New regularization using domain wall

Abstract

We present a new regularization method, for d-dimensional (Euclidean) quantum field theories in the continuum formalism, based on the domain wall configuration in (1+ d)-dimensional space-time. It is inspired by the recent progress in chiral fermions on a lattice. The wall “thickness” is given by 1/ M, where M is a regularization mass parameter and appears as a (1+ d)-dimensional Dirac fermion mass. The present approach gives a thermodynamic view on the domain wall or overlap formalism in 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 evolution”, 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 dimensions (domain wall formation). From the point of regularization, the limit Mt→0 regularizes the infra-red behavior whereas the condition on the momentum ( k μ ) integral, | k μ |⩽ M, regularizes the ultra-violet behavior. To check that the new regularization works correctly, we take four-dimensional QED and two-dimensional chiral gauge theory as examples. Especially the consistent and covariant anomalies are correctly obtained. The choice of solutions of the higher dimensional Dirac equation characterizes 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 this new regularization procedure.