A new approach to the 3-momentum regularization of the in-medium one and two fermion line integrals with applications to cross sections in the Nambu--Jona-Lasinio model

We propose the 3-momentum sphere intersection regularization applied to the one and two fermion line integrals at finite temperature and chemical potential. The quark-antiquark polarization function in this new regularization approach is equivalent to the usual 3-momentum regularization, when the absolute value of the external 3-momentum of the polarization is zero. Additionally, it respects the particle-antiparticle symmetry of meson states in the Nambu$-$Jona-Lasinio (NJL) model for all values of temperature and chemical potential. Without this symmetry, in-medium cross sections calculated in the 3-momentum regularized NJL model are not consistent. In order to demonstrate the difference between the usual 3-momentum regularization with the one proposed in this work, we study the quark-quark and quark-antiquark cross sections in both regularization schemes. To this end we use the standard $SU(3)$ NJL model, with four and six quark interactions. We observe major quantitative and qualitative differences when comparing quark-quark cross sections in both schemes. The quark-antiquark cross sections, on the other hand, are very similar in both regularizations, owning to the equivalence between the regularizations when the absolute value of the external 3-momentum is zero.