# Feynman integrals and iterated integrals of modular forms

## TL;DR

This paper demonstrates that specific Feynman integrals can be represented as iterated integrals of modular forms, providing explicit formulas and alphabets for the sunrise and kite integrals, advancing mathematical understanding in quantum field theory.

## Contribution

It introduces a novel representation of Feynman integrals using iterated integrals of modular forms, including explicit formulas for key integrals.

## Key findings

- Feynman integrals expressed as iterated integrals of modular forms
- Explicit alphabet of letters for sunrise and kite integrals
- Compact all-orders formula for the sunrise integral

## Abstract

In this paper we show that certain Feynman integrals can be expressed as linear combinations of iterated integrals of modular forms to all orders in the dimensional regularisation parameter $\varepsilon$ . We discuss explicitly the equal mass sunrise integral and the kite integral. For both cases we give the alphabet of letters occurring in the iterated integrals. For the sunrise integral we present a compact formula, expressing this integral to all orders in $\varepsilon$ as iterated integrals of modular forms.