Basis decompositions of genus-one string integrals

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.

Advanced tools for basis decompositions of genus-one string integrals

“…In the case of products of Ω without cycles, such as those represented by the last graph in figure 1, these can be algebraically expanded into basis elements using Fay identities (2.5) or (2.10) in practice. In the previous companion paper [33], we found a general formula to decompose an arbitrary Ω-cycle into basis elements: Further, our previous findings demonstrated the recursive application of single-cycle formulae to dissect products of two Ω-cycles. This paper aims to broaden this methodology to encompass products involving an arbitrary number of cycles as indicated in (1.3).…”

“…Rather than presenting a rigorous mathematical proof, we offer compelling evidence for (1.1) forming an F-IBP basis by decomposing a range of Kronecker-Eisenstein series into the chain form, thereby bolstering the credibility of the conjectural basis. In a companion paper [33], Rodriguez, Schlotterer and the author have made notable strides in advancing the IBP methodology for one-loop string integrals of the Koba-Nielsen type. Specifically, we have transformed cyclic products of the Kronecker-Eisenstein series denoted as…”

“…We will use the single-cycle formulae derived in [33] recursively to decompose (1.3) to chain basis at the cost of introducing some total Koba-Nielsen derivative terms.…”

“…In this companion paper to [33], we provide a succinct overview of the genus-one string integrands and present the closed-form formulae for decomposing a cycle product of the Kronecker-Eisenstein series. While our goal is to ensure this paper is self-contained, we direct readers to [33] and the associated references for a more in-depth exploration.…”

“…We already see the application of such simplification in the decomposition of double cycles in the companion paper [33]. We call the last line of (3.8) as a fusion of two cycles and denote JHEP05(2024)255…”