More on soft theorems: Trees, loops, and strings

Abstract

We study soft theorems in a broader context, their universality in effective field theories and string theory, as well as continue the analysis of their fate at loop level. In effective field theories with ${F}^{3}$ and ${R}^{3}$ interactions, the soft theorems are not modified. However, for gravity theories with ${R}^{2}\ensuremath{\phi}$ interactions, the sub-subleading order soft graviton theorem, which is beyond what is implied by the extended Bondi, van der Burg, Metzner, and Sachs symmetry, requires modifications at tree level for nonsupersymmetric theories and at loop level for $\mathcal{N}\ensuremath{\le}4$ supergravity due to anomalies. For open and closed superstrings at finite ${\ensuremath{\alpha}}^{\ensuremath{'}}$, via explicit calculation for lower-point examples as well as world sheet operator product expansion analysis for arbitrary multiplicity, we show that scattering amplitudes satisfy the same soft theorem as their field-theory counterpart. This is no longer true for closed bosonic or heterotic strings due to the presence of ${R}^{2}\ensuremath{\phi}$ interactions. We also consider loop corrections to gauge theories in the planar limit, where we show that tree-level soft gluon theorems are respected at the integrand level for $1\ensuremath{\le}\mathcal{N}\ensuremath{\le}4$ SYM. Finally, we discuss the fate of soft theorems for finite loop amplitudes in pure Yang-Mills theory and gravity.