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Precise Determination of the Strong Coupling Constant from Dijet Cross Sections up to the Multi-TeV Range

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Dijet cross sections from the LHC and HERA yield alpha_s(mZ)=0.1178±0.0022 and, for the first time, probe the strong coupling up to 7 TeV in agreement with QCD running.

desk verdict A careful, genuinely new full-color NNLO alpha_s extraction from LHC dijets with a wide running test; the central value holds up, and the main soft spots are scheme-approximation and selection-choice concerns rather than flaws in the fit. read the letter →

arxiv 2412.21165 v2 pith:A4PN5CHK submitted 2024-12-30 hep-ph hep-ex

Fazila Ahmadova , Daniel Britzger , Xuan Chen , Johannes Gä{ss}ler , Aude Gehrmann-De Ridder , Thomas Gehrmann , Nigel Glover , Claire Gwenlan
show 6 more authors
Gudrun Heinrich Alexander Huss Lucas Kunz João Pires Klaus Rabbertz Mark Sutton
This is my paper · ORCID
classification hep-phhep-ex PACS 12.38.Qk13.87.-a
keywords strongcouplingconstantalpha_srunningdijetproductionNNLOQCDrenormalizationgroupequationLHCHERApartondistributionfunctions
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 extracts the strong coupling constant of QCD from dijet cross sections measured at the LHC and at HERA, using next-to-next-to-leading-order (NNLO) predictions that include all subleading color contributions for the first time in a jet-based determination. From the five LHC data sets alone it obtains $\alpha_s(m_Z) = 0.1178 \pm 0.0022$, and adding HERA dijet data gives $0.1180$ with similar total uncertainty, both consistent with the world average. By fitting separate $\alpha_s(m_Z)$ parameters in twenty intervals of the renormalization scale, the analysis probes the running of the coupling from about 7 GeV to 7 TeV — the first determination up to 7 TeV — and finds agreement with the three-loop renormalization group equation across more than three orders of magnitude. The result matters because it tests the asymptotic behavior of QCD at the highest energy scales where the strong coupling has been measured.

What carries the argument

The load-bearing object is the complete next-to-next-to-leading-order QCD calculation for inclusive dijet production, including all subleading color contributions, together with an interpolation grid that stores the hard-scattering coefficients independently of the value of $\alpha_s$ and of the parton distribution functions. The renormalization scale is identified with the dijet invariant mass $m_{jj}$ at the LHC and with $\sqrt{Q^2 + \langle p_T \rangle_{1,2}^2}$ at HERA; these same scales group the data into twenty intervals, and the fit assigns one $\alpha_s(m_Z)$ parameter to each interval while the three-loop running in the modified minimal-subtraction scheme is used to evolve to $\mu_R$. A single least-squares fit over all intervals exploits correlations among experimental and PDF uncertainties, yielding both the individual $\alpha_s(m_Z)$ values and their full correlation matrix. The top quark is treated in the decoupling limit with five active flavors throughout.

What would settle it

Recompute the multi-TeV $\alpha_s(\mu_R)$ values using a scheme that includes top-quark running effects (for example, a matched six-flavor evolution) or assign a dedicated uncertainty for the decoupling approximation by comparing schemes; if the resulting $\alpha_s(\mu_R)$ points shift beyond the quoted uncertainties or acquire a trend with $\mu_R$, the claimed verification of the renormalization group running is not robust.

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Extended reading notes

Core claim

The paper's central claim is that complete NNLO QCD predictions for dijet production, evaluated from 367 LHC data points (and 486 when HERA data are included), determine $\alpha_s(m_Z) = 0.1178 \pm 0.0022$ from the LHC subset and $0.1180$ from the combined fit, both consistent with the world average of $0.1180 \pm 0.0010$. It further claims that a fit with one $\alpha_s(m_Z)$ parameter per renormalization-scale interval yields values of $\alpha_s(\mu_R)$ that follow the three-loop renormalization group running from $\mu_R = 7.4$ GeV to $\mu_R = 7.04$ TeV, with no significant deviation in any interval. The authors present this as the first determination of $\alpha_s$ at scales up to 7 TeV and as the first use of full-colour NNLO dijet predictions, including subleading color contributions, in an $\alpha_s$ extraction from LHC jet data.

