Recursion-free Solution for Two-Loop Vacuum Integrals with “collinear” Masses

We investigate the structure of a particular class of massive vacuum Feynman integrals at two loops. This class enjoys the linear relation m1 + m2 = m3 between its three propagator masses, corresponding to zeros of the associated Källén function. Apart from having applications in thermal field theory, the integrals can be mapped onto one-loop three-point functions with collinear external momenta, suggesting the term “collinear” masses. We present a closed-form solution for these integrals, proving that they can always be factorized into products of one-loop cases, for all integer-valued propagator powers.