Regularization of covariant calculations of meson decay amplitudes at one-loop order: Properties of thea0(980)resonance

The unitary quark model of Törnqvist and Roos provides an extremely interesting description of the properties of the scalar meson nonet. The model is quite phenomenological and has six parameters. In this work we are interested in seeing whether a theoretical foundation for the unitary quark model can be created using our generalized Nambu–Jona-Lasinio (NJL) model, which includes a covariant model of confinement. To make contact with Törnqvist’s work, it is necessary to go beyond the leading vacuum polarization diagram of the NJL model, ${J(P}^2),$ which is of order $n_c,$ and calculate the (complex) function ${K(P}^2),$ which is of order 1, and which describes the decay of $q\overline{q}$ states to the continuum of two-meson states. The central issue is the creation of a model for the regularization of the one-loop amplitudes with three vertices. We choose a regulator that has been used in our recent study of $f_0$ mesons. The same regularization is used for the πη, $K\overline{K},$ and $\pi {\eta }^{\prime }$ channels, when studying the properties of the $a_0\mathrm{}\left(980\right)\mathrm{}$ resonance. The coupling to the decay channels described by ${K(P}^2)$ is an important feature of the model, since the mass of the $a_0$ mass would be 1090 MeV in the absence of such coupling. The peak width obtained for the $a_0\mathrm{}\left(980\right)\mathrm{}$ resonance is 23 MeV, which is smaller than the values of 50–100 MeV usually quoted. (Larger values may be obtained in our calculation with another form of the regulator.) While our model is not parameter free, we believe we have made significant progress toward putting the unitary quark model on a firmer foundation.