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Next-to-leading order evolution of structure functions without PDFs

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper derives a closed set of NLO evolution equations for six measurable DIS structure functions, replacing parton distribution functions with the observables themselves.

desk verdict 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. read the letter →

arxiv 2412.09589 v3 pith:2VVPJ6O6 submitted 2024-12-12 hep-ph

Tuomas Lappi , Heikki Mäntysaari , Hannu Paukkunen , Mirja Tevio This is my paper · ORCID
classification hep-ph
keywords DGLAPevolutionstructurefunctionsphysicalbasisfactorizationschemeindependencepartondistributionsnext-to-leadingorderdeepinelasticscattering
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $fF'_{L}(1) = -0.003\,\frac{\alpha_s}{2\pi} fF'_2(1) + O(\alpha_s^2)$.

What would settle it

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 $\alpha_s^2$.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

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.

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 (3)
  1. [Appendix C, Eq. (C1)] 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.
  2. [Footnote 1] 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.
  3. [Section III] 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.
minor comments (6)
  1. [Appendix A, Eq. (A3)] 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].
  2. [Section III] The numerical method is not described (x-grid, interpolation for derivatives, treatment of plus-distributions and endpoint terms). A brief description would improve reproducibility.
  3. [Figure 2 caption] The phrase 'colourful curves' is informal; consider 'colored curves' or a more precise description of the lines and markers.
  4. [Footnote 1] 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.
  5. [Appendix C, Eq. (C17)] 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.
  6. [General] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the NLO physical-basis kernels are an algebraic rearrangement of standard MS coefficient functions and DGLAP splitting functions, benchmarked against CT18; self-citations are not load-bearing.

full rationale

The derivation chain starts from standard, externally published NLO MS coefficient functions (Refs. [38,39]) and the conventional DGLAP splitting functions, inverts them in a perturbative expansion (Eqs. (24)-(28), Appendix B), and substitutes into the DGLAP equations to obtain Eqs. (32)-(37). No parameter is fitted to the comparison; the numerical comparison uses CT18NLO NF3 as an external input to set initial conditions and as an independent benchmark. The only self-citations are to the authors' previous LO paper [37] for the LO inverse operator and to [33] for scale-variation consistency; the LO inverse is rederived explicitly (Eqs. (17)-(19), (B1)-(B16)), so the citation is not load-bearing. The boundary relation Eq. (C1), fF'L(1) = -0.003 (alpha_s/2pi) fF'2(1) + O(alpha_s^2), is derived from the delta-function term of the external NLO coefficient function C^(2)_FLNS, not fitted or posited as a prediction; it is a possible correctness risk (as is the unresolved discrepancy with Ref. [32] noted in footnote 1), but not a circular step. The central claim of scheme/scale independence is a consequence of the standard factorization argument, not an input assumed from the authors' prior work. Overall score 1 reflects only the presence of minor, non-load-bearing self-citations.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central evolution equations contain no fitted parameters; all inputs are standard coefficient functions, splitting functions, and the CTEQ strong coupling. The only ad hoc parameter in the paper is the 0.2635 rescaling of the s-quark PDF used in the Fig. 3 comparison. The main domain assumptions restrict applicability to the 3-flavour scheme. No new particles, forces, or fields are introduced. The boundary relation (C1) is the most paper-specific element of the construction.

free parameters (1)
  • s-quark PDF rescaling factor in Fig. 3 = 0.2635
    Applied to the analytical s quark PDF to enforce the momentum sum rule in the illustrative comparison of Fig. 3; not used in the central evolution equations.
assumptions (5)
  • standard math NLO DGLAP splitting functions in MS scheme (Ref. [39])
    Used in Eqs. (29)-(31) to derive the physical-basis evolution equations; taken as external input.
  • standard math NLO DIS coefficient functions in MS scheme (Refs. [38,39])
    Define the structure functions (6)-(11) and underpin the perturbative inversion of PDFs.
  • domain assumption The differential operator P-hat is the exact inverse of the LO coefficient function C^(1)_FLg
    Equation (26); exact only for massless quarks and the specific form C^(1)_FLg = 4TR x(1-x).
  • ad hoc to paper Boundary relation Eq. (C1) between fF'L(1) and fF'2(1)
    Used to cancel boundary terms when partially integrating P-hat; specific to this paper's kernel construction.
  • domain assumption Fixed 3-flavour massless quark scheme with s = sbar and no CKM mixing
    Limits the framework to Q^2 values where charm and heavier flavours can be neglected.

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Pith. "Pith review of Next-to-leading order evolution of structure functions without PDFs." pith.science (2026). https://pith.science/paper/2VVPJ6O6

@misc{pith2026241209589,
  author       = {Pith},
  title        = {Pith review of: Next-to-leading order evolution of structure functions without PDFs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2VVPJ6O6}},
  note         = {Machine review of arXiv:2412.09589}
}
read the original abstract

We formulate and numerically solve the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi~(DGLAP) evolution equations at next-to-leading order in perturbation theory directly for a basis of 6 physical, observable structure functions in deeply inelastic scattering. By expressing the evolution in the physical basis one evades the factorization scale and scheme dependence. Working in terms of observable quantities, rather than parametrizing and fitting unobservable parton distribution functions (PDFs), provides an unambiguous way to confront predictions of perturbative Quantum Chromodynamics with experimental measurements. We compare numerical results for the DGLAP evolution for structure functions in the physical basis to the conventional evolution with PDFs.

Figures

Figures reproduced from arXiv: 2412.09589 by the authors.

Figure 1
Figure 1. FIG. 1: The [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The NLO quark singlet and the gluon PDF counterparts in the physical basis (dashed colourful lines) where the [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The momentum sum rule and the baryon number conservation rules, for the [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The NLO structure function [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

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