Determination of $\alpha_s(M_Z)$ via a high-precision effective coupling $\alpha^{g_1}_s(Q)$

Hua Zhou; Jian-Ming Shen; Qing Yu; Xing-Gang Wu

REVIEW 3 major objections 7 minor 100 references

Determination of $\alpha_s(M_Z)$ via a high-precision effective coupling $\alpha^{g_1}_s(Q)$

T0 review · 3 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that the Bjorken-sum-rule effective coupling, with PMC∞ perturbation theory matched to the light-front holographic QCD model, fixes α_s(M_Z)=0.1191±0.0012∓0.0006.

desk verdict A competent, incrementally novel PMC∞+V-scheme extraction of α_s from the Bjorken sum rule, whose central value is anchored to the LFHQCD infrared model and whose quoted errors omit uncertainty in that model. read the letter →

arxiv 2501.15525 v1 pith:C24WZ6V4 submitted 2025-01-26 hep-ph

Qing Yu , Xing-Gang Wu , Hua Zhou , Jian-Ming Shen This is my paper · ORCID

classification hep-ph PACS 12.38.-t

keywords strongcouplingconstantBjorkensumruleprincipleofmaximumconformalitylight-frontholographicQCDeffectivechargerenormalizationscaleambiguityVschemealpha_s(M_Z)

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 proposes that the strong coupling of QCD, $\alpha_s(Q)$, can be determined at every scale from a single effective coupling defined through the Bjorken sum rule. For the perturbative part it uses the PMC∞ scale-setting procedure in the $V$-scheme, which turns the series into a conformal one and removes conventional scheme-and-scale ambiguities. For the infrared it uses the light-front holographic QCD (LFHQCD) model, whose coupling freezes to a conformal value at $Q\to 0$. Matching the two at a critical scale $Q_0$ reproduces the measured spin-structure data with a $p$-value near 99% and yields $\alpha_s(M_Z)=0.1191\pm0.0012\mp0.0006$. The result matters because it offers a parameter-light all-scale running coupling and an $\alpha_s(M_Z)$ consistent with the world average but with far smaller theoretical uncertainty from scale choice.

What carries the argument

The central object is the effective charge $a_s^{g_1}(Q)=\alpha_s^{g_1}(Q)/\pi$ defined by the Bjorken sum rule $\Gamma_1^{p-n}(Q)=|g_A/g_V|\,(1/6)[1-a_s^{g_1}(Q)]$. The paper rewrites its pQCD expansion in the $V$-scheme and applies the PMC∞ procedure, which groups the series into four conformal subsets with separate scales $\mu_1^V$, $\mu_2^V$, $\mu_3^V$, $\mu_4^V$; setting each subset's $\beta$-terms to zero removes the first kind of residual scale dependence and leaves only the last scale $\mu_4^V$ uncertain. The infrared side is the LFHQCD Gaussian $a_s^{g_1}(Q)=e^{-Q^2/4\kappa^2}$, which gives the conformal limit $a_s^{g_1}(0)=1$. Matching value and first derivative at $Q_0$ connects the two regimes and determines $Q_0$ and $\Lambda_{\rm QCD}$; with $\kappa$ fixed by spectroscopy, $\Lambda_{\rm QCD}$ determines $\alpha_s(M_Z)$ through the four-loop running equation.

What would settle it

A lattice or DSE calculation of the Bjorken-sum-rule effective charge for $Q<1$ GeV that deviates from the LFHQCD exponential $e^{-Q^2/4\kappa^2}$ beyond the stated $\kappa$ uncertainty would falsify the matching; so would an independent high-precision $\alpha_s(M_Z)$ measurement that falls outside $0.1191\pm0.0013$.

Watch

Extended reading notes

Core claim

The central discovery is that the effective charge of the Bjorken sum rule, $a_s^{g_1}(Q)$, can be made precise across all scales by combining the PMC∞ conformal series with the LFHQCD exponential form $a_s^{g_1}(Q)=e^{-Q^2/4\kappa^2}$. Working in the $V$-scheme, the authors match the value and first derivative of the two descriptions at a critical scale $Q_0$, which fixes $Q_0=1.1782$ GeV and $\Lambda_V^{n_f=3}=449$ MeV when $\kappa=0.523\pm0.024$ GeV is taken from hadron spectroscopy. Evolving this $\Lambda$ to the $Z$ mass gives $\alpha_s(M_Z)=0.1191\pm0.0012(\Delta\kappa)\mp0.0006(\mathrm{theory})$, consistent with the 2024 world average. The fit to 79 experimental points from HERMES, COMPASS, SLAC and JLab gives $\chi^2/\mathrm{d.o.f}\approx 0.6$, corresponding to a $p$-value of order 99%, and all PMC scales lie above $Q_0$, so the earlier self-consistency problem of MS-scheme matching disappears.

