Re d"sin q cosh m!K, Im d"!;/4!cos q sinh m

(134)

and the parameter p is given by p"1 for k ((2p/¸)(l!), p"!1 for k '(2p/¸)(l!). (135)     Note that the equation f "0 is equivalent to Eq. (42) for j"3.  To solve this set of coupled equations by Method 2, we need a proper initial guess. We employ the ideal strings given by the discrete Takahashi equations as initial approximation. The fact that the ideal strings provide a good estimate for the true solution is crucial to reach the correct answer. This is because the f 's are rather singular functions having many diverging points. Very often, the G true solution is very close to a divergent point. We cannot approach the solution from a point beyond a branch cut using an iterative way like the Brent method. 5.3.2. Numerical examples of k}K string solutions We present some numerical solutions for one k}K two string and one real k. The parameters in the examples below are N"3 (three electrons), ¸"10 (10 sites) and ;"5. (1) m"10, l"1 (I "0, J"):   Householder E"4.46666961980768 , Bethe ansatz E"4.46666961980768, m"1.05674954466496, K"0.245232889885225,

q"2.98893848049280 ,

k "0.305308346193987 ,  Re d"!6.42851441900183;10\ ,

Im d"2.84511548476196;10\ . (2) m"7, l"1 (I "0, J"):   Householder E"3.48250616148668 , Bethe ansatz E"3.48250616148668, m"1.99506896270533, K"3.51250692222577,

q"1.92454676428806 ,

k "0.549136186449599 ,  Re d"2.08765360554253;10\ ,

Im d"4.99459629210719;10\ .

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Let us discuss possible numerical errors for the above solutions. Their numerical errors may be evaluated by the residual, f , given for these solutions as follows: G (1) m"10, l"1 (I "0, J"):   f "!1.24900090270330;10\, f "!3.99680288865056;10\ ,   f "5.25259688971173;10\, f "4.53592718940854;10\   with p"#1. (2) m"7, l"1 (I "0, J"):   f "!1.33226762955019;10\, f "1.42108547152020;10\ ,   f "!4.66698631842825;10\, f "!7.67983898697366;10\   with p"!1. In comparison to other error values f the number f has a rather large value. However, in G  f , we have an expression like e/e (eK0,eK0). So this value contains a larger cancellation  error for smaller Re d and Im d. Note again that relative errors for the energy are always O(10\). 5.3.3. A complete list of solutions for N"3 and M"1 We can present complete lists of eigenstates for all "nite systems tractable by our numerical technique. As a further example, we consider all eigenstates for N"3 and M"1 in a 6-site system (¸"6). The list is shown in Appendix C. It con"rms again the completeness of the Bethe ansatz. Let us brie#y discuss the numbers of eigenstates of di!erent types. First, we note that there are in total 6()"90 eigenstates. Inspection of the tables in Appendix C shows that they can be classi"ed  into the following four types: (i) Forty eigenstates with three real charge momenta k , k , k and one real spin rapidity K.    (ii) Twenty four eigenstates with one k}K string with k "k "q!im, k "k "q#im, and  \  > K and k real.  (iii) Twenty eigenstates with S" and S " belonging to spin quartets. X   (iv) Six eigenstates with one g-pair and one real charge momentum. Let us now con"rm that these numbers agree with Takahashi's counting, (28)}(30): case (i) was considered in Section 3.1 below Lemma 3.2. There we showed that Takahashi's counting predicts a number of 2(*) real solutions for three electrons and one down spin. Inserting ¸"6 we have  2()"40 eigenstates, which is in accordance with our numerical calculation. Case (ii) was con sidered below Lemma 3.5. The number of eigenstates obtained there by Takahashi's counting was ¸(¸!2), which for ¸"6 gives as desired 6;4"24. Let us consider case (iii). These states are the second highest states (S ") in spin quartets. They are obtained from regular Bethe states with X  N"3, M"0 by multiplication with the spin lowering operator f?. Since N"3 and M"0, we have ()"20 states of this type. The g-pair in case (iv) is obtained by acting with g> on regular  Bethe states with N"1 and M"0. Hence, Takahashi's counting gives six eigenstates of this type on a 6-site lattice.

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5.4. Summary In this section we have presented a thorough numerical study of the Hubbard model. We calculated, in particular, all eigenstates and eigenvectors for a 6-site lattice with two and three electrons and one down spin by direct numerical diagonalization of the Hamiltonian. These data were compared with data obtained by numerical solution of the Lieb}Wu equations. Both sets of data are in perfect numerical agreement and con"rm once again the results of our analytical investigation in the previous sections. The structure of our numerical data is fully consistent with Takahashi's string hypothesis. The number and classi"cation of the eigenstates is consistent with Takahashi's counting (28)}(30). Thus, our numerical data con"rm not only the existence of k}K strings but also the completeness of the Bethe ansatz. Without k}K strings the Bethe ansatz would be incomplete.

6. Thermodynamics in the Yang}Yang approach and excitation spectrum in the in5nite volume Let us now turn to the determination of thermodynamic quantities and the zero-temperature excitation spectrum in the in"nite volume. A convenient way to construct the spectrum was pioneered by Yang and Yang for the case of the delta-function Bose gas [69]. The starting point are the Bethe ansatz equations in the "nite volume. They are used to derive a set of coupled, non-linear integral equations called thermodynamic Bethe ansatz (TBA) equations, which describe the thermodynamics of the model at "nite temperatures. The quantities entering these equations have a natural interpretation in terms of dressed energies of elementary excitations. Yang and Yang's formalism is a natural generalization of the thermodynamics of the free Fermi gas to interacting systems. In what follows, we review Takahashi's derivation of the TBA equations for the case of the repulsive Hubbard model [27]. The analogous calculations for the attractive case can be found in [85]. Our starting point are the discrete Takahashi Eqs. (25)}(27) and expressions for energy (33) and momentum (32) for very large but "nite ¸. A very important property of (25)}(27) is that as we approach the thermodynamic limit ¸PR, N/¸ and M/¸ "xed ("nite densities of electrons and spin-down electrons), the roots of (25)}(27) become dense k

H>

!k "O(¸\), KL !KL "O(¸\), H ?> ?

K L!K L"O(¸\). ?> ?

(136)

We now de"ne the so-called counting functions y, z , z as follows: L L











sin k!KL sin k!K L  +L  +L ? ! h ? , k¸"¸y(k)! h n;/4 n;/4 L ? L ?











K!KK  +K ,\+Y K!sin k H "¸z (K)# H @ , h L LK ;/4 n;/4 K @ H

(137)

(138)

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,\+Y K!sin k H ¸[arcsin(K#ni;/4)#arcsin(K!ni;/4)]"¸z (K)# h L n;/4 H  K!K K  +K @ . # H (139) LK ;/4 K @ By de"nition the counting functions satisfy the following equations when evaluated for a given solution of the discrete Takahashi equations:






y(k )"2pI /¸ , z (KL )"2pJ L/¸, z (KL )"2pJL /¸ . (140) H H L ? ? L ? ? In the next step we de"ne the so-called root densities, which are related to the counting functions as follows. By de"nition the counting functions `enumeratea the Bethe ansatz roots, e.g. ¸[y(k )!y(k )]"2p(I !I ) . (141) H L H L For a given solution of (25)}(27) certain of the (half-odd) integers between I and I will be H L &occupied', i.e. there will be a corresponding root k, whereas others will be omitted. We describe the corresponding k-values in terms of a root density o(k) for &particles' and a density oF(k) for &holes'. In a very large system we then have by de"nition (here property (136) is very important) ¸o(k) dk"number of ks in dk , ¸oF(k) dk"number of holes in dk .

(142)

It is then clear that in the thermodynamic limit we have 2p[o(k)#oF(k)]"dy(k)/dk .

(143)

The analogous equations for the other roots of (25)}(27) are 2p[p (K)#pF (K)]"dz (K)/dK, 2p[p (K)#p F(K)]"dz (K)/dK . (144) L L L L L L In the thermodynamic limit the discrete Takahashi equations can now be turned into coupled integral equations involving both counting functions and root densities








sin k!K   [p (K)#p (K)], dK h k"y(k)! L L n;/4 L \ p K!sin k  K!K  dk h dK H o(k)"z (K)# p (K), LK ;/4 L K n;/4 \p \ K arcsin(K#ni;/4)#arcsin(K!ni;/4)





























(145) (146)

  K!sin k K!K o(k)# p (K). (147) dK H K LK ;/4 n;/4 \p K \ As we are interested in the Hubbard model at "nite temperatures we need to express the entropy in terms of the root densities. This is achieved by observing that, e.g. the number of vacancies for k's in "z (K)# L

p

dk h

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the interval [k, k#dk] is simply ¸(o(k)#oF(k)) dk. Of these &vacancies' ¸o(k) dk are occupied. The corresponding contribution dS to the entropy is thus formally e1"(¸[o(k)#oF(k)] dk)!/(¸o(k) dk)!(¸oF(k) dk)! ,

(148)

where ! denotes the factorial. The di!erential dS is obtained via Stirling's formula. After integration we obtain the following expression for the total entropy density of the Hubbard model:



S/¸"

p

\p

dk+[o(k)#oF(k)]ln[o(k)#oF(k)]!o(k)ln o(k)!oF(k)ln oF(k),



  dK+[p (K)#pF (K)]ln[p (K)#pF (K)] # L L L L \ L ! p (K)ln p (K)!pF (K)ln pF (K), L L L L   # dK+[p (K)#p F(K)]ln[p (K)#p F(K)] L L L L L \ ! p (K)ln p (K)!p F(K)ln p F(K), . L L L L The Gibbs free energy per site is thus



(149)

f"(E!kN!2BSX!¹S)/¸





  dk[!2 cos k!k!;/2!B]o(k)# dK 2nBp (K) L \p \ L   (150) # dK[4 Re(1!(K!in;/4)!2nk!n;]p (K)!¹ S/¸ . L L \ Here k is a chemical potential, B is a magnetic "eld and ¹ is the temperature. The alert reader will have realized that we are still missing a set of equations that allows us to completely determine the root densities and counting functions (the thermodynamic limit (145)}(147) of the discrete Takahashi equations are clearly insu$cient). This is the topic of the following subsection. "

p



6.1. Takahashi's thermodynamic equations Let us start by di!erentiating (145)}(147), which yields

  

  n;/4 p (K)#p (K) 1 L L , dK o(k)#oF(k)" #cos k p (n;/4)#(sin k!K) 2p L \ p n;/4 o(k)  dk pF (K)"! A *p # , L LK K K p (n;/4)#(sin k!K) \p K 1  1 ! A *p p F(K)" Re LK K K L p (1!(K!in;/4) K p n;/4 o(k) dk ! . p (n;/4)#(sin k!K) \p





(151)

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Here A is an integral operator acting on a function f as LK x!y d  dy H f (y). (152) A *f " "d f (x)# LK V LK 2p LK ;/4 dx \ Eqs. (151) can be used to express the densities of holes in terms of densities of particles. A second set of equations is obtained by considering the Gibbs free energy density (150) as a functional of the root densities. In thermal equilibrium f is stationary with respect to variations of the root densities






0"df



df df  df df do(k)# doF(k)# dp (K)# dpF (K) " dp (K) L doF(k) dpF (K) L do(k) L L L df df dp (K)# dp F(K) , (153) # dp F(K) L dp (K) L L L where we need to take into account (151) as constraint equations. In this way one obtains a set of equations for the ratios f"oF/o, g "pF /p and g "p F/p L L L L L L ln f(k)



 

1 !2 cos k!k!;/2!B   n; # dK ¹ 4p (n;/4)#(sin k!K) L \ 1 1 ; ln 1# !ln 1# . g (K) g (K) L L p n;/4 1 cos k ln(1#g (K))# ln 1# dk L f(k) p (n;/4)#(sin k!K) \p 2nB  1 " # A * ln 1# . LK ¹ g K K K 1 p cos k n;/4 ln 1# ln(1#g (K))# dk L f(k) p (n;/4)#(sin k!K) \p "





















(154)




(155)








4 Re(1!(K!in;/4)!2nk!n;  1 # A * ln 1# . (156) LK ¹ g K K K Note that (151) together with (154)}(156) completely determine the densities of holes and particles in the state of thermal equilibrium. The Gibbs free energy per site is given in terms of solutions of (154)}(156) as "

f"!¹















1   dK 1 1 p dk ln 1# !¹ ln 1# Re . f(k) g (K) 2p p (1!(K!in;/4) L \p L \

(157)

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Following Takahashi we de"ne i(k)"¹ ln(f(k)), e (K)"¹ ln(g (K)), e (K)"¹ ln(g (K)) . (158) L L L L As was "rst shown by Yang and Yang for the delta-function Bose gas [69], the quantities de"ned in this way describe the dressed energies of elementary excitations in the zero-temperature limit. Before turning to this we will give a brief summary on how to calculate thermodynamic quantities in the framework of Takahashi's approach. 6.2. Thermodynamics The expression for the Gibbs free energy density (157) can be simpli"ed [31]



f"E !k!;/2!¹ 

p

\p



dk o (k) ln(1#f(k))# 



\



dK p (K) ln(1#g (K)) ,  

(159)

where



p (K)" 

p

1 1 o (k) , dk ; cosh(2p/;)(K!sin k) 

\p  dK ;/4 1 o (k)" #cos k p (K) ,  p (;/4)#(sin k!K)  2p \ p  J (u)J (u)  . (160) E "!2 dk cos(k)o (k)"!4 du    1#exp(u;/2) \p  We note that o , p and E are the root density for real k's, the root density for real K's and the    ground state energy for the half-"lled repulsive Hubbard model, respectively. J (u) and J (u) are   Bessel functions. Since the occurrence of quantities related to the half-"lled Hubbard model in (159) may be surprising, we would like to emphasize that (159), (160) holds for all negative values of the chemical potential k, i.e. for all particle densities between zero and one. Representation (159) is convenient as it shows that the Gibbs free energy is determined by the dressed energies for real k's and real K's only. In order to derive (159) the following identities are useful:





 

p

\p p



o (k) 1 1 n;  " Re , dk 4p (n;/4)#(sin k!K) p (1!(K!in;/4)



n; o (k)  "A * p . dk L  K 4p (n;/4)#(sin k!K)

(161) \p At very low temperatures ¹;B it is possible to determine the Gibbs free energy by using an expansion of the TBA equations (154)}(156) for small ¹ [31]. The TBA equations essentially reduce two only two coupled equations for f and g in this limit. For generic values of B and arbitrary  temperatures one needs to resort to a numerical solution of (154)}(156). In order to do so, one needs

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to truncate the in"nite towers of equations for K and k}K strings at some "nite value of their respective lengths. In [79,80] such a truncated set of equations was solved by iteration. The integrals were discretized by using of the order of 50 (100) points for the k (K) integrations. The results of these computations are compared to the results of the quantum transfer matrix approach in the next section. 6.3. Zero-temperature limit For the remainder of this section we set the magnetic "eld to zero B"0. For ¹P0 the thermodynamic equations (154)}(156) then reduce to [27]



 dK ;/4 e (K) , (162) p (;/4)#(sin k!K)  \ / n;/4 cos k i(k)!A * e , (163) (1!d )e (K)" dk L  K L L p (n;/4)#(sin k!K) \/ / n;/4 cos k i(k) . e (K)"4 Re(1!(K!in;/4)!2nk!n;# dk L p (n;/4)#(sin k!K) \/ (164) i(k)"!2 cos k!k!;/2#







Note that i($Q)"0, i(k)(0 for "k"(Q and e (K)(0. Using Fourier transform, Eqs. (162) and  (163) can be simpli"ed further with the result



i(k)"!2 cos k!k!;/2#



e (K)" 

/

/

dk cos kR(sin k!sin k)i(k) ,

\/

cos k 1 i(k) , dk ; cosh(2p/;)(K!sin k)

\/ e (K)"0, n"2,3,2 , L where

(165)



 du exp(iux) . (166) 2p 1#exp(;"u"/2) \ The vanishing of the dressed energies of K-strings of lengths greater than one, i.e. e (K)"0 for L n52 is due to the absence of a magnetic "eld. For "nite magnetic "elds all e (K) will be non-trivial L functions. In order to characterize the excitation spectrum we need to determine the dressed momenta in addition to the dressed energies. This can be done by considering the zero-temperature limit of (151) R(x)"



/ 1 dk cos k R(sin k!sin k)o(k), "k"4Q; o(k)" # 2p \/

o(k)"0, "k"'Q ,

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/ 1 oF(k)" # dk cos k R(sin k!sin k)o(k), "k"'Q; oF(k)"0, "k"4Q , 2p \/ / 1 1 dk o(k) , p (K)"  ; cosh (2p/;)(K!sin k) \/ p (K)"0, n"2,3,2, pF (K)"0"p (K), m"1,2,2 , L K K / o(k) 1 1 n;/4 ! . p F(K)" Re dk L p (1!(K!in;/4) p (n;/4)#(sin k!K) \/





(167)

Eqs. (167) describe the ground state of the repulsive Hubbard model at zero temperature and zero magnetic "eld. There is one Fermi sea of k's (charge degrees of freedom) with Fermi rapidity $Q and a second Fermi sea of K's (spin degrees of freedom), which are "lled on the entire real axis. The total momentum (32) can be rewritten by using (25)}(27) in the following useful manner:






 +K 2p ,\+Y  +L  +L I # JK! JL #p (n#1) P" H @ ? ¸ H K @ L ? L ?  ,\+Y  +K  +L  +L " y(k )# z (KK)! z (KL )#p (n#1) . (168) H K @ L ? H K @ L ? L ? Using this expression for the total momentum we now can identify the dressed momenta of various types of excitations. We "nd that an additional real k with "k"'Q (&particle') or hole in the sea of k's ("k"(Q) carry momentum $p(k) respectively, where



p(k)"y(k)"2p

I

dk [o(k)#oF(k)] .

(169)

 Similarly, the dressed momentum of a hole in the sea of K's is

 


p (K)"!z (K)"2p  



/



K

dK p (K)!z (R)  



2p dk arctan exp ! (K!sin k) ;

N o(k)!p . 2¸

(170) \/ This result was "rst obtained by Coll [37]. Finally, a k}K string of length n has dressed momentum "2

p (K)"!z (K)#p(n#1) L L



"!2Re arcsin(K!in;/4)#

/




dk 2 arctan



K!sin k o(k)# p(n#1) , n;/4

(171)

\/ where the second line is obtained from the ¹P0 limit of (147). We are now in a position to completely classify the excitation spectrum at zero temperature. The dispersion relations of all elementary excitations follow from (164), (165) and (169)}(171). These equations involve only the two unknown functions, o(k) and i(k), which are solutions of linear Fredholm integral equations.

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239

6.4. Ground state for a less than half-xlled band The integral equations describing the root densities of the ground state are (167)



/ 1 dk cos k R(sin k!sin k) o(k), "k"4Q , o(k)" # 2p \/ / 1 1 p (K)" o(k) , dk  ; cosh(2p/;)(K!sin k) \/ where





/

dk o(k)"N/¸,





dK p (K)"M /¸"N/2¸ .   \/ \ The ground state energy per site is given by



(E!kN)/¸"

/

dk(!2 cos k!k!;/2)o(k) .

(172)

(173)

(174)

\/

6.5. Excitations for a less than half-xlled band There are three di!erent kinds of elementary excitations: E The "rst type of elementary excitation is gapless and involves the charge degrees of freedom only. It corresponds to adding a particle to or making a hole in the distribution of k's. Such excitations have dressed energy Gi(k) and dressed momentum p (k).  E The second type of elementary excitation is gapless and carries spin but no charge. It corresponds to a hole in the distribution of K's. Such excitations are called spinons. They have dressed energy !e (K) and dressed momentum p (K).   E There is an in"nite number of di!erent types of gapped excitations that carry charge but no spin. They correspond to adding a k}K string of length n to the ground state. Their dressed energies are e (K), their dressed momenta are p (K). L L Let us emphasize that this is a classi"cation of elementary excitations in the repulsive Hubbard model below half-"lling. It is important to distinguish these from &physical' excitations, which are the permitted combinations of elementary excitations. In other words, not any combination of particle}hole excitations, spinons and k}K strings is allowed, but only those consistent with the selection rules (28)}(30). To illustrate how this works and relate our "ndings to known results in the literature we consider several examples. We introduce the following terminology: we call the set +N, M , M " n"1,2,R, of the numbers of real k's, K-strings of length n and k}K-strings of L L length n occupation numbers of the corresponding excitation. This is in contrast to our usage of the term quantum numbers, which is reserved for the eigenvalues of energy, momentum, SX, S ) S, gX and g ) g. Example 1 (Particle}hole excitation). This is a two-parametric gapless physical excitation with spin and charge zero, i.e. its quantum numbers as well as its occupation numbers are the same as

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Fig. 5. Particle}hole excitation for ;"2.0 and densities n"0.6 and 0.8. Shown are the lower and upper boundaries of the continuum.

the ones of the ground state. It is obtained by removing a spectral parameter k with "k "(Q from   the ground-state distribution of k's and adding a spectral parameter k with "k "'Q. Its energy   and momentum are E "i(k )!i(k ) ,    P "p(k )!p(k ) . (175)    This excitation is allowed by the selection rules (28)}(30) as in the ground state only the (half-odd) integers "I "4(N!1)/2 are occupied and thus the possibility of removing a root corresponding to H "I "4(N!1)/2 and adding a root corresponding to "I "'(N!1)/2 exists. This excitation was   "rst studied by a di!erent approach in [37]. In Fig. 5 we show the particle}hole spectrum for densities n"0.6 and 0.8. As we approach half-"lling the phase space for particles shrinks to zero.

Example 2a (Spin-triplet excitation). Let us consider an excitation involving the spin degrees of freedom next. If we change the number of down spins by one while keeping the number of electrons "xed we obtain an excitation with spin 1. Recalling that in the ground state we have N electrons out of which M "N/2 have spin down, the excited state will have occupation numbers N and  M "N/2!1. The selection rules (28)}(30) then read  !¸/2(I 4¸/2, "J"4N/4 . (176) H ? The "rst condition is irrelevant as we are below half-"lling, but the second one tells us that there are two more vacancies than there are roots. In other words, #ipping one spin leads to two holes in the distribution of K's. There is one more subtlety we have to take care of: changing the number of down spins by one, while keeping the number of electrons "xed leads to a shift of all I in (25) by H

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241

Fig. 6. Spin-triplet excitation for ;"2.0 and densities n"0.6 and n"0.8. Shown are the lower and upper boundaries of the continuum for the positive branch. The negative branch is obtained by reversing the sign of the momentum. Note that we have not folded back to the "rst Brillouin zone.

either  or !. The consequence of this shift is a constant contribution of $pN/¸ to the   momentum of the excited state. This leads to two &branches' of the same excitation. The physical excitation obtained in this way is a gapless two-spinon scattering state with energy and momentum E "!e (K )!e (K ) ,       P "p (K )#p (K )$pN/¸ . (177)       In Fig. 6 we show the spin-triplet spectrum for densities n"0.6 and 0.8. In the Hubbard model the spin-triplet excitations were "rst studied by Ovchinnikov [36] and by Coll [37]. The situation encountered here is similar to the spin- Heisenberg chain [29] in the sense  that the spin-triplet excitation is a scattering continuum of two spin- objects. Furthermore, there is  a spin-singlet excitation, which is precisely degenerate with the triplet (see Example 2b below). This "ts nicely into a picture based on spin- objects: scattering states of two spinons give precisely one  spin 1 and one spin 0 multiplet "10. Finally, when we approach half-"lling, the spin-triplet   continuum constructed above goes over into the S"1 two-spinon scattering continuum of the half-"lled Hubbard model [34,39]. On the other hand, there are di!erences as well: in the less than half-"lled Hubbard model the Fermi momentum is generally incommensurate, which leads to incommensurabilities in the spin excitations (see Fig. 6). More importantly, it is always possible to combine any type of excitation with a particle}hole excitation. It is therefore not possible to distinguish the two-parametric spin-triplet excitation constructed above from the special case of a four-parametric excitation, where a particle}hole excitation sits &on top' of the spin-triplet excitation and where the momenta of the particle and the hole are "xed at the Fermi rapidity. In other words, due to the presence of gapless particle}hole

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Fig. 7. Dispersion of a k}K string excitation of length 2 for several values of ; and density n.

excitations there is an inherent ambiguity in the interpretation of the excitation spectrum on the basis of an O(1) calculation of energy eigenvalues of the Hamiltonian. Example 2b (Spin-singlet excitation). Let us now choose the occupation numbers as N, M "N/2!2, M "1. The corresponding state has the same quantum numbers as the ground   state. From (28)}(30) we "nd that there are N/2 vacancies for real K's and thus two holes corresponding to rapidities K and K . In other words, the excitation considered involves two   spinons. As far as the 2-string is concerned we "nd that the associated integer must be zero J "0.  The same shift as in Example 2a occurs for the I 's. Using (165) and (168) we obtain the energy and H momentum of the associated excitation E "!e (K )!e (K ) ,       P "p (K )#p (K )$pN/¸ . (178)       We see that the spin singlet is precisely degenerate with the spin triplet considered above. This is a consequence of the spin SU(2) symmetry of the Hamiltonian in zero magnetic "eld. Example 3 (k}K string of length 2). Let us consider the simplest excitation involving a k}K string. One possibility is to choose the occupation numbers as N, M "N/2!1, M "1. In addition we   keep the distribution of I "xed in such a way that I !I "1. It is easily checked that this H H> H excitation is allowed by (28)}(30). Its energy and momentum are E K "e (K), P K "p (K) , (179) I\  I\  where K3(!R,R). In Fig. 7 the dispersion of a k}K string of length 2 is shown for ;"0.5 and 2.0 and several values of density n. We see that the range of momenta collapses to zero as we

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243

Fig. 8. Dispersion of a k}K string excitation of length 2 for several values of ; and density n, where the contribution !2k!; has been subtracted.

approach half-"lling. At the same time the dressed energy approaches zero. This is in agreement with the results for a half-"lled band [39], where both the dressed energy and the range of momentum are identically zero. In order to further exhibit this collapse we subtract the o!set !2k!;. The resulting curves are displayed in Fig. 8. It is worth noting that equations for dispersion curves of k}K strings were considered earlier in [86].

6.6. Excitations in the half-xlled band In the limit of a half-"lled band kP0 the Fermi-rapidity Q tends to p and the excitation spectrum simpli"es drastically [32}34,38,39,87]. We "nd that e (K)"0 ∀n and the only nonL vanishing dressed energies are



du J (u) cos(uK)  , u cosh(u;/4)  du J (u)cos(u sin k)e\S3  i(k)"!2 cos k!;/2!2 . (180) u cosh(u;/4)  The charge excitations are now gapped as a result of the Mott}Hubbard transition. The complete spectrum of physical excitations is derived in detail in [39] (see in particular the appendix). It is given in terms of spinon and holon scattering states forming representations of SO(4). e (K)"!2 



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6.7. Relation to the root-density formalism There exists a second standard approach to the zero-temperature excitation spectrum in Bethe ansatz solvable models, which in the case of the Hubbard model has been employed for example in [36,37]. We call this the root-density formalism (RDF). In this approach one speci"es a distribution of (half-odd) integers I , JL , JL in the discrete Takahashi equations (25)}(27). One then takes the H ? ? thermodynamic limit using that the distribution of roots becomes dense, so that (25)}(27) turn into a set of coupled linear integral equations for root densities. From these one calculates energy and momentum via the thermodynamic limit of (32) and (33). We will now show how to relate the results of the Yang}Yang approach we implemented above to the RDF. For de"niteness we consider the example of a k}K string excitation of length n. We start by rewriting the integral equation for i(k) in the following way:



(1!K)*i" " I

/

dk [d(k!k)!cos kR(sin k!sin k)]i(k)

\/

"!2 cos k!k!;/2 .

(181)

Here the kernel of K is given by K(x, y)"cos yR(sin y!sin x). Eq. (181) is solved by the Neumann series



 / i(k)"! dk(2cos k#k#;/2)KJ(k, k) , (182) J \/ where KJ(x, y) denotes the l-fold convolution of the kernel K(x, y). Using (182) in (164) we obtain the following representation for the dressed energy of a k}K string of length n: e (K)"4 Re(1!(K!in;/4)!2nk!n; L

 

/ n;/4  / cos k KJ(k, k) (2 cos k#k#;/2) . (183) ! dk dk p (n;/4)#(sin k!K) \/ J \/ Let us now de"ne a function o@Q(k"K) by L



/

\/

cos k n;/4 . dk[d(k!k)!K(k, k)]o@Q(k"K)" L p (n;/4)#(sin k!K)

(184)

The solution of (184) is given by the Neumann series



 / dk n;/4 o@Q(k"K)" KJ(k, k)cos k . L p (n;/4)#(K!sin k) \/ J Using (185) in (183) we "nally obtain



e (K)"4 Re(1!(K!in;/4)!2nk!n;! L

/

\/

dk(2 cos k#k#;/2) o@Q(k"K) . L

(185)

(186)

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6.8. Summary In this section we have reviewed the Yang}Yang approach [69] to the thermodynamics of the Hubbard model [27]. We have shown how to express the Gibbs free energy per site (157) in terms of the solutions of the in"nite set of coupled non-linear integral equations (154)}(156). By taking the zero-temperature limit of the thermodynamic equations we have obtained the complete classi"cation of the spectrum of elementary excitations of the Hubbard model below half-"lling for vanishing magnetic "eld. We emphasize that this approach is based on Takahashi's string hypothesis [27] (see Section 2.3). It does not only provide the dispersion curves of all elementary excitations in the zerotemperature limit, but also a set of &selection rules' (25)}(27). These rules determine the set of physical excitations, i.e. the allowed combinations of elementary excitations (see Section 6.5 and, for the half-"lled case, [38,39]). The numerical calculation of the Gibbs free energy from (157) or (159) is di$cult, since it involves the solution of an in"nite number of coupled non-linear integral equations (154)}(156). A truncation scheme has to be introduced [79,80], which restricts the numerical accuracy of the calculation. In the next section we present a di!erent approach to the thermodynamics of the Hubbard model, which circumvents such di$culties.

7. Thermodynamics in the quantum transfer matrix approach In this section we present a treatment of the thermodynamic properties of the Hubbard chain in a lattice path integral formulation with a subsequent eigenvalue analysis of the matrix describing transfer along the chain direction (quantum transfer matrix, QTM). This approach has several advantages. First, the analysis is very simple as only the largest eigenvalue of the QTM is necessary in order to calculate the free energy. This has to be compared with the traditional TBA [27] (see Section 6) where all eigenvalues of the Hamiltonian have to be taken into account. Second, the number of &density functions' and integral equations obtained below is xnite in contrast to [27] (see Section 6.1). Finally, the "nite-temperature correlation lengths can be derived through a calculation of next-largest eigenvalues of the QTM [47,48]. This however, will not be explored in this report. Within the QTM approach we obtain several results of which we want to point out the conceptual achievements. First, the data obtained for various physical quantities agree with those obtained in [79,80] based on Takahashi's string hypothesis. Judging from this we do not see any evidence for a failure of Takahashis's formulation based on strings. Second, in the low-temperature limit our integral equations quite naturally yield the Tomonaga}Luttinger liquid picture of separate spin and charge contributions to the free energy as suggested in [88,89]. Mathematically, the dressed energy formalism known from the ground-state analysis is recovered. 7.1. The classical counterpart There are many direct path integral formulations of the Hubbard model, see for instance [90,91]. For our purposes we want to keep the integrability structure as far as possible. To this end it

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proved useful to employ the exactly solvable classical model corresponding to the Hubbard chain, namely Shastry's model which is a two-sublattice six-vertex model with decoration [56]. More precisely, the vertex weights of the classical model for the case ;"0 are given by the product of the vertex weights of two six-vertex models (with components p and q): l (u)"  lN (u)lO (u), where   lN (u)"(cos(u)#sin(u))#(cos(u)!sin(u))pX pX #(p>p\#p\p>) . (187)          Taking account of the ;O0 interactions, the following local vertex weight operator (denoted by S) was found [56]: S (v, u)"cos(u#v)cosh(h(v, ;)!h(u, ;))l (v!u)   # cos(v!u)sinh(h(v, ;)!h(u, ;))l (u#v)pX qX , (188)    where sinh(2h(u, ;)) " : (;/4)sin(2u). The Yang}Baxter equation for triple S matrices was conjectured [56], but only recently proved in [54]. The so-called ¸-operator is related to S by ¸ (u)"S (u,0) , (189) GE GE where i and g are indices referring to the ith lattice site and the auxiliary space, respectively. For the ¸-operator the proof of the Yang}Baxter equation with S as intertwiner was already given in [56]. The commutativity of the row-to-row transfer matrix  T(u) " : tr“ ¸ (u) (190) GE G is a direct consequence. Next, we de"ne R (u, v)"P S (v, u)" , where P is the permutation matrix. The    3\3  matrix elements RIJ (u, v) will be considered as the local Boltzmann weights associated with vertex ?@ con"gurations a, b, k, l on the lower, upper, left, and right bond, where the spectral parameters u and v &live' on the vertical and horizontal bonds, respectively. For later use we introduce RM (u, v) and RI (u, v) (u and v associated with the vertical and horizontal bond) by clockwise and anticlockwise 903 rotations of R, or in matrix notation RM IJ (u, v)"R@? (v, u), RI IJ (u, v)"R?@ (v, u) . (191) ?@ IJ ?@ JI Similar to (189) and (190) we can associate a row-to-row transfer matrix with RM . We note the Hamiltonian limits T(u)"exp(iP#uH#O(u)) and T M (u)"exp(!iP#uH#O(u)). Consequentially, the partition function of the Hubbard chain at "nite temperature ¹"1/b is given by Z" lim tr e\@&" lim lim tr [T(u)T M (u)]," . (192) S@, * * , We regard the resulting system as a "ctitious two-dimensional model on a ¸;N square lattice. Here N is the extension in the "ctitious (imaginary time) direction, sometimes referred to as the Trotter number. The lattice consists of alternating rows each being a product of only R weights or of only RM weights, respectively. Now by looking at the system in a 903 rotated frame which turns

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RM and R weights into R and RI weights, it is natural to de"ne the &quantum transfer matrix' (QTM) by , G\ IG (!u, v) R +T (u, v),++@,, " : “ RIG\ I IGG IGG> (u, v) , (193) ? ? @G\ ? @ /2+ + , I G which is identical to the column-to-column transfer matrix of the square lattice for v"0. The interchangeability of the two limits (¸, NPR) [40,81] leads to the following expression for the partition function: Z" lim lim tr [T (u"b/N,0)]* . (194) /2+ , * There is a gap between the largest and the second largest eigenvalues of T (u,0) for "nite b. /2+ Therefore the free energy per site is expressed just by the largest eigenvalue K (u,0) of T (u,0),

 /2+ f"!¹ lim ln K (u"b/N,0) . (195)

 , It is relatively simple to see that (193) is integrable, i.e. a family of commuting operators for variable v and "xed u. A non-vanishing chemical potential k and magnetic "eld B can be incorporated as they merely lead to trivial modi"cations due to twisted boundary conditions for the QTM (cf. [48]). 7.2. Diagonalization of the quantum transfer matrix Here we summarize the main results of [44] where the diagonalization of (193) on the basis of an algebraic Bethe ansatz was performed. Note that the general expression for the eigenvalue of the quantum transfer matrix is quite complicated [44], but simpli"es considerably at v"0 and uP0, K K(v"0)"e@3(1#e@I> )(1#e@I\ )u, “ z . (196) H H The numbers z are charge rapidities satisfying Bethe ansatz equations which are most transparH ently written in terms of the related quantities s "(1/2i)(z !1/z ) . (197) H H H For these rapidities s and additional rapidities w the coupled eigenvalue equations read H ? e\@I\  (s )"!q (s !i;/4)/q (s #i;/4) , H  H  H e\@Iq (w #i;/2)/q (w !i;/2)"!q (w #i;/4)/q (w !i;/4) , (198)  ?  ?  ?  ? where we have employed the abbreviations for products over rapidities q (s)"“ (s!s ), q (s)"“ (s!w ) .  H  ? H ?

(199)

 We note that our conventions for the magnetic "eld in this article are di!erent from [44]. In [44] the magnetic "eld was denoted by H"2B.

