Towards the n-point one-loop superstring amplitude III: One-loop correlators and their double-copy structure

TL;DR

This paper constructs supersymmetric one-loop correlators in pure-spinor superspace, revealing a Lie polynomial-like symmetry and exploring their double-copy structure, with explicit results up to seven points and insights into higher-point correlators.

Contribution

It introduces a novel symmetry structure in one-loop correlators and provides explicit expressions up to seven points, advancing the understanding of superstring amplitude calculations.

Findings

Correlators exhibit Lie polynomial-like symmetry structure.

Explicit one-loop correlators are presented up to seven points.

The eight-point correlator captures worldsheet dependence but leaves some coefficients undetermined.

Abstract

In this final part of a series of three papers, we will assemble supersymmetric expressions for one-loop correlators in pure-spinor superspace that are BRST invariant, local, and single valued. A key driving force in this construction is the generalization of a so far unnoticed property at tree level; the correlators have the symmetry structure akin to Lie polynomials. One-loop correlators up to seven points are presented in a variety of representations manifesting different subsets of their defining properties. These expressions are related via identities obeyed by the kinematic superfields and worldsheet functions spelled out in the first two parts of this series and reflecting a duality between the two kinds of ingredients. Interestingly, the expression for the eight-point correlator following from our method seems to capture correctly all the dependence on the worldsheet punctures but leaves undetermined the coefficient of the holomorphic Eisenstein series ${\rm G}_4$. By virtue of chiral splitting, closed-string correlators follow from the double copy of the open-string results.