New Regularization Using Domain Wall

TL;DR

This paper introduces a novel continuum regularization method for quantum field theories using domain wall configurations inspired by lattice chiral fermion progress, effectively controlling infra-red and ultra-violet divergences.

Contribution

It proposes a new regularization approach based on domain walls in (1+d) dimensions, linking lattice and continuum formalisms, and demonstrates its effectiveness with QED and chiral gauge theories.

Findings

Correctly reproduces anomalies in 4D QED and 2D chiral gauge theory.

Provides a thermodynamic interpretation of the domain wall formalism.

Ensures regularization of infra-red and ultra-violet divergences.

Abstract

We present a new regularization method, for d dim (Euclidean) quantum field theories in the continuum formalism, based on the domain wall configuration in (1+d) dim space-time. It is inspired by the recent progress in the chiral fermions on the lattice. The wall "height" is given by 1/M, where M is a regularization mass parameter and appears as a 1+d dim Dirac fermion mass. The present approach gives a thermodynamic view to the domain wall or the overlap formalism in the lattice field theory. We will show qualitative correspondence between the present continuum results and those of the lattice. The extra dimension is regarded as the (inverse) temperature t. The domains are defined by the directions of the "system movement", not by the sign of M as in the original overlap formalism. Physically the parameter M controls both the chirality selection and the dimensional reduction to d dimension. From the point of regularization, the limit $Mt\ra 0$ regularize the infra-red behaviour whereas the condition on the momentum ($k^\m$) integral, $|k^\m|\leq M$, regularize the ultra-violet behaviour. To check the new regularization works correctly, we take the 4 dim QED and 2 dim chiral gauge theory as examples. Especially the consistent and covariant anomalies are correctly obtained. The choice of solutions of the higher dim Dirac equation characterize the two anomalies. The projective properties of the positive and negative energy free solutions are exploited in calculation. Some integral functions, the incomplete gamma functions and the generalized hypergeometric functions characteristically appear in the regularization procedure.