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Calculation of hadronization, decay or scattering processes at non-zerotemperatures and densities within the Nambu-Jona-Lasinio-like models requiressome techniques for computation of Feynmann diagrams. Decomposition of Feynmandiagrams at the one loop level leads to the appearance of elementary integralswith one, two, three, and four fermion lines. For example, evaluation of the$\pi\pi$ scattering amplitude requires calculating a box diagram with fourfermion lines. In this work, the real and imaginary parts of the box integralat the one loop level are provided in the form suitable for numericalevaluation. The obtained expressions are applicable to any value oftemperature, particle mass, and chemical potential. We pay special attention tothe conditions for the existence of the appearing improper integrals andcorrect the results \cite{Klevansky} for the three fermion lines. As a result,we have obtained constraints on possible values of particle momenta. Among theexpressions for the box integral, the general formulas for the integral with anarbitrary number of lines are derived for the case with zero or collinearfermion momenta.

Calculation of the one loop box integral at Finite Temperature and Density