String Representation and Nonperturbative Properties of Gauge Theories. Dissertation.
Dmitri Antonov *
Institute of Theoretical and Experimental Physics,
B. Cheremushkinskaya 25, RU-117 218 Moscow, Russia
* E-mail address: antonovSvxitep. itep.ru.
Contents
1 Introduction 3
1.1 General Ideas and Motivations........................................................ 3
1.2 Wilson’s Criterion of Confinement, and the Problem of String Representation of Gauge Theories........................................................................... 7
1.3 Method of Field Correlators in QCD: Theoretical Foundations......................... 11
2 String Representation of QCD in the Framework of the Method of Field Correlators 17
2.1 GLuodynamics String Effective Action from the Wilson Loop Expansion................. 17
2.2 Incorporation of Perturbative Corrections........................................... 25
2.3 A Hamiltonian of the Straight-Line QCD String with Spinless Quarks.................. 29
3 String Representation of Abelian-Projected Theories 32
3.1 The Method of Abelian Projections................................................... 32
3.2 Nonperturbative Field Correlators and String Representation of the Dual Abelian Higgs Model........................................................................... 38
3.2.1 London Limit............................................................... 38
3.2.2 Dual Abelian Higgs Model beyond the London Limit............................. 45
3.2.3 Dynamical Chiral Symmetry Breaking within the Stochastic Vacuum Model 47
3.3 Nonperturbative Field Correlators and String Representation of the SU(3)-Gluodynamics within the Abelian Projection Method....................................... 51
3.3.1 String Representation for the Partition Function of the Abelian-Projected SU(3)-Gluodynamics................................................................. 51
3.3.2 String Representation of Field and Current Correlators....................... 53
3.4 Representation of Abelian-Projected Theories in Terms of the Monopole Currents 56
4 Ensembles of Topological Defects in the Abelian-Projected Theories and String
Representation of Compact QED 59
4.1 Vacuum Correlators and String Representation of Compact QED . . . ................. 60
4.1.1 3D-case........................................è............................. 60
4.1.2 4D-case...................................................................... 67
4.2 A Method of Description of the String World-Sheet Excitations ...................... 73
4.3 Abelian-Projected Theories as Ensembles of Vortex Loops ............................ 75
5 Conclusion and Outlook 83
6 Acknowledgments 87
7 Appendices 88
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1 Introduction
1.1 General Ideas and Motivations
Nowadays, there is 110 doubt that strong interactions of elementary particles are adequately described by Quantum Chromodynamics (QCD) [1] (see Ref. [2] for recent monographs). Unfortunately, usual field-theoretical methods are not adequate to this theory itself. That is because in the infrared (IR) region, the QCD coupling constant becomes large, which makes the standard Feynman diagrammatic technique in this region unapplicable. However, it is the region of the strong coupling, which deals with the physically observable colourless objects (hadrons), whereas the standard perturbation theory is formulated in terms of coloured (unphysical) objects: quarks, gluons, and ghosts. This makes it necessary to develop special techniques, applicable for the evaluation of effects beyond the scope of perturbation theory. The latter are usually referred to as nonperturbative phenomena. Up to now, those are best of all studied in the framework of the approach based on lattice gauge theory [3], which provides us with a natural nonperturbative regularization scheme. Various ideas and methods elaborated on in the lattice field theories during the two last decades, together with the development of algorithms for numerical calculations and progress in the computer technology, have made these theories one of the most powerful tools for evaluation of nonperturbative characteristica of QCD (see Ref. [4] for a recent review). However, despite obvious progress of this approach, there still remain several problems. Those include e.g. the problem of simultaneous reaching the continuum and thermodynamic limits. Indeed, physically relevant length scales lie deeply inside the region between the lattice spacing and the size of the lattice. However, due to the asymptotic freedom of QCD. in the weak coupling limit, not only the lattice spacing, but also the size of the lattice (for a fixed number of sites) becomes small, as well as the region between them. However, in order to achieve the thermodynamic limit, the size of the lattice should increase. This makes it necessary to construct large lattices, which in particular leads to the technical problem of critical slowing down of simulations on them. As far as the problem of reaching the continuum limit alone is concerned, recently some progress in the solution of this problem has been achieved by making use of the conception of improved lattice actions [5] in Ref. [6]. Another problem of the lattice formulation of QCD is the appearance of so-called fermion doublers (i.e., additional modes appearing as relevant, dynamical degrees of freedom) in the definition of the fermionic action 011 the lattice due to the Nielson-Ninomiya theorem [7]. According to this No-Go theorem, would we demand simultaneously hermiticity, locality, and chiral symmetry (which will be discussed later on) of the lattice fermionic action, the doublers unavoidably appear, which means that all these three physical requirements cannot be achieved together. This makes it necessary to introduce fermionic actions which violate one of these properties (e.g. Wilson fermions [8j, violating chiral symmetry for a finite lattice or staggered (Kogut-Susskind) fermions (9), violating locality for a single flavour fermion), checking afterwards lattice artefacts associated with a particular choice of the action. Notice however, that recently a significant progress in the solution of this problem has been achieved (for a review >ee [10] and Refs, therein). Finally, there remains the important problem of reaching the chiral iimt, which becomes especially hard if one accounts for dynamical fermions. That was just one of ■he reasons why the main QCD calculations on large lattices have been performed in the quenched ipproximation, i.e., when the creation/annihilation of dynamical quark pairs is neglected.
