]>PLB33313S0370-2693(17)30894-810.1016/j.physletb.2017.11.002The Author(s)PhenomenologyFig. 1Reduced cross section for charm production in deep-inelastic scattering [14] as a function of the Bjorken scaling variable xBj for different values of photon virtuality Q2 (points). The measurements are grouped into six subsets in Q2, as indicated by the six rows, and detailed in Table 1. The curve shows the global NLO QCD fit for mc(mc)=1.26 GeV described in the text.Fig. 1Fig. 2χ2 of the comparison of the FFNS NLO QCD prediction to the charm reduced cross sections in the first Q2 interval, 2.5–7 GeV2, for different values of the charm-quark mass mc(mc) in the MS‾ running mass scheme (points). The line shows a parabolic fit.Fig. 2Fig. 3Charm-quark mass mc(mc) in the MS‾ running mass scheme determined from the charm data independently at six different scales μ. The outer error bars show the fit uncertainty combined with all model, parametrisation and theoretical systematic uncertainties added in quadrature. The inner error bars show the same uncertainties excluding the uncertainties arising from the variation of the QCD scales. The filled square at scale mc is the PDG world average [12] and the associated band shows its uncertainty.Fig. 3Fig. 4Charm-quark mass mc(μ) determined in the MS‾ running mass scheme as a function of the scale μ (black points). The error bars correspond to the inner error bars shown in Fig. 3. The red point at scale mc is the PDG world average [12] and the band shows the uncertainty and its expected running according to Eq. (3). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)Fig. 4Table 1Values of mc(mc) at different scales μ, determined from six different subsets, and corresponding values of mc(μ). The first uncertainty (fit) corresponds to the uncertainty δfitexp added in quadrature with the symmetrised systematic uncertainties δ1−δ6. The second uncertainty (scale) of mc(mc) corresponds to the scale variation uncertainty δ7. No scale uncertainty is quoted for mc(μ) (see text). The range of Q2 values contributing to the six data subsets shown in Fig. 1 is given. Also given is the corresponding logarithmic average scale μ for each subset according to Eq. (2), and the number Ndat of charm data points contributing to each measurement.Table 1SubsetNdatQ2 range [GeV2]μ [GeV]mc(mc) [GeV]fitscalemc(μ) [GeV]fit

1152.5–73.31.256−0.070+0.078−0.000+0.0540.984−0.076+0.085

21212–184.51.192−0.073+0.075−0.000+0.0430.867−0.075+0.077

31332–607.01.208−0.088+0.092−0.000+0.0450.830−0.085+0.089

47120–20012.71.344−0.131+0.130−0.074+0.0730.90±0.12

54350–65021.91.14−0.22+0.22−0.16+0.130.68±0.19

61200044.81.05−0.76+0.68−0.15+0.400.56±0.56

Table 2Summary of the systematic uncertainties in the mc(mc) determinations. The definitions of the uncertainty sources, the meaning of the symbols in the first and second row and related details are given in the text. In cases where opposite variations of a variable yield uncertainties with the same sign, only the larger one is considered for the uncertainty combination in Table 1. Except for δ7, these uncertainties also apply to mc(μ), before evolution to the appropriate scale.Table 2Subsetδfitexp [%]δ1 (mb) [%]δ2 (αs) [%]δ3 (fs) [%]δ4 (Q0) [%]δ5 (Qmin2) [%]δ6 (param.) [%]δ7 (scale) [%]

