Connecting the Chiral and Heavy Quark Limits : Full Mass Dependence of Fermion Determinant in an Instanton Background

TL;DR

This paper calculates the exact mass dependence of the fermion determinant in an instanton background, smoothly connecting the chiral and heavy quark limits, using a numerical method based on radial symmetry and Gelfand-Yaglom theorem.

Contribution

It introduces a computational approach that extends the Gelfand-Yaglom theorem to higher dimensions for calculating fermion determinants with full mass dependence in instanton backgrounds.

Findings

Provides a smooth interpolation between chiral and heavy quark limits.

Develops an efficient numerical method using partial wave decomposition.

Extends Gelfand-Yaglom theorem to higher-dimensional operators.

Abstract

This talk reports work done in collaboration with Jin Hur, Choonkyu Lee and Hyunsoo Min concerning the computation of the precise mass dependence of the fermion determinant for quarks in the presence of an instanton background. The result interpolates smoothly between the previously known chiral and heavy quark limits of extreme small and large mass. The computational method makes use of the fact that the single instanton background has radial symmetry, so that the computation can be reduced to a sum over partial waves of logarithms of radial determinants, each of which can be computed numerically in an efficient manner using a theorem of Gelfand and Yaglom. The bare sum over partial waves is divergent and must be regulated and renormalized. We use the angular momentum cutoff regularization and renormalization scheme. Our results provide an extension of the Gelfand-Yaglom result to higher dimensional separable differential operators. I also comment on the application of this approach to a wide variety of fluctuation determinant computations in quantum field theory.