Revisiting NLO QCD corrections to total inclusive $J/$$\psi$ and $\Upsilon$ photoproduction cross sections in lepton-proton collisions

We revisit inclusive $J/$$\psi$ and $\Upsilon$ photoproduction at lepton-hadron colliders, namely in the limit when the exchanged photon is quasi real. Our computation includes the next-to-leading-order (NLO) $\alpha_s$ corrections to the leading-order contributions in v. Similarly to the case of NLO charmonium-hadroproduction processes, the resulting cross sections obtained in the $\overline{MS}$ factorisation scheme are sometimes found to be negative. We show that the scale fixing criterion which we derived in a previous study of $\eta_c$ production successfully solves this problem from the EicC all the way up to the FCC-eh energies. We then elaborate on how to study a scale uncertainty akin to that derived by scale variations when one fixes a scale. In turn, we investigate where both $J/$$\psi$ and $\Upsilon$ photoproduction could be used to improve our knowledge of gluon content of the proton at scales as low as a couple of GeV