Dynamic masses or quarks in quantum chromodynamics

Abstract

Using Dyson-Schwinger equations we obtain an ultraviolet asymptotics for the dynamical mass of quark in QCD mD(p) = 〈qq〉 3 ḡ2(p2/μ2, g2 R) p2 ( ḡ2(p2/μ2, g2 R) g2 R )−d . We also determine a numerical value for the π meson decay constant fπ. Electronic version of the paper published in Russ. Phys. J. 25, 55 (1982). As is well known, strong interactions at high energies are approximately invariant with respect to the chiral symmetry group SUL(N) ⊗ SUR(N) (N is the number of different quark species). The Lagrangian of QCD – the most popular field-theoretical model of strong interactions – has the chiral symmetry group SUL(N)⊗ SUR(N) in approximation of massless quarks. The success of current algebra and PCAC hypothesis shows that the chiral symmetry SUL(N) ⊗ SUR(N) is spontaneously broken down to SU(N) symmetry. The spontaneous violation of the chiral symmetry reveals itself in nonvanishing vacuum expectation values 〈qq〉 and also in the appearance of dynamical mass for originally massless quarks. The problem of dynamical mass generation in massless theories has a long history. First, in refs. [1, 2] it has been shown in some concrete models that the generation of the dynamical mass can take place in quantum field theory. However, the consistent study of the problem of dynamical mass generation requires going beyond perturbation theory. It is obvious that this problem is rather nontrivial in the field theory because of the absence of the consistent methods different from perturbation theory. The purpose of the present paper is to investigate a possibility of generation of the dynamical mass of quarks in QCD. Using the Dyson-Schwinger equations we show that one can reliably determine the dependence of the dynamical mass of quarks m(p) on the momentum p in the ultraviolet domain due to the property of asymptotic freedom of strong interactions. In the following we will work in Euclidean space-time and in the transverse gauge (Landau gauge). The use of this gauge is convenient since the propagator of the massless quark is not renormalized in the leading logarithmic approximation. The Dyson-Schwinger equation for the quark propagator G(p) has the form G(p) = S(p) + g (2π)4 ∫ dk Γμ(p, k)G(k)Dμν(p− k)γν(λ /2). (1) Here S(p) is the propagator of free quarks, λ/2 are the generators of the fundamental representation of the group SU(N). Equation (1) is written for bare quantities. After performing the renormalization we find Z 2 G −1 R (p) = S (p) + g R (2π)4 ∫ dk Γ μ (p, k)GR(k)D R μν(p− k)γν(λ /2), where Z2 is the renormalization constant of the quark propagator. At the lowest order in g 2 R in the Landau gauge the renormalization constant of the quark propagator is equal to one Z2 = 1+O(g 4 R). Therefore in the leading logarithmic approximation we can set G(p) = m(p)−p. For the quantity m(p) we obtain the equation of the form m(p) = 4 (2π)4 ∫ dk m(k)ḡ((p− k)) (m2(k2) + k2)(p− k)2 where ḡ(p) = ḡ(p/μ, g R) = g 2 R/ ( 1 + (11− 2N/3) g R 16π2 ln p μ2 ) is the effective quark-gluon coupling constant. We split the region of integration into two subdomains determined by the requirements k > p (sub-domain (I)) and k < p (sub-domain (II))