New superconvergence relations for spin and tensor structure functions of $γγ$ fusion

TL;DR

The paper develops a unified dispersive framework for exact sum rules in spin and tensor structure functions probed by photons. By exploiting the LT-degeneracy at the Siegert point $\nu^2=-Q^2$ and the high-energy suppression of longitudinal photons, it re-derives the Burkhardt-Cottingham sum rule and obtains a new class of superconvergence relations for doubly virtual $\gamma^{*}\gamma^{*}$ fusion, valid for arbitrary photon virtualities. In the real-photon limit, these yield established photon-spin sum rules $\int_0^1 dx\, g_1^{\gamma}(x,Q^2)=0$ and $\int_0^1 dx\, g_2^{\gamma}(x,Q^2)=0$ and provide constraints on meson-transition form factors. The results open avenues for applying these relations to higher-spin targets, richer tensor structures, and potential gravitational analogue structure functions.

Abstract

The Burkhardt--Cottingham sum rule is an exact superconvergence relation for a spin-structure function, derived from general principles of light absorption and scattering, and valid at any momentum transfer $Q^2$. I illustrate how a class of such relations emerges from the Siegert point, an unphysical kinematical point where both the probe and the target are at rest. From light-by-light scattering, new sum rules for $γ^\ast γ^\ast$ fusion are emerging, valid for arbitrary photon virtualities. Regarding the convergence of these relations, there is a simple argument for the suppression of longitudinal photon polarizations at high energy. Among its consequences is the prediction of $σ_L/ σ_T \to 0$ at high energy, for the ratio of unpolarized nucleon photoabsorption cross sections.