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Global QCD Analysis of Parton Structure of the Nucleon: CTEQ5 Parton Distributions

An up-to-date global QCD analysis of high energy lepton-hadron and hadron-hadron interactions is performed to better determine the gluon and quark parton distributions in the nucleon. Improved experimental data on inclusive jet production, in conjunction with precise deep inelastic scattering data, place good constraints on the gluon over a wide range of x; while new data on asymmetries in Drell-Yan processes contribute to better determine the d/u ratio. Comparisons with results of other recent global analyses are made, and the differences are described. Open issues and the general problem of determining the uncertainties of parton distributions are discussed.


Introduction
The structure of hadrons represented by parton distributions is an essential part of our knowledge of the elementary particle physics world. The interpretation of existing experimental data in terms of the Standard Model (SM), the precision measurements of SM parameters, as well as the direct search for signals for physics beyond the SM, all rely heavily on calculations based on Quantum Chromodynamics (QCD) and the QCD-parton picture, with the parton distribution (and fragmentation) functions as essential input. The (non-perturbative) parton distribution functions at some given momentum scale are currently determined phenomenologically by a global analysis of a wide range of available hard scattering processes involving initial-state hadrons, using the perturbative QCD-parton framework.
The global analysis of parton distributions requires a continuing effort. As new experimental and theoretical advances occur, the parton distributions can be determined with increasing accuracy and assurance. Although current knowledge of the parton distributions, based on several generations of global analyses, is far more quantitative than in the early years of the parton model, gaps still remain, as will be discussed at the end of this paper. In addition to filling these gaps, there are two important motivations for the vigorous pursuit of global analysis: (i) a comprehensive global analysis constitutes an important test of the consistency of perturbative QCD when available experimental constraints exceed the degrees of freedom inherent in the non-perturbative shape parameters, and provides a powerful tool to discover the boundaries of applicability of the conventional theory, hence to discover hints for the need of new physics tools or ideas; and (ii) since the global analyses inevitably involve both experimental and theoretical uncertainties, it is important to quantify the uncertainties in the resulting parton distributions. Some important efforts on precision measurement of SM parameters such as the W mass, as well as the determination of signals and background for new physics searches, are limited by uncertainties on parton distributions. This paper extends the series of global QCD analyses of the CTEQ group [1,2] to include significant new experimental results of the last two years. Sec. 2 summarizes these new experimental developments. Sec. 3 discusses issues that arise in a quantitative global QCD analysis of these data, and our approach to these issues. Sec. 4 describes the main results of our analysis, in the form of several sets of new CTEQ5 parton distributions, chosen to meet the needs of different types of applications as required by the consistent use of QCD theory. In Sec. 5 we compare our results with those of recent parallel efforts and point out the origin of the differences. Finally, in Sec. 6 we describe the remaining open problems in the global analysis of parton distributions, as well as various sources of uncertainties in the parton distribution parameters and the prospect for quantifying the uncertainties.
In the Appendix we summarize some issues regarding the choice of renormalization and factorization schemes for the treatment of heavy quarks, which are relevant for quantitative global analysis of increasingly precise data in many processes.

New experimental information and their use in global analysis
Since the publication of the last round of CTEQ global QCD analysis [2], improved and new experimental data have become available for many processes. These are summarized here. The use of these data in the global analysis depends on theoretical considerations which we discuss in the next section.
Deep inelastic scattering: The NMC and CCFR collaborations have finished and published analyses of their respective data on muon-nucleon [3] and neutrinonucleus [4] scattering. These new results lead to subtle changes in their implications for α s and parton distribution determination. The H1 and ZEUS collaborations at HERA have published more extensive and more precise data on the total inclusive structure function F p 2 [5,6]. 1 These results provide tighter constraints on the quark distributions, as well as on the gluon distribution, mainly through the Q-evolution of the structure functions. The HERA experiments also present new data on semiinclusive F c 2 , with charm particles in the final state [7,8]. The analysis of the F c 2 data will be discussed in the next section. Lepton-pair production (p/d) asymmetry: The E866 collaboration has measured the ratio of lepton-pair production (Drell-Yan process) in pp and pd collisions over the x range 0.03 -0.35 [9], thus expanding greatly the experimental constraint on the ratio of parton distributionsd/ū (compared to the single point of NA51 at x = 0.18 [10]). This data set has the most noticeable impact on the new round of global analysis.