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The function is de"ned by

(s)"[(1!z /z(s))(1!z /z(s))/(1#z /z(s))(1#z /z(s)), , \ > \ > z(s)"is(1#((1!1/s)) ,

(200)

where z(s) possesses two branches. The standard (&"rst') branch is chosen by the requirement z(s)K2is for large values of s, and the branch cut line [!1,1]. The numbers z are de"ned by ! z "exp(a)(tan u)!, where sinh(a)"!(;/4) sin 2u. ! As usual, equations like (198) which are identical in structure to (7) and (8) have many solutions. In our approach to the thermodynamics just the largest eigenvalue of the QTM matters. The corresponding distribution of rapidities is relatively simple. For k"B"0 the rapidities are all situated on the real axis. Naturally, for "nite k and B we have modi"cations which, however, do not a!ect qualitative aspects of the distribution. In a similar way a path integral formulation of the Hubbard chain and a diagonalization of the corresponding QTM was employed in [41}43,92]. In [92] the eigenvalue equations were studied numerically for "nite Trotter number and the case of half-"lling. For "nite doping the direct solution of the Bethe ansatz equations is much more involved as the rapidities deviate from the real axis and move into the complex plane. Here severe stability problems arise in numerical studies. In [43] an attempt was undertaken to study the limit of in"nite Trotter number analytically and to derive a set of non-linear integral equations. Unfortunately, these equations were rather ill posed with respect to numerical evaluations. In a rather recent publication [44] all of these problems were solved by the derivation of a set of well-posed non-linear integral equations which allow for the study of "nite doping and the limit of in"nite Trotter number. This is the topic of the next section. 7.3. Non-linear integral equations In this section we are concerned with the derivation of well-posed integral equations equivalent to (198) for the largest eigenvalue of the QTM and thus for the free energy per site (see (195)). We introduce a set of auxiliary functions (b, c, and c ) described in more detail in (207) below. These auxiliary functions are complex functions, however mostly evaluated close to the real axis. In terms of the auxiliary functions the Gibbs free energy per site is expressed in several ways ¹ ; f"!k! ! 2pi 4 #

¹ 4pi

 

ln

L

¹ ; " ! 2pi 4

L



¹ [ln z(s)] ln(1#c#c ) ds# 4pi L



ln

L



z(s!i;/2)  ln(1#b(s)) ds z(s)



z(s#i;/2)  ln(1#1/b(s)) ds z(s)

[ln z(s)] ln

1#c#c ¹ ds! c 2pi

For yet another expression see [44].

(201)



L

[ln z(s!i;/2)] ln(1#c(s)) ds .

(202)

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Eqs. (201) and (202) have to be compared with Eqs. (157) and (159) which give the free energy in the string-based approach of Takahashi. In contrast to the string-based approach, the auxiliary functions b, c and c entering (201) and (202) satisfy a closed set of "nitely many (non-linear) integral equations, ln b"!2bB#K 䊐 ln(1#b)!K
ln(1#1/c ) ,   ln c"!b;/2#b(k#B)#u!KM 䊐 ln(1#1/b)!KM
ln(1#c ) ,   ln c "!b;/2!b(k#B)!u#K 䊐 ln(1#b)#K
ln(1#c) .  

(203)

Here we have used the de"nition u(x)"!2bix(1!1/x

(204)

and have introduced the notation



(g
f)(s)"

g(s!t)f(t) dt

(205)

L

for the convolution of two functions g and f with contour L surrounding the real axis at in"nitesimal distance above and below in anticlockwise manner. The de"nition of 䊐 is similar, with integration contour surrounding the real axis at imaginary parts $;/4. The kernel functions are rational functions, K (s)"(;/4)p/s(s#i;/2), 

K (s)";/4p/s(s!i;/2) , 

K (s)"(;/2)p/(s#;/4) . 

(206)

Next, we want to point out that the function b will be evaluated on the lines Im s"$;/4. The functions c and c need only be evaluated on the real axis in"nitesimally above and below the interval [!1,1]. Also the convolutions involving the &c functions' in (203) can be restricted to a contour surrounding [!1,1] as these functions are analytic outside. Lastly, we want to comment on the derivation of (202) and (203). The explicit expressions of the functions b, c, c are b"(l #l #l #l )/(l #l #l #l ) ,         l #l l #l #l #l      , c"  l #l l #l #l #l #l #l #l #l          

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l #l l #l #l #l      c "  , (207) l #l l #l #l #l #l #l #l #l           where the functions l and l are H H l (s)"j (s!i;/4)e@ >(s) \(s), l (s)"j (s#i;/4) (280) H H H H and the j are de"ned in terms of the q and q functions, i.e. in terms of the Bethe ansatz rapidities H   j (s)"e@I>  (s!i;/4)/q (s!i;/4), j (s)"e@Iq (s!i;/2)/q (s)q (s!i;/4) ,       j (s)"q (s#i;/2)/q (s)q (s#i;/4) ,     j (s)"e@I\ / (s#i;/4) q (s#i;/4) . (209)   The functions de"ned in (207) are proven to satisfy a set of closed functional equations which can be transformed into integral form (203), cf. [44]. Also (202) follows from (207) after a lengthy yet direct calculation. The merit of (202) and (203) is that this formulation does no longer make any reference to the Bethe ansatz equations! Hence, the calculation of an in"nite set of discrete rapidities is replaced by the computation of analytic functions for which much more powerful tools are available. 7.4. Analytical solutions of the integral equations Before numerically studying various thermodynamic properties for general temperatures and particle concentrations we want to give some analytic treatments of limiting cases of the Hubbard model. 7.4.1. Strong-coupling limit In the strong-coupling limit ;PR at half-"lling (k"0) the Hubbard model is expected to reduce to the Heisenberg chain. Indeed, in the strong-coupling limit we "nd that c, c P0. Hence, the only non-trivial function determining the eigenvalue of the QTM is b, see the "rst expression of (202). The integral equation for b as obtained from (203) is identical to that obtained directly for the thermodynamics of the Heisenberg model [44]. 7.4.2. Free-fermion limit Next, let us consider the limit ;P0 leading to a free-fermion model, however representing a non-trivial consistency check of the equations. Indeed, the auxiliary functions can be calculated explicitly. Finally, the free energy per site reads



¹ p f"! ln+[1#exp((k#B#2 cos k)/¹)] [1#exp((k!B#2 cos k)/¹)], dk , 2p \p which, as desired, is the result for free tight binding electrons.

(210)

7.4.3. Low-temperature asymptotics The low-temperature regime is the most interesting limit as the system shows Tomonaga}Luttinger liquid behavior. We want to describe the relation of the non-linear integral

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equations to the known dressed energy formalism [88,89] of the Hubbard model. This represents a further and in fact the most interesting consistency check. For ¹"1/bP0 we can simplify the non-linear integral equations as they turn into linear integral equations, however in the generic case (B and kO0) with xnite integration contours. This can be seen as follows. To be speci"c let us adopt "elds B'0, k40 (particle density n41). In the low-temperature limit several auxiliary functions tend to zero, namely b(s)P0 on the line Im s"!;/4, c(s)P0 above and below the real axis, and 1/c (s)P0 just below the real axis, i.e. on the line Im s"!o. The remaining non-trivial functions are b on the line Im s"#;/4 and 1/c just above the real axis for which we introduce the notation b(j)"b(j#i;/4), c(j)"1/c (j#ie) .

(211)

We note that "b","c"<1 for "x"(j ,p and "b","c";1 for "x"'j ,p (212)     for certain crossover values j , p . The slopes for the crossover are steep, so that the following   approximations to the integral equations (203) are valid:

 

ln b" ! @ ln c" # A

H

\H H



k (j!j) ln b(j) dj# 

I

\I

k (j!sin k) cos k ln c(k) dk , 

k (sin k!j) ln b(j) dj . 

(213) (214)

\H where k (j)"K (j!i;/4)"K (j#i;/4) and k (j)"K (j). In order to facilitate comparison      with the dressed energy formalism we also introduced a new integration variable k leading to a change of the boundaries of integration, p Pk "arcsin p . The driving terms in (213) and (214)    are related to the bare energie e"B, e"!2 cos k!k!;/2!B Q A

(215)

by

"!be#O(1/b) and "!be#O(1/b) . (216) @ Q A A Therefore, we "nd the following connections between auxiliary functions and the dressed energy functions: ln b"!b e #O(1/b), ln c"!b e #O(1/b) . (217) Q A For a comparison with [88,89] note the di!erent normalization of the chemical potential. For a comparison with the results in Section 6 we set the magnetic "eld equal to zero. Then e"B"0, and it can be seen that j "R. Setting Q"k we can identify (213) and (214) in the Q   zero-temperature limit with (165). We "nd !lim ¹ ln b(j)"lim e (j)"lim ¹ ln g (j) ,   2 2 2

(218)

!lim ¹ ln c(j)"lim i(k)" lim ¹ ln f(k) . 2 2 2

(219)

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The free energy also admits the above approximation scheme, yielding up to O(¹)-terms the low-temperature expansion f"e !(p/6)(1/v #1/v )¹ . (220)    Here the de"nitions of the sound velocities and the ground-state energy are standard. The additive occurrence of 1/v and 1/v on the right-hand side of (220) is a manifestation of spin}charge   separation in the one-dimensional Hubbard model, due to which each elementary excitation contributes independently to (220). The velocities v and v typically take di!erent values.   7.4.4. High-temperature limit Finally, we consider the high-temperature limit ¹PR with B, ; as well as bk "xed ensuring a "xed particle density n. The integral equations turn into algebraic equations which are easily solved, resulting in the high-temperature limit S"2 ln(2/(2!n))!n ln (n/(2!n))

(221)

for the entropy, as expected by counting the degrees of freedom per lattice site. Especially at half-"lling, n"1, this equals to S"ln(4). 7.5. Numerical results Here we show numerical results for speci"c heat C, magnetic susceptibility s , and charge

susceptibility s for half-"lling and small doping, see Figs. 9}11. In addition, we aim at a compari son of results [79,80] obtained within the thermodynamic Bethe ansatz [27] based on the string hypothesis, and results [44] obtained within the quantum transfer matrix approach. In essence, we observe convincing agreement of the data obtained within the two absolutely di!erent approaches. We conclude that there is no indication for any failure of Takahashi's formulation of thermodynamics based on the string hypothesis. Of course rather small di!erences exist in the sets of data. A more thorough comparison shows deviations of the data presented in Figs. 9 and 10. This is nothing but expected due to the truncation procedure adopted in [79,80]. Instead of dealing with an in"nite set of integral equations for density functions (151), (154)}(156) for string excitations of spin rapidities (22) and charge rapidities (23), a "nite subset was taken into account. In the case of complex charge rapidities, strings describe excitations with gap. These degrees of freedom are less sensitive to errors introduced by the truncation than strings involving only spin rapidities which describe gapless excitations. In fact, the agreement of the data for the charge susceptibility s is best, small deviations are observed in C and s . 

The advantage of the QTM approach over the traditional TBA is threefold. First, in [44] the Hubbard chain with very strong doping was analyzed (apparently not possible in the traditional TBA) and quite unexpected structures in the susceptibility data were found. In the magnetic susceptibility only traces of spinon excitations were visible. The charge susceptibility, however, exposed holon signatures and additional maxima due to spinon excitations indicating deviations from the concept of spin}charge separation. Second, the accuracy of numerical data obtained within the QTM approach is much higher as it is more e$cient to deal with a set of integral equations which is strictly "nite from beginning. Finally, even correlation lengths can be calculated within the QTM approach although not presented in detail yet.

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Fig. 9. Speci"c heat C versus temperature for ;"8 and particle densities n"1,0.9,0.8 and 0.7. The left graph shows results obtained within the string based approach, the right graph shows results obtained by the quantum transfer matrix method.

Fig. 10. Magnetic susceptibility s versus temperature for ;"8 and particle densities n"1,0.9,2,0.5. The left graph

shows results obtained within the string based approach, the right graph shows results obtained by the quantum transfer matrix method.

7.6. Equivalence of string-based thermodynamics with QTM approach Here we want to address the fundamental and perhaps puzzling question how two exact, however completely di!erent formulations of the thermodynamics of the Hubbard model as presented in Sections 6 and 7 can exist. It is very important to understand this problem as the

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Fig. 11. Charge susceptibility s versus temperature for ;"8 and particle densities n"1,0.9,0.8 and 0.7. The left graph  shows results obtained within the string-based approach, the right graph shows results obtained by the quantum transfer matrix method.

means of derivation and the mathematical properties of both approaches are seemingly lacking any similarities. In the remainder of this section we restrict ourselves to a sketch of the general mathematical structures underlying the relation of the traditional thermodynamical formulation based on typically in"nitely many density functions and the new formulation with strictly "nitely many auxiliary functions. This relation has been worked out for a number of models including spin chains [47] and correlated fermion models like t}J systems [93], not yet however for the Hubbard chain. In the following, we want to introduce the general technique for spin- Heisenberg models  and related systems. The starting point is an integrable Hamiltonian corresponding to an exactly solvable classical model as discussed in Section 7.1. For this model a path integral representation is formulated leading to an integrable QTM T(v) with eigenvalue equation ¹(v)"( f (v!j)q(v#j)#f (v)q(v!j))/q(v) ,

(222)

where f (v) is a known function and q(v) is a polynomial or a product over trigonometric functions with zeros exactly at the Bethe ansatz rapidities. By a line of reasoning similar to that presented in Section 7.3 one non-linear integral equation can be derived which yields full information on the free energy of the model. The key to understanding the relation of the standard TBA to the above approach is the notion of fusion algebras and inversion identities. By repeated use of the fusion procedure a set of transfer matrices T(v)"T(v),T(v),T(v),2, is generated satisfying TO(v)TO(v#j)"f (v!j) f (v#j)I#T O>(v)T O\(v#j) ,

(223)

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where q takes integer values, q"1,2,2, and T (v) is proportional to the identity operator. Introducing operators YO(v)"T O\(v#j)TO>(v)/f (v!j) f (v#qj) ,

(224)

we "nd the extremely useful inversion identity hierarchy YO(v)YO(v#j)"[I#YO\(v#j)][I#YO>(v)] .

(225)

This set of functional equations can be transformed into a set of non-linear integral equations for the largest eigenvalue of T(t). These integral equations appear to be identical to the equations obtained along the traditional TBA approach! Several remarks are in order. Relations (225) were derived in [94,95] starting with the TBA equations, however the connection to the fusion hierarchies was not made. In contrast to the TBA integral equations the functional equations (225) (as of course Eq. (222)) admit more than one solution. The physical meaning of these solutions becomes clear only through the microscopic construction of transfer matrices. The next-leading eigenvalues describe the correlation lengths of static correlation functions at "nite temperature. Such applications are investigated in current research. Some results have been reported in [93,96}98]. In summary, the traditional TBA equations are not in contradiction with the new thermodynamics based on the QTM. Rather on the contrary, the QTM approach represents a uni"ed approach to both sets of integral equations. The central object is the study of the largest eigenvalue of the QTM. This can be achieved by two di!erent methods, either by a Bethe ansatz (222) or by use of the fusion hierarchy (223). In physical terms, the appearance of the fusion hierarchy is related to the pole structure of the intertwiner, i.e. the R-matrix satisfying the YBE. In turn the R-matrix is related to the S-matrix of particles, the poles are related to bound states (strings) which in turn are the central objects of the traditional analysis of the thermodynamics of integrable systems. In this way we observe mathematical and physical connections between the initially so di!erently looking approaches.

8. Conclusions In this article we considered fundamental questions concerning the exact solution of the one-dimensional Hubbard model. In Appendices A and B we presented a detailed derivation of the wave function. We studied solutions of the Lieb}Wu equations. We concentrated our attention on k}K strings. Unlike K strings they were never carefully studied in the literature before. We arrived to the conclusion that, for "xed coupling and large lattice size, k}K strings deviate from their ideal positions in a controllable way. In the in"nite chain limit k-K strings approach their ideal positions. We also reviewed the thermodynamics of the model. There are two di!erent approaches to thermodynamics: one is based on strings, whereas the other one is not. Both approaches show convincing agreement for the calculation of the bulk thermodynamic properties of the Hubbard model. Passing to the zero-temperature limit we obtained the dispersion curves of all elementary excitations at zero magnetic "eld, below half-"lling.

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We would like to conclude with pointing out some interesting open problems: (i) Perhaps the most fundamental open problem is the calculation of the norm of the wave function (2)}(6). Formulae for norms of wave functions are known for Bethe ansatz solvable models with only one level of Bethe ansatz equations, like the Bose gas with delta interaction [83,99] or the XXX and XXZ spin! chains [99] (see also [82]). For these models the norms  are expressed as determinants of the Jacobians of the Bethe ansatz equations. The calculation of the norm of the Bethe ansatz wave functions may be considered as a "rst step towards the calculation of determinant representations of correlation functions [82]. (ii) Another interesting open problem which we already mentioned above is the calculation of the "nite-temperature correlation length within the quantum transfer matrix approach. This requires to calculate the second largest eigenvalue of the quantum transfer matrix. One has to overcome certain technical subtleties coming from the fact that the Hubbard model is a model of fermions. A formalism capable of calculating the correlation length of integrable fermionic systems has recently been developed [97,98,100]. (iii) At zero temperature some of the correlation functions of the Hubbard model show a power-law decay. Conformal "eld theory naturally describes these powers (the set of conformal dimensions) [88]. Still, there are open problems within the conformal approach. For example, since the expansion of the lattice operators in terms of conformal "elds is not known explicitly, the resulting expressions contain unknown amplitudes. Some of these amplitudes may actually vanish. Recently [102] the vanishing of the amplitudes corresponding to density correlations for the half-"lled model was shown by use of the SO(4) symmetry. For a more thorough understanding of correlation functions an identi"cation of the operators which are of interest for the Hubbard model with the standard operators of conformal "eld theory would be important. Recent work on a scaling limit of the Hubbard model [103}105] may prove to be useful in this context. (iv) The predictions of the conformal approach to correlation functions are limited to large distances, corresponding to very low energies. Theoretically as well as from the point of view of recent experiments on quasi-one-dimensional structures in solids (e.g. [13,14]) it would be highly appreciable to have a method for the calculation of correlation functions at all energy scales. This problem was recently tackled in [106] within the form factor approach [107}109], which was originally designed for integrable (1#1)-dimensional quantum "eld theories. In [106] it was argued that the form factor approach might as well apply to the half-"lled Hubbard model, and a formula for the two-spinon form factor of the spin-operator S> was H presented. It would be interesting to extend the result to form factors of electronic creation and annihilation operators, as such kind of extension could be directly applied to the interpretation of the angle-resolved photo-emission spectroscopy data of [13,14]. (v) Despite the progress in the understanding of the mathematical structure of the Hubbard model, which was achieved over the past few years and which we brie#y discussed in the introduction, we still feel uncomfortable with the present stage of our knowledge. Shastry's R-matrix [55}57], which is the key for our present understanding of the algebraic structure behind the

 In the context of condensed matter physics conformal "eld theory is equivalent to Luttinger liquid theory [101].

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Hubbard model, is unusual as compared to R-matrices of other integrable models. It does not possess the so-called di!erence property, i.e. it is not a function of the di!erence of the spectral parameters alone. The S-matrix at half-"lling [38], on the other hand, possesses the di!erence property and can therefore be associated with a Y(su(2))Y(su(2)) Yangian [61]. The precise relation between R- and S-matrix is only understood in the rather simple situation of an empty band [62,63]. Because of the lack of the di!erence property we can neither "nd a boost operator for the Hubbard model by the reasoning of [110] nor can we associate a spectral curve with it. Another problem is the dimension of the elementary ¸-operator related to Shastry's R-matrix, which is 4;4 (rather than 3;3 as one could guess naively from the fact that the Bethe ansatz for the Hubbard model has two levels). For this reason there are too many candidates for creation and annihilation operators in the algebraic Bethe ansatz [56,58,59]. Again this redundancy has only been partially understood in the empty band case [63]. The known algebraic Bethe ansatz [58,59] is of involved structure and hopefully will be simpli"ed in the future.

Acknowledgements We would like to thank C. M. Hung for drawing "gures 1-3 for us. We are indebted to H. Frahm, N. Kawakami, B. M. McCoy, A. Schadschneider, J. Suzuki, A. M. Tsvelik and M. Takahashi for helpful and stimulating discussions. This work was supported by the EPSRC (F.H.L.E), by the Deutsche Forschungsgemeinschaft under grant numbers Go 825/2-1 (F.G.) and Kl 645/3 (A.K.) and by the National Science Foundation under grant number PHY-9605226 (V.E.K). A.K. acknowledges support by the Sonderforschungsbereich 341, KoK ln-Aachen-JuK lich.

Appendix A. Derivation of the wave function A.1. General setting The expression (2)}(6) for the Bethe ansatz wave function of the Hubbard model was "rst presented by Woynarovich [32]. The purpose of this appendix is to give a detailed derivation of Woynarovich's wave function. We will make use of results obtained in Refs. [5,25] and of the quantum inverse scattering method [82,70]. Other derivations of the Bethe ansatz wave functions for the Hubbard model can be found for example in [111,112]. The outline of this appendix is as follows. In Section A.1 we de"ne the wave function t(x ,2, x ), for N electrons and derive the "rst quantized SchroK dinger's equation for the Hubbard  , model. In Sections A.2 and A.3 we present the explicit solution of the SchroK dinger equation with periodic boundary conditions for cases of 2 and 3 electrons, respectively. We treat these cases in considerable detail for pedagogical reasons. Finally, in Section A.4 we discuss the general case of N electrons. An important ingredient in the construction of the wave function is the exact solution of an inhomogeneous spin- Heisenberg model. The essence of the nested Bethe ansatz procedure 

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employed in constructing wave functions for the Hubbard Hamiltonian is the reduction of this problem to a simpler one, which involves only the spin degrees of freedom. The dynamics of the spin degrees of freedom are described by an inhomogeneous Heisenberg model, and its exact solution constitutes the &nesting' of the Bethe ansatz procedure. We summarize the algebraic Bethe ansatz solution of the inhomogeneous Heisenberg model in Appendix B. Let us recall the explicit form of the Hamiltonian * * H"! (c> c #c> c )#; (n !)(n !) . (A.1) HN H>N H>N HN Ht  Hs  H Nts H As the total number of electrons N and the number of electrons with spin-down M are good quantum numbers, we can use them to label eigenstates of (A.1) (A.2) "N, M2" t(x 2x ;p ,2,p )c-   2c- , , "02 . V N  ,  , V N + , + , NH VI Here + H , denotes summation over all N!/((N!M)!M!) possible spin con"gurations with M down N spins. Due to the anticommutation relations between the fermion operators, we may assume, without loss of generality, that the amplitudes t are totally antisymmetric t(x  ,2, x , ;p  ,2,p , )"sign(P)t(x ,2, x ;p ,2,p ) , . . .  ,  , .

(A.3)

where P"(P , P ,2, P ) is a permutation of the labels +1,2,2, N,, i.e. an element of the   , symmetric group S . The antisymmetry property (A.3) implies that the summation over spin , con"gurations in (A.2) is redundant. Indeed one "nds that (A.4) "N, M2"N!/(N!M)!M! t(x ,2, x ;p ,2,p )c-   2c- , , "02 , V N  ,  , V N + , VH where (p ,2,p )3S is arbitrary. In order to derive the SchroK dinger equations it is therefore  , , convenient to work with the following simpli"ed expression for general eigenstates of H: "N, M;r2" t(x ,2, x ;p ,2,p )c-   2c- , , "02 . V N  ,  , V N + , VH It is now straightforward to show, that the eigenvalue problem H"N, M;r2"E"N, M;r2 ,

(A.5)

(A.6)

implies the following SchroK dinger's equation for the wave function t [5]: , ! t(x ,2, x #s,2, x ;r)#; d(x , x )t(x ,2, x ;p ,2,p )  H , H I  ,  , H Q! HI "(E#;N/2!;¸/4)t(x ,2, x ;p ,2,p ) .  ,  , Here d(a, b) denotes the Kronecker delta.

(A.7)

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259

A.2. Two electrons Let us now explicitly construct the wave function for the case of two electrons, N"2. The SchroK dinger equation is of the form !t(x !1, x ;p ,p )!t(x #1, x ;p ,p )!t(x , x !1;p ,p )!t(x , x #1;p ,p )                 # ;d(x , x )t(x , x ;p ,p )"(E#;!;¸/4)t(x , x ;p ,p ) . (A.8)           As long as x (x or x 'x (A.8) reduces to the SchroK dinger equation for free electrons on     a lattice and its solutions are therefore just superpositions of plane waves. When the electrons occupy the same site, they interact. This can be thought of in terms of a scattering process. Due to integrability this scattering is purely elastic, which means that the momenta of the two electrons are individually conserved. Thus, the most that can happen is that the electrons exchange their momenta. These considerations lead to the famous &nested' Bethe ansatz form for the wave functions, which we will discuss next. Let Q be a permutation of the labels of coordinates, i.e. Q"(Q , Q )3+(1,2),(2,1),. In the §or'   Q de"ned by the condition x  4x  the nested Bethe ansatz for the wave function is / /  t(x , x ;p ,p )" sign(PQ)A / / (k  , k  )exp i k x . (A.9)     N N . . .H /H .Z1 H Substituting (A.9) into (A.8) for the case x Ox we obtain   E"!(2 cos k #2 cos k )!;#;¸/4 . (A.10)   When x "x we have to &match' the wave function de"ned in the two sectors Q"(12) and (21).   This requires single valuedness






t(x, x;p ,p )"[A   (k , k )!A   (k , k )]exp(i[k #k ]x) NN       NN   "[A   (k , k )!A   (k , k )]exp(i[k #k ]x) . (A.11) NN   NN     In addition, the SchroK dinger Eq. (A.8) for x"x "x needs to be ful"lled, which yields the   condition !e\ I A   (k , k )#e\ I A   (k , k )#e I A   (k , k ) NN   NN   NN   ! e I A   (k , k )!e I A   (k , k )#e I A   (k , k ) NN   NN   NN   # e\ I A   (k , k )!e\ I A   (k , k ) NN   NN   #;[A   (k , k )!A   (k , k )] NN   NN   "!2(cos k #cos k )[A   (k , k )!A   (k , k )] . (A.12)   NN   NN   By means of (A.12) and (A.11) we can express two of the four amplitudes A / / (k  , k  ) in terms of N N . . the other two. A short calculation gives A   (k , k )"[!(;/2i)/(sin k !sin k !;/2i)]A   (k , k ) NN     NN   # [(sin k !sin k )/(sin k !sin k !;/2i)]A   (k , k ) .     NN  

(A.13)

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Eq. (A.13) has a natural interpretation in terms of a scattering process of two particles. In order to see this we rewrite it as (A.14) A   (k , k )" SN O (k , k )A   (k , k ) , NO   OO   NN   O O where S(k , k ) is the two-particle S-matrix with elements   SN O (k , k )"[!(;/2i)/(sin k !sin k !;/2i)]PN O NO   NO   # [(sin k !sin k )/(sin k !sin k !;/2i)]IN O . (A.15)     NO Here I is the identity operator IN O "d   d   and P is a permutation operator PN O " N O N O NO NO d  d  . N O N O In the next step we want to impose periodic boundary conditions on the wave function t(¸#1, x ;p ,p )"t(1, x ;p ,p ) ,       t(0, x ;p ,p )"t(¸, x ;p ,p ) ,       t(x , ¸#1;p ,p )"t(x ,1;p ,p ) ,       t(x ,0;p ,p )"t(x , ¸;p ,p ) .       A short calculation shows that this imposes the following conditions on the amplitudes:

(A.16)

A / / (k  , k  )"exp(ik  ¸)A / / (k  , k  ) , (A.17) N N . . . N N . . where P, Q3S are arbitrary. In principle, we now could simply solve (A.17) and thus determine  the quantization conditions for the momenta k . Rather than proceeding in this way, we will  introduce some seemingly unnecessary formalism, which however will be very useful for treating the case of more than two electrons. We de"ne an auxiliary spin model on a lattice with N sites, i.e. a lattice formed by the electrons. On every site j there are two allowed con"gurations "!2 and " 2 , corresponding to spin-up and H H -down respectively. Next, we de"ne spin operators S!X acting on the resulting Hilbert space as H follows: S\" 2 "0"S>"!2 , S\"!2 "" 2 , S>" 2 ""!2 , H H H H H H H H H H SX" 2 "!" 2 , SX"!2 ""!2 . H H  H H H  H We now de"ne a particular set of states in the spin model in the following way [70]:

(A.18)

, l (A.19) "k  ,2, k , 2" A  2 , (k  ,2, k , ) “ (S\ l )\N "02 . N . . . . N l 2 N N, !  Here we have used conventions where p "! corresponds to 1 and p " corresponds to !1. The H H set of Eqs. (A.14), due to the SchroK dinger equation, now induces the following between states in the spin model: "k  , k  2">(sin k  ,sin k  )"k  , k  2 , . . . . . .

(A.20)

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where the operator >HI is given by >HI(v , v )"[!(;/2i)/(v !v !;/2i)]I#[(v !v )/(v !v !;/2i)]P HI . (A.21)         Here I is the identity operator over the Hilbert space of the spin model and P HI is a permutation operator P HI"(I#4S ) S ) . (A.22)  H I Here SVWX are the spin operators de"ned in (A.18), where S!"SV$iSW . In order to derive (A.20) I I I I one simply uses the explicit form (A.15) of the S-matrix and the following properties of permutation operators: P"02""02, PS\P"S\, PS\P"S\ . (A.23)     The >-operators (which are related to the S-matrix (A.15) via multiplication by a permutation operator) were "rst introduced by Yang in his seminal 1967 paper [25]. On the level of the auxiliary spin model the periodic boundary conditions (A.17) translate into "k  , k  2"exp(ik  ¸)P"k  , k  2 . . . . . . With the use of (A.20) this is then transformed into

(A.24)

"k  , k  2"exp(ik  ¸)X(sin k  ,sin k  )"k  , k  2 , . . . . . . . where the operator XHI is given by

(A.25)

XHI(v , v )"P HI>HI(v , v )     "[!(;/2i)/(v !v !;/2i)]P HI#[(v !v )/(v !v !;/2i)]I . (A.26)       In other words, X is the S-matrix (A.15) viewed as an operator in the auxiliary spin model. We now make the crucial observation that the operator X is precisely the transfer matrix of an inhomogeneous spin- Heisenberg model on a 2-site lattice  X(sin k  ,sin k  )"q(sin k  " +sin k  ,sin k  ,) , (A.27) . . . . . where q is given by (B.10) with N"2. This is shown in full generality in (B.16). The periodic boundary conditions (A.17) can thus be rewritten as an eigenvalue problem for the transfer matrix q(K"sin k  " +sin k  ,sin k  ,) of an inhomogeneous Heisenberg model on 2-site . . . lattice (A.28) "k  , k  2"e *I. q(sin k  "+sin k  ,sin k  ,) " k  , k  2 . . . . . . . . The diagonalization of the transfer matrix q(K " +sin k  ,sin k  ,) is carried out in Appendix B. From . . the point of view of constructing eigenstates of the Hubbard Hamiltonian with periodic boundary conditions, we have succeeded in reducing the problem to a much simpler one, namely diagonalizing the transfer matrix of an inhomogeneous Heisenberg model. This is the essence of Yang's nested Bethe ansatz procedure. For the problem at hand we need to distinguish two cases, depending on the spins of the two electrons: E Two electrons with spin up: Here the appropriate eigenstate of q(K " +sin k  ,sink  ,) is found in . . the sector with no overturned spins, i.e. it is the ferromagnetic state with all spins-up. From (B.15)

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we see that its eigenvalue is equal to 1. The periodic boundary conditions for the case of two electrons with spin-up thus take the simple form e IH *"1,

j"1,2 .

(A.29)

These are precisely the Lieb}Wu Eqs. (7) and (8) for the case N"2, M"0. Using that the appropriate eigenstate of q(K " +sin k  ,sin k  ,) is equal to "02, we infer by comparison with (A.19) . . that all amplitudes are equal to 1. This then implies the following explicit form for the wave function in sector Q:






 . (A.30) t(x , x ;p ,p )" sign(PQ)exp i k x .H /H     H .Z1 This agrees with Woynarovich's result (4). E One electron with spin-up, one with spin-down: Now the appropriate eigenstate of q is found in the sector with one overturned spin. Using (B.15) we obtain the corresponding eigenvalue (A.31) 1!i;/[2(sin k  !j #i;/4)]"(sin k  !j !i;/4)/(sin k  !j #i;/4) ,  .  .  . where j ful"lls the Bethe ansatz equations of the inhomogeneous Heisenberg model on a 2-site  lattice  j !sin k #i;/4 H . (A.32) 1" “  j !sin k !i;/4 H H  Inserting (A.31) into (A.28) we obtain the following quantization conditions for the momenta k due H to periodic boundary conditions: exp(i¸k  )"(j !sin k  !i;/4)/(j !sin k  #i;/4) for P3S . (A.33) .  .  .  Eqs. (A.32) and (A.33) coincide with the Lieb}Wu equations (7) and (8) for the case N"2 and M"1. Let us also determine an explicit expression for the amplitudes A   (k , k ). Comparing result NN   (B.35) for the eigenstate of q(sin k  " +sin k  ,sin k  ,) with (A.19) we see that . . . 1 W\ j !sin k H !i;/4  . , (A.34) A   (k  , k  )" “ NN . . j !sin k H #i;/4 j !sin k W #i;/4  .  . H where y is the position of the down spin in the sequence p p . The wave functions (A.9) with   amplitudes (A.34) coincide with Woynarovich's result (4) for the case N"2, M"1.






A.3. Three electrons Let us now explicitly construct the wave function for the case of three electrons N"3. The SchroK dinger equation is of the form ! [t(x #s, x , x ;p ,p ,p )#t(x , x #s, x ;p ,p ,p )             Q! # t(x , x , x #s;p ,p ,p )]#; d(x , x )t(x , x , x ;p ,p ,p )       H I       HI

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263



3; ;¸ " E# ! t(x , x , x ;p ,p ,p ) .       2 4

(A.35)

In the sector de"ned by x  4x  4x  , where Q is a permutation of the labels +1,2,3,, the Bethe / / / ansatz wave function reads






 t(x , x , x ;p ,p ,p )" sign(PQ)A / / / (k  , k  , k  )exp i k x . (A.36)       N N N . . . .H /H .Z1 H Substituting (A.36) into (A.35) for the case x Ox Ox Ox we obtain the following expression     for the energies: E"!(2 cos k #2 cos k #2 cos k )!3;/2#;¸/4 . (A.37)    Single valuedness of the wave function now leads to a larger number of relations between the amplitudes. We have to consider the three cases t(x, x, x ;p ,p ,p ), t(x, x , x;p ,p ,p ),         t(x , x, x;p ,p ,p ). A simple calculation yields the following conditions on the amplitudes:     A / / / (k  , k  , k  )!A / / / (k  , k  , k  ) N N N . . . N N N . . . (A.38) " A / / / (k  , k  , k  )!A / / / (k  , k  , k  ) , . . . N N N . . . N N N where Q and P are arbitrary permutations of +1,2,3, and Q"Q ( j, j#1), P"P ( j, j#1). Our notations are such that for any permutation of N elements S"(S ,2, S )  , S ( j, j#1)"(S ,2, S , S , S , S ,2, S ) . (A.39)  H\ H> H H> , Let us consider a speci"c example of (A.38) in more detail. The wave function t(x, x, x ;p ,p ,p )     with x 'x can be expressed alternatively in sector Q"(1,2,3) or in sector Q"(2,1,3). Equating  the respective expressions (A.36) and using the orthogonality property of plane waves we obtain A    (k  , k  , k  )!A    (k  , k  , k  ) NNN . . . NNN . . . (A.40) " A / / / (k  , k  , k  )!A / / / (k  , k  , k  ) . N N N . . . N N N . . . This indeed coincides with the general result (A.38). We note that the Bethe ansatz wave function (A.36) by construction is antisymmetric under simultaneous exchange of spin and space variables. It is easy to see that this fact assures the SchroK dinger equation (A.35) to be satis"ed, when the three electron are occupying the same site. Moreover, this reasoning readily generalizes to the case of an arbitrary number of electrons in the following subsection. The only non-trivial case left to consider is the case of two electrons at the same site. Let us start with x "x "x(x . Using (A.36) in (A.35) we obtain the following condition on    the amplitudes: (k , k , k )#[e\ I. #e I. ]A    (k  , k  , k  ) N N N . . . NNN . . . # [e I. #e\ I. ]A    (k  , k  , k  )![e I. #e\ I. ]A    (k  , k  , k  ) NNN . . . NNN . . . # ;[A    (k  , k  , k  )!A    (k  , k  , k  )] NNN . . . NNN . . . "!2(cos k  #cos k  )[A    (k  , k  , k  )!A    (k  , k  , k  )] . . NNN . . . NNN . . . .

