{"id":"4fb64aad-cbe0-4655-be9d-be9e025792a6","arxiv_id":"2412.09589","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The authors build a complete NLO momentum-space DGLAP evolution for a six-observable physical basis of DIS structure functions and compare its numerical results with conventional PDF-based evolution.","lead":"This paper derives and solves the next-to-leading-order DGLAP evolution equations for six deep-inelastic-scattering structure functions directly, without any parton distribution functions. The work is a step toward QCD predictions that are free of factorization scale and scheme choices, relevant for future electron-ion collider data.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The NLO physical-basis kernels hinge on Eq. (C1), a paper-specific boundary relation that is not independently checked; the unresolved discrepancy with Ref. [32] in the two-observable limit means the central claim requires a direct consistency test.","rationale":"The reader identified Eq. (C1) as the weakest assumption, and my reading confirms this is the most load-bearing point. The paper's central claim is that Eqs. (32)-(37) together with the Appendix C kernels give a closed, scheme-independent NLO DGLAP evolution of six structure functions. The derivation necessarily involves partial integration of the differential operator \\hat{P}, and the boundary terms at x=1 are removed using the paper-specific relation (C1). This relation is not derived in detail, and the paper's own footnote 1 reports an unresolved discrepancy with the two-observable limit of Ref. [32], which could indicate an error in the boundary treatment rather than in the reference. The numerical comparisons in Figs. 1-2 show differences of order 10%, which the authors attribute entirely to higher-order terms, but no explicit test is given that the residual is of the expected order in alpha_s. Because the correctness of every kernel that involves fF'_2 or fF'_L depends on Eq. (C1), this is the single point where the argument could fail. The reader's CONDITIONAL verdict is therefore appropriate: the derivation is plausible and technically demanding, but it should be accepted only after an independent check of the boundary handling, for example by comparing the operator form and the kernel form of the evolution as described in the concrete test. My concern does not change the verdict, so I recommend UNCHANGED.","tokens_in":30574,"tokens_out":8342,"duration_ms":76151,"concrete_test":"Implement the evolution in the operator form, keeping \\hat{P} acting on structure functions as in Eqs. (32)-(37) with G(0), G(1) from Eqs. (18), (28), using a numerical method that evaluates derivatives (e.g., Chebyshev or finite differences on a fine grid) and handles the plus-distributions exactly. Compare the resulting dF_i/d log Q^2 with the kernel form (41)-(46) using the same input structure functions from CT18NLO NF3 at Q^2=1.69 GeV^2. If the two forms agree to the expected O(alpha_s^2) truncation error, Eq. (C1) and the partial integration are correct. In addition, vary the coefficient -0.003 in Eq. (C1) by ±20%; if the physical-basis evolution changes by more than the truncation uncertainty, the result is sensitive to this paper-specific input.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result, Eqs. (32)-(37) and the explicit kernels in Appendix C, is obtained by inserting inverted PDFs (which contain the differential operator \\hat{P} of Eq. (19) acting on structure functions) into the standard DGLAP equations and then partially integrating \\hat{P} off the structure functions. This generates boundary terms at x=1 involving \\tilde{F}'_L(1) and \\tilde{F}'_2(1). The paper eliminates these using Eq. (C1), \\tilde{F}'_L(1) = -0.003 (alpha_s/2pi) \\tilde{F}'_2(1) + O(alpha_s^2), whose coefficient is taken from the delta-function term in the NLO coefficient function C^{(2)}_{FLNS}. The relation is asserted, not derived in the text, and it is exactly this relation that converts the operator form (32)-(37) into the usable kernel form (40)-(46). If the coefficient or the O(alpha_s) structure of (C1) is wrong—for instance if additional boundary contributions from the plus-distributions in the splitting functions or from delta-function terms in other coefficient functions are missed—then every kernel that involves fF'_2 or fF'_L would be incorrect, and the equations would not reproduce NLO DGLAP. The paper's footnote 1 reports a disagreement with the two-observable limit of Ref. [32] and attributes it to