Lepton charge asymmetry in W-production: The CDF collaboration has improved the accuracy and extended the y range of the measurement of the asymmetry between W → ℓ ± ν at the Tevatron [11]. This provides additional constraints on d/u.
Inclusive large p T jet production: The D0 collaboration has recently finished the final analysis of their inclusive jet production data, including information on the correlated systematic errors [12]. The CDF collaboration also has presented new results from their RunIB data set [13]. Systematic errors in these data sets dominate the experimental uncertainty over much of the measured p T range. The correlated systematic errors provide important information on the shape of the differential cross-section, dσ/dp T , and constrain the parton distributions accordingly.
Direct photon production: The E706 collaboration at Fermilab has published the highest energy fixed-target direct photon production data available to date [14]. The measured cross-sections lie a factor of 2 − 3 above the traditional next-to-leading (NLO) QCD calculation, thus posing a real challenge for their theoretical interpretation and their use in global analysis.

Global analysis issues and procedures
In this section, we consider various physics issues relevant to incorporating the new experimental data in the global analysis of parton distribution functions, and describe the specific inputs to the CTEQ5 analysis.
Charge asymmetry data and quark flavor differentiation: Most inclusive processes are not sensitive to differences between the quark parton flavors, since contributions from them are summed in the cross-section. In global analysis, these differences represent "fine structure" that can be resolved by including physical quantities asymmetric in the various flavors. In particular, the difference between the u and d quarks is determined by differences between crosssections with proton/neutron targets in DIS and Drell-Yan processes, or with W ± final states (manifested by the decay leptons) inpp collisions. As mentioned in the previous section, new data from E866 and CDF have an immediate impact on flavor differentiation in current global analyses. These new data are complemented by the final results from the very precise measurement of F d 2 /F p 2 by the NMC experiment.
In our analysis, information from the E866 σ pd /σ pp Drell-Yan experiment is maximized by treating the data sets above and below the Υ peak separately (rather than integrating over the invariant mass of the lepton pair, with an x-dependent Υ gap). 2 For the CDF W-lepton asymmetry data, we use the resummed NLO calculation provided by the ResBos program [54]. 3 Resummation has an effect on the theoretical calculation at large rapidity of the lepton, due to experimental cuts on the lepton p T .
An important source of uncertainty in the study of quark flavor dependence arises from the necessity of using DIS and Drell-Yan data on a deuteron target, in lieu of a neutron target. On general grounds, the impulse approximation of considering the deuteron cross-section as the incoherent sum of those of proton and neutron is expected to be good at small x. In the large x region, there have been studies of the "deuteron correction" factor needed to extract the neutron cross-2 Data in the Υ region are excluded since they involve different physics. We thank Paul Reimer of E866 for providing the detailed information on the measurement in the two separate regions which makes this treatment possible. 3 We thank Csaba Balazs for assistance in this calculation. section from deuteron data; but there is no universally accepted theory on the size and shape of this correction. Furthermore, there are additional complications in the large x region such as "higher-twist" effects of various origins, including target mass effects. Recently, these problems have been revisited by two phenomenological studies [15,16]. Using approximate (i.e. modified MRSA) quark-distributions, Ref. [16] advanced the case for an unconventional behavior of the d/u ratio at large x, as the result of specific deuteron-and target-mass corrections; and pointed out the importance of studying this issue in a full global QCD analysis. In performing such an analysis, we have found that equally consistent descriptions of all current data can be obtained with or without applying the corrections of [15,16] to the DIS deuteron cross-sections. Consequently, for the general purpose CTEQ5 parton distributions, to be used in usual applications, we follow the conventional practice of treating deuteron cross-sections as an incoherent sum of proton and neutron ones. A more specific study of this problem, including recent theoretical development of higher twist effects and possible ways to distinguish between the alternative behaviors of the d/u ratio at large x in HERA measurements, will be presented in a forthcoming paper. Neutral current (NC) and charged current (CC) DIS scattering (initiated by charged leptons and neutrinos respectively) are sensitive to different combinations of quark flavors, hence also provide information on their differences. Here one encounters a separate set of inter-related uncertainties: heavy-target correction for neutrino experiments, origin of the apparent disagreement of measured ratio of CC and NC structure functions with the classic "charge ratio" (5/18 rule) in the region x < 0.1, the strange quark fraction, and the validity of charge symmetry for parton distributions (i.e. f u p = f d n . . . ). For some recent investigations, see [17]. These open questions deserve further study. In the absence of compelling reasons to do otherwise, we follow the practices of previous CTEQ (and MRS) analyses [2,1] on these issues.