![e\ I. #e I. ]A

(A.41)

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Now making use of the fact that (A.41) and (A.38) are of the same structure as (A.12) and (A.11), we conclude that the following relation between amplitudes holds: (A.42) A    (k  , k  , k  )" SN O (k  , k  ) A    (k  , k  , k  ) , NO . . OON . . . NNN . . . O O where S is the two-particle S-matrix (A.15). All other cases of coinciding coordinates can be analyzed in exactly the same way. The "nal result is O (k , k )A (k , k , k ) , (A.43) A / / / (k  , k  , k  )" SN//HH> . . . N O .H .H> N/ 2N/H\ O O N/H> 2N/ . . . N N N O O where Q and P are arbitrary permutations and Q"Q( j, j#1), P"P( j, j#1). Eq. (A.43) has important consequences. In order to exhibit these clearly, it is convenient to express (A.43) in the framework of the auxiliary spin model introduced above. Inserting (A.43) into (A.19) we obtain (A.44) "k  , k  , k  2">HH>(sin k H ,sin k H> )"k  , k  , k  2 , . . . . . . . . where >HI is given by (A.21). There are altogether (N!1)N!"12 equations (A.44) and all of them have to be consistent with one another! This puts severe constraints on the operators >HI. Let us explain this in more detail. We recall that the symmetric group S is generated by , the identity id and the transpositions of nearest neigbors ( j, j#1), j"1,2, N!1, modulo the relations ( j, j#1)( j#1, j#2)( j, j#1)"( j#1, j#2)( j, j#1)( j#1, j#2) ,

(A.45)

( j, j#1)(k, k#1)"(k, k#1)( j, j#1) for " j!k"'1 ,

(A.46)

( j, j#1)( j, j#1)"id .

(A.47)

Eq. (A.45) is called the braid relation. Note that (A.46) is non-trivial only for N'3. By inspection of (A.44) we see that the >-operators act on the states "k , k , k 2 by exchanging    neighboring components of the vector k"(k , k , k ), that determines the state "k , k , k 2 of our       auxiliary spin system. Hence all states "k  , k  , k  2, P3S , can be obtained from "k , k , k 2 by . . .     repeated use of (A.44). Equivalently, (A.44) allows us to obtain the state corresponding to any permutation PM 3S from a state corresponding to any other permutation P3S .   It follows from (A.45)}(A.47) that a representation of a permutation as a product of transpositions of nearest neighbors is not unique. For the case at hand, N"3, we have for instance, (1,3)"(1,2)(2,3)(1,2)"(2,3)(1,2)(2,3). We are thus facing a consistency problem for Eqs. (A.44): Relations (A.45)}(A.47) impose consistency conditions on the >-operators. Let us study these consistency conditions. For N"3 we only have to consider (A.45) for the case j"1 and (A.47). Thus (A.45) implies that "k M  , k M  , k M  2">(sin k  ,sin k  )>(sin k  ,sin k  )>(sin k  ,sin k  )"k  , k  , k  2 , . . . . . . . . . . . . ">(sin k  ,sin k  )>(sin k  ,sin k  )>(sin k  ,sin k  )"k  , k  , k  2 , . . . . . . . . . (A.48)

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where PM "P(1,3)"P(1,2)(2,3)(1,2)"P(2,3)(1,2)(2,3)"(P , P , P ). Assuming "k , k , k 2 to be ar      bitrary, we conclude that >(s , s )>(s , s )>(s , s )">(s , s )>(s , s )>(s , s ) , (A.49)             where s are arbitrary complex numbers. This is the famous Yang}Baxter equation. It is now  crucial that (A.49) can be veri"ed by direct calculation. Therefore (A.44) is consistent with (A.45). Similarly, the use of Eq. (A.47) in (A.44) yields the requirement (>HH>(u, v))\">HH>(v, u) .

(A.50)

It is easy to verify by direct calculation that the operators >HI de"ned in (A.21) indeed ful"ll (A.50). Hence (A.44) is consistent with (A.47), and we conclude that the entire set of Eq. (A.44) is consistent. Note that the above considerations easily generalize to the case of arbitrary N, which will be treated in the next subsection. In addition to (A.45) and (A.47) we then will have to consider (A.46) which leads to the condition > (s , s )> (s , s )"> (s , s )> (s , s ) for " j!k"'1 , HH>   II>   II>   HH>   which is trivially satis"ed. Let us now impose periodic boundary conditions on the wave function t(0, x , x ;p ,p ,p )"t(¸, x , x ;p ,p ,p ) ,           t(1, x , x ;p ,p ,p )"t(¸#1, x , x ;p ,p ,p ) ,           t(x ,0, x ;p ,p ,p )"t(x , ¸, x ;p ,p ,p ) ,           t(x ,1, x ;p ,p ,p )"t(x , ¸#1, x ;p ,p ,p ) ,           t(x , x ,0;p ,p ,p )"t(x , x , ¸;p ,p ,p ) ,           t(x , x ,1;p ,p ,p )"t(x , x , ¸#1;p ,p ,p ) .           Inserting (A.36) into (A.52) yields

(A.51)

(A.52)

A / / / (k  , k  , k  )"exp(ik  ¸)A / / / (k  , k  , k  ) , (A.53) N N N . . . . N N N . . . where Q and P are arbitrary permutations of +1,2,3,. In terms of the auxiliary spin model (A.53) is expressed as "k  , k  , k  2"exp(ik  ¸)PP"k  , k  , k  2 . . . . . . . "exp(ik  ¸)PP>(sin k  ,sin k  ) . . . ;>(sin k  ,sin k  )"k  , k  , k  2 . . . . . "exp(ik  ¸)PX(sin k  ,sin k  )P . . . ;X(sin k  ,sin k  )"k  , k  , k  2 . . . . . "exp(ik  ¸)X(sin k  ,sin k  ) . . . ;X(sin k  ,sin k  )"k  , k  , k  2 , . . . . .

(A.54)

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where we have used identities (B.17). Using (B.16) we now obtain in complete analogy with the 2-electron case "k  , k  , k  2"e *I. q(sin k  " +sin k H ;j"1,2,3,) "k  , k  , k  2 . (A.55) . . . . . . . . We again can use the results for the diagonalization of the inhomogeneous transfer matrix q(sin k  " +sin k H ;j"1,2,3,) derived in Appendix B. We need to distinguish two cases, depending . . on the spins of the two electrons (recall that we consider only states for which the number of down spins not larger than the number of up spins, all other states are obtained by using the spin-reversal symmetry). E Three electrons with spin-up: Here the appropriate eigenvector of q(K " +sin k H ;j"1,2,3,) is the . ferromagnetic state, with eigenvalue 1. The periodic boundary conditions for the case of three electrons with spin up thus take the form e IH *"1,

j"1,2,3 .

(A.56)

These are precisely the Lieb}Wu equations (7) and (8) for the case N"3, M"0. Using (A.19) we see that all amplitudes are trivial A    (k  , k  , k  )"1 . (A.57) NNN . . . The corresponding wave function (A.36) coincides with Woynarovich's result (4). E Two electrons with spin-up, one with spin-down: Now the appropriate eigenvector of q is found in the sector with one overturned spin. From (B.15) its eigenvalue is given by 1!i;/[2(sin k  !j #i;/4)]"(sin k  !j !i;/4)/(sin k  !j #i;/4) , (A.58) .  .  .  where j ful"lls   j !sin k #i;/4 H 1" “  . (A.59) j !sin k !i;/4 H H  Inserting (A.58) into (A.55) we obtain the following quantization conditions due to periodic boundary conditions: (A.60) exp(i¸k  )"(j !sin k  !i;/4)/(j !sin k  #i;/4) for P3S .  .  .  . Eqs. (A.59) and (A.60) are precisely the Lieb}Wu equations (7) and (8) for the case N"3 and M"1. An explicit expression for the amplitudes again is obtained from (A.19) and (B.35), with the result






1 W\ j !sin k H !i;/4  . , (A.61) A    (k  , k  , k  )" “ NNN . . . j !sin k H #i;/4 j !sin k W #i;/4 .  . H  where y is the position of the down spin in the sequence p p p . Inserting (A.61) into (A.36) we    obtain (4) for the case N"3, M"1.

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A.4. N electrons It is now clear how to generalize the above results to the case of N electrons. The Bethe ansatz for the solution t of the SchroK diger equation (A.7) in the sector Q with x  4x  424x , is / / / , . (A.62) t(x ,2, x ;p ,2,p )" sign(PQ)A / 2 /, (k  ,2, k , )exp i k x N N . . .H /H  ,  , H .Z1, Substituting (A.62) into (A.7) for the case x Ox ( j, k"1,2, N; jOk) we obtain the following H I expression for the energies:






, N; ;¸ # . (A.63) E"!2 cos k ! H 4 2 H Using the single valuedness of the wave function and solving the matching conditions at the sector boundaries, i.e. the SchroK dinger equation for the cases where two of the coordinates coincide, we obtain the following set of equations: O (k , k ) A / 2 /H\   /H> 2 /, (k  ,2, k , ) , (A.64) A / 2 /, (k  ,2, k , )" SN//HH> N N O .H .H> N . . N N OON . . N   O O where Q and P are arbitrary permutations and Q"Q( j, j#1), P"P( j, j#1). In terms of the auxiliary spin model (A.64) reads (A.65) "k  ,2, k , 2">HH>(sin k H ,sin k H> )"k  ,2, k , 2 . . . . . . . The mutual consistency of Eqs. (A.65) follows from (A.49) and (A.50). We now impose periodic boundary conditions on the wave function t(x ,2, x ,0, x ,2, x ;p ,2,p )"t(x ,2, x , ¸, x ,2, x ;p ,2,p ) ,  H\ H> ,  ,  H\ H> ,  , t(x ,2, x ,1, x ,2, x ;p ,2,p )"t(x ,2, x , ¸#1, x ,2, x ;p ,2,p ) ,  H\ H> ,  ,  H\ H> ,  , (A.66) where j"1,2, N. Inserting (A.62) into (A.66) yields A / 2 /, (k  ,2, k , )"exp(ik  ¸)A / 2 /, / (k  ,2, k , , k  ) , (A.67) N N . . . N N N . . . where Q, P3S are arbitrary. In terms of the auxiliary spin model (A.67) is expressed as , "k  ,2, k , 2"exp(ik  ¸)PP2P,\,"k  ,2, k , , k  2 . . . . . . "exp(ik  ¸)PP2P,\, . ,\ “ >,\K\,\K(sin k  ,sin k ,\K ) "k  ,2, k , 2 . . . . K "exp(ik  ¸)X,(sin k  ,sin k , )X,\(sin k  ,sin k ,\ )2 . . . . . ;2X(sin k  ,sin k  )X(sin k  ,sin k  )"k  ,2, k , 2 , (A.68) . . . . . .





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where we have used identities (B.17). Using (B.16) we now obtain in complete analogy with the 2 and 3 electron cases (A.69) "k  ,2, k , 2"e *I. q(sin k  "+sin k H ; j"1,2, N,)"k  ,2, k , 2 . . . . . . . Next, we again use the results for the diagonalization of the inhomogeneous transfer matrix q(sin k  "+sin k H ;j"1,2, N,) derived in Appendix B. We now need to distinguish [N/2]#1 cases . . ([x] is the integer part of x), corresponding to the possible values of M. In the sector with M down spins the eigenvalue of q(sin k  "+sin k H ; j"1,2, N,) is given by (B.15) . . + sin k  !j !i;/4 . H “ , (A.70) sin k  !j #i;/4 H . H where the rapidities j ful"ll H , j !sin k #i;/4 + j !j #i;/2 J I “ H "“ H . (A.71) j !sin k !i;/4 I j !j !i;/2 J H I J H I$H Inserting this result into (A.69) we "nally obtain + j !sin k  !i;/4 . exp(i¸k  )" “ H for P3S . (A.72) . , j !sin k  #i;/4 . H H Eqs. (A.71) and (A.72) are precisely the Lieb}Wu equations (7) and (8) for the case of N electrons, M of which have spin down. In order to obtain an explicit expression for the amplitudes we now need to make use of the general result (B.18) for eigenstates of the transfer matrix q(sin k  "+sin k H ;j"1,2, N,) of the inhomogeneous Heisenberg model. We "nd . . 1 WR \ j R !sin k Q !i;/4 + . “ L , (A.73) A  2 , (k  ,2, k , )" A “ . L N N . j R !sin k WR #i;/4 j R !sin k Q #i;/4 . . Q L LZ1+ R L where 14y (y (2(y+ 4N are the positions of the down spins in the sequence p 2p  , and A is given by L j J !j I !i;/2 L L A " “ . (A.74) L j J !j I L L XJIX+ The resulting explicit expression for the wave function (A.62) coincides with Woynarovich's result (4).










Appendix B. Inhomogeneous Heisenberg model B.1. Algebraic Bethe ansatz Our starting point is a < < 2< , where <   , H 0 1 0 qV" , qW" 1 0 i




and q!"(qV$iqW). 




lattice of N spin-s. The corresponding Hilbert space is  is isomorphic to ". We de"ne the Pauli matrices s"(qV,qW,qX) by



!i 0




, qX"

1

0

0 !1



(B.1)

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The central object of the quantum inverse scattering method is the R-matrix, which is a solution of the Yang}Baxter equation. For the case of the spin- Heisenberg model it is of the form  f (k,j) 0 0 0




R(j,k)"

where

0

g(k,j) 1

0

1

g(k,j) 0

0

0

0

f (k,j)"1!i;/2(k!j),

0



,

(B.2)

g(k,j)"!i;/2(k!j) .

(B.3)

f (k,j)

R(j,k) acts on the tensor product space < < , where < is isomorphic to ". The Yang}Baxter    equation for R is an equation on the space < < < and can be written as    R(j,k)R(j,l)R(k,l)"R(k,l)R(j,l)R(j,k) , (B.4) where the superscript indicates in which spaces the R-matrix acts non-trivially. We now de"ne an ¸-operator acting on the tensor product between a &matrix space' < and a &quantum-space' < ,  L which is identi"ed with the Hilbert space over the nth site of our lattice of spins, by ¸ (j)"[j/(j#i;/2)]I#[(i;/2)/(j#i;/2)]PL , L j#(1#qX )i;/4 q\i;/2 1 L L " . (B.5) j#i;/2 q>i;/2 j#(1!qX )i;/4 L L The Yang}Baxter equation (B.4) implies the following intertwining relations for the ¸-operator:






R(j,k)(¸ (j)¸ (k))"(¸ (k)¸ (j))R(j,k) , (B.6) L L L L where the tensor product is between matrix spaces, i.e. (B.6) is a relation over the space < < < .   L Next we note, that the intertwiner for the ¸-operator (B.6) still holds, if we shift both spectral parameters j and k by an arbitrary amount l , i.e. L R(j,k)(¸ (j!l )¸ (k!l ))"(¸ (k!l )¸ (j!l ))R(j,k) . (B.7) L L L L L L L L We now construct an inhomogeneous monodromy matrix in the following way: ¹(k"+a ,)"¸ (k!a )¸ (k!a ) ¸ (k!a ) H , , ,\ ,\ 2   A(k) B(k) . " C(k) D(k)






(B.8)

Here a ,2, a are N arbitrary complex constants. Intertwiner (B.7) can be lifted to the level of the  , monodromy matrix R(j,k)(¹(j"+a ,)¹(k"+a ,))"(¹(k"+a ,)¹(j"+a ,))R(j,k) . H H H H

(B.9)

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By tracing (B.8) over the matrix space < one then "nds that the transfer matrices  q(k"+a ,)"Tr (¹(k"+a ,))"A(k)#D(k) (B.10) H  H commute for any values of spectral parameter k, i.e. [q(k"+a ,),q(l"+a ,)]"0. That implies that the H H transfer matrix is the generating functional of an in"nite number of mutually commuting conserved quantum operators (via expansion in powers of k). Eigenstates of the transfer matrix are constructed by means of the algebraic Bethe ansatz. Starting point is the choice of a reference state, which is a trivial eigenstate of q(k"+a ,). We choose H the saturated ferromagnetic state "02""!!!2!2", "!2 . L L

(B.11)

The action of the ¸-operator (B.5) on "!2 can be easily calculated and implies the following actions L of the matrix elements of the monodromy matrix: A(k)"02"a(k)"02, a(k)"1 , k!a , H , D(k)"02"d(k)"02 , d(k)" “ k!a #i;/2 H H C(k)"02"0, B(k)"02O0 .

(B.12)

From (B.12) we see that B(j) play the role of creation operators. Acting with B corresponds to #ipping a spin. States with M down spins can be constructed as + F(j ,2,j )" “ (!2i/;)B(j !i;/4)"02 , (B.13)  + H H where we have shifted the spectral parameters and introduced a particular normalization for later convenience. The requirement that states (B.13) ought to be eigenstates of the transfer matrix puts constraints on the values j : the set +j , must be a solution of the following system of Bethe ansatz L H equations: , j !a #i;/4 + j !j #i;/2 I J “ H "“ H , j"1,2, M . j !a !i;/4 J j !j !i;/2 I H J I H J$H The corresponding eigenvalues of the transfer matrix are



(B.14)



+ + q(k"+a ,) F(j ,2,j )" a(k) “ f (k,j !i;/4)#d(k) “ f (j !i;/4,k) F(j ,2,j ) . H  + H H  + H H (B.15) For our present purposes we need to consider the transfer matrix evaluated at the "rst inhomogeneity. We "nd q(a "+a ,)"Tr [¸ (a !a )¸ (a !a ) ¸ (a !a )P]  H  ,  , ,\  ,\ 2    "Tr [PP¸ (a !a )PP¸ (a !a )  ,  , ,\  ,\ ;PP2P¸ (a !a )P]   

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"Tr [PX,(a , a )X,\(a , a ) X(a , a )]   ,  ,\ 2   "X,(a , a )X,\(a , a ) X(a , a ) , (B.16)  ,  ,\ 2   where XHI is de"ned in (A.26). Here we have "rst used the explicit form of the ¸-operator (B.5) and then the identities P HIP HLP HI"PIL,

P HIP HI"I .

(B.17)

B.2. Explicit expressions for the eigenstates In this subsection we derive the following expression for the eigenstates (B.13):





1 WR \ K R !a Q !i;/4 + + . F(K ,2,K )" A “ “ L “ q\H "02 , (B.18)  + W L K R !a WR #i;/4 K R !a Q #i;/4 + , . . G + L L Q H R W LZ1 where the summation extends over 14y (y (y (2(y 4N, P is a permutation of    + N elements, and the A is given by L K J !K I !i;/2 L L A " “ . (B.19) L K J !K I L L XJIX+ Our discussion closely parallels [70]. The basic tool for proving (B.18) is the &n-site generalized model' [113,114] (see also [82, pp. 151f, 171]). Generalized two-site model: Let us "rst divide the product of ¸-operators in (B.8) into two parts ¹ (K)"¸ (K!a )2¸ (K!a ) ¸ (K!a ) , ' L L     ¹ (K)"¸ (K!a )2¸ (K!a )¸ (K!a ). '' , , L> L> L> L> Clearly, we have

(B.20)

¹(K)"¹ (K) ¹ (K) . (B.21) '' ' Both ¹ (K) and ¹ (K) are 2;2 matrices '' ' A (K) B (K) A (K) B (K) ' '' ¹ (K)" ' , ¹ (K)" '' . (B.22) ' '' C (K) D (K) C (K) D (K) ' ' '' '' By construction the matrix elements of ¹ (K) commute with the matrix elements of ¹ (K). The ' '' commutation relations of the matrix elements of the same ¹-operator are as in (B.9)











R(K ,K )(¹ (K )¹ (K ))"(¹ (K )¹ (K ))R(K ,K ), a"I,II . (B.23)   ?  ?  ?  ?    The matrix elements of ¹ can be expressed in terms of the matrix elements of ¹ and ¹ , e.g., ' '' B(K)"A (K) B (K)#B (K) D (K) . (B.24) '' ' '' ' It is also possible to express the vectors “+ B(K ) "02 in terms of B and B . In order to do this we H H ' '' will use that C (K)"02"0, A (K)"02"a (K)"02 , ? ? ? D (K)"02"d (K)"02, a"I,II . ? ?

(B.25)

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Here a (K)"1, '

L K!a H d (K)" “ , ' K!a #i;/2 H H

(B.26)

and , K!a H a (K)"1, d (K)" “ . '' '' K!a #i;/2 H HL> In [113,114] it was proved that

(B.27)

+ “ B(K ) "02" “ “ a (K' )d (K'' ) f (K' ,K'' ) B (K'' ) B (K' )"02 . (B.28) H '' I ' K I K '' K ' I S S K' S K'' S ' '' I ' K '' H  Z Z On the right-hand side of (B.28) we have summations with respect to partitions of the set of all K 's H into two subsets S "+K' , and S "+K'' ,. Here k labels di!erent K in the subset S and m labels ' I '' K ' di!erent K in the subset S . The above model is called generalized two-site model because ¹ is '' represented as a product of two factors. This is not su$cient for our purposes however. Let us therefore now consider the so-called Generalized k-site model: Let us represent ¹ as a product of k factors ¹(K)"¹ (K)2¹ (K)¹ (K) . (B.29) I   Here each ¹ (K) is a string of ¸-operators like in (B.20). The commutation relations of the matrix ? elements of each of the ¹ (K) is given by the same intertwiner as in (B.9). ? The matrix elements of ¹ (K) commute with the matrix elements of ¹ (K) if aOb and like for @ ? the two-site model we have






A (K) B (K) ? , (B.30) ¹ (K)" ? ? C (K) D (K) ? ? C (K)"02"0, A (K)"02"a (K)"02, D (K)"02"d (K)"02 . (B.31) ? ? ? ? ? By explicitly multiplying the matrices in (B.29) we can express B(K) in terms of matrix elements of the ¹ (K). Iteration of (B.28) leads to the following expression for the eigenfunctions of the transfer ? matrix ¹(K):






+ I “ B(K )"02" “ “ B (K? ? )"02 H ? K S 2S K? S    I ? K? Z ? H




;



“ “ “ a (K? ? )d (K@@ ) f (K? ? ,K@@ ) . (B.32) @ K ? I K I K? ? S K@@ S X?@XI K Z ? I Z @ Here the summation is with respect to the partitions of the set of all K 's into k disjoint subsets H S ,b"1,2, k. The index m enumerates di!erent K in the subset S and the index k enumerates @ ? ? @ di!erent K in the subset S . Eq. (B.32) was "rst proved in [114]. @

T. Deguchi et al. / Physics Reports 331 (2000) 197}281

273

We now consider the special case of the generalized N-site model, where N is the length of the underlying lattice. This means that each factor ¹ (K) in our generalized N-site model is identi"ed ? with an individual ¸-operator in (B.8). The eigenvalues in (B.31) are given by a (K)"1, d (K)"(K!a )/(K!a #i;/2) . ? ? ? ? The B-operators are given by

(B.33)

B (K)"[(i;/2)/(K!a #i;/2)] q\ (B.34) ? ? ? and have the important feature that B (K )B (K )"0. This implies that each set S in (B.32) ?  ?  ? consists of maximally one element. We are now in the position to write down explicit expressions for eigenstates (B.13). (a) One overturned spin: Let us "rst consider the eigenfunctions in the sector with one overturned spin. Application of (B.32) yields W\ K!a !i;/4 , 1 G “ . (B.35) F(K)"(!2i/;)B(K!i;/4)"02" q\"02 W K!a #i;/4 K!a #i;/4 G W G W Here all sets S in (B.32) except one are empty. This one (one-element) set is S "+K!i;/4,. ? W (a) Two overturned spins: Now application of (B.32) leads to 1 1 F(K ,K )" q\ q\ "02   W W K  !a  #i;/4 K  !a  #i;/4 L L W W XW W X, LZ1 W \ K  !a !i;/4 W \ K  !a !i;/4 H J “ L ; “ L f (K  ,K  ) . L L K  !a #i;/4 K  !a #i;/4 H J H L J L Here n is a permutation of two elements 1,2 and









(B.36)

f (K  ,K  )"(K  !K  !i;/2)/(K  !K  ) . (B.37) L L L L L L In this case only two subsets S are non-empty. Each of them consists of one element ? (S  "+K  !i;/4, and S  "+K  !i;/4,). Eq. (B.36) is of the desired form (B.18) if we L W L W identify A "(K  !K  !i;/2)/(K  !K  ) , (B.38) L L L L L which is in complete agreement with (B.19). (c) M overturned spins: The result for two overturned spins generalizes straightforwardly to M overturned spins. The non-empty subsets S in (B.32) (each of which consists of exactly one ? element) are S  "K  !i;/4, S  "K  !i;/4,2,S + "K + !i;/4, where n is some perL W L W L W mutation of M elements. A straightforward calculation then yields (B.18) and (B.19).

Appendix C. The spectrum of three electrons with one-down spin for L" "6 In Table 2 we present a complete list of eigenstates of the Hubbard Hamiltonian (119) in Section 5 for the case N"3 and M"1 for a 6-site system (¸"6) and ;"5. The energy levels are listed in increasing order.

274

T. Deguchi et al. / Physics Reports 331 (2000) 197}281

Table 2 Eigenstates of the Hubbard Hamiltonian No. 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

Energy

S

P/(n/3)

Types

!4.19862084914891 1/2 5 Real I "0.5,!0.5,!1.5, J"0.5 H k "0.542662224387082,!0.315837723840216,!1.27402205174346, K "0.587983554411128 H ? !4.19862084914891 1/2 1 Real I "1.5,0.5,!0.5, J"!0.5 H k "1.27402205174346,0.315837723840216,!0.542662224387082, K "!0.587983554411128 H ? !4.00000000000000 3/2 0 Quartet I "1,0,!1 H !3.35402752146807 1/2 4 Real I "0.5,!0.5,!1.5, J"!0.5 H k "0.231206350487730,!0.682137186480516,!1.64346426640041, K "!1.27426628748753 H ? !3.35402752146807 1/2 2 Real I "1.5,0.5,!0.5, J"0.5 H k "1.64346426640041,0.682137186480516,!0.231206350487730, K "1.27426628748753 H ? !2.63449090541439 1/2 0 Real I "1.5,!0.5,!1.5, J"0.5 H k "1.52991315867874,!0.279871354037999,!1.25004180464074, K "0.845079113394529 H ? !2.63449090541439 1/2 0 Real I "1.5,0.5,!1.5, J"!0.5 H k "1.25004180464074,0.279871354037999,!1.52991315867874, K "!0.845079113394529 H ? !2.21221243379665 1/2 4 Real I "0.5,!0.5,!2.5, J"0.5 H k "0.554943548673639,!0.307417851034944,!2.34192080003189, K "0.644785906059544 H ? !2.21221243379665 1/2 2 Real I "2.5,0.5,!0.5, J"!0.5 H k "2.34192080003189,0.307417851034944,!0.554943548673640, K "!0.644785906059544 H ? !2.17072407234717 1/2 5 Real I "1.5,!0.5,!1.5, J"!0.5 H k "1.22622874924835,!0.660511074050219,!1.61291522639472, K "!1.15790499577381 H ? !2.17072407234717 1/2 1 Real I "1.5,0.5,!1.5, J"0.5 H k "1.61291522639472,0.660511074050219,!1.22622874924835, K "1.15790499577381 H ? !2.00000000000000 3/2 1 Quartet I "2,0,!1 H !2.00000000000000 3/2 5 Quartet I "1,0,!2 H !2.00000000000000 3/2 3 Quartet I "2,1,0 H !2.00000000000000 3/2 3 Quartet I "0,!1,!2 H !1.68312858421806 1/2 3 Real I "0.5,!0.5,!2.5, J"!0.5 H k "0.261468569600159,!0.628328469155396,!2.77473275403456, K "!0.993984117066770 H ? !1.68312858421806 1/2 3 Real I "2.5,0.5,!0.5, J"0.5 H k "2.77473275403456,0.628328469155396,!0.261468569600159, K "0.993984117066770 H ? !1.00000000000000 3/2 2 Quartet I "2,1,!1 H !1.00000000000000 3/2 4 Quartet I "1,!1,!2 H !1.00000000000000 3/2 2 Quartet I "3,0,!1 H !1.00000000000000 3/2 4 Quartet I "3,1,0 H !0.867143471169408 1/2 3 Real I "0.5,!1.5,!2.5, J"0.5 H k "0.515010200239397,!1.28773972583561,!2.36886312799358, K "0.460329439565352 H ? !0.867143471169409 1/2 3 Real I "2.5,1.5,!0.5, J"!0.5 H k "2.36886312799358,1.28773972583561,!0.515010200239397, K "!0.460329439565352 H ? !0.763936521983257 1/2 2 Real I "!0.5,!1.5,!2.5, J"0.5 H k "!0.404815536059732,!1.33874801493405,!2.44522665379261, K "0.071452164223840 H ? !0.763936521983257 1/2 4 Real I "2.5,1.5,0.5, J"!0.5 H k "2.44522665379261,1.33874801493405,0.404815536059732,K "!0.07.1452164223840 H ? !0.724935621196484 1/2 5 Real I "1.5,0.5,!2.5, J"!0.5 H k "1.27598401405461,0.318718889893781,!2.64190045514499, K "!0.568959086784317 H ? !0.724935621196484 1/2 1 Real I "2.5,!0.5,!1.5, J"0.5 H k "2.64190045514499,!0.318718889893781,!1.27598401405461, K "0.568959086784317 H ?

T. Deguchi et al. / Physics Reports 331 (2000) 197}281

275

Table 2 Continued S

P/(n/3)

No.

Energy

28

!0.633335023683704 1/2 5 Real I "1.5,!0.5,!2.5, J"0.5 H k "1.54060965024643,!0.274638622892597,!2.31316857855043, K "0.886033905628400 H ? !0.633335023683705 1/2 1 Real I "2.5,0.5,!1.5, J"!0.5 H k "2.31316857855043,0.274638622892597,!1.54060965024643, K "!0.886033905628400 H ? !0.441363635441783 1/2 4 Real I "1.5,!0.5,!2.5, J"!0.5 H k "1.24802157741378,!0.602680979199234,!2.73973570060774, K "!0.869103625350605 H ? !0.441363635441782 1/2 2 Real I "2.5,0.5,!1.5, J"0.5 H k "2.73973570060774,0.602680979199234,!1.24802157741378, K "0.869103625350605 H ? !0.164932552352930 1/2 0 Real I "1.5,0.5,!2.5, J"0.5 H k "1.61980144017864,0.665436431311280,!2.28523787148992, K "1.18390417778921 H ? !0.164932552352929 1/2 0 Real I "2.5,!0.5,!1.5, J"!0.5 H k "2.28523787148992,!0.665436431311280,!1.61980144017864, K "!1.18390417778921 H ? 0.00000000000000 3/2 0 Quartet I "2,0,!2 H 0.00000000000000 3/2 1 Quartet I "3,1,!1 H 0.041639350031250 1/2 2 Real I "0.5,!1.5,!2.5, J"!0.5 H k "0.242333631055016,!1.61486765220468,!2.81625618363673, K "!1.16526625596158 H ? 0.041639350031250 1/2 4 Real I "2.5,1.5,!0.5, J"0.5 H k "2.81625618363673,1.61486765220468,!0.242333631055016, K "1.16526625596158 H ? 0.599066446144297 1/2 1 Real I "!0.5,!1.5,!2.5, J"!0.5 H k "!0.702579425111547,!1.67294123942593,!2.86046709144551, K "!1.39028884293989 H ? 0.599066446144296 1/2 5 Real I "2.5,1.5,0.5, J"0.5 H k "2.86046709144551,1.67294123942593,0.702579425111547,K "1.39028884293989 H ? 0.625362070788712 1/2 4 Real I "1.5,!1.5,!2.5, J"0.5 H k "1.49854259024225,!1.26094870462919,!2.33198898800626, K "0.722114430777064 H ? 0.625362070788713 1/2 2 Real I "2.5,1.5,!1.5, J"!0.5 H k "2.33198898800626,1.26094870462919,!1.49854259024225, K "!0.722114430777064 H ? 1.00000000000000 3/2 1 Quartet I "3,1,0 H 1.00000000000000 3/2 5 Quartet I "3,0,!1 H 1.00000000000000 3/2 1 Quartet I "2,1,!1 H 1.00000000000000 3/2 5 Quartet I "1,!1,!2 H 1.24466526322010 1/2 3 Real I "1.5,!1.5,!2.5, J"!0.5 H k "1.23359323035324,!1.58513579306981,!2.79005009087323, K "!1.05370338517746 H ? 1.24466526322010 1/2 3 Real I "2.5,1.5,!1.5, J"0.5 H k "2.79005009087323,1.58513579306981,!1.23359323035324, K "1.05370338517746 H ? 1.26651528335317 1/2 0 Real I "2.5,!0.5,!2.5, J"0.5 H k "2.65775114820892,!0.311801389555702,!2.34594975865322, K "0.614983937128585 H ? 1.26651528335317 1/2 0 Real I "2.5,0.5,!2.5, J"!0.5 H k "2.34594975865322,0.311801389555702,!2.65775114820892, K "!0.614983937128585 H ? 1.55774931070832 1/2 5 Real I "2.5,!0.5,!2.5, J"!0.5 H k "2.31151260511968,!0.609456221165668,!2.74925393515061, K "!0.901701357480205 H ? 1.55774931070832 1/2 1 Real I "2.5,0.5,!2.5, J"0.5 H k "2.74925393515061,0.609456221165668,!2.31151260511968, K "0.901701357480206 H ? 2.00000000000000 3/2 0 Quartet I "3,2,1 H 2.00000000000000 3/2 0 Quartet I "3,!1,!2 H 2.00000000000000 3/2 2 Quartet I "3,1,!2 H 2.00000000000000 3/2 4 Quartet I "3,2,!1 H 2.59087887855288 1/2 4 Real I "2.5,!1.5,!2.5, J"0.5 H

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

Types

(continued on next page)

276

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Table 2 Continued

No.

