Functional Integral Approaches to the Bosonization of Effective Multi-Quark Interactions with $U_{\mathrm{A}}(1)$ Breaking

Low energy hadron phenomenology involving the (u,d,s) quarks is often approached through effective multi-quark Lagrangians with the symmetries of QCD. A very successful approach consists in taking the four-quark Nambu–Jona-Lasinio Lagrangian with the chiral $U_L(3)\times U_R(3)$ symmetry in the massless limit, combined with the $U_{\mathrm{A}}(1)$ breaking six-quark flavour determinant interaction of 't Hooft. We review the present status and some very recent developments related to the functional integration over the cubic term in auxiliary mesonic variables that one introduces to bosonize the system. Various approaches for handling this functional, which cannot be integrated exactly, are discussed: the stationary phase approximation, the perturbative expansion, the loop expansion, their interrelation and importance for the evaluation of the effective action. The intricate group structure rules out the method of Airy's integral. The problem of the instability of the vacuum is stated and a solution given by including eight-quark interactions.