{"id":"e1d070af-049c-48fb-ada4-8657a170b831","arxiv_id":"2501.18289","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"All common NLO factorisation schemes for proton PDFs are special cases of a single discrete family, and scheme choice can shift LHC jet predictions by up to 20-30%.","lead":"Proton parton distribution functions can be defined in several different 'factorisation schemes' at next-to-leading order. This paper collects those schemes into one notation and shows that they reduce to a few simple choices, which can change LHC predictions by more than the usual scale uncertainty.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The quantitative scheme-uncertainty claim rests on method (iii), and self-consistent scheme evolution could materially change the reported spreads.","rationale":"The reader's weakest assumption correctly identifies the method-(iii) dependence of the numerical comparisons. I agree that this is the most load-bearing point for the quantitative claim, because the analytical unification result is independent of how PDFs are obtained and is well supported by explicit kernels, coefficient-function tables, and two independent numerical implementations. The numerical scheme-uncertainty estimates, however, are conditional on transforming MS-evolved PDFs locally at each scale; a self-consistent scheme definition would require transformation at the input scale plus evolution with the scheme-specific splitting functions. Since the transformation is non-perturbative at low scales for some schemes, method (iii) may systematically understate or distort the true scheme dependence. The proposed test would settle whether method (ii) changes the PDFs and cross-section ratios enough to invalidate the Sec. 4 conclusions. Until such a test is done, the conditional verdict is appropriate; I do not see grounds to reject the paper, whose analytical core is a fair and useful synthesis.","tokens_in":53707,"tokens_out":21804,"duration_ms":227444,"concrete_test":"Take CT18NLO MS input at Q0 and, for the Krk and Mpos schemes, generate PDFs by method (ii): transform at Q0 with the kernels of Tables 1-4, then evolve to Q = 100 GeV using the FS DGLAP kernels with PFS(2) from Eq. (23), which differs from MS through the beta K and [K,P] terms. Compare these to the method-(iii) grids used in Figs. 1, 14 and 15. If the PDFs or the Z/H+jet cross-section ratios change by more than about 10% relative to the MS-scheme scale-variation band, the Sec. 4 claim is method-dependent rather than a property of the schemes themselves.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative conclusion that factorisation-scheme variation can exceed scale uncertainty (Sec. 4) is supported only by PDF grids made with method (iii): MS PDFs evolved in the MS scheme and then transformed locally at the target scale via Eq. (5). The three methods for obtaining PDFs in a given FS are inequivalent (Sec. 3), and the paper acknowledges in Sec. 5 that conclusions may not extend to methods (i)/(ii). What makes this load-bearing is that method (iii) is the only one that never applies the transformation kernel at the low input scale, where the O(alpha_s) correction is not small: Sec. 3.2.3 reports the NLO gluon contribution comparable to LO for Krk/Mpos at Q = 2 GeV, and Appendix D shows the perturbative inversion fails badly at such scales. A PDF fitted or evolved in the scheme would carry those large low-scale corrections through evolution to LHC scales, so the 20-30% cross-section shifts and the order-of-magnitude low-x gluon spread may be partly artefacts of applying the kernel only at high scales. The analytical unification claim (Eqs. 69-72) is not affected; the numerical spread is.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper presents a systematic comparison of NLO factorisation schemes for proton PDFs, collecting the definitions of the Dis, Krk/KrkDy, Dpos/Pos/Mpos/Mposδ, A versa, and Phys schemes in a common notation and expressing each transformation kernel in a unified decomposition of distributions, logarithms, rational functions, and delta-function terms (Tables 1–4). The main analytical result is that, after neglecting the polynomial piece P(z), all considered kernels are special cases of a common parametric form (Eqs. 69–72) with a handful of discrete coefficients. The paper further derives the corresponding coefficient functions for DIS, Drell–Yan and Higgs production (Tables 5–7), studies the momentum and number sum rules, examines positivity using an adaptation of