# The vacuum seagull: evaluating a 3-loop Feynman diagram with 3 mass scales

## TL;DR

This paper analyzes a complex 3-loop Feynman integral with multiple mass scales using symmetry methods, deriving differential equations and explicit solutions in special functions, advancing computational techniques in quantum field theory.

## Contribution

It introduces a symmetry-based approach to evaluate a 3-loop Feynman diagram with multiple mass scales, including potential numerators, and provides explicit solutions.

## Key findings

- Determined the symmetry group G for the integral.
- Reduced the problem to a line integral over simpler diagrams.
- Presented explicit solutions in special functions for three mass scales.

## Abstract

We study a 3-loop 5-propagator Feynman Integral, which we call the vacuum seagull, with arbitrary masses and spacetime dimension using the Symmetries of Feynman Integrals method. It is our first example with potential numerators. We determine the associated group $G \subset GL(3)$ which happens to be 5 dimensional and the associated set of 5 differential equations. $G$ is determined by a geometric approach which we term "current freedom". We find the generic $G$-orbit to be co-dimension 0 and hence the method is maximally effective, and the diagram reduces to a line integral over simpler diagrams. For a reduced parameter space with 3 mass scales we are able to present explicit results in terms of special functions. This might be the first such example.