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Natural Selection Rules: New Positivity Bounds for Massive Spinning Particles

Joe Davighi, Scott Melville, Tevong You

TL;DR

This work derives new EFT positivity bounds for elastic $2\to2$ scattering of massive spinning particles by combining unitarity, causality, locality, and Lorentz invariance with sharper angular-momentum selection rules in the $s$- and $u$-channels. The authors introduce a regulated amplitude $\hat{\mathcal{A}}$ and a helicity-aware framework that yields stronger bounds on $t$-derivatives than forward-limit results, including a novel bound that remains robust for large total helicities. They show how these bounds constrain dimension-6 and higher operators under various UV-growth assumptions (Froissart, super-Froissart, sub-Froissart) and illustrate with concrete examples: spinor-scalar, vector-scalar, and four-fermion interactions. The results provide a more stringent bridge between IR EFT coefficients and general UV properties, with practical implications for SMEFT analyses and the search for UV completions, while suggesting several avenues for further strengthening via full crossing, upper bounds, and the moment problem. Overall, the paper advances the theoretical toolkit for connecting low-energy EFT isotypes to fundamental UV principles in the presence of spin and crossing symmetry.

Abstract

We derive new effective field theory (EFT) positivity bounds on the elastic $2\to2$ scattering amplitudes of massive spinning particles from the standard UV properties of unitarity, causality, locality and Lorentz invariance. By bounding the $t$ derivatives of the amplitude (which can be represented as angular momentum matrix elements) in terms of the total ingoing helicity, we derive stronger unitarity bounds on the $s$- and $u$-channel branch cuts which determine the dispersion relation. In contrast to previous positivity bounds, which relate the $t$-derivative to the forward-limit EFT amplitude with no $t$ derivatives, our bounds establish that the $t$-derivative alone must be strictly positive for sufficiently large helicities. Consequently, they provide stronger constraints beyond the forward limit and can be used to constrain dimension-6 interactions with a milder assumption about the high-energy growth of the UV amplitude.

Natural Selection Rules: New Positivity Bounds for Massive Spinning Particles

TL;DR

This work derives new EFT positivity bounds for elastic scattering of massive spinning particles by combining unitarity, causality, locality, and Lorentz invariance with sharper angular-momentum selection rules in the - and -channels. The authors introduce a regulated amplitude and a helicity-aware framework that yields stronger bounds on -derivatives than forward-limit results, including a novel bound that remains robust for large total helicities. They show how these bounds constrain dimension-6 and higher operators under various UV-growth assumptions (Froissart, super-Froissart, sub-Froissart) and illustrate with concrete examples: spinor-scalar, vector-scalar, and four-fermion interactions. The results provide a more stringent bridge between IR EFT coefficients and general UV properties, with practical implications for SMEFT analyses and the search for UV completions, while suggesting several avenues for further strengthening via full crossing, upper bounds, and the moment problem. Overall, the paper advances the theoretical toolkit for connecting low-energy EFT isotypes to fundamental UV principles in the presence of spin and crossing symmetry.

Abstract

We derive new effective field theory (EFT) positivity bounds on the elastic scattering amplitudes of massive spinning particles from the standard UV properties of unitarity, causality, locality and Lorentz invariance. By bounding the derivatives of the amplitude (which can be represented as angular momentum matrix elements) in terms of the total ingoing helicity, we derive stronger unitarity bounds on the - and -channel branch cuts which determine the dispersion relation. In contrast to previous positivity bounds, which relate the -derivative to the forward-limit EFT amplitude with no derivatives, our bounds establish that the -derivative alone must be strictly positive for sufficiently large helicities. Consequently, they provide stronger constraints beyond the forward limit and can be used to constrain dimension-6 interactions with a milder assumption about the high-energy growth of the UV amplitude.

Paper Structure

This paper contains 51 sections, 164 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Summary of Main Results
  3. Notation and Conventions
  4. Two-particle states:
  5. $\bm{2 \to 2}$ amplitudes:
  6. $\bm{s}$-channel kinematics:
  7. $\bm{u}$-channel kinematics:
  8. Unitarity Bounds with Angular Momentum
  9. Unitarity and the Optical Theorem:
  10. Partial wave expansion:
  11. Forward limit:
  12. First $\bm{t}$ derivative:
  13. Crossing Relation with Spin
  14. $\bm{u}$-channel unitarity:
  15. Crossing relation:
  16. ...and 36 more sections

Figures (1)

  • Figure 1: The contour $C_\mu$ used to define $\hat{\mathcal{A}}^{(2N)}_{h_u}$ in \ref{['eqn:An']}.