On the Feynman Graph Integrals on Elliptic Curves

TL;DR

This paper develops a framework for regularized integrals on elliptic curves, explores their holomorphic anomaly, and provides a general computational method for these graph integrals.

Contribution

It introduces regularized graph integrals on elliptic curves and formulates a holomorphic anomaly equation, offering a new approach to compute these integrals.

Findings

Regularized integrals produce almost holomorphic functions.

Holomorphic anomaly equation describes anti-holomorphic dependence.

A general computational method for graph integrals is provided.

Abstract

We introduce the regularized integrals for decorated graphs on elliptic curves, which produces an almost holomorphic function on upper half plane. Then we give the graph version of holomorphic anomaly equation to study the anti-holomorphic dependence of the graph integral. Finally, we give a general method to compute the graph integral.