Curing the high-energy perturbative instability of vector-quarkonium-photoproduction cross sections at order α α ₛ³ with high-energy factorisation

We cure the perturbative instability of the total-inclusive-photoproduction cross sections of vector S -wave quarkonia observed at high photon-proton-collision energies ( √sγ p s γ p ) in Next-to-Leading Order (NLO) Collinear-Factorisation (CF) computations. This is achieved using High-Energy Factorisation (HEF) in the Doubly-Logarithmic Approximation (DLA), which is a subset of the Leading-Logarithmic Approximation (LLA) of HEF which resums higher-order QCD corrections proportional to α ₛⁿ ln ⁿ⁻¹ (ŝ/M²) α s n ln n - 1 ( s ^ / M 2 ) in the Regge limit ŝ M² s ^ ≫ M 2 with M² M 2 being the quarkonium mass and ŝ s ^ is the squared partonic-center-of-mass energy. Such a DLA is strictly consistent with the NLO and NNLO DGLAP evolutions of the parton distribution functions. By improving the treatment of the large- ŝ s ^ asymptotics of the CF coefficient function, the resummation cures the unphysical results of the NLO CF calculation. The matching is directly performed in ŝ s ^ space using the Inverse-Error Weighting matching procedure which avoids any possible double counting. The obtained cross sections are in good agreement with data. In addition, the scale-variation uncertainty of the matched result is significantly reduced compared to the LO results. Our calculations also yield closed-form analytic limits for ŝ M² s ^ ≫ M 2 of the NLO partonic CF and numerical limits for contributions to those at NNLO scaling like α ₛ² ln (ŝ/M²) α s 2 ln ( s ^ / M 2 ) .