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REVIEW 3 major objections 6 minor 29 references

NNLO compatibility between pQCD theory and phenomenology in determination of the $b$-quark pole and \MSbar running masses

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that separately fitted b-quark pole and running masses agree with the NNLO perturbative QCD relation to within 0.02 GeV.

desk verdict The advertised NNLO compatibility check evaluates a relation defined at the b-quark scale with alpha_s(M_Z), so the claimed 99.98% agreement is a scale artifact; the underlying PDF fit is competent but the paper's central claim fails. read the letter →

arxiv 1909.00796 v2 pith:X2GJVPBV submitted 2019-09-02 hep-ph

A. Vafaee , K. Javidan , A. B. Shokouhi This is my paper · ORCID
classification hep-ph PACS 12.38.Bx13.60.Hb14.65.Fy
keywords b-quarkpolemassMSbarrunningnext-to-next-to-leadingorderHERAbeautyproductionpartondistributionfunctionsgluonQCDrenormalizationstrongcouplingconstant
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 attempts to show that two standard ways of quoting the b-quark mass, the pole mass (the propagator-pole mass) and the MSbar running mass (a scale-dependent short-distance mass), satisfy the next-to-next-to-leading-order conversion relation of perturbative QCD. The authors derive the NNLO formula from the mass renormalization constants and then fit both masses from HERA inclusive and beauty-production data. The pole mass comes out $4.66 \pm 0.14$ GeV and the running mass $4.40 \pm 0.10$ GeV, and inserting these into the pQCD relation predicts $4.38$ GeV, an agreement to $0.02$ GeV. They also report that freeing the b-quark mass in the PDF fit improves the uncertainty band and shifts the gluon and valence distributions. A sympathetic reader would care because a genuine 0.02 GeV agreement would validate the NNLO conversion and show that beauty data can pin down a Standard Model parameter directly.

What carries the argument

The carrying identity is the NNLO conversion formula derived from the ratio of the on-shell and MSbar mass renormalization factors, $Z^{\mathrm{OS}}_m / Z^{\overline{\mathrm{MS}}}_m$. With the standard color factors $C_F = 4/3$, $C_A = 3$, $T_R = 1/2$ and one heavy flavor, the relation becomes $\overline{m}_b(M_b) = M_b\left[1 - \frac{4}{3}\left(\frac{\alpha_s}{\pi}\right) + \left(\frac{\alpha_s}{\pi}\right)^2(1.0414\,N_L - 14.3323)\right]$. The paper inserts the phenomenological masses into this identity as a consistency check. The companion machinery is a QCD fit over HERA inclusive and beauty data with the b mass treated as an extra free parameter.

What would settle it

Recompute Eq. (20) with $\alpha_s$ evaluated at $\mu = M_b \approx 4.66$ GeV, where $\alpha_s \approx 0.21$, instead of $\alpha_s(M_Z) = 0.118$; the right-hand side then gives about 4.0 GeV rather than 4.38 GeV, which differs from the fitted 4.40 GeV by roughly 0.4 GeV and removes the claimed 99.98% compatibility.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is that the separately fitted pole mass and MSbar running mass satisfy the NNLO pQCD relation: $4.40 \simeq 4.66\left[1 - \frac{4}{3}\left(\frac{\alpha_s}{\pi}\right) + \left(\frac{\alpha_s}{\pi}\right)^2 (1.0414\,N_L - 14.3323)\right] = 4.38$. The paper calls this compatibility up to approximately $99.98\%$, with a difference of $|4.40 - 4.38| = 0.02$ GeV. It further claims that the running-mass fit is preferred: the $\chi^2$ improves by about $0.8\%$ and the mass uncertainty shrinks by about $4\%$ relative to the pole-mass fit, and that the gluon distribution is sensitive to the mass choice.

Load-bearing premise

The load-bearing premise is that the strong coupling in the NNLO conversion formula can be evaluated at the Z-boson scale, $\alpha_s(M_Z)=0.118$, even though the formula itself is defined at the b-quark mass scale; replace it with $\alpha_s(M_b)\approx 0.21$ and the claimed agreement disappears.

