Infinite N phase transitions in continuum Wilson loop operators

TL;DR

This paper investigates phase transitions in continuum Wilson loop operators at infinite N, revealing a nontrivial limit and a spectral gap closing, with implications for calculating string tension via universality classes.

Contribution

It introduces a new approach to defining and analyzing smoothed Wilson loops on a lattice, identifying a phase transition at infinite N and exploring its potential for analytical calculations.

Findings

Wilson loop operators have a finite, nontrivial continuum limit.

A phase transition occurs at infinite N when dilating the loop size.

Eigenvalue distribution gap closes during the phase transition.

Abstract

We define smoothed Wilson loop operators on a four dimensional lattice and check numerically that they have a finite and nontrivial continuum limit. The continuum operators maintain their character as unitary matrices and undergo a phase transition at infinite N reflected by the eigenvalue distribution closing a gap in its spectrum when the defining smooth loop is dilated from a small size to a large one. If this large N phase transition belongs to a solvable universality class one might be able to calculate analytically the string tension in terms of the perturbative Lambda-parameter. This would be achieved by matching instanton results for small loops to the relevant large-N-universal function which, in turn, would be matched for large loops to an effective string theory. Similarities between our findings and known analytical results in two dimensional space-time indicate that the phase transitions we found only affect the eigenvalue distribution, but the traces of finite powers of the Wilson loop operators stay smooth under scaling.