Load-bearing premise

The result stands on the assumption that the five-flavor massless scheme, with the top quark decoupled and bottom quarks treated as massless, is accurate at every fitted scale from about 7 GeV to 7 TeV.

Editorial extensions

If this is right

  • The combined value $\alpha_s(m_Z) = 0.1180$ from LHC and HERA dijets adds an independent, high-scale constraint on the strong coupling with precision comparable to the world average.
  • The measured running from 7 GeV to 7 TeV extends the direct test of QCD's renormalization group equation beyond any previous single analysis, covering the multi-TeV region where the coupling is smallest.
  • Including subleading color contributions in the NNLO predictions is shown to be feasible and relevant for $\alpha_s$ determinations from jet data, moving beyond the leading-color approximation used in earlier LHC jet fits.
  • The consistency between HERA and LHC dijet data in a single correlated fit ($\chi^2/\mathrm{ndof} = 0.88$) strengthens confidence in the NNLO predictions and in the parton distribution functions used.

Reading between the lines

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

  • A natural next step would be to apply the same scale-binned fitting technique to inclusive jet, three-jet, or event-shape measurements at the LHC; those observables have different scale choices and could provide independent confirmation of the 7 TeV point, which currently carries the largest uncertainty.
  • The published correlation matrix for the twenty $\alpha_s$ parameters would allow future global analyses, such as PDF determinations, to reuse these results as external constraints without re-fitting the raw dijet cross sections.
  • Because the 7.4 GeV interval lies below the two-bottom-quark threshold, a four-flavor calculation for that single interval would test whether the five-flavor prediction biases the low-scale end of the reported running.
  • If future high-luminosity data reveal a deviation above 1 TeV, separating it from large-$x$ gluon PDF effects will be the principal challenge; the rapidity cuts in this analysis already reduce that entanglement, but a dedicated multi-TeV PDF study would sharpen any such claim.
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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. The manuscript determines the strong coupling constant and tests its running using inclusive dijet cross sections from ATLAS and CMS at 7, 8 and 13 TeV, together with HERA dijet data, all confronted with complete (full-color) NNLO QCD predictions obtained with NNLOJET and stored in APPLfast grids. A fit to the five LHC data sets (367 points after rapidity cuts) yields alpha_s(m_Z) = 0.1178 +/- 0.0014 (fit,PDF) +/- 0.0001 (mu0) +/- 0.0017 (mu_R,mu_F) with chi2/ndof = 0.92, and the combined LHC+HERA fit (486 points) gives 0.1180 +/- 0.0024 total with chi2/ndof = 0.88. In a second step the data are grouped into 20 renormalization-scale intervals and one alpha_s(m_Z) parameter is fitted per interval, giving the running from mu_R = 7.4 GeV to 7.04 TeV; the resulting values are claimed to agree with the RGE expectation, which the authors present as a test of the QCD running over almost three orders of magnitude. The appendix contains a consistency study with seven PDF sets, split by rapidity and mass bins, and the full correlation matrix of the 20 scale-interval parameters.

Significance. The central alpha_s(m_Z) result is credible and well cross-checked: variants of the fit (CMS 2D vs 3D data, individual vs combined data sets, seven PDF choices) return values between 0.1172 and 0.1181, all consistent with the world average, and the global fit quality (chi2/ndof = 0.88-0.92) is good. The paper appears to deliver the first full-color NNLO alpha_s extraction from LHC dijet data and extends the scale reach well beyond previous LHC jet determinations, with competitive precision (about 1-2% per point) in the 0.25-3.5 TeV interval. The explicit reporting of the correlations of the 20 fitted alpha_s(m_Z) parameters (Table V), the tests with seven PDF sets, and the careful separation of fit/PDF, mu0, and scale uncertainties are strengths. The headline multi-TeV and wide-scale running claims, however, rest on the highest- and lowest-scale points, where the paper itself flags validity issues (five-flavor scheme below 2m_b and top decoupling far above m_t); these claims need additional support before the paper can be accepted as stated.