that reference; this is a red flag that the boundary handling in the reduced case is not settled. Since no independent numerical check (e.g., comparing the \\hat{P}-form and the kernel-form of the evolution) is provided, the correctness of Eq. (C1) is the load-bearing assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper constructs a physical-basis DGLAP evolution at next-to-leading order for six DIS structure functions. Starting from the MS expressions for the structure functions in terms of PDFs, the authors invert these relations iteratively to express PDFs in terms of structure functions, substitute into the NLO DGLAP equations, and obtain a closed set of evolution equations (32)-(37) for the structure functions. After partial integration of the differential operator P-hat, they present explicit evolution kernels in Appendix C. The numerical solution for CT18NLO-based initial conditions is compared with standard PDF evolution, with differences typically at the 10-20% level. The paper also discusses inverted PDFs, sum-rule violations, and the computation of a non-basis structure function. The central claim is that the physical-basis evolution is factorization-scale and scheme independent by construction.","tokens_in":30882,"tokens_out":8224,"duration_ms":73376,"significance":"If correct, this work provides the most extensive NLO physical-basis evolution to date, with potential applications to global fits of DIS data and to cross-section predictions in terms of physical observables. A notable strength is the explicit listing of all evolution kernels and the honest discussion of limitations, including fixed-flavor scheme, massless quarks, and the violation of sum rules by the perturbatively inverted PDFs. The paper also explicitly footnotes a discrepancy with the two-observable limit of Ref. [32], which is commendable but, as detailed below, needs to be resolved. The algebraic derivation is presented in detail and the comparison with CT18 provides a sanity check, but the validation is incomplete due to the asserted boundary relation and the absence of a direct consistency test of the operator and kernel forms.","major_comments":[{"comment":"The boundary relation fF'_L(1) = -0.003 (alpha_s/2pi) fF'_2(1) + O(alpha_s^2) is asserted with only a parenthetical remark that the factor -0.003 multiplies the delta function in C^(2)_FLNS. The text states that the evolution kernels are obtained using this relation, so it is load-bearing for the central result. However, the relation involves the x-derivatives of the structure functions at the endpoint, and it is not shown how the delta-function coefficient alone determines these derivatives; the regular part of C^(2)_FLNS and the O(alpha_s) correction to Xi could contribute to fF'_L(1). Please provide a complete derivation from Eq. (11), including a distributional treatment of the endpoint, and verify the relation numerically on a test structure-function input. Without this, the kernels in Appendix C are not fully justified.","section":"Appendix C, Eq. (C1)"},{"comment":"The paper reports an unresolved discrepancy with the two-observable (F2, FL) limit of Ref. [32], attributing it to a possible inconsistency in that reference. Since the two-observable limit is the minimal independent check of the physical-basis construction, and since the same boundary handling is involved, the authors should either demonstrate the inconsistency in Ref. [32] (for example by reproducing its Mellin-space computation) or reconcile the two results. As it stands, the discrepancy leaves the NLO derivation without an independent limiting-case validation.","section":"Footnote 1"},{"comment":"The numerical solution is presented only via the kernel form (41)-(46), with no independent check that these kernels reproduce the original operator equations (32)-(37). Given the nontrivial partial integration and the use of Eq. (C1), a consistency test comparing dF_i/d log Q^2 computed directly from (32)-(37) (with explicit derivatives of the structure functions) and from the Appendix C kernels for the same input would materially strengthen the validation. Please add such a test, at least for a representative structure function (e.g., F2 or FL) over the x-range used in the figures.","section":"Section III"}],"minor_comments":[{"comment":"The notation 'log(1 - x)^2' is ambiguous; please use log^2(1-x) or [log(1-x)]^2 to avoid confusion with log[(1-x)^2].","section":"Appendix A, Eq. (A3)"},{"comment":"The numerical method is not described (x-grid, interpolation