Editorial extensions

If this is right

  • If the compatibility is genuine, HERA beauty-production data alone can fix the b-quark mass in either scheme with an NNLO uncertainty below 0.15 GeV.
  • The small $\chi^2$ improvement and reduced uncertainty for the running mass imply that future fits should treat the beauty mass as scale dependent rather than as a fixed pole mass.
  • The gluon distribution and valence ratios change noticeably when the beauty mass is freed, so combined inclusive-plus-beauty fits must report the mass parameter alongside the PDFs.
  • Agreement with the world average supports using this approach as a cross-check on the b mass.

Reading between the lines

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

  • The authors do not state this, but if the NNLO relation is evaluated at the b-quark scale rather than at $M_Z$, the extracted pole and running masses are not compatible; the 0.02 GeV agreement depends on inserting $\alpha_s(M_Z) = 0.118$.
  • A decisive extension would be a simultaneous fit of the pole mass and the strong coupling so that the scale dependence becomes an explicit fit parameter rather than an external input.
  • The modest $\chi^2$ preference for the running mass may reflect PDF-mass correlations rather than a genuinely better description; separating the two requires fits at different mass scales and comparing the pulls of the beauty data.
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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 b-quark pole mass and MSbar running mass from HERA combined inclusive and beauty production data using xFitter at NNLO (the HBPoleMass and HBRunMass fits), and it derives a standard NNLO relation between the pole and MSbar masses (Eqs. (15)-(18)). The fits give Mb = 4.66 ± 0.14 GeV and mb = 4.40 ± 0.10 GeV. The paper then claims that these two fitted values satisfy the NNLO relation at the 0.02 GeV level, quoting a "compatibility up to approximately 99.98%", and it studies the impact of the b-quark mass on the gluon and valence PDFs.

Significance. If the central compatibility claim were correct, this would be a useful cross-check of the NNLO pole-to-MSbar mass relation using HERA beauty data. The paper has strengths: it uses standard open-source tools (xFitter, QCDNUM, APFEL), it reports fit-quality numbers and uncertainties, and the PDF-sensitivity study concerning the b-quark mass is a potentially useful byproduct. The fitted mass values are plausible and broadly consistent with the PDG range. However, the central quantitative claim is invalid: Eq. (20) evaluates a relation defined at the b-quark mass scale with alpha_s(M_Z), and a correct evaluation gives a discrepancy of roughly 0.4 GeV rather than 0.02 GeV. The advertised compatibility is therefore not established.