the argument of [46], and presents numerical results for PDFs transformed from CT18NLO, NNPDF40MC and MSHT20nlo PDFs. These are used to compute LO Z+jet and H+jet cross-sections, finding that scheme-to-scheme variation can exceed the conventional MS factorisation-scale uncertainty, especially at low pT.","tokens_in":53831,"tokens_out":8945,"duration_ms":86094,"significance":"The analytical unification, if correct, is a valuable contribution: it reduces the space of scheme choices to a small set of discrete parameters and provides a common language for future scheme proposals. The tabulated kernels and coefficient functions will serve as a reference. The numerical work is carefully validated: two independent convolution codes agree to order 10^{-5}, the momentum sum rule is reproduced to approximately 10^{-6}, and the Krk and Phys implementations are checked against original codes. The paper also gives a fair presentation of the limitations of its numerical study, noting explicitly that only method (iii) for defining scheme PDFs is used. The positivity analysis extends the argument of [46] to alternative schemes and is appropriately labelled preliminary. Taken together, the paper is a useful and reliable resource, provided the phenomenological claims are properly scoped.","major_comments":[{"comment":"The central quantitative conclusion of Sec. 4, that factorisation-scheme variation can exceed the factorisation-scale uncertainty, is obtained exclusively with PDFs defined by method (iii): MS PDFs evolved in MS and transformed locally at each scale via Eq. (5). The three methods are inequivalent (Sec. 3), and the paper explicitly acknowledges in Sec. 5 that the conclusions may not apply to methods (i) and (ii). This is not a mere technicality: Sec. 3.2.3 reports that the NLO gluon contribution is comparable to the LO term for the Krk and Mpos schemes at Q = 2 GeV, and Appendix D shows that the perturbative inversion fails at Q = 1.3 GeV. A PDF fitted or evolved in the scheme would carry these large low-scale corrections through the DGLAP evolution to LHC scales, so the 20–30% cross-section shifts in Figs. 14 and 15 may be partly artefacts of applying the transformation only at high scales. To support the Sec. 4 claim as stated, the authors should either perform a consistency test using method (ii) (transform at the input scale and evolve with the modified DGLAP kernels) for at least the most extreme schemes (Krk, Mpos), or substantially soften the abstract and Sec. 4 conclusions to state clearly that the numerical estimates apply only to method (iii). The analytical unification is unaffected, but the phenomenological claim is load-bearing and currently under-supported.","section":"Sec. 3 (methods), Sec. 4, Sec. 5"},{"comment":"The unified form of Eqs. (69–72) is obtained after setting the polynomial piece P(z) to zero. For the Mpos scheme, P(z) is not a numerically suppressed detail: the qq and gg kernels in Tables 1 and 4 consist solely of the soft-function terms 350/3 z^2(1−z)^2, which are the scheme's defining momentum-conservation mechanism, and Fig. 8b shows that this choice produces an O(1) difference in the momentum-fraction distribution relative to Mposδ. Consequently, the statement that Mpos is a special case of Eq. (69) is only an approximation that omits the term that distinguishes the scheme. The conclusion that the 'domain of interest for factorisation-scheme variation within the literature is much smaller than might initially be assumed' therefore needs either a quantitative demonstration that P(z) is negligible for all considered schemes (which Fig. 8b contradicts for Mpos), or a more cautious wording that the unified form captures only the distributional and logarithmic content of the kernels.","section":"Sec. 5, Eqs. (69–72)"}],"minor_comments":[{"comment":"The sentence 'we show the decomposition of the transformed gluon PDFs in in Fig. 4' contains a duplicated 'in'.","section":"Sec. 3.2.3"},{"comment":"The sign convention for the delta-function term (the decomposition writes '- Δ δ(1−z)' but the tables list positive values under a '-δ(1−z)' heading) should be explained explicitly the first time the decomposition is used.","section":"Eq. (33), Tables 1–4"},{"comment":"The scheme name is spelled inconsistently: 'A versa' in the text and tables, 'Aversa' in figure captions and the appendix; please unify.","section":"Throughout"},{"comment":"The modified cumulant of Eq. (49) places the absolute value inside the integral, which is a strengthening of the