FERMIONIC AND BOSONIC DETERMINANTS IN THE SEMIBOSONIZED NAMBU-JONA-LASINIO MODEL

For the semibosonized SU(2) Nambu-Jona-Lasinio model an expansion of t he classical action is made up to the second order in the fermionic an d bosonic fluctuations. Integrating out both types of fluctuations and performing the Legendre transform one obtains an effective action wit h classical meson fields and classical, anticommuting fermion fields. An explicit form for the effective potential is derived as well as exp ressions for the renormalization of the coupling constants, fermion ma ss and wave functions. The effects on meson and quark masses and the e ffective potential is of order N-c(-1) and numerically turn out to be small in the physical region of coupling constants. It is shown that o ne can obtain the contributions from exchange diagrams in four-fermion theories within the present path integral quantization. The results d iffer from investigations in the literature by the inclusion of wave f unction renormalization. The gradient expansion method, well known fro m fermion determinants, can be generalized such that it can be applied to bosonic determinants. If such an expansion is performed in the pre sent case one can identify terms which also appear in the Weinberg Lag rangian. Nevertheless altogether the influence of the bosonic loop on the axial vector coupling constant g(A) Of Constituent quarks vanishes and we obtain the classical value g(A) = 1. The contribution of the b osonic loop to the isosinglet axial vector current of the quarks is co nsidered for the case of a U(2)R X U(2)L symmetry and is actually nonv anishing.