Z. Phys. C 74, 463–468 (1997) ZEITSCHRIFTFUR PHYSIK Cc© Springer-Verlag 1997

Charm production and parton distributions ?

H.L. Lai, W.K. Tung

Michigan State University, E. Lansing, MI 48824, USA

Received: 23 February 1997

Abstract. Recent accurate data onF2(x,Q) and onF c2 (x,Q)

from HERA at small-x require a more precise treatment ofcharm production in the global analysis of parton distribu-tions. We improve on existing global QCD analyses by im-plementing the leptoproduction formalism of Aivazis et al.which represents a natural generalization of the conventionalzero mass QCD parton framework to include heavy quarkmass effects. We also perform analyses based on the fixed-flavor-number scheme, which is widely used in the literature,and demonstrate their uses and limitations. We discuss theimplications of the improved treatment of heavy quark masseffect in practical applications of PQCD and compare ourresults with recent related works.

1 Introduction

The QCD parton picture, based on perturbative QuantumChromodynamics (PQCD), has furnished a remarkably suc-cessful framework to interpret a wide range of high energylepton-lepton, lepton-hadron, and hadron-hadron processes.Through global analysis of these processes, detailed infor-mation on the parton structure of hadrons, especially thenucleon, has been obtained. Existing global analyses havemostly been performed within the traditional zero-mass par-ton picture since, up to now, “heavy quark” partons (charm,bottom and top) have played a relatively minor role in mea-sured observables used in the analyses. With the advent ofprecise data on inclusiveF2 [1] and the direct measure-ment of the charm componentF c

2 [2] from HERA, this isno longer the case. The latter comprises about 25% of theinclusive structure function at small-x. It is now necessaryto sharpen the formulation of the theory for heavy flavorproduction used in these global analyses.

The conventional QCD parton model is formulated inthe zero-mass parton limit. Since the early 1980’s, whenthe parton picture became the essential tool in calculatingall high energy processes in the Standard Model and in thesearch for “New Physics”, it has been generalized to theform of a master (factorization) formula? This work is partially supported by the National Science Foundation

σN→X (s,Q) =∑a

faN (x, µ)⊗ σa→X (s, Q, µ) (1)

whereN denotes one or two hadrons in the initial state;X, a set of inclusive or semi-inclusive final states con-sisting of ordinary or new particles; “a”, the parton label;faN (x, µ), the parton distribution at the factorization scaleµ;σa→X , the perturbatively calculable partonic cross-section;and the parton label “a” is to be summed over all possi-ble active parton species. “Active” partons, according tothis widely accepted credo, include all quanta which canparticipate effectively in the dynamics at the relevant en-ergy scale [3, 4], here denoted generically byQ (e.g., Qin deep inelastic scattering,MW in W production,pt indirect photon or jet production,...). Thus, for heavy par-ticle production (W, Z, Higgs, SUSY particles...),a = {g, u, d, s, c, b, (t)} . This general picture has beenadopted by most global analysis work [5–7], resulting inseveral generations of parton distributions which include allthe parton flavors. The charm, bottom (and top) quark distri-butions all contribute to the calculation of high energy pro-cesses above energy scales higher than the respective quarkmasses.

Viewed from this perspective, the QCD theory for heavyquark flavor production poses a special challenge. If a heavyquark, say charmc,1 is produced as part of the final stateX,(i) should one treat this process literally like the productionof other heavy particles (W, Z, Higgs, ...), i.e., differenti-atec from the other partons, and exclude it from the initialstate (by restricting the sum over “a” (1) to only the lightquarks) [8, 9]; or, (ii) based on the very physical ideas behindthe factorization formula, should one still countc among theinitial state partons because (unlike the electroweak heavyparticles) it certainly is an active participant in strong inter-action dynamics, including its own production, provided theenergy scale is high enough [4, 10]? In principle, the twoalternatives can be regarded as two different but equivalentschemesfor organizing the perturbation series in PQCD. Inpractice, since the perturbation series is truncated after oneor two terms, the effectiveness (i.e., accuracy) of the two

1 For clarity, and for the specific applications of this paper, we shallfocus on charm in most of our discussions. The same considerations applyto bottom quark production with the substitutionc→ b everywhere

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approaches can be quite different in different kinematic re-gions.

