Soft Algebra for $\mathcal{N}=4$ SYM

Abstract

Scattering amplitudes of $n$ particles in nonabelian gauge theories admit factorizations of the general form ${\mathcal{A}}_n={\mathcal{A}}_n^{\mathrm{soft}}\times {\mathcal{A}}_n^{\mathrm{hard}}$, where ${\mathcal{A}}_n^{\mathrm{soft}}$ is IR divergent, while ${\mathcal{A}}_n^{\mathrm{hard}}$ is IR finite and encodes the higher loop corrections to scattering. We specify a particular all-orders definition of this factorization for planar $\mathcal{N}=4$ super Yang-Mills (SYM) and argue that the resulting ${\mathcal{A}}_n^{\mathrm{hard}}$ obeys an uncorrected tree-level soft theorem. Moreover it furnishes a representation of the undeformed tree-level $\mathcal{S}$-algebra generated by a tower of soft gluons. The results follow from several commonly invoked assumptions for $\mathcal{N}=4$ SYM, including BDS one-loop exponentiation of the splitting function and amplitude/Wilson-loop duality.

41 pages