Double-logarithms in N=8 supergravity: impact parameter description and mapping to 1-rooted ribbon graphs

Abstract

The set of double-logarithmic (DL) contributions $(\alpha \, t \ln^2 {s})^n$ to the 4-graviton amplitude in ${\cal N}$$=$ 8 supergravity (SUGRA), with $\alpha$ being the gravitational coupling and $(s,t)$ the Mandelstam invariants, is studied in impact parameter ($\rho$) representation. This sector of the amplitude shows interesting properties which shed light on the nature of quantum corrections in gravity. Besides having a convergent behaviour as $s$ increases, which is not present in ${\cal N}$$<$ 4 SUGRA theories, there exists a critical line $\rho_c(s)$ above which the Born amplitude prevails. The short distance region $\rho < \rho_c(s)$ is dominated by the DL terms. As a consequence, when studied in terms of an eikonal approach in the forward limit, the scattering angle linked to the bending of the semiclassical trajectory of the graviton shows a transition from attractive gravity at large distances to a region at small $\rho$ characterized by a repulsive DL contribution to the gravitational potential due to the gravitino content of the theory. In the complex angular momentum plane, this DL high energy asymptotics is driven by the rightmost pole singularity of a parabolic cylinder function. The resummation of DL quantum corrections in ${\cal N}$$=$ 8 SUGRA can be understood in terms of the counting of 1-rooted maps on orientable surfaces.