All these problems together with the necessity of getting deeper theoretical insights into non->erturbative phenomena require to develop analytical nonperturbative techniques in QCD arid
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other theories displaying such phenomena. This is the main motivation for the present work.
It is worth realizing that up to now a systematic way of analytical investigation of nonper-turbative phenomena in QCD is still lacking. Instead of that, there exist various approaches, enabling one to take them into account phenomenologically for describing hadron interactions. Those approaches include e.g. the potential and bag models [11, 12], the large-N expansion methods [13], the effective Lagrangian approach [14], the QCD sum rule approach [15], and later on its generalization, the so-called Method of Field Correlators [16, 17, 18, 19]. It is natural to compare the situation in QCD appearing due to the absence of a systematic approach for the investigation of nonperturbative phenomena with nonrelativistic Quantum Mechanics. One convinces himself that the latter is the correct theory of atomic spectra by studying simple objects, like the hydrogen atom, though the practical calculation of the spectrum of a certain composite atom within this theory is quite complicated. Unfortunately, in QCD there exists no analogy of an isolated “hydrogen atom”, which forces us to study the theory of strong interactions at quite sophisticated examples.
The most fundamental problem associated with the IR dynamics of QCD, which is known to be one of the most important problems of modern Quantum Field Theory, is the problem of explanation and description of confinement (for a review see e.g. [20, 19]). In general, by confinement one implies the phenomenon of absence in the spectrum of a certain field theory of the physical |in) and |out) states of some particles, whose fields are however present in the fundamental Lagrangian. It is this phenomenon in QCD, which forces quarks and antiquarks to combine into colourless hadronic bound states. The important characteristic of confinement is the existence of string-like field configurations leading to a linearly rising quark-antiquark potential as expressed by the Wilson’s area law (see the next Subsection). Such a field configuration emerging between external quarks is usually referred to as QCD string. In the present Dissertation, we shall demonstrate that the properties of this string can be naturally studied in the framework of the Method of Field Correlators and derive by virtue of this method the corresponding string Lagrangian. Notice, that the advantage of the Method of Field Correlators is that it deals directly with QCD, and therefore enables us to express the coupling constants of this Lagrangian in terms of the fundamental QCD quantities, which are the gluonic condensate and the so-called correlation length of the vacuum. This approach will also allow us to incorporate quarks and derive a Hamiltonian of the QCD string with quarks in the confining QCD vacuum.
Another fundamental phenomenon of nonperturbative QCD is the spontaneous breaking oj chiral symmetry, i.e., the U (Nj) x U (N/)-symmetry of the massless QCD action. Indeed, though one could expect this symmetry to be observed on the level of a few MeV, it does not exhibit itself in the hadronic spectra. Were this symmetry exact, one would expect parity degeneracy of all hadrons, whereas in reality parity partners are generally split by a few hundred MeV. Such a spontaneous symmetry breaking has far-reaching consequences. In particular, it implies that there exist massless Goldstone bosons, which are identified with pions. The signal for chiral symmetry breaking is the appearance of a nonvanishing quark condensate, which plays an important role in nany nonperturbative approaches [21, 22:.
The non-Abelian character of the gauge group SU{3) makes it especially difficult to study ;he problems of confinement, and chiral symmetry breaking in the QCD case. To explain the nechanisms of these phenomena microscopically, a vast amount of models of the QCD vacuum las been proposed (see [19] for a recent review). Those are based either on an ensemble of classical ield configurations (e.g. instantons [23, 24], see [25] for recent reviews) or on quantum background ields [26, 27]. The most general demand made on all of them was to reproduce two characteristic
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quantities of the QCD vacuum, which are nonzero quark [21] and gluon [15] condensates, related to the chiral symmetry breaking and confinement, respectively. However, at least the semiclassical scenario possesses several weak points. First of all, since the topological charge of the QCD vacuum as a whole is known to vanish, this vacuum cannot be described by a certain unique classical configuration, but should be rather built out of a superposition of various configurations, e.g. instantons and antiinstantons. However, such a superposition already does not satisfy classical equations of motion and is, moreover, unstable w.r.t. annihilation of the objects with the opposite topological charge. Secondly, in order to reproduce the phenomenological gluon condensate [15], classical configurations should be dense packed (about one configuration per (fm)4), which leads to a significant distorsion of the solutions corresponding to these configurations within the original superposition ansatz [28] And last but not least, a further counterargument against semiclassical models of the QCD vacuum is that not all of them, once being simulated in the lattice experiments, yield the property of confinement (see discussion in Ref. [19]).