major comments (3)
  1. [Sec. IV and Sec. VI, Eq. (20)] Equation (20) evaluates the NNLO mass relation of Eq. (17) with alpha_s(M_Z) = 0.118, but Eqs. (15)-(17) define alpha_s as alpha_s^{(N_L+N_H)}(M_b), i.e. the strong coupling renormalized at the b-quark pole-mass scale. At this scale alpha_s is approximately 0.21, not 0.118. Replacing 0.118/pi in Eq. (20) by alpha_s(M_b)/pi changes the bracket from roughly 0.93 to roughly 0.86, so the theoretical prediction for m_b(M_b) from M_b = 4.66 GeV becomes approximately 4.0 GeV. The difference from the fitted m_b = 4.40 ± 0.10 GeV is then about 0.4 GeV, far outside the quoted uncertainty. The claimed compatibility therefore rests on evaluating a mu = M_b relation at mu = M_Z and fails when the relation is evaluated at its proper scale.
  2. [Sec. VI, Eq. (20) and following text] The text states a "precision of 1 part in 10^2" and a "compatibility up to approximately 99.98%". These statements are arithmetically inconsistent: |4.40 - 4.38| = 0.02 GeV relative to 4.40 GeV is 0.45%, not 0.02%, so the compatibility is about 99.55% even if the value 4.38 were correct. Moreover, evaluating Eq. (17) with alpha_s(M_Z) = 0.118 and N_L = 3 gives 4.35 GeV rather than 4.38 GeV, making the numerical agreement still weaker. These arithmetic errors compound the scale error identified above.
  3. [Secs. V-VI and Table II] The compatibility check compares two mass values extracted from the same HERA data set with the same NNLO FONLL implementation. The HBRunMass variant (FONLL-C RUNMON) presumably uses a running-mass scheme internally, and the manuscript does not document whether that scheme conversion is independent of the perturbative relation that Eq. (20) is testing. Without such documentation, the agreement between the two fitted masses is a weaker cross-check than the text implies, even setting aside the scale error.
minor comments (6)
  1. [Sec. I] The sentence beginning "Measurements of open b-quark production in DIS..." is repeated nearly verbatim in the first two paragraphs of the Introduction.
  2. [Sec. V] In the bullet defining the heavy-quark scale, "mu_r = mu_f = mu_r = sqrt(Q^2 + 4m_b^2)" contains a typo; the last scale should presumably be a heavy-quark scale such as mu_b.
  3. [Sec. II, Eqs. (3)-(5)] The notation for beauty contributions is inconsistent: sigma_red^{b bar b} is used in Eqs. (3)-(4), while F^{bb} appears in Eq. (5) and the surrounding text.
  4. [Sec. VI, Figs. 2-5] The figure captions refer to the "PPDs analysis" where "PPDFs analysis" is meant.
  5. [Sec. VI, Figs. 2-5] The quantities delta(x d_v)/(x d_v) and similar ratios shown in the figures are not defined in the text; a definition should be given in the captions or in Sec. VI.
  6. [Sec. IV, Eq. (18)] Equation (18) is presented as the inverse of Eq. (17), but the scale and flavor-counting conventions used there are not stated; the text should specify that alpha_s is evaluated at mu = m_b in that formula.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the NNLO mass relation is an external pQCD input and the two fitted masses are separate fit outputs; the scale mismatch in Eq. (20) is a correctness issue, not a circularity.

full rationale

The claimed derivation chain is not circular. The pole-to-running-mass relation in Eqs. (15)-(18) is obtained from the standard mass-renormalization constants Z_m^MS and Z_m^OS in Eqs. (7)-(14), with coefficients quoted as pQCD results; it is not defined in terms of the fitted masses. The pole mass M_b=4.66 GeV and the running mass m_b=4.40 GeV are outputs of two separate xFitter fits (HBPoleMass and HBRunMass) to the HERA combined inclusive data and the H1+ZEUS beauty data. The compatibility check in Eq. (20) evaluates that independent theoretical relation using the fitted numbers; there is no fitted parameter that is renamed as a prediction. The paper's self-citations (Refs. [8], [14]-[20]) concern the fitting framework and PDF parametrization, not the mass relation, so they are not load-bearing. One important non-circularity concern is that Eq. (20) inserts alpha_s(M_Z)=0.118 into Eq. (17), whose alpha_s is defined in Eq. (15) at mu=M_b; this is a scale-consistency/correctness flaw that would substantially change the right-hand side (the bracket becomes about 0.86 if alpha_s(M_b)~0.21, giving ~4.0 GeV instead of 4.38 GeV), but it does not make the argument circular. No circular step is therefore established.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim depends on two fitted mass values, 4.66 GeV and 4.40 GeV, and on 14 nuisance PDF shape parameters. No new physical entities are introduced. The theory relation uses standard QCD color factors and renormalization constants, and the FONLL-C scheme is an external tool assumption. The critical unstated premise is the use of alpha_s(M_Z) in Eq. (20); that is an error in execution rather than a stated axiom, and it is captured in the red flags.

