Non-abelian $Z$-theory: Berends-Giele recursion for the $\alpha'$-expansion of disk integrals

TL;DR

This paper introduces a recursive computational method for the $ ext{α'}$-expansion of disk integrals in open string scattering, leveraging Berends-Giele recursion, and provides explicit results up to order $ ext{α'}^7$.

Contribution

It develops a novel recursive approach to compute the $ ext{α'}$-expansion of disk integrals in string theory, connecting it with $Z$-theory and providing explicit higher-order results.

Findings

Recursive method successfully computes $ ext{α'}$-expansion up to order $ ext{α'}^7$.

Connection established between Berends-Giele recursion and $Z$-theory equations of motion.

Explicit computational implementation made publicly available.

Abstract

We present a recursive method to calculate the $\alpha'$-expansion of disk integrals arising in tree-level scattering of open strings which resembles the approach of Berends and Giele to gluon amplitudes. Following an earlier interpretation of disk integrals as doubly partial amplitudes of an effective theory of scalars dubbed as $Z$-theory, we pinpoint the equation of motion of $Z$-theory from the Berends-Giele recursion for its tree amplitudes. A computer implementation of this method including explicit results for the recursion up to order $\alpha'^7$ is made available on the website http://repo.or.cz/BGap.git