Load-bearing premise

The extraction rests on the assumption that the true infrared behavior of the Bjorken-sum-rule coupling is exactly the LFHQCD exponential $e^{-Q^2/4\kappa^2}$; if that curve is wrong, the matched scale $Q_0$ and the resulting $\Lambda$ (hence $\alpha_s(M_Z)$) shift by an amount not included in the quoted errors.

Editorial extensions

If this is right

The extracted $\alpha_s(M_Z)=0.1191\pm0.0012\mp0.0006$ agrees with the 2024 world average $0.1180\pm0.0009$, with the residual-scale error smaller by more than an order of magnitude.
The PMC∞ series converges faster than both the conventional MS series and the PMCs series even at the charm scale, where the conventional N$^3$LO term is larger than the N$^2$LO term.
All three PMC∞ scales exceed the matched critical scale $Q_0\approx 1.18$ GeV in the $V$-scheme, so the earlier self-consistency problem seen in the MS-scheme analysis is resolved.
The resulting all-scale $\alpha_s^{g_1}(Q)$ is a precise input for any QCD observable needing a running coupling in the low- and medium-energy domain.
Varying the last uncomputed scale $\mu_4^V$ from $Q$ to $2Q$ (and even to $8Q$ at charm mass) changes $\alpha_s(M_Z)$ by only $\mp0.0006$, so the remaining pQCD uncertainty is negligible.

Reading between the lines

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

The quoted $\pm0.0012$ error covers only the $\kappa$ variation; the systematic uncertainty from assuming the LFHQCD exponential form itself is not included, so a comparison of low-$Q$ lattice or DSE results with $e^{-Q^2/4\kappa^2}$ would quantify the missing model error.
The same PMC∞+LFHQCD matching could be applied to other effective charges, such as the Gross–Llewellyn Smith sum rule, to test whether the extracted $\alpha_s(M_Z)$ is specific to the Bjorken sum rule or reflects a universal infrared coupling.
Since the $p$-value near 99% is partly a reflection of the sizeable data errors, a sharper test of the method would compare the slope of $\alpha_s^{g_1}(Q)$ across $Q_0$ once more precise low-$Q$ spin-structure data or lattice data become available.
The use of the $V$-scheme and four-loop $\beta$-functions means that when five-loop $V$-scheme coefficients and a fifth-order perturbative coefficient for the Bjorken sum rule appear, the residual $\mu_4^V$ uncertainty should shrink further and the method's precision could improve beyond the present level.

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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

The punchline: this is a solid, craftsmanlike application of PMC∞ in the V-scheme to the Bjorken sum rule. The specific combination—V-scheme running, PMC∞ multi-scales, and matching to LFHQCD—has not appeared before, and the resulting all-scale effective coupling and α_s(M_Z)=0.1191 are new. But the central value is not a pure pQCD extraction; it is forced to agree with the LFHQCD exponential in both value and slope at Q0, and that model choice is not in the error budget.

What the paper does well: it lays out the full V-scheme transformation of the N3LO coefficients, compares PMCs and PMC∞ explicitly, shows heavy-quark mass effects are negligible, and demonstrates that the previous 'self-consistency problem' disappears because the V-scheme PMC scales stay above Q0. The residual scale dependence is genuinely small. The authors are transparent about their inputs and even flag in a footnote that the conventional p-value is artificially high due to large scale errors—an honest touch.

Where it is soft: the load-bearing step is Eq. (26), a_s^{g1}(Q)=e^{-Q^2/4κ^2}, taken as the IR truth. The two matching conditions fix Q0 and Λ_V, and α_s(M_Z) then follows from evolution. The quoted errors propagate only Δκ and the second-kind residual scale dependence; there is no term for the Gaussian form itself. DSE/lattice effective charges shown in Fig. 4 have a different infrared curvature and freezing value, so the functional-form uncertainty is not academic. A different but equally plausible IR parameterization would shift the extracted α_s(M_Z) by an amount that could exceed ±0.0012. The p-value ~99% should be retired: with χ²/d.o.f ≈ 0.6 the fit is fine, but as the conventional case demonstrates, tiny χ² is an artifact of large theory errors, not a sign of precision. The abstract's 'matches well with p-value 99%' is overinterpreted.

Who should read it: people working on effective charges and PMC scale-setting. It is a reasonable alternative extraction of α_s, consistent with PDG, and worth having in the literature.