Direct photons, inclusive jets, and the gluon distribution: Since the recently published E706 direct photon data [14], measured at 530 and 800 GeV, cover a wide range of x, and report comparatively small statistical and systematic errors, one might hope to determine the gluon distribution directly from this process over the full range covered by this and earlier experiments. However, the measured cross-section by E706 is roughly a factor of 2 − 3 larger than the conventional NLO QCD calculation. This result strengthens a previous suggestion [18] that initial state parton k T broadening, due to multiple soft-gluon radiation, greatly enhances the steeply falling photon p T spectrum [14]. However, to make use of these data to determine the gluon distribution with any confidence, one needs a theory capable of predicting the needed large theoretical correction factor with considerable accuracy. Does that theory exist? In a recent paper [19], it is shown that a phenomenological treatment of the k T broadening effect based on 4 conventional Gaussian smearing, with the amount of smearing determined by data from related processes such as di-photon and photon+jet, can consistently describe a wide range of existing data on direct photon and related pion production. However, this study also reveals the expected fact that the results of such calculations are still rather model-dependent. A variety of uncertainties associated with the choices of parameters, phenomenological procedures, scales, ... etc. described in this study show that the shape as well as normalization of the p T spectrum can be significantly affected by choices made in the model calculation. 4 The wide range of uncertainty is underlined by another recent study [21], which argues that aside from E706, k T broadening is not necessarily required to reconcile NLO QCD and earlier fixed-target and ISR direct photon experiments.
In short, one finds that the consistency between existing fixed-target experiments on direct photon is still open to question, the problem being partly dependent on the theoretical framework used to compare experiments at different energies. Furthermore, the QCD theory for direct photon production in the p T range of these experiments is very much in a state of active development: Are resummation effects beyond NLO as large as a factor of 2∼3 at E706 energies? How quantitative can the resummation theory become? We refer the reader to Ref. [19,21,22] for detailed discussions on these issues. Under the present circumstances, it is impossible to incorporate these experiments in the global QCD analysis without introducing subjective choices of experimental data, as well as model-dependent theoretical procedures. We note that, direct photon production has also been measured at hadron colliders [23]. The cross-section at high p T agree rather well with NLO QCD calculations; however, at the low p T end, one also observes an enhenced cross-section compared to theory. The statistics for these experiments are currently too low to make these data useful for the global analysis of parton distributions.
Inclusive large p T jet production at the Tevatron, on the other hand, provides a much more reliable experimental constraint on the gluon distribution, since the NLO QCD theory has been shown to be rather stable [24,25] in the region p T > 40 GeV where measurements exist. This energy scale is considerably higher than that of fixed-target direct photon discussed in the previous paragraph. Multi-soft gluon effects are insignificant for data in this range. Preliminary CDF and D0 data were used in the previous CTEQ4 analysis. It was shown that for the determination of the gluon distribution, the inclusive jet data supplement very well the precise, but indirect, constraints implied by the Q-dependence of DIS data. Now that these experimental results have been finalized [12,13], it is natural to take full advantage of them in the new global analysis. An exhaustive study, based on all available data, confirms the previous finding that the combination of jet and DIS data constrain the gluon distribution quite well in the range 0.05 < x < 0.25. We will test the parton distributions obtained in this way against the direct photon data to see whether we get a consistent picture of the gluon.
The strong coupling α s : For this study, we have kept the value of α s (m Z ) fixed at 0.118. As is well known, the value of α s is strongly coupled to the gluon distribution in analyzing lepton-hadron and hadron-hadron processes. In particular, this correlation has been examined in Ref. [2], and presented in the "A-series" CTEQ4 parton distributions which span a range of α s around the world average. When α s is left as a free parameter in the current fit, we find a range of values of α s (m Z ), including 0.118, which give almost equally good fits. Since there are many more constraints on α s from processes beyond those used in parton distribution analysis, many of which are independent of the uncertainties on the gluon distribution, we have chosen to use a fixed α s (m Z ) in the final CTEQ5 analysis. We will comment on the effect of varying α s in the next section. The range of variation of parton distributions due to a variation in α s can still be inferred from Ref. [2].
