Light-by-light scattering sum rule for radiative transitions of bottomonia

TL;DR

This paper extends a light-by-light scattering sum rule to heavy quarkonium radiative transitions, applies it to bottomonium states, and finds it holds within errors, offering a new tool for studying exotic states.

Contribution

It generalizes a sum rule to heavy quarkonium, applies it to bottomonium, and tests its validity using experimental data and theoretical estimates.

Findings

Sum rule is satisfied for low-lying bottomonium states within errors.

The approach can be used to investigate exotic states in quarkonium spectra.

Theoretical estimates help fill gaps where experimental data is lacking.

Abstract

We generalize a forward light-by-light scattering sum rule to the case of heavy quarkonium radiative transitions. We apply such sum rule to the bottomonium states, and use available data on radiative transitions in its evaluation. For the transitions that are not known experimentally, we provide theoretical estimates within a potential model, and consider the spread between similar approaches in the literature as an estimate for the model error. For the $\Upsilon(1S)$, $\Upsilon(2S)$, and $\Upsilon(3S)$ states we observe that, due to a cancellation between transitions involving $\chi_{b0}, \chi_{b1}$, and $\chi_{b2}$ states, the sum rule is satisfied within experimental and theoretical error estimates. Having tested this sum rule for the low-lying bottomonium states, it may be used as a tool to investigate the nature of exotic states in the charmonium and bottomonium spectrum through the corresponding radiative transitions.