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

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 $N\to \infty$, 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 $m^2>m_{\mathrm{crit}}^2(=d)$. In $d=1$ it exhibits critical behaviour as $m^2\to m_{\mathrm{crit}}^2$. 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 $m^2>m_{\mathrm{crit}}^2$ 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.

31 pages

References in corpus (9)

• Induced QCD at Large N
• One-Dimensional String Theory with Vortices as Upside-Down Matrix Oscillator
• Exact Solution of Induced Lattice Gauge Theory at Large $N$
• Area Law and Continuum Limit in "Induced QCD"
• Induced QCD and Hidden Local ZN Symmetry
• Exact solution of d=1 Kazakov-Migdal induced gauge theory
• 1/N Expansion and Particle Spectrum in Induced QCD
• The Problem of Large-N Phase Transition in Kazakov-Migdal Model of Induced QCD
• Large-N Reduction, Master Field and Loop Equations in Kazakov-Migdal Model

Cited by in corpus (10)

• Integrability and Matrix Models
• Generation of Matrix Models by W-operators
• BGWM as Second Constituent of Complex Matrix Model
• The Kazakov-Migdal Model as a High Temperature Lattice Gauge Theory
• Kazakov--Migdal Model with Logarithmic Potential and the Double Penner Matrix Model
• A Study of the $N=2$ Kazakov-Migdal Model
• Gross-Witten-Wadia Phase Transition in Induced QCD on the Graph
• Symmetry and Observables in Induced QCD
• Large Representation Recurrences in Large N Random Unitary Matrix Models
• Itzykson-Zuber correlators from character expansion