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Hidden Zeros in Massive Theories

Mariana Carrillo González, Freddie Ward

Abstract

We investigate whether the hidden zeros and associated factorisations found for massless colour-ordered amplitudes persist under massive deformations. Using the kinematic mesh construction, we show that hidden zeros survive only for symmetry controlled mass generation. For massive $\text{Tr} Φ^3$ with a uniform mass, the zeros and their factorisation patterns are inherited after a massive shift of planar variables, and an analogous statement holds for Kaluza-Klein reductions where the relevant non-planar variables are modified by conserved mode numbers. For the non-linear sigma model (NLSM), a naive pion mass term generically spoils hidden zeros, while a spurion induced potential restores them. This allows factorisation near zeros, including odd point channels described by an appropriately mass deformed NLSM + $φ^3$ theory, and leads to a hidden zero based on-shell recursion for massive NLSM amplitudes. For spin-one, a simple massive Yang-Mills theory fails to exhibit hidden zeros, while spontaneously broken gauge theories preserve them.

Hidden Zeros in Massive Theories

Abstract

We investigate whether the hidden zeros and associated factorisations found for massless colour-ordered amplitudes persist under massive deformations. Using the kinematic mesh construction, we show that hidden zeros survive only for symmetry controlled mass generation. For massive with a uniform mass, the zeros and their factorisation patterns are inherited after a massive shift of planar variables, and an analogous statement holds for Kaluza-Klein reductions where the relevant non-planar variables are modified by conserved mode numbers. For the non-linear sigma model (NLSM), a naive pion mass term generically spoils hidden zeros, while a spurion induced potential restores them. This allows factorisation near zeros, including odd point channels described by an appropriately mass deformed NLSM + theory, and leads to a hidden zero based on-shell recursion for massive NLSM amplitudes. For spin-one, a simple massive Yang-Mills theory fails to exhibit hidden zeros, while spontaneously broken gauge theories preserve them.
Paper Structure (18 sections, 77 equations, 3 figures)

This paper contains 18 sections, 77 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Hidden Zeros and the Kinematic Mesh
  3. Massive phi3 Theory
  4. Dimensional Reduction of Tr phi3
  5. Massive Non-linear Sigma Model
  6. Simple Massive NLSM
  7. Massive NLSM through Spurions
  8. Hidden Zeros
  9. Factorisation
  10. Double Copy
  11. On-Shell Recursion
  12. Dimensional Reduction of NLSM
  13. Massive Yang-Mills
  14. Massive YM Theory
  15. Spontaneously Broken YM Theory
  16. ...and 3 more sections

Figures (3)

  • Figure 1: Top: Tiles that make up the kinematic mesh. Bottom: Examples of causal diamonds on the $n=6$ kinematic mesh. Setting all $c_{i,j}=0$ inside the causal diamonds makes the corresponding 6-point amplitude vanish.
  • Figure 2: General scheme for constructing the factorisation in Eq. (\ref{['generalfactorisation']}). The two triangular regions formed by extending the causal diamond to the mesh boundary define regions with variables defining lower-point processes. The planar variables in the green lower and upper edges of the "up" and "down" regions respectively are replaced by planar variables opposite to them in the encircled edge.
  • Figure 3: Summing the tiles in one cycle on any edge column of a kinematic mesh with alternating relative signs, shown here for $2n=6$, is a visual way to view the consistency requirement on $c_{i,j}$/$r_{i,j}$ that results from the one-dimensional kernel of the $c$-equations. The sum fully telescopes, leaving a relation solely in terms of $c_{i,j}$/$r_{i,j}$.