Two-Loop Superstrings in Hyperelliptic Language III: the Four-Particle Amplitude

TL;DR

This paper explicitly computes the four-particle amplitude in superstring theories using hyperelliptic language, revealing a simple, insertion point-independent expression and demonstrating the vanishing of certain perturbative corrections at two loops.

Contribution

It provides a new explicit calculation of the four-particle amplitude in superstring theory using hyperelliptic methods and confirms the vanishing of specific two-loop corrections.

Findings

Derived a simple expression for the four-particle amplitude.

Proved the amplitude's independence from insertion points.

Showed the perturbative correction to the R^4 term vanishes at two loops.

Abstract

We compute explicitly the four-particle amplitude in superstring theories by using the hyperelliptic language and the newly obtained chiral measure of D'Hoker and Phong. Although the algebra of the intermediate steps is a little bit involved, we obtain a quite simple expression for the four-particle amplitude. As expected, the integrand is independent of all the insertion points. As an application of the obtained result, we show that the perturbative correction to the $R^4$ term in type II superstring theories is vanishing point-wise in (even) moduli space at two loops.