[Paper Review] Bounds on photon scattering

Kelian Häring, Aditya Hebbar+3 more|arXiv (Cornell University)|Nov 9, 2022

Particle physics theoretical and experimental studies4 citations

TL;DR

This paper establishes non-perturbative bounds on 2-to-2 scattering amplitudes of massless spin-1 particles (like photons) in four-dimensional Minkowski space using full nonlinear unitarity and crossing symmetry. It derives novel numerical constraints on low-energy Wilson coefficients in the effective field theory description, revealing that certain coefficients cannot be bounded due to the absence of a lower bound in the unitarity constraints, while also recovering known positivity bounds via alternative methods.

ABSTRACT

Numerical data for scattering amplitudes of photons in d=4 obtained by solving various optimisation problems. The data is stored in .txt files. Mathematica notebook is provided for loading and plotting the data.

Motivation & Objective

• To derive non-perturbative constraints on low-energy scattering amplitudes of massless spin-1 particles, such as photons, in four-dimensional spacetime.
• To identify which Wilson coefficients in the effective field theory expansion are bounded or unbounded under full nonlinear unitarity.
• To compare bounds derived from full unitarity with those from positivity conditions, clarifying their respective strengths and limitations.
• To systematically analyze the structure of 2-to-2 scattering amplitudes under Lorentz invariance, unitarity, analyticity, and crossing symmetry.
• To explain why certain scattering observables cannot be bounded, resolving a long-standing ambiguity in the S-matrix bootstrap program for photons.

Proposed method

• The authors define five independent scattering amplitudes (Φ₁ to Φ₅) for 2-to-2 photon scattering in the center-of-mass frame, exploiting parity and particle identity to reduce the number of independent amplitudes.
• They expand the amplitudes in powers of Mandelstam variables (s, t, u), identifying polynomial and logarithmic terms that encode low-energy Wilson coefficients.
• Using full nonlinear unitarity, they derive constraints on the coefficients of logarithmic terms (L₁, L₂) in terms of the polynomial coefficients (g₂, f₂, etc.), enforcing the positivity of spectral functions.
• They construct an ansatz for the partial wave amplitudes using a basis of monomials in ρ(s), χ(s), and their products, and impose unitarity conditions on partial waves of all spins.
• They analyze the large-energy (s→∞) behavior of the amplitudes to derive asymptotic constraints, including vanishing of constant and 1/√s terms in the leading-order expansion.
• They compare results from full unitarity with those from positivity bounds, showing that full unitarity yields stronger and more complete constraints in some cases.

Experimental results

Research questions

• RQ1 Which Wilson coefficients in the low-energy effective theory of photon scattering are bounded by non-perturbative unitarity constraints?
• RQ2 Why do some Wilson coefficients remain unbounded despite the application of full nonlinear unitarity and crossing symmetry?
• RQ3 How do bounds derived from full unitarity compare with those obtained from the standard positivity bounds in the S-matrix bootstrap program?
• RQ4 What is the role of the logarithmic terms in the amplitude expansion, and how are they fixed by unitarity?
• RQ5 Can the asymptotic behavior of scattering amplitudes at high energy be used to derive stronger constraints on the low-energy coefficients?

Key findings

• The paper derives novel numerical bounds on Wilson coefficients g₂, f₂, g₃, f₃, g₄, f₄, h₃, and g₄′ using full nonlinear unitarity, which are stronger than those from positivity alone.
• It identifies that the coefficient h₃ in the Φ₅ amplitude (proportional to stu) cannot be bounded by any unitarity constraint, as it does not appear in the spectral functions of any partial wave.
• The logarithmic terms L₁ and L₂ in the amplitude expansions are fully determined by the polynomial coefficients via unitarity, with explicit expressions involving β coefficients that depend on g₂ and f₂.
• The leading-order asymptotic behavior of the amplitudes requires the sum of coefficients in the ansatz to vanish, ensuring that constant terms in the partial waves do not violate unitarity.
• At the 1/√s order, additional constraints emerge from the imaginary parts of partial waves, requiring specific linear combinations of coefficients to be non-positive, which further restrict the allowed parameter space.
• The analysis confirms that the positivity bounds are recovered as a subset of the full unitarity constraints, but full unitarity provides additional, non-trivial constraints not captured by positivity alone.