Phenomenological implications of a predictive formulation of the Nambu–Jona-Lasinio model having tensor couplings and isospin symmetry breaking terms

In the present work we consider the phenomenological consequences of a predictive formulation of the Nambu–Jona-Lasinio (NJL) model at the one loop level of perturbative calculations. The investigation reported here can be considered as an extension of previously made ones on the same issue. In the study made in this work we have included vector and tensor couplings, simultaneously, as well as $SU(2)$ isospin symmetry breaking terms. As a consequence of the last ingredient mentioned, there are different masses in the model amplitudes. In spite of this, within the context of the adopted procedure, we verify that it is possible to eliminate unphysical dependencies on the arbitrary choices for the routing of internal lines momenta as well as Ward identities violating contributions and scale ambiguous terms, from the corresponding one loop amplitudes, through the simple and universal Consistency Relations. The total content of divergence of the amplitudes is reduced to only two basic divergent objects. They are related to two inputs of the model in a way that, due to their scale properties, an unique arbitrariness remains. However, due to the critical condition found in the mechanism which generates the constituent quark mass, within our approach, this arbitrariness is also removed turning the model predictive in the sense that its phenomenological consequences is not dependent in possible choices made in intermediary steps of the calculations, as occurs in usual treatments. In this scenario, we investigate the most typical static properties of the scalar, pseudoscalar, vector and axial-vector mesons at low-energy. Special attention is given to the consequences of the $SU(2)$ isospin symmetry breaking for the phenomenological predictions. The implications of the tensor couplings for the model observables, which can be considered an original contribution of the present work, at the level of the content and not only in the form, is analyzed in a detailed way. The found values are in good accordance with the expectations and are globally consistent, having the obvious advantage that the predictions are not dependent in parameters aliens to the model Lagrangian as occurs in traditional approaches based in regularizations.

Figure 1

The possible physical solutions of Eq. (53) for $M$ as a function of $C$ for $f_{{\pi }^{+}}=92.4\mathrm{MeV}$, $m_{{\pi }^{+}}=139.6\mathrm{MeV}$, and $⟨\overline{\psi }\psi ⟩=(-250.0\mathrm{MeV}{)}^3$.

Figure 2

Behavior of $M_u$ and $M_d$ as function of $⟨\overline{d}d⟩$ keeping fix $⟨\overline{u}u⟩$.

Figure 3

Denominator of the charged pion propagator ($D_{{\pi }^{+}}$) as function of $p^2$ in the Euclidian region. Each curve, as is indicated in the legend, corresponds to different values of $G_V$ which are given in ${\mathrm{GeV}}^{-2}$.

Figure 4

Denominator of the charged rho propagator ($D_{{\rho }^{+}}$) as function of $p^2$ in the Euclidian region. Each curve, as is indicated in the legend, corresponds to different values of $G_V$ which are given in ${\mathrm{GeV}}^{-2}$.