Much of the recent specific literature on heavy quark pro-duction is based on the first approach ((i) above) [11, 12], ex-tended to next-to-leading order (NLO), including in the ini-tial states only the light partons –{g, u, d, s} for charm pro-duction. These are referred to asfixed-flavor-number(FFN)calculations – 3-flavor (nf = 3) for charm. This schemeis conceptually simple. However, when the ratio of en-ergy scalesQmc

becomes large,i.e., when the charm quarkbecomes relatively light compared to the prevalent energyscale, its reliability comes into question because of the pres-ence of large lnQmc

factors – both in terms of the givenorder and in terms of higher orders which are left out. Formeasurements at HERA (and for both charm and bottomproduction at the Tevatron), this consideration becomes in-creasingly relevant. In fact, tell-tale signs of the inadequacyof these calculations have been known for some time: (i)the calculated cross-sections have unacceptably large depen-dence on the renormalization and factorization scaleµ [13];and (ii) the theoretical cross-sections, in most cases, fall wellbelow the experimental values for all reasonable choices ofµ [14].

On the other hand, as mentioned above, most work onglobal analysis of inclusive structure functions (which con-tain heavy quark final state contributions) use the secondapproach – but with a simplification: once a massive quarkis turned on above its threshold, it is treated as massless,on the same footing as the other light flavors. This is anapproximation which is strictly correct only in the asymp-totic regionQ� mc. In practice, this approximation makeslittle difference when charm production only contributes asmall fraction of the measured structure functions used inthe global analysis. This is no longer the case.

A consistent formulation of PQCD with massive quarks,representing a natural implementation of the physical prin-ciples behind the general formula, (1), has been given byAivazis et al. [10]. It reduces to the appropriate limits – FFNscheme near the threshold region (Q ' mc), and the simplemassless QCD formalism in the high energy limit (Q� mc)– and provides a unified description of charm productionover the full energy range. This formalism has been appliedto the analysis of charged current lepto-production of charm[15]. Some recent papers have attempted to compare thisapproach, referred to as the variable-flavor-number (VFN)scheme – because the active number of flavors depends onthe scale, with the FFN scheme [16]. Results are not con-clusive, since comparable parton distributions in the variousschemes are not available in the literature for a consistentcomparison.2

In this paper, we apply the more complete variable-flavor-number formalism to the global analysis of partondistributions. To carry out a systematic comparison, we alsoperform equivalent new analyses in the FFN scheme usingboth nf = 3 andnf = 4. We compare these results with

2 In addition, the proper implementation of the VFN scheme requires adelicate cancellation between three inter-related terms. (See next section.)When the parton distributions near the threshold region are not preciselygenerated to match the scheme, the cancellation will fail. This hampersmost of these references

each other, and with those obtained previously in the zero-mass approximation (CTEQ4) [7]. We find that: (i) the morecomplete formalism gives the best fit to the global data,further confirming the robustness of the PQCD theory; (ii)the nf = 3 FFN scheme has difficulty accommodating thehadron collider data included in the global analysis, whereasthe nf = 4 fit appears to be acceptable; and (iii) the recentprecision data from HERA at small-x are also sensitive tothe small differences among the various schemes and ap-proximations. In the concluding section, we discuss the im-plications of the improved treatment of heavy quark masseffect in practical applications of PQCD and compare ourresults with recent related works.

2 Treatment of heavy quark mass effectand new global analysis

We will extend the global QCD analysis to incorporate heavyquark mass effects according to the formalism of Aivazis etal. [10] (referred to as ACOT in the following). In compar-ison to recent analyses using the massless approach, signif-icant differences only arise in deep inelastic structure func-tions at small-x where the charm contribution to the mea-sured cross-section becomes a significant fraction. (See laterpart of this section for more details.) Thus, we will focuson this aspect of the calculation in most of the followingdiscussions.

