{"id":"fae9a993-cc0a-4ad7-9c06-7f866800c4b4","arxiv_id":"2501.15525","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Using the PMC∞ scale-setting method in the V-scheme and matching to the LFHQCD model, the paper extracts α_s(M_Z)=0.1191±0.0012∓0.0006 from the Bjorken sum rule effective coupling.","lead":"The authors combine a scale-setting scheme for perturbative QCD with a holographic model for the low-energy region to produce a smooth strong coupling across all energies, and extract α_s(M_Z)=0.1191 ± 0.0012. The result is consistent with the world average, but it relies heavily on the assumed low-energy model and on matching the two descriptions at one point.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central value is anchored to the LFHQCD exponential form Eq. (26), and the quoted error bars omit any uncertainty in that functional form; a wrong IR parameterization would shift the extracted α_s(M_Z).","rationale":"I read the paper as a proposal to obtain α_s(M_Z) by joining a PMC∞-resummed pQCD prediction for the Bjorken-sum-rule effective charge to the LFHQCD infrared form at a matching point Q0. The crucial step is Eq. (26): the Gaussian LFHQCD form is not derived for the BSR effective charge, and because both Q0 and Λ_QCD are determined by value-plus-derivative matching, any error in the shape of the IR curve propagates directly into the extracted α_s(M_Z). This is exactly the reader's weakest assumption, so I agree with that diagnosis. I did not find an internal algebraic inconsistency in the PMC∞ construction or in the V-scheme transformation; the resolution of the previous self-consistency problem appears internally coherent. The reported p-value is overinterpreted as a validation of the method, since χ^2/d.o.f≈0.6 with 79 points already suggests that the uncertainties entering Eq. (27) are enlarged, but the p-value is not what fixes the central value. The decisive missing item is a quantified sensitivity of α_s(M_Z) to the assumed IR functional form. The paper already contains the DSE band, so the required check is inexpensive. Unless that test shows the extracted α_s(M_Z) to be stable under replacement of Eq. (26), the paper should remain CONDITIONAL, with the condition being explicit inclusion of IR-model-form uncertainty in the final error budget.","tokens_in":22032,"tokens_out":13097,"duration_ms":123509,"concrete_test":"Replace Eq. (26) in the matching of Sec. IIIB/IIIC with the DSE process-independent effective charge a_PI(Q) of Refs. [26,27] (or, as a minimal alternative, a one-parameter form a_s(Q)=A/(1+Q^2/κ^2) with A=a_s(0)=π that shares the same freezing value), keeping the identical PMC∞ pQCD series and solving the two matching conditions for Q0 and Λ_QCD. If the resulting α_s(M_Z) moves by more than ~0.002 relative to 0.1191, the omitted model-form systematics dominate the quoted ±0.0012 and the paper's error budget is incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The final α_s(M_Z)=0.1191±0.0012∓0.0006 is not obtained from pQCD alone. In Sec. IIIB/IIIC the PMC∞ series is matched to the LFHQCD coupling a_s^{g1}(Q)=e^{-Q^2/4κ^2} (Eq. (26)) by requiring equal values and equal derivatives at Q0. These two conditions fix both Q0 and Λ_QCD, and α_s(M_Z) is then evolved from that Λ_QCD. The quoted errors propagate only Δκ=±0.024 GeV and the second-kind residual scale dependence (Eq. (30)); there is no term for the uncertainty in the assumed Gaussian/confining form itself. The LFHQCD expression is a model of an approximately conformal AdS/QCD theory, not a derived result for the Bjorken-sum-rule effective charge, and the DSE/lattice process-independent charge shown in Fig. 4 has a different infrared shape and freezing value (~0.97π). Since the matching conditions include the first derivative, the Q-dependent curvature of the IR model enters the fitted Λ_QCD directly. A wrong functional form therefore biases α_s(M_Z) by an amount that can exceed the stated total error, and the χ^2/d.o.f≈0.6 / p≈99% statistic in Sec. IIIB does not resolve this because the same model choice shapes the theoretical error band.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22435,"tokens_out":5864,"duration_ms":52567,"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":[{"comment":"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).","section":"Sec. IIIB/IIIC, Eq. (26)"},{"comment":"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.","section":"Sec. IIIB, Eq. (27) and footnote 1"},{"comment":"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.","section":"Secs. IIIB and IIIC, Fig. 4 and Table II"}],"minor_comments":[{"comment":"The phrase 'the critical scale MZ' is presumably a typo for 'the reference scale MZ'; the critical scale elsewhere in the paper is Q0.","section":"Abstract and Sec. IIIC"},{"comment":"The text says 'CREN' where it should say 'CERN'.","section":"Sec. I"},{"comment":"The sentence beginning 'Different form the conventional results' should read 'Different from the conventional results'.","section":"Sec. IIIB"},{"comment":"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.","section":"Eq. (28)"},{"comment":"The notation 'the.' and the mixed use of ± and ∓ signs for the errors are confusing; the caption should define every error source explicitly.","section":"Table II caption"},{"comment":"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.","section":"Fig. 2 caption"},{"comment":"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.","section":"Sec. IIIB, Eq. (27)"}],"recommendation":"major_revision","confidential_remarks":"The main result is conditional on the LFHQCD model, and the current error bars do not include that model uncertainty. I do not see this as a reason for rejection, provided the authors reframe the claim as a model-dependent extraction and add a model-uncertainty estimate. I would also ask them to soften the p-value statement, since the reported χ2/d.o.f actually indicates overestimated errors rather than a particularly good fit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nWhere 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.