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Low energy hadron phenomenology involving the (u,d,s) quarks is oftenapproached through effective multi-quark Lagrangians with the symmetries ofQCD. A very successful approach consists in taking the four-quarkNambu--Jona-Lasinio Lagrangian with the chiral $U_L(3)\times U_R(3)$ symmetryin the massless limit, combined with the $U_\A(1)$ breaking six-quark flavourdeterminant interaction of 't Hooft. We review the present status and some veryrecent developments related to the functional integration over the cubic termin auxiliary mesonic variables that one introduces to bosonize the system.Various approaches for handling this functional, which cannot be integratedexactly, are discussed: the stationary phase approximation, the perturbativeexpansion, the loop expansion, their interrelation and importance for theevaluation of the effective action. The intricate group structure rules out themethod of Airy's integral. The problem of the instability of the vacuum isstated and a solution given by including eight-quark interactions.

Functional Integral Approaches to the Bosonization of Effective Multi-Quark Interactions with UA(1) Breaking

Functional Integral Approaches to the Bosonization of Effective Multi-Quark Interactions with U_A(1) Breaking