Superstring amplitudes from the field-theory limit: an n-point map at one loop
Ricardo Monteiro, Lecheng Ren
TL;DR
This paper develops a complete genus-one, $n$-point worldsheet basis for one-loop superstring correlators, showing that the non-cusp part of the chiral integrand encodes BCJ numerators of maximal SYM/SUGRA and that the entire string amplitude is fixed by the field-theory limit up to cusp-form ambiguities that arise at high multiplicity. By constructing monodromy-invariant building blocks and a regularisation scheme Reg that excludes closed cycles while preserving modularity, the authors map string-theory coefficients to field-theory BCJ numerators, thereby generalising tree-level BCJ constructions to one loop and high multiplicities. They further reveal new constraints on the field-theory limit induced by modularity, such as the absence of $G_2(\tau)$-type contributions to the chiral integrand, yielding relations among BCJ numerators starting from six points. The work establishes a framework to express one-loop superstring amplitudes purely in terms of their field-theory data, clarifying the role of modular forms and cusp forms and opening avenues for higher-loop and broader-structure explorations, including connections to KLT relations and ambitwistor-string formulations.
Abstract
Are perturbative superstring amplitudes for massless external states just an $α'$ dressing of super-Yang-Mills/supergravity? This is the case at tree level, where the worldsheet correlators at $n$ points can be written in a natural worldsheet basis, such that the kinematic coefficients are BCJ numerators of super-Yang-Mills/supergravity amplitudes, with the non-trivial $α'$ dependence carried only by the Koba-Nielsen factor. Motivated by this construction, we present for the first time a complete worldsheet basis of one-loop superstring correlators at $n$ points. All the kinematic coefficients associated to non-cusp basis elements are identified with pieces of one-loop BCJ numerators of super-Yang-Mills/supergravity. This determines the superstring correlators up to 15 points in terms of field theory. Starting at 16 points (modular weight 12), the worldsheet basis may include cusp forms, which vanish in the field-theory degeneration, such that the associated coefficients cannot be fixed in this manner. Therefore, the one-loop answer to our initial question is determined, at high multiplicity, by whether the coefficients of cusp basis elements vanish or not. As a by-product of our construction, we present new constraints on the field-theory limit that result from string modularity. These are expressed as additional relations among one-loop BCJ numerators in maximal super-Yang-Mills/supergravity starting at 6 points.