# Wick rotations, Eichler integrals, and multi-loop Feynman diagrams

## TL;DR

This paper employs contour deformations and modular forms to evaluate complex Feynman integrals in two-dimensional quantum field theory, connecting them to modular $L$-series and confirming recent conjectures.

## Contribution

It introduces a novel approach combining contour deformations and modular forms to compute multi-loop Feynman diagrams explicitly.

## Key findings

- Explicit evaluation of certain Bessel moments as constants or $L$-series values
- Verification of recent conjectures by Broadhurst
- Establishment of a link between Feynman integrals and modular forms

## Abstract

Using contour deformations and integrations over modular forms, we compute certain Bessel moments arising from diagrammatic expansions in two-dimensional quantum field theory. We evaluate these Feynman integrals as either explicit constants or critical values of modular $L$-series, and verify several recent conjectures of Broadhurst.