major comments (3)
  1. [V, Table III (rows 4880, 7040), Appendix A] The multi-TeV running claim rests on the two highest-scale entries of Table III, but the theoretical framework there treats the top quark in the decoupling limit: Appendix A states that alpha_s evolution is performed with n_f = 5 'in particular also beyond the top-quark mass threshold' and that the NNLO calculation 'does not include top-quark effects.' In the inclusive dijet phase space at m_jj of several TeV, top-pair production followed by jetty W decays can populate the selected final state, and because the partonic luminosity and the 2->2 matrix elements are of the same order as for massless dijet production at these x values, the relative contamination is not obviously negligible. No estimate or uncertainty for this effect is given; a few-percent top-quark contribution would shift the alpha_s(m_Z) values in the 4.88 and 7.04 TeV rows by amounts comparable to their quoted (fit,PDF) and scale uncertainties (0.0031 and 0.0034, and 0.0128 and 0.0037, respectively). I request that the authors either (i) quantify the top-quark-initiated contribution to the selected bins, e.g., with a parton-level t tbar + jets calculation in the experimental phase space, and subtract it or assign an uncertainty, or (ii) state the multi-TeV determination with an explicit caveat limiting the running claim to scales where the decoupling approximation is safe.
  2. [V, Table III, Table V, Fig. 2] The RGE test is currently asserted visually ('excellent agreement') without a quantitative statistic. The authors' own machinery provides the natural test: comparing the one-parameter fit of Section IV (chi2/ndof = 0.88 over 486 points) with the 20-parameter per-interval fit of Section V, the difference chi2(1 param) - chi2(20 params) on 19 degrees of freedom directly tests the running hypothesis and should be reported. In addition, the claim that 'the assumption of the QCD running enters in each interval only within a very limited range' is hard to reconcile with the fact that each fitted alpha_s(m_Z) is converted to alpha_s(mu_R) by evolving over the full interval from m_Z to mu_R; the actual test is the consistency of the 20 extracted alpha_s(m_Z) values, and this should be stated explicitly. Since the (mu0) and (mu_R,mu_F) uncertainties are declared fully correlated across intervals (Appendix E), the cleanest statement would be a chi2 of the 20 alpha_s(m_Z) values against the world average, evaluated with the full covariance, with an indication of what such a test can exclude given the correlations in Table V.
  3. [II and Supplementary material (Fig. 4, Sec. E.1)] The nominal selection cuts y* < 2.0 and y_b < 1.0 were adopted after the consistency study in the supplementary material showed elevated chi2/ndof at large rapidity, and the text states the cuts were chosen to 'avoid a possible bias' and to 'reduce some moderate tensions.' Because this is a data-driven selection that removes about 26% of the LHC data points, the quoted central value and uncertainty could carry a selection bias, and no demonstration of stability under the cut choice is provided. I ask the authors to report alpha_s(m_Z) for at least one tighter (e.g., y* < 1.5) and one looser (e.g., y* < 2.5) selection, or to include the high-y bins with the per-bin alpha_s values shown; this is needed to confirm that the result is stable at the level of the quoted total uncertainty of 0.0022.
minor comments (6)
  1. [Abstract and Section V] The claim of spanning 'more than three orders of magnitude' is not strictly met by the reported endpoints: 7.04 TeV / 7.4 GeV = 951, which is below 1000, and the 7.4 GeV point is itself flagged in the text as below the 2m_b threshold. Recommend rewording to 'nearly three orders of magnitude' and stating the range without the 7.4 GeV point (from about 10 GeV) when the scheme-validity caveat is applied.
  2. [V] The sentence 'the assumption of the QCD running enters in each interval only within a very limited range' is misleading for the multi-TeV intervals, since the RGE is used to relate alpha_s(m_Z) to alpha_s(mu_R) across the full scale range; please clarify the wording.
  3. [Appendix B] The bin-to-bin correlation of 0.5 assumed for the non-perturbative uncertainties is adopted from Ref. [42] without a sensitivity study; please state the effect of this assumption on the fit uncertainty (e.g., repeats with correlation 0 and 1).
  4. [Appendix A, Abstract, Supplementary material] Typos and referencing: 'with with n_f=5' (Appendix A), 'mutli-TeV' (Abstract), 'previosuly' (Supplementary material). Also, Ref. [41] is a self-reference to the inline supplementary material and should be replaced by a direct citation of the appendix.
  5. [Fig. 2 and Fig. 3 captions] The captions describe the shaded/hatched band as 'the value of alpha_s(m_Z) from LHC dijet data and its running'; since the upper panels show alpha_s(mu_R), please clarify which uncertainty is propagated to the band and how the correlation of the scale uncertainty is treated.
  6. [Table III] The 7.04 TeV row has a (fit,PDF) uncertainty of 0.0128 on alpha_s(m_Z) and about 6% relative uncertainty on alpha_s(mu_R); a sentence stating this precision would prevent the title's 'precise determination up to the multi-TeV range' from being overread as applying uniformly to all scales.

