The standard choice is the modified minimal subtraction scheme, known as MS-bar, which has become the universal language of modern quantum chromodynamics, the theory of the strong force. It earned that status through sheer practical convenience: its mathematical structure keeps fixed-order calculations tidy, and the enormous computational infrastructure built up during the so-called NNLO revolution, from global PDF fits to Monte Carlo generators, was constructed entirely around it. Early pioneers of PDF fitting sometimes provided results in both the MS-bar scheme and the older DIS scheme, but as calculations grew more sophisticated, the alternatives fell away. Today, essentially every PDF set, every evolution code and every prediction tool on the market speaks only MS-bar. That monopoly has an uncomfortable consequence: nobody actually knows how large the theoretical uncertainty associated with this particular convention might be.

The new paper attacks that blind spot head-on. The authors derive the next-to-leading-order DGLAP splitting functions, the kernels of the coupled integro-differential equations that describe how quark and gluon distributions change with the energy scale at which the proton is probed, in a family of alternative factorisation schemes. These include the Krk scheme, which underpins the KrkNLO method for matching precise calculations to parton shower simulations, and the Phys scheme, designed to strip out certain infrared contributions that linger in MS-bar from long-distance interactions between massless partons. Crucially, they also introduce a parametrised ‘generalised’ scheme, dubbed GEN, whose adjustable coefficients span a physically motivated subspace of the entire space of possible factorisation schemes. Setting the parameters to zero recovers MS-bar; other choices reproduce the named schemes, and still other choices explore territory no one has mapped before.

The technical machinery behind the result is a Mathematica package called DGLAPFS, built by the authors to handle the Mellin convolutions and commutator structures that arise when transformation kernels act on the known MS-bar splitting functions. The calculations were verified in several independent ways: the resulting kernels satisfy the momentum and valence-quark sum rules numerically to a tolerance of ten to the minus seventieth, colour factor by colour factor, and the DIS-scheme results reproduce previously published commutators. The package even yields closed-form analytic expressions for the DIS scheme splitting functions that, remarkably, had never before appeared in the literature despite the scheme’s historical significance.

What emerges from the plots is striking. In the interior of the interval between zero and one, where the momentum fraction x takes moderate values, the different schemes agree reasonably well, because the perturbative transformation kernels are bounded there. But at the edges, the schemes diverge dramatically. As x approaches one, corresponding to a parton carrying nearly all of the proton’s momentum, the splitting functions can develop increasingly severe singularities of the form of logarithms of one minus x divided by one minus x. As x approaches zero, relevant to the high-energy frontier, singularities of the form of logarithms of x divided by x appear. Scheme transformations change not just the size but the degree of these singularities, meaning that formally higher-order corrections, though suppressed by powers of the strong coupling, can balloon in these kinematic limits.

The authors trace this behaviour to a handful of parameters in their generalised scheme. In the large-x limit, the parameter a controls whether double-logarithmic threshold terms appear in the evolution; in the small-x limit, the parameter c plays the analogous role for high-energy logarithms. The MS-bar scheme turns out to be atypical in both directions. Its splitting functions never contain plus-distributions more singular than the simplest one, to all perturbative orders, and long-understood ‘accidental’ cancellations tame its small-x behaviour at NLO. In other words, the perturbative convergence physicists have grown accustomed to is a special property of one convention, not a general feature of QCD evolution.

Perhaps the most elegant result of the paper is the physical interpretation of these leading corrections. The dominant scheme-dependent terms are all proportional to the beta function, which governs how the strong coupling runs with scale, and they can be reinterpreted as what happens when the coupling is evaluated not at the nominal factorisation scale but at an x-dependent effective scale. In the large-x limit, that effective scale takes the form of the nominal scale multiplied by a power of one minus x, with the power set by the scheme parameter. The MS-bar scheme corresponds to no shift at all; the DIS and Phys schemes correspond to an effective scale of the nominal scale times the square root of one minus x; and the Krk scheme corresponds to the nominal scale times one minus x. This hierarchy mirrors the familiar scale structure of soft-collinear effective theory, matching the hard, jet and soft scales respectively. At small x, an analogous interpretation holds, with the Krk scheme effectively evolving the gluon at a scale divided by the square root of x, reminiscent of the angular-ordering hierarchy of coherent parton evolution.

The practical implications reach into resummation, the art of summing infinite towers of large logarithms to all orders. Threshold resummation at large x and BFKL-style high-energy resummation at small x both depend on the asymptotic behaviour of the splitting functions, and the new results show exactly how that behaviour changes from scheme to scheme. Numerical studies of a benchmark valence-quark distribution show that evolution in a Krk-like scheme suppresses the distribution at large momentum fractions by an amount comparable to the entire NLO correction in MS-bar, a genuinely large effect. On the small-x side, comparisons with the resummed HELL 4.0 calculations show that over the experimentally relevant region the resummed results sit comfortably within the envelope spanned by the named schemes, suggesting that evolution in alternative schemes should be numerically well-behaved even where the asymptotics differ qualitatively.

The work is explicitly a foundation-laying exercise rather than a phenomenological verdict. Evolving and fitting PDFs directly in alternative schemes, the authors note, is a prerequisite for quantifying factorisation scheme uncertainty in the same way physicists already quantify factorisation scale uncertainty, by varying the convention and observing how predictions move. Their results also extend naturally to timelike evolution of fragmentation functions through a known conversion, opening the door to scheme studies on the final-state side as well. The authors suggest that the dominant effect of scheme variation can even be captured in existing evolution codes simply by promoting the scale of the running coupling to a function of x, which would simultaneously resum the associated higher-order terms. What the study ultimately delivers is a map: a reduced, physically interpretable space of factorisation schemes, each with a transparent effective-scale meaning, within which systematic scheme-variation studies at NLO and beyond suddenly look feasible. For a field whose precision ambitions at future colliders hinge on taming every last theoretical uncertainty, that map may prove one of the most quietly consequential documents of the decade.

Subject of Research: Next-to-leading-order DGLAP splitting functions and PDF evolution in alternative QCD factorisation schemes

Article Title: PDF evolution in alternative factorisation schemes

Article References: Delorme, S., Kusina, A., Siódmok, A., & Whitehead, J. (2026). PDF evolution in alternative factorisation schemes. The European Physical Journal C, 86(9), Article 1111. https://doi.org/10.1140/epjc/s10052-026-16303-w

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16303-w

Keywords: parton distribution functions, DGLAP evolution, factorisation schemes, splitting functions, quantum chromodynamics, MS-bar scheme, KrkNLO, threshold resummation, small-x resummation, anomalous dimensions, running coupling, perturbative QCD