Polylogarithms, Multiple Zeta Values and Superstring Amplitudes

TL;DR

The paper develops a comprehensive framework to compute tree-level open superstring amplitudes at arbitrary multiplicity and order in α', revealing a deep kinship between world-sheet disk integrals and color-ordered Yang-Mills subamplitudes. By organizing disk integrals into KK- and BCJ-like structures, and employing KLT-inspired representations alongside integration-by-parts, the authors identify multiple equivalent bases of integrals and systematically extract both pole residues and regular polylogarithmic contributions. Regular parts are evaluated using multiple polylogarithms and motivic multiple zeta values, enabling efficient α'-expansions up to high weights for N up to 7, with explicit constructions and parity/cyclicity shortcuts. The approach unifies field-theory patterns with string-theoretic corrections, offering a practical, scalable path to exact open-string amplitudes and exposing rich algebraic structures, including Hopf-algebraic descriptions of MZVs and their motivic lifts. This framework paves the way for deeper mathematical explorations and applications to string amplitudes and their transcendental structures.

Abstract

A formalism is provided to calculate tree amplitudes in open superstring theory for any multiplicity at any order in the inverse string tension. We point out that the underlying world-sheet disk integrals share substantial properties with color-ordered tree amplitudes in Yang-Mills field theories. In particular, we closely relate world-sheet integrands of open-string tree amplitudes to the Kawai-Lewellen-Tye representation of supergravity amplitudes. This correspondence helps to reduce the singular parts of world-sheet disk integrals -including their string corrections- to lower-point results. The remaining regular parts are systematically addressed by polylogarithm manipulations.

Figures (10)

• Figure 1: Each of the $N-2$ integration-by-parts equivalent representations of the basis functions $F_\Pi{}^\sigma$ can be mapped to another $(N-3)!$ basis of KK integrals $Z_\Pi(1,\ldots,N-1)$. Depending on the position $\nu+1$ of leg $N$ in the $Z_\Pi$, the transformation matrix is given by a product of $(\nu-1)$- and $(N-2-\nu)$-particle momentum kernels.
• Figure 4: Pole structure of the function $Z[s_{12}]$.
• Figure 5: Pole structure of the function $Z[s_{12}s_{34}]$.
• Figure 6: Pole structure of the function $Z[s_{12}s_{123}]$.
• Figure 7: Pole structure of the function $Z[s_{12}s_{123}s_{45}]$.