for derivatives, treatment of plus-distributions and endpoint terms). A brief description would improve reproducibility.","section":"Section III"},{"comment":"The phrase 'colourful curves' is informal; consider 'colored curves' or a more precise description of the lines and markers.","section":"Figure 2 caption"},{"comment":"The important caveat about the discrepancy with Ref. [32] is placed in a footnote; it would be better moved to the main text or an appendix, since it bears directly on the validation of the results.","section":"Footnote 1"},{"comment":"The notation (C ⊗ G^(1))_{Fi} is introduced but not fully explained for all Fi in the list; a short definition of how these terms arise after partial integration would aid the reader.","section":"Appendix C, Eq. (C17)"},{"comment":"The paper would benefit from a statement on the availability of the numerical code or input files; as it stands, reproducing the figures requires reimplementation of long expressions.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is technically ambitious and honest about limitations. The main risks are the asserted boundary relation (C1) and the unresolved discrepancy with Ref. [32]; both are fixable with additional derivations and checks. The paper is within the scope of the journal. I recommend major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is the NLO momentum-space DGLAP evolution for a six-observable physical basis of DIS structure functions, with all evolution kernels written out in Appendix C. That is a real step beyond the earlier two-observable bases and the authors' own LO work, and the derivation is systematic: invert the PDF relations through an iterative αs expansion, insert into the standard DGLAP equations, and partially integrate the differential operator P-hat off the structure functions. The scheme and scale cancellation is argued plausibly, and the numerical comparison to CT18 shows differences of a few to roughly ten percent, which is consistent with higher-order truncation. No fitted parameters enter; the inputs are standard coefficient functions and splitting functions. Credit where due: the paper is honest about sum-rule breaking in the physical-basis inverted PDFs and about the limited 3-flavour applicability.\n\nThe soft spots are real but not fatal. The main one is Eq. (C1), the boundary relation fF'_L(1) = -0.003 (αs/2π) fF'_2(1) + O(αs^2). This relation is asserted rather than derived, and it is exactly what converts the operator-form equations (32)-(37) into the usable kernel form (40)-(46). The coefficient is read off from the delta-function term in C^(2)_FLNS, so it is not pulled from thin air, but the boundary handling deserves a derivation or at least a numerical consistency check. The footnote about the disagreement with Ref. [32] in the two-observable limit is unresolved; the authors blame that reference, but without a detailed cross-check I read that as a yellow flag. There is also a clear typo in Eq. (36) where P^(0)_qg should read P^(0)_gq in the term multiplying C^(1)_F2g. And there is no code or data release, so the kernel list is the only artifact.\n\nNone of this invalidates the central derivation on its face. The right next step is an independent verification of Eq. (C1) and ideally a released implementation or a direct comparison between the operator form and the kernel form. This paper deserves serious refereeing; the referee should push on the boundary relation, the Ref. [32] discrepancy, and the typos. I would cite it as the current state of the art in physical-basis evolution, and I'd bring it to a reading group precisely because the boundary-term issue is instructive.","headline":"First NLO six-observable physical-basis DGLAP evolution in momentum space, carefully derived but with a load-bearing boundary relation (C1) that needs an independent check before I'd fully trust the kernels.","tokens_in":31453,"tokens_out":2283,"would_cite":true,"duration_ms":21802,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives a closed set of NLO evolution equations for six measurable DIS structure functions, replacing parton distribution functions with the observables themselves.","keywords":["DGLAP evolution","structure functions","physical basis","factorization scheme independence","parton distributions","next-to-leading order","deep inelastic scattering"],"falsifier":"Take any NLO PDF set, compute $F_2$ and $F_L$ at several scales, and evaluate the combination $fF'_{L}(1) + 0.003\\,\\frac{\\alpha_s}{2\\pi} fF'_2(1)$; if it does