The Nambu, Jona-Lasinio model and low energy effective meson Lagrangians

Using standard techniques, the Nambu-Jona-Lasinio Lagrangian with its chirally invariant four-fermion couplings is transformed into a bosonized form which includes scalar, pseudo-scalar, vector and axial-vector fields. Then, after eliminating the scalar and axialvector fields in an appropriate way, we obtain the Lagrangian given by Bando et al. based on the "hidden" local symmetry approach, except that in the present approach the free parameter a of their model is now uniquely given by a=(1- M_V²/M_A²)^-1 in terms of the vector and axial-vector meson masses M_V and M_A. The value of a=2, which led them to remarkable phenomenological successes is therefore directly connected with the Weinberg mass relation M_A=√2M_V in this model. The formal equivalence between the hidden symmetry approach and the massive Yang-Mills scheme is clearly demonstrated in the present approach, which derives an effective meson theory starting from a Lagrangian at the underlying quark level.