Another natural way of investigation of the nonperturbative phenomena in QCD might lie in the simplification of the problem under study by considering some solvable theories displaying the same type of phenomena. In this way, the problem of chiral symmetry breaking is best of all analytically studied in the so-called Nambu-Jona-Lasinio (NJL) type models, which are models containing local four-quark interactions [30, 31, 32, 33] (see Ref. [34] for a recent review). These models lead to a gap equation for the dynamical quark mass, signalling spontaneous breaking of chiral symmetry. After applying the so-called bosonization procedure (which can be performed either by making use of the standard Hubbard-Stratonovich transformation or within the field strength approach [35]) as well as a derivative expansion of the resulting quark determinant at low energies, this leads to the construction of nonlinear chiral meson Lagrangians [14]. The advantage of the latter ones is that they summarize QCD low-energy theorems, which is the reason why these Lagrangians are intensively used in the modern h&dronic physics [36, 34]. The techniques developed for NJL type models have been in particular applied to the evaluation of higher-order derivative terms in meson fields [32, 33], which enabled one to estimate the structure constants of the effective chiral Lagrangians introduced in Ref. [36]. Furthermore, in this way in Refs. [31, 32, 33, 34] it has been demonstrated that the low-energy properties of light pseudoscalar, vector, and axial-vector mesons are well described by effective chiral Lagrangians following from the QCD-motivated NJL models. In addition, the path-integral bosonization of an extended NJL model including chiral symmetry breaking of light quarks and heavy quark symmetries of heavy quarks has been performed [37] (see the second paper of Ref. [38] for a review), which yielded the effective Lagrangians of pseudoscalar, vector, and axial-vector D or B mesons, interacting with light tt, p, and fli mesons.
As far as the theories possessing the property of confinement are concerned, those firstly include compact QED and the 3D Georgi-Glashow model [20] and, secondly, the so-called Abelian-projected theories [39] 2. In the present Dissertation, we shall concentrate ourselves on the confining properties of the above mentioned non-supersymmetric theories. In all of them, confinement occurs due to the expected condensation of Abelian magnetic monopoles, after which the vacuum structure of these theories becomes similar to that of the dual superconductor (the so-•alled dual Meissner scenario of confinement) [41]. Such a vacuum then leads to the formation >f strings (flux tubes) connecting external electric charges, immersed into it. These strings are
1 Recently, some progress in the solution of this problem has been achieved in Ref. [29].
2 In what follows, we shall not consider recently discovered supersymmetric theories, also possessing the property f confinement [40].
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dual to the (magnetic) Abrikosov-Nielsen-Olesen strings [42]. The latter ones emerge as classical field configurations in the Abelian Higgs Model, which is the standard relativistic version of the Ginzburg-Landau theory of superconductivity. It turns out that the properties of electric strings in the dual superconductor are similar to the ones of the realistic strings in QCD, which connect quarks with antiquarks and ensure confinement. It is worth noting that this analogy based on the ’t Hooft-Mandelstam scenario led to several phenomenological dual models of QCD (see Ref. [43] for a review). Thus, the properties of the QCD string can be naturally studied in the framework of the Abelian projection method. Moreover, this approach turns out to provide us with the representations for the partition functions of effective Abelian-projected theories of the SU(2)-and SC/(3)-gluodynamics in terms of the integrals over string world-sheets. Such an integration, which is absent in the approach to the QCD string based on the Method of Field Correlators, appears nowT from the integration over the singular part of the phase of the magnetic Higgs field. The reformulation of the integral over the singularities of this field into the integral over string world-sheets is possible due to the fact that such singularities just take place at the world-sheets. In particular, an interesting string picture emerges in the St/(3)-gluodynamics, where after the Abelian projection there arise three types of magnetic Higgs fields, leading to three types of strings, which (self)interact via the exchanges of two massive dual gauge bosons. An exact procedure of the derivation of the string representations for the partition functions of Abelian-projected theories in the language of the path-integral, the so-called path-integral duality transformation, will be described in details below. In the framework of this approach, we shall also investigate field correlators in the Abelian-projected theories and find them to parallel those of QCD, predicted by the Method of Field Correlators and measured in the lattice simulations. Topological properties of Abelian-projected theories will be also discussed. Furthermore, the effects brought, about by the summation over the grand canonical ensembles of small vortex loops, built out of the paired electric Abrikosov-Nielsen-Olesen strings, in these theories will be studied. In particular in the dilute gas approximation, the effective potential of such vortex loops will be derived and employed for the evaluation of their correlators. Besides that, we shall study the string representation and field correlators of compact QED in 3D and 4D. Notice that due to the absence of the Higgs field in this theory, the integration over the string world-sheets is realized there in another way than in Abelian-projected theories. Namely, it results from the summation over the branches of the multivalued effective monopole potential, which turns out. to have the same form as the 3D version of the above discussed potential of vortex loops. Finally, similar forms of the string effective actions in QCD, Abelian-projected theories, and compact QED will enable us to elaborate for all these theories a unified method of description of the string world-sheet excitations, based on the methods of nonlinear sigma models, known from the standard string theory.
