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.
Highlights
Several model-independent sum rules were derived for the forward light-by-light scattering, and were exactly verified at leading order in scalar and spinor QED [1,2,3,4,5]
We considered the spread between similar approaches in the literature as an estimate for the model error
We checked the potential model on the known Υð2SÞ → γχbJð1PÞ and Υð3SÞ → γχbJð2PÞ transitions and found that the theoretical estimates agree with experiment to within 15%
Summary
Several model-independent sum rules were derived for the forward light-by-light scattering, and were exactly verified at leading order in scalar and spinor QED [1,2,3,4,5]. [9,10,11] for some recent reviews and references therein Such sum rule relations have the potential to reveal how much of the radiative decay strength from or into vector quarkonium states results from possible exotic mesons.
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