Rigorous Bounds on Light-by-Light Scattering

TL;DR

This work derives rigorous, two-sided bounds on EFT coefficients governing $2\to 2$ photon scattering in four dimensions by combining unitarity, analyticity, crossing symmetry, dispersion relations, and a photon-specific partial-wave analysis. The low-energy amplitudes are organized into a polynomial basis with coefficients $f_2,g_2,f_3,g_3,f_4,g_{4,1},g_{4,2}$, which are related to high-energy spectral densities through positive brackets; null constraints from crossing symmetry sharpen the bounds. Analytic results yield bounds such as $|f_2|\le g_2$, $g_3/g_2$ bounded by $r_{\min}/M^2 \le g_3/g_2 \le 1/M^2$ with $r_{\min} \approx -4.9789$, and eight-derivative coefficients confined to a triangular region in the $(g_{4,1},g_{4,2})$ plane, all cross-checked numerically via semidefinite programming. Comparisons with partial UV completions show that tree-level scalar and axion exchanges saturate several bounds, while graviton and loop completions lie inside the allowed region, offering insights into possible unknown UV structures and signaling where further UV data could tighten or refine the bounds.

Abstract

We bound EFT coefficients appearing in $2 \to 2$ photon scattering amplitudes in four dimensions. After reviewing unitarity and positivity conditions in this context, we use dispersion relations and crossing symmetry to compute sum rules and null constraints. This allows us to derive new rigorous bounds on operators with four, six, and eight derivatives, including two-sided bounds on their ratios. Comparing with a number of partial UV completions, we find that some of our bounds are saturated by the amplitudes that arise from integrating out a massive scalar or axion, while others suggest the existence of unknown amplitudes.

Figures (7)

• Figure 1: The dispersion relation used in \ref{['eq:doubledisp']} starts from the contour at infinity (dashed). The contour is deformed inwards, picking up contributions from the three poles indicated and from two cuts on the real $s'$ axis, starting at $M^2$ and $-M^2-u$ respectively. Based on the assumed symmetry $s'\leftrightarrow t=-s'-u$, the integrals over the two cuts can be combined.
• Figure 2: Bounds on the six-derivative terms $f_3/g_2$ and $g_3/g_2$ as a function of $f_2/g_2$. The dots refer to the values in the partial UV completions discussed in section \ref{['sec:partialUV']}: massive axion ($a$), scalar ($\phi$)and graviton ($h$), and QED ($e$), scalar QED ($\tilde{e}$) and $W^\pm$ sector ($W$).
• Figure 3: Allowed region in the plane $(g_3/g_2,\, f_3/g_2)$ for fixed values $f_2/g_2 = k$. The blue, purple, green, yellow and red regions correspond respectively to $k = 0, 0.25, 0.5, 0.75, 1$.
• Figure 4: Showing the triangle with the vertices given by \ref{['eq:triangle']}, and, in orange, the bound \ref{['eq:EFTHedron']} from Arkani-Hamed:2020blm, where allowed region is above the graph.
• Figure 5: Bounds in the plane $\frac{g_2}{M^2}e_1=g_3$, $\frac{g_2}{M^4}e_2=g_{4,1}+2g_{4,2}$ normalized to $g_2$. The blue, red regions corresponds to $f_2/g_2=0,\pm 1$. Intermediate allowed regions can be obtained by linear interpolation.