Pure spinor computation towards open string three-loop

Abstract

Using the recent results in the pure spinor formulation, we lay out a groundwork 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 amplitude and reproduce the result that was obtained by a symmetry argument before. 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. It is noted that the terms in higher power in $ \frac{1}{\uplambda \uplambda } $ 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.