Charm production in DIS: Preliminary results on charm production at HERA [7,8] have highlighted the need for a more careful treatment of heavy quarks in the perturbative QCD (PQCD) formalism. Although theories for heavy quark production exist [26,27,28,29], and the CTEQ4 analysis provided several sets of parton distributions which incorporate charm quark mass effects (CTEQ4HQ,4F3,4F4), it is important to bear in mind the limitations of the current state of the art on this subject. Experimentally, one can only measure the cross-sections for producing D and D * mesons in certain kinematic ranges. Extracting F c 2,exp requires: (i) an extrapolation of D-and D * -production data to the full phase space to obtain F D,D * 2 ; and (ii) a procedure to infer F c 2,exp from F D,D * 2 involving, among other uncertainties, the not so well-known fragmentation functions for D, D * . On the theoretical side one faces a different, but related, dilemma. On one hand, among the existing schemes for treating heavy quarks in PQCD, F c 2,th is in principle defined only in the fixed-three-flavor scheme (with u, d, s being the only quark partons); but this scheme is not suitable for quantitative treatment of high energy (i.e. collider) inclusive processes which are essential for global QCD analyses. On the other hand, in generalized MS schemes admitting a non-zero mass charm quark as an active parton at high energies 5 , which are suitable for global analyses at high energies, "F c 2 " is not a well-defined 5 Some background information on the available theoretical schemes for treating heavy quarks in PQCD, useful for this discussion and that of the following section, is provided in the Appendix. 6 quantity in principle because the naive "F c 2 " contains large logarithms of the same type that are resummed into charm parton distributions. Only in F tot 2 , and in F D,D * 2 , do these logarithms cancel between contributions from all parton flavors to yield infrared safe quantities that are suitable for comparison with experiment. For these experimental and theoretical reasons, the emerging charm production measurements can best be used as a testing ground for further development of both, rather than as a mature input to quantitative parton distribution analysis. 6 Thus, we shall not use the preliminary data on charm production in DIS in the global analysis; but we will present a comparison of the new parton distributions with available data, using order α s formulas in the generalized 4-flavor MS scheme.

CTEQ5 Parton Distributions
Based on the considerations discussed above, we carried out an extensive round of global analyses, using DIS data sets from BCDMS [30], NMC [3], H1 [5], ZEUS [6], CCFR [4], E665 [31]; Drell-Yan data from E605 [32], E866 [9]; W-leptonasymmetry data from CDF [11]; and inclusive jet data from D0 [12] and CDF [13]. The kinematic ranges spanned by the various experiments, and the wide scope of the overall coverage, are shown in Fig. 1. (Data points below Q 2 = 4 GeV 2 are not included in this twist-two QCD analysis; hence are absent on this plot.) The complementary roles of the fixed-target, HERA, and Tevatron experiments are clearly illustrated in this plot.
Various theoretical and phenomenological issues described in the previous section are explored by systematically studying the effect of reasonable variations of the known uncertainties in each case. The initial (non-perturbative) parton distributions are parametrized at Q 0 = 1 GeV; cuts on the kinematic variables Q, W, p T . . . on data points used are generally of the order of 2 -4 GeV, the same as in [2]. Our results are not sensitive to the specific choices.
The following sets of CTEQ5 parton distributions, representative of our best fits, are provided for general purpose use in applications of perturbative QCD to calculate high energy processes, as well as for special purpose applications as specified. These sets are summarized in Table I large-x gluon enhanced on-mass-shell heavy quark sets CTEQ5HQ

CTEQ5 sets in the conventional schemes:
The CTEQ5M set is defined in the MS scheme, matched with conventional NLO hard cross-sections calculated in the zero-quark-mass approximation for all active flavors, including charm and bottom. This set is the most convenient one to use for general calculations, as the vast majority of available hard cross-sections in the literature and in existing programs have been calculated in this limit. It represents an updated version of the CTEQ4M distribution set.