1±5.4−0.4+0.1+2.6−1.2+0.2−0.4+0.5+1.4+0.5+4.3+3.1

2±6.0−0.5+0.2+0.7−0.9+0.2−0.5+0.3+1.0+0.9+3.6+2.4

3±7.2−0.7+0.3+0.3−0.4+0.3−0.8+1.7+0.3+1.8+3.7+0.1

4±9.6−0.8+0.5−0.6+0.7+0.5−0.8+0.5−1.2+0.1+5.4−5.5

5±19.2−1.2+0.5−1.8+1.6+0.5−1.2−0.5+2.1−1.7+11.6−14.3

6±63.8−2.9−7.4−5.7+5.9−7.6−3.0+6.5−33.3+9.5−14.2+38.1

Running of the charm-quark mass from HERA deep-inelastic scattering dataA.GizhkoiA.Geiseri⁎Achim.Geiser@desy.deS.MochgI.AbtqO.BehnkeiA.BertolintJ.BlümleinaaD.BritzgeriR.BrugnerauA.BuniatyanbP.J.BusseyfR.CarlinuA.M.Cooper-SarkarsK.DaumzS.DusinitE.Elseni3L.FavartaJ.FeltesseeB.Fosterh1A.GarfagniniuM.GarzelligJ.GayleriD.HaidtiJ.HladkỳvA.W.Jungj2M.KapichinedI.A.KorzhavinapB.B.LevchenkopK.LipkaiM.Lisovyii5A.LonghintS.MikockilTh.NaumannaaG.NowaklE.PaulcR.PlačakytėgK.RabbertzkS.SchmittiL.M.Shcheglovap6Z.SiwH.SpiesbergernoL.StancotP.TruölabT.TymienieckayA.VerbytskyiqK.Wichmanni11M.Wingm4A.F.ŻarneckixO.ZenaieviZ.ZhangraInter-University Institute for High Energies ULB-VUB, Brussels and Universiteit Antwerpen, Antwerpen, Belgium77Supported by FNRS-FWO-Vlaanderen, IISN-IIKW and IWT and by Interuniversity Attraction Poles Programme, Belgian Science Policy.Inter-University Institute for High Energies ULB-VUBBrussels and Universiteit AntwerpenAntwerpenBelgiumbSchool of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom88Supported by the UK Science and Technology Facilities Council.School of Physics and AstronomyUniversity of BirminghamBirminghamUnited KingdomcPhysikalisches Institut der Universität Bonn, Bonn, Germany1010Supported by the German Federal Ministry for Education and Research (BMBF), under contract No. 05 H09PDF.Physikalisches Institut der Universität BonnBonnGermanydJoint Institute for Nuclear Research, Dubna, RussiaJoint Institute for Nuclear ResearchDubnaRussiaeIrfu/SPP, CE-Saclay, Gif-sur-Yvette, FranceIrfu/SPPCE-SaclayGif-sur-YvetteFrancefSchool of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom8School of Physics and AstronomyUniversity of GlasgowGlasgowUnited KingdomgII. Institute for Theoretical Physics, Hamburg University, Hamburg, GermanyII. Institute for Theoretical PhysicsHamburg UniversityHamburgGermanyhInstitut für Experimentalphysik, Universität Hamburg, Hamburg, GermanyInstitut für ExperimentalphysikUniversität HamburgHamburgGermanyiDeutsches Elektronen-Synchrotron DESY, Hamburg, GermanyDeutsches Elektronen-Synchrotron DESYHamburgGermanyjKirchhoff-Institut für Physik, Universität Heidelberg, Heidelberg, Germany1212Supported by the Bundesministerium für Bildung und Forschung, FRG, under contract number 05H09GUF.Kirchhoff-Institut für PhysikUniversität HeidelbergHeidelbergGermanykKarlsruher Institut für Technologie, Karlsruhe, GermanyKarlsruher Institut für TechnologieKarlsruheGermanylThe Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Krakow, Poland1515Partially supported by Polish Ministry of Science and Higher Education, grant DPN/N168/DESY/2009.The Henryk Niewodniczanski Institute of Nuclear PhysicsPolish Academy of SciencesKrakowPolandmPhysics and Astronomy Department, University College London, London, United Kingdom8Physics and Astronomy DepartmentUniversity College LondonLondonUnited KingdomnPRISMA Cluster of Excellence, Institut fur Physik, Johannes Gutenberg-Universität, Mainz, GermanyPRISMA Cluster of ExcellenceInstitut fur PhysikJohannes Gutenberg-UniversitätMainzGermanyoCentre for Theoretical and Mathematical Physics and Department of Physics, University of Cape Town, Rondebosch 7700, South AfricaCentre for Theoretical and Mathematical PhysicsDepartment of PhysicsUniversity of Cape TownRondebosch7700South AfricapLomonosov Moscow State University, Skobeltsyn Institute of Nuclear Physics, Moscow, Russia1414Partially supported by RF Presidential grant NSh-7989.2016.2.Lomonosov Moscow State UniversitySkobeltsyn Institute of Nuclear PhysicsMoscowRussiaqMax-Planck-Institut für Physik, München, GermanyMax-Planck-Institut für PhysikMünchenGermanyrLAL, Université Paris-Sud, CNRS/IN2P3, Orsay, FranceLALUniversité Paris-SudCNRS/IN2P3OrsayFrancesDepartment of Physics, University of Oxford, Oxford, United Kingdom8Department of PhysicsUniversity of OxfordOxfordUnited KingdomtINFN Padova, Padova, Italy99Supported by the Italian National Institute for Nuclear Physics (INFN).INFN PadovaPadovaItalyuDipartimento