Bottom line: it deserves peer review, but with a request to quantify the LFHQCD model-form uncertainty (e.g., vary the functional form or compare with DSE-informed shapes) and to soften the p-value rhetoric. As is, I would treat the central value as model-dependent at the few-per-mille level.

Referee Report

3 major / 7 minor

Summary. The manuscript proposes a determination of α_s(M_Z) from the Bjorken-sum-rule effective coupling α_s^{g1}(Q). The leading-twist part is reorganized with the PMC∞ scale-setting procedure in the V-scheme, and the infrared behavior is taken from the LFHQCD exponential model, Eq. (26). Matching the value and derivative of the two descriptions at a critical scale Q0 fixes Q0 and the V-scheme QCD scale Λ_V^{nf=3}; using κ=0.523±0.024 GeV from hadron spectroscopy then yields Q0=1.1782 GeV, Λ_V^{nf=3}=449 MeV, and α_s(M_Z)=0.1191±0.0012∓0.0006, consistent with the PDG average. The paper also reports a p-value of about 99% for the agreement of the resulting coupling with 79 experimental points.

Significance. The paper supplies a useful set of V-scheme coefficients and demonstrates that the PMC∞ procedure removes the first-kind residual scale dependence and resolves the previously noted self-consistency problem, with a final value in agreement with world data. The central limitation is that the extraction is not data-driven in the infrared: it is anchored to the LFHQCD exponential form, and the quoted errors omit the uncertainty of that model. The goodness-of-fit claim is also weaker than stated. With these caveats addressed, the method could constitute a legitimate precision determination, but in its present form the result is conditional on an external model assumption.

major comments (3)

[Sec. IIIB/IIIC, Eq. (26)] The central value is obtained by imposing that the PMC∞ series and the LFHQCD expression a_s^{g1}(Q)=exp(-Q^2/4κ^2) meet with equal value and derivative at Q0; the final errors in Eqs. (29)-(30) propagate only Δκ and the second-kind residual scale dependence. No term accounts for the uncertainty in the assumed Gaussian/confining functional form itself. Since the derivative condition makes the extracted Λ_V^{nf=3} sensitive to the curvature of the model, a different but equally reasonable IR parameterization (for instance one that freezes near 0.97π as the DSE/lattice charge shown in Fig. 4) would shift α_s(M_Z) by an amount that can exceed the stated total error. The authors should either quantify this model uncertainty by repeating the matching with alternative forms or clearly label the result as conditional on Eq. (26).
[Sec. IIIB, Eq. (27) and footnote 1] The claim of a p-value of about 99% is not supported as evidence for the model. With 79 data points, χ2/d.o.f≈0.6 indicates that the assigned theory errors are too large, or that the data are correlated, rather than that the fit is exceptionally good. The conventional prediction has χ2/d.o.f≈0.1 and p~100% for the opposite reason, as the footnote itself admits, so the statistic does not discriminate between the two approaches. The authors should remove or strongly qualify the p-value claim and report the fit quality using error bars that include the model uncertainty.
[Secs. IIIB and IIIC, Fig. 4 and Table II] The role of κ changes between the two sections. In Fig. 4 the matching conditions fix both κ and Q0, giving κ=0.501^{+0.030}_{-0.028} GeV and Q0=1.130^{+0.066}_{-0.059} GeV; in Table II the final result is obtained by instead fixing κ=0.523±0.024 GeV from spectroscopy and solving for Q0 and Λ_V^{nf=3}. The paper should specify which procedure defines the central result and explain the difference, since the extracted Λ_V^{nf=3} and the resulting α_s(M_Z) depend on this choice.

minor comments (7)

[Abstract and Sec. IIIC] The phrase 'the critical scale MZ' is presumably a typo for 'the reference scale MZ'; the critical scale elsewhere in the paper is Q0.
[Sec. I] The text says 'CREN' where it should say 'CERN'.
[Sec. IIIB] The sentence beginning 'Diﬀerent form the conventional results' should read 'Different from the conventional results'.
[Eq. (28)] The ratio D(Q)/D(Q) appears to be a typo; the numerator and denominator should be distinct functions, or one of them should be defined explicitly.
[Table II caption] The notation 'the.' and the mixed use of ± and ∓ signs for the errors are confusing; the caption should define every error source explicitly.
[Fig. 2 caption] The caption lists red dashed, purple dotted, and green solid curves; if the published figure is grayscale, the curves should also be distinguished by line style in the plot itself.
[Sec. IIIB, Eq. (27)] If κ is an input rather than a fitted parameter in the final extraction, the denominator n-2 should be justified; otherwise the paper should state that Q0 and Λ_V^{nf=3} are the two fitted parameters.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the LFHQCD matching is a disclosed extraction assumption, and the data comparison is an independent consistency check, not a fit that reduces to the input.