free parameters (3)
  • b-quark pole mass M_b = 4.66 +/- 0.14 GeV
    Fitted as the extra mass parameter in the HBPoleMass analysis; it is the input used in the compatibility check of Eq. (20).
  • b-quark MSbar running mass m_b = 4.40 +/- 0.10 GeV
    Fitted as the extra mass parameter in the HBRunMass analysis; it is the phenomenology value against which the pQCD prediction is compared.
  • 14 HERAPDF shape parameters = See Table II
    The PDF functional form in Eq. (19) has 14 free parameters fitted together with the mass; they act as nuisance parameters for the mass extraction and are not the focus of the central claim.
assumptions (4)
  • domain assumption QCD factorization and DGLAP collinear evolution describe the HERA inclusive and beauty cross sections at NNLO.
    The fit uses QCDNUM and APFEL to evolve the PDFs and FONLL-C to include heavy-quark contributions; if these tools mis-implement NNLO, the fitted masses are biased.
  • domain assumption The combined H1 and ZEUS data sets are statistically compatible and have correctly reported uncertainties.
    Both data sets enter the simultaneous fit and dominate the mass extraction; the paper does not independently validate their systematic uncertainties.
  • standard math The standard SU(3) color factors and the quoted NNLO renormalization constants in Sec. IV are correct.
    CF=4/3, CA=3, TR=1/2 and the d_k coefficients are taken from established QCD results, not re-derived in the paper.
  • domain assumption The FONLL-C and FONLL-C RUNMON variants in APFEL correctly convert between pole and MSbar masses at NNLO.
    The two fits inherit whatever scheme conversion the code uses; if the conversion differs from Eq. (17), the compatibility test is partly built into the tool.

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Cite this review

Pith. "Pith review of NNLO compatibility between pQCD theory and phenomenology in determination of the $b$-quark pole and \MSbar running masses." pith.science (2026). https://pith.science/paper/X2GJVPBV

@misc{pith2026190900796,
  author       = {Pith},
  title        = {Pith review of: NNLO compatibility between pQCD theory and phenomenology in determination of the $b$-quark pole and \MSbar running masses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X2GJVPBV}},
  note         = {Machine review of arXiv:1909.00796}
}
abstract

This contribution attempts to determine the $b$-quark pole mass $M_b$ and \MSbar running mass $\overline{m}_b$ with two different approaches at the next-to-next-to-leading order (NNLO) corrections. At the first approach, we derive a relation between the $b$-quark pole mass $M_b$ and its \MSbar running mass $\overline{m}_b$ at the NNLO corrections based on the perturbative Quantum Chromo Dynamics (pQCD) predictions. At the second approach, we extract numerical values of the $b$-quark pole and \MSbar running masses based on the NNLO phenomenology of H1 and ZEUS Collaborations combined beauty vertex production experimental data. Then we discuss about the compatibility between the pQCD theory results and phenomenology approach in determination of the $b$-quark pole and \MSbar running masses at the NNLO corrections. Also, we investigate the role and influence of the $b$-quark mass as an extra degree of freedom added to the input parameters of the Standard Model Lagrangian, on the improvement of the uncertainty band of the proton parton distribution functions (PDFs) and particularly on the gluon distribution.

Figures

Figures reproduced from arXiv: 1909.00796 by the authors.

Figure 1
Figure 1. Comparison of pure impact of the b-quark pole mass Mb (blue color) and MS running mass mb (yellow color) on the gluon distribution as a function of x at Q2 = 1.9, 5.0, 8.0 and 10 GeV2 . 14 [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Comparison of the d-valence and dv-ratio of the proton PDFs without (PPDs analysis with blue color) and with (HBPoleMass analysis with purple color) inclusion of the b-quark pole mass Mb as an extra degree of freedom added to the input parameters of the Standard Model Lagrangian. x −4 10 −3 10 −2 10 −1 10 1 ) 2 (x,Q V xd 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 2 = 100 GeV 2 Q PPDFs HBRunMass x −4 10 −3 10 −2 10 −1 10 … view at source ↗
Figure 3
Figure 3. Comparison of the d-valence and dv-ratio of the proton PDFs without (PPDs analysis with blue color) and with (HBRunMass analysis with yellow color) inclusion of the b-quark MS running mass mb as an extra degree of freedom added to the input parameters of the Standard Model Lagrangian. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Comparison of the u-valence and uv−dv-ratio of the proton PDFs without (PPDs analysis with blue color) and with (HBPoleMass analysis with red color) inclusion of the b-quark pole mass Mb as an extra degree of freedom added to the pQCD Lagrangian. x −4 10 −3 10 −2 10 −1…
Figure 5
Figure 5. Figure 5: Comparison of the u-valence and uv−dv-ratio of the proton PDFs without (PPDs analysis with red color) and with (HBRunMass analysis with blue color) inclusion of the b-quark MS running mass mb as an extra degree of freedom added to the pQCD Lagrangian. 16 [PITH_FULL_IM…

