CONTINUUM LIMITS OF “ INDUCED QCD ” : LESSONS OF THE GAUSSIAN MODEL AT d = 1 AND BEYOND

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 → ∞, only the conventional large-N limit is relevant for its continuum limit. We also give a mean–field analysis of the N = 2 model in any d and show that a saddle point always exists in the region m 2 > m 2 crit (= d). In d = 1 it exhibits critical behaviour as m 2 → m 2 crit. 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 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.