A Determination of ${\alpha}_s(m_Z)$ at ${\bf aN^3LO_{QCD}}\otimes {\bf {NLO}_{QED}}$ Accuracy from a Global PDF Analysis

Andrea Barontini; Emanuele R. Nocera; Felix Hekhorn; Juan Cruz-Martinez; Juan Rojo; Roy Stegeman; Stefano Forte; The NNPDF Collaboration: Richard D. Ball

REVIEW 4 major objections 5 minor 2 cited by

A Determination of ${\alpha}_s(m_Z)$ at ${\bf aN^3LO_{QCD}}\otimes {\bf {NLO}_{QED}}$ Accuracy from a Global PDF Analysis

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The authors determine $\alpha_s(m_Z)=0.1194^{+0.0007}_{-0.0014}$ simultaneously with the proton's parton distribution functions at approximate N$^3$LO QCD with NLO QED accuracy, using two closure-validated methods.

desk verdict A technically strong alpha_s extraction whose central value still carries an unquantified positivity bias, despite a careful closure-test program. read the letter →

arxiv 2506.13871 v2 pith:MQCXPTC2 submitted 2025-06-16 hep-ph hep-ex

The NNPDF Collaboration: Richard D. Ball , Andrea Barontini , Juan Cruz-Martinez , Stefano Forte , Felix Hekhorn , Emanuele R. Nocera , Juan Rojo , Roy Stegeman This is my paper

classification hep-phhep-ex

keywords strongcouplingconstantpartondistributionfunctionsapproximateN3LOQCDNLOQEDcorrectionsmissinghigherorderuncertaintiesclosureteststheorycovariancemethodcorrelatedreplica

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

This paper presents a simultaneous determination of the strong coupling $\alpha_s(m_Z)$ and the proton's parton distribution functions (PDFs) from a global dataset, at approximate N$^3$LO QCD accuracy combined with NLO QED corrections. The central result is $\alpha_s(m_Z)=0.1194^{+0.0007}_{-0.0014}$, which the authors find consistent with the current world average and with recent lattice determinations. The paper is aimed at showing that the extraction is reliable: two independent methods that fully correlate $\alpha_s$ with the PDFs agree after a closure test, and known biases from multiplicative uncertainties and positivity constraints are detected and corrected. A sympathetic reader would care because this is among the most precise PDF-based $\alpha_s$ determinations and the first to include correlated missing higher order uncertainties, QED effects, and closure-test validation in one error budget.

What carries the argument

The argument rests on two simultaneous extraction methods and on closure tests. The correlated replica method fits PDFs to each Monte Carlo data replica at a grid of fixed $\alpha_s$ values and interpolates the loss to find the best $\alpha_s$ per replica. The theory covariance method (TCM) treats $\lambda=\alpha_s-\alpha_s^0$ as a Bayesian nuisance parameter with a Gaussian prior, adds a covariance contribution $S_{ij}=\beta_i\beta_j(\Delta\alpha_s)^2$ built from the linearized dependence of predictions on $\alpha_s$, and obtains the posterior mean and variance from closed-form expressions. Closure tests, in which synthetic data are generated from a known true $\alpha_s$, validate the error budget and identify two biases: an $\alpha_s$-dependent covariance matrix biases the result upward, and positivity constraints bias it upward, so the paper fixes the covariance matrix and adds the positivity shift as a one-sided linear uncertainty.

What would settle it

Run a closure test with a known true $\alpha_s(m_Z)=0.1194$, keep the positivity constraints, and repeat with enough runs of the universe to resolve the mean well below $0.0007$; if the extracted mean is not offset from the truth by the $0.0007$ shift assumed, or if the no-positivity TCM and CRM disagree beyond their statistical uncertainties when both are run with large replica samples, the quoted central value and one-sided uncertainty are not reliable.

