A new look at one-loop integrals in string theory

TL;DR

This paper introduces new methods for evaluating one-loop modular integrals in string theory that maintain T-duality and handle unphysical tachyons, simplifying calculations of BPS-saturated couplings.

Contribution

It presents a modular invariant approach to one-loop integrals, extending the Rankin-Selberg-Zagier method and introducing techniques for cases with unphysical tachyons.

Findings

Recovered known results for d=2 in simplified steps

Developed a modular invariant regularisation technique

Connected integrals to Epstein Zeta series

Abstract

We revisit the evaluation of one-loop modular integrals in string theory, employing new methods that, unlike the traditional 'orbit method', keep T-duality manifest throughout. In particular, we apply the Rankin-Selberg-Zagier approach to cases where the integrand function grows at most polynomially in the IR. Furthermore, we introduce new techniques in the case where `unphysical tachyons' contribute to the one-loop couplings. These methods can be viewed as a modular invariant version of dimensional regularisation. As an example, we treat one-loop BPS-saturated couplings involving the $d$-dimensional Narain lattice and the invariant Klein $j$-function, and relate them to (shifted) constrained Epstein Zeta series of O(d,d;Z). In particular, we recover the well-known results for d=2 in a few easy steps.