Parity-conserving extended Nambu and Jona-Lasinio model in terms of Abelian gauge field with a non-zero bare mass

We develop a dynamical theory of spontaneous symmetry breakdown of global chiral gauge invariance in terms of Abelian gauge field with a non-zero bare mass based on the parity-conserving interaction f anti phi (x) γ5γ sup(μ) phi(x)A sub(μ) (x) in the framework of manifestly covariant indefinite-metric quantum field theory. The formula 1-m02L>0 and the mass formula L=Zsub(chi)(M2+m02Zsub(chi))-1=Zm-2+∫sub(+0)sup(infinity)dss-1σ(s) are derived where M=2f<0+anti phi(x)phi(x)+0>not equal 0 and m0, m, Zsub(chi),Z and σ(s) are a non-zero bare mass of Abelian gauge field A sub(μ)(x), a physical mass of A sub(μ)(x), a wave-function renormalization constant of Goldstone boson field, a wave-function renormalization constant of A sub(μ)(x) and the spectral function of A sub(μ)(x), respectively. Our theory indicates that the Goldstone boson field always has a positive norm. (author)