57 58 59 60 61

62

63 64 65 66

67

68

69

70

71

72

Energy

S

P/(n/3)

Types

k "2.60714264000049,!1.28680359174393,!2.36753659945316, K "0.468661303303178 H ? 2.59087887855288 1/2 2 Real I "2.5,1.5,!2.5, J"!0.5 H k "2.36753659945316,1.28680359174393,!2.60714264000049, K "!0.468661303303178 H ? 3.00000000000000 1/2 3 g-Pairing I "0 H 3.24859982950560 1/2 4 Real I "2.5,!1.5,!2.5, J"!0.5 H k "2.29420005680492,!1.59212356246732,!2.79647159673079, K "!1.07985902893246 H ? 3.24859982950560 1/2 2 Real I "2.5,1.5,!2.5, J"0.5 H k "2.79647159673079,1.59212356246732,!2.29420005680492, K "1.07985902893246 H ? 3.53450070903252 1/2 4 Complex m"10.0,l"6.0, J"1.5,p"!1.0 q"2.18262982829371,m"1.51998108466021, k "6.10671585537857, K"1.96074971503998  Re d"0.138234567206297;10\,Im d"0.236218958183487;10\ 3.53450070903252 1/2 2 Complex m"8.0,l"1.0, J"!1.5,p"1.0 q"4.10055547888588,m"1.51998108466021, k "0.176469451801019, K"!1.96074971503998  Re d"!0.138234567211404;10\,Im d"0.236218958176826;10\ 4.00000000000000 3/2 3 Quartet I "3,2,!2 H 4.00000000000000 1/2 4 g-Pairing I "1 H 4.00000000000000 1/2 2 g-Pairing I "!1 H 4.13147838656489 1/2 5 Complex m"5.0,l"1.0, J"1.5,p"!1.0 q"2.23719190707960,m"1.45368482839940, k "0.761603941823783, K"1.77325189299191  Re d"0.307921735414718;10\,Im d"0.266758568093550;10\ 4.13147838656489 1/2 1 Complex m"13.0,l"6.0, J"!1.5,p"1.0 q"4.04599340009998,m"1.45368482839940, k "5.52158136535580, K"!1.77325189299191  Re d"!0.307921735405836;10\,Im d"0.266758568103764;10\ 4.33177213111493 1/2 5 Complex m"11.0,l"6.0, J"0.5,p"!1.0 q"2.77491534511596,m"1.10073629130283, k "5.96934237293065, K"0.601275696875128  Re d"!0.273302194108371;10\,Im d"!0.199420060157429;10\ 4.33177213111493 1/2 2 Complex m"7.0,l"1.0, J"!0.5,p"1.0 q"3.50826996206362,m"1.10073629130283, k "0.313842934248937, K"0.601275696875128  Re d"!0.119981837180916;10,Im d"!0.199420060157118;10\ 4.57467498439465 1/2 0 Complex m"6.0,l"1.0, J"0.5,p"1.0 q"2.88935479499487,m"1.07299907022393, k "0.504475717189853, K"0.411558289383258  Re d"!0.399317226400675;10\,Im d"0.222938603299694;10\ 4.57467498439465 1/2 0 Complex m"12.0,l"6.0, J"!0.5,p"!1.0 q"3.39383051218472,m"1.07299907022393, k "5.778709589989730, K"0.411558289383258  Re d"!0.819123406502514;10,Im d"0.222938603298584;10\ 4.71197559752276 1/2 3 Complex m"9.0,l"5.0, J"1.5,p"!1.0 q"2.16082673394224,m"1.54894763250639, k "5.10312449288489, K"2.04356190829635  Re d"0.892857558656424;10\,Im d"0.211992582481946;10\

T. Deguchi et al. / Physics Reports 331 (2000) 197}281

277

Table 2 Continued No. 73

74

75

76

77

78 79 80

81

82

83

84 85

86

Energy 4.71197559752276

S

P/(n/3)

1/2

3

Types

Complex m"9.0,l"2.0, J"!1.5,p"1.0 q"4.12235857323734,m"1.54894763250639, k "1.18006081429470, K"!2.04356190829635  Re d"!0.892857558563165;10\,Im d"0.211992582494824;10\ 5.58600548103459 1/2 4 Complex m"10.0,l"5.0, J"0.5,p"!1.0 q"2.72669142318074,m"1.11641164129461, k "5.01859266560450, K"0.683372926842314  Re d"!0.186687577849687;10\,Im d"!0.244844007284084;10\ 5.58600548103459 1/2 2 Complex m"8.0,l"2.0, J"!0.5,p"1.0 q"3.55649388399885,m"1.11641164129461, k "1.26459264157509, K"0.683372926842313  Re d"!0.136487897790614;10,Im d"!0.244844007284262;10\ 5.95823319001949 1/2 0 Complex m"6.0,l"2.0, J"1.5,p"!1.0 q"2.27376322866332,m"1.41368501399761, k "1.73565884985295, K"1.66056016294642  Re d"0.455885386176691;10\,Im d"0.245749596857303;10\ 5.95823319001949 1/2 0 Complex m"12.0,l"5.0, J"!1.5,p"1.0 q"4.00942207851627,m"1.41368501399761, k "4.54752645732663, K"!1.660560162946420  Re d"!0.455885386181354;10\,Im d"0.245749596852640;10\ 6.00000000000000 1/2 5 g-Pairing I "2 H 6.00000000000000 1/2 1 g-Pairing I "!2 H 6.03334376215914 1/2 1 Complex m"7.0,l"2.0, J"0.5,p"1.0 q"2.96456730023144,m"1.06124089913747, k "1.40124825791330, K"0.288686041886590  Re d"!0.375435197170576;10\,Im d"0.208597324485038;10\ 6.03334376215914 1/2 5 Complex m"11.0,l"5.0, J"!0.5,p"!1.0 q"3.31861800694814,m"1.06124089913747, k "4.88193704926629, K"0.288686041886588  Re d"!0.573617731801462,Im d"0.208597324485194;10\ 6.70841301401077 1/2 2 Complex m"8.0,l"4.0, J"1.5,p"!1.0 q"2.16393969303817,m"1.54471884698899, k "4.04970102349645, K"2.03142493862260  Re d"0.956259360065381;10\,Im d"0.215691965483877;10\ 6.70841301401077 1/2 4 Complex m"10.0,l"3.0, J"!1.5,p"1.0 q"4.11924561414142,m"1.54471884698899, k "2.23348428368314, K"!2.031424938622590  Re d"!0.956259360207490;10\,Im d"0.215691965477882;10\ 7.00000000000000 1/2 0 g-Pairing I "3 H 7.48332665113181 1/2 1 Complex m"7.0,l"3.0, J"1.5,p"!1.0 q"2.20081321450656,m"1.49694330556893, k "2.92875642936307, K"1.89535079698079  Re d"0.187321491183834;10\,Im d"0.252337664304214;10\ 7.48332665113181 1/2 5 Complex m"11.0,l"4.0, J"!1.5,p"1.0 q"4.08237209267303,m"1.49694330556893, k "3.354428877816510, K"!1.89535079698079  Re d"!0.187321491188275;10\,Im d"0.252337664298441;10\ (continued on next page)

278

T. Deguchi et al. / Physics Reports 331 (2000) 197}281

Table 2 Continued

No.

Energy

S

P/(n/3)

87

7.59363119464461

1/2

3

88

89

90

Types

Complex m"9.0,l"4.0, J"0.5,p"!1.0 q"2.74080575567842,m"1.11156342668968, k "3.94316644941255, K"0.659158265098385 Re d"!0.212814421543628;10\,Im d"!0.234942161399343;10\ 7.59363119464461 1/2 3 Complex m"9.0,l"3.0, J"!0.5,p"1.0 q"3.54237955150117,m"1.11156342668968, k "2.34001885776704, K"0.659158265098384 Re d"!0.131618838598134;10,Im d"!0.234942161399543;10\ 8.02701965828632 1/2 2 Complex m"8.0,l"3.0, J"0.5,p"1.0 q"2.89722498813356,m"1.07154335884761, k "2.58313043330567, K"0.398665945793610 Re d"!0.401340764685176;10\,Im d"0.414789194807641;10\ 8.02701965828632 1/2 4 Complex m"10.0,l"4.0, J"!0.5,p"!1.0 q"3.38596031904603,m"1.07154335884761, k "3.70005487387392, K"0.398665945793609 Re d"!0.793318483940372,Im d"0.414789194806753;10\

The energy eigenvalues obtained by direct numerical diagonalization of the Hamiltonian (the Householder-QR method) and by the Bethe ansatz method coincide within an error of O(10\). In the table only the energy eigenvalues obtained by Bethe ansatz are listed. The last digit for each numerical value has a rounding error. The symbols S and P denote the spin and momentum of the eigenstate, respectively. There are 90 eigenstates for N"3 and M"1, as we enumerated in Section 5.3.3. The numerical results shown in the table con"rm the completeness of the Bethe ansatz.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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GRAVITATIONAL WAVE EXPERIMENTS AND EARLY UNIVERSE COSMOLOGY

Michele MAGGIORE INFN, sezione di Pisa, and Dipartimento di Fisica, Universita` di Pisa, via Buonarroti 2, I-56127, Italy

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

Physics Reports 331 (2000) 283}367

Gravitational wave experiments and early universe cosmology Michele Maggiore INFN, sezione di Pisa, and Dipartimento di Fisica, Universita% di Pisa, via Buonarroti 2, I-56127 Pisa, Italy Received August 1999; editor: R. Petronzio Contents 1. Introduction 2. Characterization of stochastic backgrounds of GWs 2.1. The energy density X ( f )  2.2. The spectral density S ( f ) and the F characteristic amplitudes h ( f )  3. The response of a single detector 3.1. Detector tensor and detector pattern functions 3.2. The strain sensitivity hI D 4. Two-detectors correlation 4.1. The overlap reduction functions c( f ), C( f ) 4.2. Optimal "ltering 4.3. The SNR for two-detectors correlations 4.4. The characteristic noise h ( f ) of L correlated detectors 5. Detectors in operation, under construction, or planned 5.1. The "rst generation of large interferometers: LIGO, VIRGO, GEO600, TAMA300, AIGO 5.2. Advanced LIGO 5.3. The space interferometer LISA 5.4. Resonant bars: NAUTILUS, EXPLORER, AURIGA, ALLEGRO, NIOBE 5.5. Some projects at a preliminary stage

286 288 288 289 291 291 295 296 296 298 299 301 302

302 306 308 311 312

6. Sensitivity of various two-detectors correlations 6.1. An ideal two-interferometers correlation 6.2. LIGO}LIGO 6.3. VIRGO}LIGO, VIRGO}GEO, VIRGO}TAMA 6.4. VIRGO}Resonant mass and Resonant mass}Resonant mass 7. Bounds on h X ( f )   7.1. The nucleosynthesis bound 7.2. The COBE bound 7.3. Pulsars as GW detectors: ms pulsars, binary pulsars and pulsar arrays 7.4. Light de#ection by GWs 8. Production of relic GWs: a general orientation 8.1. The characteristic frequency 8.2. The form of the spectrum 8.3. Characteristic intensity 9. Ampli"cation of vacuum #uctuations 9.1. The computation of Bogoliubov coe$cients 9.2. Ampli"cation of vacuum #uctuations in in#ationary models 9.3. Pre-big-bang cosmology 10. Other production mechanisms 10.1. Phase transitions

E-mail address: [email protected] (M. Maggiore) 0370-1573/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 1 0 2 - 7

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M. Maggiore / Physics Reports 331 (2000) 283}367 10.2. Cosmic strings 10.3. Production during reheating 11. Stochastic backgrounds of astrophysical origin 11.1. Supernovae collapse to black hole 11.2. GWs from hydrodynamic waves in rotating neutron stars

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11.3. GWs from mass multipoles of rotating neutron stars 11.4. Unresolved galactic and extragalactic binaries 12. Conclusions Note added in proof Acknowledgements References

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Abstract Gravitational-wave experiments with interferometers and with resonant masses can search for stochastic backgrounds of gravitational waves of cosmological origin. We review both experimental and theoretical aspects of the search for these backgrounds. We give a pedagogical derivation of the various relations that characterize the response of a detector to a stochastic background. We discuss the sensitivities of the large interferometers under constructions (LIGO, VIRGO, GEO600, TAMA300, AIGO) or planned (Avdanced LIGO, LISA) and of the presently operating resonant bars, and we give the sensitivities for various two-detectors correlations. We examine the existing limits on the energy density in gravitational waves from nucleosynthesis, COBE and pulsars, and their e!ects on theoretical predictions. We discuss general theoretical principles for order-of-magnitude estimates of cosmological production mechanisms, and then we turn to speci"c theoretical predictions from in#ation, string cosmology, phase transitions, cosmic strings and other mechanisms. We "nally compare with the stochastic backgrounds of astrophysical origin.  2000 Elsevier Science B.V. All rights reserved. PACS: 04.30.!w; 04.30.Db; 04.80.Nn; 98.80.Cq Keywords: Gravitational wave detectors; Stochastic backgrounds; Early universe cosmology

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1. Introduction A large e!ort is presently under way in the scienti"c community for the construction of the "rst generation of large-scale interferometers for gravitation wave (GW) detection. The two LIGO detectors are under construction in the US, and the Italian}French interferometer VIRGO is presently under construction near Pisa, Italy. The detectors are expected to start taking data around the year 2002, with a sensitivity where we might "nd signals from known astrophysical sources. These detectors are also expected to evolve into second generation experiments, with better sensitivities. At the same time, two other somewhat smaller interferometers are being built: GEO600 in Germany and TAMA300 in Japan, and they will be important also for developing the techniques needed by second generation experiments. To cover the low-frequency region, where many interesting signals are expected, but which on the Earth is unaccessible because of seismic noise, it is planned to send an interferometer into space: this extraordinary experiment is LISA, which has been selected by the European Space Agency has a cornerstone mission in his future science program Horizons 2000 Plus, and might #ight somewhere around 2010}2020, with a sensitivity levels where many sources are virtually guaranteed. Finally, although at a lowersensitivity level, cryogenic resonant bars are in operation since 1990, and more sensitive resonant masses, e.g. of spherical shape, are under study. A possible target of these experiments is a stochastic background of GWs of cosmological origin. The detection of such a background would have a profound impact on early Universe cosmology and on high-energy physics, opening up a new window and exploring very early times in the evolution of the Universe, and correspondingly high energies, that will never be accessible by other means. The basic reason why relic GWs carry such a unique information is that particles which decoupled from the primordial plasma at time t&t , when the Universe  had a temperature ¹ , give a &snapshot' of the state of the Universe at ¹&¹ . All informations   on the Universe when the particle was still in thermal equilibrium has instead been obliterated by the successive interactions. The weaker the interaction of a particle, the higher is the energy scale when they drop out of thermal equilibrium. Hence, GWs can probe deeper into the very early Universe. To be more quantitative, the condition for thermal equilibrium is that the rate C of the processes that maintain equilibrium be larger than the rate of expansion of the Universe, as measured by the Hubble parameter H. The rate (in units "c"1) is given by C"np"v" where n is the number density of the particle in question, and for massless or light particles in equilibrium at a temperature ¹, n&¹; "v"&1 is the typical velocity and p is the cross-section of the process. Consider for instance the weakly interacting neutrinos. In this case the equilibrium is maintained, e.g., by electron}neutrino scattering, and at energies below the = mass p&G 1E2&G ¹ where G is $ $ $ the Fermi constant and 1E2 is the average energy squared. The Hubble parameter during the radiation dominated era is related to the temperature by H&¹/M . Therefore, . G ¹ K(¹/1 MeV) . (1) (C/H) & $   ¹/M . Even the weakly interacting neutrinos, therefore, cannot carry informations on the state of the Universe at temperatures larger than approximately 1 MeV. If we repeat the above computation for gravitons, the Fermi constant G is replaced by Newton constant G"1/M , where $ .

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M &10 GeV is the Planck mass. At energies below M , . . (C/H) &(¹/M ) . (2)    . The gravitons are therefore decoupled below the Planck scale. It follows that relic gravitational waves are a potential source of informations on very high-energy physics. Gravitational waves produced in the very early Universe has not lost memory of the conditions in which they have been produced, as it happened to all other particles, but still retain in their spectrum, typical frequency and intensity, important informations on the state of the very early Universe, and therefore on physics at correspondingly high energies, which cannot be accessed experimentally in any other way. It is also clear that the property of gravitational waves that makes them so interesting, i.e. their extremely small cross section, is also responsible for the di$culties of the experimental detection. In this review we will examine a number of experimental and theoretical aspects of the search for a stochastic background of GWs. The paper is organized as follows. We start from the experimental side; in Sections 2}4 we present the basic concepts for characterizing a stochastic background and for describing the detector. This seems to be a subject where everybody has its own de"nitions and notations, and we made an e!ort to clean up many formulas appearing in the literature, to present them in what we consider the clearest form, and to give the most straightforward derivation of various relations. In Section 5 we examine the various detectors under construction or already operating; we show the sensitivity curves of LIGO, VIRGO, GEO600, Advanced LIGO, LISA and NAUTILUS (kindly provided by the various collaborations), and we discuss the limiting sources of noise and the perspectives for improvement in the near and mid-term future. To detect a stochastic background with Earth-based detectors, it turns out that it is mandatory to correlate two or more detectors. The sensitivities that can be reached with two-detectors correlations are discussed in Section 6. We will show graphs for the overlap reduction functions between VIRGO and the other major detectors, and we will give the estimate of the minimum detectable value of the signal. In Section 7 we examine the existing bounds on the energy density in GWs coming from nucleosynthesis, COBE, msec pulsars, binary pulsar orbits, pulsar arrays. Starting from Section 8 we move toward more theoretical issues. First of all, we discuss general principles for order of magnitude estimate of the characteristic frequency and intensity of the stochastic background produced by cosmological mechanisms. With the very limited experimental informations that we have on the very high-energy region, M :E:M , and on the history of %32 . the very early Universe, it is unlikely that theorists will be able to foresee all the interesting sources of relic stochastic backgrounds, let alone to compute their spectra. This is particularly clear in the Planckian or string theory domain where, even if we succeed in predicting some interesting physical e!ects, in general we cannot compute them reliably. So, despite the large e!orts that have been devoted to understanding possible sources, it is still quite possible that, if a relic background of gravitational waves will be detected, its physical origin will be a surprise. In this case it is useful to understand what features of a theoretical computation have some general validity, and what are speci"c to a given model, separating for instance kinematical from dynamical e!ects. The results of this section provide a sort of benchmark, against which we can compare the results from the speci"c models discussed in Sections 9 and 10.

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In Section 9 we discuss the most general cosmological mechanism for GW production, namely the ampli"cation of vacuum #uctuations. We examine both the case of standard in#ation, and a more recent model, derived from string theory, known as pre-big-bang cosmology. In Section 10 we examine a number of other production mechanisms. Finally, in Section 11 we discuss the stochastic background due to many unresolved astrophysical sources, which from the point of view of the cosmological background is a &noise' that can compete with the signal (although, of course, it is very interesting in its own right).

2. Characterization of stochastic backgrounds of GWs A stochastic background of GWs of cosmological origin is expected to be isotropic, stationary and unpolarized. Its main property will be therefore its frequency spectrum. There are di!erent useful characterization of the spectrum: (1) in terms of a (normalized) energy density per unit logarithmic interval of frequency, h X ( f ); (2) in terms of the spectral density of the ensemble   avarage of the Fourier component of the metric, S ( f ); (3) or in terms of a characteristic amplitude F of the stochastic background, h ( f ). In this section we examine the relation between these  quantities, that constitute the most common description used by the theorist. On the other hand, the experimentalist expresses the sensitivity of the apparatus in terms of a strain sensitivity with dimension Hz\, or thinks in terms of the dimensionless amplitude for the GW, which includes some form of binning. The relation between these quantities and the variables S ( f ), h X ( f ), h ( f ) will be discussed in Section 3. F    2.1. The energy density X ( f )  The intensity of a stochastic background of gravitational waves (GWs) can be characterized by the dimensionless quantity [67] X ( f )"(1/o )do /d log f , (3)    where o is the energy density of the stochastic background of gravitational waves, f is the  frequency (u"2pf ) and o is the present value of the critical energy density for closing the  Universe. In terms of the present value of the Hubble constant H , the critical density is given by  o "3H /8pG . (4)   The value of H is usually written as H "h ;100 km/(s}Mpc), where h parametrizes the     existing experimental uncertainty. Ref. [195] gives a value 0.5(h (0.85. In the last few years  there has been a constant trend toward lower values of h and typical estimates are now in the  range 0.55(h (0.60 or, more conservatively, 0.50(h (0.65. For instance Ref. [211], using   the method of type IA supernovae, gives two slightly di!erent estimates h "0.56$0.04 and  h "0.58$0.04. Ref. [229], with the same method, "nds h "0.60$0.05 and Ref. [155], using   a gravitational lens, "nds h "0.51$0.14. The spread of values obtained gives an idea of the  systematic errors involved. It is not very convenient to normalize o to a quantity, o , which is uncertain: this uncertainty   would appear in all the subsequent formulas, although it has nothing to do with the uncertainties

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on the GW background itself. Therefore, we rather characterize the stochastic GW background with the quantity h X ( f ), which is independent of h . All theoretical computations of    primordial GW spectra are actually computations of do /d log f and are independent of the  uncertainty on H . Therefore the result of these computations is expressed in terms of h X ,    rather than of X .  2.2. The spectral density S ( f ) and the characteristic amplitudes h ( f ) F  To understand the e!ect of the stochastic background on a detector, we need however to think in terms of amplitudes of GWs. A stochastic GW at a given point x"0 can be expanded, in the transverse traceless gauge, as

 

 (5) h (t)" df dXK hI ( f, XK )e\p DRe (XK ) ,  ?@ ?@ >" \ where hI (!f, XK )"hI H ( f, XK ). XK is a unit vector representing the direction of propagation of the wave   and dXK "d cos h d . The polarization tensors can be written as e> (XK )"m( m( !n( n( , e" (XK )"m( n( #n( m( , ?@ ? @ ? @ ?@ ? @ ? @ with m( , n( unit vectors ortogonal to XK and to each other. With these de"nitions,

(6)

e (XK )eY?@(XK )"2dY . (7) ?@ For a stochastic background, assumed to be isotropic, unpolarized and stationary (see [8,12] for a discussion of these assumptions) the ensemble average of the Fourier amplitudes can be written as (8) 1hI H ( f, XK )hI ( f , XK )2"d( f!f )(1/4p)d(XK , XK )d S ( f ) ,  Y Y  F where d(XK , XK )"d( ! )d(cos h!cos h). The spectral density S ( f ) de"ned by the above equaF tion has dimensions Hz\ and satis"es S ( f )"S (!f ). The factor  is conventionally inserted in F F  the de"nition of S in order to compensate for the fact that the integration variable f in Eq. (5) F ranges between !R and #R rather than over the physical domain 04f(R. The factor 1/(4p) is a choice of normalization such that



dXK dXK 1hI H ( f, XK )hI ( f ,XK )2"d( f!f )d S ( f ) .  Y Y  F

(9)

Using Eqs. (5) and (8) we get



1h (t)h?@(t)2"2 ?@



\



d f S ( f )"4 F

D d(log f ) f S ( f ) . F D

(10)

 This simple point has occasionally been missed in the literature, where one can "nd the statement that, for small values of H , X is larger and therefore easier to detect. Of course, it is larger only because it has been normalized using   a smaller quantity.  Our convention for the Fourier transform are g ( f )" dt exp+2pift,g(t), so that g(t)" df exp+!2pift,g ( f ). \ \

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We now de"ne the characteristic amplitude h ( f ) from  D 1h (t)h?@(t)2"2 d(log f )h( f ) . (11) ?@  D Note that h ( f ) is dimensionless, and represents a characteristic value of the amplitude, per unit  logarithmic interval of frequency. The factor of two on the right-hand side of Eq. (11) is part of our de"nition, and is motivated by the fact that the left-hand side is made up of two contributions, given by 1hI H hI 2 and 1hI H hI 2. In an unpolarized background these contributions are equal, while > > " " the mixed term 1hI H hI 2 vanishes, Eq. (8). > " Comparing Eqs. (10) and (11), we get



h( f )"2f S ( f ) . (12)  F We now relate h ( f ) and h X ( f ). The starting point is the expression for the energy density of    gravitational waves, given by the 00-component of the energy}momentum tensor. The energy} momentum tensor of a GW cannot be localized inside a single wavelength (see e.g. Ref. [188, Sections 20.4 and 35.7] for a careful discussion), but it can be de"ned with a spatial averaging over several wavelengths: o "(1/32pG)1hQ hQ ?@2 . (13)  ?@ For a stochastic background, the spatial average over a few wavelengths is the same as a time average at a given point, which, in Fourier space, is the ensemble average performed using Eq. (8). We therefore insert Eq. (5) into Eq. (13) and use Eq. (8). The result is



D 4 d(log f ) f (2pf )S ( f ) , o " F  32pG D so that

(14)

do /d log f"(p/2G) f S ( f ) .  F Comparing Eqs. (15) and (12) we get the important relation

(15)

do /d log f"(p/4G) f h( f ) ,   or, dividing by the critical density o ,  X ( f )"(2p/3H ) f h( f ) .    Using Eq. (12) we can also write

(16) (17)

X ( f )"(4p/3H ) f S ( f ) . (18)   F Some of the equations that we will "nd below are very naturally expressed in terms of h X ( f ).   This will be true in particular for all theoretical predictions. Conversely, all equations involving the signal-to-noise ratio and other issues related to the detection are much more transparent when written in terms of S ( f ). Eq. (18) will be our basic formula for moving between the two F descriptions. Inserting the numerical value of H , Eq. (17) gives  (19) h ( f )K1.263;10\(1 Hz/f )(h X ( f ) .   

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Actually, h ( f ) is not yet the most useful dimensionless quantity to use for the comparison with  experiments. In fact, any experiment involves some form of binning over the frequency. In a total observation time ¹, the resolution in frequency is *f"1/¹, so one does not observe h X ( f ) but   rather



D> D *f (20) d(log f )h X ( f )K h X ( f )   f   D and, since h X ( f )&h( f ), it is convenient to de"ne    h ( f, *f )"h ( f )(*f/f ) . (21)   Using 1/(1 yr)K3.17;10\ Hz as a reference value for *f, and 10\ as a reference value for h X ,   Eqs. (19) and (21) give h ( f, *f )K2.249;10\(1 Hz/f )(h X ( f )/10\)(*f/(3.17;10\ Hz))    or, with reference value that will be more useful for LISA,

(22)

h ( f, *f )"7.111;10\(1 mHz/f )(h X ( f )/10\)(*f/(3.17;10\ Hz)) . (23)    Finally, we mention another useful formula which expresses h X ( f ) in terms of the number of   gravitons per cell of the phase space, n(x, k). For an isotropic stochastic background n(x, k)"n D depends only on the frequency f""k"/(2p), and





dk  2pfn "16p d(log f ) f n . o "2 D  D (2p)  Therefore, do /d log f"16pn f   D

(24)

(25)

and h X ( f )K3.6(n /10)( f/1 kHz) . (26)   D This formula is useful in particular when one computes the production of a stochastic background of GW due to ampli"cation of vacuum #uctuations, since the computation of the Bogoliubov coe$cients (see Section 9.1) gives directly n . D As we will discuss below, to be observable at the LIGO/VIRGO interferometers, we should have at least h X &10\ between 1 Hz and 1 kHz, corresponding to n of order 10 at 1 kHz and   D n &10 at 1 Hz. A detectable stochastic GW background is therefore exceedingly classical, D n <1. I 3. The response of a single detector 3.1. Detector tensor and detector pattern functions The quantities X ( f ), h ( f ), S ( f ) discussed above are all equivalent characterization of the   F stochastic background of GWs, and have nothing to do with the detector used. Now, we must

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make contact with what happens in a detector. The total output of the detector S(t) is in general of the form S(t)"s(t)#n(t) ,

(27)

where n(t) is the noise and s(t) is the contribution to the output due to the gravitational wave. For instance, for an interferometer with equal arms of length ¸ in the x}y plane, s(t)"(d¸ (t)!d¸ (t))/¸, where d¸ are the displacements produced by the GW. The relation V W VW between the scalar output s(t) and the gravitational wave h (t) in the transverse-traceless gauge has ?@ the general form [112,108] s(t)"D?@h (t) . ?@ D?@ is known as the detector tensor. Using Eq. (5), we get

(28)

 

 s(t)" d f dXK hI ( f, XK )e\p DRD?@e (XK ) .  ?@ >" \ It is therefore convenient to de"ne the detector pattern functions F (XK ),  F (XK )"D?@e (XK )  ?@ so that

 

 s(t)" df dXK hI ( f, XK )F (XK )e\p DR   >" \ and the Fourier transform of the signal, s ( f ), is

(29)

(30)

(31)



s ( f )" dXK hI ( f, XK )F (XK ) . (32)   >" The pattern functions F depend on the direction XK "(h, ) of arrival of the wave. Furthermore,  they depend on an angle t (hidden in e ) which describes a rotation in the plane orthogonal to XK , ?@ i.e. in the plane spanned by the vectors m( , n( , see Eq. (6). Once we have made a de"nite choice for the vectors m( , n( , we have chosen the axes with respect to which the # and ; polarizations are de"ned. For an astrophysical source there can be a natural choice of axes, with respect to which the radiation has an especially simple form. Instead, for a unpolarized stochastic background, there is no privileged choice of basis, and the angle t must cancel from the "nal result, as we will indeed check below. For a stochastic background the average of s(t) vanishes and, if we have only one detector, the best we can do is to consider the average of s(t). Using Eqs. (31), (30) and (8),



1s(t)2"F



\



d f S ( f )"F  F





dfS (f) , F

(33)

where



F,

dXK 4p

F(XK , t)F(XK , t) . >"

(34)

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Note that, while the value of 1s(t)2 is obtained summing over all stochastic waves coming from all directions XK , the factor 1/(4p) in Eq. (8) produces an average of F #F over the directions. The > " factor F gives a measure of the loss of sensitivity due to the fact that the stochastic waves come from all directions, compared to the sensitivity of the detector for waves coming from the optimal direction. To compute the pattern functions, we need the explicit expressions for the vectors m( , n( that enters the de"nition of Eq. (6). We use polar coordinates centered on the detector. Then XK "(sin h sin , sin h cos , cos h)

(35)

and a possible choice for m( , n( is n( "(!cos h sin ,!cos h cos , sin h),

m( "(!cos , sin , 0) .

(36)

The most general choice is obtained with the rotation n( Pn( cos t#m( sin t , m( P!n( sin t#m( cos t ,

(37)

so that n( "(!cos h sin cos t!cos sin t,!cos h cos cos t#sin sin t, sin h cos t) , m( "(!cos cos t#cos h sin sin t, sin cos t#cos h cos sin t,!sin h sin t) .

(38)

We now compute the explicit expression for F and F in the most interesting cases. 3.1.1. Interferometers For an interferometer with arms along the u( and v( directions (not necessarily orthogonal) (39) D?@"(u( ?u( @!v( ?v( @) .  Using the de"nition of F , Eq. (30), together with Eq. (6) for e and with the above expressions for  ?@ m( , n( one obtains, restricting now to an interferometer with perpendicular arms [99,113,216,226], F (h, , t)"(1#cos h) cos 2 cos 2t!cos h sin 2 sin 2t , >  F (h, , t)"(1#cos h) cos 2 sin 2t#cos h sin 2 cos 2t . "  The factor F is then given by



dXK 4p

(40)

2 F(XK , t)F(XK , t)" . (41) 5 >" Note also that 1s(t)2 depends on the F only through the combination F #F which is  > " independent of the angle t, as can be checked from the explicit expressions, in agreement with the argument discussed above. The same will be true for resonant bars and spheres. Therefore below we will often use the notation F (XK ) instead of F (XK , t).   F,

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It is interesting to see how these results are modi"ed if the arms are not perpendicular. If a is the angle between the two arms, we "nd F (h, , t)"(sin a)[(1#cos h) sin (a#2 ) cos 2t#cos h cos (a#2 ) sin 2t] , >  F (h, , t)"(sin a)[(1#cos h) sin (a#2 ) sin 2t!cos h cos (a#2 ) cos 2t] . "  Therefore,

(42)

(43) F" sin a .  For a"p/2 we recover results (40) and (41) and the sensitivity is maximized, while for a"0 the sensitivity of the interferometric detection of course vanishes. 3.1.2. Cylindrical bars For a cylindrical bar with axis along the direction lK , one has instead D?@"lK ?lK @ .

(44)

Since D?@ is contracted with the traceless tensor h , it is de"ned only apart from terms &d?@, so ?@ that we can equivalently use the traceless tensor (45) D?@"lK ?lK @!d?@ .  To compute F it is convenient, compared to the case of the interferometer, to perform a rede"ni tion tPt#p/2, and de"ne h as the polar angle measured from the bar direction lK . (Of course, when we will discuss interferomer-bar correlations in Section 4.3, we will be careful to use the same de"nitions in the two cases.) Then F (h, , t)"sin h cos 2t , > F (h, , t)"sin h sin 2t . " and the angular factor F is



F,

dXK 4p

8 F(XK , t)F(XK , t)" . 15 >"

(46)

(47)

3.1.3. Spherical detectors Finally, it is interesting to give also the result for a detector of spherical geometry. For a spherical resonant-mass detector there are "ve detection channels [147] corresponding to the "ve degenerates quadrupole modes. The functions D?@ and F for each of these channel are given in Ref. [243]:  one de"nes the real quadrupole spherical harmonics as > ,> ,   1 (> !> ), > , > A (2 \ i > , (> #> ), Q (2 \ >

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1 > , (> #> ), A (2 \ > i (> !> ). > , > Q (2 \

(48)

The pattern functions associated to the 0, 1c, 1s, 2c, 2s channels are (3 sin h, F (h, )" > 2

F

"

(h, )"0 ,

F (h, )" sin 2h cos , F (h, )"!sin h sin , >A  "A F (h, )"!sin h cos , F (h, )"! sin 2h sin , "Q >Q  F (h, )"(1#cos h) cos 2 , F (h, )"!cos h sin 2 , >A  "A F (h, )"!cos h cos 2 . (49) F (h, )"!(1#cos h) sin 2 , "Q >Q  Here the angle t has been set to zero; as discussed above, this corresponds to a choice of the axes with respect to which the polarization states are de"ned. With this choice, at h"0, only F ,F are non-vanishing, consistently with the fact GWs have elicities $2. The same value as Q A for interferometers, F", is obtained for the sphere in the channel m"0, as well as for the  channels m"$1 and $2. 3.2. The strain sensitivity hI D The ensemble average of the Fourier components of the noise satis"es (50) 1n H( f )n ( f )2"d( f!f )S ( f ) .  L The above equation de"nes the functions S ( f ), with S (!f )"S ( f ) and dimensions Hz\. The L L L factor  is again conventionally inserted in the de"nition so that the total noise power is obtained  integrating S ( f ) over the physical range 04f(R, rather than from !R to R, L  df S ( f ) . (51) 1n(t)2" L  The function S is known as the square spectral noise density. L Equivalently, the noise level of the detector is measured by the strain sensitivity hI D hI ,(S ( f ) , (52) D L where now f'0. Note that hI is linear in the noise and has dimensions Hz\. D



 Unfortunately, there is not much agreement about notations in the literature. The square spectral noise density, that we denote by S ( f ) following e.g. Ref. [108], is called P( f ) in Ref. [8]. Other authors use the notation S ( f ), which we L F instead reserve for the spectral density of the signal. To make things worse, S is sometime de"ned with or without the L factor  in Eq. (50). 

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Comparing Eqs. (51) and Eq. (33), we see that in a single detector a stochastic background will manifest itself as an excess noise, and will be observable at a frequency f if S ( f )'(1/F)S ( f ) , (53) F L where F is the angular e$ciency factor, Eq. (34). Using Eq. (18) and S ( f )"hI  , we can express this L D result in terms of the minimum detectable value of h X , as   h X ( f )K(1/F);10\( f/100 Hz)(hI /10\ Hz\) . (54)   D In Section 4 we will use this result to compute the sensitivity of various single detectors to a stochastic background.

