Pure spinor computation towards open string three-loop

TL;DR

This paper explores the computation of open superstring three-loop amplitudes using the pure spinor formalism, revisiting lower-loop results and testing regulators, with indications that the three-loop amplitude may vanish or be obtainable.

Contribution

It introduces a method to compute three-loop open superstring amplitudes in the pure spinor framework, analyzing regulator effects and proposing a potential three-loop amplitude.

Findings

Two-loop amplitude matches previous symmetry-based results.

Using the AB regulator suggests the three-loop amplitude may vanish.

The proposed three-loop amplitude is a plausible candidate based on current analysis.

Abstract

Using the recent results in the pure spinor formulation, we lay out a ground-work towards the full momentum space amplitudes of open superstrings at three-loop. After briefly reviewing the one-loop amplitude, we directly work out the two-loop and reproduce the result that was obtained by a symmetry argument. For the three-loop, first we use the two-loop regulator as a warm-up exercise. The result vanishes. We then employ the regulator that has been recently proposed by Aisaka and Berkovits (AB). It is noted that the terms in higher power in $\frac{1}{\lambda\bar{\lambda}}$ that render the two-loop regulator disqualified for the three-loop do not contribute. This with a few other indications suggests a possibility that the AB regulator might also lead to a vanishing result. Nevertheless, we argue that it is possible to acquire the three-loop amplitude, and present a result that we anticipate to be the three-loop amplitude.