научная статья по теме CHIRAL-SYMMETRY BREAKING AND THE QCD STRING Физика

Текст научной статьи на тему «CHIRAL-SYMMETRY BREAKING AND THE QCD STRING»

ЯДЕРНАЯ ФИЗИКА, 2008, том 71, № 2, с. 342-355

= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ

CHIRAL-SYMMETRY BREAKING AND THE QCD STRING

© 2008 A. V. Nefediev, Yu. A. Simonov

Institute of Theoretical and Experimental Physics, Moscow, Russia Received July 16,2007

The Lorentz nature of the effective interquark interaction is investigated in a heavy—light quarkonium. The approach of the Dyson—Schwinger-type equation and the quantum-mechanical Hamiltonian method of the QCD string with quarks at the ends are employed to demonstrate that the effective scalar interaction, which appears due to chiral-symmetry breaking, is responsible for the QCD-string formation. The Hamiltonian of the QCD string with quarks at the ends arizes naturally, if this effective scalar interaction dominates. If, on the contrary, chiral symmetry is manifest, the effective interquark interaction remains vectorial, and the corresponding bound-state equation is incompatible with the QCD-string Hamiltonian. We conclude therefore that the genuine Lorentz nature of the QCD string is scalar.

PACS:12.38.Aw, 12.39.Ki, 12.39.Pn

1. INTRODUCTION

We address one of the long-standing problems in QCD, namely, the problem of the Lorentz nature of the interquark interaction, which results in the formation of the QCD string. The phenomenologi-cal picture of hadrons consisting of quarks and/or gluons attached at the ends of the Nambu—Goto string appears rather successful in various studies of hadronic properties. This picture allows one to take into account several important phenomena celebrated in QCD, such as confinement, proper dynamics of the gluonic degrees of freedom both in the form of the string rotation as well as in the form of hybrid string excitations. The mechanism of the quark constituentlike mass formation due to interaction also finds its natural explanation in the given approach. An important step was made in the framework of the Vacuum Correlators Method (VCM) [1] in which the Lagrangian of the QCD string with quarks at the ends can be derived naturally starting from the fundamental QCD Lagrangian [2]. In addition, the nonperturbative spin-dependent forces in heavy and light quarkonia were found, following the formalism established in [3]. Thus, for the spin—orbit interaction, a celebrated representation,

4m|) (r dr ^ r <9r ) ^ ^

| 1 , x _(rl_)l<W2

2mqmq q q q q r dr '

was introduced in [3], where e(r) is the static confining potential, and a general relation (Gromes rela-

tion [4]) is valid,

e' + V{ - V2 = 0. (2)

For a purely scalar interaction, one obtains at large r's:

v{ = -e, V2 = 0, (3)

and this was demonstrated explicitly for the Gaussian approximation for the field correlators in [5]. For the case of vector confinement, for example, for the Coulomb potential, one would find

V/ = 0, V2 > 0, (4)

and the coefficient at the spin—orbit term would have the opposite sign. Phenomenology of the heavy-quarkonia spectrum favors the first possibility, Eq. (3), so that one has an evidence that Nature prefers scalar interquark interaction, at least for heavy quarks. On the lattice, numerous data also support the first possibility (see [6] for recent results and the vast bibliography). In the meantime, the QCD-string method, as a quantum-mechanical approach, meets severe problems with the description of spontaneous breaking of chiral symmetry. An obvious reason of this failure is related to the fact that the time-forward and backward motions of quarks are considered separately — quantum-mechanical models being simply unable to describe particles and antiparticles simultaneously. In the meantime, for example, in the chiral pion, both types of the quark motion are equally important. It is therefore necessary to go beyond the ordinary quantum-mechanical treatment for such systems and employ genuine quantum-field-theory-based approaches. An example of such a treatment, also based on the VCM,

is given by the Dyson—Schwinger-type approach to heavy—light quarkonia suggested in [7]. The problem of the Lorentz nature of confinement for heavy quarks was addressed in this formalism in [7—9]. In the meantime, a similar problem of the Lorentz nature of the effective interaction for light quarks still remains open, in particular, the question of the interrelation between chiral-symmetry breaking and the QCD-string formation. A progress in understanding this connection can be achieved on the way of merging the quantum-mechanical approach of the QCD string with quarks at the ends and the quantum-field-theory-inspired Dyson—Schwinger-type approach to quarkonia. In particular we demonstrate that, starting from the fundamental QCD Lagrangian and using the Gaussian approximation for the background field correlators, one can derive a Dyson—Schwinger-type equation for the heavy—light quarkonium, which, at large interquark distances, reduces to a Dirac-like equation with an effective interquark interaction which contains a dynamically generated scalar part, as a consequence of chiral-symmetry breaking. We also prove that, at the same time, a Schrodinger-like equation with the Hamiltonian of the QCD string with quarks at the ends (in the form of the Salpeter equation) arizes naturally from the same Dyson—Schwinger-type equation [10]. We conclude, therefore, that this is the dynamical scalar interaction responsible for the QCD-string formation. This result is quite general and holds regardless of the explicit form of the quark interaction kernel, suffices it is confining and thus leads to spontaneous breaking of chiral symmetry. This is yet another important outcome of our work since the vast results obtained in the literature in the framework of potential quark models [11, 12] are valid for our situation as well — what we do is approaching the same problem from another side.

