Exploring transcendentality in superstring amplitudes

Abstract

Abstract It is well known that the low energy expansion of tree-level superstring scattering amplitudes satisfies a suitably defined version of uniform 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 $$\mathrm{\mathcal{M}}$$ ℳ R surrounding the cusp and its complement $$\mathrm{\mathcal{M}}$$ ℳ L , and an exact expression is obtained for the contribution to the amplitude from $$\mathrm{\mathcal{M}}$$ ℳ R . The low-energy expansion of the $$\mathrm{\mathcal{M}}$$ ℳ R contribution is proven to be free of irreducible multiple zeta-values to all orders. The contribution to the amplitude from $$\mathrm{\mathcal{M}}$$ ℳ L is computed in terms of modular graph functions up to order D 12 $$\mathrm{\mathcal{R}}$$ ℛ 4 in the low-energy expansion, and general arguments are used beyond this order to conjecture the transcendentality properties of the $$\mathrm{\mathcal{M}}$$ ℳ L contributions. Uniform transcendentality of the full amplitude holds provided we assign a non-zero weight to certain harmonic sum functions, an assumption which is familiar from transcendentality assignments in quantum field theory amplitudes.</jats:p