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Extended reading notes

Core claim

At aN$^3$LO$_{\rm QCD}\otimes$NLO$_{\rm QED}$ accuracy, the paper claims $\alpha_s(m_Z)=0.1194^{+0.0007}_{-0.0014}$, with the central value taken from the theory covariance method and the asymmetric uncertainty obtained by adding, linearly and only on the lower side, the estimated shift due to positivity constraints. The authors show that the two extraction methods give mutually consistent results at the permille level, that the result is stable under variations of the evolution solution, the top-quark mass, and the data replica generation, and that without missing higher order uncertainties the NNLO and aN$^3$LO values would disagree at the four-to-five $\sigma$ level. QED corrections raise the central value by a few permille, an effect the paper notes is absent from other simultaneous determinations. The paper also reports that closure tests expose a bias from recomputing the covariance matrix as $\alpha_s$ varies, and a bias from positivity constraints, and that both must be handled to obtain faithful uncertainties.

Load-bearing premise

The quoted asymmetric uncertainty rests on the assumption that the difference between the TCM result with positivity constraints ($0.1194$) and without them ($0.1187$) is a valid estimate of a non-Gaussian systematic bias that should be added linearly to the lower uncertainty; the paper itself states that this size cannot be reliably estimated because the two methods no longer agree when positivity is removed.

Editorial extensions

If this is right

At aN$^3$LO$_{\rm QCD}\otimes$NLO$_{\rm QED}$ accuracy, the extracted value is $\alpha_s(m_Z)=0.1194^{+0.0007}_{-0.0014}$, consistent with the current world average and with recent lattice determinations.
Without correlated missing higher order uncertainties, the NNLO and aN$^3$LO results disagree at the four-to-five sigma level; with them, the two orders agree within one sigma.
QED corrections and the photon PDF raise $\alpha_s$ by a few permille, so omitting them biases any simultaneous PDF-based extraction.
The release of correlated PDF sets at fixed $\alpha_s$ values lets a user propagate the $\alpha_s$ uncertainty into hadronic cross-section predictions.
All baseline results are stable under variations of the evolution equation, the top-quark mass, and the data replica generation procedure.

Reading between the lines

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

If the positivity bias found here is generic, other PDF-based $\alpha_s$ extractions that enforce positivity may be biased upward by a comparable amount; a direct test would be to repeat those fits without positivity.
The theory covariance method is not tied to $\alpha_s$: it could be iterated to extract other parameters, such as the top-quark mass, simultaneously with the PDFs, with closure tests extended to the joint posterior.
Treating the positivity shift as a one-sided linear error implies the final uncertainty is non-Gaussian; an alternative honest presentation would quote the no-positivity interval and list positivity as a theory-choice uncertainty bracketing the central value from both sides.
The finding that the covariance matrix must be held fixed as $\alpha_s$ varies suggests that older profile-$\chi^2$ methods that recompute the $t_0$ matrix at each coupling value may overstate the precision.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear X,

The headline: this is a careful, technically impressive alpha_s extraction from a global NNPDF4.0 fit at aN3LO QCD x NLO QED, and the central value 0.1194 is plausible. But the paper's own closure tests show the value they quote as central comes from a fit with a known upward bias, and their way of dealing with that bias—adding a one-sided systematic to the lower error—leaves the central value uncorrected. That is the soft spot, and it is real.

What is actually new: first simultaneous PDF+alpha_s determination at aN3LO QCD with NLO QED and a photon PDF; first application of the TCM to alpha_s; first closure-test validation of an alpha_s extraction. The two methods (CRM and TCM) agree to the permille level, closure tests are done properly and expose the multiplicative-covariance and positivity biases, MHOUs are included, and the LHAPDF grids are released. That is solid, reproducible work, and the comparison with MSHT, lattice, and PDG is balanced. The dataset-dependence study is a nice extra.

The soft spot is the positivity handling in Sec. 4.1. In the closure test, imposing positivity pulls alpha_s up by ~2-3 sigma (Table 3.1), and removing it eliminates the bias. That is a clear signal. Yet the real-data central value is taken from the TCM with positivity (0.1194), and the bias is not subtracted; instead the shift from removing positivity (0.0007 for TCM, zero for CRM) is added only to the lower uncertainty. The paper admits the size of the bias is not reliably estimable because TCM and CRM disagree without positivity. To make things harder, the closure-test bias is only about 0.0001, five times smaller than the 0.0007 shift they apply in real data. So the quoted asymmetric error depends on one method's behavior where the two methods diverge. This is not fatal—they are transparent about it, and the lower-side inflation is conservative—but it means the central value could carry an unquantified upward shift of up to ~0.0007 or more.

