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On a class of Feynman integrals evaluating to iterated integrals of modular forms

Luise Adams, Stefan Weinzierl

TL;DR

The work identifies a sub-class of Feynman integrals whose all-order $\varepsilon$-expansion can be expressed as iterated integrals of modular forms associated with a single elliptic curve and a single scale $x$. By transforming their differential equations to an $\varepsilon$-form with modular kernels, these integrals admit efficient evaluation via $q$-expansions and exhibit a clear elliptic-curve structure revealed through Picard-Fuchs analysis and maximal cuts. The paper details how to construct master integrals, identify the elliptic curve, and formulate the problem in terms of modular forms with weights $2$, $3$, and $4$ for specific congruence subgroups, using a sunrise-like two-loop example. This approach extends the scope of analytically tractable Feynman integrals beyond harmonic polylogarithms and provides a practical framework for numerical precision calculations in single-scale, elliptic cases.

Abstract

In this talk we discuss a class of Feynman integrals, which can be expressed to all orders in the dimensional regularisation parameter as iterated integrals of modular forms. We review the mathematical prerequisites related to elliptic curves and modular forms. Feynman integrals, which evaluate to iterated integrals of modular forms go beyond the class of multiple polylogarithms. Nevertheless, we may bring for all examples considered the associated system of differential equations by a non-algebraic transformation to an $\varepsilon$-form, which makes a solution in terms of iterated integrals immediate.

On a class of Feynman integrals evaluating to iterated integrals of modular forms

TL;DR

The work identifies a sub-class of Feynman integrals whose all-order -expansion can be expressed as iterated integrals of modular forms associated with a single elliptic curve and a single scale . By transforming their differential equations to an -form with modular kernels, these integrals admit efficient evaluation via -expansions and exhibit a clear elliptic-curve structure revealed through Picard-Fuchs analysis and maximal cuts. The paper details how to construct master integrals, identify the elliptic curve, and formulate the problem in terms of modular forms with weights , , and for specific congruence subgroups, using a sunrise-like two-loop example. This approach extends the scope of analytically tractable Feynman integrals beyond harmonic polylogarithms and provides a practical framework for numerical precision calculations in single-scale, elliptic cases.

Abstract

In this talk we discuss a class of Feynman integrals, which can be expressed to all orders in the dimensional regularisation parameter as iterated integrals of modular forms. We review the mathematical prerequisites related to elliptic curves and modular forms. Feynman integrals, which evaluate to iterated integrals of modular forms go beyond the class of multiple polylogarithms. Nevertheless, we may bring for all examples considered the associated system of differential equations by a non-algebraic transformation to an -form, which makes a solution in terms of iterated integrals immediate.

Paper Structure

This paper contains 9 sections, 89 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Periodic functions and periods
  3. Elliptic curves
  4. Modular forms
  5. Iterated integrals
  6. Precision calculations
  7. Picard-Fuchs operators
  8. Feynman integrals evaluating to iterated integrals of modular forms
  9. Conclusions

Figures (3)

  • Figure 1: The periods $(1,\tau)$ and $(1,\tau')$ generate the same lattice.
  • Figure 2: If for some exponent we have $\nu_j=0$, the corresponding propagator is absent and the topology simplifies.
  • Figure 3: Examples of Feynman integrals evaluating to iterated integrals of modular forms. Internal solid lines correspond to a propagator with mass $m^2$, internal dashed lines to a massless propagator. External dashed lines indicate a light-like external momentum.