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

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.