Solving the QCD non-perturbative flow equation as a partial differential equation and its application to the dynamical chiral symmetry breaking

TL;DR

This paper develops a direct partial differential equation approach to solve the non-perturbative QCD flow equation, enabling accurate calculation of chiral symmetry breaking phenomena beyond traditional approximations.

Contribution

It introduces a novel method to solve the flow equation directly as a PDE, overcoming divergence issues and improving gauge independence in chiral condensate calculations.

Findings

Successful calculation of dynamical mass and chiral condensates

Reduced gauge dependence in beyond-ladder approximations

Demonstration of the method's effectiveness in non-perturbative QCD

Abstract

Non-perturbative renormalization group approach to the dynamical chiral symmetry breaking is an effective method which can accommodate beyond the ladder (mean filed) approximation. The usual method relying on the field operator expansion suffers explosive behaviors of the 4-fermi coupling constant, which prevent us from evaluating the physical quantities in the broken phase. In order to overcome this difficulty, we solve the flow equation directly as a partial differential equation and calculate the dynamical mass and the chiral condensates. Also we formulate a beyond the ladder equation and it gives almost gauge independent results for the chiral condensates.