di Fisica e Astronomia dell' Università and INFN, Padova, Italy9Dipartimento di Fisica e Astronomia dell' UniversitàINFNPadovaItalyvInstitute of Physics, Academy of Sciences of the Czech Republic, Praha, Czech Republic1616Supported by the Ministry of Education of the Czech Republic under the project INGO-LG14033.Institute of PhysicsAcademy of Sciences of the Czech RepublicPrahaCzech RepublicwShandong University, Jinan, Shandong Province, PR ChinaShandong UniversityJinanShandong ProvincePR ChinaxFaculty of Physics, University of Warsaw, Warsaw, PolandFaculty of PhysicsUniversity of WarsawWarsawPolandyNational Centre for Nuclear Research, Warsaw, PolandNational Centre for Nuclear ResearchWarsawPolandzFachbereich C, Universität Wuppertal, Wuppertal, GermanyFachbereich CUniversität WuppertalWuppertalGermanyaaDeutsches Elektronen-Synchrotron DESY, Zeuthen, GermanyDeutsches Elektronen-Synchrotron DESYZeuthenGermanyabPhysik-Institut der Universität Zürich, Zürich, Switzerland1717Supported by the Swiss National Science Foundation.Physik-Institut der Universität ZürichZürichSwitzerland⁎Corresponding author.1Alexander von Humboldt Professor; also at DESY and University of Oxford.2Now at Fermilab, Chicago, USA, and Purdue University, West Lafayette, USA.3Now at CERN, Geneva, Switzerland.4Also supported by DESY and the Alexander von Humboldt Foundation.5Now at Physikalisches Institut, Universität Heidelberg, Germany.6Also at University of Bristol, UK.11Supported by the Alexander von Humboldt Foundation.Editor: G.F. GiudiceAbstractCombined HERA data on charm production in deep-inelastic scattering have previously been used to determine the charm-quark running mass mc(mc) in the MS‾ renormalisation scheme. Here, the same data are used as a function of the photon virtuality Q2 to evaluate the charm-quark running mass at different scales to one-loop order, in the context of a next-to-leading order QCD analysis. The scale dependence of the mass is found to be consistent with QCD expectations.1IntroductionThe Standard Model of particle physics is based on Quantum Field Theory, which can provide predictions that rely on a perturbative approach. In the MS‾ renormalisation scheme of perturbative quantum chromodynamics (pQCD), the values of all basic QCD parameters depend on the scale μ at which they are evaluated. The most prominent example is the scale dependence, i.e. running, of the strong coupling constant αs, a by now well established property of pQCD. It has, for example, been determined from measurements of hadronic event shapes or jet production at e+e− colliders [1,2], and from measurements of jet production at HERA [3], Tevatron [4] and LHC [5].The scale dependence of the mass mQ of a heavy quark in the MS‾ scheme can likewise be evaluated perturbatively, using the renormalisation group equation(1)μ2ddμ2mQ(μ)=mQ(μ)γmQ(αs), which is governed by the mass anomalous dimension γmQ(αs) known up to five-loop order [6] in perturbation theory. The running of the MS‾ beauty-quark mass has already been successfully investigated from measurements at the LEP e+e− collider [7]. Heavy-flavour production in deep-inelastic scattering (DIS) at HERA is particularly sensitive to heavy-quark pair production at the kinematic threshold. A recent determination of the beauty-quark mass mb(mb) [8] by the ZEUS experiment at HERA was reinterpreted as a measurement of mb(μ=2mb) using the solution of Eq. (1) at one loop. The comparison [9–11] of this result with the measurements from LEP and the PDG world average [12,13] shows consistency with the expected running of the beauty-quark mass.An explicit investigation of the running of the charm-quark mass has not been performed yet. Combined HERA measurements [14] on charm production in deep-inelastic scattering have already been used for several determinations of the charm-quark mass mc(μ=mc) in the MS‾ renormalisation scheme [14–18]. Fig. 1 shows the measured reduced cross section for charm production [14] as a function of the Bjorken variable xBj in 12 bins of photon virtuality Q2 in the range 2.5 GeV2<Q2<2000 GeV2. In this paper, these data are used to investigate the running of the charm-quark mass with the same treatment of the uncertainties of the combination as in Ref. [14]. The fixed flavour