The setting sun diagram with complex external momenta

TL;DR

This paper explores the analytic continuation of Feynman integrals with complex external momenta, focusing on the setting sun diagram in two dimensions, and discusses alternative approaches beyond the traditional Källén-Lehmann representation.

Contribution

It provides an analysis of analytic continuation methods for Feynman integrals with complex momenta, highlighting potential differences from the standard Källén-Lehmann approach.

Findings

Analysis of the setting sun diagram in 2D with real mass

Discussion of alternative continuation methods

Implications for complex mass cases

Abstract

We revisit the issue of analytically continuing Feynman integrals from Euclidean to Minkowski signature, allowing for generic complex momenta. Although this is well-known in terms of the K\"all\'{e}n-Lehmann representation, we consider potential alternative takes on the same problem and discuss how these are not necessarily equivalent to the K\"all\'en-Lehmann integral outcome. We present our analysis for a simple enough case -- the setting sun diagram in $d=2$ with a real mass -- but already with an eye out to the more general case with complex masses which will further complicate matters.