Continuum Limits of ``Induced QCD": Lessons of the Gaussian Model at d=1 and Beyond

TL;DR

This paper investigates the continuum limits of the Kazakov--Migdal model of induced QCD, focusing on the scalar field sector in one dimension and beyond, revealing the role of Gaussian and non-Gaussian terms in critical behavior and continuum limit formation.

Contribution

It provides a detailed mean-field analysis of the scalar sector in the Kazakov--Migdal model across different dimensions, highlighting the conditions for critical behavior and continuum limits.

Findings

Large-N limit is relevant for the continuum limit in d=1.

Critical behavior occurs at m^2 approaching m_crit^2 in d=1.

Adding a logarithmic term to the scalar potential induces critical behavior in d>1.

Abstract

We analyze the scalar field sector of the Kazakov--Migdal model of induced QCD. We present a detailed description of the simplest one dimensional {($d$$=$$1$)} model which supports the hypothesis of wide applicability of the mean--field approximation for the scalar fields and the existence of critical behaviour in the model when the scalar action is Gaussian. Despite the ocurrence of various non--trivial types of critical behaviour in the $d=1$ model as $N\rightarrow\infty$, only the conventional large-$N$ limit is relevant for its {\it continuum} limit. We also give a mean--field analysis of the $N=2$ model in {\it any} $d$ and show that a saddle point always exists in the region $m^2>m_{\rm crit}^2(=d)$. In $d=1$ it exhibits critical behaviour as $m^2\rightarrow m_{\rm crit}^2$. However when $d$$>$$1$ there is no critical behaviour unless non--Gaussian terms are added to the scalar field action. We argue that similar behaviour should occur for any finite $N$ thus providing a simple explanation of a recent result of D. Gross. We show that critical behaviour at $d$$>$$1$ and $m^2>m^2_{\rm crit}$ can be obtained by adding a $logarithmic$ term to the scalar potential. This is equivalent to a local modification of the integration measure in the original Kazakov--Migdal model. Experience from previous studies of the Generalized Kontsevich Model implies that, unlike the inclusion of higher powers in the potential, this minor modification should not substantially alter the behaviour of the Gaussian model.