Non-perturbative computation of lattice correlation functions by differential equations

TL;DR

This paper introduces a novel non-perturbative method for calculating lattice correlation functions using differential equations and twisted cohomology, revealing insights into the series' asymptotic nature.

Contribution

It transfers perturbative calculation techniques to non-perturbative lattice computations, deriving differential equations for correlation functions with respect to coupling parameters.

Findings

Differential equations exhibit an essential singularity at zero coupling.

The perturbative series is shown to be asymptotic.

Method demonstrates potential for non-perturbative lattice calculations.

Abstract

We show that methods developed in the context of perturbative calculations can be transferred to non-perturbative calculations. We demonstrate that correlation functions on the lattice can be computed with the method of differential equations, supplemented with techniques from twisted cohomology. We derive differential equations for the variation with the coupling or -- more generally -- with the parameters of the action. Already simple examples show that the differential equation with respect to the coupling has an essential singularity at zero coupling and a regular singularity at infinite coupling. The properties of the differential equation at zero coupling can be used to prove that the perturbative series is only an asymptotic series.