't Hooft Determinant: fluctuations and the structure of the vacuum

摘要

The't Hooft six quark flavor mixing interaction (N_f = 3) is bosonized by the path integral method. The considered complete Lagrangian is constructed on the basis of the combined 't Hooft and U(3) x U(3) extended chiral four fermion Nambu ― Jona-Lasinio interactions. The method of the steepest descents is used to derive the effective mesonic Lagrangian. Additionally to the known lowest order stationary phase (SP) result of Reinhardt and Alkofer we obtain the contribution from the small quantum fluctuations of bosonic configurations around their stationary phase trajectories. Fluctuations give rise to multivalued solutions of the gap equations. Using the gap equations we construct the effective potential, from which the structure of the vacuum can be settled. We obtain that from the several extremal solutions, only one is a minimum. The others are either maxima or a saddle point. The effective potential reveals furthermore the existence of logartithmically divergent attractive wells at certain field expectation values, signalizing caustic regions.