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Heavy-quark deep-inelastic scattering with a running mass

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S. Alekhin, S. Moch

High Energy Physics - Phenomenology · Submitted 26 Nov 2010 · 42 references

Pith's verdict: A real, checkable first extraction of the MS-bar charm mass from DIS; the central value is fine, but the ±0.03 GeV theory error is smaller than the NLO→NNLO shift it is supposed to cover.

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arXiv ID1011.5790 (version 1)
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CitationS. Alekhin, S. Moch. "Heavy-quark deep-inelastic scattering with a running mass." arXiv:1011.5790 (2010). https://pith.science/paper/Q4TRSUOV
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The paper

The paper claims

Replacing the pole mass with a running MS-bar mass makes heavy-quark deep-inelastic predictions more stable and yields a charm mass of $m_c(m_c)=1.01$ GeV from a global fit to fixed-target and HERA data.

The reading

Heavy-quark production in deep-inelastic scattering is one of the places where the value of the charm mass feeds directly into the parton distributions used at the LHC, yet every global fit still parametrizes that mass as a pole mass. The authors argue that this is the wrong parameter: the pole mass is beset by ambiguities of order $1/m$ and produces a perturbative series that grows rather than settles when higher orders are added. They convert the known NLO and approximate NNLO structure functions from the pole mass to the running $MS$-bar mass $m(m)$, using the three-loop relation between the two schemes, and show that the corrections shrink and the scale-variation band narrows, especially for neutral-current charm and bottom production. On that basis they refit fixed-target neutrino data and HERA collider data and extract $m_c(m_c) = 1.01$ GeV, close to the accepted world average but obtained from space-like rather than $e^+e^-$ kinematics. If the argument holds, heavy-quark mass uncertainties in global fits can be reduced to the few-percent level and the resulting shifts in predicted $W^+$ and $Z$ rates at the LHC can be brought under control.

What carries the argument

The load-bearing device is the perturbative scheme conversion from pole mass to running $MS$-bar mass inside the DIS structure functions. Because the coefficient functions depend on the mass both explicitly and through the threshold variable $η(z)$ and the ratio $ξ = Q^2/m^2$, the derivative $∂_m$ can be rewritten as a $z$-derivative acting on the parton distributions plus explicit $ξ$- and $μ_f$-derivative terms, with a boundary contribution at $z=z_{max}$ that vanishes for the Born term and can be removed to all orders by integration by parts. This turns the mass shift into a convolution of the PDFs with derivatives of the coefficient functions, so the whole conversion costs no new hard scattering computation. A second ingredient is the scale-setting choice $μ_r^2 = μ_f^2 = Q^2 + κ m_c^2$ with $κ ∈ [0,8]$, used both as the central scale and as the estimator of theoretical uncertainty, since the usual $μ=Q$ choice would erase the mass sensitivity the fit relies on.

Extended reading notes

Core claim

The paper's central claim is that the running $MS$-bar mass, not the pole mass, is the correct mass parameter for heavy-quark deep-inelastic scattering, and that using it gives both a better-behaved perturbative expansion and a first-principles determination of the charm mass from DIS data. Concretely, the authors take the existing NLO neutral-current and charged-current coefficient functions and the threshold-enhanced approximate NNLO terms, and apply the scheme transformation of Eq. (8): derivatives of the structure functions with respect to the mass, evaluated with the three-loop pole-to-$MS$-bar coefficients $d^{(1)}, d^{(2)}$, with the matching scale set to $μ_r = m(m)$. The resulting predictions show relative corrections dropping from 32% at NLO and 13% at NNLO for a 1.5 GeV pole mass to 17% and 6% for $m_c(m_c)=1.3$ GeV, and a visibly smaller band under independent variation of $μ_r$ and $μ_f$. Fitting the same 25 parameters as the ABKM analysis, with $m_b(m_b)$ fixed to the PDG value because the fit shows no sensitivity to it, they obtain $m_c(m_c) = 1.26 ± 0.09 (exp) ± 0.11 (th)$ GeV at NLO and $m_c(m_c) = 1.01 ± 0.09 (exp) ± 0.03 (th)$ GeV at NNLOapprox, the latter quoted as the best estimate and consistent with the PDG average within about $1.5σ$.