To be consistent with the treatment of all the processesused in the NLO global analysis, the charm production con-tribution will be calculated to orderαs – the same as thelight-flavor contributions.3 The partonic processes includedin the calculation are, for charged current interactions:

α0s FE : W +(W−) + s(s) → c(c)

α1s FC : W +(W−) + g → s(s) + c(c)

(2)

and, for neutral current interactions:

α0s FE : γ∗(Z) + c(c) → c(c)

α1s FC : γ∗(Z) + g → c + c

(3)

As pointed out in [10], theflavor excitationα0s term and the

flavor creation(or gluon fusion) α1s term cannot be simply

added because there is an overlapping region where they rep-resent the same physics. The formalism provides a consistentprocedure to subtract out the contribution of the collinearconfiguration in the gluon fusion process which is alreadyincluded in the DGLAP evolved quark distribution in theflavor excitation process. See Fig. 1 for an illustration. Thisprocedure eliminates the double counting.4 By keeping thequark massmc in the hard cross-sections for all terms, thisformalism reduces to the gluon fusion results of the FFN

3 To keep the following discussion simple, we will in fact only focus onthe most important (both physically and numerically) orderαs contribu-tions. Not shown explicitly in the following is the orderαs quark scatteringprocess, e.g.,W (γ∗) + c → g + s(c) for CC(NC), along with virtual cor-rections to the orderα0

s terms. These terms give contributions which arenumerically comparable to the orderα2

s flavor creation processes (afterthe large logarithms have been removed according to the ACOT proceduredescribed below). We have confirmed this point in a separate calculation

4 This subtraction has its counter-part in the subtraction of the1ε

pole inthe dimensional regularization of collinear singularities in massless PQCD

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Fig. 1. Charm production in NC deep inelastic scattering in PQCD: flavorexcitation and gluon fusion production mechanisms and their overlap whichhas to be subtracted. The mark′′ on the internal quark line indicates it ison-mass-shell and collinear to the gluon.

scheme near threshold because the flavor excitation and sub-traction terms cancel each other in that region. This repre-sents the correct physics for that region. In the other limit,whenQ� mc, the subtraction term cancels the mass singu-larity of the gluon fusion term, the theory is free from largelogarithms of the FFN scheme, the cross-section is dom-inated by the flavor excitation mechanism which providesa good approximation to the correct physics in that region(with only small orderαs flavor creation corrections). Fordetails, see [10]. For the purpose this paper, the charm quarkdistributionf c(x,Q) is assumed to be zero at the thresholdQ = mc = 1.6 GeV; and it is dynamically generated by QCDevolution to higher scales. The general approach does allowthe option of including the presence of intrinsic charm in-side the nucleon at threshold, which is excluded in the FFNscheme by definition. This option is of particular interestin the dedicated study ofF c

2 (x,Q) [2]; it will be exploredelsewhere.

Before performing a new global fit, it is instructive toquantify the difference between the above treatment of thepartonic processes for charm production and the approxima-tion made in previous CTEQ global analyses. Thus, we firstconvolute the existing CTEQ4M distributions with the im-proved hard-scattering cross-section described above to cal-culate the structure functions, and compare the results withthe experimental data which were used in the CTEQ4 analy-sis. As is known from previous studies5, very little differenceis found for all the fixed-target experiments. However, thedifferences at small-x for the HERA experiments are com-parable to the current experimental errors; hence they domatter. This is illustrated for threeQ2 data bins in Fig. 2.The structure functionF2(x,Q) calculated in the massiveformalism lies below that from the zero-mass results be-cause the latter clearly over-estimates the cross-section byignoring kinematic threshold suppression factors.