Circularity Check

1 steps flagged · score 6.0 of 10

The central α_s(m_Z) extraction is self-contained, but the headline running plot is partially circular: the α_s(µ_R) values in Table III and Fig. 2 are obtained by evolving the per-interval α_s(m_Z) fit parameters with the same RGE used inside the NNLO predictions, so the displayed agreement with RGE running is enforced by construction.

  1. self definitional [Section V, 'Running of the strong coupling', Fig. 2 and Table III]
    "The technical fit parameter of α_s(m_Z) in each interval is evolved to the appropriate scale value α_s(µ_R) as needed for the computation of the NNLO prediction. ... The α_s(m_Z) values are evolved to the central value of each µ_R interval, illustrating the running of the strong coupling. Overall, excellent agreement with the expectation from the RGE running (when using the world average value for α_s(m_Z)) is observed over the entire range from about 7 GeV up to 7 TeV."

    The column α_s(µ_R) is not an independent measurement of the coupling at µ_R: it is the per-interval fit parameter α_s(m_Z) mapped through the same three-loop, n_f=5 renormalization-group evolution (CRunDec) that is used inside the NNLO predictions to compute the cross sections. Plotting these evolved values against µ_R and comparing them with the RGE curve therefore compares the evolution with itself; the agreement in the upper panel of Fig. 2 is enforced by construction. The genuine test is the lower panel, which shows the 20 fitted α_s(m_Z) values and checks whether they are consistent across scales; that part is not circular. The central α_s(m_Z) = 0.1178 ± 0.0022 result is not affected by this construction.

full rationale

The paper's primary determination of α_s(m_Z) from LHC and HERA dijet data is a well-posed least-squares fit of complete NNLO predictions, including sub-leading color matrix elements, against external experimental data. The PDF4LHC21 input embeds α_s(m_Z) near 0.118, but the fit returns 0.1178 ± 0.0022 using a free α_s, and the authors explicitly test the PDF/α_s interdependence with α_s(m_Z) ± 0.001 PDF variants and µ_0 variations; this is a standard mitigation, not a circular reduction. The dominant circular element is in the running claim: Section V defines the α_s(µ_R) values by evolving the fitted α_s(m_Z) parameters with the same QCD RGE used to produce the NNLO predictions, so the upper panel of Fig. 2 and the statement of 'excellent agreement with the expectation from the RGE running' compare the evolution to itself. The lower panel, where the 20 fitted α_s(m_Z) values are shown to be consistent, is the non-trivial test of the running and is not forced. The paper's acknowledged use of the five-flavor scheme with the top quark in the decoupling limit, including the µ_R = 7.4 GeV point below 2m_b, is a scheme-validity and robustness concern rather than a circularity. Overall, the central α_s(m_Z) determination is independent, but one headline 'prediction' — the displayed running and its agreement with RGE — reduces by construction, giving a partial circularity score of 6.

Assumptions & free parameters 4 free parameters · 8 assumptions · 0 invented entities

The paper fits one parameter, alpha_s(m_Z), on top of a large imported toolbox: NNLOJET matrix elements, PDF4LHC21, Apfel++, CRunDec, experimental NP/EW corrections, and assumed correlation models. The 20 per-interval alpha_s(m_Z) parameters in the running study are the same parameter refit in disjoint scale bins. The most fragile imported assumptions are the nf=5 scheme at all scales and the PDF-alpha_s interplay, both of which the paper addresses only partially.