Who this is for: anyone doing precision QCD at the LHC or averaging alpha_s. It deserves a serious referee. I would send it to review with a request that the authors either (a) correct the central value for the positivity bias, (b) present a central value based on the no-positivity fit as a cross-check with the positivity shift treated as a symmetric systematic, or (c) give a quantitative argument for why the with-positivity central value is preferred despite the closure evidence. As is, the result is conditionally acceptable.

Referee Report

4 major / 5 minor

Summary. The manuscript presents a simultaneous determination of the strong coupling alpha_s(m_Z) and PDFs using the NNPDF4.0 methodology, at NNLO and approximate N3LO QCD accuracy, with NLO QED corrections and a photon PDF. Two extraction methods are used: the correlated replica method (CRM) and the theory covariance method (TCM). The work reports extensive closure tests, which identify two biases: an upward bias from recomputing the t0 covariance matrix as a function of alpha_s, and a positive bias from positivity constraints. The final result is alpha_s(m_Z)=0.1194^{+0.0007}_{-0.0014} at aN3LO_QCD x NLO_QED, obtained from the TCM fit with positivity constraints, with the positivity shift added linearly to the lower uncertainty. The paper also studies perturbative convergence, QED corrections, MHOU, top-quark mass dependence, and methodological stability, and it releases PDF sets in LHAPDF format.

Significance. If fully valid, the result is a significant step in PDF-based alpha_s determination: it is one of the first to include approximate N3LO QCD with NLO QED, correlated MHOUs, and two cross-checked extraction methods, and the closure-test validation is unusually thorough. The public release of correlated PDF sets and the explicit detection of the fixed-covariance and positivity biases are strengths. However, the final central value is taken from a procedure that the paper's own closure tests show to be biased, and the asymmetric uncertainty is based on a single-method shift that the paper admits is not reliably estimable; these issues are load-bearing for the quoted value and uncertainty.

major comments (4)

[Sec. 4.1, 'Impact of positivity'; Tables 3.1, 4.1, 4.2] The central value 0.1194 is taken from the TCM fit with positivity constraints, but the closure tests in Table 3.1 show that these constraints introduce a positive bias: with fixed covariance the pull is 3.4 for TCM and 2.2 for CRM with positivity, versus 0.39 and 0.38 without. The paper does not correct the central value for this detected bias; it only adds the shift to the lower uncertainty. Since the closure test identifies the with-positivity estimate as biased, reporting it as the best value is not a faithful interpretation of the validation. The text itself states that "it is not easy to estimate reliably the size of the bias," which means the final central value is not supported by a validated estimator.
[Sec. 4.1, Tables 4.1 and 4.2] The entire asymmetric lower uncertainty at the baseline order comes from the TCM shift (0.1194 to 0.1187) when positivity is removed, while the CRM value is unchanged at 0.1194. The paper does not explain why the TCM no-positivity shift is a reliable bias estimate when the CRM no-positivity result is described as unstable, with outliers and persistent non-Gaussianity. A single-method, non-reproduced shift is not a sufficient basis for the quoted -0.0014 lower uncertainty; the authors should either provide a cross-method estimate of the bias or fold the method discrepancy into a larger systematic uncertainty.
[Sec. 4.1, 'Impact of positivity'] The statement that "the TCM and the CRM in the absence of positivity no longer agree" is not supported by Table 4.2: at aN3LO_QCD x NLO_QED the two central values, 0.1187 and 0.1194, differ by 0.0007, which is less than one combined standard deviation (about 0.0012). If the real concern is the non-Gaussianity of the CRM replica distribution rather than the central values, that should be quantified; as written, the rationale for not estimating the bias reliably is weakened.
[Sec. 3.2, 'Closure Results' and Sec. 4.1] The closure tests validate the no-positivity procedures, but the final procedure adds a linear shift to the lower uncertainty of the with-positivity result. There is no closure-test check of the coverage of the final interval 0.1194^{+0.0007}_{-0.0014}; the authors should demonstrate in the closure setup that this interval contains the true value at approximately 68% confidence. Without such a check, the asymmetric uncertainty is an ad hoc construction rather than a validated error estimate.

