The Structure of n-Point One-Loop Open Superstring Amplitudes

TL;DR

This paper derives the worldsheet integrand for one-loop open superstring amplitudes with any number of massless states, revealing combinatorial structures that unify string and gauge theory results.

Contribution

It provides a detailed construction of the one-loop integrand for open superstrings with arbitrary external states, highlighting the combinatorial basis and its relation to gauge theory.

Findings

The integrand's polarization dependence uses BRST invariant kinematic combinations.

The basis size for kinematics and integrals is given by the Stirling number S_3^{n-1}.

The same combinatorial structures appear in pure Yang Mills theory and string amplitude color tensors.

Abstract

In this article we present the worldsheet integrand for one-loop amplitudes in maximally supersymmetric superstring theory involving any number n of massless open string states. The polarization dependence is organized into the same BRST invariant kinematic combinations which also govern the leading string correction to tree level amplitudes. The dimensions of the bases for both the kinematics and the associated worldsheet integrals is found to be the unsigned Stirling number S_3^{n-1} of first kind. We explain why the same combinatorial structures govern on the one hand finite one-loop amplitudes of equal helicity states in pure Yang Mills theory and on the other hand the color tensors at quadratic alpha prime order of the color dressed tree amplitude.