Low energy expansion of the four-particle genus-one amplitude in type II superstring theory

TL;DR

This paper develops a diagrammatic method to compute low-energy expansion coefficients of the genus-one four-particle amplitude in type II superstring theory, revealing their structure in terms of zeta values and analyzing their behavior under compactification.

Contribution

It introduces a novel diagrammatic expansion approach for genus-one amplitudes and computes coefficients up to order s^6R^4, including their rational and zeta value structure.

Findings

Coefficients are rational multiples of multiple zeta values.

Disentanglement of analytic and threshold terms is achieved.

Results provide boundary data for non-perturbative amplitude formulations.

Abstract

A diagrammatic expansion of coefficients in the low-momentum expansion of the genus-one four-particle amplitude in type II superstring theory is developed. This is applied to determine coefficients up to order s^6R^4 (where s is a Mandelstam invariant and R^4 the linearized super-curvature), and partial results are obtained beyond that order. This involves integrating powers of the scalar propagator on a toroidal world-sheet, as well as integrating over the modulus of the torus. At any given order in s the coefficients of these terms are given by rational numbers multiplying multiple zeta values (or Euler--Zagier sums) that, up to the order studied here, reduce to products of Riemann zeta values. We are careful to disentangle the analytic pieces from logarithmic threshold terms, which involves a discussion of the conditions imposed by unitarity. We further consider the compactification of the amplitude on a circle of radius r, which results in a plethora of terms that are power-behaved in r. These coefficients provide boundary `data' that must be matched by any non-perturbative expression for the low-energy expansion of the four-graviton amplitude. The paper includes an appendix by Don Zagier.