number scheme (FFNS) is used at next-to-leading order (NLO) with nf=3 active flavours. This scheme gives a very good description of the charm data [14,19], as shown in Fig. 1. Calculations of next-to-next-to-leading order corrections with massive coefficient functions [18,19] have not yet been completed, and are therefore not used in this paper.2Principle of the mc(μ) determinationThe theoretical reduced cross section for charm production is obtained from a convolution of charm-production matrix elements with appropriate parton density functions (PDFs). The latter are obtained from inclusive DIS cross sections, which include a charm contribution. Thus both, matrix elements and PDFs, depend on the value of the charm-quark mass. The scale dependence of the charm-quark mass is evaluated by subdividing the charm cross-section data [14] into several subsets corresponding to different individual scales, as indicated by different rows in Fig. 1. In contrast, in the evaluation of the PDFs, data spanning a large scale range such as the inclusive HERA DIS data [20,21] must be used in order to get significant PDF constraints. A subdivision into individual scale ranges is thus not possible for the PDF determination. On the other hand, it has been established that, apart from the strong constraint which the charm measurements impose on the charm-quark mass [14], their influence on a combined PDF fit of both inclusive and charm data is small [21]. Therefore, the PDFs extracted from inclusive DIS can be used for investigations of charm-quark properties, provided that the same charm-quark mass is used throughout, recognising that thereby some correlation between the mass and PDF extractions is induced. The influence of this correlation on the determination of the charm-quark mass running is minimised as described in section 4.To obtain the charm-quark mass at different scales, the charm data are subdivided into six kinematic intervals according to the virtuality of the exchanged photon. Each measurement in a given range in Q2, as listed in Table 1 and shown in Fig. 1, is performed with charm data originating from collisions at a typical scale of μ=Q2+4mc2. The actual scale used for each interval is defined according to(2)logμ=〈log(Q2+4mc2)〉, where the brackets indicate the logarithmic average of the considered range. The resulting value for each Q2 range is also listed in Table 1.Technically, a value of mc(mc) is extracted separately from a fit to each interval. The value of mc(mc) is obtained assuming the running of both αs and mc as predicted by QCD. To that end, Eq. (1) is solved using the one-loop dependence on the scale μ, as relevant in a NLO calculation, as(3)mQ(μ)=mQ(mQ)×(αs(μ)αs(mQ))c0, where c0=4/(11−2nf/3)=4/9 as appropriate for QCD with nf=3 for the number of light quark flavours. Equation (3) is used to evaluate the mass running in all results of this work.Expanded and truncated to leading order in powers of αs, this can also be expressed in the form (not used here)(4)mQ(μ)=mQ(mQ)(1+αs(μ)πlog(μ2mQ2)+O(αs2)). This illustrates that the scale dependence is logarithmic and justifies the logarithmic average in Eq. (2).According to Eq. (3) the mass has actually been determined at the scale μ, and was extrapolated to the scale mc when expressed as mc(mc). If each determination of mc(mc) is reinterpreted in terms of a value of mc(μ) using Eq. (3), the mass determinations are reverted to their unextrapolated value, and the effect of the initial assumption of QCD running on the interpretation of their value is minimised for the final result.3QCD predictions and systematic uncertaintiesQCD predictions for the reduced charm cross sections are obtained at NLO in pQCD (O(αs2)) using the OPENQCDRAD package [22] as available in HERAFitter1818Recently renamed xFitter. [20,23]. These predictions are based on the ABM implementation [24] of charm cross-section calculations in the 3-flavour FFNS. The renormalisation and factorisation scales are always taken to be identical. In the calculations, the same settings and parametrisations are chosen as those used for the earlier measurement of mc(mc) [14]. In addition, scale variations were applied as in Ref. [15]. For all explicit calculations of charm-quark mass running, an implementation of the