Load-bearing premise

The approximate NNLO prediction, built from threshold-enhanced terms plus exact scale and scheme-conversion pieces, is accurate enough in the low-$Q$, small-$x$ region where the fit has leverage that the quoted $0.03$ GeV theory error really covers the missing two-loop corrections.

Editorial extensions

If this is right

  • Global PDF fits can adopt the running mass and propagate the charm and bottom mass errors directly into the parton distributions, with no extra assumptions about scheme dependence.
  • The few-percent shifts in predicted $W^\pm$ and $Z$ production at the LHC that follow from 0.15 GeV changes in the pole charm mass can be largely removed once a short-distance mass is used.
  • The charm mass extracted from space-like DIS kinematics agrees with the $e^+e^-$ and $B$-decay determinations that feed the PDG average, so the two kinematic regimes can be combined into a single consistent value.
  • The same conversion can be carried through in variable-flavor-number schemes, where the massive operator matrix elements would then also be expressed in the running-mass scheme.
  • The publicly released numerical code makes the running-mass structure functions available for direct use in future fits and in comparisons with HERA Run II data.

Reading between the lines

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

  • If the threshold-enhanced approximation is later replaced by the full NNLO coefficient functions, the central value is likely to move by an amount comparable to or larger than the quoted $\pm 0.03$ GeV theory error, since the approximation captures only the logarithmically enhanced part of the two-loop correction.
  • The fact that the NLO and NNLOapprox fits differ by about 0.25 GeV while each quotes a theory error of 0.11 and 0.03 GeV respectively suggests the theory-error estimate is more an indicator of scale sensitivity than a full account of missing higher orders.
  • Because the fit shows no sensitivity to $m_b(m_b)$, a bottom-mass determination along these lines would need bottom-tagged structure function data at higher $Q^2$ rather than the currently available samples.
  • The improvement in apparent convergence with a running mass is a general feature of short-distance observables, so the same conversion should pay off in other processes where the pole mass is still the default, such as charged-current heavy-quark production at future neutrino facilities.
Original abstract

