A new look at one-loop integrals in string theory

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.

D6ℛ4 amplitudes in various dimensions

“…(0,0) at d = 2 can be further assigned to a simple pole in the one-loop contribution, 42) producing the correct anomalous term on the r.h.s. in the second line of (2.23).…”

“…These large radius expansions can be checked from the Eisenstein series representations (2.14)-(2.16), or just as well from the modular integral representation (2.7)-(2.9), using e.g. the techniques in [42].…”

“…In defining these divergent integrals we use the renormalization prescription of [42,43], and normalize the integration measure dµ h as in [34]. At two-loop, the corrections to R 4 couplings vanish, but the corrections to D 4 R 4 and D 6 R 4 are given by modular integrals over the fundamental domain F 2 of the Siegel upper-half plane of degree 2, which parametrizes genus 2 Riemann surfaces, [22,33],…”

“…is the non-holomorphic Eisenstein series of SL(2, Z), in the normalization of [42]. In defining these divergent integrals we use the renormalization prescription of [42,43], and normalize the integration measure dµ h as in [34].…”

Thermodynamics of string gas

“…New techniques for evaluating the one-loop modular integrals are still developing. Recent progress can be found in [64].…”