The contrast between the fixed-target and HERA casesoriginates from the fact that charm production comprises afew percent of the measuredF2 in the former; but it rises toabout 25% at small-x for the latter. This is explicitly shownin Fig. 3: here the ratioF c

2 /F2 is plotted againstx for bothtype of experiments with the points representing the averagevalue over theQ2 range covered by the experiments. At thefew percent level, a relatively large theoretical uncertainty onF c

2 can be tolerated in previous comparisons to fixed-targetdata, since experimental errors themselves are in the fewpercent range. This is no longer the case when the fractionalcontribution rises to 25% with experimental errors in the fewpercent range. Of course, it will be even more important

5 CTEQ notes for the CTEQ3 analysis, 1994 (unpublished)

Fig. 2. Comparison of H1 data in the small-x region with calculations usingCTEQ4M parton distributions in the (original) massless QCD scheme (solidline) and with massive hard cross-section formulas of Sect. 2 (dashed line).Also shown is the result of the new fit CTEQ4HQ

Fig. 3. Fractional contribution onF2 from the charm quark for differentexperimental (x,Q) range. The data shown are calculated using CTEQ4Mwith average on the Q bins of that particular experiment at fixedx

to have the theory formulated accurately in the study ofmeasuredF c

2 itself.

These results clearly imply the need to perform newglobal analyses to account for the correct physics underlyingthe recent measurements. Thus, we repeat the CTEQ4 globalanalysis [7], using the improved theory for heavy quark pro-duction. Charged current and neutral current DIS processesare treated consistently as described above. We find the over-all χ2 for the global fit is improved from the previous best fitCTEQ4M – 1278 vs. 1320 for 1297 data points, as shownin Table 1 where both the overallχ2 and its distributionamong the DIS and D-Y data sets are presented.6 The smallimprovement inχ2 is spread over both the fixed-target andHERA DIS data sets. The comparison of this new fit to thesmall-x data of H1 is included in Fig. 2. We shall refer to

6 Direct photon and jet-production data were also used to constrain thefits. Because of the difficulty in quantifying current theoretical and exper-imental uncertainties for these processes, the specificχ2 values are dif-ficult to interpret; hence they are not explicitly presented. (See [7] fordiscussions)

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Fig. 4. Comparison of CTEQ4M and CTEQ4HQ parton distributions

Table 1. Total χ2 values and their distribution among the DIS and DYexperiments: The first columnFT − NC stands for fixed target neutralcurrent experiments which includes BCDMS, NMC and E665

Expt. #pts CTEQ4M CTEQ4HQ CTEQ4F4 CTEQ4F3FT −NC 691 725 714 726 726CCFR 126 130 119 122 127HERA 351 362 343 375 418CDFAW 9 4 5 4 10NA51 1 0.6 0.8 0.8 0.9E605 119 98 96 98 98Total 1297 1320 1278 1326 1380

this new set of parton distributions as CTEQ4HQ (HQ forheavy quark).7

As expected, the deviation of CTEQ4HQ distributionsfrom CTEQ4M are rather minor, and they are most notice-able at small-x. Fig. 4 shows the comparison of these partondistributions. Interestingly, the differences are more obvi-ous for the light quarks than for the gluon and charm; andthey consist of an increase in these distributions through-out the small-x range (which is most visible in the plot).This is because the improved theory for heavy quark pro-duction reduces the predicted cross-section which has to bebrought back to the experimental values by increased partondistributions. This can be done most efficiently by a smallfractional increase in the light quark flavors. Of course, ifthe momentum sum rule is to be preserved, there has to be aslight decrease in the parton distributions at largex. This isindeed the case, even though it is barely visible in this plot.Proportionally, this decrease is extremely small, because themomentum sum-rule integral strongly suppresses differencesat small-x.

3 Global analyses in the fixed-flavor-number scheme

Since much of the current literature on heavy flavor produc-tion in recent years adopts the fixed-flavor-number schemedescribed in the introduction [8, 9, 12, 16], it is useful tohave available up-to-date parton distributions in this schemeand to study its efficacy in describing existing global data.By comparing results from such an analysis with those ob-tained above, one can also draw more definitive conclusions

7 Computer code for this and the following parton sets will be availableat the CTEQ Web site http://www.phys.psu.edu:80/˜cteq/

concerning the effectiveness of the FFN scheme versus themore complete variable-flavor-number formalism.

