Large deformations of Tr(${\Phi }^3$) and the world at infinity

Abstract

The amplitudes of the non-linear sigma model can be obtained from those of Tr(${\Phi }^3$) theory by sending the kinematic (Mandelstam) variables to infinity in a certain direction. In this paper we characterize the behavior of Tr(${\Phi }^3$) amplitudes under a general class of large kinematic shifts called $g$-vector shifts. The objects that live in this world at infinity retain certain key amplitude-like properties, most notably factorization, and admit descriptions in terms of polytopes, but they are not generally amplitudes of any cognizable theory. We identify particular $g$-vector shifts that lead at infinity to mixed amplitudes involving two pions and any number of scalars, allowing us to provide polytopal descriptions of these amplitudes.

25 pages, 14 figures, v2 minor corrections