4. Two-detectors correlation 4.1. The overlap reduction functions c( f ), C( f ) To detect a stochastic GW background the optimal strategy consists in performing a correlation between two (or more) detectors, since, as we will see, the signal is expected to be far too low to exceed the noise level in any existing or planned single detector (with the exception of the planned space interferometer LISA). The strategy for correlating two detectors has been discussed in Refs. [187,71,108,239,8]. We write the output S (t) of the ith detector as S (t)"s (t)#n (t), where i"1, 2 labels the detector, and G G G G we have to face the situation in which the GW signal s is much smaller than the noise n . At G G a generic point x we rewrite Eq. (5) as





 df dXK hI ( f, XK )e\p DR\XK  xAe (XK ) , h (t, x)"  ?@ ?@ >" \ so that

(55)

 

 (56) s (t)" df dXK hI ( f, XK )e\p D R\XK  xG AF(XK ) ,  G G >" \ where F are the pattern functions of the ith detector. The Fourier transform of the scalar signal in G the ith detector, s ( f ), is then related to hI ( f, XK ) by G 



dXK hI ( f, XK )ep DXK  xG AF(XK ) . s ( f )"  G G >" We then correlate the two outputs de"ning



2



2

(57)

dt S (t)S (t)Q(t!t) , (58)   \2 \2 where ¹ is the total integration time (e.g. 1 yr) and Q a real "lter function. The simplest choice would be Q(t!t)"d(t!t), while the optimal choice will be discussed in Section 4.2. At any rate, S " 

dt

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Q(t!t) falls rapidly to zero for large "t!t"; then, taking the limit of large ¹, Eq. (58) gives



>

d f SI H( f )SI ( f )QI ( f ) . (59)   \ Taking the ensemble average, the contribution to 1S 2 from the GW background is  > d f1s H( f )s ( f )2 QI ( f ) 1s 2,    \ > " df dXK dXK ep DAXK  x \XK Y x  F(XK )FY(XK ) 1hI H ( f, XK )hI ( f, XK )2 QI ( f )    Y \ Y  "¹ (60) d f S ( f )C( f )QI ( f ) ,  F \ where we have used Eq. (8) and d(0)"2 dt"¹. In the last line we have de"ned \2 dXK *x F(XK )F(XK ) exp 2pifXK ) , (61) C( f ),   4p c  where * x is the separation between the two detectors. Note that FF is independent of t, even    if the two detectors are of di!erent type, e.g. an interferometer and a bar, as can be checked from the explicit expressions, and therefore again we have not written explicitly the t dependence. We introduce also S " 

 



 



 





dXK F(XK )F(XK )" , (62) F ,     4p  where the subscript means that we must compute F taking the two detectors to be perfectly  aligned, rather than with their actual orientation. Of course if the two detectors are of the same type, e.g. two interferometers or two cylindrical bars, F is the same as the constant F de"ned in  Eq. (34). The overlap reduction function c( f ) is de"ned by [71,108] c( f )"C( f )/F . (63)  This normalization is useful in the case of two interferometers, since F " already takes into   account the reduction in sensitivity due to the angular pattern, already present in the case of one interferometer, and therefore c( f ) separately takes into account the e!ect of the separation * x between the interferometers, and of their relative orientation. With this de"nition, c( f )"1 if the separation *x"0 and if the detectors are perfectly aligned. This normalization is instead impossible when one considers the correlation between an interferometer and a resonant sphere, since in this case for some modes of the sphere F "0,  as we will see below. Then one simply uses C( f ), which is the quantity that enters directly Eq. (60). Furthermore, the use of C( f ) is more convenient when we want to write equations that hold independently of what detectors (interferometers, bars, or spheres) are used in the correlation.

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4.2. Optimal xltering As we have seen above, in order to correlate the output of two detectors we consider the combination



S " 

>

d f SI H( f )SI ( f )QI ( f ) .  

(64) \ We now "nd the optimal choice of the "lter function QI ( f ), [187,71,108,239,8] following the discussion given in Ref. [8]. Since the spectral densities are de"ned, in the unphysical region f(0, by S ( f )"S (!f ), S ( f )"S (!f ), we also require QI ( f )"QI (!f ) and QI ( f ) real (so that F F L L QI ( f )"QI H(!f ) and Q(t) is real). We consider the variation of S from its average value,  N,S !1S 2 . (65)   By de"nition 1N2"0, while



1N2"1S 2!1S 2"  



df df  QI ( f )QI H( f )

\ [1SI H( f )SI ( f )SI ( f )SI H( f )2!1SI H( f )SI ( f )21SI H( f )SI ( f )2] . (66)         We are now interested in the case n


1N2K



\

df df  QI ( f )QI H( f )1SI H( f )SI ( f )21SI H( f )SI ( f )2    

(67)

and 1SI H( f )SI ( f )2K1n H( f )n ( f )2"d( f!f )SG( f ) . G G  L G G Using d(0)"¹, one therefore gets



¹  1N2" df "QI ( f )"S( f ) , L 4 \ where we have de"ned S ( f )"[S( f )S( f )] . L L L We now de"ne the signal-to-noise ratio (SNR),

(68)

(69)

(70)

SNR"[1s 2/1N2] . (71)  Note that we have taken an overall square root in the de"nition of the SNR since s is quadratic  in the signal, and so the SNR is linear in the signal (this di!ers from the convention used in Ref. [8], where it is quadratic); 1s 2 is the contribution to 1S 2 from the GW, given in   Eq. (60).

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We now look for the function QI ( f ) that maximizes the SNR. For two arbitrary complex functions A( f ), B( f ) one de"nes the (positive de"nite) scalar product [8]



(A, B)"



\

df AH( f )B( f )S( f ) . L

(72)

Then 1N2"¹(QI , QI )  and, from Eq. (60),

(73)

1s 2"¹(QI , (S C/S)) .   F L Then we have to maximize

(74)

(SNR)"1s 2/1N2"¹(QI , CS /S)/(QI , QI ) .  F L The solution of this variational problem is standard,

(75)

QI ( f )"cC( f )S ( f )/S( f ) (76) F L with c an arbitrary normalization constant. Note that, since we used C( f ) rather then c( f ), this formula holds true independently of the detectors used in the correlations. It is also important to observe that the optimal "lter depends on the signal that we are looking for, since S ( f ) enters Eq. (76). This means that one should perform the data analysis considering F a set of possible "lters, chosen either in order to encompass a range of typical behaviours, e.g. chosing S ( f )&f ? for di!erent values of the parameter a, or using the theoretical predictions F discussed in later chapters. Of course this point is not relevant for correlations involving resonant bars, because of their narrow bandwidth. 4.3. The SNR for two-detectors correlations We can now write down the value of the SNR obtained with the optimal "lter: (SNR)"¹(CS /S, CS /S) , F L F L

(77)

or

 



S( f )  . df C( f ) F S( f )  L In particular, for two interferometers C( f )"()c( f ) and  8  S( f )  ¹ df c( f ) F . (SNR) " \  25 S( f )  L For two cylindrical bars, C( f )"(  )c( f ) and  128  S( f )  df c( f ) F . ¹ (SNR) "
\
 225 S( f )  L SNR" 2¹



 



 



(78)

(79)

(80)

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For the correlation between an interferometer and a cylindrical bar, again C( f )"()c( f ), so that we  get again Eq. (79). In these formulas, the e!ect due to the fact that the two detectors will have in general a di!erent orientation is fully taken into account into c( f ). If however one wants to have a quick order of magnitude estimate of the e!ect due to the misalignement without performing each time numerically the integral that de"nes c( f ), one can note, from Eq. (61), that in the limit 2pfd;1, where d""* x" is the distance between the detectors,

 



dXK F(XK )F(XK ) , (81)   4p  where of course the pattern functions F relative to the two detectors are computed with their actual orientation. Therefore in the limit 2pfdP0, the ratio between the value of C that takes into account the actual orientation and the value for aligned detectors is equal to the ratio between F computed with the actual orientation and F for aligned detectors.   In the de"nition of c enters F computed for aligned detectors, Eq. (62). Then a simple estimate  of the e!ect due to the misalignement is obtained substituting in Eq. (78) C( f )KF c( f ) where  F is now computed for the actual orientation of the detectors and c( f ) is the overlap reduction  function for the perfectly aligned detectors. We apply this procedure to estimate the e!ect of misalignement on the SNR, in the correlation between an interferometer and a cylindrical bar. In the limit 2pfd;1 we can also neglect the e!ect of the Earth curvature. Then, in the frame where the interferometer arms are along the x, y axes, the bar also lies in the (x, y) plane and its direction is in general given by lK "(sin a, cos a, 0); the angle a measures the misalignement between the interferometer y axis and the bar. In this frame, computing the detector pattern function of the bar we get C( f )K

F
"!cos h cos( !a) cos 2t#sin( !a) cos 2t#cos h sin 2( !a) sin 2t , > F
"!cos h cos( !a) sin 2t#sin( !a) sin 2t!cos h sin 2( !a) cos 2t . (82) " To compare with Eq. (46) note that here h is measured from the z-axis, and the bar is in the (x, y) plane, while in Eq. (46) h is measured for the direction of the bar. For the interferometer the functions F are given in Eq. (40). A straightforward computation shows that  1 F
F " (1#cos h) cos 2 [sin( !a)!cos h cos( !a)]   2  !cos h sin 2( !a) sin 2 . (83) Note that again the dependence on the angle t has cancelled. Then



dXK 2 F
F "! cos 2a . (84) F "    4p 5  The overall sign of F is irrelevant since C( f ) enters quadratically in the SNR. Therefore we get  SNR(a)K"cos 2a"SNR(0) . (85) Of course, the SNR is maximum if the bar is aligned along the y-axis (u"0) of along the x-axis (u"p/2); the correlation becomes totally ine!ective when a"p/4.

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Finally, it is interesting to consider the correlation between an interferometer and a sphere [27]. Using Eqs. (49) and (40) with t"0, one "nds in particular that for the correlation with the m"0 channel of the sphere, (3 sin h(1#cos h) cos 2

(86) FF "   4  and therefore F vanishes because of the integral over . Similarly for the channel 1s, 1c one gets  an integral over cos cos 2 , which again vanishes. In this case of course c( f ) cannot be de"ned and one simply uses C( f ). Instead for each of the two modes 2s, 2c, F is just one-half of the value  for the interferometer}interferometer correlation, so that summing over all modes of the sphere one gets F ".   4.4. The characteristic noise h ( f ) of correlated detectors L We have seen in Section 2.2 that there is a very natural de"nition of the characteristic amplitude of the signal, given by h ( f ), which contains all the informations on the physical e!ects, and is  independent of the apparatus. We will now associate to h ( f ) a corresponding noise amplitude  h ( f ), that embodies all the informations on the apparatus, de"ning h ( f )/h ( f ) in terms of the L  L optimal SNR. If, in the integral giving the optimal SNR, Eq. (78), we consider only a range of frequencies *f, such that the integrand is approximately constant, we can write SNRK[2¹*f (C( f )S( f )/S( f ))]"[¹*f C( f )h( f )/2f S( f )] , (87) F L  L where we have used Eq. (12). The right-hand side of Eq. (87) is proportional to h ( f ), and we can  therefore de"ne h ( f ) equating the right-hand side of Eq. (87) to h ( f )/h ( f ), so that L  L h ( f ),[1/((1/2)¹*f )]( f S ( f )/C( f )) . (88) L L In the case of two interferometers approximately at the same site,

 



dXK F(XK , t)F(XK , t) "F , 4p  and we recover Eq. (66) of Ref. [226], C( f )K

(89)

h ( f )"[1/(1/2¹*f )][ f S ( f )/F] . (90) L L For comparison, note that the quantity denoted by S ( f ) in Ref. [226] is called here S ( f )/2, and F L F"1F 2#1F 2"21F 2. > " > From the derivation of Eq. (88) we can better understand the limitations implicit in the use of h ( f ). It gives a measure of the noise level only under the approximation that leads from Eq. (78), L which is exact (at least in the limit h ;n ), to Eq. (87). This means that *f must be small enough G G compared to the scale on which the integrand in Eq. (78) changes, so that C( f )S ( f )/S ( f ) must be F L approximately constant. In a large bandwidth this is non-trivial, and of course depends also on the form of the signal; for instance, if h X ( f ) is #at, as in many examples that we will "nd in later   sections, then S ( f )&1/f . For accurate estimates of the SNR at a wideband detector there is no F

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substitute for a numerical integration of Eq. (78). However, for order of magnitude estimates, Eq. (19) for h ( f ) together Eq. (88) for h ( f ) are simpler to use, and they have the advantage of  L clearly separating the physical e!ect, which is described by h ( f ), from the properties of the  detectors, that enter only in h ( f ). L Eq. (88) also shows very clearly the advantage of correlating two detectors compared with the use of a single detector. With a single detector, the minimum observable signal, at SNR"1, is given by the condition S ( f )5S ( f )/F. This means, from Eq. (12), a minimum detectable value for h ( f ) F L  given by h "(2f S ( f )/F) . (91)

 L The superscript 1d reminds that this quantity refers to a single detector. From Eq. (88) we "nd for the minimum detectable value with two interferometers in coincidence, h "h ( f ). For close

 L detectors of the same type, C( f )KF, so that





1 1 Hz  1 yr  1 h ( f )K1.12;10\h ( f ) . (92) h ( f )K



 *f ¹ (¹*f ) (2   Of course, the reduction factor in the noise level is larger if the integration time is larger, and if we increase the bandwidth *f over which a useful coincidence is possible. Note that h X is quadratic   in h ( f ), so that an improvement in sensitivity by two orders of magnitudes in h means four orders   of magnitude in h X .   5. Detectors in operation, under construction, or planned In this section we will discuss the characteristic of the detectors existing, under construction, or planned, and we will examine the sensitivities to stochastic backgrounds that can be obtained using them as single detectors. The results will allow us to fully appreciate the importance of correlating two or more detectors, at least as far as stochastic backgrounds are concerned. 5.1. The xrst generation of large interferometers: LIGO, VIRGO, GEO600, TAMA300, AIGO A great e!ort is being presently devoted to the construction of large-scale interferometers. The Laser Interferometer Gravitational-Wave Observatory (LIGO) is being developed by an MITCaltech collaboration. LIGO [4] consists of two widely separated interferometers, in Hanford, Washington, and in Livingston, Louisiana. The commissioning of the detectors will begin in 2000, and the "rst data run is expected to begin in 2002 [29]. The arms of both detectors will be 4 km long; in Hanford there will be also a second 2 km interferometer implemented in the same vacuum system. The sensitivity of the 4 km interferometers is shown in Fig. 1. The sensitivity is here given in meters per root Hz; to obtain hI , measured in 1/(Hz, one must divide by the arm length in meters, D i.e. by 4000. The sensitivity in 1/(Hz is shown in Fig. 5, next subsection, together with the sensitivity of advanced LIGO. The LIGO interferometer already has a 40 m prototype at Caltech, which has been used to study sensitivity, optics, control, and even to do some work on data analysis [9].

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Fig. 1. The planned LIGO sensitivity curve, in meters/(Hz (courtesy of the LIGO collaboration).

A comparable interferometer is VIRGO [66,238,50] which is being built by an Italian}French collaboration supported by INFN and CNRS, with 3 km arm length, and is presently under construction in Cascina, near Pisa. These experiments are carried out by very large collaborations, comparable to the collaborations of particle physics experiments. For instance, VIRGO involves about 200 people. Fig. 2 shows, in terms of hI , the expected sensitivity of the VIRGO interD ferometer. By the year 2000 it should be possible to have the "rst data from a 7 m prototype, which will be used to test the suspensions, vacuum tube, etc. Interferometers are wide-band detectors, that will cover the region between a few Hertz up to approximately a few kHz. Figs. 1 and 2 show separately the main noise sources. At very low frequencies, f(2 Hz for VIRGO and f(40 Hz for LIGO, the seismic noise dominates and sets the lower limit to the frequency band. Above a few Hz, the VIRGO superattenuator reduces the seismic noise to a neglegible level. Then, up to a few hundreds Hz, thermal noise dominates, and "nally the laser shot noise takes over. The various spikes are mechanical resonances due to the thermal noise: "rst, in Fig. 2, the high-frequency tail of the pendulum mode (in this region LIGO is still dominated by seismic noise), then a series of narrow resonances due to the violin modes of the wires; "nally, the two rightmost spikes in the "gures are the low-frequency tail of the resonances due to internal modes of the two mirrors. All resonances appear in pairs, at frequencies close but non-degenerate, because the two mirrors have di!erent masses. The width of these spikes is of the order of fractions of Hz. Note that the clustering of the various resonances in the kHz region is due to the logarithmic

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Fig. 2. The planned VIRGO sensitivity curve (from Ref. [62]).

scale in Figs. 1 and 2. The resonances are actually evenly spaced and narrow, so that in this frequency range the relevant curve is mostly the one given by the shot noise. Under construction is also GEO600 [174], a collaboration between the Max-Planck-Institut fuK r Quantenoptik in Garching and the University of Glasgow. It is being built near Hannover, with arms 600 m in length. GEO600 is somewhat smaller than LIGO and VIRGO, but will use techniques that will be important for the advanced detectors. In particular, the signal recycling of GEO600 provides an opportunity to change the spectral characteristics of the detector response, especially those due to the shot noise limitation, therefore in the high-frequency range of the available bandwidth. By choosing low or high mirror re#ectivities for the signal-recycling mirror, one can use the recycling either to distribute the improvement in the sensitivity on a wideband, or to improve it even further on a narrow band. This is the so-called narrow-banding of an interferometer. The sensitivity of GEO600 broad-band is shown in Fig. 3, and with narrowbanding at 600 Hz in Fig. 4. The frequency of maximum sensitivity is tunable to the desired value by shifting the signalrecycling mirror, and thus changing the resonance frequency of the signal-recycling cavity. In its tunable narrow-band mode, GEO600 might well be able to set the most stringent limits of all for a while. TAMA300 (Japan) [153] is the other large interferometer. It is a "ve-year project (1995}2000) that aims to develop the advanced techniques that will be needed for second generation experiments, and catch GWs that may occur by chance within out local group of galaxies. It has 300 m arm length. It is hoped that the project will evolve into the proposed Laser Gravitational Radiation Telescope (LGRT), which should be located near the SuperKamiokande detector. Furthermore, Australian scientists have joined forces to form the Australian Consortium for Interferometric Gravitational Astronomy (ACIGA) [212]. A design study and research with

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Fig. 3. The planned GEO sensitivity curve, without narrow-banding (courtesy of the GEO collaboration).

Fig. 4. The planned GEO sensitivity curve, with narrow-banding at 600 Hz (courtesy of the GEO collaboration).

particular emphasis on an Australian detector is nearing completion. The detector, AIGO, will be located north of Perth. Its position in the southern hemisphere will greatly increase the baseline of the worldwide array of detectors, and it will be close to the resonant bar NIOBE so that correlations can be performed. The detector will use sapphire optics and sapphire test masses, a material that LIGO plans to use only in its advanced stage, see Section 5.2.

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All these detectors will be used in a worldwide network to increase the sensitivity and the reliability of a detection. Using the values of hI given in the "gures and the value 1/F"2.5 for interferometers, we see D from Eq. (54) that VIRGO, GEO600 or any of the two LIGOs, used as single detectors, can reach a minimum detectable value for h X of order 10\ or at most a few times 10\, at f" 100 Hz.   Unfortunately, at this level current theoretical expectations exclude the possibility of a cosmological signal, as we will discuss in Sections 7 and 8. As we will see, an interesting sensitivity level for h X should be at least of order 10\. To reach such a level with a single interferometer we   need, e.g., hI ( f"100 Hz)(10\ Hz\ D

(93)

or hI ( f"1 kHz)(3;10\ Hz\ . (94) D We see from the "gures that such small values of hI are very far from the sensitivity of D "rst-generation interferometers, and are in fact even well below the limitation due to quantum noise. Of course, one should stress that theoretical prejudices, however well founded, are no substitute for a real measurement, and that even a negative result at the level h X &10\ would   be interesting. 5.2. Advanced LIGO It is important to have in mind that the interferometers discussed in the previous section are the "rst generation of large-scale interferometers, and in this sense they represent really a pioneering e!ort. At the level of sensitivity that they will reach, they have no guaranteed source of detection. However, they will open the way to second generation interferometers, with much better sensitivity. GEO600 and TAMA300 are important for testing the technique that will be needed for second generation experiments, while the larger interferometers VIRGO and LIGOs should evolve into second generation experiments. The LIGO collaboration has presented a recommended program for research and development, that will lead to the Advanced LIGO project [167]. The results are shown in Fig. 5, together with the changes in a number of parameters that are responsible for the various improvements shown in the "gure. The improvement program is divided into three stages: LIGOII near term, LIGOII medium term and LIGOIII. Curve 1 in Fig. 5 shows the sensitivity of LIGOI, while LIGOII near term is shown in curve 2. The changes leading to LIGOII near term should be incorporated by the end of 2004, and are based on engineering developments of existing technology or modest stretches from present day systems. They would result in signi"cant improvements of the sensitivity, and are also necessary to gain full advantage of subsequent changes. In particular, a signi"cant reduction in thermal noise will be obtained using fused silica "bers in the test mass suspension; a moderate improvement in seismic isolation will move the seismic noise below this new thermal noise #oor, and an increase in the laser power to &100 W results in a decrease of shot noise.

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Fig. 5. The planned advanced LIGO sensitivity curves, and the parameters relative to the various improvements, from Ref. [167] (courtesy of the LIGO collaboration).

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The goal of the medium term program are more ambitious technically, and it is estimated that they could be incorporated in LIGO by the end of 2006. They include signal recycling. As discussed above, this could be used either to obtain a broad-band redistribution of sensitivity (curve 3), or to improve the sensitivity in some narrow band (curve 4). Another possible improvement is the change of the test masses from fused silica to sapphire or other similar materials. Sapphire has higher density and higher quality factor, and this should result in a signi"cant reduction in thermal noise. For an optical con"guration without recycling, the sensitivity is given by curve 5. With signal recycling, possible responses are shown in curves 6 and 7. These changes require signi"cant advances in material technology, and so are considered possible but ambitious for LIGOII. Fig. 5 does not show performances for LIGOIII detectors, which are expected to provide possibly another factor of 10 of improvement in hI , but of course are quite di$cult to predict D reliably at this stage. The overall improvement of LIGOII can be seen to be, depending on the frequency, one or two orders of magnitude in hI . This is quite impressive, since two order of magnitudes in hI means four D D order of magnitudes in h X ( f ) and therefore an extremely interesting sensitivity, even for a single   detector, without correlations. 5.3. The space interferometer LISA The space interferometer LISA [33,90,143] was proposed to the European Space Agency (ESA) in 1993, in the framework of ESA's long-term space science program Horizon 2000. The original proposal involved using laser interferometry between test masses in four drag-free spacecrafts placed in a Heliocentric orbit. In turn, this led to a proposal for a six space-crafts mission, which has been selected by the European Space Agency has a cornerstone mission in its future science program Horizon 2000 Plus. This implies that, in principle, the mission is approved and that funding for industrial studies and technology development is provided right away. The launch year however depends on the availability of fundings. With reasonable estimates it is then expected that it will not be launched before 2017, and possibly as late as 2023. In Feb. 1997 the LISA team and ESA's Fundamental Physics Advisory Group proposed to carry out LISA in collaboration with NASA. If approved, this could make possible to launch it between 2005 and 2010. The design has also been somewhat simpli"ed, with three drag-free spacecrafts. The spacecrafts will be in a Heliocentric orbit, at a distance of 1 AU from the Sun, 203 behind the Earth. This design has lower costs, and it is likely to be adopted for future studies relevant to the project [143]. The mission is planned for 2 yr, but it could last up to 10 yr without exausting on-board supplies. LISA has three arms, "rst of all for redundancy. Thus, it can be thought of as two interferometers sharing a common arm. Of course, this means that the two interferometers will have a common noise. However, most signals are expected to have a signal-to-noise ratio so high that the noise will be neglegible. Then, the output from the two interferometers can be used to obtain extra informations on the polarization and direction of a GW. For the stochastic background, the third arm will help to discriminate backgrounds as those produced by binaries or by cosmological e!ects from anomalous instrumental noise [33]. Going into space, one is not limited anymore by seismic and gravity-gradient noises; LISA could then explore the very low-frequency domain, 10\ Hz (f(1 Hz. At the same time, there is also

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the possibility of a very long path length (the mirrors will be freely #oating into the spacecrafts at distances of 5;10 km from each other!), so that the requirements on the position measurment noise can be relaxed. The goal is to reach a strain sensitivity [33] hI "4;10\ Hz\ (95) D at f"1 mHz. At this level, one expects "rst of all signals from galactic binary sources, extra-galactic supermassive black holes binaries and super-massive black hole formation. Concerning the stochastic background, Eq. (54) shows clearly the advantage of going to low frequency: the factor f  in Eq. (54) gives an extremely low value for the minimum detectable value of h X . The sensitivity of Eq. (95) would correspond to   h X ( f"1 mHz)K1;10\ . (96)   Since LISA cannot be correlated with any other detector, to have some con"dence in the result it is necessary to have a SNR su$ciently large. Certainly one cannot work at SNR"1.65, as we will do when considering the correlation between two detectors. The standard choice made by the LISA collaboration is SNR"5, and Eq. (96) refers to this choice. Note that the minimum detectable value of h X is proportional to (SNR), since this SNR refers to the amplitude, and   h X ( f )&h( f ). Furthermore, Eq. (96) takes into account the angles between the arms, a"603,    and the e!ect of the motion of LISA, which together results in a loss of sensitivity by a factor approximately equal to 1/(5. Eq. (96) shows that LISA could reach a truly remarkable sensitivity in h X .   LISA will have its best sensitivity between 3 and 30 mHz. Above 30 mHz, the sensitivity degrades because the wavelength of the GW becomes shorter than twice the arm-length of 5;10 km. At low frequencies, instead, the noise curve rises because of spurious forces on the test masses. At some frequency below 0.1 mHz the accelerometer noise will increase rapidly, and the instrumental uncertainty would increase even more rapidly with decreasing frequency, setting a lower limit to the frequency band of LISA. Below a few mHz it is expected also a stochastic background due to compact white-dwarf binaries, that could cover a cosmological background. The sensitivity curve of LISA to a stochastic background, together with the estimated white-dwarf binaries background, is shown in Fig. 6. On the vertical axis is shown h ,h ( f, *f"3;10\ Hz), de"ned in Eq. (23), which is written in [33]    in the equivalent form h ( f, *f"3;10\ Hz)"5.5;10\(X /10\)(1 mHz/f )(H /75 km s\ Mpc\) . (97)    Two lines of constant X , equal to 10\ and 10\, are also shown in the "gure.  To compare with interferometers, Fig. 7 shows the sensitivity to the GW amplitude for both LISA and the advanced LIGO, together with a number of signals expected from astrophysical sources. It is apparent from the "gure that ground-based and space-borne interferometers are complementary, and together can cover a large range of frequencies, and, in case of a detection of a cosmological signal, together they can give crucial spectral informations. The sensitivities shown are presented by the collaboration as conservatives because [33] 1. The errors have been calculated realistically, including all substantial error sources that have been thought of since early studies of drag-free systems, and since the "rst was #own over

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Fig. 6. The sensitivity of LISA to a stochastic background of GWs after 1 yr of observation, and SNR"5 (from Ref. [33]; courtesy of the LISA collaboration).

Fig. 7. The sensitivity of LISA for periodic sources, and of advanced LIGO for burst sources, together with some expected astrophysical signals (from Ref. [33]; courtesy of the LISA collaboration).

25 years ago, and in most cases (except shot noise) the error allowance is considerably larger than the expected size of the error and is more likely an upper bound. 2. LISA is likely to have a signi"cantly longer lifetime than 1 yr, which is the value used to compute these sensitivities.

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3. The sensitivity shown refers only to one interferometer. Using three arms could increase the SNR by perhaps 20%. An interesting feature of LISA is that, as it rotates in its orbit, its sensitivity to di!erent directions changes. Then, LISA can test the isotropy of a stochastic background. This can be quite important to separate a cosmological signal from a stochastic background of galactic origin, which is likely to be concentrated on the galactic plane. By comparing two 3-month stretches of data, LISA should have no di$culty in identifying this e!ect. Furthermore, if the mission lasts 10 yr, LISA will be close to the sensitivity needed to detect the dipole anisotropy in a cosmological background due to the motion of the solar system. If the GW background turned out not to have the same dipole anisotropy as the cosmic microwave background, one would have found evidence for anisotropic cosmological models. 5.4. Resonant bars: NAUTILUS, EXPLORER, AURIGA, ALLEGRO, NIOBE Cryogenic resonant antennas have been taking date since 1990 (see e.g. Ref. [201] for review). Fig. 8 shows the sensitivity of NAUTILUS [16}18,76,78], the resonant bar located in Frascati, near Rome. It is an ultracryogenic detector, operating at a temperature ¹"0.1 K. This "gure is based on a 2 h run, but the behaviour of the apparatus is by now quite stationary over a few days, with a duty cycle limited to 85% by cryogenic operations. Compared to the interferometers, we see that bars are narrow-band detectors, and work at two resonances. The bandwidth is limited basically by the noise in the ampli"er. The value of the resonance can be slightly tuned, at the level of a few Hz, working on the electronics. In the "gure, the resonances are approximately at 907 and

Fig. 8. The sensitivity curve of NAUTILUS (courtesy of the NAUTILUS collaboration).

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922 Hz, with half-height bandwidths of about 1 Hz. At these frequencies the strain sensitivity is hI K5;10\ , (98) D about a factor of 5 higher than the sensitivity of the interferometers at the same frequencies. With a thermodynamic temperature of 0.1 K, the experimental data are in very good agreement with a Gaussian distribution, whose variance gives an e!ective temperature ¹ "4.1 mK. The  e!ective temperature ¹ is related to the thermodynamic temperature ¹ by ¹ K4¹C, where   C is inversely proportional to the quality factor Q of the resonant mode, so that C;1. With improvements in the electronics, the collaboration plans to reach in a few years the target sensitivity hI K8.6;10\ Hz\ , (99) D and ¹ &10 lK, over a bandwidth of 10 Hz [17,18].  The cryogenic resonant bar EXPLORER [194] is in operation at CERN since 1990, cooled at a temperature ¹"2.9 K, and has taken data at approximately the same frequencies as NAUTILUS, 905 and 921 Hz; its sensitivity is approximately the same as NAUTILUS. Similar sensitivities are obtained by ALLEGRO, (Louisiana) [182,135]. ALLEGRO has been taking data since 1991. The resonances are at 897 and 920 Hz. The resonant bar NIOBE [42,43,137,138] is located in Perth, Australia, and it has operated since 1993. The resonant frequencies are at 694.6 and 713 Hz. It is made of niobium, instead of aluminium as the other bars, since it has a higher mechanical quality factor. This has allowed to reach a noise temperature of less than 2 mK, while the typical noise temperature is around 3 mK. Improvement in the transducer system should allow to lower ¹ to a few microkelvin, in  a bandwidth of 70 Hz. This would allow to reach a sensitivity comparable to interferometers over a large bandwidth, approximately 650}750 Hz [138]. AURIGA [69,203,204] is located in Legnaro, near Padua, Italy. It began full operations in 1997. The bar is cooled at 0.2 K, and the e!ective temperature is ¹ "7 mK. The resonances are at 911  and 929 Hz. The spectral sensitivity reached [204] is similar to that of NAUTILUS. All these bars are designed to operate in a well-coordinated way, in order to improve the chances of reliable detection. From Eq. (99) we see that, without performing correlations between di!erent detectors, the sensitivity that one could get from resonant bars is between at most h X ( f )&O(1) and   h X ( f )&O(10) and therefore quite far from a level where one can expect a cosmological signal.   In Section 6 we will discuss the level that can be reached with correlations between bars, and between a bar and an interferometer. 5.5. Some projects at a preliminary stage A number of other projects for gravitational wave detection have been put forward and are presently at the stage of testing prototypes.  There is an International Gravitational Event Collaboration, an agreement between the bar detector experiments presently in operation, signed at CERN on July 4, 1997.

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An especially interesting project involves resonant mass detectors with truncated icosahedron geometry [147,185,186,77,73,74,39,220]. A spherical detector, in fact, has a larger mass compared to a bar with the same resonance frequency, and therefore a larger cross section. Furthermore, it has the informations about the directions and polarization of a GW that could only be obtained with "ve di!erent resonant bars. The problem of attaching mechanical resonators to the detector suggests a truncated icosahedral geometry, rather than a sphere. A prototype called TIGA (Truncated Icosahedral Gravitational Wave Antenna) has been built at the Lousiana State University [185]. It has its "rst resonant mode at 3.2 kHz. The Rome group has started the SFERA project [24]. A working group has been formed to carry out studies and measurements in order to de"ne a project of a large spherical detector, 40}100 tons of mass, competitive with large interferometers but with complementary features. A correlation between an interferometer and a sphere could in particular have quite interesting sensitivities for the stochastic background. These spheres could reach approximately one order of magnitude better in the sensitivity hI D [77,171] compared to resonant bars. Furthermore, they have di!erent channels, see Eqs. (48) and (49), that are sensitive to di!erent spin content [113,241,147]. This would allow to discriminate between di!erent theories of gravity, separating for instance the e!ects of scalar GWs, which could originate from a Brans}Dicke theory [240,149,171,51,39,40,192,178]. Scalars "elds originating from string theory, as the dilaton and the moduli of compacti"cations, are however a much more di$cult target: to manifest themselves as coherent scalar GWs they should be extremely light, m(10\ eV [178]. Actually, there is the possibility to keep the dilaton and moduli "elds extremely light or even massless, with a mechanism that has been proposed by Damour and Polyakov [89]. In fact, assuming some form of universality in the string loop corrections, it is possible to stabilize a massless dilaton during the cosmological evolution, at a value where it is essentially decoupled from the matter sector. In this case, however, the dilaton becomes decoupled also from the detector, since the dimensionless coupling of the dilaton to matter (a in the notation of [89]) is smaller than 10\ (see also [87]). Such a dilaton would then be unobservable at VIRGO, although it could still produce a number of small deviations from general relativity which might, in principle, be observable improving by several orders of magnitude the experimental tests of the equivalence principle [85]. Recently, hollow spheres have been proposed in Ref. [75]. The theoretical study suggests a very interesting value for the strain sensitivity, even of order hI & a few;10\ . (100) D The resonance frequency, depending on the material used and other parameters, can be between approximately 200 Hz and 1}2 kHz, and the bandwidth can be of order 20 Hz. Another idea which is discussed is an array of resonant masses [114,115,106], each with a di!erent frequency f, and a bandwidth *f&f/10, which together would cover the region from below the kHz up to a few kHz. Although not of immediate applicability to GW search, we "nally mention the proposal of using two coupled superconducting microwave cavities to detect very small displacements. The idea goes back to the works [197}199,68,145]. These detectors have actually been constructed in Ref. [206], using two coupled cavities with frequencies of order of 10 GHz; the coupling induces a shift in the levels of order of 1 MHz, and transitions between these levels could, in principle, be induced by

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GWs. However, Eq. (54) shows clearly the problem with working at such a high frequency, f&1 MHz: to reach an interesting value for h X ( f ) of order 10\, one needs hI &10\ Hz\,   D very far from present technology. There is however the proposal [37] to repeat the experiment with improved sensitivity, so to reach hI &6;10\ Hz\ at f"1 MHz, and, if this should work, D then one would try to lower the resonance frequency down to, possibly, a few kHz. This second step would mean to move toward much larger cavities, say of the kind used at LEP.

6. Sensitivity of various two-detectors correlations In the previous section we have seen that with a single detector (and with the exception of LISA) we cannot reach sensitivities interesting for cosmological backgrounds of GWs. On the other hand, we have found in Sections 4.3 and 4.4 that many orders of magnitudes can be gained correlating two detectors. In this section we examine the sensitivities for various two-detector correlations. It is also important to stress that performing multiple detector correlations is crucial not only for improving the sensitivity, but also because in a GW detector there are in general noises which are non-Gaussian, and can only be eliminated correlating two or more detectors. An example of a non-Gaussian noise in an interferometer is the creep, i.e. the sudden energy release in the superattenuator chain, and most importantly in the wires holding the mirrors, and due to accumulated internal stresses in the material. Indeed, it is feared that this and similar nonstochastic noises might turn out to be the e!ective sensitivity limit of interferometers, possibly exceeding the design sensitivity [93]. Some sources of non-stochastic noises have been identi"ed, and there are techniques for neutralizing or minimizing them. However, it is impossible to guarantee that all possible sources of non-stochastic noises have properly been taken into account, and only the cross correlation between di!erent detectors can provide a reliable GW detection. 6.1. An ideal two-interferometers correlation We now discuss the sensitivities that could be obtained correlating the major interferometers. In order to understand what is the best result that could be obtained with present interferometer technology, we "rst consider the sensitivity that could be obtained if one of the interferometers under constructions were correlated with a second identical interferometer located at a few tens of kilometers from the "rst, and with the same orientation. This distance would be optimal from the point of view of the stochastic background, since it should be su$cient to decorrelate local noises like, e.g., the seismic noise and local electromagnetic disturbances, but still the two interferometers would be close enough so that the overlap reduction function does not cut o! the high-frequency range. For this exercise we use the data for the sensitivity of VIRGO, Fig. 2.

 Correlations between two interferometers have already been carried out using prototypes operated by the groups in Glasgow and at the Max}Planck-Institute for Quantum Optics, with an e!ective coincident observing period of 62 h [191]. Although the sensitivity of course is not yet signi"cant, they demonstrate the possibility of making long-term coincident observations with interferometers.

