A new look at one-loop integrals in string theory

We revisit the evaluation of one-loop modular integrals in string theory, employing new methods that 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 and, furthermore, we introduce new techniques in the case where `unphysical tachyons' do contribute to the one-loop couplings. These methods can be viewed as a modular invariant version of dimensional regularisation, unlike the conventional orbit decomposition that obscures the underlying T-duality. As an example, we specialise to the study of one-loop BPS-saturated couplings involving the d-dimensional Narain lattice and the invariant Klein j-function. We relate them to (shifted) constrained Epstein Zeta series of O(d,d;Z), and recover the well-known result for Gamma_{(2,2)} in a few easy steps.