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We cure the perturbative instability of the total-inclusive-photoproductioncross sections of vector $S$-wave quarkonia observed at highphoton-proton-collision energies ($\sqrt{s_{\gamma p}}$) in Next-to-LeadingOrder (NLO) Collinear-Factorisation (CF) computations. This is achieved usingHigh-Energy Factorisation (HEF) in the Doubly-Logarithmic Approximation (DLA),which is a subset of the Leading-Logarithmic Approximation (LLA) of HEF whichresums higher-order QCD corrections proportional to $\alpha_s^n \ln^{n-1}(\hat{s}/M^2)$ in the Regge limit $\hat{s}\gg M^2$ with $M^2$ being thequarkonium mass and $\hat{s}$ is the squared partonic-center-of-mass energy.Such a DLA is strictly consistent with the NLO and NNLO DGLAP evolutions of theParton Distribution Functions. By improving the treatment of thelarge-$\hat{s}$ asymptotics of the CF coefficient function, the resummationcures the unphysical results of the NLO CF calculation. The matching isdirectly performed in $\hat{s}$ space using the Inverse-Error Weightingmatching procedure which avoids any possible double counting. The obtainedcross sections are in good agreement with data. In addition, thescale-variation uncertainty of the matched result is significantly reducedcompared to the LO results. Our calculations also yield closed-form analyticlimits for $\hat{s}\gg M^2$ of the NLO partonic CF and numerical limits forcontributions to those at NNLO scaling like $\alpha_s^2 \ln(\hat{s}/M^2)$.

Curing the high-energy perturbative instability of  vector-quarkonium-photoproduction cross sections at order $\alpha \alpha_s^3$  with high-energy factorisation