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Let us "rst give a rough estimate of the sensitivity using h ( f ), h ( f ). From Fig. 2 we see that we  L can take, for our estimate, hI &10\ Hz\ over a bandwidth *f&1 kHz. Using ¹"1 yr, D Eq. (88) gives h ( f )&4.5;10\( f/100 Hz)(hI /10\ Hz\) . (101) L D We require for instance SNR"1.65 (this corresponds to 90% con"dence level; a more precise discussion of the statistical signi"cance, including the e!ect of the false alarm rate can be found in Ref. [12]). Then the minimum detectable value of h ( f ) is h ( f )"1.65h ( f ), and from Eq. (19) we   L get an estimate for the minimum detectable value of h X ( f ),   h X ( f )&3;10\( f/100 Hz)(hI /10\ Hz\) . (102)   D This suggests that correlating two VIRGO interferometers we can detect a relic spectrum with h X (100 Hz)&3;10\ at SNR"1.65, or 1;10\ at SNR"1. Compared to the case of   a single interferometer with SNR"1, Eq. (54), we gain "ve orders of magnitude. As already discussed, to obtain a precise numerical value one must however resort to Eq. (79). This involves an integral over all frequencies, (that replaces the somewhat arbitrary choice of *f made above) and depends on the functional form of h X ( f ). If for instance h X ( f ) is independent of the     frequency, using the numerical values of hI plotted in Fig. 2 and performing the numerical integral, D we have found for the minimum detectable value of h X   h X K2;10\(SNR/1.65)(1 yr/¹) (h X ( f )"const.) . (103)     This number is quite consistent with the approximate estimate (102), and with the value 2;10\ reported in Ref. [79]. Stretching the parameters to SNR"1 (68% c.l.) and ¹"4 yr, the value goes down at (3}4);10\. This might be considered an absolute (and quite optimistic) upper bound on the capabilities of "rst-generation experiments. It is interesting to note that the main contribution to the integral comes from the region f(100 Hz. In fact, neglecting the contribution to the integral of the region f'100 Hz, the result for h X  changes only by approximately 2%. Also, the lower part of the accessible frequency   range is not crucial. Restricting for instance to the region 20 Hz 4f4200 Hz, the sensitivity on h X degrades by less than 1%, while restricting to the region 30 Hz4f4100 Hz, the sensitivity   on h X degrades by approximately 10%. Then, from Fig. 2 we conclude that by far the most   important source of noise for the measurement of a #at stochastic background is the thermal noise. In particular, the sensitivity to a yat stochastic background is limited basically by the mirror thermal noise, which dominates in the region 40 Hz:f:200 Hz, while the pendulum thermal noise dominates below approximately 40 Hz. The sensitivity depends however on the functional form of X ( f ). Suppose for instance that in  the VIRGO frequency band we can approximate the signal as X ( f )"X (1 kHz)( f/1 kHz)? . (104)   For a"1 we "nd that the spectrum is detectable at SNR"1.65 if h X (1 kHz)K3.6;10\. For   a"!1 we "nd (taking f"5 Hz as lower limit in the integration) h X (1 kHz)K6;10\. Note   however that in this case, since a(0, the spectrum is peaked at low frequencies, and h X (5 Hz)K1;10\. So, both for increasing or decreasing spectra, to be detectable h X must     have a peak value, within the VIRGO band, of order a few ;10\ in the case a"$1, while

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a constant spectrum can be detected at the level 2;10\. Clearly, for detecting increasing (decreasing) spectra, the upper (lower) part of the frequency band becomes more important, and this is the reason why the sensitivity degrades compared to #at spectra, since for increasing or decreasing spectra the maximum of the signal is at the edges of the accessible frequency band, where the interferometer sensitivity is worse. 6.2. LIGO}LIGO Let us now see what can be done with existing interferometers. The two LIGO detectors are under construction at a large distance from each other, d&3000 km. This choice optimizes the possibility of detecting the direction of arrival of GWs from astrophysical sources, but it is not optimal from the point of view of the stochastic background, since the overlap reduction function cuts o! the integrand in Eq. (78) at a frequency of the order of 1/(2pd). The overlap function c( f ) for the LIGO}LIGO correlation has been computed in Ref. [108] and it has its "rst zero at fK64 Hz. Furthermore, the arms of the two detectors are not exactly parallel, and therefore "c(0)""0.89 rather than 1. The sensitivity to a stochastic background for the LIGO}LIGO correlation has been computed in Refs. [187,71,108,8,12]. The result, as we have discussed, depends on the functional form of h X . For X ( f ) independent of f, the minimum detectable value is    h X K5;10\ (105)   for the initial LIGO. However, we have seen that second generation interferometers could result in much better sensitivities. The sensitivity of the correlation between two advanced LIGO is estimated in Ref. [8] to be h X K5;10\ , (106)   which is an extremely interesting level. These numbers are given at 90% c.l. in [8], and a detailed analysis of the statistical signi"cance is given in [12]. 6.3. VIRGO}LIGO, VIRGO}GEO, VIRGO}TAMA In Fig. 9 we show the overlap reduction functions for the correlation of VIRGO with the other major interferometers. Using these functions, one can compute numerically the integral in Eq. (79) and obtain the minimum detectable value of h X . These values are shown in Table 1 (from   Ref. [27]), together with the value for LIGO}LIGO computed in Ref. [8]. All these numbers are at 90% con"dence level. We see that the correlation between VIRGO and any of the two LIGO is suppressed by the overlap reduction function above, say, 30}40 Hz. In the case of the VIRGO}GEO correlation, instead, c( f ) cuts the integrand only above say 200 Hz. However, the sensitivity of GEO is below 200 Hz is lower than LIGO, so that the result are basically the same as for VIRGO}LIGO, see Table 1. To obtain the precise numbers for the sensitivity, of course one has to perform the integral in Eq. (79). However, it is easy to have an understanding of the numbers that come out. As we have found in Section 6.1, the minimum detectable value of h X , using two identical VIRGO  

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Fig. 9. The overlap reduction functions c( f ) for the correlation of VIRGO with the other major interferometers (from Ref. [27]). Table 1 The sensitivity of various two-interferometers correlation Correlation

h X  

LIGO}WA * LIGO}LA VIRGO * LIGO}LA VIRGO * LIGO}WA VIRGO * GEO600 VIRGO * TAMA300

5;10\ 4;10\ 5;10\ 5.6;10\ 1;10\

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detectors, degrades only by less than 10% if, from the full VIRGO bandwidth, we restrict to the region 20 Hz(f(100 Hz; in this region "c( f )" for the VIRGO}GEO correlation is constant to a good accuracy, and of order 0.1. From Eqs. (79) and (18) we see that the minimum detectable value of h X scales with c( f ) as "c( f )"\. Therefore, for the VIRGO}GEO correlation we get   about a factor of 10 worse than the ideal result (103). Comparing Figs. 2 and 4, and recalling that h X &hI  , we see that we lose about another factor of 3 in h X compared to an ideal   D   VIRGO}VIRGO correlation, so that one gets the estimate h X &6;10\(SNR/1.65)(1 yr/¹) (h X ( f )"const.) , (107)     quite in agreement with the result of the more accurate computation shown in Table 1. The situation with the VIRGO}TAMA300 correlation is instead worse, as can be seen from the overlap reduction function and from the value in Table 1. 6.4. VIRGO}Resonant mass and Resonant mass}Resonant mass The correlation between two resonant bars and between a bar and an interferometer has been considered in Refs. [239,79,20}22,27]. Fig. 10 shows the overlap reduction functions for the correlation between VIRGO and one of the three resonant bars NAUTILUS, EXPLORER, AURIGA. These overlap reduction functions have been computed assuming that the bars have been reoriented so to achieve the maximum correlation with VIRGO, which is technically feasible. One should also note that there is the danger that one of the spikes due to the violin modes in the VIRGO sensitivity curve comes close to a resonant frequency of the bar; surprisingly, this apparent unlikely event is just what happens with the data used to draw Fig. 2; in fact, according to these data there is a pair of violin modes at 922.6 and 977.2 Hz. The "rst one happens to fall exactly at the resonance at 922 Hz of NAUTILUS! However, these data for the violin modes are not yet "nal. For instance, the VIRGO collaboration is presently considering the possibility of using silica instead of steel for the wires, which would change the position of the resonances. Furthermore, the resonance frequency of the bars can be tuned within a few Hz with the electronics; this would be quite su$cient, since the violin modes have a very high Q and so are much narrower than one Hz. The minumum detectable values for h X for some bar}bar and bar}interferometer correla  tions are given in Table 2 (from Ref. [27]), for 1 yr of observation and 90% con"dence level (SNR"1.65). A three detectors correlation AURIGA}NAUTILUS}VIRGO, with present orientations, would reach h X K3;10\ ,   at 90% c.l., while with optimal orientation,

(108)

h X K1.6;10\ . (109)   Although the improvement in sensitivity in a bar}bar}interferometer correlation is not large compared to a bar}bar or bar}interferometer correlation, a three-detectors correlation would be important in ruling out spurious e!ects [239].

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Fig. 10. The overlap reduction function c( f ) for the correlation of VIRGO with NAUTILUS (solid line), AURIGA (dashed line) and EXPLORER (dotted line) in the case in which the bars have been reoriented so to be aligned with VIRGO (from Ref. [27]).

Table 2 The sensitivity of the correlation of VIRGO with resonant bars, and of two resonant bars (from Ref. [27]), but setting the con"dence level to 90%) Correlation

h X  

VIRGO * AURIGA VIRGO * NAUTILUS AURIGA * NAUTILUS

4;10\ 7;10\ 5;10\

In the case of correlations involving resonant bars, obviously there is no issue of optimal "ltering, since they are narrow-band detector, and the sensitivity does not depend on the shape of the spectrum. As we have seen in Section 5, using resonant optical techniques, it is possible to improve the sensitivity of interferometers at special values of the frequency, at the expense of their broad-band sensitivity. Since bars have a narrow-band anyway, narrow-banding the interferometer improves the sensitivity of a bar}interferometer correlation by about one order of magnitude [79]. Thus, the limit of bar}bar}interferometer correlation, with narrow banding of the interferometer, is of order h X &a few;10\.   Cross-correlation experiments have already been performed using NAUTILUS and EXPLORER [23]. The bars are oriented so that they are parallel, and then the overlap reduction function C( f ) results in a reduction of sensitivity of about a factor of 6, compared to ideal same site detectors. The detectors are tuned at the same resonance frequency f"907.2 Hz, with an

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overlapping band $0.05 Hz, and the overlapped data cover a period of approximately 12 h. Of course, with such a short coincidence time, the bound on h X is not yet signi"cant. Running for   one year, it is expected to reach h X (1. Cross-correlations, searching for bursts, have also been   done between ALLEGRO and EXPLORER [19]. While resonant bars have been taking data for years, spherical detectors are at the moment still at the stage of theoretical studies (although prototypes might be built in the near future), but could reach extremely interesting sensitivities. In particular, two spheres with a diameter of 3 m, made of Al5056, and located at the same site, could reach a sensitivity h X &4;10\ [239]. This "gure   improves using a more dense material or increasing the sphere diameter, but it might be di$cult to build a heavier sphere. Another very promising possibility is given by hollow spheres [75]. The theoretical studies of Ref. [75] suggest that correlating two hollow spheres one could reach the h X K10\;( f/200 Hz)(hI /10\ Hz\)(20 Hz/*f )(10 s/¹) . (110)   D With the value of hI and *f suggested by Ref. [75], it could be possible to reach an extremely D interesting value, h X &10\.   7. Bounds on h20 X ( f )  In this section we discuss various experimental bounds on h X ( f ). We will be interested not   only on the bounds at values of f where interferometers or resonant bars can operate, but also at all possible frequencies. The reason will become apparent when we will discuss the spectra from various speci"c cosmological mechanisms for the production of relic GWs. These spectra depend of course on the parameters of the cosmological model. Often the frequency dependence is, to a "rst approximation, completely determined, but the overall value of h X depends on some para  meters of the model. In some case, and especially for the ampli"cation of vacuum #uctuations (Section 9.1) these spectra extend over a huge range of frequencies, ranging from frequencies as small as 10\ Hz (corresponding to wavelength of the order of the present Hubble radius of the Universe) up to possibly the GHz region. It is therefore important to see what are the experimental constraint, at any frequency, on h X , since they automatically imply bounds on the parameters of   the model that enter in the spectrum, and therefore on its the value at frequencies accessible to interferometers or resonant masses. The various limits discussed in this section are summarized in Fig. 11. 7.1. The nucleosynthesis bound Nucleosynthesis successfully predicts the primordial abundances of deuterium, He, He and Li in terms of one cosmological parameter g, the baryon to photon ratio. In the prediction enter also parameters of the underlying particle theory, which are therefore constrained in order not to spoil the agreement. In particular, the prediction is sensitive to the e!ective number of species at time of nucleosynthesis, g "g (¹KMeV). With some simpli"cations, the dependence on g can be H H understood as follows. A crucial parameter in the computations of nucleosynthesis is the ratio of the number density of neutrons, n , to the number density of protons, n . As long as thermal   equilibrium is maintained we have (for non-relativistic nucleons, as appropriate at ¹&MeV, when

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Fig. 11. The various bounds on h X ( f ) discussed in the text.  

nucleosynthesis takes place) n /n "exp(!Q/¹) where Q"m !m K1.3 MeV. Equilibrium is     maintained by the process pe  nl, with width C , as long as C 'H. When the rate drops CJ CJ below the Hubble constant H, the process cannot compete anymore with the expansion of the Universe and, apart from occasional weak processes, dominated by the decay of free neutrons, the ratio n /n remains frozen at the value exp(!Q/¹ ), where ¹ is the value of the temperature     at time of freeze-out. This number therefore determines the density of neutrons available for nucleosynthesis, and since practically all neutrons available will eventually form He, the "nal primordial abundance of He is exponentially sensitive to the freeze-out temperature ¹ . Let us  take for simplicity C &G ¹ (which is really appropriate only in the limit ¹
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a decrease in g, and therefore we also need a lower limit on g, which is provided by the comparison with the abundance of deuterium and He. Rather than g , it is often used an &e!ective number of neutrino species' N de"ned as follows. In H J the standard model, at ¹& a few MeV, the active degrees of freedom are the photon, e!, neutrinos and antineutrinos, and they have the same temperature, ¹ "¹. Then, for N families of light G J neutrinos, g (N )"2#()(4#2N ), where the factor of 2 comes from the two elicity states of J H J  the photon, 4 from e! in the two elicity states, and 2N counts the N neutrinos and the J J N antineutrinos, each with their single elicity state; the factor () holds for fermions. In the J  Standard Model with N "3, g ". So we can de"ne an &e!ective number of neutrino species' J H  N from J ¹  7 ¹  7 43 G # G "2# (4#2N ) # g g (112) G ¹ G ¹ J 8 8 4 G
 G   or













¹  7 ¹  7 G # G " (N !3) . g g (113) G ¹ G 8 ¹ 4 J G
 G   One extra species of light neutrino, at the same temperature as the photons, would contribute one unit to N , but all species, weighted with their energy density, contribute to N , which of course J J in general is not an integer. For i"gravitons, we have g "2 and (¹ /¹)"o /o , where G G  A o "2(p/30)¹ is the photon energy density. If gravitational waves give the only extra contribuA tion to N , compared to the standard model with N "3, then J J ¹  o G "2  (114) g G ¹ o A G
 and therefore




(o /o ) "(N !3) , (115)  A ,1  J where the subscript NS reminds that this equality holds at time of nucleosynthesis. If more extra species, not included in the standard model, contribute to g (N ), then the equal sign in the above H J equation is replaced by lower or equal. The same happens if there is a contribution from any other form of energy present at the time of nucleosynthesis and not included in the energy density of radiation, like, e.g., primordial black holes. To obtain a bound on the energy density at the present time, we note that from the time of nucleosynthesis to the present time o scaled as 1/a, while the photon temperature evolved  following g (¹)¹a"const. (i.e. constant entropy) and o &¹&1/(ag). Therefore 1 A 1 (o /o ) "(o /o ) (g (¹ )/g (1 MeV))"(o /o ) (3.913/10.75) , (116)  A   A ,1 1  1  A ,1  To compare with Eq. (56) of Ref. [8], note that while we use o , in Ref. [8] the bound is written in terms of the total A energy density in radiation at time of nucleosynthesis, which includes also the contribution of e!, neutrinos and antineutrinos, (o ) "[1#()(2#3)](o ) .   ,1  A ,1

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where the subscript zero denotes present time. Therefore, we get the nucleosynthesis bound at the present time, (o /o ) 40.227(N !3) . (117)  A  J Of course this bound holds only for GWs that were already produced at time of nucleosynthesis (¹&MeV, t&s). It does not apply to any stochastic background produced later, like backgrounds of astrophysical origin. Note that this is a bound on the total energy density in gravitational waves, integrated over all frequencies. Writing o "d(log f ) do /d log f, multiplying both o and o in    A Eq. (117) by h /o and inserting the numerical value h o /o K2.481;10\ [195], we get    A  D d(log f )h X ( f )45.6;10\(N !3) . (118)   J D The bound on N from nucleosynthesis is subject to various systematic errors in the analysis, which J have to do mainly with the issues of how much of the observed He abundance is of primordial origin, and of the nuclear processing of He in stars, and as a consequence, over the last "ve years have been quoted limits on N ranging from 3.04 to around 5. The situation has been reviewed in J Ref. [84]. The conclusions of Ref. [84] is that, until current astrophysical uncertainties are clari"ed, N (4 is a conservative limit. Using extreme assumptions, a meaningful limit N (5 still exists, J J showing the robustness of the argument. Correspondingly, the right-hand side of Eq. (118) is, conservatively, of order 5;10\ and anyway cannot exceed 10\. If the integral cannot exceed these values, also its positive de"nite integrand h X ( f ) cannot   exceed it over an appreciable interval of frequencies, * log f&1. One might still have, in principle, a very narrow peak in h X ( f ) at some frequency f, with a peak value larger than say 10\, while   still its contribution to the integral could be small enough. But, apart from the fact that such a behaviour seems rather implausible, or at least is not suggested by any cosmological mechanism, in the detection at broadband detectors like VIRGO, one must also take into account that, even if we gain in the height of the signal, this is partially compensated by the fact that we lose because of the reduction of the useful frequency band *f, see Eq. (78). It is sometimes useful to think in terms of the dimensionless amplitude h ( f ) instead of h X .    From Eq. (19), we see that a bound h X ( f )(5;10\, valid independently of the frequency f,   translates into a frequency-dependent bound on h ( f ),  h ( f )(2.82;10\(1 Hz/f ) , (119)  so that, for instance, h ( f )(10\ around 200}300 Hz, where ground-based interferometers have  their highest sensitivity, and h ( f )(10\ at the lower range accessible to LISA, f&10\ Hz. 



7.2. The COBE bound Another important constraint comes from the COBE measurement of the #uctuation of the temperature of the cosmic microwave background radiation (CMBR). The basic idea is that a strong background of GWs at very long wavelengths produces a stochastic redshift on the frequencies of the photons of the 2.7 K radiation, and therefore a #uctuation in their temperature. This is known as the Sachs}Wolfe e!ect [209,158]. We "rst give a qualitative understanding of this bound, and then quote more precise numbers.

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It is conventional to expand the CMBR temperature #uctuations on the sky in spherical harmonics,  J d¹(XK ) " a (r)> (XK ) . (120) JK JK ¹ J K\J The dipole term is dominated by the Earth motion, and therefore is omitted. The multipole amplitudes a depends on the observer position r, and "a (r)", averaged over all available JK JK positions, is the angular power spectrum. The rms #uctuations of the temperature, averaged over the sky, are given by


 

d¹   2l#1 " 1"a "2= , (121) JK J ¹ 4p J where = depends only on the detector. COBE measures a temperature di!erence between two J antennas separated by an angle h, and (122) = "2[1!P (cos h)]e\J>N , J J where P (cos h) are the Legendre polynomials and p is the "nite width of the beam, which e!ectively J cuts the resolution at l&1/p. For the quadrupole moment, one de"nes





 1"a "2  K . (123) 4p K\ Assuming a #at Harrison}Zeldovich spectrum, i.e. spectral index n"1 (see Section 9.2.2), the COBE result is Q"¹

Q"18$1.6 lK ,

(124)

while "tting also n the best "t gives n"1.2$0.3 and Q"15.3 lK [36]. In any case, from the quadrupole alone we get d¹/¹K(5!6);10\ . It is not di$cult to see that the COBE bound on X X ( f )((H /f )(d¹/¹) .   and holds for 3;10\ Hz(f(10\ Hz .

(125) 

has qualitatively the form (126)

(127)

Let us "rst discuss why this is the relevant frequency range. The limitation f'3;10\ Hz is just the condition that these waves are inside the horizon today. The upper limit instead comes out as follows. It is important to distinguish between large- and small-angle #uctuations. The dividing line is given by the angle h such that #uctuations on angular scales h(h corresponds to *11 *11 comoving lengths that were inside the horizon at time of last scattering, while h'h corresponds *11 to super-horizon sized scales at time of last scattering. Small-scale #uctuations are sensitive to microphysical processes taking place at time of last scattering; large-scale #uctuations are instead insensitive to microphysics, because they refer to regions that were not in causal contact at that

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time, and on these angular scales the Sachs}Wolfe e!ect, due either to scalar perturbations or to tensor perturbations, i.e. GWs, is the dominant contribution to the CMBR anisotropy. To compute h , one "rst de"nes conformal time today, g , and at the last scattering surface, *11  g as *11 R dt "2H\ , g "   a(t)  g R*11 dt  K , (128) g " *11 a(t) (1#z  *11 where we assumed a #at matter-dominated Universe today. The quantities cg and cg are the  *11 comoving size of the horizon today and at last scattering, respectively; the comoving distance to the last scattering surface is





d

(129) "cg (1!1/(1#z )Kcg .  *11  *11 In the standard picture, where recombination occurs at z K1300, then z K1100. In non *11 standard scenarios, z could be as low as O(100) [225]. The angle subtended by the horizon at last *11 scattering is [233] h

&g /g &1/(z . (130) *11 *11  *11 For a non-#at Universe, X O1, this generalizes to h &(X /z ). For z <100 this is  *11  *11 *11 a good approximation to the exact formula. The deviations for smaller redshift can be found in [225]. For z "1100, Eq. (130) gives h &23. *11 *11 The temperature #uctuations over an angle h are related to a by JK (d¹/¹)&l1"a "2 , (131) F JK for l&2003/h .

(132)

Therefore, multipoles up to, say, l&30, are due to large-angle #uctuations, and the corresponding anisotropies a are dominated by the Sachs}Wolfe e!ect, due either to scalar perturbations or JK to GWs. The waves that were outside the horizon at the last scattering, today have a frequency, (133) f((z H K10\ Hz , *11  where in the last equality we have taken z "O(10). For a non-standard reionization scenario *11 with z &100 the bound becomes approximately 3;10\ Hz. *11 Let us now explain how Eq. (126) comes out. A GW with a wave number k(h)&(2003/h)g\ , (134)  subtends an angle h on the last scattering surface and creates temperature #uctuation on this angular scale given approximately by [233] (d¹/¹) &h[z , k(h)] , (135) F *11 where h[z , k(h)] is the characteristic amplitude that this GW had at time of last scattering. *11

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We therefore must connect the value of the amplitude at time of last scattering with the value of h ( f ) today. One then observes that, as long as the GW is outside the horizon, its amplitude is  constant, while, after it enters the horizon, it redshifts with the FRW scale factor a(t) as 1/a. This can be derived from the fact that the energy density of the GW is o &f h ( f ), Eq. (16); f redshifts as   1/a and, after entering the horizon, o &1/a, as for any relativistic particle, so that h ( f )&1/a.   Therefore, denoting by a the value of a at horizon crossing,  h (f) "(a /a )h ( f ) "a h /a z . (136)       *11  *11 *11 *11 During matter dominance, the scale factor depends on conformal time as a(g)&g. Then, for waves that enter the horizon during matter dominance, a /a K(g f )\. Using g K  *11 *11 *11 g /(z "2/(H (z ), one "nally gets  *11  *11 h (f) K(H /2f )h ( f ) &(H /2f )(d¹/¹) . (137)      *11  Using Eq. (17), we therefore get X ( f )"(2p/3H ) f h( f )&(H /f )(d¹/¹) . (138)     Of course, the value of (d¹/¹) induced by GWs cannot exceed the observed value, Eq. (125), and this gives the bound on h X .   Using the more quantitative analysis of Refs. [11,8], where the e!ect of multipoles with 24l430 is included, this bound reads h X ( f )(7;10\(H /f ), (3;10\ Hz(f(10\ Hz) . (139)    This bound is stronger at the upper edge of its range of validity, f&30H &10\ Hz, where it  gives h X ( f )(10\, ( f&10\ Hz) . (140)   Another issue is whether GWs can saturate this bound, or whether instead the dominant contribution to d¹/¹ comes from scalar rather than tensor perturbations; the answer is in general model-dependent, and we will discuss examples in Section 9.2.2. The important point that we can draw is the following: from the explicit computations discussed below, we will "nd cosmological spectra that extends from very low frequencies, f&H K  3;10\ Hz, up to the frequencies accessible at GW experiment and higher. This will be typically the case of the spectra predicted by the ampli"cation of vacuum #uctuations. The frequency dependence can often be computed reliably. Then, in order for these spectra to be observable in the 1 Hz}1 kHz region, they must grow fast enough, so that they can satisfy Eq. (140), and still have a sizeable value of h X at higher frequencies. The requirement becomes even more stringent if we   combine it with the nucleosynthesis bound: the spectra cannot keep growing until they reach the cuto!, say at 1 GHz, because otherwise the NS bound on the integral of h X would be violated.   Therefore, it is necessary that there is at some stage a change of regime, so that the spectrum #attens or starts decreasing. Thus, it is clear that the combination of the NS bound and the COBE bound, together with the request of having a sizeable value of h X at, say, 1 kHz, restricts considerably the cosmological   models that can produce an interesting spectrum of vacuum #uctuations. Quite remarkably, there

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is however one candidate, theoretically well motivated, that passes these severe restrictions; this is the string cosmology model of Gasperini and Veneziano, and we will discuss it in Section 9.3. 7.3. Pulsars as GW detectors: ms pulsars, binary pulsars and pulsar arrays Pulsars are rapidly rotating, highly magnetised neutron stars formed during the supernova explosion of stars with 5}10 solar masses (for a recent review see Ref. [173]). Soon after their discovery, it became clear that pulsars are excellent clocks. Actually, single pulses have a rather herratic pro"le in time, but after integrating over a number of pulses one "nds an integrated pro"le that is extremely stable, and can be used as a clock. Most of the known pulsars have a period of the order of one second, while a growing fraction of this sample has periods in the range 1.5}30 ms. These millisecond pulsar are an extremely impressive source of high precision measurements. For instance, the observations of the "rst ms pulsar discovered, B1937#21, after 9 yr of data, give a period of 1.557 806 468 819 794 5$0.000 000 000 000 000 4 ms. The speed up of the orbital period of the double neutron star system B1913#16 provided experimental evidence for the existence of GWs and a test of General Relativity at the level of 1% [144,223,224,86]. This phenomenal stability makes pulsar competitive with atomic clocks (for instance PSR B1855#09 becomes competitive with atomic clocks after 3 yr of observations), and has even allowed to detect orbiting bodies smaller than the Earth from the radial acceleration that they induce on the pulsar [173]. Pulsars are also a natural detector of GWs [213,94,207,136,38,46,222,152]. In fact, a GW passing between us and the pulsar causes a #uctuation in the time of arrival of the pulse, proportional to the GW amplitute h ( f ). Consider a GW traveling along the z-axis, and let h be the polar angle of  the Earth}pulsar direction, measured from the z-axis. Then the Doppler shift at time t induced by the GW is [94,111] (141) *l(t)/l"(1!cos h)[cos 2t h (t!(s/c))#sin 2t h (t!(s/c))] , > "  where, as in Section 3, t is a rotation in the plane orthogonal to the direction of propagation of the wave (in this case the (x, y) plane), corresponding to a choice of the axes to which the # and ; polarization are referred. The factor ()(1!cos h) is the standard angular dependence of the  Doppler shift, and s is the distance along the path, so that t!s/c is the time when the GW crossed the Earth}pulsar direction. If the uncertainty in the time of arrival of the pulse is e and the total observation time is ¹, this &detector' would be sensitive to h ( f )&e/¹, for frequencies f&1/¹. The highest sensitivities can  then be reached for a continuous source, as a stochastic background, after one or more years of integration, and therefore for f&10\}10\ Hz. Based on the data from PSR B1855#09 and correcting an error from a previous analysis, Ref. [228] gives a limit, at f,f "4.4;10\ Hz, H h X ( f )(1.0;10\, (95% c.l.) , (142)   H or h X ( f )(4.8;10\, (90% c.l.) . (143)   H Since the error on h ( f ) is proportional to 1/¹ and h X &h, the bound for f'f is (at 90% c.l.)     H h X ( f )(4.8;10\( f/f ) (144)   H

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and therefore it is quite signi"cant (better than the NS bound) also for f&10f . For f(f , instead, H H the pulsar provides no limit at all. These limits are likely to continue to improve since a number of recently discovered ms pulsars appear to have a timing stability comparable or better than PSR B1855#09. These limits are especially signi"cant for cosmic strings, as we will see in Section 10.2. Another important limit comes from pulsars in binary systems, e.g. with a white dwarf or neutron star companion. These systems have a second clock, which is the binary orbit itself, and are very clean. The change in the orbital period P can be computed from General Relativity and in many cases is negligibly small. This has the advantage that it is not necessary to make a "t to an unknown derivative of the orbital period, PQ , and subtract it to isolate the contribution of GWs. This gives a limit that holds for a large range of frequencies, c/D(f(1/¹ ,

(145)

where ¹ is the observation time and D the distance to the pulsar. A detailed analysis of this limit has been performed in [159], correcting previous estimates. In particular, the limit from B1855#09 turns out to be the most stringent and gives [159] h X (2.7;10\ , (146)   in the frequency range 10\(f(4.4;10\ Hz. Finally, one can observe that these limits have been set using a pulsar as a single detector. From the analysis of Section 3 it is clear that if one can use two or more pulsars as coincident detectors one could obtain a much better sensitivity [136,111,173]. From Eq. (141) we see that the Doppler shift from the ith pulsar has the general form *l (t) G "a s(t)#n (t) , (147) G G l G where s(t) is the GW signal, a a geometrical factor depending on the line-of-sight direction, and G n (t) contains the noise intrinsic to the pulsar timing. The cross-correlation between two pulsars G then gives a a 1s(t)2#a 1sn 2#a 1sn 2#1n n 2 (148) G H G G H H G H and, since the noises from di!erent pulsars are uncorrelated, increasing the observation time this tends to a a 1s(t)2. Furthermore, the correlation between many pulsars allows to subtract the G H errors due to terrestrial clocks, since there is now the possibility of timing multiple pulsars against each other. Furthermore, precise timing also require to subtract the e!ect of the Earth motion, and the precision of pulsar timing measurement is now near the limit of models of the Earth motion. Again, multiple pulsar timing allows to subtract this e!ect. Performing pulsar coincidences using the present database of long-term timing observation does not improve on the limits obtained with single pulsars [173]. The basic reason is that, in order to e!ectively subtract the error due to the Earth motion, one needs three pulsars widely separated across the sky. A fourth pulsar is needed as a clock, and a "fth to measure the GW amplitude [111]. However, the galactic ms pulsars known around 1990, for which we now have many years of data taking, were all concentrated in the same region of the sky (as a consequence of the fact that they were all found at Arecibo); in this situation one cannot subtract e!ectively the error due to the

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Earth motion. However, the ms pulsars known today are distributed much more uniformly across the sky, and therefore the continued timing of these pulsar in the next few years will greatly improve the sensitivity of pulsar arrays as GW detectors. Finally, we observe that the limit on h X from pulsar timing holds for all sort of GW   backgrounds, not necessarily of primordial origin, contrarily to the nucleosynthesis bound discussed in Section 7.1 that of course holds only for GWs that were already present at time of nucleosynthesis. 7.4. Light deyection by GWs We "nally mention that there has been quite some activity in trying to obtain bounds on h X   from the e!ect that a stochastic background of GWs would have on the propagation of light from distant sources [245,48]. The subject is somewhat controversial. Examining the proper motion of quasars, Ref. [134] sets a limit DH df h X (0.11, where f &2;10\ Hz is the inverse of the   H observation time, and states that\ considerable improvement in this limit should be possible in the next decade. Note that the limit is weaker than the one given by nucleosynthesis, but, if correct, it would still be useful because it is not restricted to GWs present before nucleosynthesis. A similar e!ect is the apparent clustering in the galaxy}galaxy correlation function which, according to Ref. [170], provides a bound h X (10\ for wavelength between a few tens of kpc to a few   hunderds Mpc, i.e. for 10\ Hz(f(10\ Hz. However, these bounds have been reanalyzed in Ref. [148] where (con"rming older computations [245]) it has been found that, to linear order in the metric perturbation, the e!ect due to GWs does not grow linearly with the distance to the source, contrarily to previous claims, but only logarithmically. Ref. [148] then analyzes a number of potentially interesting e!ects: proper motion of distant sources, galaxy clustering, lateral displacement of caustic networks, weak-lensing of distant objects, and concludes that no useful bound on h X can be set. This result is valid for stochastic GWs,   and does not apply immediately to the de#ection of light by GWs from localized sources. However, in this case the analysis of Ref. [88] con"rms the conclusion that these e!ects are too small to be of observational interest.

8. Production of relic GWs: a general orientation Theoretical predictions of relic GW backgrounds are necessarily subject to large uncertainties, due either to the fact that we must use physics beyond the Standard Model, or to uncertainties in the details of the cosmological mechanisms. It is therefore important to understand what features of a given result are relatively general, and what are speci"c to a given model. In this section we discuss results on the characteristic frequency, the form of the spectrum and the typical intensity, that we will consider as a sort of &benchmark'. The results from various speci"c computations can then be better understood from the comparison with these benchmarks, and in particular, one can get a feeling of what results are relatively general and model independent, and what results are really peculiar to a speci"c computation.

