Linear lattice gauge theory

TL;DR

This paper investigates linear lattice gauge theory with arbitrary matrix link variables, exploring its continuum limit, phase structure, and potential advantages over traditional formulations for non-perturbative studies.

Contribution

It introduces the linear formulation with matrix link variables, analyzes its phase structure, and discusses the flow of the link potential minimum for understanding continuum limits.

Findings

Linear and non-linear formulations share the same universality class.

The link potential minimum flow distinguishes between perturbative and confinement regimes.

Linear formulation reveals additional excitations beyond gauge fields.

Abstract

Linear lattice gauge theory is based on link variables that are arbitrary complex or real $N\times N$ matrices, in distinction to the usual (non-linear) formulation with unitary or orthogonal matrices. For a large region in parameter space both formulations belong to the same universality class, such that the continuum limits of linear and non-linear lattice gauge theory are identical. We explore if the linear formulation can help to find a non-perturbative continuum limit formulated in terms of continuum fields. Linear lattice gauge theory exhibits excitations beyond the gauge fields. In the linear formulation the running gauge coupling corresponds to the flow of the minimum of a ``link potential''. This minimum occurs for a nonzero value of the link variable $l_0$ in the perturbative regime, while $l_0$ vanishes in the confinement regime. We discuss a flow equation for the scale dependent location of the minimum $l_0(k)$.