Hidden zeros for particle/string amplitudes and the unity of colored scalars, pions and gluons

TL;DR

This work uncovers a hidden unity among colored scalars, pions, and gluons by analyzing zeros and factorization of tree-level Tr$(\phi^3)$ amplitudes through the ABHY associahedron geometry. The key mechanism is that zeros arise when a maximal causal diamond in the kinematic mesh collapses, and turning on a single non-planar Mandelstam inside the diamond produces precise factorization into lower-point amplitudes with geometry-determined kinematic shifts. Remarkably, the same zero/factorization structures survive in the stringy extension and generalize to the Non-linear Sigma Model and Yang-Mills theory, once viewed through a universal stringy deformation called the $\delta$-deformation, which uniquely preserves the nonplanar data. This deformation interpolates between Tr$(\phi^3)$, NLSM, and YM, and provides a framework in which gluons can be viewed as scaffolded by pairs of scalars, with factorization patterns that emanate from the underlying kinematic geometry. Collectively, the results point to a unified, geometry-driven description of widely different colored theories and open avenues to harness zeros for amplitude reconstruction and loop-level extensions.

Abstract

Recent years have seen the emergence of a new understanding of scattering amplitudes in the simplest theory of colored scalar particles - the Tr$(φ^3)$ theory - based on combinatorial and geometric ideas in the kinematic space of scattering data. In this paper we report a surprise: far from the toy model it appears to be, the ''stringy'' Tr$(φ^3)$ amplitudes secretly contain the scattering amplitudes for pions, as well as non-supersymmetric gluons, in any number of dimensions. The amplitudes for the different theories are given by one and the same function, related by a simple shift of the kinematics. This discovery was spurred by another fundamental observation: the tree-level Tr$(φ^3)$ field theory amplitudes have a hidden pattern of zeros when a special set of non-planar Mandelstam invariants is set to zero. Furthermore, near these zeros, the amplitudes simplify, by factoring into a non-trivial product of smaller amplitudes. Remarkably the amplitudes for pions and gluons are observed to also vanish in the same kinematical locus. These properties further generalize to the ''stringy'' Tr$(φ^3)$ amplitudes. There is a unique shift of the kinematic data that preserves the zeros, and this shift is precisely the one that unifies colored scalars, pions, and gluons into a single object. We will focus in this paper on explaining the hidden zeros and factorization properties and the connection between all the colored theories, working for simplicity at tree-level. Subsequent works will describe this new formulation for the Non-linear Sigma Model and non-supersymmetric Yang-Mills theory, at all loop orders.

Figures (18)

• Figure 1: 6-point momentum polygon.
• Figure 2: Kinematic mesh for 6-point.
• Figure 3: Triangulations and corresponding mesh subregions for 6-point kinematics.
• Figure 4: 5-point ABHY associahedron and respective Minkowski summands.
• Figure 5: 5-point factorization near zeros.