# Light-by-light forward scattering sum rules for charmonium states

## TL;DR

This paper applies light-by-light scattering sum rules to charmonium states, revealing cancellations between bound states and continuum contributions, and predicts the unmeasured $b3^* b3$ coupling of the $b1_{c1}(1P)$ state, testable at colliders.

## Contribution

It introduces the application of sum rules to charmonium, predicting an unmeasured coupling and analyzing the interplay between bound states and continuum contributions.

## Key findings

- Sum rules imply cancellations between charmonium states and continuum contributions.
- Predicted the $b3^* b3$ coupling of $b1_{c1}(1P)$, testable at colliders.
- Provided duality estimates for continuum contributions above $D ar D$ threshold.

## Abstract

We apply three forward light-by-light scattering sum rules to charmonium states. We show that these sum rules imply a cancellation between charmonium bound state contributions, which are mostly known from the $\gamma \gamma$ decay widths of these states, and continuum contributions above $D \bar D$ threshold, for which we provide a duality estimate. We also show that two of these sum rules allow to predict the yet unmeasured $\gamma^\ast \gamma$ coupling of the $\chi_{c1}(1P)$ state, which can be tested at present high-luminosity $e^+ e^-$ colliders.