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Feynman integrals and iterated integrals of modular forms

Luise Adams, Stefan Weinzierl

TL;DR

This paper tackles the challenge of expressing elliptic Feynman integrals beyond multiple polylogarithms by recasting equal-mass sunrise and kite integrals as linear combinations of iterated integrals of modular forms, valid to all orders in the dimensional regulator ε. The authors establish a framework using modular forms on congruence subgroups, Eichler integrals, and eta quotients to encode integration kernels, and derive a compact all-orders formula for the sunrise integral through a generating-function approach. They present two elliptic-curve routes—one from the Feynman parameter representation yielding Γ_1(12) forms and another from the maximal cut yielding Γ_1(6) forms—related by a quadratic transformation, and show how period choices govern the modular structure. The results connect high-precision Feynman integral evaluations with modular-forms techniques, enabling systematic all-orders constructions and suggesting deeper coaction structures in elliptic settings.

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.

Feynman integrals and iterated integrals of modular forms

TL;DR

This paper tackles the challenge of expressing elliptic Feynman integrals beyond multiple polylogarithms by recasting equal-mass sunrise and kite integrals as linear combinations of iterated integrals of modular forms, valid to all orders in the dimensional regulator ε. The authors establish a framework using modular forms on congruence subgroups, Eichler integrals, and eta quotients to encode integration kernels, and derive a compact all-orders formula for the sunrise integral through a generating-function approach. They present two elliptic-curve routes—one from the Feynman parameter representation yielding Γ_1(12) forms and another from the maximal cut yielding Γ_1(6) forms—related by a quadratic transformation, and show how period choices govern the modular structure. The results connect high-precision Feynman integral evaluations with modular-forms techniques, enabling systematic all-orders constructions and suggesting deeper coaction structures in elliptic settings.

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 . 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 as iterated integrals of modular forms.

Paper Structure

This paper contains 21 sections, 3 theorems, 244 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review of basic definitions and facts
  3. Modular forms
  4. Modular forms for congruence subgroups
  5. Dirichlet characters
  6. Eisenstein series
  7. An example
  8. Eta quotients
  9. Iterated integrals of modular forms
  10. Eichler integrals
  11. The two-loop sunrise integral
  12. The elliptic curve
  13. Integration kernels
  14. Modular transformations of the first term in the $\varepsilon$-expansion
  15. A compact expression to all orders
  16. ...and 6 more sections

Key Result

Theorem 2.1

Suppose $K$ is a positive integer, the Dirichlet characters $\phi$, $\psi$ are as above and $k$ is a positive integer such that $\phi(-1) \psi(-1) = (-1)^k$. For $k=1$ we require in addition $\phi(-1)=1$ and $\psi(-1)=-1$. Except when $k=2$ and $\phi=\psi=1$, the Eisenstein series $E_k(K \tau;\phi,\

Figures (3)

  • Figure 1: A set of master integrals for the sunrise family. A dot on a propagator indicates, that this propagator is raised to the power two.
  • Figure 2: The kite integral. Solid lines correspond to massive propagators, dashed lines to massless propagators.
  • Figure 3: The lattices generated by the periods $(\psi_{1},\psi_{2})$ (left) and $(\psi_{1,C},\psi_{2,C})$ (right). The lattice generated by $(\psi_{1},\psi_{2})$ is rectangular for $t<0$. It is a sub-lattice of the one generated by $(\psi_{1,C},\psi_{2,C})$.

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3