Струнное представление и непертурбативные свойства калибровочных теорий

• Автор: Антонов, Дмитрий Владимирович
• Шифр специальности: 01.04.02
• Научная степень: Кандидатская
• Год защиты: 1999
• Место защиты: Москва
• Количество страниц: 100 с.

String Representation and Nonperturbative Properties of Gauge Theories. Dissertation.
Dmitri Antonov *
Institute of Theoretical and Experimental Physics,
B. Cheremushkinskaya 25, RU-117 218 Moscow, R,ussia
Contents
1 Introduction
1.1 General Ideas and Motivations
1.2 Wilson’s Criterion of Confinement and the Problem of String Representation of Gauge Theories
1.3 Method of Field Correlators in QCD: Theoretical Foundations
2 String Representation of QCD in the Framework of the Method of Field Correlators
2.1 Gluodynamics String Effective Action from the Wilson Loop Expansion
2.2 Incorporation of Perturbative Corrections
2.3 A Hamiltonian of the Straight-Line QCD String with Spinless Quarks
3 String Representation of Abelian-Projected Theories
3.1 The Method of Abelian Projections
3.2 Nonperturbative Field Correlators and String Representation of the Dual Abelian Higgs Model
3.2.1 London Limit
3.2.2 Dual Abelian Higgs Model beyond the London Limit
3.2.3 Dynamical Chiral Symmetry Breaking within the Stochastic Vacuum Model
3.3 Nonperturbative Field Correlators and String Representation of the SU(3)-Gluodynamics within the Abelian Projection Method
3.3.1 String Representation for the Partition Function of the Abelian-Projected SU(3)-Gluodynamics
3.3.2 String Representation of Field and Current Correlators
3.4 Representation of Abelian-Projected Theories in Terms of the Monopole Currents
4 Ensembles of Topological Defects in the Abelian-Projected Theories and String
Representation of Compact QED
4.1 Vacuum Correlators and String Representation of Compact QED
4.1.1 3D-case
4.1.2 4D-case
4.2 A Method of Description of the String World-Sheet Excitations
4.3 Abelian-Projected Theories as Ensembles of Vortex Loops
5 Conclusion and Outlook
6 Acknowledgments
7 Appendices

1 Introduction
1.1 General Ideas and Motivations
Nowadays, there is no 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 on the lattice due to the Nielsen-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 [8], 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 see [10] and Refs, therein). Finally, there remains the important problem of reaching the chiral imit, 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-rerturbative phenomena require to develop analytical nonperturbative techniques in QCD and

where a = 1,2. Consequently, one has for the field strength tensor
F/J.V = [A + C = [A] + (D |y4] A C)/iU — ig [Cp, CJ, (71)
where
(° A G)pa, = - OuG», and [A] = dp — ig [Ap, ].
Eq. (71) can be straightforwardly rewritten as follows
Fpp = (fp„ + Cpp)T3 + Sap„Ta.
Here, fpp = (d A a) and CpV = gsab3ApAbv stand for the contributions of diagonal and off-diagonal components of the gluon field to the diagonal part of the field strength tensor, respectively, and SpV = (vab A H6) is the off-diagonal part of the field strength tensor with "Z?“6 = dp5ab — geab3ap. This yields the following decomposition of the action (70) taken now on the gauge transformed fields,
SVm [A'}] ={dix (/„, + Cpp + + fd4x (s% + (i)°)2,
where [F y
Next, since our aim will be the investigation of the confining (i.e., infrared) properties of the Abelian-projected St/(2)-gluodynamics (rather than the problems of its renormalization, related to the region of asymptotic freedom), we shall disregard the “-dependent terms [108]. Within this approximation, the resulting effective action takes the form
SW. [<*„, Cs] =jdAx (fpV + , (72)
where we have denoted for brevity /™g- = ) . The monopole current is defined via the
modified Bianchi identities as
j? = dp {hu + Cs') = epVx„dp}f
with fpv = epvpfp. Thus the obtained effective theory (72) can be regarded as a 17(1) gauge theory with magnetic monopoles. To proceed with the investigation of the monopole ensemble, it is reasonable to cast the partition function under study, Z = f Zl/™g'Ila/1e_Soff-, to the dual form 15. This can be done by making use of the first-order formalism, i.e., linearizing the square fpV in Eq. (72) by introducing an integration over an auxiliary antisymmetric tensor field BpV as follows

DdpDBpp exp - / d4x
{-/<
V + BpVfpV + (cs):
(73)
15Notice that the gauge fixing term for the Abelian field is assumed to be included into the integration measure Da