Finely tuned Nambu–Jona-Lasinio and linear σ models

It is argued anew that the Nambu–Jona-Lasinio (NJL) model is renormalizable in the sense that all cutoff dependence (ignoring inverse powers of the cutoff) can be absorbed into a renormalized mass parameter and renormalized Yukawa and four-scalar coupling constants G and λ, respectively. These couplings are not independent, although the relation between them depends upon subtle details of the cutoff procedure. At energies very much less than the cutoff the NJL model is equivalent to the linear σ model. From the renormalization group equations of the latter model, it is shown that at scales very much less than the cutoff the couplings G and λ are small and the ratio λ/${\mathit{G}}^2$ is approximately determined to within an error which for NN= number of ‘‘colors’’) is less than the corrections coming from one loop. The average σ field v, the σ mass ${\mathit{M}}_{\mathrm{\sigma }}$, and the fermion mass m are calculated in the one-loop (fermion or scalar) approximation.