The paper is organized as follows. In Section 2 we consider the exactly solvable two-dimensional model for QCD in the limit of an infinite number of colors — the 't Hooft model [13]. Although the underlying mechanisms of confinement are totally different in two and four dimensions — the Coulomb interaction being confining in two dimensions — we find the example of the 't Hooft model to be quite instructive in explaining the method of the chiral angle, which is used then, in Section 3, to investigate the essentially nonlinear Dyson—Schwinger-type bound-state equation for the heavy—light quarkonium and to establish the above-mentioned interrelation between chiral-symmetry breaking and the QCD-string formation in the heavy—light quark—antiquark system. We conclude in Section 4.

2. TWO-DIMENSIONAL QCD (THE't HOOFT MODEL)

In this section, as a warm-up, we consider an instructive example of QCD in two dimensions and demonstrate how an effective scalar interquark interaction appears as a result of spontaneous breaking of chiral symmetry. Two-dimensional QCD, as a toy model for strong interactions, was suggested by 't Hooft in 1974 [13] and since then it has been studied by a large number of authors in a variety of approaches (see, for example, [14—18]). The key features of this model are: (i) an infinite number of quark colors necessary in order to bypass the Coleman no-go theorem for two-dimensional theories [19]; (ii) an instantaneous interquark interaction, as a consequence of absence of propagating degrees of freedom of the two-dimensional gluon; (iii) the confining potential growing linearly with the interquark separation; (iv) spontaneous breaking of chiral symmetry; (v) "Regge trajectories" for the spectrum of bound states — the quark—antiquark mesons.

We start from the Lagrangian of two-dimensional QCD,

L(x 0,x) = 0,x) +

(5)

+ y] qf (xo,x)(iD - mf )qf (xo,x), f

D = (d,u - igAlta)^,

where f is the flavor index, and our convention for y matrices is

Yo = / = a3, Yi = ia2, Y5 = a = Y0Y1 =

We fix the axial gauge

A?(xo ,x) = 0. (6)

It was demonstrated in [16] that a self-consistent procedure of regularization of infrared divergent integrals, in this gauge, is given by the principal value prescription, which we also assume below.

In the axial gauge, the central object of the VCM, the Gaussian correlator of gluonic fields, takes the form

i{0!TFiao (xo,x)Fbw (yo ,y)\0) = (7)

= -g25ab5(2) (x - y),

which formally corresponds to the limit of a vanishing gluonic correlation length Tg in the general Euclidean formula [1]

(F;Axo,x)FbXp(y0,y)) = 5ab x (8)

C

xo - yo x - y

x D ( —-—, j (S^xSvp ~

where the profile D is a decreasing function of its arguments. In the aforementioned limit of Tg — 0 — also known as the string limit of QCD — Eq. (8) can be written in Minkowski space in the form of Eq. (7). In what follows, in this section, we work in Minkowski space. Notice that, in two dimensions and in gauge (6), all higher order irreducible correlators of gluonic fields vanish, the Gaussian dominance being exact. In analogy with the four-dimensional case one can introduce the string tension as [1]

u = 2

oo oo

j dr j d\D(r, X) = (9)

0 0

L

4

2 / !

Nc-

Nc

Nc~>oo'

>g2Nc 4 '

A%(xo, 0) =0,

(11

which, together with the axial-gauge condition (6), forms the two-dimensional analog of the so-called modified Fock—Schwinger gauge [21]. In this gauge, the static antiquark becomes the center of force without its own degree of freedom, so that the Dyson-type series of self-energy-like diagrams for the light quark can be summed and a one-particle-like Dyson— Schwinger-type equation can be derived for the lightquark Green's function, the latter coinciding with the full quark—antiquark Green's function [7]. Indeed, the

Green's function of the colorless qQ system can be written in the following form:

Sgq(x,y) = D^DA^ x (12)

x exp \ - - / <PxF$ +

+ i J d2xip(id — m — Atyj x

x tp(x)SQ(x — y)^{y), with the static-quark propagator being simply

The large-Nc limit implies that a, also playing the role of the effective coupling constant, remains finite. It is important to keep in mind that, in order to p

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