not vanish at order $\\alpha_s^2$ as a function of $Q^2$, the boundary handling is incorrect, and the physical-basis evolution will differ from conventional PDF evolution by a term of order $\\alpha_s$ rather than $\\alpha_s^2$.","tokens_in":30331,"feed_emoji":"⚛️","tokens_out":5481,"duration_ms":51684,"temperature":0.7,"pith_summary":"The paper derives a closed set of next-to-leading-order evolution equations that take six measurable deep-inelastic-scattering structure functions directly from one scale $Q^2$ to another, bypassing parton distribution functions (PDFs) entirely. Because the evolution is written for observables rather than for unobservable PDFs, the factorization scale and factorization scheme dependence cancel by construction, leaving only the renormalization scheme of $\\alpha_s$ as an unphysical choice. Numerically, the physical-basis evolution reproduces the results of conventional PDF-based evolution to within about ten percent for most structure functions, with the valence-type observables agreeing to a few percent. This gives a way to confront perturbative QCD with data that does not require fitting or parametrizing PDFs.","feed_headline":"Six DIS structure functions now evolve at NLO without PDFs","feed_subtitle":"By rewriting DGLAP evolution in a physical basis, scheme and scale dependence drop out of the kernels.","key_machinery":"The machinery is the six-observable physical basis together with the differential operator $\\hat P(x) = x^2 \\frac{d^2}{dx^2} - 2x \\frac{d}{dx} + 2$, which is the exact inverse of the leading-order gluon coefficient function $C^{(1)}_{FLg}$ in the longitudinal structure function. This operator lets the paper invert the PDF–structure-function relation perturbatively, expressing the gluon and quark distributions as series in $\\alpha_s$ of structure functions and their derivatives. Substituting these inverted PDFs into the standard NLO DGLAP equations and using partial integration to move the derivatives onto the structure functions produces the evolution kernels listed in Appendix C; the boundary terms left by partial integration are removed using the relation $fF'_{L}(1) = -0.003\\,\\frac{\\alpha_s}{2\\pi} fF'_2(1) + O(\\alpha_s^2)$.","core_discovery":"The central result is the closed system of equations (32)–(37), accurate to next-to-leading order in $\\alpha_s$, for the six structure functions $F_2$, $F_3$, $\\Delta F_2^W$, $F_3^{W^-}$, $F_{2c}^{W^-}$, and $F_L$. The paper obtains these equations by inverting the NLO coefficient-function relations to express PDFs as power series in $\\alpha_s$ whose coefficients are structure functions and their $x$-derivatives, substituting these inverted PDFs into the conventional DGLAP splitting functions, and partially integrating the differential operator that arises from the inversion. The factorization-scale and scheme dependence cancel between coefficient functions and splitting functions, so the resulting evolution kernels are factorization-scale and scheme independent by construction. When evolved numerically from initial conditions given by a standard NLO PDF set, the physical-basis structure functions track the PDF-evolved ones with differences that are formally higher order in $\\alpha_s$; the deviations are typically below ten percent and smallest for the valence-dominated observables.","pith_inferences":["If the boundary relation at $x=1$ is robust, the same partial-integration strategy could be carried to NNLO, but the appearance of up to three nested differential operators at that order makes accurate numerical handling of high derivatives the main technical challenge.","The negative large-$x$ values of the physical-basis inverted NLO gluon suggest that positivity constraints derived for MS PDFs do not transfer directly to physical-basis PDF counterparts; the paper notes this undercuts a positivity argument made elsewhere for MS PDFs.","The physical basis offers a concrete way to estimate factorization-scheme uncertainties in LHC cross sections: comparing physical-basis predictions with conventional MS predictions quantifies scheme dependence without needing to construct alternate PDF fits.","A natural testable extension is to apply the physical-basis construction to Drell-Yan or other hadronic processes, checking that the scheme and scale dependences cancel in those cross sections exactly as they do for DIS structure functions."],"forward_implications":["Physical-basis DGLAP evolution is factorization-scale and factorization-scheme independent, so the only