We study the production of heavy quarks in deep-inelastic scattering within perturbative QCD. As a novelty, we employ for the first time the running mass definition in the MSbar scheme for deep-inelastic charm and bottom production. We observe an improved stability of the perturbative expansion and a reduced theoretical uncertainty due to variations of the renormalization and factorization scales. As our best estimate we extract from a global fit to fixed-target and HERA collider data for the charm-quark an MSbar mass of m_c(m_c) = 1.01 +- 0.09 (exp) +- 0.03 (th) GeV.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,654 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.95 · states limitation 0.21 · human participants 0.01 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.48 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We study the production of heavy quarks in deep-inelastic scattering within perturbative QCD. As a novelty, we employ for the first time the running mass definition in the MSbar scheme for deep-inelastic charm and bottom production. We observe an improved stability of the perturbative expansion and a reduced theoretical uncertainty due to variations of the renormalization and factorization scales. As our best estimate we extract from a global fit to fixed-target and HERA collider data for the charm-quark an MSbar mass of m_c(m_c) = 1.01 +- 0.09 (exp) +- 0.03 (th) GeV.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study the production of heavy quarks in deep-inelastic scattering within perturbative QCD. contribution 0.61 0.85 0.03 asserted 0.93
s1 As a novelty, we employ for the first time the running mass definition in the MSbar scheme for deep-inelastic charm and bottom production. contribution 0.99 0.95 0.10 asserted 1.00
s2 We observe an improved stability of the perturbative expansion and a reduced theoretical uncertainty due to variations of the renormalization and factorization scales. result 0.92 0.95 0.32 asserted 1.00
s3 As our best estimate we extract from a global fit to fixed-target and HERA collider data for the charm-quark an MSbar mass of m_c(m_c) = 1.01 +- 0.09 (exp) +- 0.03 (th) GeV. result 1.00 0.96 0.67 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.54 · evidence from analysis of existing data 0.98.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The mass dependence of the NC charm structure function F p 2 for a proton with Q 2 = 10GeV2 , x = 10−3 and µr = µf = p Q2 +4m2 c using the PDFs of [21]. The charm-quark mass is taken in the on-shell scheme (left) and in the MSscheme (right) at LO (blue), NLO (green) and NNLOapprox (red). F2 charm (ABKM_3 NNLO) Q 2 = 10 GeV2 , x = 10-3 scale unc. pole mass mc (GeV) LO NLO NNLOapprox 0 0.1 0.2 0.3 0.4 0.5 0.6 1 1.5 2 F2 charm (ABKM_3 NNLO) Q 2 = 10 GeV2 , x = 10-3 scale unc. running mass mc (mc ) (GeV) LO NLO NNLOapprox 0 0.1 0.2 0.3 0.4 0.5 0.6 1 1.5 2
Figure 2. Same as in Fig.1. The band denotes the independent variation of the scales µr , µf = κ p Q2 +4m2 c in the range κ ∈ [1/2,2]. F2 bottom (ABKM_4 NNLO) Q 2 = 50 GeV2 , x = 10-3 µ 2 = Q2 + 4 mb 2 pole mass mb (GeV) LO NLO NNLOapprox 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 3.5 4 4.5 5 F2 bottom (ABKM_4 NNLO) Q 2 = 50 GeV2 , x = 10-3 µ 2 = Q2 + 4 mb 2 running mass mb (mb ) (GeV) LO NLO NNLOapprox 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 3.5 4 4.5 5
Figure 3. The mass dependence of the NC bottom structure function F p 2 for a proton with Q 2 = 50GeV2 , x = 10−3 and µr = µf = p Q2 +4m2 c . The bottom-quark mass is taken in the on-shell scheme (left) and in the MSscheme (right) at LO (blue), NLO (green) and NNLOapprox (red). 13
Figure 4. Same as in Fig.3. The band denotes the independent variation of the scales µr , µf = κ p Q2 +4m2 c in the range κ ∈ [1/2,2]. F2 charm (ABKM_3 NLO) Q 2 = 10 GeV2 , x = 10-1 µ 2 = Q2 + mc 2 pole mass mc (GeV) LO NLO 0 0.05 0.1 0.15 0.2 0.25 0.3 1 1.5 2 F2 charm (ABKM_3 NLO) Q 2 = 10 GeV2 , x = 10-1 µ 2 = Q2 + mc 2 running mass mc (mc ) (GeV) LO NLO 0 0.05 0.1 0.15 0.2 0.25 0.3 1 1.5 2
Figure 5. The mass dependence of the CC charm structure function F N 2 for a nucleon with Q 2 = 10GeV2 , x = 10−1 and µr = µf = p Q2 + m2 c using the PDFs of [21]. The charm-quark mass is taken in the on-shell scheme (left) and in the MSscheme (right) at LO (blue) and NLO (green). F2 charm (ABKM_3 NLO) Q 2 = 10 GeV2 , x = 10-1 scale unc. pole mass mc (GeV) LO NLO 0 0.05 0.1 0.15 0.2 0.25 0.3 1 1.5 2 F2 charm (ABKM_3 NLO) Q 2 = 10 GeV2 , x = 10-1 scale unc. running mass mc (mc ) (GeV) LO NLO 0 0.05 0.1 0.15 0.2 0.25 0.3 1 1.5 2
Figure 6. Same as in Fig.5. The band denotes the independent variation of the scales µr , µf = κ p Q2 + m2 c in the range κ ∈ [1/2,2]. 14
Figure 7. The light-quark (left) and gluon (right) PDFs obtained in the global fit: The dotted (red) lines denote the ±1σ band of absolute uncertainties resulting from the fit of this paper and the solid (red) line indicates the central prediction with with running masses 1.18 GeV (charm) and 4.19 GeV (bottom). For comparison the shaded (grey) area represents the results of ABKM [21]. ZEUS 10 -2 10 -1 1 10 10 2 10 3 10 4 10 5 1 10 10 2 10 3 mc (mc )=1.18 GeV x=3e-5 7e-5 1.8e-4 3.5e-4 6e-4 0.001 0.0015 0.003 0.006 0.012 0.03 Q 2 (GeV2 ) F c 2 4 i
Figure 8. The predictions of the global fit for the charm structure function F c 2 compared to data of Ref. [32] on F c 2 from Run I of HERA. The running mass mc(mc) = 1.18 GeV has been obtained in the variant of the fit with the PDG value Eq. (26) as an additional constraint. 15

