Dynamical chiral symmetry breaking and weak nonperturbative renormalization group equation in gauge theory

TL;DR

This paper investigates dynamical chiral symmetry breaking in gauge theories using a nonperturbative renormalization group equation, introducing a weak solution concept to handle singularities and evaluate physical quantities.

Contribution

It introduces the notion of a weak solution for the nonlinear PDE in NPRGE, enabling analysis beyond singularities in chiral symmetry breaking.

Findings

Weak solutions can be used to analyze the NPRGE beyond critical scales.

The approach allows evaluation of physical quantities in the presence of non-analytic singularities.

The method provides a new way to understand nonperturbative phenomena in gauge theories.

Abstract

We analyze the dynamical chiral symmetry breaking in gauge theory with the nonperturbative renormalization group equation (NPRGE), which is a first order nonlinear partial differential equation (PDE). In case that the spontaneous chiral symmetry breaking occurs, the NPRGE encounters some non-analytic singularities at the finite critical scale even though the initial function is continuous and smooth. Therefore there is no usual solution of the PDE beyond the critical scale. In this paper, we newly introduce the notion of a weak solution which is the global solution of the weak NPRGE. We show how to evaluate the physical quantities with the weak solution.