full rationale

I walked the derivation chain from Eq. (2) through the PMC∞ series of Sec. II to the matching conditions of Sec. IIIB and the α_s(M_Z) extraction of Sec. IIIC. The central value α_s(M_Z)=0.1191 is obtained by solving the matching equations that require the PMC∞ pQCD series and the LFHQCD model a_s^{g1}(Q)=e^{-Q^2/4κ^2} to have equal values and equal first derivatives at Q0, with κ=0.523±0.024 GeV taken from hadron spectroscopy rather than from the BSR data. This is a parameter extraction from an explicitly labeled model input, not a prediction whose output is identical to its input by construction; a wrong LFHQCD form would shift the result, but that is model dependence and omitted model-form uncertainty, not circularity. The p-value ~99% and χ^2/d.o.f≈0.6 are computed after the fact against 79 experimental points and are not used to fix the reported α_s(M_Z), so the data comparison is a genuine external check. The paper's self-citations to the PMC∞ formalism [51,57] are load-bearing in the sense that the method comes from prior work by overlapping authors, but those are published, externally checkable derivations of scale-setting identities rather than an unverified premise that is equivalent to the present conclusion. No equation in the paper is reduced to itself by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper's central extraction rests on the LFHQCD model input (κ and the exponential form), the matching conditions, and the PMC∞ machinery from prior work. No new free entities are introduced beyond the fitted scales.

free parameters (3)

κ (LFHQCD confinement scale) = 0.523 ± 0.024 GeV (input from hadron spectroscopy, Ref [70])
The central α_s(M_Z) error is dominated by Δκ; the paper does not derive this value from its own analysis.
Q0 (critical matching scale) = 1.1782^{+0.0511}_{-0.0512} GeV for PMC∞
The splicing point where the pQCD and LFHQCD curves are forced to agree in value and derivative; it is a matching parameter, not a first-principles quantity.
Λ_V^{nf=3} (V-scheme QCD scale in the three-flavor region) = 449 ± 22 MeV
Determined by the smooth-matching conditions in Sec. IIIC; this is the fitted quantity that fixes α_s(M_Z).

assumptions (5)

ad hoc to paper The LFHQCD coupling a_s^{g1}(Q)=exp(-Q^2/4κ^2) describes the full nonperturbative BSR effective coupling for Q<Q0.
Introduced in Eq. (26) and used to fix Q0 and Λ in Sec. III; no direct derivation from QCD is given.
domain assumption The PMC∞ decomposition of the pQCD series into conformal subsets with r_{i,IC}=r_{i,0} is valid.
Standard PMC∞ assumption from Refs. [50,51,57], invoked in Sec. II.
domain assumption The V-scheme is the appropriate expansion basis to avoid the self-consistency problem.
Choice made in Sec. II and justified by comparison with the MS-scheme in Ref. [36].
ad hoc to paper The matching conditions requiring equality of value and derivative at Q0 uniquely determine a physically meaningful transition.
Used in Sec. IIIB and IIIC; the two conditions fix two parameters but the choice of functional form for the IR region is assumed.
domain assumption The four-loop RGE running with decoupling at mc=1.67 GeV and mb=4.78 GeV is the correct description above Q0.
Standard pQCD ingredient, Eq. (21); the quark masses are taken as fixed inputs.

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Cite this review

Pith. "Pith review of Determination of $\alpha_s(M_Z)$ via a high-precision effective coupling $\alpha^{g_1}_s(Q)$." pith.science (2026). https://pith.science/paper/C24WZ6V4

@misc{pith2026250115525,
  author       = {Pith},
  title        = {Pith review of: Determination of $\alpha_s(M_Z)$ via a high-precision effective coupling $\alpha^g_1_s(Q)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C24WZ6V4}},
  note         = {Machine review of arXiv:2501.15525}
}