The FFN scheme is simpler to implement, since it in-cludes only light partons{g, u, d, s} in the initial state. Thus,for neutral current deep inelastic scattering as an example,the only partonic process contributing (to orderαs ) is thegluon fusion mechanismγ∗(Z) + g → c + c. There is noneed for any subtraction, since the charm mass regulates thecollinear singularity, and there is no double counting. Usingthis scheme consistently for all processes, one can performa global analysis in thenf = 3 FFN scheme, and obtain anappropriate set of parton distributions. As in the previoussection, the treatment of data sets and the global analysisprocedures are the same as in the previous CTEQ4 analysis[7]. We shall call this the CTEQ4F3 set.

Although we have emphasized charm production due toits immediate relevance, the same physics issues apply tobottom quark production. At HERA,b production is quitenegligible in the totalF2. But at LEP, the Tevatron and LHC,it is not. For this reason, and due to the fact thatmb isgenuinely “heavy” whereasmc is really on the borderline,we also carry out a global analysis in thenf = 4 FFNscheme for completeness. For this case, charm is countedas an active parton and treated above threshold on the samefooting as the other light partons, but bottom is treated asa heavy quark in the spirit of the FFN scheme. The set ofparton distributions obtained this way is called CTEQ4F4.

An overview of the comparison between these FFNscheme global fits to the ones described in the previous sec-tion is included in Table 1. We see that the overallχ2 for thetwo FFN scheme fits are 1380 and 1326 (fornf = 3, 4 re-spectively) compared to 1278 for CTEQ4HQ. A substantialpart of the difference is due to the HERA data (particu-larly the ZEUS data set). Other than that, thenf = 4 FFNfit (CTEQ4F4) is very close to CTEQ4M. This is expectedsince the only differences lie in the treatment of the smallbottom quark contribution to the measured quantities, and inthe slight difference in theαs function above bottom thresh-old (corresponding tonf = 4, 5 respectively).

More revealing is thenf = 3 FFN fit (CTEQ4F3) whichshows signs of difficulty in achieving a good fit. The majordifference between this fit and the others is the marked in-crease inχ2 for the HERA data, along with a slight increasefor fixed-target data compared to CTEQ4HQ. This result isunderstandable in the context of a precise global analysis:the physics that is missing from thenf = 3 fixed-flavorformalism matters the most in the small-x region; to makeup for this, the global fitting procedure makes compensatingchanges in the parton distributions which indirectly affectthe fits to the largerx region as well.

Comparison of thenf = 3 FFN scheme with hadron col-lider data at the Tevatron is a delicate matter because, atsuch a high energies (a typical energy scales areQ2 ∼M2

W ,or p2

t ∼ 1000− 100000 GeV2), it is clearly not a naturalPQCD scheme to use. To get any reasonable results, onemust include some higher order terms. [9] We have not triedto do this systematically in the current study because: (i) therequired higher-order hard cross-sections (including mass ef-fects) do not exist for all processes (e.g. jet production); and(ii) the gain in accuracy will in any case be just temporaryif we keep in mind the need to go to even high energies (say

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LHC). In the simple calculations we have done, there is aslight increase in theχ2 on the CDFW -lepton asymmetrydata [17], although this is still within about one standarddeviation from experiment.

With these provisoes, the CTEQ4F3 and CTEQ4F4 dorepresent the most up to date parton distributions to be usedwith FFN scheme calculations. CTEQ4F3 is in the samescheme as GRV94 [9]. CTEQ4F3 gives better fits to theglobal data, and are comparable to the other CTEQ4 dis-tributions for the purpose of studying scheme dependences.As mentioned, these distributions could be further improvedwith the inclusion of orderα2

s hard cross-sections and withmore recent data.8

4 Concluding remarks

Recent HERA data both on precision measurement of theinclusiveF2(x,Q) and onF c

2 (x,Q) require a more carefultheoretical treatment of heavy quark production in PQCD.This entails, in turn, both a clearer definition of the perturba-tive scheme used and the careful choice of a scheme whichis appropriate for the full energy range probed by the exper-iments in question. We have described the salient featuresof most of the schemes which have been used (explicitly orimplicitly) in current literature, and made a systematic com-parison in the context of the global QCD analysis of hardprocesses to extract the universal parton distributions.

