Accessing the real part of the forward $J/\psi$-p scattering amplitude from $J/\psi$ photo-production on protons around threshold

We provide an updated analysis of the forward $J/\psi$-p scattering amplitude, relating its imaginary part to $\gamma p \to J/\psi p$ and $\gamma p \to c \bar c X$ cross section data, and calculating its real part through a once-subtracted dispersion relation. From a global fit to both differential and total cross section data, we extract a value for the spin-averaged $J/\psi$-p s-wave scattering length $a_{\psi p} = 0.046 \pm 0.005$ fm, which can be translated into a $J/\psi$ binding energy in nuclear matter of $B_\psi = 2.7 \pm 0.3$ MeV. We estimate the forward-backward asymmetry to the $\gamma p \to e^- e^+ p$ process around the $J/\psi$ resonance, which results from interchanging the leptons in the interference between the $J/\psi$ production and the Bethe-Heitler mechanisms. We show that to good approximation this asymmetry depends linearly on $a_{\psi p}$, and can reach values around -25% for forthcoming $J/\psi$ threshold production experiments at Jefferson Lab. Its measurement can thus provide a very sensitive observable for a refined extraction of $a_{\psi p}$.