criterion in [46]; the text notes this only in a footnote, but since the subsequent numerical comparison uses this modified criterion, its role should be highlighted in the main discussion.","section":"Sec. 3.5.1, Eq. (49)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within scope for EPJC and the analytical content is solid. The main risk is that the abstract and Sec. 4 overstate the numerical results, which are conditional on method (iii); a revision that tempers these claims or adds a method-(ii) consistency check would make the paper acceptable. I see no problems with the citation pattern; reference [60] is a private communication that could be expanded. The paper fits the journal's remit and, after the requested revisions, would be a useful reference."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The real payoff here is the unified treatment. The paper collects the published NLO factorisation schemes into one notation, lists the kernels in Tables 1-4, and then shows that almost all of them are special cases of a small discrete parameter space (Eqs. 69-72). That observation is new and it does real work: it tells the community that scheme variation within the existing literature is narrower than it looks. The coefficient-function tables and the Mellin-space comparison in Sec. 3.6 are also useful reference material.\n\nThe paper is careful. Two independent convolution codes agree to ~1e-5, the momentum sum rule is recovered to ~1e-6, and the Krk and Phys implementations are checked against the original codes. The authors are explicit about the three inequivalent ways to define PDFs in an alternative scheme, and they state plainly that they only use method (iii), local transformation of MS-evolved PDFs. They also flag in Sec. 5 that their conclusions may not carry over to fits or evolution performed directly in the scheme.\n\nThe soft spot is exactly that. The quantitative scheme-uncertainty claim in Sec. 4 - that scheme variation can exceed factorisation-scale variation - rests entirely on method (iii). The stress-test note is right that this matters. For Krk and Mpos the NLO gluon correction is comparable to the LO term at Q=2 GeV (Sec. 3.2.3), and Appendix D shows the perturbative inversion fails badly at low scales. A PDF fitted or evolved in the scheme would carry those large low-scale corrections through DGLAP evolution, so the 20-30% shifts and the order-of-magnitude low-x gluon spread are, to an unknown extent, an artefact of applying the kernel only at high scales. The analytic unification claim is untouched; the numerical spread is conditional. The paper is honest about this, so it is a fair limitation rather than a hidden one.\n\nMinor: no code is released, but the validation appendix is detailed enough to reproduce the numerics.\n\nWho should read this: anyone fitting PDFs, matching NLO calculations to parton showers, or estimating theory uncertainties from scheme choice. It deserves a serious referee. The analytical core should be published; the numerical section needs either a softening of the scheme-uncertainty claim or a demonstration that method (ii) gives similar spreads for at least one scheme. I would accept it for peer review.","headline":"A genuinely useful synthesis of NLO PDF factorisation schemes, with a solid analytic core and a conditional numerical tail that the authors themselves flag.","tokens_in":54447,"tokens_out":2793,"would_cite":true,"duration_ms":32234,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["12.38.-t","12.38.Bx"],"model":"deepseek-v4-flash","headline":"All the major NLO factorisation schemes for proton PDFs reduce to one general form with a few discrete coefficients, and switching schemes can shift LHC cross-section shapes by more than the usual scale-variation uncertainty does.","keywords":["factorisation scheme","parton distribution functions","NLO QCD","MS scheme","DGLAP evolution","PDF positivity","LHC phenomenology","threshold logarithms"],"falsifier":"Recompute the transformed PDFs and the Z-plus-jet and Higgs-plus-jet ratios using the alternative construction the paper does not use — transform the $\\overline{\\mathrm{MS}}$ input at the starting scale, then evolve each PDF with that scheme's modified DGLAP kernels — and compare the between-scheme spread with the local-transformation results of Figs. 14 and 15; if the spread drops below the scale-variation band or the low-$x$ gluon enhancement disappears, the reported scheme dependence is an artifact of the transformation prescription rather than a