minor comments (5)

[Sec. 4.1, last paragraph before 'Final results'] The sentence contains a duplicated article: "estimate reliably the the size of the bias" should read "estimate reliably the size of the bias."
[Table 3.1 and Eqs. (3.4)-(3.6)] The column header "<sigma_alpha>/sqrt(N_r)" is described as the uncertainty of the mean, but Eq. (3.5) defines <sigma_alpha> as a weighted uncertainty; the distinction between the two is easy to miss and would benefit from an explicit sentence in the caption.
[Sec. 2.2, Eq. (2.24)] The quantity lambda^(0) is used in Eq. (2.24) but is not explicitly defined there; it would be clearer to state that lambda^(0) is the central value of the replica ensemble of nuisance parameters, given by Eq. (2.16).
[Table 4.4, ATLAS row] The accuracy label "N3LO x N4LLa QCD" uses an undefined superscript 'a'; please define it in the footnotes or in the text.
[Fig. 4.1] The y-axis label "normalized frequency" is not self-explanatory; state explicitly that the histograms are normalized so that the integral is one.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: alpha_s is extracted from external experimental data through an explicitly linearized derivative and validated by closure tests; the positivity-bias treatment raises statistical-consistency concerns but does not reduce outputs to inputs.

full rationale

The determination of alpha_s is data-driven and not circular in the derivation-chain sense. Alpha_s enters the theory predictions through the linearized derivative beta = dT^(0)/dalpha_s in Eq. (2.20), and the TCM posterior, Eq. (2.24), depends on the experimental residual D - T^(0) through beta^T (C+S)^{-1}(D - T^(0)) Delta alpha_s. No equation defines the output alpha_s in terms of itself or in terms of the final quoted value. The CRM and TCM are validated by closure tests against a known truth value alpha_s(m_Z) = 0.118 (Table 3.1), including explicit prior-independence checks with priors 0.117 and 0.119 yielding posteriors 0.11801 and 0.11811. Self-citations to NNPDF4.0 methodology, EKO, PineAPPL, and the theory covariance formalism are code-reproduced and their assumptions do not include the target value of alpha_s. The positivity-bias handling in Sec. 4.1 is a methodological and statistical vulnerability: the paper itself says 'because the TCM and the CRM in the absence of positivity no longer agree, it is not easy to estimate reliably the size of the bias,' and it adopts the with-positivity TCM central value even though its own closure tests show a positive bias. However, this is an internal-consistency and uncertainty-estimation concern, not a circular reduction: the difference between with- and without-positivity fits is used as an extra uncertainty, not as the definition of the central value. The core extraction therefore does not reduce a prediction to its inputs by construction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The extraction rests on standard factorization and Monte Carlo replica assumptions, the NNPDF neural network parametrization, the fixed t0 covariance choice, and the positivity-bias correction. No invented physical entities are introduced. The only numerical quantity used as a correction is the TCM positivity shift; the TCM prior width is chosen but shown to be prior-independent.

free parameters (2)

Positivity-bias systematic shift = 0.0007 at aN3LO_QCD x NLO_QED (0.1194 with positivity vs 0.1187 without, TCM)
Used to enlarge the lower uncertainty of the final alpha_s; the paper does not correct the central value. The shift is derived from the same data and depends on the TCM choice.
TCM prior width Delta alpha_s = 0.002
Chosen for linearization of the theory covariance matrix and checked for prior independence by iterating and by using priors at 0.117 and 0.119 in the closure test.

assumptions (5)

domain assumption QCD factorization and DGLAP evolution, including QED and a photon PDF, describe all included data within the stated MHOUs.
Used throughout; the entire global fit assumes perturbative QCD and factorization.
domain assumption The NNPDF neural network parametrization is sufficiently flexible to represent the true PDFs, and hyperoptimization at a single alpha_s value transfers to other alpha_s values.
Section 2.3 notes that hyperoptimization is performed only at alpha_s=0.118; extrapolation to other alpha_s values is assumed.
ad hoc to paper The t0 covariance matrix should be held fixed when alpha_s is varied, even though the true matrix may depend on alpha_s.
Section 3.2 shows that varying the covariance with alpha_s in closure tests biases alpha_s upward; the paper adopts the fixed-covariance choice for real data.
ad hoc to paper The difference between TCM fits with and without positivity constraints is a valid estimate of a non-Gaussian systematic bias, added linearly to the lower uncertainty only.
Section 4.1 'Impact of positivity'; the paper acknowledges the size is not reliably estimable because TCM and CRM disagree without positivity.
domain assumption Data uncertainties are Gaussian and Monte Carlo replica generation from the covariance matrix is valid; positivity constraints then produce a bias from non-Gaussian tails at kinematic boundaries.
Section 3.2 explains the positivity bias mechanism in terms of non-Gaussian tails near kinematic boundaries.