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8.1. The characteristic frequency First of all, we discuss what is the typical frequency where we can expect to "nd today a signal of cosmological origin. Obviously, part of the answer depends on the dynamics of the production mechanism, and therefore is model dependent, and part is kinematical, depending on the redshift from the production era. First of all, it is useful to separate the kinematics from the dynamics. To specify the kinematics, we need a cosmological model. We consider the standard Friedmann}Robertson}Walker (FRW) cosmological model, consisting of a radiation-dominated (RD) phase followed by the present matter-dominated (MD) phase, and we call a(t) the FRW scale factor. The RD phase goes backward in time until some new regime sets in. This could be an in#ationary epoch, e.g. at the grand uni"cation scale, or the RD phase could go back in time until Planckian energies are reached and quantum gravity sets in, i.e., until t&t K5;10\ s. . A graviton produced with a frequency f , at a time t"t within the RD phase, has today (t"t ) H H  a redshifted frequency f given by f "f a(t )/a(t ). To compute the ratio a(t )/a(t ) one uses the   H H  H  fact that during the standard RD and MD phases the Universe expands adiabatically. The entropy per unit comoving volume is S"const.g (¹)a(t)¹ , (149) 1 where g (¹) is de"ned by [158]. 1 ¹  7 ¹  G # G . (150) g g (¹)" g G ¹ 1 G ¹ 8 G  G
 Here g counts the internal degrees of freedom of the ith particle (spin, colour, etc.). G Thus g (¹) is a measure of the e!ective number of degrees of freedom at temperature ¹, as far as 1 the entropy is concerned. Another useful quantity is













¹  7 ¹  G # G g(¹)" g , (151) g G ¹ G ¹ 8 G
 G  which counts the e!ective number of degrees of freedom at temperature ¹, as far as the energy is concerned, and that entered our discussion of nucleosynthesis, Section 7.1. In the early Universe most particles had a common temperature, and g(¹) and g (¹) are indistinguishable. In the 1 standard model, at ¹9300 GeV, they become constant and have the value g "g"106.75, while 1 today, assuming three neutrino species, g (¹ )K3.91 and g (¹ )K3.36 [158]. 1  1  The sum in Eqs. (150) and (151) runs over relativistic species. This holds if a species is in thermal equilibrium at the temperature ¹ . If instead it does not have a thermal spectrum (which in general G is the case for gravitons) we can still use the above equation, where for this species ¹ does not G represent a temperature but is de"ned (for bosons) by o "g (p/30)¹, where o is the energy G G G G density of this species. Using g (¹ )a(t )¹ "g (¹ )a(t )¹ 1 H H H 1    and ¹ "2.728$0.002K [107], one "nds [160]  f "f a(t )/a(t )K8.0;10\f (100/g (¹ ))(1 GeV/¹ ) .  H H  H 1 H H

(152)

(153)

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We must now ask what is the characteristic value of the frequency f of a gravition produced at H time t , when the temperature was ¹ . Here of course the dynamics enters, but still some general H H discussion is possible. One of the relevant parameters in this estimate is certainly the Hubble parameter at time of production, H(t ),H . This comes from the fact that H\ is the size of the H H H horizon at time t . The horizon size, physically, is the length scale beyond which causal microH physics cannot operate (see e.g. [158, Chapter 8.4]), and therefore, for causality reasons, we expect that the characteristic wavelength of gravitons or any other particles produced at time t will be of H order H\ or smaller. H Therefore, we write j "eH\ . H H The above argument suggests e41. During RD, H "(8p/3)Go . Then H  8pg ¹ H H , H " H 90M . and, using f ,H /e, Eq. (153) can be written as [160] H H 1 f K1.65;10\ (¹ /1 GeV)(g /100) Hz .  H e H

(154)

(155)

(156)

This simple equation allows to understand a number of important points concerning the energy scales that can be probed in GW experiments. Basically, the e!ects of the dynamics have been isolated into the parameter e. The relation between time and temperature during the RD phase then tells us how far back in time are we exploring the Universe, when we observe a graviton produced at temperature ¹ , H t K(2.42/g)(MeV/¹ ) s (157) H H H and therefore, detecting a GW that today has a frequency f , we are looking back at the Universe at  time t K6.6;10\ 1/e(1 Hz/f )(100/g ) s . H  H

(158)

 On a more technical side, the deeper reason has really to do with the invariance of general relativity under coordinate transformations, combined with the expansion over a "xed, non-uniform, background. Consider for instance a scalar "eld (x) and expand it around a given classical con"guration, (x)" (x)#d (x). Under a general coordinate  transformation xPx, by de"nition a scalar "eld transforms as (x)P (x)" (x). However, when we expand around a given background, we keep its functional form "xed and therefore under xPx, (x)P (x), which for a non  constant "eld con"guration, is di!erent from (x). It follows that the perturbation d (x) is not a scalar under general  coordinate transformations, even if (x) was a scalar. The e!ect becomes important for the Fourier components of d (x) with a wavelength comparable or greater than the variation scale of the background (x). (We are discussing a scalar  "eld for notational simplicity, but of course the same holds for the metric tensor g .) In a homogeneous FRW IJ background the only variation is temporal, and its timescale is given by the H\. Therefore, modes with wavelength greater than H\ are, in general, plagued by gauge artefacts. This problem manifests itself, for instance, when computing density #uctuations in the early Universe. In this case one "nds spurious modes which can be removed with an appropriate gauge choice, see e.g. Ref. [158, Section 9.3.6] or Ref. [189].

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Table 3 The production time t and the production temperature ¹ for GWs observed today at frequency f , if at time of H H  production they had a wavelength of the order of the horizon length Detection frequency (Hz)

Production time (s)

Production temperature (GeV)

10\ 1 100 10

7;10\ 7;10\ 7;10\ 7;10\

600 6;10 6;10 6;10

The simplest estimate are obtained taking e"1 in Eq. (156). Table 3 gives some representative values for the production time t and the temperature of the Universe ¹ corresponding to H H frequencies relevant to LISA and to ground based interferometers, setting e"1. However, one should keep in mind that the estimate j &H\, i.e. e&1, can sometimes be H H incorrect, even as an order of magnitude estimate. Below we will illustrate this point with some speci"c examples. The values in Table 3 are more correctly interpreted as a starting point for understanding how the result is a!ected by the dynamics of the speci"c production mechanism. An important question is whether GW experiments can explore the very high-energy regime, when the Universe had temperatures of the order of a typical grand uni"cation scale, or of the order of the Planck scale. The scale of quantum gravity is given by the Planck mass, related to Newton constant by G"1/M . More precisely, since in the gravitational action enters the combination 8pG, we expect . that the relevant scale is the reduced Planck mass M /(8p)K2.44;10 GeV. Using Eq. (156) . with ¹ "M /(8p) and e"1 gives H . (159) f &400(g /100) GHz, (¹ "M /(8p) .  H H . The dependence on g is rather weak because of the power  in Eq. (156). For g "1000, H  H f increases by a factor &1.5 relative to g "100. Instead, for ¹ "M &10 GeV, and  H H %32 g K220 (likely values for a supersymmetric uni"cation), H f &2 GHz, (¹ "M ). (160)  H %32 Using instead the typical scale of string theory, M , the characteristic frequency is between 1 these two values, since M is expected to be approximately in the range 10\:M /M :10\ 1 1 . [151]. Thus, the expected peak frequency of GW spectra produced at these very early epochs is in the GHz region. In this region no present or future experiment can operate, and the reason is easily seen from Eq. (22): considering that the maximum value of h X is "xed by nucleosynthesis, the   required sensitivity in the GW amplitude goes as f \. At f"1 GHz one needs a value of hI smaller by a factor 10, compared to a value that would give the same sensitivity in h X at D   f"1 kHz. However, one should not hurry toward pessimistic conclusions. The characteristic frequency that we have discussed is the value of the cuto! frequency in the graviton spectrum. Above

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this frequency, the spectrum decreases exponentially, and no signal can be detected. Below this frequency, however, the form of the spectrum is not "xed by general arguments. We discuss the issue in the next section. 8.2. The form of the spectrum For a thermal spectrum, the occupation number per cell of the phase space, n , is given by the D Bose}Einstein distribution and therefore, below the characteristic frequency, n "1/(epDI2!1)&k¹/2pf . (161) D Then, from Eq. (26) we then "nd that at low frequencies a thermal spectrum corresponds to X ( f )&f  . (162)  Of course, if we have a thermal spectrum with a cuto! frequency in the GHz region, the value of h X ( f ) at, say, f"1 kHz is utterly negligible. However, we have seen in the Introduction that   below the Planck scale gravitons interact too weakly to thermalize, and therefore there is no a priori reason for a &f  dependence. The gravitons will retain the form of the spectrum that they had at time of production, and this is a very model-dependent feature. However, from a number of explicit examples and general arguments that we will discuss below, we learn that spectra #at or almost #at over a large range of frequencies seem to be not at all unusual in cosmology. In particular: 1. The ampli"cation of vacuum #uctuations always give spectra that extend over a huge range of frequency, from &3;10\ Hz, corresponding to wavelength equal to the present size of the horizon, up to a cuto! in the GHz region. The spectrum found in some cases in string cosmology is particularly interesting from this point of view, since it can satisfy the COBE and nucleosysnthesis bound, and still have a large intensity at the frequency accessible to experiments. The basic reason for this behaviour is the fact that in the problem there is only one relevant length scale, given by the value of the Hubble parameter. 2. Another example of relic GW spectrum which is almost #at over a very large range of frequencies is provided by GWs produced by the decay of cosmic strings, discussed in Section 10.2. In this case the spectrum has a peak around f&10\ Hz, where h X can reach a few   times 10\, and then it is almost #at from f&10\ Hz to the GHz region, where it has the cuto! ("xed by the arguments previously discussed, considering that the relevant scale for cosmic string is M ). The basic reason for such a behaviour is the scaling property of the string %32 network, which says that a single length scale, the Hubble length, characterizes all properties of the string network [236,8]. The network of strings evolves toward a self-similar con"guration, with small loops being chooped o! very long strings, and the typical radiation emitted by a single loop has a wavelength related to the length of the loop. 3. Phase transitions can also give spectra that extend over a large range of frequencies; in this case, as we will see in Section 10.1.3, the reason is that in an expanding Universe a "eld cannot have a correlation length larger than the horizon size, basically because of causality. These facts have potentially important consequences. They suggest that, even if a spectrum of gravitons produced during the Planck era has a cutow at frequencies much larger than the range of

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frequencies accessible to interferometers, still in this range we can hope to observe the low-frequency part of these spectra. 8.3. Characteristic intensity The numbers discussed in Section 7.1 give a "rst idea of what can be considered an interesting detection level for h X ( f ), which should be at least   h X :5;10\ , (163)   especially considering that bound (118) refers not only to gravitational waves, but also to all possible sources of energy which have not been included, like particles beyond the standard model, primordial black holes, etc. The next question is whether it is reasonable to expect that some cosmological production mechanism saturates the nucleosynthesis bound. This of course depends on the production mechanism, but some relatively general considerations are possible. First of all, it is clear that, if GWs are produced at the Planck scale by collisions and decays together with the photons that we observe today in the CMBR, and there is not an in#ationary phase at later time, we expect roughly o &o . Since h o /o K2.481;10\, in this case bound  A  A  (115) is approximately saturated. More precisely, if at some time t"t both the photons that we H observe in the CMBR and gravitons were produced, and with a total fraction of the energy density in gravity waves X , then, taking into account the di!erent redshifts of GWs and photons due to H the fact that GWs decouple immediately, at the present time we would have [160] h X K1.67;10\(100/g )X . (164)   H H If instead the mechanism that produces GWs is di!erent from the mechanism that produces the photons in the CMBR, often it is still possible to relate the respective energy densities, simply because the scales of the two processes are related. A few examples will illustrate the point. 1. In Section 9.3 we will discuss the spectrum produced in string cosmology by the ampli"cation of vacuum #uctuations. However, the typical value of the characteristic frequency and the peak value of h X can be understood using only general arguments, following Ref. [54].   The production temperature is "xed by the criterium H &M , since in this model the H 1 Hubble parameter is stabilized by stringy corrections, and therefore at the string scale, see Section 9.3. Using H &¹ /M , this corresponds to an e!ective &temperature' at time of H H . production ¹ &(M M )"(M /M )M . Redshifting ¹ we get a characteristic freH 1 . 1 . . H quency today f &(M /M )¹ ,  1 .  and a peak energy density

(165)

do /d log f&f &(M /M ) do /d log f . (166)   1 . A Therefore o is related to o , although with a suppression factor (M /M ) which is expected  A 1 . to range between 10\ and 10\, and numerical factors.

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2. Another mechanism that we will discuss is the production of GWs through bubble collisions when in#ation terminates with a "rst order phase transition. In this case there is only one energy scale, the vacuum energy density M during in#ation (so that M is the false-vacuum energy density). The value of M "xes the reheating temperature, and therefore o , from A (pg /30)¹ KM, and "xes also the energy liberated in gravitational waves. Therefore, the H  energy density in photons and in gravitons are related. Writing, as in Section 8.1, j "eH\, H H where j , H are the typical wavelength produced and the Hubble parameter at time of H H production, the computation of Section 10.1.1 shows that, in a strongly "rst-order phase transition, o &eo , (167)  A so that again o and o are related.  A From these explicit examples we see that, independently of the production mechanism, an equation of the form Eq. (167) is quite general. The scale for the intensity of a relic GW background is indeed "xed in many cases by o , which therefore gives a "rst order of magnitude estimate of the A e!ect. However, there are also suppression factors, like the factor e in Eq. (236) or (M /M ) 1 . in Eq. (166), that, together with the exact numerical coe$cients, are crucial for the detection at present experiments. A value e&0.1 would allow detection at the level h X &10\ while   e&10\ would require h X &10\, which is beyond the possibilities of "rst generation   experiments. Furthermore, another suppression factor is present if we have a spectrum of relic GWs with a total energy density o &o , and a cuto! in the GHz, and we want to observe it in the kHz  A region. In this case, as we discussed, we must hope that the spectrum is practically #at between the kHz and the GHz. While we have seen that there are various examples of such a behaviour, the fact that the energy density is spread over such a wide interval of frequencies diminishes the maximum allowed value of h X ( f ) at a given frequency, since the nucleosynthesis bound is a limit on the   integral of h X ( f ) over all frequencies. The nucleosynthesis bound (118) then gives a maximum   value at 1 kHz h X (1 kHz)4(5.6;10\(N !3))/(log(1 GHz/1 kHz))K4;10\(N !3) . (168)   J J If the spectrum extends further toward lower frequencies, the maximum value of h X decreases   accordingly, with a factor log(1 GHz/f ) instead of log(1 GHz/1 kHz).

 9. Ampli5cation of vacuum 6uctuations 9.1. The computation of Bogoliubov coezcients The ampli"cation of vacuum #uctuations is a very general mechanisms, "rst discussed in a cosmological setting in Refs. [131,218] and then in many other papers, see e.g. [208,101,2,3,1, 7,164,132]. Specializing immediately to a FRW metric, it is convenient to introduce conformal time g, related to cosming time t by dg"dt/a(t), so that ds"a(g)(!dg#dx) .

(169)

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(In the following we set c"1.) In the presence of a classical GW in a FRW background, we write g "a(g)(g #h ) IJ IJ IJ

(170)

with g "(!,#,#,#) and, in the TT gauge, IJ k (g) e k  xe (XK ) , (171) ?@ >" k Note that x are the comoving coordinates; the physical coordinates are x "a(t)x. Correspond ingly, k is a comoving momentum, and the physical momentum is k "k/a(t).  The Einstein equations, linearized over the FRW background, give an equation for  k, h (g, x)"(8pG ?@ ,

k #2 (a/a) k #k k "0 ,

(172)

where the prime is the derivative with respect to g, and we omitted the index A. This equation is just the Klein}Gordon equation for a homogeneous scalar "eld in a FRW background, 䊐 ,(1/(!g)R ((!ggIJR ) "0 . I J

(173)

The factor (8pG in Eq. (171) is a convenient choice of normalization, such that the action of ,

 derived from the linearization of the Einstein}Hilbert action corresponds to the action of two (A"#,;) canonically normalized scalar "elds. Therefore, one can use the standard technique of Bogoliubov coe$cients for a scalar "eld [41]. One considers the situation in which, over a time scale *¹, there is a change of regime in the cosmological evolution, between two regimes that we denote as I and II (for instance from an in#ationary regime to a standard radiation dominated era). Let t be the time of transition between the two regimes, and consider a mode whose physical H frequency, at time t , is f . For values of f such that 2pf *¹;1, the change of regime can be H H H H considered abrupt, while if 2pf *¹<1 the change of regime is adiabatic. It is convenient to de"ne H a new variable tk (g)"(1/a) k (g) .

(174)

In terms of k , Eq. (172) becomes tk #(k!a/a)tk "0 .

(175)

Let us denote by fk (g) the solution of this equation with the scale factor of phase I, and Fk (g) for phase II. Then the mode expansion of h in phase I is ?@



dk 1 h "(8pG [a (k) fk (g)e k  x#aR (k) fkH(g)e\ k  x]e (XK ) ?@ ,   ?@ a(g)  (2p)(2k

(176)

(we have used the fact that our e are real, Eq. (6); a (k), aR (k) are the creation and annihilation ?@   operators in this phase, so that the vacuum in phase I is de"ned by a (k)"02 "a (k)"02 "0 > ' " '

(177)

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and we can construct the Fock space appropriate to this phase acting with aR(k). Similarly, in phase II, one has the expansion



dk 1 k x [A (k)Fk (g)e k  x#AR (k)FH K) (178) h "(8pG k (g)e\  ]e (X  ?@ ?@ , a(g)  (2p)(2k  and a new vacuum state "02 such that '' A (k)"02 "A (k)"02 "0 . (179) > '' " '' Since both the ( fk , fkH) and the (Fk , FHk ) are a complete set, we can express one in terms of the others; of course, since the fk alone are not a complete set, the expansion of Fk will involve both fk and fkH, and therefore there will be a mixing of positive and negative frequency modes. The relation between Fk and ( fk , fkH) is known as a Bogoliubov transformation, (180) Fk " (akk fk #bkk fkH) . Y Y Y Y k Y Inserting this relation into Eq. (178) and comparing with Eq. (176) one "nds the relation between creation and annihilation operators (we omit hereafter the index A"#,;), a(k)" [ak k A(k)#bHk k AR(k)] Y Y k Y

(181)

and A(k)" [aH kk a(k)!bH kk aR(k)] . Y Y k Y The Bogoliubov coe$cients satisfy

(182)

[ak k aH k k !bk k bH k k ]"dk1 k2 ,   

(183)

[ak k bk k !bk k ak k ]"0

(184)

k

k

If the cosmological background metric is time dependent but isotropic and spatially homogeneous, the gravitational "eld can give energy to the particles, but not momentum, so that bkk "b dkk . (185) akk "a dkk Y D Y Y D Y Eqs. (181) and (182) give the formal relation between the Fock spaces constructed in phases I and II. Now, the basic physical idea is the following: suppose that just before the transition the quantum state was "s2. For instance, we can express "s2 in terms of the occupation numbers n , relative to the D creation operators aR a . Then, for modes such that the transition is sudden, that is 2pf *¹;1, the D D H physical state does not have time to change during the transition, so that it is still given by "s2""+n ,2 just after the transition. However, we now must express the occupation numbers with D respect to N "AR A . Using the Bogoliubov transformation, one immediately "nds D D D (186) N "n #2"b "(n #) . D D D D 

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This shows, "rst of all, that any preexisting value of n is ampli"ed, with an ampli"cation factor D 1#2"b ". And furthermore, even the &half-quantum' due to vacuum #uctuations is ampli"ed: D due to the mixing between positive and negative frequencies, even the vacuum state of phase I is a multiparticle state in the Fock space appropriate to phase II. The above discussion refers to frequencies f such that 2pf *¹;1. In the opposite H H limit 2pf *¹<1, the mode whose physical frequency, at t"t , is f , sees the transition between H H H the two regimes as adiabatic, and the quantum state has the time to follow smoothly the evolution of the scale factor. In this case, therefore, there is no particle production, or more precisely the ampli"cation factor approaches one exponentially [41]. Therefore, for any practical purpose, there is a cuto! in the spectrum produced by the ampli"cation of vacuum #uctuations, at f &1/2p*¹ . (187) H To obtain the value of the cuto! frequency today, one then redshifts f  from time of production H t"t to the present time t . In a cosmological setting, the scale of time variation is given by the H  Hubble constant, so that *¹&H\; then the condition 2pf *¹<1 is met when the reduced H physical wavelength j /(2p) is smaller than the horizon; thus, modes that at time of transition H where outside the horizon are ampli"ed. Instead, no ampli"cation occurs for sub-horizon sized modes. It is also interesting to note, from Eq. (186), that even if we have an in#ationary phase, the occupation numbers N after in#ation are very sensitive to the occupation numbers n before D D in#ation. This is in contrast with what happens in most cases after an in#ationary phase: usually in#ation practically erases the information about a previous era. This does not happen for the ampli"cation of vacuum #uctuations, and the technical reason behind this is that the quantity which is ampli"ed is a number of particle per unit cell of the phase space. The volume of a cell of the phase space, dx dk/(2p), is una!ected by the expansion of the Universe, contrarily to a physical spatial volume dx&a(t). 9.2. Amplixcation of vacuum yuctuations in inyationary models We now compute the Bogoliubov coe$cients for the transition between an in#ationary phase and the radiation-dominated (RD) phase, followed by the matter-dominated (MD) phase. We start from the simplest case of exact De Sitter in#ation and then we compute the modi"cations for slow-roll in#ation. The ampli"cation of vacuum #ucutations at the in#ation}RD transition is the most studied example of this general phenomenon [131,218,208,105,3,1,7,11,210,132,133,221], but, as we will see, it is by no means the most promising from the point of view of the detection at interferometers or resonant masses. Still, we start with this example because it is the simpler setting to understand the computational technique. 9.2.1. De Sitter inyation We start with a FRW cosmological model with ds"dt!a(t) dx, and we concentrate for the moment on the particle production at the De Sitter}RD transition. The scale factors are a(t)&e&R for !R(t(t , and a(t)&t for t (t(t , where t is the time when MD sets in. In terms    

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of conformal time g, a(g)"!1/Hg

(188)

for !R(g(g , with g (0, and   a(g)"(1/Hg )(g!2g ) , (189)   for g (g(g . We have "xed the constants so that a(g) and a(g) are continuous across the   transition. Here we are considering an istantaneous transition at t"t . More realistically, we  should consider a transition that takes place in a time *t. As already discussed in the previous section, as long as we consider frequencies such that 2pf *t;1, the approximation of an H instantaneous transition gives negligible errors. When 2pf *t&1, the approximation breaks H down, and there is an exponential suppression of the particle production. The solution of Eq. (175) with these scale factors is easily found. In general the solution is a superposition of a term that oscillates as e\ IE and a term e IE. For g(g , we impose the  boundary condition that only the positive energy solution, i.e. the term &e\ IE, is present. This is not an assumption on the state of the system at g(g , but simply a trick to compute the  Bogoliubov coe$cient, which can be de"ned by matching a solution with only positive frequencies at g(g with the most general solution at g'g . Then, the solutions are  

(g)"(1!i/kg)e\ IE (190) I for !R(g(g , with g (0, and  

(g)"(a e\ IE#b e IE) (191) I   for g (g(g . Requiring that the ,  are continuous across the transition gives   I I a "1!i/kg !1/2kg , b "1/2kg . (192) I   I  Note that "a "!"b ""1. The number of particles produced at the De Sitter}RD transition, per I I cell of the phase space, is given by Eq. (186); in particular, if before the transition the system was in the vacuum state, n "0, after the transition I N ""b ""1/4kg . (193) I I  As discussed before, k is the comoving momentum and is related to the physical momentum k by k "k/a(t). We must now express N in terms of physical quantities observed today,   I taking into account the redshift during the RD and MD phases. Of course, during the RD}MD transition there will be a further creation of quanta; we will discuss it below, and for the moment we express the spectrum of particles produced at the De Sitter}RD transition in terms of physical quantities measured today. We reserve the notation f for the physical frequency observed today, so that 2pf"(k ) "k/a(t ) ,    where t is the present value of cosmic time. Therefore,  k"g ""2pfa(t )"g ""(2pf/H)a(t )/a(t )"(2pf/H)(t /t )(t /t ) ,         

(194)

(195)

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where in the second equality we have used a(t )"1/(H"g "); we now use [158]   (t /t ),1#z "2.32;10(X h )¹\ K2.40;10(X h ) , (196)          where X is the total density of the Universe in units of the critical density, and we have  taken ¹"2.728 K. For the time of matter}radiation equilibrium we have [158] t "4.3608;10(X h )\(¹ )K4.1;10 X\h\s.        From Eq. (195) we see that the parameter g can be traded for a more meaningful parameter  f de"ned by  k"g ""f/f , (197)   so that f "(H/2pz )(t /t ) . (198)     Note that here t is not a free parameter: during RD, H(t)"1/(2t), so that H(t )"1/(2t ), since    t is the time when RD begin; and since the scale factor and its derivative are continuous across the  transition, H(t ) is equal to the constant value H during the De Sitter phase, so that t "1/(2H).   Then, inserting the numerical values, f K10(H/10\M ) Hz . (199)  . Note that both h and X cancel in f . We have chosen 10\M as a useful reference value for H,    . for reasons that will be clear below. In terms of the physical frequency observed today, f, and of the parameter f ,  (200) N "( f /f ) , D   if before the transition n "0. Inserting this value into Eq. (26) we see that f cancels and we get D a #at spectrum, h X ( f )K10\(H/10\M ) . (201)   . This is the contribution of the De Sitter}RD transition. A further ampli"cation comes from the RD}MD transition. The mode that, at the time of the transition between RD and MD, had a physical frequency much bigger than the typical timescale of the transition, are not subject to further ampli"cation. The typical time scale of the transition is given by the inverse of the Hubble constant at time of equilibrium, which is H(t )"4((2!1)/(3t ) (see [158], p. 60). A mode that,   at t"t , had a physical momentum k "H(t ), today has a physical frequency f given by     1 4((2!1) 1 f " K10\X h Hz . (202)  2p   3t z   The modes with f'f are not further ampli"ed at the RD}MD transition. Another useful way to  express this result is the statement that modes that, at the time of transition, had a physical wavelength much smaller than the horizon are not ampli"ed. Therefore, the spectrum at f'f is  given by Eq. (201). For 3;10\ Hz(f(f , instead, we must compute the Bogoliubov coe$cients for the  RD}MD transition, and the total Bogoliubov transformation is obtained with the composition of the De Sitter}RD and RD}MD transformations. As always, the condition f'3;10\ Hz comes from requiring that the mode is inside the horizon today.

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A straightforward computation, similar to the one performed above, shows that in this frequency range the spectrum now goes like 1/f  rather than being #at. So the "nal result is h X ( f )K10\( f /f )(H/10\M ) , (203)    . if 3;10\ Hz(f(f K10\ Hz, and  h X ( f )K10\(H/10\M ) (204)   . for f (f(f . Finally, from the de"nition of f , Eq. (197), we see that a mode that today has f'f ,     at the time of transition between De Sitter and RD had a wavelength smaller than the horizon; therefore these modes are never ampli"ed, so that f is the upper cuto! of the spectrum. The  spectrum is shown in Fig. 12. Note that the behaviour 1/f  in the region 3;10\ Hz(f(10\ Hz is the same as the COBE bound, and is in fact due to the same physical reason, i.e. to the further ampli"cation of modes that are inside the horizon at time of matter}radiation equilibrium. The comparison of Eq. (203) with the COBE bound, Eq. (139), then gives an upper limit on the Hubble constant during De Sitter in#ation; from a precise statistical analysis, the limit at 95% c.l. is [164] H(6;10\M . .

(205)

9.2.2. Slow-roll inyation In the above section we have considered a simpli"ed model with an exact De Sitter in#ation. If in#ation is driven by a scalar "eld slowly rolling in a potential <( ), the Hubble parameter during

Fig. 12. The spectrum of ampli"cation of vacuum #uctuations produced by a phase of De Sitter in#ation (solid line), with a value of H that saturates the COBE bound. The nucleosynthesis bound (dotted line) and the pulsar bound (triangle shaped) of Fig. 11 are also shown for comparison.

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in#ation is not exactly constant, and this results in a small tilt in the spectrum. Furthermore, scalar perturbations will also contribute to the COBE anisotropy, and the ratio of tensor to scalar perturbation is "xed by the choice of <( ). Then, in the region f (f(f , one "nds that h X ( f )     is not #at, as in Eq. (204), but has a frequency dependence (206) h X ( f )&f L2 ,   where n is called the spectral index for tensor perturbations. If < is the value of the in#ationary 2 H potential when the present horizon scale (k"H ) crossed the horizon during in#ation, and < is  H the "rst derivative of the potential at that point, then, to lowest order in the deviation from scale invariance, and for single "eld models with a smooth potential [168,231,175] n "!(M /8p)(< /< ) . (207) 2 . H H Therefore n (0, and the spectrum is decreasing rather than #at. Since the slow-roll condition is 2 just "M /8p(< /< )";1, then "n ";1. It is convenient to de"ne the scalar (S) and tensor (¹) . H H 2 contributions to the quadrupole anisotropy, 51"a1 "2 2.2< /M K " H . , S, 4p (M < /< ) . H H 51"a2 "2 K "0.61< /M , ¹, H . 4p

(208) (209)

where the second equalities holds for slow-roll in#ation. From Eqs. (207)}(209) it follows that n "!¹/S . (210) 2  To obtain the precise frequency dependence, we must also take into account the fact that the behaviours h X &1/f  and h X &const. in Eqs. (203) and (204) hold only in the limits f;f      and f





















The relation between n , ¹ and S, Eq. (210), goes in the wrong direction for the detection at 2 interferometers: the higher the contribution of tensor perturbations at COBE, and therefore the maximum allowed value of the GW spectrum at, say, f"10\ Hz, the steeper is the decrease in the spectrum with the frequency. Furthermore, when extrapolating through a very large range of frequencies, one must also take into account that n itself is not constant, but rather [162] 2 dn /d log k"!n (M /4p)(< /< ) . (213) 2 2 . H H

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Combining these e!ects, one "nds that, at a "xed frequency f"10\ Hz, relevant for LISA, the spectrum is maximized for n K!0.025 [231], or ¹/SK0.175. Values of this order are realized 2 in various models of in#ation, although there are as well models that predict ¹/S&10\. As discussed in Section 7.2, GWs can contribute to the anisotropy only on angular scales larger than approximately 13. Combining the results of COBE with other observations on smaller angular scales, one can place upper limits on ¹/S, and the analysis of Ref. [244] gives ¹/S(0.5, at 95% c.l. Even with the value of n that maximizes the signal at f"10\ Hz, the expected signal at LISA 2 is of order 10\, therefore various order of magnitudes below the expected sensitivity. 9.3. Pre-big-bang cosmology 9.3.1. The model In recent years, a cosmological model derived from the low-energy e!ective action of string theory has been proposed [235,125]. The simplest starting point is the e!ective action in the metric-dilaton sector; at lowest order in the derivatives and in e( it is given by



S & d"x(!g[e\((R#R RI )] , I 

(214)

where is the dilaton "eld. This action receives two kinds of perturbative corrections: corrections parametrized by e(, which are just higher loops in the "eld theory sense. And corrections proportional to a, where j "(2a is the string length, which are genuinely string e!ects, and take  into account the "nite size of the string. Since a has dimension of (length), it is associated with higher derivative operators as for instance the square of the Riemann tensor, R . The remarkIJMN able fact observed in Refs. [235,125] is that there exists a regime in the early Universe evolution of this model, where the perturbative approach is well justi"ed, and in a "rst approximation one can use Eq. (214), neglecting the corrections. This regime occurs if we take as initial condition a Universe at weak coupling and low curvatures. Then, a generic inhomogeneous string vacuum shows a gravitational instability [59]. Specializing for simplicity to a homogeneous model, the solution of the equations of motion derived from action (214) reads, for generic anisotropic FRW scale factors a (t), i"1,2, D!1, G a (t)"(!t)AG , G

(t)" #c log(!t) G  

(215)

with "\ "\ c"1, c "1#c . (216) G G  G G This is just a generalization in the presence of the dilaton of the Kasner solution of general relativity. Considering for simplicity an isotropic model, one has, in particular, a solution with c "!1/(D!1, c "!1!(D!1. This corresponds to a superin#ationary evolution: the  G Hubble parameter H"a /a grows until it reaches the string scale &1/j , and "nally formally  diverges at a singularity. When H&1/j , of course, the description based on the lowest-order  action is not anymore adequate. Much work has been devoted recently to understanding how string theory cures the apparent singularity, and if it is possible to match this &pre-big-bang' phase

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to standard post-big-bang cosmology [57,58,150,96,83,123,176,109,56,110]. Apart from the phenomenological aspects concerning GWs that we will discuss, it is clear that the problem has great conceptual interest, since it is a theoretically well motivated attempt to cure the Big-Bang singularity. One can then investigate whether pertubative corrections succeed in turning the regime of pre-Big-Bang accelerated expansion into the standard decelerated expansion. Indeed, the inclusion of a corrections can turn the unbounded growth of the curvature into a De Sitter phase with linearly growing dilaton [123]. Since in this new regime the dilaton keeps growing linearly, "nally also e( gets large, even if the evolution started at very weak coupling; at this stage loop corrections are important, and they can trigger the exit from the De Sitter phase [56,110]. An example of this behaviour, using loop corrections derived from orbifold compacti"cations of string theory, has been found in Ref. [110], and is shown in Fig. 13. These results are encouraging, but various technical limitations in the analysis have also indicated that most probably the true resolution of the singularity problem lies outside the domain of validity of the perturbative approach [110]. In this case one must take into account that at strong coupling and large curvature new light states appear and then the approach based on the e!ective supergravity action plus string corrections breaks down. The light modes are now di!erent and one must turn to a new e!ective action, written in terms of the new relevant degrees of freedom. In particular, at strong coupling D-branes [196] are expected to play an important role, since their mass scales like the inverse of the string coupling, &1/g and they are copiously produced by gravitational "elds [179]. For the purpose of computing the ampli"cation of vacuum #uctuations in this model, the approach that can be taken is to consider a model with three phases: "rst an isotropic superin#ationary evolution described by Eq. (215), with the value of the constants c , c determined by G  Eq. (216). Then this phase is matched to a De Sitter phase with linearly growing dilaton, which may be considered as representative of a typical solution in the large curvature regime; the constant

Fig. 13. The evolution of the Hubble parameter H and of Q in string cosmology with a and loop corrections (from Ref. [110]).

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values of H, Q in this phase are taken as free parameters, of the order of the string scale. Finally, we match this phase to the standard RD era. Of course, this model can only be taken as indicative of possible behaviours in string cosmology, since what happens in the large curvature phase is certainly much more complicated. However, we will see that some important characteristics of the GW spectrum depend only on the low curvature regime, where we can use the lowest order e!ective action and therefore is under good theoretical control. Variants of this model can of course be constructed. In particular, Ref. [184] considers the e!ect of a second burst of in#ation, or of the possibility that the Universe becomes temporarily dominated by long-lived massive particles, as for instance the moduli "elds of string theory, which then decay, restoring a standard RD phase [130]. 9.3.2. Production of GWs We can now discuss the production of GWs through ampli"cation of vacuum #uctuations in this model [126,121,53] (see also Refs. [60,54,52,10,177,180,118}120,81,122,124,127]). We specialize for illustration to an isotropic model compacti"ed to four dimensions. Writing the scale factors and the dilaton in terms of conformal time, from Eq. (215), we "nd that for !R(g(g (with g (0),   we have a dilaton-dominated regime with a(g)"!(1/H g )(g!(1!a)g /ag )\? (217)    

(g)" !c log(g!(1!a)g /g ) (218)    and the Kasner conditions Eq. (216) give a"1/(1#(3), c"(3. At a value g"g the curvature becomes of the order of the string scale, and we assume a  De Sitter expansion with linearly growing dilaton; in terms of conformal time, this means a(g)"!1/H g,

(g)" !2b log(g/g ) . (219)    The parameter H is of the order of the string scale, while b is a free parameter of the model. Note  that 2b" Q /H , where Q is the derivative with respect to cosmic time.  The stringy phase lasts for g (g(g (again g (0), and at g we match this phase to     a standard RD era. This gives, for g (g(g , (with g '0)  P P a(g)"(1/H g )(g!2g ),

" . (220)     After that, the standard matter dominated era takes place. We have chosen the additive and multiplicative constants in a(g) in such a way that a(g) and da/dg (and therefore also da/dt) are continuous across the transitions. The equation for the Fourier modes of metric tensor perturbations for the two physical polarizations in the transverse traceless gauge is [121] dt /dg#[k!<(g)]t "0 , I I <(g)"(1/a)e((d/dg)ae\( .