\n\nWho 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.\n\nBottom 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.","headline":"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.","tokens_in":22921,"tokens_out":3011,"would_cite":false,"duration_ms":26473,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["12.38.-t"],"model":"deepseek-v4-flash","headline":"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.","keywords":["strong coupling constant","Bjorken sum rule","principle of maximum conformality","light-front holographic QCD","effective charge","renormalization scale ambiguity","V scheme","alpha_s(M_Z)"],"falsifier":"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 $0.1191\\pm0.0013$.","tokens_in":21842,"feed_emoji":"🎯","tokens_out":12030,"duration_ms":93333,"temperature":0.7,"pith_summary":"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.","feed_headline":"Bjorken sum rules and holographic QCD fix alpha_s(M_Z)=0.1191","feed_subtitle":"Matching the PMC∞ perturbative series to the infrared model reproduces 79 data points and nearly removes scale error.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the Bjorken sum rule whose effective coupling $a_s^{g_1}(Q)$ is the central observable of the paper.","marker":"[5, 6]"},{"why":"Supplies the four-loop (N$^3$LO) perturbative coefficients of the leading-twist part of the effective coupling.","marker":"[16–20]"},{"why":"Gives the charm- and bottom-quark decoupling mass corrections that are included in the perturbative series.","marker":"[21]"},{"why":"Introduces the PMC∞ scale-setting procedure that produces the order-by-order conformal series and removes the first kind of residual scale dependence.","marker":"[51]"},{"why":"Applied the earlier PMCm treatment to the Bjorken sum rule and exposed the self-consistency problem that the present paper resolves.","marker":"[49]"},{"why":"Showed that switching from the MS-scheme to the V-scheme avoids the self-consistency problem, motivating the scheme choice here.","marker":"[36]"},{"why":"Provides the light-front holographic QCD model, including the Gaussian form used for the infrared coupling.","marker":"[25]"},{"why":"Fixes the LFHQCD mass scale $\\kappa=0.523\\pm0.024$ GeV from hadron spectroscopy; this input produces the dominant quoted error.","marker":"[70]"},{"why":"Provides the Jefferson Lab spin-structure data that, with other experiments, define the 79-point dataset used for the fit and $p$-value.","marker":"[10–14]"},{"why":"Supplies the V-scheme $\\beta$-functions and the coefficient conversion needed to transform the series from the MS-scheme.","marker":"[58, 59]"}],"fun_headline_variants":["Holographic QCD and PMC∞ yield precise alpha_s=0.1191","Bjorken sum rules + holography fix alpha_s(M_Z)=0.1191","New alpha_s precision: 0.1191 from Bjorken rule and LFHQCD","Eliminating scale ambiguities: alpha_s(M_Z)=0.1191 precise","PMC∞ and LFHQCD combo sharpens alpha_s to 0.1191"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Holographic QCD and PMC∞ yield precise alpha_s=0.1191","Bjorken sum rules + holography fix alpha_s(M_Z)=0.1191","New alpha_s precision: 0.1191 from Bjorken rule and LFHQCD","Eliminating scale ambiguities: alpha_s(M_Z)=0.1191 precise","PMC∞ and LFHQCD combo sharpens alpha_s to 0.1191"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001359,"raw_usage":{"total_tokens":5625,"prompt_tokens":1164,"completion_tokens":4461,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":780,"completion_tokens_details":{"reasoning_tokens":4346}},"tokens_in":780,"tokens_out":4461,"duration_ms":31020,"temperature":1.0,"reasoning_tokens":4346,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:12:16.247778+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $0.1191\\pm0.0013$.","supporting_citations":[{"cited_title":"Fischler, Nucl","cited_arxiv_id":null,"evidence_quote":"Fixes the LFHQCD mass scale $\\kappa=0.523\\pm0.024$ GeV from hadron spectroscopy; this input produces the dominant quoted error."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the PMC∞ scale-setting procedure that produces the order-by-order conformal series and removes the first kind of residual scale dependence."},{"cited_title":"Peterman, Phys","cited_arxiv_id":null,"evidence_quote":"Applied the earlier PMCm treatment to the Bjorken sum rule and exposed the self-consistency problem that the present paper resolves."},{"cited_title":"Bl¨ umlein, G","cited_arxiv_id":null,"evidence_quote":"Showed that switching from the MS-scheme to the V-scheme avoids the self-consistency problem, motivating the scheme choice here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the light-front holographic QCD model, including the Gaussian form used for the infrared coupling."}],"review_version":1}