Basis decompositions of genus-one string integrals

TL;DR

This paper develops a method to decompose complex genus-one string integrals into a conjectured basis, simplifying calculations of one-loop amplitudes across various string theories and supporting further mathematical research.

Contribution

It explicitly decomposes genus-one integrands into a conjectural chain basis, validating its structure and enabling simplified amplitude computations.

Findings

Validated the chain basis for genus-one integrals

Derived explicit expansion coefficients for specific cases

Facilitated simplification of multiparticle amplitudes

Abstract

One-loop scattering amplitudes in string theories involve configuration-space integrals over genus-one surfaces with coefficients of Kronecker-Eisenstein series in the integrand. A conjectural genus-one basis of integrands under Fay identities and integration by parts was recently constructed out of chains of Kronecker-Eisenstein series. In this work, we decompose a variety of more general genus-one integrands into the conjectural chain basis. The explicit form of the expansion coefficients is worked out for infinite families of cases where the Kronecker-Eisenstein series form cycles. Our results can be used to simplify multiparticle amplitudes in supersymmetric, heterotic and bosonic string theories and to investigate loop-level echoes of the field-theory double-copy structures of string tree-level amplitudes. The multitude of basis reductions in this work strongly validate the recently proposed chain basis and stimulate mathematical follow-up studies of more general configuration-space integrals with additional marked points or at higher genus.