$$ {\mathcal{R}^4} $$ counterterm and E 7(7) symmetry in maximal supergravity

Abstract

The coefficient of a potential $$ {\mathcal{R}^4} $$ counterterm in $$ \mathcal{N} = 8 $$ supergravity has been shown previously to vanish in an explicit three-loop calculation. The $$ {\mathcal{R}^4} $$ term respects $$ \mathcal{N} = 8 $$ supersymmetry; hence this result poses the question of whether another symmetry could be responsible for the cancellation of the three-loop divergence. In this article we investigate possible restrictions from the continuous coset symmetry E 7(7)/SU(8), exploring the limits as a single scalar becomes soft, as well as a double-soft scalar limit relation derived recently by Arkani-Hamed et al. We implement these relations for the matrix elements of the $$ {\mathcal{R}^4} $$ term that occurs in the low-energy expansion of closed-string treelevel amplitudes. We find that the matrix elements of $$ {\mathcal{R}^4} $$ that we investigated all obey the double-soft scalar limit relation, including certain non-maximally-helicity-violating sixpoint amplitudes. However, the single-soft limit does not vanish for this latter set of amplitudes, which suggests that the E 7(7) symmetry is broken by the $$ {\mathcal{R}^4} $$ term.