free parameters (4)
  • alpha_s(m_Z) = 0.1178 (LHC), 0.1180 (LHC+HERA)
    The fitted strong coupling at the Z pole, the central output of the analysis.
  • Per-interval alpha_s(m_Z) parameters = 20 values, 0.1105 to 0.1232 (Table III)
    One alpha_s(m_Z) fit parameter per renormalization-scale interval in the running study; the consistency of these 20 values is the running test.
  • DGLAP starting scale mu_0 = 90 GeV
    Chosen starting scale for PDF evolution, not fitted to dijet data; varied by factors of 0.5 and 2 to estimate the (mu_0) uncertainty.
  • NP bin-to-bin correlation coefficient = 0.5
    Assumed bin-to-bin correlation of non-perturbative correction uncertainties, taken from Ref. [42] rather than fitted to dijet data.
assumptions (8)
  • domain assumption QCD factorization for hadronic and DIS jet cross sections
    Appendix A, Eq. A1: the cross section is a convolution of PDFs and partonic hard scattering; standard, unproven-in-the-paper framework.
  • standard math DGLAP evolution with 3-loop kernels
    Eqs. A4-A5, implemented in Apfel++ [54-57]; standard QCD input.
  • standard math 3-loop MS beta function for alpha_s running via CRunDec
    Appendix A, refs. [51-53]; the RGE running used to relate alpha_s(m_Z) to the probed scales.
  • domain assumption nf=5 active flavors at all scales, top quark decoupled
    Appendix A; the calculation treats top in the decoupling limit; the authors flag low scales as 'at the edge of their validity' (Sec. V) and cut HERA data below 2m_b (Appendix D), but no uncertainty is assigned at scales far above m_top.
  • domain assumption NNLO full-color dijet predictions from NNLOJET are correct
    Refs. [7-9,33-37]; the hard matrix elements and phase-space integration are imported from prior work and are not machine-checked here.
  • domain assumption PDF4LHC21 PDFs are a valid base with alpha_s embedded
    Section II and Appendix B; the fit uses PDFs that assume alpha_s near 0.118; the mu_0-variation and alpha_s +/- 0.001 variants are the stated mitigation.
  • domain assumption NP and EW correction factors from experimental papers
    Appendix A, Eq. A6; c_NP and c_EW are taken from refs. [10-14] rather than recomputed.
  • domain assumption Cross-experiment systematic uncertainties are uncorrelated
    Appendix B; correlation between ATLAS/CMS/H1/ZEUS systematic uncertainties assumed zero, justified by CMS PAS SMP-24-007 [26].

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Pith. "Pith review of Precise Determination of the Strong Coupling Constant from Dijet Cross Sections up to the Multi-TeV Range." pith.science (2026). https://pith.science/paper/A4PN5CHK

@misc{pith2026241221165,
  author       = {Pith},
  title        = {Pith review of: Precise Determination of the Strong Coupling Constant from Dijet Cross Sections up to the Multi-TeV Range},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A4PN5CHK}},
  note         = {Machine review of arXiv:2412.21165}
}
abstract

We determine the value of the strong coupling $\alpha_\text{s}$ and study its running over a wide range of scales as probed by the dijet production process at hadron colliders, based on an NNLO QCD analysis of LHC dijet data. From a large subset of these data a value of $\alpha_\text{s} (m_\text{Z}) = 0.1178 \pm 0.0022$ is obtained for the strong coupling at the scale of the Z-boson mass $m_\text{Z}$, using the invariant mass of the dijet system to select the scale where $\alpha_\text{s}$ is probed. The combination of different data sets enhances the reach and precision of the analysis in the mutli-TeV range and allows for the first determination of $\alpha_\text{s}$ up to scales of 7 TeV. Complementing the LHC data with dijet cross sections measured at the HERA electron-proton collider, the kinematic range is extended to test the running of the strong coupling towards smaller scales. Our results exhibit excellent agreement with predictions based on the renormalization group equation of QCD, and represent a comprehensive test of the asymptotic behavior of QCD, spanning more than three orders of magnitude in energy scale.

Figures

Figures reproduced from arXiv: 2412.21165 by the authors.

Figure 1
Figure 1. FIG. 1. Comparison of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Running of the strong coupling as a function of the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Running of the strong coupling as a function of the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Post-fit [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Left: Post-fit values of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Left: Post-fit values of [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Precision determination of $\alpha_\text{s}$ from Dijet Cross Sections in the Multi-TeV Range

    hep-ph 2025-07 conditional novelty 5.0 of 10

    The strong coupling αs is measured from NNLO dijet cross sections as 0.1178 ± 0.0022 at the Z mass, with its running tested from about 7 GeV to 7 TeV.

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