Path integral bosonization of the ‘t Hooft determinant: quasi-classical corrections

Abstract

The many-fermion Lagrangian which includes the ‘t Hooft six-quark flavor mixing interaction ( N f = 3) and the U L(3) x U R(3) chiral symmetric four-quark Nambu-Jona-Lasinio- (NJL-) type interactions is bosonized by the path integral method. The method of the steepest descents is used to derive the effective quark-mesonic Lagrangian with linearized many-fermion vertices. We obtain, additionally to the known lowest order stationary phase result of Reinhardt and Alkofer, the next to leading order (NLO) contribution arising from quantum fluctuations of auxiliary bosonic fields around their stationary phase trajectories (the Gaussian integral contribution). Using the gap equation we construct the effective potential, from which the structure of the vacuum can be settled. For some set of parameters the effective potential has several extrema, that in the case of SU(2) I x U(1) Y flavor symmetry can be understood on topological grounds. With increasing strength of the fluctuations the spontaneously broken phase gets unstable and the trivial vacuum is restored. The effective potential reveals furthermore the existence of logarithmic singularities at certain field expectation values, signalizing caustic regions.