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UV complete me: Positivity Bounds for Particles with Spin

Claudia de Rham, Scott Melville, Andrew J. Tolley, Shuang-Yong Zhou

TL;DR

The paper develops a comprehensive framework to constrain low-energy EFTs with spinful particles by deriving an infinite set of positivity bounds beyond the forward limit. Central to the approach is the transversity formalism, which diagonalizes crossing and yields dispersion relations with positive left- and right-hand-cut discontinuities for regularized amplitudes. By combining unitarity, analyticity, and crossing, the authors produce both simple leading bounds and general higher-order derivatives that constrain EFT coefficients for arbitrary spin, including scenarios with multiple mass eigenstates. The results substantially strengthen EFT viability tests and have broad implications for theories of gravity and beyond, providing a robust tool to exclude EFTs incompatible with any local, Lorentz-invariant UV completion.

Abstract

For a low energy effective theory to admit a standard local, unitary, analytic and Lorentz-invariant UV completion, its scattering amplitudes must satisfy certain inequalities. While these bounds are known in the forward limit for real polarizations, any extension beyond this for particles with nonzero spin is subtle due to their non-trivial crossing relations. Using the transversity formalism (i.e. spin projections orthogonal to the scattering plane), in which the crossing relations become diagonal, these inequalities can be derived for 2-to-2 scattering between any pair of massive particles, for a complete set of polarizations at and away from the forward scattering limit. This provides a set of powerful criteria which can be used to restrict the parameter space of any effective field theory, often considerably more so than its forward limit subset alone.

UV complete me: Positivity Bounds for Particles with Spin

TL;DR

The paper develops a comprehensive framework to constrain low-energy EFTs with spinful particles by deriving an infinite set of positivity bounds beyond the forward limit. Central to the approach is the transversity formalism, which diagonalizes crossing and yields dispersion relations with positive left- and right-hand-cut discontinuities for regularized amplitudes. By combining unitarity, analyticity, and crossing, the authors produce both simple leading bounds and general higher-order derivatives that constrain EFT coefficients for arbitrary spin, including scenarios with multiple mass eigenstates. The results substantially strengthen EFT viability tests and have broad implications for theories of gravity and beyond, providing a robust tool to exclude EFTs incompatible with any local, Lorentz-invariant UV completion.

Abstract

For a low energy effective theory to admit a standard local, unitary, analytic and Lorentz-invariant UV completion, its scattering amplitudes must satisfy certain inequalities. While these bounds are known in the forward limit for real polarizations, any extension beyond this for particles with nonzero spin is subtle due to their non-trivial crossing relations. Using the transversity formalism (i.e. spin projections orthogonal to the scattering plane), in which the crossing relations become diagonal, these inequalities can be derived for 2-to-2 scattering between any pair of massive particles, for a complete set of polarizations at and away from the forward scattering limit. This provides a set of powerful criteria which can be used to restrict the parameter space of any effective field theory, often considerably more so than its forward limit subset alone.

Paper Structure

This paper contains 53 sections, 265 equations, 3 figures.

Table of Contents

  1. Introduction
  2. From Helicity to Transversity
  3. Helicity Formalism
  4. Spherical wave states:
  5. Plane wave states:
  6. Partial wave expansion:
  7. Kinematical singularities:
  8. Unitarity:
  9. Crossing:
  10. Transversity Formalism
  11. Transversity Amplitudes:
  12. Crossing:
  13. Kinematical Singularities:
  14. Positivity Bounds
  15. Unitarity and the Right Hand Cut
  16. ...and 38 more sections

Figures (3)

  • Figure 1: The difference between the helicity and transversity formalism. The horizontal plane ($xz$ plane) is the particle interaction plane. In the helicity formalism particle spins are projected onto the direction of motion, while in the transversity formalism particle spins are projected in the vertical direction, which is transverse to the interaction plane.
  • Figure 2: The scattering amplitude can be analytically continued to the entire complex $s$ plane, with the poles at $s=m^2$ and $3m^2-t$ and branch cuts along the real axis from $-t$ to $-\infty$ and from $4m^2$ to $\infty$.
  • Figure 3: The scattering amplitude on the real $s$ axis in a theory with two massive states. We can draw integration contours analogous to $C$ and $C'$ from Figure \ref{['fig:1']}, providing that none of the poles overlap with the branch cuts.