Upsilon photo-production on the proton at the Electron-Ion Collider

We present a dispersive analysis with the aim to extract the $\Upsilon$-p scattering length from $\gamma p \to \Upsilon p$ experiments. In this framework, the imaginary part of the $\Upsilon$-p forward scattering amplitude is obtained from $\gamma p \to \Upsilon p$ cross section measurements, and is constrained at high energies from existing HERA and LHC data. Its real part is calculated through a once-subtracted dispersion relation, and the subtraction constant is proportional to the $\Upsilon$-p scattering length. We perform a feasibility study for $\Upsilon$ photo-production experiments at an Electron-Ion Collider and discuss the sensitivity and precision that can be reached in the extraction of the $\Upsilon$-p scattering length.