An All-loop Soft Theorem for Pions

Abstract: In this letter, we discuss a generalization of the Adler zero to loop integrands in the planar limit of the $SU(N)$ non-linear sigma model (NLSM). While possible to maintain at one-loop, the Adler zero for integrands is violated starting at the two-loop order and is only recovered after integration. Here we propose a non-zero soft theorem satisfied by loop integrands with any number of loops and legs. This requires a generalization of NLSM integrands to an off-shell framework with certain deformed kinematics. Defining an algebraic soft limit', we identify a particularly simple non-vanishing soft behavior of integrands, which we call thealgebraic soft theorem'. We find that the proposed soft theorem is satisfied by the `surface' integrand of Arkani-Hamed, Cao, Dong, Figueiredo and He, which is obtained from the shifted ${\rm Tr}\phi^3$ surfacehedron integrand. Finally, we derive an on-shell version of the algebraic soft theorem that takes an interesting form in terms of propagator renormalization factors and lower-loop integrands in a mixed theory of pions and scalars.