abstract

We propose a novel method to determine the strong coupling of quantum chromodynamics (QCD) and fix its running behavior at all scales by using the Bjorken sum rules (BSR). The BSR defines an effective coupling $\alpha^{g_1}_s(Q)$ which includes the nonperturbative high-twist corrections and perturbative QCD (pQCD) corrections to the leading-twist part. For the leading-twist part of $\alpha^{g_1}_s(Q)$, we adopt the infinite-order scale-setting procedure of the principle of maximum conformality ($\rm{PMC}_\infty$) to deal with its pQCD corrections, which reveals the intrinsic conformality of series and eliminates conventional renormalization scheme-and-scale ambiguities. Using the $\rm{PMC}_\infty$ approach, we not only eliminate \textit{the first kind of residual scale dependence} due to uncalculated higher-order terms, but also resolve the previous ``self-consistence problem". The holographic light-front QCD model is used for $\alpha^{g_1}_s(Q)$ in the infrared region, which also reveals a conformal behavior at $Q\to 0$. As a combination, we obtain a precise $\alpha^{g_1}_s(Q)$ at all scales, which matches well with the known experimental data with $p$-value $\sim99\%$, we determine the strong coupling constant at the critical scale $M_Z$, $\alpha_s(M_Z)=0.1191\pm{0.0012}\mp0.0006$, where the first error comes from $\Delta\kappa$ of LFHQCD model and the second error is from \textit{the second kind of residual scale dependence} that is negligible.

Figures

Figures reproduced from arXiv: 2501.15525 by the authors.

**Figure 2.** FIG. 2. The PMC scales for the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. The matching of the LFHQCD model of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. The matching of the LFHQCD model of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

∆ κ” stems from ∆ κ = ±0. 024 GeV and the second error “the

As for the PMC ∞ prediction, we ob- tain χ 2/d.o.f ≃ 0. 6. The smaller χ 2/d.o.f -value obtained by conventional pQCD series is mainly due to large µ r- dependence in low and medium Q-region. Diﬀerent form the conventional results, the PMC predictions are µ r- independent, and the good ﬁt of the PMC to the experi- mental data further indicates the two par...

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This can be done with the help of the relation between the V -scheme and the MS-scheme RGEs [ 58, 59]

can be written as ag1 s (Q) = n∑ i=1 rV i (Q, µ r)aV,i s (µ r), = rV 1, 0aV s (µ r) + ( rV 2, 0 + β0rV 2, 1 ) aV, 2 s (µ r) + ( rV 3, 0 + β1rV 2, 1 + 2β0rV 3, 1 + β 2 0 rV 3, 2 ) aV, 3 s (µ r) + ( rV 4, 0 + β V 2 rV 2, 1 + 2β1rV 3, 1 + 5 2 β0β1rV 3, 2 + 3β0rV 4, 1 +3β 2 0rV 4, 2 + β 3 0rV 4, 3 ) aV, 4 s (µ r) + O(aV, 5 s ), (4) where aV s (µ r) = α V s (µ...

[4] [4]

self-consistency problem

into the V -scheme and the heavy quark mass eﬀects are also considered in the perturbation coeﬃcient r2 [21], when mc = 1 . 67 GeV and mb = 4 . 78 GeV are taken. At the typical momentum Q = mc, the eﬀective coupling ag1 s (mc) takes the form ag1 s (mc)|Conv. = aV s (mc) + 1. 429aV, 2 s (mc) − 1. 586 ×aV, 3 s (mc) + 40. 442aV, 4 s (mc), (22) in which µ r i...

[5] [5]

181− 0. 045 − 0. 012 (∆ QV ∗ ), indicating that the residual scale de- pendence for PMCs is about ( − 24. 9% − 6. 6% ) . Diﬀerent from the PMCs scale QV ∗ , the PMC ∞ eﬀec- tive scales are ﬁxed order-by-order and free of perturba- tive nature. Its uncertainty comes from the unknown last PMC scale µ V 4 in Eq.( 23), which needs a higher-order rV 5 - term t...

[6] [6]

self-consistency problem

024 GeV [ 70]. In infrared region, the LFHQCD model gives ag1, LFHQCD s (0) = 1. To achieve a smooth scale- running behavior of α g1 s (Q) across the whole Q-range, one can match the low-energy LFHQCD model with its behavior in perturbative behavior by ﬁxing proper val- ues for the critical scale Q0 and κ. Their values can be ﬁxed by two criterions, e.g. ...

[7] [7]

456γ m 2 + 26

633 − 15 16 γ ns 1 + 3 8 K ns 1 + 9 16 Π ns 1 ) , (44) ˆrV 3, 2 = 25 12 γ ns 1 − 10 3 K ns 1 − 5 2 Π ns 1 , (45) ˆrV 4, 0 = −10 + 35. 456γ m 2 + 26. 592γ ns 2 + 9 2 γ ns 3 + 3 4 γ ns 4 −4 × 10− 14K ns 1 − 2 × 10− 14Π ns 1 +γ ns 1 [

[8] [8]

832 − 27 4 γ ns 2 − 9 8 γ ns 3 ] − 9 16 (γ ns 2 )2 −(γ ns 1 )2 [