Explicit Modular Invariant Two-Loop Superstring Amplitude Relevant for R^4

TL;DR

This paper derives an explicit, convergent, modular invariant formula for the two-loop four-point superstring amplitude, focusing on the R^4 correction in the effective action, with implications for understanding superstring interactions.

Contribution

It provides the first explicit modular invariant formula for the two-loop four-point superstring amplitude in terms of a complex integral, relevant for R^4 corrections.

Findings

Derived a convergent, modular invariant integral formula

Focused on the leading term of the graviton amplitude at zero momenta

The expression may be zero but is not proven to vanish

Abstract

In this note we derive an explicit modular invariant formula for the two loop 4-point amplitude in superstring theory in terms of a multiple integral (7 complex integration variables) over the complex plane which is shown to be convergent. We consider in particular the case of the leading term for vanishing momenta of the four graviton amplitude, which would correspond to the two-loop correction of the R^4 term in the effective Action. The resulting expression is not positive definite and could be zero, although we cannot see that it vanishes.