String theory integrands and supergravity divergences

Abstract

At low energies, interactions of massless particles in type II strings compactified on a torus $T^d$ are described by an effective Wilsonian action $\mathcal{S}(Λ)$, consisting of the usual supergravity Lagrangian supplemented by an infinite series of higher-derivative vertices, including the much studied $\nabla^{4p+6q} \mathcal{R}^4$ gravitational interactions. Using recent results on the asymptotics of the integrands governing four-graviton scattering at genus one and two, I determine the $Λ$-dependence of the coefficient of the above interaction, and show that the logarithmic terms appearing in the limit $Λ\to 0$ are related to UV divergences in supergravity amplitudes, augmented by stringy interactions. This provides a strong consistency check on the expansion of the integrand near the boundaries of moduli space, in particular it elucidates the appearance of odd zeta values in these expansions. I briefly discuss how these logarithms are reflected in non-analytic terms in the low energy expansion of the string scattering amplitude.

40 pages; v2: after fixing a factor of 2 mistake in Eq. (2.41), all divergent terms now agree with SUGRA predictions. Added a note at end of Sec 1 on the definition of the truncated moduli space M_{h,n}(Λ)