Non--analytic terms from nested divergences in maximal supergravity

Abstract

The $D^4{\mathcal{R}}^4$ and $D^6{\mathcal{R}}^4$ coefficient functions in the effective action of type II string theory compactified on $T^d$ contain terms of the form ${\mathcal{E}}_1\ln g_d$ and ${\mathcal{E}}_2(\ln g_d{)}^2$ in specific dimensions, where $g_d$ is the T--duality invariant string coupling, and ${\mathcal{E}}_1$ and ${\mathcal{E}}_2$ are U--duality invariant coefficient functions. We derive these non--analytic terms from nested ultraviolet divergences in two and three loop maximal supergravity. For the $D^4{\mathcal{R}}^4$ coupling, the contribution involves ${\mathcal{E}}_{{\mathcal{R}}^4}\ln g_d$, while for the $D^6{\mathcal{R}}^4$ coupling, it involves ${\mathcal{E}}_{{\mathcal{R}}^4}\ln g_d$, ${\mathcal{E}}_{D^2{\mathcal{R}}^4}(\ln g_d{)}^2$ and ${\mathcal{E}}_{D^4{\mathcal{R}}^4}\ln g_d$; where ${\mathcal{E}}_{{\mathcal{R}}^4}$, ${\mathcal{E}}_{D^2{\mathcal{R}}^4}$ and ${\mathcal{E}}_{D^4{\mathcal{R}}^4}$ are the ${\mathcal{R}}^4$, $D^2{\mathcal{R}}^4$ and $D^4{\mathcal{R}}^4$ coefficient functions respectively. The contribution from ${\mathcal{E}}_{D^2{\mathcal{R}}^4}$, the coefficient function of an amplitude that vanishes onshell, arises from a two loop nested subdivergence of the three loop amplitude.

34 pages, LaTeX, 7 figures

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