Schwinger-Dyson equation for quarks in a QCD inspired model

TL;DR

This paper formulates QCD in Minkowski space, deriving a Schwinger-Dyson equation for quarks that reveals a critical coupling for chiral symmetry breaking, using operator product expansion and normal ordering.

Contribution

It introduces a Minkowski-space formulation of QCD with a novel derivation of the Schwinger-Dyson equation for quarks, highlighting the role of the gluon condensate and a critical coupling.

Findings

Identifies a critical coupling constant $rac{4}{\pi}$ for chiral symmetry breaking

Derives a Schwinger-Dyson equation for a massless quark in a QCD-inspired model

Performs numerical and analytical analysis of the quark propagator

Abstract

We discuss formulation of QCD in Minkowski-spacetime and effect of an operator product expansion by means of normal ordering of fields in the QCD Lagrangian. The formulation of QCD in the Minkowski-spacetime allows us to solve a constraint equation and decompose the gauge field propagator in the sum of an instantaneous part, which forms a bound state, and a retarded part, which contains the relativistic corrections. In Quantum Field Theory, for a Lagrangian with unordered operator fields, one can make normal ordering by means of the operator product expansion, then the gluon condensate appear. This gives us a natural way of obtaining a dimensional parameter in QCD, which is missing in the QCD Lagrangian. We derive a Schwinger-Dyson equation for a quark, which is studied both numerically and analytically. The critical value of the strong coupling constant $\alpha_s = 4/{\pi}$, above which a nontrivial solution appears and a spontaneous chiral symmetry breaking occurs, is found. For the sake of simplicity, the considered model describes only one flavor massless quark, but the methods can be used in more general case. The Fourier-sine transform of a function with log-power asymptotic was performed.