It is worth realizing, that the string theories, to be derived below, should be treated as effective, rather than fundamental ones. The actions of all of them turn out to have the form of an interaction between the elements of the string world-sheet, mediated by certain (nonperturbative) gauge field >ropagators. Being expanded in powers of the derivatives w.r.t. world-sheet coordinates, these ictions yield as a first term of such an expansion the usual Nambu-Goto action. The latter me is known to suffer from the problem of conformal anomaly in D £ 26 appearing during its juantization, which will not be discussed below. It this sense, throughout the present Dissertation, ve shall treat the obtained string theories as effective 4D ones. It, is also worth noting that within >ur approach only pure bosonic strings without supersymmetric extensions appear. As far as ;uperstrings arc concerned, during the last fifteen years, a great progress has been achieved in heir development (see e.g. [44] for comprehensive monographs). Among the achievements of the
superstring theory it is worth mentioning such ones as the calculation of the critical dimension of the space-time, inclusion of gravity in a common scheme, and, presumably, the absence of divergencies for some of these theories. The aim of all the superstring theories is the unification of all the four fundamental interactions. In another language, one should eventually be able to derive from them both the Standard Model and gravity, whereas all the auxiliary heavy modes should become irrelevant. Therefore, the final strategy of superstring theories is a derivation of the known field theories out of them. Contrary to this ideology, the aim of the present Dissertation is the derivation of effective string theories from gauge field theories possessing string-like excitations. As it has been discussed above, such string-like field configurations naturally appear in the confining phases of gauge theories.
Another possible direction of investigation of confinement and chiral symmetry breaking in QCD is based on a derivation of self-coupled equations for gauge-invariant vacuum amplitudes starting directly from the QCD Lagrangian and seeking for solutions allowing for these properties [45, 46, 47, 48]. Recently, this approach turned our to be quite useful for the investigation of the problem of interrelation between these two phenomena [48]. Once such an interrelation takes place, there should exist a relation between quark and gluon condensates as well, which has just been established in Ref. [48]. We shall briefly demonstrate the method of derivation of such a relation later on.
The organization of the Dissertation is as follows. In the next Subsection of the Introduction, we shall introduce the main quantitative parameters for the description of confinement in gauge theories and quote the criterion of this phenomenon in the sense of Wilson’s area law. This criterion will then serve as our starting point in a derivation of certain string effective actions in various gauge theories. In the last Subsection, we shall consider theoretical foundations of the Method of Field Correlators. Section 2 is devoted to a derivation and investigation of the QCD string effective action within this method. In Section 3, we investigate the problem of string representation of QCD from the point of view of Abelian-projected theories and demonstrate a correspondence between the Abelian projection method and the Method of Field Correlators. In Section 4, we study the string representation and vacuum correlators of compact QED in 3D mid 4D. All these investigations eventually bring us to the conclusion that both QCD within the Method of Field Correlators, Abelian-projected theories, and compact QED have similar string representations. Such an observation then enables us to consider strings in these theories from the same point of view and elaborate for them a unified mechanism of description of string excitations. Besides that in Section 4, by virtue of the techniques developed for the investigation of the grand canonical ensemble of monopoles in compact QED, collective effects in similar ensembles of topological defects emerging in the Abelian-projected theories will be studied. Finally, we discuss the main results of the Dissertation as well as possible future developments in the Conclusion and Outlook. In four Appendices, some technical details of transformations performed in the main text arc outlined.
L.2 Wilson’s Criterion of Confinement and the Problem of String Representation of Gauge Theories
\s a most natural characteristic quantity for the description of confinement one usually considers he so-called Wilson loop. For example in the case of QCD, this object has the following form
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