one-loop formula [25], Eq. (3), is used, which is consistent with that used implicitly in OPENQCDRAD.These predictions are fitted to the data. The fit uncertainty δfitexp is determined by applying the criterion △χ2=1 with the same formalism as in Ref. [14]. It contains the experimental uncertainties, the extrapolation uncertainties and the uncertainties of the default PDF parametrisation. In addition, the result has uncertainties attributed to the choices of extra model parameters, additional variations of the PDF parametrisation and uncertainties on the perturbative QCD parameters as listed in terms of δ1 to δ7 below.The following additional parameters are used in the calculations, presented with the variations performed to estimate their systematic uncertainties•δ1: MS‾ running mass of the beauty quark, mb(mb)=4.75 GeV, varied within the range mb(mb)=4.3 GeV to mb(mb)=5.0 GeV, to be consistent with [14];•δ2: strong coupling constant αsnf=3,NLO(MZ)=0.105±0.002, corresponding to αsnf=5,NLO(MZ)=0.116±0.002, as in [14];•δ3: strangeness suppression factor fs=0.31, varied within the range fs=0.23 to fs=0.38, as in [14];•δ4: evolution starting scale Q02=1.4 GeV2, varied to Q02=1.9 GeV2, as in [14];•δ5: minimum Q2 of inclusive data in the fit Qmin2. For the PDF extraction, the minimum Q2 of the inclusive data was set to Qmin2=3.5 GeV2 and varied to Qmin2=5 GeV2, as in [14];•δ6: the parametrisation of the proton structure is described by a series of FFNS variants of the HERAPDF1.0 PDF set [20] at NLO, evaluated for the respective charm-quark mass, for αsnf=3,NLO(MZ)=0.105±0.002, consistent with δ2.The additional PDF parametrisation uncertainties are calculated according to the HERAPDF1.0 prescription [20], by freeing three extra PDF parameters Duv,DD¯ and DU¯ in the fit;•δ7: renormalisation and factorisation scales μf=μr=Q2+4mQ2(mQ)=μ, varied simultaneously up (upper value) or down (lower value) by a factor of two for the massive quark (charm and beauty) parts of the calculation, as in [15].The numerical values for each bin are shown in Table 2. The dominant uncertainties are those arising from δfitexp, followed by those from the scale variations δ7.4ResultsIn order to minimise the correlated contribution from inclusive data to the charm-mass determinations, and in particular from the implicit charm-mass scale dependence therein, a set of PDFs in the 3-flavour FFNS is extracted from a QCD fit to inclusive DIS HERA data [20]. This extraction uses exactly the same setup as that used in a previous publication [14], but allows for different charm-quark masses. The charm-quark mass as a function of scale is then extracted from a fit to the charm data only. When this analysis was originally performed [10,11], the inclusive HERA II DIS data [21] were not yet available. The use of the earlier inclusive data [20] has been retained for several reasons. Firstly, all systematic uncertainties can be treated exactly as in the corresponding previous global mc(mc) determination [14]. Secondly, the newer and more precise inclusive data are more strongly sensitive [17] to the assumed charm-quark mass and its running than the earlier inclusive data. This is actually counterproductive for the purpose of this paper in which the cross-correlations to the inclusive data, which cannot be subdivided into scale intervals, need to be minimised. Thirdly, the uncertainties on the determination of charm-quark mass running arising from the PDF uncertainties are already small (Table 2) compared to other uncertainties. For the purpose of this paper, the conceptual advantage of minimising the mass-related correlations between the charm and inclusive data sets therefore outweighs the potential gain from a higher PDF precision.For each charm-quark mass hypothesis, predictions for the reduced charm cross sections are obtained using the corresponding PDF and are compared to one of the six subsets of the charm data listed in Table 1 and shown in Fig. 1. The MS‾ running mass of the charm quark is varied within the range mc(mc)=1 GeV to mc(mc)=1.5 GeV in several steps. The χ2 distribution of this comparison is used to extract the value of the charm-quark mass mc(mc). An example of such a distribution for the first Q2 interval is shown in Fig. 2, together with a parabolic