SIRENA -- Sum-Integral REductioN Algorithm

TL;DR

SIRENA is a software package that automates the reduction of multi-loop sum-integrals in finite-temperature quantum field theory, extending existing techniques to higher loops and fermionic cases.

Contribution

It introduces a new implementation of the Laporta algorithm for finite-temperature sum-integrals, including novel reductions for 3-loop fermionic integrals and an analytic factorization formula for 2-loop fermionic integrals.

Findings

Successfully reproduces known 3-loop results

Provides first-time reductions for 3-loop fermionic sum-integrals

Derives an analytic factorization formula for 2-loop fermionic integrals

Abstract

We present SIRENA, a Python and C++ implementation of the Laporta algorithm for the automatic reduction of multi-loop sum-integrals via integration-by-parts identities. The method builds on established techniques for zero-temperature Feynman integrals and extends them to finite-temperature quantum field theory by consistently accounting for the Matsubara sum structure. We validate the framework by reproducing several known results from the literature up to 3-loop order, and we further provide, for the first time, reductions for selected 3-loop fermionic sum-integrals. In addition to the package, we derive an analytic factorization formula for arbitrary 2-loop fermionic sum-integrals, extending on a previous result for the bosonic case.