Figures reproduced from arXiv:1011.5790 by the authors.

References
How later papers used it

50 later papers cite it

Evidence: substantial. 30 of the citing papers share no author with it, and 12 of those carry a confident label from Pith's citation classifier.

Pith read 25 of the 94 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 6 of 10 papers with this label · 4 by the same authors

By the same authors

  • Applied the formalism developed in this paper and fit the charm mass to cross section data together with all other non-perturbative parameters.

    Precise charm-quark mass from deep-inelastic scattering (2012) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

  • Matched its 4(5)-flavor ABM11 PDFs with the 3(4)-flavor ones at the charm and bottom mass scales, employing massive OMEs taken in the running-mass definition of this paper.

    PDF fit in the fixed-flavor-number scheme (2012) (same authors) · 0.96 · paraphrased: the licence does not allow quoting

  • Employed the running-mass definition in the MS scheme for the heavy quarks from this paper in its treatment of the heavy-quark contributions in DIS.

    Parton distribution functions and benchmark cross sections at NNLO (2012) (same authors) · 0.93 · paraphrased: the licence does not allow quoting

  • Employed the heavy-quark Wilson coefficients with the MS definition for the charm- and bottom-quark masses, as suggested in this paper.

    ABM11 parton distributions and benchmarks (2012) (same authors) · 0.89 · paraphrased: the licence does not allow quoting

Compares with it 1 of 7 papers with this label · 1 by the same authors

By the same authors

  • Compared the predictions based on the variant of the NNLO ABKM09 fit performed with the running-mass scheme of this paper to demonstrate that the FFN treatment of charm production is applicable at large transverse momentum.

    PDF fit in the fixed-flavor-number scheme (2012) (same authors) · 0.95 · paraphrased: the licence does not allow quoting

How the literature uses it

49 arXiv papers cite this in 93 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 59 sentences without a confident label

Builds on it 1 paper
Uses its method or tool 10 papers
Compares with it 7 papers
Cites it as support 5 papers
  • The massive Wilson coefficients re-calculated in terms of the running mass also demonstrate improved perturbative stability and the reduced renormalization/factorization scale sensitivity at the HERA kinematics [ref] .

    PDF fit in the fixed-flavor-number scheme · Introduction · 0.99

  • This demonstrates that the in principle unambiguous fixed-order perturbative predictions for heavy-quark production within the 3-flavor FFNS are sufficient for describing and explaining present experiments [ref] .

    Delineating parton distributions and the strong coupling · Quantitative Standard and Dynamical Results · 0.96

  • As a benefit, predictions for the heavy-quark structure functions in terms of the -mass display better convergence properties and greater perturbative stability at higher orders [ref] , thus reducing the inherent theoretical uncertainty.

    Parton distribution functions and benchmark cross sections at NNLO · Perturbative QCD · 0.98

Cites it as background 7 papers

What it cites · 29 sentences without a confident label

Uses the method or tool of 4 papers
Compares with 2 papers
Cites as background 3 papers

Discussion

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