Recursion-free solution for two-loop vacuum integrals with "collinear" masses

TL;DR

This paper presents a closed-form, recursion-free solution for a specific class of two-loop vacuum integrals with collinear masses, simplifying their evaluation by factorizing them into products of one-loop integrals.

Contribution

The authors derive a novel closed-form solution for two-loop vacuum integrals with collinear masses, enabling straightforward factorization into one-loop integrals for all integer propagator powers.

Findings

Integrals can be expressed in closed form without recursion.

Factorization into one-loop integrals is always possible.

Applicable to integrals with mass relation m1 + m2 = m3.

Abstract

We investigate the structure of a particular class of massive vacuum Feynman integrals at two loops. This class enjoys the linear relation $m_1+m_2=m_3$ between its three propagator masses, corresponding to zeros of the associated K\"all\'en 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.