The propagator seagull: general evaluation of a two loop diagram

TL;DR

This paper evaluates a complex two-loop propagator diagram using the SFI method, providing general formulas, symmetry analysis, and explicit solutions across different energy scales and dimensions.

Contribution

It introduces the SFI group and equations for this diagram, offering a comprehensive analysis and explicit expressions in various parameter regimes.

Findings

Derived general solutions for the two-loop propagator diagram.

Identified the singular locus and diagram values on it.

Provided explicit formulas in different energy scale sectors.

Abstract

We study a two loop diagram of propagator type with general parameters through the Symmetries of Feynman Integrals (SFI) method. We present the SFI group and equation system, the group invariant in parameter space and a general representation as a line integral over simpler diagrams. We present close form expressions for three sectors, each with three or four energy scales, for any spacetime dimension $d$ as well as the $\epsilon$ expansion. We determine the singular locus and the diagram's value on it.