All-loop soft theorem for pions
Christoph Bartsch, Karol Kampf, Jiří Novotný, Jaroslav Trnka
Abstract
In this paper we discuss a generalization of the Adler zero to loop integrands in the planar limit of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>S</a:mi><a:mi>U</a:mi><a:mo stretchy="false">(</a:mo><a:mi>N</a:mi><a:mo stretchy="false">)</a:mo></a:math> nonlinear sigma model (NLSM). The Adler zero for integrands is violated starting at the two-loop order and is only recovered after integration. Here we propose a 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 , we identify a simple nonvanishing soft behavior of integrands, which we call the . We find that the proposed soft theorem is satisfied by the “surface” integrand of Arkani-Hamed and Figueiredo [], which is obtained from the shifted <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>Tr</e:mi><e:msup><e:mi>φ</e:mi><e:mn>3</e:mn></e:msup></e:math> surfacehedron integrand. Finally, we derive an on shell version of the algebraic soft theorem that takes an interesting form in terms of self-energy factors and lower-loop integrands in a mixed theory of pions and scalars. Published by the American Physical Society 2024