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Cite this review

Pith. "Pith review of A Determination of ${\alpha}_s(m_Z)$ at ${\bf aN^3LO_{QCD}}\otimes {\bf {NLO}_{QED}}$ Accuracy from a Global PDF Analysis." pith.science (2026). https://pith.science/paper/MQCXPTC2

@misc{pith2026250613871,
  author       = {Pith},
  title        = {Pith review of: A Determination of $\alpha_s(m_Z)$ at $\bf aN^3LO_QCD\otimes \bf NLO_QED$ Accuracy from a Global PDF Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MQCXPTC2}},
  note         = {Machine review of arXiv:2506.13871}
}

abstract

We present a determination of the strong coupling $\alpha_s(m_Z)$ from a global dataset including both fixed-target and collider data from deep-inelastic scattering and a variety of hadronic processes, with a simultaneous determination of parton distribution functions (PDFs) based on the NNPDF4.0 methodology. This determination is performed at NNLO and approximate N$^3$LO (aN$^3$LO) perturbative QCD accuracy, including QED corrections and a photon PDF up to NLO accuracy. We extract $\alpha_s$ using two independent methodologies, both of which take into account the cross-correlation between $\alpha_s$ and the PDFs. The two methodologies are validated by closure tests that allow us to detect and remove or correct for several sources of bias, and lead to mutually consistent results. We account for all correlated experimental uncertainties, as well as correlated theoretical uncertainties related to missing higher order perturbative corrections (MHOUs). We study the perturbative convergence of our results and the impact of QED corrections. We assess individual sources of uncertainty, specifically MHOUs and the value of the top quark mass. We provide a detailed appraisal of methodological choices, including the choice of input dataset, the form of solution of evolution equation, the treatment of the experimental covariance matrix, and the details of Monte Carlo data generation. We find $\alpha_s(m_Z)=0.1194^{+0.0007}_{-0.0014}$ at aN$^3$LO$_{\rm QCD}\otimes {\rm NLO}_{\rm QED}$ accuracy, consistent with the latest PDG average and with recent lattice results.

Figures

Figures reproduced from arXiv: 2506.13871 by the authors.

**Figure 3.1.** The best-fit values of α (j) s and the associated one standard deviation uncertainties σ (j) α obtained using the TCM in the Nr = 100 individual runs of the closure tests (left), and the corresponding distribution of normalized bias R (j) bv Eq. (3.2) (right). For reference, a univariate zero-mean Gaussian is also displayed in the right panels. Results obtained both when imposing positivity (top) and when not imposi… view at source ↗

**Figure 3.2.** Same as [PITH_FULL_IMAGE:figures/full_fig_p011_3_2.png] view at source ↗

**Figure 4.1.** Histogram of the values of the Nrep best-fit values α (k) s obtained with the TCM and CRM when applied to the experimental data entering the NNPDF4.0 global fit. In both cases, results shown correspond to the fits carried out at aN3LOQCD⊗NLOQED accuracy and accounting for the positivity of physical observables, see the bottom row of [PITH_FULL_IMAGE:figures/full_fig_p014_4_1.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.2.** Figure 4.2: The values of αs(mZ) extracted at aN3LOQCD⊗NLOQED accuracy from the TCM applied to the partial χ 2 evaluated for separate groups of processes. In all cases, uncertainties shown correspond to 68% CL intervals. The dashed vertical line corresponds to the best-fit value…

**Figure 4.3.** Figure 4.3: Graphical representation of the results of [PITH_FULL_IMAGE:figures/full_fig_p018_4_3.png]

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