Exploring transcendentality in superstring amplitudes

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 ${\mathcal{M}}_R$ surrounding the cusp and its complement ${\mathcal{M}}_L$, and an exact expression is obtained for the contribution to the amplitude from ${\mathcal{M}}_R$. The low-energy expansion of the ${\mathcal{M}}_R$ contribution is proven to be free of irreducible multiple zeta-values to all orders. The contribution to the amplitude from ${\mathcal{M}}_L$ is computed in terms of modular graph functions up to order $D^{12}{\mathcal{R}}^4$ in the low-energy expansion, and general arguments are used beyond this order to conjecture the transcendentality properties of the ${\mathcal{M}}_L$ 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.

65 pages, 4 figures; typos corrected, reference added, minor edits in version 2; factor of 4 corrected in theorem 4.1 in version 3

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