Superstring loop amplitudes from the field theory limit
Yvonne Geyer, Ricardo Monteiro, Ricardo Stark-Muchão
TL;DR
The paper tackles constructing loop-level moduli-space integrands for the type II superstring from the field-theory limit. Its core approach fuses BCJ double-copy gravity data with ambitwistor-string localization on a nodal sphere and modular uplift to higher genus, enabling a direct link between field theory amplitudes and string amplitudes. The authors reproduce the known two-loop, four-point result and present a three-loop conjecture for the chiral half-integrand that aligns with a proposed genus-3 chiral measure, supported by detailed degeneration analyses. This work provides a practical framework to import high-loop supergravity results into string computations and suggests pathways to extend the method to heterotic strings and higher-point amplitudes. The results hinge on modular invariance, hyperelliptic degeneration, and the structure of the genus-3 chiral measure, illustrating a fruitful bridge between field theory techniques and string-theoretic amplitudes.
Abstract
We propose a procedure to determine the moduli-space integrands of loop-level superstring amplitudes for massless external states in terms of the field theory limit. We focus on the type II superstring. The procedure is to: (i) take a supergravity loop integrand written in a BCJ double-copy representation, (ii) use the loop-level scattering equations to translate that integrand into the ambitwistor string moduli-space integrand, localised on the nodal Riemann sphere, and (iii) uplift that formula to one on the higher-genus surface valid for the superstring, guided by modular invariance. We show how this works for the four-point amplitude at two loops, where we reproduce the known answer, and at three loops, where we present a conjecture that is consistent with a previous proposal for the chiral measure. Useful supergravity results are currently known up to five loops.