Lattice worldline representation of correlators in a background field

TL;DR

This paper introduces a lattice worldline approach to compute continuum limits of local operator expectation values in a background field, specifically applied to scalar fields in non-Abelian gauge backgrounds, using random walk sums.

Contribution

It develops a novel discrete worldline representation method to analyze continuum limits of correlators in background fields, with explicit formulas involving random walks.

Findings

Coefficients expressed as sums over random walks.

Results valid for anisotropic lattices.

Simplification of coefficients into integrals using combinatorial identities.

Abstract

We use a discrete worldline representation in order to study the continuum limit of the one-loop expectation value of dimension two and four local operators in a background field. We illustrate this technique in the case of a scalar field coupled to a non-Abelian background gauge field. The first two coefficients of the expansion in powers of the lattice spacing can be expressed as sums over random walks on a d-dimensional cubic lattice. Using combinatorial identities for the distribution of the areas of closed random walks on a lattice, these coefficients can be turned into simple integrals. Our results are valid for an anisotropic lattice, with arbitrary lattice spacings in each direction.