Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Softness and Amplitudes' Positivity for Spinning Particles

Brando Bellazzini

TL;DR

The paper derives universal positivity bounds for forward scattering amplitudes of spinning particles by combining unitarity, analyticity and crossing. It shows that EFTs cannot be arbitrarily soft: the leading energy growth must be at least $O(p^4)$, with positivity translating into constraints on Wilson coefficients and their couplings to light fields. The Goldstino and a fermionic shift-symmetric fermion exemplify maximally soft yet consistent IR behavior, while dispersion-relation arguments reveal nontrivial obstructions to truly supersoft UV completions, including connections to Galileon-like theories and massive gravity. Overall, the work clarifies the IR/UV interplay in spinful EFTs and provides precise criteria for viable UV completions under fundamental S-matrix axioms.

Abstract

We derive positivity bounds for scattering amplitudes of particles with arbitrary spin using unitarity, analyticity and crossing symmetry. The bounds imply the positivity of certain low-energy coefficients of the effective action that controls the dynamics of the light degrees of freedom. We show that low-energy amplitudes strictly softer than $O(p^4)$ do not admit unitary ultraviolet completions unless the theory is free. This enforces a bound on the energy growth of scattering amplitudes in the region of validity of the effective theory. We discuss explicit examples including the Goldstino from spontaneous supersymmetry breaking, and the theory of a spin-1/2 fermion with a shift symmetry.

Softness and Amplitudes' Positivity for Spinning Particles

TL;DR

The paper derives universal positivity bounds for forward scattering amplitudes of spinning particles by combining unitarity, analyticity and crossing. It shows that EFTs cannot be arbitrarily soft: the leading energy growth must be at least , with positivity translating into constraints on Wilson coefficients and their couplings to light fields. The Goldstino and a fermionic shift-symmetric fermion exemplify maximally soft yet consistent IR behavior, while dispersion-relation arguments reveal nontrivial obstructions to truly supersoft UV completions, including connections to Galileon-like theories and massive gravity. Overall, the work clarifies the IR/UV interplay in spinful EFTs and provides precise criteria for viable UV completions under fundamental S-matrix axioms.

Abstract

We derive positivity bounds for scattering amplitudes of particles with arbitrary spin using unitarity, analyticity and crossing symmetry. The bounds imply the positivity of certain low-energy coefficients of the effective action that controls the dynamics of the light degrees of freedom. We show that low-energy amplitudes strictly softer than do not admit unitary ultraviolet completions unless the theory is free. This enforces a bound on the energy growth of scattering amplitudes in the region of validity of the effective theory. We discuss explicit examples including the Goldstino from spontaneous supersymmetry breaking, and the theory of a spin-1/2 fermion with a shift symmetry.

Paper Structure

This paper contains 15 sections, 98 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Crossing Symmetry
  3. Crossing two or more particles
  4. Forward scattering and crossing as $s\leftrightarrow u$
  5. Unitarity constraints and positivity bounds
  6. Soft limits
  7. Soft limits for spin-1/2
  8. Coupling light fields and the Goldstino
  9. Supersoft amplitudes
  10. Fermionic shift symmetry
  11. Obstructions, loopholes and massive gravity
  12. Conclusions
  13. Polarizations
  14. Linear polarizations and crossing symmetric amplitudes
  15. Sum rules, lower subtractions and mixed states

Figures (2)

  • Figure 1: Scattering processes $X+\Psi \rightarrow \mathrm{out}$ and $X\rightarrow \left(\mathrm{out}+\overline\Psi\right)$ which are related by crossing symmetry according to Eq. (\ref{['scattering0']}) and (\ref{['scattering0cross']}).
  • Figure 2: Contours in the complex $s$-plane in the forward elastic scattering with $t=0$. The branch-cut of the square-root type represented by a cyan saw-like line in the interval $\mathcal{I}_\rho=(u_{IR},s_{IR})$ may arise only from the polarizations of massive half-integer spins, and only for certain parity violating interactions, see main text and Eq. (\ref{['Eq:examplediscfer']}). The standard discontinuities of the scattering amplitudes are represented by a red saw-like lines $\mathcal{I}_{\mathcal{M}}$; they come from the amputated correlation functions. At the branch-points $s_{IR}=(m_1+m_2)^2$ and $u_{IR}=(m_1-m_2)^2$, associated with the 2-particle elastic thresholds, the amplitude and the discontinuities vanish.