Factorisation schemes for proton PDFs

Beyond leading-order, perturbative QCD requires a choice of factorisation scheme to define the parton distribution functions (PDFs) and hard-process cross-section. The modified minimal-subtraction ($\overline{\mathrm{M}\mathrm{S}}$) scheme has long been adopted as the default choice due to its simplicity. Alternative schemes have been proposed with specific purposes, including, recently, PDF positivity and NLO parton-shower matching. In this paper we assemble these schemes in a common notation for the first time. We perform a detailed comparison of their features, both analytically and numerically, and estimate the resulting factorisation-scheme uncertainty for LHC phenomenology.