Area Law and Continuum Limit in "Induced QCD"

TL;DR

This paper explores the continuum limit of the Kazakov-Migdal matrix model, revealing a unique phase transition at zero temperature and connections to a $c>1$ string theory, through analysis of filled Wilson loops and associated statistical systems.

Contribution

It provides a novel analysis of the continuum limit in the Kazakov-Migdal model, linking it to a critical point at zero temperature and a $c>1$ string theory analogy.

Findings

The string tension equals the free energy density of a related statistical system.

The continuum limit occurs at a zero-temperature critical point in the large N limit.

The model's continuum limit differs drastically from Wilson and adjoint theories, resembling a $c>1$ string theory.

Abstract

We investigate a class of operators with non-vanishing averages in a D-dimensional matrix model recently proposed by Kazakov and Migdal. Among the operators considered are ``filled Wilson loops" which are the most reasonable counterparts of Wilson loops in the conventional Wilson formulation of lattice QCD. The averages of interest are represented as partition functions of certain 2-dimensional statistical systems with nearest neighbor interactions. The ``string tension" $\alpha'$, which is the exponent in the area law for the ``filled Wilson loop" is equal to the free energy density of the corresponding statistical system. The continuum limit of the Kazakov--Migdal model corresponds to the critical point of this statistical system. We argue that in the large $N$ limit this critical point occurs at zero temperature. In this case we express $\alpha'$ in terms of the distribution density of eigenvalues of the matrix-valued master field. We show that the properties of the continuum limit and the description of how this limit is approached is very unusual and differs drastically from what occurs in both the Wilson theory ($S\propto({\rm Tr}\prod U +{\rm c.c.})$) and in the ``adjoint'' theory ($S\propto\vert{\rm Tr}\prod U\vert^2$). Instead, the continuum limit of the model appears to be intriguingly similar to a $c>1$ string theory.