property of the schemes themselves.","tokens_in":53418,"feed_emoji":"⚛️","tokens_out":15221,"duration_ms":125516,"temperature":0.7,"pith_summary":"Beyond leading order, perturbative QCD leaves a genuine choice in how the short-distance partonic cross-section and the long-distance parton distribution functions (PDFs) are separated; that choice is the factorisation scheme, with $\\overline{\\mathrm{MS}}$ the default. The paper assembles the main alternative schemes in the literature — the DIS scheme, the Krk scheme built for NLO parton-shower matching, the positivity-motivated Pos/Mpos family, the Aversa scheme, and the Phys scheme — into one common notation, and shows that at NLO every one of them is a special case of a single general form for the four scheme-transformation kernels, differing only in a few discrete coefficients. If true, the space of scheme choices actually explored in the literature is far smaller than it appears, and a systematic survey of the remaining choices becomes feasible. The paper also estimates the practical cost of the choice: for Z-plus-jet and Higgs-plus-jet kinematics at the LHC, the spread between schemes can exceed the conventional factorisation-scale uncertainty, so scale variation alone may understate the theoretical uncertainty of an NLO calculation.","feed_headline":"One formula spans all major proton-PDF schemes","feed_subtitle":"A unified form covers all major NLO schemes, and scheme choice shifts LHC rates beyond the usual scale-variation band.","key_machinery":"The load-bearing object is the transformation kernel $K^{\\overline{\\mathrm{MS}}\\to\\mathrm{FS}}_{ab}(z)$, which converts $\\overline{\\mathrm{MS}}$ PDFs into another factorisation scheme by a convolution integral performed at each scale. Every kernel is decomposed into standard pieces — the plus-distributions $D_k$, $\\log(1-z)$ and $\\log z$ terms, a rational polynomial $P(z)$, and a $\\delta(1-z)$ term — and the paper shows that all studied schemes are recovered from one general four-kernel template (Eqs. 69–72) by choosing small discrete values for the coefficients $a$, $b$, $c$ and fixing the diagonal delta terms through the momentum sum rule. The template captures the threshold logarithms that dominate the differences: schemes with the largest $a_{qq}$ and $a_{gg}$ values absorb the double-log $D_1$ soft-gluon terms into the PDFs, leaving the Drell–Yan and Higgs coefficient functions asymptotically constant in Mellin space, while schemes with smaller coefficients leave those terms in the coefficient functions.","core_discovery":"The paper's central claim is unification: at NLO, every factorisation scheme it studies is obtained from $\\overline{\\mathrm{MS}}$ by convolution with kernels $K_{qq}$, $K_{qg}$, $K_{gq}$, $K_{gg}$ that share one common structure (Eqs. 69–72), and the differences between schemes reduce to the values of a few discrete coefficients ($a_{qq}, a_{gg} \\in \\{0,1,2\\}$; $a_{qg}, a_{gq} \\in \\{1,2\\}$; the remaining $b$ and $c$ coefficients in $\\{0,1\\}$), with the diagonal delta-function terms fixed by the momentum sum rule. Two exceptions are noted: the DIS-scheme gluon kernels, fixed by a local momentum-conservation convention, and the Aversa gluon kernel, which uses a single rational term. On the authors' reading, 'the domain of interest for factorisation-scheme variation within the literature is much smaller than might initially be assumed' (Sec. 5). The supporting numerical claim is that the choice matters at the LHC: the transformed low-$x$ gluon can be an order of magnitude larger at low scales, and for Z-plus-jet and Higgs-plus-jet production the scheme-to-scheme spread reaches roughly 30% and 20% respectively in some regions, exceeding the factorisation-scale variation band.","pith_inferences":["Because all studied schemes sit in one small discrete parameter space, a systematic scan of that space would define a complete scheme-envelope uncertainty for LHC processes; the paper samples only the schemes that happen to exist in the literature, not the full space its template spans.","The reported 20–30% shifts are tied to the paper's transformation-at-each-scale prescription; transforming at the input scale and evolving in the scheme instead could redistribute the effect between the PDFs and the DGLAP evolution, and checking whether the shifts survive is the most direct test of how intrinsic to the schemes they are.","The heavy-quark negativity just above threshold suggests a pragmatic remedy the paper mentions only in passing: moving