Integrating three-loop modular graph functions and transcendentality of string amplitudes

TL;DR

This paper evaluates three-loop modular graph functions' integrals in string theory, revealing that their transcendental structure aligns with uniform transcendentality up to weight seven, but deviations at higher weights involve Eisenstein series products, challenging existing conjectures.

Contribution

The work explicitly computes three-loop modular graph function integrals and analyzes their transcendentality, providing new insights into the structure of string amplitude expansions.

Findings

Up to weight seven, integrals are consistent with uniform transcendentality.

At weight eight and above, deviations involve triple products of Eisenstein series.

Violations of uniform transcendentality may cancel in specific combinations relevant to string amplitudes.

Abstract

Modular graph functions (MGFs) are $\mathrm{SL}(2,\mathbb{Z})$-invariant functions on the Poincar\'e upper half-plane associated with Feynman graphs of a conformal scalar field on a torus. The low-energy expansion of genus-one superstring amplitudes involves suitably regularized integrals of MGFs over the fundamental domain for $\mathrm{SL}(2,\mathbb{Z})$. In earlier work, these integrals were evaluated for all MGFs up to two loops and for higher loops up to weight six. These results led to the conjectured uniform transcendentality of the genus-one four-graviton amplitude in Type II superstring theory. In this paper, we explicitly evaluate the integrals of several infinite families of three-loop MGFs and investigate their transcendental structure. Up to weight seven, the structure of the integral of each individual MGF is consistent with the uniform transcendentality of string amplitudes. Starting at weight eight, the transcendental weights obtained for the integrals of individual MGFs are no longer consistent with the uniform transcendentality of string amplitudes. However, in all the cases we examine, the violations of uniform transcendentality take on a special form given by the integrals of triple products of non-holomorphic Eisenstein series. If Type II superstring amplitudes do exhibit uniform transcendentality, then the special combinations of MGFs which enter the amplitudes must be such that these integrals of triple products of Eisenstein series precisely cancel one another. Whether this indeed is the case poses a novel challenge to the conjectured uniform transcendentality of genus-one string amplitudes.