Numerator Seagull and Extended Symmetries of Feynman Integrals

TL;DR

This paper extends the Symmetries of Feynman Integrals (SFI) method to include irreducible numerators, enabling new analytical solutions for complex seagull diagrams with multiple loops and masses.

Contribution

The paper introduces an extended SFI (xSFI) framework that incorporates irreducible numerators into the analysis of vacuum and propagator seagull diagrams.

Findings

Extended SFI system includes two new equations for numerators.

Derived closed-form solutions and epsilon expansions for specific mass configurations.

Achieved novel evaluations extending previous numerator-free results.

Abstract

The Symmetries of Feynman Integrals (SFI) method is extended for the first time to incorporate an irreducible numerator. This is done in the context of the so-called vacuum and propagator seagull diagrams, which have 3 and 2 loops, respectively, and both have a single irreducible numerator. For this purpose, an extended version of SFI (xSFI) is developed. For the seagull diagrams with general masses, the SFI equation system is found to extend by two additional equations. The first is a recursion equation in the numerator power, which has an alternative form as a differential equation for the generating function. The second equation applies only to the propagator seagull and does not involve the numerator. We solve the equation system in two cases: over the singular locus and in a certain 3 scale sector where we obtain novel closed-form evaluations and epsilon expansions, thereby extending previous results for the numerator-free case.