# Exploring transcendentality in superstring amplitudes

## TL;DR

This paper extends the concept of maximum transcendentality to genus-one superstring amplitudes, demonstrating that certain contributions are free of complex zeta-values and conjecturing the transcendentality properties of remaining parts.

## Contribution

It introduces a natural extension of transcendentality to genus-one superstring amplitudes and proves the absence of irreducible multiple zeta-values in a key region.

## Key findings

- The ${
m M}_R$ contribution is free of irreducible multiple zeta-values.
- The ${
m M}_L$ contribution is computed up to order $D^{12} {
m R}^4$ in the low-energy expansion.
- Maximal transcendentality holds with a specific harmonic sum weight assignment.

## Abstract

It is well known that the low energy expansion of tree-level superstring scattering amplitudes satisfies a suitably defined version of maximum transcendentality. In this paper it is argued that there is a natural extension of this definition that applies to the genus-one four-graviton Type II superstring amplitude to all orders in the low-energy expansion. To obtain this result, the integral over the genus-one moduli space is partitioned into a region ${\cal M}_R$ surrounding the cusp and its complement ${\cal M}_L$, and an exact expression is obtained for the contribution to the amplitude from ${\cal M}_R$. The low-energy expansion of the ${\cal M}_R$ contribution is proven to be free of irreducible multiple zeta-values to all orders. The contribution to the amplitude from ${\cal M}_L$ is computed in terms of modular graph functions up to order $D^{12} {\cal R}^4$ in the low-energy expansion, and general arguments are used beyond this order to conjecture the transcendentality properties of the ${\cal M}_L$ contributions. Maximal transcendentality of the full amplitude holds provided we assign a non-zero weight to certain harmonic sums, an assumption which is familiar from transcendentality assignments in quantum field theory amplitudes.