Schwinger–Dyson Equation for Quarks in a QCD Inspired Model

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 \({{\alpha }_{s}} = {4 \mathord{\left/ {\vphantom {4 \pi }} \right. \kern-0em} \pi }\), 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.