CTEQ5D is the corresponding set in the DIS scheme, obtained by an independent fit (rather than making a theoretical transformation from CTEQ5M), 8 using identical experimental input and fitting procedure. Likewise, CTEQ5L is the corresponding set in leading order QCD, which should be appropriate for simple calculations and for use in many Monte Carlo programs. Fig. 2 shows an overview of the parton distributions of the proton in the CTEQ5M set. Compared to the previous generation of distributions, such as CTEQ4M, the most noticeable changes are in the difference ofū andd quarks, due to the influence of the new data of E866, NMC, and CDF W-lepton asymmetry. Fig. 3 and Fig. 4 show the combinationsd/ū andd−ū respectively, in each case comparing CTEQ5M with CTEQ4M. These parton distributions give excellent fits to about 1000 data points of DIS, 140 of DY, and 57 of jet experiments. We shall bypass plots showing the excellent fit to the well-known data sets; and focus on comparisons to data that are new or have changed since the CTEQ4 analysis [2]. Fig. 5, Fig. 6, and Fig. 7 show comparisons of NLO QCD calculations based on the CTEQ5M parton distributions to the experimental data of NMC on the DIS deuteron to proton ratio, of CDF in the W-lepton asymmetry, and of E866 on the Drell-Yan deuteron to proton ratio respectively. Excellent agreement is observed in all cases. There is no obvious need for a different treatment of the deuteron 8 Performing a transformation from one scheme to another by the order α s perturbative formula [33] can lead to large errors for the gluon and the sea quarks in kinematic regions of (x, Q) where the "leading order" term is small compared to the correction term: e.g. valence quark corrections to the gluon, and gluon correction to the sea quarks at large x.
8 data as suggested in Ref. [16], although we find that the alternative scenario is also allowed by the global analysis.
The shifts in the quark distributions, plus the influence of the new data on inclusive jets, also results in some shift in the gluon distribution from CTEQ4M. Fig. 8 shows the CTEQ5M G(x, Q) compared to CTEQ4M at Q = 2, 5, and 80 GeV. These functions are uniformly scaled by x −1.5 (1 − x) 3 in order to make the differences both at large and small x visible on a linear scale. One can see that the difference between the two distributions diminishes as Q increases as the result of QCD evolution. This well-known feature of parton distributions has many phenomenological consequences. The bottom plots in Figs. 9 and 10 show the measured D0 and CDF inclusive jet production cross-sections, compared to NLO QCD calculations based on CTEQ5M (and CTEQ5HJ, to be described later), and the top plots show the same in the ratio form, Data/Theory. (The cross-section plot is more appropriate for comparing the same data to different theories, as we will do later; we scale the cross-section by p 7 t , so that it becomes practical to show the comparison on a linear scale. The ratio plot is used often in experimental papers.) The data are systematic error limited in most regions, except at very large p T . The known correlated systematic errors, which constrain the shape of the differential distribution, are incorporated in the global fit. The normalization factors in these comparison plots are 1.04 for D0 and 1.00 for CDF. This difference in normalization factor is consistent with a known 3-5% difference in their luminosity calibration. We will return to more discussions (and comparison) of these jet data in a latter section on the CTEQ5HJ parton distribution set.
One obvious question about our determination of G(x, Q) is whether we sacrificed useful information on the gluon distribution by leaving out the direct photon data in the analysis? The answer is no, due to the very large theoretical uncertainties discussed in Sec. 3. This situation can be illustrated by using our new parton distributions to calculate direct photon cross-section in the existing theoretical frameworks and comparing them with available data. Fig. 11 shows the comparison of the WA70 data with a NLO QCD calculation, using the CTEQ5M parton distributions. A scale parameter µ = p T /2 is used. A normalization factor of 1.08 is found to bring about a perfect agreement between theory and experiment. This is not surprising since the CTEQ5 distributions are not so different from the previous generation of CTEQ4 and MRSR distributions, which fit the WA70 data well. It is known, of course, that the straight NLO QCD result falls well below the recent E706 data, at somewhat higher energies. The introduction of initial state parton k T broadening effect can account for the difference [18], although the implementation of this effect is phenomenological and model dependent. Fig.12 shows the comparison of E706 data with a NLO QCD calculation, with Gaussian k T broadening by an amount (1.2-1.3 GeV) determined from differential distributions measured in the same experiment, cf. [14,19]. The same scale parameter µ = p T /2 is used and no other parameters are adjusted. The agreement is seen to be perfect. One should note, however, if k T broadening is similarly introduced in the comparison with WA70 data, using an amount seen in the WA70 di-photon momentum imbalance say, then the agreement shown in the previous plot, Fig.11, would no longer hold. Therefore, as discussed in Sec. 3, until the relevant experimental and theoretical issues are resolved, direct photon data cannot be unambiguously utilized to determine the gluon distribution.
