The Migdal Recursion Equation as a Probe for Crossover Points: Lattice Gauge Theory on Discrete Groups

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Abstract
Four dimensional lattice gauge theories (LGT) on discrete groups, Z(N) and the subgroups of SU(2), Q, ~T, ~O and ~I, are investigated on the basis of the Migdal approximation. In the recursion equation, the partition function is given in somewhat general way by the irreducible characters. For Z(2), Z(3) and Z(4) LGT the critical point βc is exactly in agreement with the value estimated from the self duality. For Z(N) (N ≥5)LGT, βc is nearly 1 and almost independent of N. As for the subgroups of SU(2), βc is near a first order critical point of Monte Carlo simulations for Q and ~T and βc is near the crossover point (β≃2) for ~O and ~I. The application of a similar parametrization of the partition function to continuous groups, U(1) and SU(2), results in βc≃0.92 and 1.9, respectively. The latter value is very close to the crossover point (βc ≃2) in Creutz's result. The relation between our results and the roughening transition is discussed.

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