Feynman integrals, elliptic integrals and two-parameter K3 surfaces

TL;DR

This work connects two multi-loop Feynman integrals to two-parameter families of $K3$ surfaces by computing their periods and mirror maps, showing they admit representations in terms of modular forms and complete elliptic integrals. The conformal two-loop $\text{traintrack}$ is analyzed through an Appell $F_2$ system, yielding a modular parametrisation on a one-parameter slice and then a full two-parameter form that makes a sunrise-elliptic symmetry manifest. The three-loop banana with three equal masses is shown to factorise into a product of two elliptic curves; its holomorphic period, mirror map, and modular parametrisation are expressed via Hauptmoduls for suitable congruence subgroups, exposing a hidden symmetry exchanging the two elliptic components. Together, the results provide constructive modular descriptions of $K3$-periods attached to Feynman integrals and suggest a pathway to fully analytic, iterated-modular representations of these and related integrals.

Abstract

The three-loop banana integral with three equal masses and the conformal two-loop five-point traintrack integral in two dimensions are related to a two-parameter family of K3 surfaces. We compute the corresponding periods and the mirror map, and we show that they can be expressed in terms of ordinary modular forms and functions. In particular, we find that the maximal cuts of the three-loop banana integral with three equal masses can be written as a product of two copies of the maximal cuts of the two-loop equal-mass sunrise integral. Our computation reveals a hidden symmetry of the banana integral not manifest from the Feynman integral representation, which corresponds to exchanging the two copies of the sunrise elliptic curve.

Figures (2)

• Figure 1: The conformal two-loop traintrack integral.
• Figure 2: The three-loop banana graph with three equal masses.