Series expansions for lattice Green functions

TL;DR

This paper derives series expansions for lattice Green functions across various dimensions, identifying singularities and providing explicit hypergeometric series in low dimensions, with a generalized algorithm for arbitrary anisotropies and masses.

Contribution

It introduces a comprehensive method to obtain series expansions of lattice Green functions in arbitrary dimensions, including singularity analysis and a generalized algorithm for anisotropic cases.

Findings

Explicit series expressions in 1D and 2D as hypergeometric functions.

Identification of singular points with different types in odd and even dimensions.

A generalized algorithm for arbitrary anisotropies and mass in lattice Green functions.

Abstract

Lattice Green functions appear in lattice gauge theories, in lattice models of statistical physics and in random walks. Here, space coordinates are treated as parameters and series expansions in the mass are obtained. The singular points in arbitrary dimensions are found. For odd dimensions these are branch points with half odd-integer exponents, while for even dimensions they are of the logarithmic type. The differential equations for one, two and three dimensions are derived, and the general form for arbitrary dimensions is indicated. Explicit series expressions are found in one and two dimensions. These series are hypergeometric functions. In three and higher dimensions the series are more complicated. Finally an algorithmic method by Vohwinkel, Luscher and Weisz is shown to generalize to arbitrary anisotropies and mass.