(221) (222)

Note that, for constant , <(g) reduces to Eq. (175). Inserting expressions (217)}(220), the potential is <(g)"(4l!1)(g!(1!a)g )\, !R(g(g ,    <(g)"(4k!1)g\, g (g(g ,    <(g)"0, g (g(g ,  P

(223)

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where 2k""2b!3", 2l""2a!c#1". From a"1/(1#(3) and c"(3, we get l"0. The exact solutions of Eq. (221) in the three regions are t (g)"("g!(1!a)g "CH(k"g!(1!a)g "), !R(g(g , I     t (g)"("g"[A H(k"g")#A H(k"g")], g (g(g , > I \ I   I



2 [B e IE!B e\ IE], g (g(g , t (g)"i \  P I pk >

(224)

where H are Hankel's functions. As we have seen in Section 9.2.1, it is convenient to trade g , J  g for more physically meaningful quantities. As before, we then write  k"g ""2pfa(t )"g ""(2pf/H )a(t )/a(t )"(2pf/H )(t /t )(t /t ) (225)            and therefore the parameter g can be traded for a parameter f de"ned by   f k"g "" , f K4.3;10 Hz;(H /0.15M )(t /j ) . (226)       f  where we have chosen a reference value H of the order of the string scale, and t Kj . Similarly,    we can introduce a parameter f instead of g , from k"g ""f/f . This parameter depends on the     duration of the string phase, and therefore, contrarily to f , is totally unknown, even as an order of  magnitude. However, since "g "("g ", we have f (f .     To summarize, the model has a parameter f with dimensions of a frequency, which can have  any value in the range 0(f (f , with f &1}100 GHz, and a dimensionless parameter k50    (or equivalently b with 2k""2b!3"). Performing the matching between the three phases one "nally gets the spectrum. The exact form is [60] X ( f )"a(k) [(2pf )/H M ] ( f /f )I>( f/f )\I"H(af/f )J ( f/f )          I  #H(af/f )J ( f/f )![(1!a)/2a] f /f H(af/f )J ( f/f )" , (227)   I     I  where a(k) is a number, which depends on k. Expanding this expression for small values of f/f one  gets a result of the form



  


 

(2pf ) f I> f   af   X ( f )& a # a log #a , (228)     H M f f 2f      where a , a , a depend on k.    In the opposite limit f( f/f )\I"[(2pf )/H M ] ( f/f )\I . (229)            Both limits reproduce the frequency dependence "rst found in Refs. [121,53]. It is important to stress that in the high frequency limit the unknown parameter f cancels.  This spectrum has quite interesting properties, depending on the two free parameters f and k.  First of all, it goes like f  at low frequencies. This result is a consequence of the behaviour of the cosmological model in the superin#ationary pre-big-bang phase, where everything is under good

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theoretical control, so this is a rather general prediction of the model. This means that h X ( f ) can   evade the COBE bound very easily, and yet be large in the range accessible to interferometers. This is very di!erent from slow-roll in#ation, where instead the spectrum is decreasing. If however the spectrum would behave as f  for all frequencies, up to the cuto! f , it would be  totally negligible at, say, 1 kHz, since the value of h X ( f ) at the cuto! frequency f is bounded by    the nucleosynthesis limit discussed in Section 7.1. However, above the unknown frequency f , the  spectrum predicted by this model changes and behaves as f \I; this re#ects the fact that the Universe changed regime and entered a phase of De Sitter in#ation with linearly growing dilaton. The parameter k measures the growth of the dilaton in units of the Hubble constant H during the  string phase, and for an almost constant dilaton, 3!2kK0. This means that, in some range for the parameters f and k, it is possible to have the optimal  situation from the point of view of interferometers: a spectrum that grows at very low frequencies, so that it can evade the COBE bound, and then #attens, at the maximum level allowed by the nucleosynthesis limit. The resulting spectrum is shown in Fig. 14, compared to the COBE, pulsar and nucleosynthesis bounds discussed in Section 11. Fig. 15 enlarges the frequency range relevant for ground-based interferometers, displaying some features of the spectrum which might be a signature of this background. Finally, it is clear that in this model GWs cannot be responsible for the COBE anisotropies, since the &f  behaviour makes h X negligible at very low frequencies. In the pre-big-bang model,   a possible seed for the CMBR anisotropies can be provided by pseudo-scalar "elds [80,82,95,61,55,183].

Fig. 14. The spectrum of GWs due to ampli"cation of vacuum #uctuations in string cosmology, compared with the COBE bound (dot}dashed), pulsar bound (asterisk) and nucleosynthesis (dotted), for a choice of the parameters f "10 Hz, k"1.5 (from Ref. [60]). 

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Fig. 15. The GW spectrum, enlarging the region between the Hz and the kHz, for f "100 Hz, k"1.4. The dashed and  dot}dashed lines are the low- and high-frequency limits (from Ref. [60]).

10. Other production mechanisms 10.1. Phase transitions In the history of the Universe are expected a number of phase transition. In particular, the QCD phase transition takes place at ¹ &150 MeV. Above this temperature quarks and gluons are H decon"ned and form a quark}gluon plasma, while below they are in the con"ned phase. Recent lattice calculations with three quark #avors and with the physical mass for the strange quark, using Wilson fermions, suggest that the phase transition is "rst order [146], but the issue is not yet settled. Around ¹ &100 GeV we expect the electroweak phase transition, when S;(2);;(1) H breaks to ;(1) . Further phase transitions could occur even earlier, at the grand uni"ed scale.  These phase transitions are dramatic events in the history of the Universe, and are good places to look for GW production. 10.1.1. Bubble collisions In "rst-order phase transitions the Universe "nds itself in a metastable state. True vacuum bubbles are then nucleated via quantum tunneling. In strongly "rst-order phase transition the subsequent bubble dynamics is relatively simple: once the bubble are nucleated, if they are smaller than a critical size their volume energy cannot overcome the shrinking e!ect of the surface tension, and they disappear. However, as the temperature drops below the critical temperature ¹ , it  becomes possible to nucleate bubble that are larger than this critical size. These &critical bubbles' start expanding, until their wall move at a speed close to the speed of light. The energy gained in the transition from the metastable state to the ground state is transferred to the kinetic energy of the

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bubble wall. As the bubble expands, more and more regions of space convert to the ground state, and the wall becomes more and more energetic. At the same time, it also becomes thinner, and therefore the energy density stored in the wall increases very fast. As long as we have a single spherical bubble, this large energy of course cannot be converted in GWs. But when two bubbles collide we have the condition for the liberation of a large amount of energy into GWs. In particular, there are two possible &combustion' modes for the two bubbles: detonation [219], that basically takes place when the boundaries propagate faster than the speed of sound, and de#agration, when instead they move slower. In the "rst case there is a large production of GWs [141,242,160]. The characteristic frequency, as seen today, of these GWs is given by Eq. (156), with ¹ equal to H the temperature of the phase transition. First of all, we must estimate the parameter e that enters Eq. (156). Plausible values of e have been discussed in detail by Hogan [141] and by Witten (Ref. [242, app. B]). In the case of the QCD and electroweak transitions e&1 is excluded because otherwise in the bubble collisions there would be overproduction of primordial black holes. Production of primordial black holes is severely constrained because the energy density of primordial black holes scales like 1/a in the RD phase, and therefore they would come to dominate the energy density of the Universe. In particular, the Universe would not be radiation dominated during nucleosynthesis. Actually, primordial black holes can evaporate through emission of Hawking radiation. However, in order to evaporate before nucleosynthesis, the scale at which the phase transition takes place must be higher than 10 GeV [234]. Therefore, e could be of order one in phase transitions which take place at these temperatures, but not in the QCD or electroweak phase transition. To estimate e it is necessary to make assumptions about how the phase transition is nucleated. If it is nucleated by thermal #uctuations, Ref. [141] suggests an upper bound e:10\. For nucleation of bubbles via quantum tunneling a detailed analysis has been done in Ref. [232]. The relevant parameter is the nucleation rate; the phase transition can be modelled assuming an exponential bubble nucleation rate per unit volume, C"C e@R . (230)  The parameter b sets the scale for the typical time variation, and therefore the frequency at time of production, f , is determined by b. In fact, Ref. [160] found that the spectrum peaks at 2pf K2b. H H Then 1/e,f /H Kb/pH . (231) H H H Of course, b is a model-dependent quantity, but for a wide class of models one typically "nds e&10\}10\ [140,141,232,161,160]. If however the transition is nucleated by impurities (which is most often the case, except in very pure and homogeneous samples) the issue is much more complicated. For instance, the impurities could be given by turbolent motion of the cosmic #uid generated prior to the QCD epoch; they would depend on the detailed spectrum of the #uid motion, or in general on the characteristic distance between impurities, and in this case e is basically impossible to estimate, even as an order of magnitude [242]. Coming back to the simplest case of nucleation by quantum or thermal #uctuations, for the electroweak phase transition, Ref. [160] "nds b/H K1.3;10\ H

(232)

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and, therefore, f K4.1;10\ Hz . (233)  This phase transition is therefore very relevant for LISA. With the same value for e, the QCD phase transition would instead peak around f &4;10\ Hz. A smaller value of e is required to bring it  into the frequency window of LISA. A "rst estimate of the value of h X , that as we will see really holds only for strongly "rst order   phase transitions, can be given as follows [234]. The typical wavelength j produced in the H collision of two bubbles will be of the order of the radius of the bubbles when they collide, which in turn is a fraction of the horizon scale H\. The energy liberated in GWs in the collision of two H bubbles is of order E &GM /j , (234) %5 H where M &o j is the energy of a typical bubble and o the di!erence in energy density from  H  the metastable state to the ground state. The fraction of the false vacuum energy that goes into GWs instead of going into o is therefore E /M . Since A %5 E /M &GM /j &Go j (235) %5 H  H and H &Go , we get, at time of production H  o &j H o "eo . (236)  H H A A Therefore, o is related to o and the basic reason is that there is essentially only one dimensionful  A parameter that enters in the estimate. If we neglect the variation in g(¹) from time of production to the present value, o /o is almost constant, and since today h o /o K2.5;10\, this quick order  A  A  of magnitude estimate gives h X ( f )&10\e. However, further suppression factors appears in    the accurate computation of Ref. [160], which gives h X ( f )K1.1;10\i(H /b)[a/(1#a)][v/(0.24#v)](100/g ) . (237)    H H Here a is the ratio of vacuum energy to thermal energy in the symmetric phase. It characterizes the strength of the phase transition; for aP0 we have a very weak "rst order and for aPR a very strong "rst-order phase transition. For the electroweak phase transition, Ref. [160] gives a"1.4;10\ .

(238)

i is an e$ciency factor quantifying the fraction of vacuum energy which goes into bubble wall kinetic energy, and in the electroweak case is estimated to be iK7.8;10\. Finally, v is the velocity of the bubble wall, and is approximately equal to the speed of sound during RD, vK1/(3. Altogether, these suppression factors give a disappointing h X &10\ . (239)   These suppression factors are a consequence of the fact that the electroweak phase transition is very weakly "rst order, if "rst order at all. It is clear that strongly "rst-order phase transitions are needed for a detectable signal. Quite interestingly, the requirement of strongly "rst-order phase transition is the same that it is necessary to make possible electroweak baryogenesis (see e.g. Ref. [205] for a recent review). In

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particular, the ways in which baryons can be produced have been separated into &local baryogenesis', when both baryon number violating and CP violating processes occur near the bubble wall, and &nonlocal baryogenesis', when only CP violating processes take place near the wall. The condition for local baryogenesis to dominate is that the speed of the wall be greater than the speed of sound, which is the same condition for detonation to dominate the bubble}bubble collisions. Partly motivated by the importance for baryogenesis, there have been many recent investigations on the strength of the phase transition in extensions of the Standard Model, and in particular in the Minimal Supersymmetric Standard Model (MSSM) [129,190,98,49]. Both analytical [65,92,97,47,172,102,91] and lattice [165,166,72] computations have shown that in the MSSM the phase transition can be of strongly "rst order if the top squark is lighter then the top quark. The observation of a relic stochastic background at LISA could therefore give us a hint for physics beyond the standard model. 10.1.2. Turbulence For weakly "rst-order phase transitions the situation is actually more complex. Bubble nucleation occurs both via quantum tunneling and with thermal #uctuations, and the subsequent evolution of the bubble depends on the interaction of the wall with the surrounding plasma; part of the energy gained in the transition from the metastable state to the ground state is used to heat up the plasma, and another fraction is converted into bulk motion of the #uid. If the Reynolds number of the Universe at the phase transition is large enough, then this results in the onset of turbulence in the plasma, and this is another powerful source of GWs. The estimates for the characteristic frequency and for h X turn out to be [160]   v (240) f K2.6;10\ Hz  (b/H )(¹ /1 GeV)(g /100) , H H H  v h X K10\(H /b)vv (100/g ) , (241)   H  H where v is the #uid velocity on the largest length scales on which the turbulence is being driven,  and v the velocity of the bubble wall. These estimates indicate that fully developed turbulence can be comparable, or even more potent than bubble collisions in generating GWs. 10.1.3. Scalar xeld relaxation Consider a global phase transition in the early Universe, associated with some scalar "eld that, below a critical temperature, gets a vacuum expectation value 1 2. For causality reasons, in an expanding Universe the scalar "eld cannot have a correlation length larger than the horizon size. So, even if the con"guration which minimizes the energy is a constant "eld, the "eld will be constant only over a horizon distance. During the RD phase the horizon expands, and the "eld will relax to a spatially uniform con"guration within the new horizon distance. This relaxation process will in general produce gravitational waves, since there is no reason to expect 1¹ 2 for the IJ classical "eld just entering the horizon to be spherically symmetric [163]. A simple estimate of the spectrum produced makes use of the fact that the characteristic scale of spatial or temporal variation of the "eld is given by the inverse of the Hubble constant, H\, at the value of time under consideration. Then, the energy density of the "eld is o&(R )&1 2H. The total energy in

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a Hubble volume is oH\&1 2H\ and the corresponding quadrupole moment is Q&(oH\)H\&1 2H\, times a constant smaller than one, which measures the non-sphericity of the con"guration. The energy liberated in GWs in a horizon time is then given by [163] *E&H\;Luminosity&H\G(dQ/dt)&G1 2H\ ,

(242)

and the energy density of GW produced is (do /d log f ) &*E/H\&G1 2H . (243)  H This is the energy density at a frequency f&H, at a value of time t"t ( f ) which is the time when H the mode with frequency f enters the horizon. The asterisk in (do/d log f ) reminds that this H quantity is evaluated at horizon crossing, therefore at di!erent values of time for di!erent frequencies. However, in the RD phase, the energy density scales like o&1/a(t)&t\ and H&t\ so that o scales as H. Therefore, Eq. (243) means that, if we evaluate do/d log f at the same value of time, for all frequencies which at this value of time are inside the horizon, the spectrum is #at, h X ( f )& const. This is a rather nice example of how a #at spectrum can follow simply from   the requirement of causality and simple dimensional estimates. The simplest way to redshift Eq. (243) to the present time is to use H"(8p/3)Go , so that Eq. (243) can be written as A (do /d log f ) &(8p/3)G1 2(o ) , (244)  H AH where both sides are evaluated at horizon crossing. Then o redshifts as 1/a, o adiabatically, Eq.  A (152), and o "g(¹)o , so that (X ) "(g /g )(X ) , and therefore Eq. (243) gives, for the  A    H  H energy density today, h X &5;10\(1 2/M )(100/g ) . (245)   . H A quite similar mechanism has been studied in Ref. [142], considering a theory with a multicomponent scalar "eld and an e!ective potential <( ) with degenerate minima at a set of points with " "" <1. The direction in "eld space with R
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10.2. Cosmic strings Cosmic strings are topological defect that might have formed during phase transitions in the early Universe (see e.g. [236,237,139] and references therein). Interest in cosmic strings has also been fueled by the fact that they were considered a candidate for seeding structure formations. The most recent comparison with CMB anisotropies show that in models with vanishing cosmological constant they do not reproduce the dependence of the anisotropy from the multipole moment [200,6]. However, with a non-vanishing cosmological constant they can still be a viable option [26,32]. The spectrum of GWs produced by cosmic strings is discussed in detail in Refs. [236,237,5,35,63, 8,64,31]. Gravitational waves are produced by the relativistic oscillations of a cosmic string, in a cosmic string network. The dynamics of this string network has been studied in detail, and it has been found that it obeys a scaling property, so that the only relevant scale is given by the Hubble length. Small loops oscillate producing GWs and disappear, while more small loops are chooped of very long strings, replacing the loops that disappear, and the network has a self-similar con"guration, with loops of all length scales. Since the wavelength of the GWs emitted is "xed by the length of the loop, the spectrum of the cosmic string network extends across a very large frequency band. The spectrum has two main features: an almost #at region that extends from f&10\ Hz up to f&10 Hz, and a peak in the region f&10\ Hz. At LIGO/VIRGO frequencies we are in the #at part of the spectrum, and the typical estimate of the intensity is of order h X &10\}10\ . (246)   (Similar results are obtained from hybrid topological defects [181]). The scale for these numbers is given "rst of all by combination (Gk), where k is the mass per unit length of the string. It comes from the fact that a loop radiates with a power P&cGk (c is a dimensionless constant) while, evaluating h X , another factor of G comes from 1/o . For strings created at the GUT scale    Gk&10\. Then, there are also large numbers related to the number of strings per horizon volume, which is thought to be of order 50, to the fact that also c&50, and to the size of the loops at formation time [8], that "nally give a value h X &10\}10\.   For the cosmic string spectrum the most relevant bound comes from the pulsars, discussed in Section 11. Pulsar timing implies a bound on the mass per unit length of the string [31] Gk/c((5.4$1.1);10\ .

(247)

For comparison, the value of k obtained "tting CMBR anisotropies in an open Universe is (Gk)/c&1.7;10\ for X "0.2 [25]. The improvement in pulsar timing expected in the next few  years, see Section 11, will therefore be crucial for testing the cosmic string scenario. 10.3. Production during reheating If we have an in#ationary phase below the Planck scale, there is no gravitational analog of the 2.7 K radiation. At the Planck scale photons and gravitons can be produced simply by thermal collisions, with probably similar production rates. However, the particles produced in this way at the Planck (or string) scale are exponentially diluted by the subsequent in#ationary phase, and the

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photons that we see today in the cosmic microwave background radiation (CMBR) have been produced during the reheating era which terminated the in#ationary phase. Since the reheating temperature ¹ (M ;M , thermal collisions are by now unable to   . produce a substantial amount of gravitons. However, in this case relic GWs can be produced through some non-equilibrium phenomenon connected with the reheating process. For a process that takes place at the reheating temperature ¹ the estimate of the characteristic frequency  is given by Eq. (156) with ¹ "¹ . In principle, the reheating temperature can be between H  a minimum value at the TeV scale (since this is the last chance for baryogenesis, via the anomaly in the electroweak theory) and a maximum value ¹ &M :3;10 GeV. In supersymmetric   theories, the gravitino problem give a further constraint ¹ :10 GeV [158,154]. More precisely,  Ref. [154] gives a bound on ¹ &10}10 GeV for a gravitino mass 100 GeV}1 TeV, and a bound  10}10 GeV independent of the gravitino mass. So, phenomena occurring at reheating can manifest themselves with cuto! frequencies between 10\ and 10 Hz, if we set e"1 in Eq. (156). In typical non-supersymmetric models the reheating temperature is large, say 10 GeV, corresponding to f &(1/e)10 Hz, and in this case inter ferometers or resonant masses could only look for their low-frequency tails. One should however keep in mind the possibility that bubble nucleation occurs before the end of in#ation, and then the cuto! frequency would be redshifted by the subsequent in#ationary evolution toward lower values, possibly within the VIRGO/LIGO frequency range [28]. If instead the reheating temperature is ¹ &10}10 GeV, non-equilibrium phenomena taking  place during reheating would give a signal just in the LIGO/VIRGO frequency band. A reheating temperature ¹&TeV would instead be relevant for LISA. A mechanism for GW generation during reheating is bubbles collision, in the case when in#ation terminates with a "rst order phase transition [234,160]. We have already discussed it in Section 10.1.1. Another possibility is that reheating occurs through the decay of the in#aton "eld. In this case, in a very general class of models, there is "rst an explosive stage called preheating, where the in#aton "eld decays through a non-perturbative process due to parametric resonance [157], and at this stage there are mechanisms that can produce GWs, see [156,30]. We believe that in these cases a reliable estimate of the characteristic frequency is really very di$cult to obtain, since the relevant parameter, which is the value of the Hubble constant at time of production, depends on the complicated dynamics of preheating and on the speci"c in#ationary model considered. Ref. [156] examines models where the characteristic frequency turns out to be in the region between a few tens and a few hundreds kHz.

11. Stochastic backgrounds of astrophysical origin The emission of GWs from a large number of unresolved astrophysical sources can create a stochastic background of GWs. This background would give very interesting informations on the state of the Universe at redshifts z&2}5. It will provide a probe of star formation rates, supernova rates, branching ratios between black hole and neutron star formations, mass distribution of black holes births, angular momentum distributions and black hole growth mechanisms [45]. At the same time, however, from the point of view of cosmological backgrounds produced in the

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primordial Universe, the astrophysical background is a &noise', which can mask the relic cosmological signal. In this section we investigate a number of astrophysical mechanisms that can produce a stochastic background of GWs. As for a general orientation into the typical intensity and frequency, we should note that GWs of astrophysical origin of course are not subject to the nucleosynthesis bound, since they were created much later, so that one of our main &benchmarks' disappears in this case. The estimates of the typical values of h X is strictly dependent on the mechanisms, and we   will see examples below. Concerning the frequency, a "rst observation is that there is a maximum frequency at which astrophysical sources can radiate. This comes from the fact that a source of mass M, even if very compact, will be at least as large as its gravitational radius 2GM, the bound being saturated by black holes. Even if its surface were rotating at the speed of light, its rotation period would be at least 4pGM, and the source cannot emit waves with a period much shorter than that. Therefore we have a maximum frequency [227], f:1/4pGM&10(M /M) Hz . (248) > To emit near this maximum frequency an object must presumably have a mass of the order of the Chandrasekhar limit &1.2M , which gives a maximum frequency of order 10 kHz [227], and this > limit can be saturated only by very compact objects (see Ref. [105] for a recent review of GWs emitted in the gravitational collapse to black holes, with typical frequencies f:5 kHz). The same numbers, apart from factors of order one, can be obtained using the fact that for a self-gravitating Newtonian system with density o, radius R and mass M"opR, there is a natural dynamical  frequency [215] f "(pGo)/2p"(3GM/16pR) . (249)  With R52GM we recover the same order of magnitude estimate apart from a factor (3/8)K0.6. This is an interesting result, because it shows that the natural frequency domains of cosmological and astrophysical sources can be very di!erent. In particular, a GW signal detected above, say, 10 kHz, would be unambiguosly of cosmological origin. However, as we have seen, it is not easy to build sensitive detectors operating at such high frequencies. In discussing a stochastic background of astrophysical origin, apart from the characteristic frequency and characteristic intensity, there is a third important parameter, which, for burst sources, is the duty cycle D. This is given by the ratio of the typical duration of the signal (e.g. 1 ms for supernovae) to the average distance between successive bursts. If D;1, the signal is not stochastic. For intermediate values, say DK0.1, we have a so-called &popcorn' noise, or shot noise. As DP1, we get a continuous, stochastic background. The value of D is important because, if D;1, we can distinguish a signal from the very early Universe form an astrophysical signal, even when the latter corresponds to higher peak amplitude. 11.1. Supernovae collapse to black hole Su$ciently massive stars, at the "nal stage of their evolution, collapse to form a black hole or a neutron star. In this explosive supernova event GWs are liberated. Depending on the rate of supernova events, this can give origin to a stochastic background. The typical time scale of

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supernovae is millisecond, and therefore the typical frequency of the GWs produced is around the kHz. These processes are therefore relevant for ground-based experiments. Earlier estimates [44,45] suggested a duty cycle D&0.3 and a value h X possibly even of   order 10\ at the kHz, but acknowledged the need for better collapse models and duty cycle informations. Recently, a careful study has been done in Ref. [103]. The crucial point is the determination of the star formation rate. However, in the last few years the understanding of the origin and evolution of galaxies has greatly improved, thanks to the spectacular data from the Hubble space telescope, Keck, and other large telescopes, that have allowed to investigate the Universe up to redshifts z&4}5. This makes possible a determination of the star formation rate density evolution, based on observations. Denote by o (z) the comoving star formation rate density, i.e. the mass of H gas that goes into stars, per unit time and comoving volume element, at redshift z. Then the rate of core-collapse supernovae event R (z), i.e. the number of events per unit time in a comoving 1, volume, at redshift z, is [103]



X



d< o (z) + H U(M) dM , (250) dz dz z#1  +  where U(M) is the Salpeter initial mass function and d reference value 25M has been used in [103], while M "125M . With these values, adopting > S > a value h "0.5 for the Hubble constant and assuming a #at cosmology, X "1, with vanishing   cosmological constant, the total number of supernovae explosions per unit time leading to black hole formation is [103] R (z)" 1,

R

"4.74 events/s . & The duty cycle due to all sources up to redshift z is de"ned as



(251)

X

dR *q (1#z) , (252) & %5  where *q is the average time duration of single bursts at the emission, and is of order 1 ms. Then %5 the total duty cycle turns out to be [103] D(z)"

D"1.57;10\ .

(253)

This implies that this GW background is not stochastic, but rather a sequence of bursts, with typical duration 1 ms and a much larger typical separation &0.2 s. This is a good news from the perspective of primordial Universe cosmology, since this astrophysical background can be distinguished from relic GWs produced at much higher redshifts. The value of h X has been computed in Ref. [103] and is of order 10\, peaked around 2 kHz.   It is shown in Fig. 16 (provided by the authors of Ref. [103], and including a recent update of the star formation rate at large z) for three di!erent values of the rotation parameter a,Jc/(GM ),  where J is the angular momentum. A black hole forms only if a is smaller than a critical value estimated in the range a &0.8}1.2; otherwise rotational energy dominates, the star bounces, and   no collapse occurs. For comparison with the other plots of h X shown in this report, note that  

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Fig. 16. X



357

against frequency, for GWs emitted in the supernovae collapse to black holes, from Ref. [103].

Fig. 16 shows X rather than h X , and that Ref. [103] uses h "0.5, so the value of h X is       obtained from Fig. 16 dividing by 4. Note also that both axes are on a linear scale, rather than a log}log scale, as the other plots show. From the "gure, we see that this background is out of reach for "rst generation experiments, while it could be detectable with advanced ground-based interferometers. It is also important to observe that the resulting GW signal is insensitive to the uncertainties that are present in the star formation rate in the high redshift region z&4}5, since the result turns out to be dominated by low-to-intermediate redshift sources. 11.2. GWs from hydrodynamic waves in rotating neutron stars In rapidly rotating neutron stars there is a whole class of instabilities driven by the emission of GWs, called CFS instabilities after Chandrasekhar [70], who discovered them, and Friedman and Schutz [117], who showed that they are generic to rotating neutron stars. Basically, they form when there is a mode that is forward-going, as seen from a distant observer, but backward going with respect to the rotation of the star. In this case, when this mode radiates away angular momentum, the star can "nd a rotation state with lower energy, than fueling further growth of the mode and therefore an instability occurs. Viscous forces, instead, tend to damp this instability. Recent studies of rapidly rotating relativistic stars have revealed the existence of a particularly interesting class of modes (r-modes) that are unstable due to the emission of GWs [13,116,169]. As the star spins down, for these modes an energy of order of 1% of a solar mass is emitted in GWs, making the process very interesting for GW detection [193,14]. Ref. [193] give a rough estimate of the spectrum assuming a comoving number density of neutron star births constant in the range 0(z(4, and zero at higher z. A more detailed analysis has been done in Refs. [104,214], using again the star formation rate determined by observation, mentioned in the previous section. The analysis is performed for three di!erent cosmological models, a #at Universe with vanishing

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cosmological constant (X "1, XK "0) and h "0.5, a #at, low-density model (X "0.3, XK " +  + 0.7, h "0.6), and an open model with X "0.4, XK "0, h "0.6. Similarly to the computation   + of R in the previous section, one can now compute the number of neutron stars formed per unit ,1 time within a comoving volume out at redshift z. Depending on the cosmological model, and on the upper cuto! in the progenitor mass, it turns out that R K(15}30) events/s . (254) ,1 Comparing with the collapse to black hole, with the same parameters, one "nds that R &4R . ,1 & The mechanism of GW emission is discussed in Refs. [169,193,14,15]. The neutron star is modelled as a #uid with a polytropic equation of state p"ko, with k chosen so that the mass of the neutron star is 1.4M and the radius is R"12.53 km. The r-mode excitation starts as a small > perturbation of the velocity "eld. As long as the amplitude a of the r-mode is small, the e!ect of the instability is such that a grows exponentially while the angular velocity X of the neutron star is nearly constant. After approximately 500 s, a becomes of order one, and we reach a regime where non-linear hydrodynamic e!ects are important. The non-linear e!ects saturate the growth of a, which stays approximately constant. This phase lasts approximately for a time q &1 yr, during ,1 which the star loses approximately  of its rotational energy by emission of GWs. Finally, the  angular velocity becomes su$ciently low so that the r-mode ceases to be unstable. Since GWs are emitted continuously over approximately 1 yr, the duty cycle for this process is [104]



D"



dR q (1#z)&10 ,1 ,1

(255)

 and therefore we de"nitely have a stochastic background. The minimum and maximum values of the frequency radiated are estimated as follows [193]. It is known that the maximum possible angular velocity of a rotating star is X &7.8;10 Hz[M/M (10 km/R)] , (256) ) > where M, R are the mass and radius of the corresponding non-rotating star. Assuming that most neutron stars are born with X close to X , one can write the initial rotational energy of the star. ) Using the fact that about  of it is radiated into GWs, one "nds  f K(2/3p)X . (257)

 ) The lower cuto! in f is associated to the fact that the r-mode instability becomes ine!ective below some critical rotational frequency, and it can be estimated that f K120 Hz . (258)

 The resulting spectrum of GWs is shown in Fig. 17, (provided by the authors of Ref. [104], and including a recent update of the star formation rate at large z and other technical improvements) for the three di!erent cosmological models de"ned above. We see that the three curves are very similar, and therefore the result is quite solid against changes in the cosmological model, and [104] h X K8;10\  

(259)

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Fig. 17. h X against frequency, for GWs produced by the r-mode of rotating neutron stars (from Ref. [104]).  

in the frequency range (500}700) Hz. This value is quite interesting, but from the discussion in Section 6 it is clear that it is still below the sensitivities that can be obtained with correlations of "rst generation interferometers. It will be instead quite accessible to second generation experiments. 11.3. GWs from mass multipoles of rotating neutron stars In the previous subsection we examined the case where GWs are emitted by rotating neutron stars through their coupling to the current multipoles associated to hydrodynamic waves on the star surface. We now discuss the emission by the simplest mechanism, which is the coupling of the GWs to the multipoles of the mass distribution of a rotating neutron star. The stochastic background from mass multipoles of rotating neutron stars has been discussed in Ref. [202]. The main uncertainty comes from the estimate of the typical ellipticity e of the neutron star, which measures its deviation from sphericity. An upper bound on e can be obtained assuming that the observed slowing down of the period of known pulsars is entirely due to the emission of gravitational radiation. This is almost certainly a gross overestimate, since most of the spin down is probably due to electromagnetic losses, at least for Crab-like pulsars. The typical frequency for these GWs can be in the range of ground-based interferometers. With realistic estimates for e, Ref. [202] gives, at f"100 Hz, a value of h ( f )&5;10\, that, using  Eq. (19), corresponds to h X (100 Hz)&10\ . (260)   This is very far from the sensitivity of even the advanced experiments. An absolute upper bound can be obtained assuming that the spin down is due only to gravitational losses, and this gives h X (100 Hz)&10\, but again this value is probably a gross overestimate.   Techniques for the detection of this background with a single interferometer using the fact that it is not isotropic and exploiting the sideral modulation of the signal have been discussed in Ref. [128].

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11.4. Unresolved galactic and extragalactic binaries These backgrounds will be relevant in the LISA frequency band. For frequencies below a few mHz, one expects a stochastic background due to a large number of galactic white dwarf binaries [100,34,227,215,202,33]. In Sections 11.1 and 11.2, in order to determine whether the superposition of burst signals from supernovae formed a stochastic background, we used the duty cycle. For a continuous signal as that produced by binaries, instead, whether we have a stochastic background or not depends on our frequency resolution. After a time ¹ of observation, we can resolve *f"1/¹; for ¹"1 yr, *f&3;10\ Hz, and the number of resolvable frequencies in the LISA frequency band is of order (1 Hz)/(3;10\ Hz)"3;10. It is estimated that most frequency bins below a critical frequency of the order of 1 mHz will contain signal from more than one galactic binary, and will therefore form a confusion-limited background. Above this critical frequency, instead, individual signals will be resolved. In particular, it is expected a contribution from compact white dwarf binaries. The computation of the intensity of this background depends on the rate of white dwarf mergers, which is uncertain. A possible estimate is shown in Fig. 6 [33]. We see that, below a few mHz, it would cover a stochastic background of cosmological origin at the level h X &10\.   It should be observed that, even if an astrophysical background is present, and masks a relic background, not all hopes of observing the cosmological signal are lost. If we understand well enough the astrophysical background, we can subtract it, and the relic background would still be observable if it is much larger than the uncertainty that we have on the subtraction of the astrophysical background. In fact, LISA should be able to subtract the background due to white dwarf binaries, since there is a large number of binaries close enough to be individually resolvable [33]. This should allow to predict with some accuracy the space density of white dwarf binaries in other parts of the Galaxy, and therefore to compute the stochastic background that they produce. Furthermore, any background of galactic origin is likely to be concentrated near the galactic plane, and this is another handle for its identi"cation and subtraction. The situation is more uncertain for the contribution of extragalactic binaries, which again can be relevant at LISA frequencies. The uncertainty in the merging rate is such that it cannot be predicted reliably, but it is believed to be lower than the galactic background [202]. In this case the only handle for the subtraction would be the form of the spectrum. In fact, even if the strength is quite uncertain, the form of the spectrum may be known quite well [33].

12. Conclusions Present GW experiments have not been designed especially for the detection of GW backgrounds of cosmological origin. Nevertheless, there are chances that in their frequency window there might be a cosmological signal. The most naive estimate of the frequency range for signals from the very early Universe singles out the GHz region, very far from the region accessible to ground based interferometers, f(a few kHz, or to resonant masses. To have a signal in the accessible region, one of these two conditons should be met: either we "nd a spectrum with a long low-frequency tail, that extends from the GHz down to the kHz region, or we have some explosive

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production mechanism much below the Planck scale. As we have discussed, both situations seem to be not at all unusual, at least in the examples that have been worked out to date. The crucial point is the value of the intensity of the background. With a very optimistic attitude, one could hope for a signal, with the maximum intensity compatible with the nucleosynthesis bound, h X &a few;10\ (or even 10\, stretching all   parameters to the maximum limit). Such an option is not excluded, and the fact that such a background is not predicted by the mechanisms that have been investigated to date is probably not a very strong objection, given our theoretical ignorance of physics at the Planck scale and the rate at which new production mechanisms have been proposed in recent years, see the reference list. However, with more realistic estimates, on general grounds it appears di$cult to predict a background that, either in the region between a few Hz and a few kHz, relevant for ground based interferometers, or in the region (10\}1) Hz relevant to LISA, exceeds the level h X &a few   ;10\, independently of the production mechanism. This should be considered the minimum detection level where a signi"cant search can start. Such a level is beyond the sensitivity of "rst generation ground-based detectors, but it is accessible both to advanced interferometers and to the space interferometer LISA. The di$culties of such a detection are clear, but the payo! of a positive result would be enormous, opening up a window in the Universe and in fundamental high-energy physics that will never be reached with particle physics experiments. Note added in proof The statement in Section 7.4 that the subject is somewhat controversial does not apply to the bound obtained from the proper motion of quasars. I thank Carl Gwinn for the information. Acknowledgements I am grateful to Adalberto Giazotto for many interesting discussions and questions, which stimulated me to write down this report. I thank for useful discussions or comments on the manuscript Danilo Babusci, Stefano Braccini, Maura Brunetti, Alessandra Buonanno, Massimo Cerdonio, Eugenio Coccia, Viviana Fafone, Valeria Ferrari, Stefano Fo!a, Maurizio Gasperini, Alberto Nicolis, Emilio Picasso, Ra!aella Schneider, Riccardo Sturani, Carlo Ungarelli, Gabriele Veneziano and Andrea Vicere`. I thank Barry Barish and Alberto Lazzarini for providing Figs. 1 and 5, Karsten Danzmann and Roland Schilling for providing Figs. 6 and 7. I thank Danilo Babusci for computing the overlap reduction functions and producing Figs. 9 and 10, Viviana Fafone for providing Fig. 8, and Ra!aella Schneider and Valeria Ferrari for Figs. 16 and 17. References [1] L. Abbott, D. Harari, Nucl. Phys. B 264 (1986) 487. [2] L. Abbott, M. Wise, Phys. Lett. B 135 (1984) 279.

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CONTENTS VOLUME 331 A. Hebecker. Di!raction in deep inelastic scattering

1

Yu.S. Kivshar, D.E. Pelinovsky. Self-focusing and transverse instabilities of solitary waves

117

T. Deguchi, F.H.L. Essler, F. GoK hmann, A. KluK mper, V.E. Korepin, K. Kusakabe. Thermodynamics and excitations of the one-dimensional Hubbard model

197

M. Maggiore. Gravitational wave experiments and early universe cosmology

283