the flavour-number transition above the quark mass would very likely restore positivity without disturbing the unification, since the transformation kernels themselves are unchanged."],"forward_implications":["The space of NLO factorisation schemes in the literature is far smaller than it appears: the studied schemes differ only through a handful of discrete coefficients in one general kernel form.","For Z-plus-jet and Higgs-plus-jet observables, the scheme-to-scheme spread exceeds the factorisation-scale variation band in several kinematic regions, so scale variation alone can understate the theoretical uncertainty of an NLO prediction.","The choice of scheme modifies the valence-quark number sum rules at $\\mathcal{O}(\\alpha_s)$ and makes them scale-dependent in most schemes, which must be handled if PDFs are fitted in those schemes.","Heavy-quark PDFs turn negative just above their mass thresholds in every studied scheme when they are generated perturbatively, so positivity cannot be assumed as a universal property of alternative factorisation schemes."],"supporting_citations":[{"why":"Supplies the original DIS-scheme definition, the earliest alternative to MS that the paper collects into its unified notation.","marker":"[1]"},{"why":"Defines the MS scheme, the baseline from which every transformation in the paper is made.","marker":"[2]"},{"why":"Defines the Aversa scheme, one member of the unified set of transformation kernels.","marker":"[13]"},{"why":"Introduces the Dpos/Pos/Mpos positivity-motivated schemes that the paper shows fit the general template.","marker":"[15]"},{"why":"Define the KrkDy and Krk schemes from Catani–Seymour subtraction dipoles, the schemes that absorb the D1 threshold terms.","marker":"[16,17]"},{"why":"Defines the Phys scheme, included in the comparison.","marker":"[22]"},{"why":"Provides the momentum-conservation convention that fixes the DIS-scheme gluon kernels in the tables.","marker":"[43]"},{"why":"Supplies the positivity argument the paper adapts to every scheme in Sec. 3.5.","marker":"[46]"},{"why":"Provides the CT18NLO MS-scheme PDF set used for the main numerical transformations.","marker":"[56]"},{"why":"Provides the NNPDF40MC MS-scheme PDF set used in the LHC phenomenology comparisons.","marker":"[57]"}],"fun_headline_variants":["One kernel unifies all NLO proton PDF schemes","All PDF schemes share a single kernel form","Scheme choice shifts LHC rates up to 30%","Proton PDF schemes collapse to one structure","All major PDF schemes share one NLO form"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The quantitative conclusions — the order-of-magnitude low-$x$ gluon enhancement and the 20–30% cross-section shifts — are computed by taking $\\overline{\\mathrm{MS}}$-fitted PDFs and transforming them locally at every scale, which is only one of three inequivalent ways to define PDFs in an alternative scheme, and the paper states that its conclusions may not apply to the other two.","fun_headline_variants_meta":{"raw":{"variants":["One kernel unifies all NLO proton PDF schemes","All PDF schemes share a single kernel form","Scheme choice shifts LHC rates up to 30%","Proton PDF schemes collapse to one structure","All major PDF schemes share one NLO form"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000834,"raw_usage":{"total_tokens":3637,"prompt_tokens":941,"completion_tokens":2696,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":2624}},"tokens_in":557,"tokens_out":2696,"duration_ms":17174,"temperature":1.0,"reasoning_tokens":2624,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T00:02:08.742710+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the transformed PDFs and the Z-plus-jet and Higgs-plus-jet ratios using the alternative construction the paper does not use — transform the $\\overline{\\mathrm{MS}}$ input at the starting scale, then evolve each PDF with that scheme's modified DGLAP kernels — and compare the between-scheme spread with the local-transformation results of Figs. 14 and 15; if the spread drops below the scale-variation band or the low-$x$ gluon enhancement disappears, the reported scheme dependence is an artifact of the transformation prescription rather than a property of the schemes themselves.","supporting_citations":[{"cited_title":"Scheme dependence, leading order and higher twist studies of MRST partons","cited_arxiv_id":"hep-ph/9808371","evidence_quote":"Provides the momentum-conservation convention that fixes the DIS-scheme gluon kernels in the tables."}],"review_version":1}