fit. The minimum yields the measured charm-quark mass, while the fit uncertainty is obtained from Δχ2=1. The corresponding distributions for the other intervals can be found in Ref. [10]. A global fit to the complete charm data set for mc(mc)=1.26 GeV, the central value obtained from the earlier global mc(mc) analysis [14], is shown as a curve in Fig. 1 for comparison. The data are well described.The values of mc(mc) extracted for each of the subsets of charm data are shown in Fig. 3 and listed in Table 1 as a function of the corresponding scale μ, together with their uncertainties. The breakdown of the uncertainties into individual sources is summarised in Table 2. The values of mc(mc) determined in the different subsets agree well within uncertainties with each other, with the value from the global analysis quoted above, and with the independent PDG world average1919The PDG2012 [12] value is used since it does not yet contain the result from [14] in the average, and is thus an independent value. The latest PDG2016 [13] value only differs very slightly from it. of 1.275±0.025 GeV [12].In order to test the stability of the mc(mc) determination, the default analysis procedure is cross-checked with an alternative method. For each Q2 interval a simultaneous PDF fit of the charm data from this interval and the full inclusive DIS data is performed. From the total χ2 obtained by these fits, the χ2 of the corresponding fits to the inclusive data only, is subtracted. These differences are then used for the determination of mc(mc) by a χ2 scan in the same way as for the standard procedure. Despite the more direct cross-correlation of the χ2 from the charm sample with that from the inclusive sample in this method, the difference between the results of both methods is found to be negligible [10], i.e. smaller than the width of the line in Fig. 2. This indicates that the residual effect of the cross-correlation is small.In the final step, the values of mc(mc) are consistently translated back to mc(μ) assuming the running of αs and mc as predicted by the QCD framework (Eq. (3)). The resulting values of mc(μ) are included in Table 1. The fractional contributions of the uncertainties for the individual sources (before the translation) are the same as those for mc(mc) as listed in Table 2, with the exception of the scale-variation uncertainties δ7 as discussed below.In Fig. 4, the resulting scale dependence of mc(μ) is shown together with the world average of mc(mc) and the expectation for the evolution of mc(μ) within the NLO QCD framework. The data are well described by the theoretical expectations. The running of the charm-quark mass as a function of the scale μ is clearly visible, if the independent PDG point obtained mainly from low scale QCD lattice calculations is included.No scale variations are shown in Fig. 4. In addition to the scale uncertainties, δ7, extracted from the fit, these would correspond to a variation in scale of the horizontal axis of the figure and/or a shift of the points along the expected scale-dependence curve, which are difficult to represent graphically. Furthermore, they are strongly correlated point by point, such that the shape of the distribution will stay essentially unchanged. In any case it is clear from Fig. 3 that their effect is not dominant.Overall, this result is a nontrivial consistency check of the charm-quark mass running. It is conceptually similar to the procedure of extracting the running of αs(μ) from jet production at different transverse energy scales [3–5] or at different e+e− centre-of-mass energies [1,2].5ConclusionsThe running of the charm-quark mass mc(μ) in the MS‾ scheme is evaluated for the first time, using the combined reduced-cross-section charm data from HERA. It is found to be consistent with the expectation from QCD. Within the limited scale range of each subset of the charm data used for the determination of mc(μ), the running of the charm-quark mass is implicitly assumed as part of the QCD theory input. Therefore this determination is not fully unbiased. However, the implicit bias of each individual mc(μ) value is much smaller than the bias of the earlier extractions of a single mc(mc) value from the complete data set. 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