remaining scheme choice in a calculation is the renormalization scheme of $\\alpha_s$.","Initial conditions for the physical basis are the same at every perturbative order, in contrast to PDFs, which must be refitted at each order and scheme.","Because the inverted PDFs are expressed in terms of structure functions, other PDF-dependent cross sections can be rewritten in the physical basis; the paper demonstrates this by computing the non-basis structure function $F_2^{W^+}$ within about twenty percent of the PDF-based value.","The iterative inversion procedure extends straightforwardly to higher orders in $\\alpha_s$, so the approach can be pushed beyond NLO, with the caveat that more nested differential operators appear at each order.","Extending the framework to a variable-flavour-number scheme and including heavy-quark masses would be needed before the physical basis can serve as a full phenomenology-ready alternative to PDF evolution."],"supporting_citations":[{"why":"Introduces the physical-basis concept for DIS structure functions, which this paper extends to a six-observable NLO basis.","marker":"[20]"},{"why":"The authors' previous work establishing the six-observable physical basis at leading order in momentum space, which the NLO construction builds on.","marker":"[37]"},{"why":"Supplies the NLO coefficient functions for the longitudinal structure function $F_L$ used in the physical-basis inversion.","marker":"[38]"},{"why":"Supplies the NLO DGLAP splitting functions and coefficient functions in the MS scheme that enter the derived evolution equations.","marker":"[39]"},{"why":"Provides a two-structure-function Mellin-space physical-anomalous-dimension comparison point for the numerical implementation.","marker":"[32]"},{"why":"Supplies the NLO PDF set used to generate the initial structure functions and to compare the physical-basis evolution against conventional PDF evolution.","marker":"[41]"},{"why":"Provides the numerical interface used to evaluate the PDF set for the initial conditions and comparisons.","marker":"[42]"}],"fun_headline_variants":["Physical basis evades scheme and scale in NLO evolution","Six structure functions evolve without PDFs at NLO","Scheme-independent NLO evolution from observables alone","Direct DGLAP for six structure functions skips PDFs","No PDFs: NLO evolution of DIS structure functions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction stands on the boundary relation $fF'_{L}(1) = -0.003\\,\\frac{\\alpha_s}{2\\pi} fF'_2(1) + O(\\alpha_s^2)$, which eliminates the surface terms generated when the differential operator is partially integrated; if this relation is wrong or misses higher-order terms, the evolution kernels listed in Appendix C no longer reproduce NLO DGLAP evolution.","fun_headline_variants_meta":{"raw":{"variants":["Physical basis evades scheme and scale in NLO evolution","Six structure functions evolve without PDFs at NLO","Scheme-independent NLO evolution from observables alone","Direct DGLAP for six structure functions skips PDFs","No PDFs: NLO evolution of DIS structure functions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000705,"raw_usage":{"total_tokens":3140,"prompt_tokens":869,"completion_tokens":2271,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":485,"completion_tokens_details":{"reasoning_tokens":2192}},"tokens_in":485,"tokens_out":2271,"duration_ms":15713,"temperature":1.0,"reasoning_tokens":2192,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T16:53:39.977051+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any NLO PDF set, compute $F_2$ and $F_L$ at several scales, and evaluate the combination $fF'_{L}(1) + 0.003\\,\\frac{\\alpha_s}{2\\pi} fF'_2(1)$; if it does not vanish at order $\\alpha_s^2$ as a function of $Q^2$, the boundary handling is incorrect, and the physical-basis evolution will differ from conventional PDF evolution by a term of order $\\alpha_s$ rather than $\\alpha_s^2$.","supporting_citations":[{"cited_title":"Furmanski and R","cited_arxiv_id":null,"evidence_quote":"Introduces the physical-basis concept for DIS structure functions, which this paper extends to a six-observable NLO basis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the NLO DGLAP splitting functions and coefficient functions in the MS scheme that enter the derived evolution equations."},{"cited_title":"Buckley, J","cited_arxiv_id":null,"evidence_quote":"Provides the numerical interface used to evaluate the PDF set for the initial conditions and comparisons."}],"review_version":1}