The advantages of both the massless approximation tothe variable-flavor-number scheme used by most previousglobal analyses (e.g., both CTEQ and MRS) and the FFNscheme used by previous heavy quark production calcula-tions (as well as by the GRV parton sets) lie in their sim-plicity in implementation. They both have limitations whichcan no longer be totally neglected. The generalized variable-flavor-number formalism contains the right physics in thefull energy range, including both the threshold and asymp-totic regions, and is free of large logarithms (except those ofthe small-x kind which we have not discussed). It is reassur-ing to see that the more complete theory does give a betterdescription of the wide range of experimental data includedin the global analysis; and, non-trivially, the less completetheory does fall short where it is expected to. This providesfurther confirmation of the soundness of the PQCD theory.

Since the completion of this study, a preprint on anew treatment of heavy quark production appeared [19].This (MRRS) approach is also in the variable-flavor-numberscheme and is the same in spirit to the formalism of ACOTwhich we adopted, hence it shares the characteristics men-tioned in the previous paragraph. The new MRRS parton setis comparable to CTEQ4HQ. The differences in implemen-tation of the basic ideas of the VFN scheme lie in the waymass effects are factorized in the parton distributions (hence

8 Since the publication of the previous CTEQ4 analysis, NMC has pub-lished results of their final data analysis [18]. The differences betweenthe old and new results are not significant. We did not replace the previousNMC data by the new ones in this analysis because, in order to demonstratethe small differences among the various schemes, including the previousCTEQ4M analysis, we have to use the same data sets. We also antici-pate new data on the CCFR measurement of the charge current structurefunctions in the near future

the splitting functions) and the hard scattering coefficients.This results in some (higher order) differences in the regionjust above threshold.

We note that, the increase in precision of experiments andin sophistication of the theory place more demand on users ofthe QCD parton formalism:for consistency, each set of newparton distributions can only be applied to hard scatteringcross-sections calculated in the same scheme. When charmand bottom mass effects matter, one must be fully aware ofwhich heavy quark mass scheme should be used in a givencalculation – this isin addition tothe familiarMS and DISschemes of the massless theory. For most lepton-hadron andhadron-hadron processes away from very small-x, data can-not distinguish among the various schemes studied in thispaper, provided the parton distributions and hard scatteringcalculations are not mismatched. In these cases, the usualparton distributions, even if not precisely the true ones of na-ture, provide an effective description of the physics probed.In this sense, the continued use of existing standard partondistribution sets (such as CTEQ [7] and MRS [6]) for theseprocesses is acceptable. However, for processes sensitive toinitial or final state heavy quarks, it will be imperative touse the more complete theory, with matching parton distri-butions, if meaningful physical quantities are to be extracted.For these processes, current theory can still be improved toinclude higher order terms; and detailed phenomenology isyet to be done when both experiment and theory mature.

Note added in proof:Since the manuscript was submitted,we have further fine-tuned the global fit which led to theCTEQ4HQ parton distribution set. A new set of parton dis-tributions is obtained which fits all the data sets described inthe text at about the same level of accuracy as CTEQ4HQ,and in addition gives a better fit to the H1F c

2 (x,Q) datawhich was not included in the original fit. This new set, de-noted by CTQ4HQ1, uses a charm quark mass of 1.3 GeV,in the middle of the current estimated range formc (cf. Par-ticle Data Group, Phys. Rev.D54, 303 (1996)), and it startsat a lower value ofQ0 of 1.0 GeV. The charm distribution inthis set is slightly higher than in the previous set. The Fortranprogram for CTEQ4HQ1, along with that forF c

2 (x,Q) andF2(x,Q) defined in our heavy quark scheme are availableon the WWW at http://www.phys.psu.edu:80/∼cteq/

Acknowledgements.We thank John Collins, Frederick Olness and CarlSchmidt for useful discussions on heavy quark physics; Joey Huston, SteveKuhlmann, Joseph Owens, Davison Soper and Harry Weerts for collab-oration on the previous CTEQ4 analysis on which the current study isbased; and Kuhlmann and Raymond Brock for detailed comments on themanuscript.

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