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Works this paper leans on

29 extracted references · 21 canonical work pages

  1. [1]

    8 % in the quality of the fit for determination of the b-quark MS running mass mb relative to the b-quark pole mass Mb

    187 ∼ 0. 8 % in the quality of the fit for determination of the b-quark MS running mass mb relative to the b-quark pole mass Mb. HERA run I + II combined and H1 and ZEUS Collaboration beauty combined da ta Experiment HBPoleMass HBRunMass HERA I +II CC e+p [1] 51 / 39 50 / 39 HERA I +II CC e−p [1] 49 / 42 49 / 42 HERA I +II NC e−p [1] 218 / 159 217 / 159 HE...

    work page

  2. [2]

    26 ± 0. 40 2. 30 ± 0. 44 B′ g 0. 005 ± 0. 058 0. 020 ± 0. 069 mb pole mass Mb = 4. 66 ± 0. 14 MS running mass mb = 4. 40 ± 0. 10 Table II: The NNLO numerical values of 15 fit parameters and their uncertainties, including 14 free central PDF parameters and 1 extra mb parameter corresponding to HBPoleMass and HBRunMass analysis. Now, if we insert our phenome...

    work page

  3. [3]

    0414 × 3 − 14

    141 ) 2 (1. 0414 × 3 − 14. 3323) ] , (20)

    work page

  4. [4]

    14 GeV and MS running mass mb = 4

    66 ± 0. 14 GeV and MS running mass mb = 4. 40 ± 0. 10 GeV, we obtain up to ∼ 4. 0 % pure improvement in the uncertainty value of the b-quark MS running mass mb relative to the b-quark pole mass Mb. In Sec. IV, we extracted the relation between the b-quark pole mass Mb and MS running mass mb at the NNLO of pQCD framework as follows: mb(Mb) = Mb [ 1 − 4 3 (...

    work page

  5. [5]

    3 · 10−2 ≤ x ≤ 4

    0 · 104 GeV2 and 1. 3 · 10−2 ≤ x ≤ 4. 0 · 10−1 at values of the inelasticity 3. 7 · 10−4 ≤ y = Q2 sx ≤ 7. 6 · 10−3 . The reduced cross sections for inclusive unpolarized CC e±p scattering are defined in terms of CC structure functions W ± 2 , W ± 3 and W ± L as follows [1]: σ ± r,CC = 2πx G2 F [ M 2 W + Q2 M 2 W ] 2 d2σ e± p CC dxdQ2 (2) = (1 + (1 − y)2) 2...

    work page 2000

  6. [6]

    66 [ 1 − 4 3 ( 0

    40 ∼ 4. 66 [ 1 − 4 3 ( 0. 118

    work page

  7. [7]

    Special Support Program for the Promotion of Scienti fic Authority

    40 ∼ 4. 38 , where according to our methodology in Sec. V, we set the stron g coupling constant at M 2 Z scale to α NNLO s (M 2 Z) = 0 . 118. If we qualify the error as: △x = |xf − xi|, we see that the difference of our numerical results extracted based on the phenomenology of experiment al data for the b-quark pole mass 12 Mb and MS running mass mb with t...

    work page

  8. [8]

    Abramowicz et al

    H. Abramowicz et al. [H1 and ZEUS Collaborations], Eur. Phys. J. C 75, no. 12, 580 (2015) [arXiv:1506.06042 [hep-ex]]

    arXiv 2015

Show all 29 references
  1. [9]

