Communications in Number Theory and Physics

Volume 6 (2012)

Number 1

A new look at one-loop integrals in string theory

Pages: 159 – 201



Carlo Angelantonj (Dipartimento di Fisica Teorica, Università di Torino, Italy)

Ioannis Florakis (Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, München, Germany)

Boris Pioline (Université Pierre et Marie Curie, Paris, France)


We revisit the evaluation of one-loop modular integrals instring theory, employing new methods that, unlike thetraditional “orbit method“, keep T-duality manifestthroughout. In particular, we apply theRankin–Selberg–Zagier approach to cases where theintegrand function grows at most polynomially in the infrared.Furthermore, we introduce new techniques in the case where“unphysical tachyons” contribute to the one-loopcouplings. These methods can be viewed as a modularinvariant version of dimensional regularization. As anexample, we treat one-loop BPS-saturated couplingsinvolving the $d$-dimensional Narain lattice and theinvariant Klein $j$-function, and relate them to (shifted)constrained Epstein Zeta series of ${\rm O} (d,d;\mathbb{Z})$. In particular, we recover the well-knownresults for $d=2$ in a few easy steps.

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