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. 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mtext>SU</a:mtext><a:mo>(</a:mo><a:mn>3</a:mn><a:mo>)</a:mo></a:mrow></a:math> 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.