    Abramowicz et al

    H. Abramowicz et al. [H1 and ZEUS Collaborations], Eur. Phys. J. C 78, no. 6, 473 (2018) [arXiv:1804.01019 [hep-ex]]

    2018 arXiv

  2. [10]

    Gluck, E

    M. Gluck, E. Reya and M. Stratmann, Nucl. Phys. B 422, 37 (1994)

    1994

  3. [11]

    R. S. Thorne and R. G. Roberts, Phys. Rev. D 57, 6871 (1998) [hep-ph/9709442]

    1998 arXiv

  4. [12]

    V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys. 15, 675 (1972) [Yad. Fiz. 15, 1218 (1972)]

    1972

  5. [13]

    Aktas et al

    A. Aktas et al. [H1 Collaboration], Eur. Phys. J. C 47, 597 (2006) [hep-ex/0605016]

    2006 arXiv

  6. [14]

    Abramowicz et al

    H. Abramowicz et al. [ZEUS Collaboration], Eur. Phys. J. C 71, 1573 (2011) [arXiv:1101.3692 [hep-ex]]

    2011 arXiv

  7. [15]

    Vafaee and A

    A. Vafaee and A. N. Khorramian, Nucl. Phys. B 921, 472 (2017) [arXiv:1709.08346 [hep-ph]]

    2017 arXiv

  8. [16]

    A. D. Martin, W. J. Stirling and R. S. Thorne, Phys. Lett. B 636, 259 (2006) [hep-ph/0603143]

    2006 arXiv

  9. [17]

    M. A. G. Aivazis, F. I. Olness and W. K. Tung, Phys. Rev. D 50, 3085 (1994) [hep-ph/9312318]

    1994 arXiv

  10. [18]

    Bertone, S

    V. Bertone, S. Carrazza and J. Rojo, Comput. Phys. Commun . 185, 1647 (2014) [arXiv:1310.1394 [hep-ph]]

    2014 arXiv

  11. [19]

    Forte, E

    S. Forte, E. Laenen, P. Nason and J. Rojo, Nucl. Phys. B 834, 116 (2010) [arXiv:1001.2312 [hep-ph]]

    2010 arXiv

  12. [20]

    http://xFitter.org [xFitter.org] [arXiv:1410.4412 [hep-ph]]

    xFitter, An open source QCD fit framework. http://xFitter.org [xFitter.org] [arXiv:1410.4412 [hep-ph]]

    arXiv

  13. [21]

    Vafaee and A

    A. Vafaee and A. B. Shokouhi, arXiv:1906.07390 [hep-ph]

    1906 arXiv

  14. [22]

    Vafaee and A

    A. Vafaee and A. B. Shokouhi, arXiv:1904.04285 [hep-ph]

    1904 arXiv

  15. [23]

    A. B. Shokouhi and A. Vafaee, Nucl. Part. Phys. Proc. 300-302, 35 (2018)

    2018

  16. [24]

    Vafaee, Nucl

    A. Vafaee, Nucl. Part. Phys. Proc. 300-302, 30 (2018)

    2018

  17. [25]

    Vafaee, arXiv:1806.07995 [hep-ph]

    A. Vafaee, arXiv:1806.07995 [hep-ph]

    arXiv

  18. [26]

    Vafaee and A

    A. Vafaee and A. Khorramian, Nucl. Part. Phys. Proc. 282-284, 32 (2017)

    2017

  19. [27]

    Vafaee, A

    A. Vafaee, A. Khorramian, S. Rostami and A. Aleedaneshv ar, Nucl. Part. Phys. Proc. 270-272, 27 (2016)

    2016

  20. [28]

    Botje, Comput

    M. Botje, Comput. Phys. Commun. 182, 490 (2011) [arXiv:1005.1481 [hep-ph]]

    2011 arXiv

  21. [29]

    K. A. Olive et al. [Particle Data Group], Chin. Phys. C 38, 090001 (2014). 19

    2014

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