Large-x gluons and the CTEQ5HJ set: The CDF RunIA inclusive jet production data [34] stimulated much interest in physics at large x, in particular the possible range of the gluon distribution in that region. The CTEQ4HJ parton distribution set, proposed two years ago [35], has served as a useful example in investigations of various large x phenomena. In a subsequent systematic study [36], we showed that the range of uncertainty of the gluon distribution is quite significant beyond x ∼ 0.2. For currently available jet production data, CDF inclusive jet p T distribution, as well as the CDF and D0 di-jet mass m jj distributions [37], continue to show a rise of the cross-section above the NLO QCD calculations based on conventional parton distributions, at large p T and m jj respectively. 9 It is therefore desirable to update the CTEQ4HJ parton distribution set, to complement the new CTEQ5M. This updated set is designated CTEQ5HJ. It gives almost as good a global fit as CTEQ5M to the full set of data on DIS and DY processes, with only marginally higher overall χ 2 , and has the feature that the gluon distribution is significantly enhanced in the large x region, resulting in improved agreement with the observed trend of jet data at high momentum scales mentioned above. The existence of excellent fits of this kind again serves to illustrate the fact that the large x region remains a fertile ground for further experimental exploration and theoretical development. Fig.13 shows the comparison between the gluon distributions of CTEQ5HJ and CTEQ5M at 2, 5, and 80 GeV. Due to the feature of QCD evolution mentioned earlier, the large difference of the two distributions at low Q represents the amplified effect of fitting jet data at an energy scale greater than 40 GeV at the Tevatron. The dashed lines in the bottom plots in Figs. 9,10 show the comparison of NLO QCD calculations based on CTEQ5HJ with D0 and CDF data; and in the top plots, they are normalized to the calculation based on CTEQ5M, along with the relevant data. In Fig. 14, we collect the ratio plots for the two experiments together; both sets of data are normalized to NLO QCD calculation based on CTEQ5HJ. This plot shows that CTEQ5HJ accounts well for both data sets, and that the two sets are in quite good agreement with each other. Note that experimental systematic errors are not included in this plot; and a relative normalization factor of 4% between the two experiments is used (Cf. the previous discussions on this factor).
Special CTEQ5 sets for Heavy Flavor physics studies: In applying perturbative QCD to processes in which heavy quarks play an important role, such as charm production at HERA, the standard renormalization and factorization schemes using zero-mass heavy-quark partons may be inadequate. See the Appendix for a fuller discussion. For this class of applications, we obtained the CTEQ5HQ set using the ACOT scheme [28] which gives a more accurate formulation of charm quark physics, valid from the threshold region (Q ≈ m c ) to the asymptotic region (Q ≫ m c ). The ACOT scheme consists of parton distributions defined in the (mass-independent) MS scheme, matched with hard cross-sections calculated using on-mass-shell (i.e. non-zero mass) heavy quarks when mass-effects are non-negligible. At very high momentum scales, it reduces to the conventional zero-mass-parton MS theory. For energy scales not far above the heavy quark masses, it gives a more accurate description of the underlying physics which is close to that of the fixed-flavor-number scheme with light flavors only [28,29]. In practice, for processes included in our global analysis, only the DIS structure functions are sensitive to the difference of the conventional (zero-mass-parton) and the ACOT (on-mass-shell parton) schemes. Thus, in extracting CTEQ5HQ, we used ACOT scheme Wilson coefficients in calculating DIS structure functions, along with available MS hard cross-sections from the literature for calculating Drell-Yan, W-, Z-boson, and jet production processes. The CTEQ5HQ set represents an updated version of CTEQ4HQ [38], and is similar in principle to the recent MRST [20] distributions, which uses a different implementation of the non-zero mass heavy parton approach.
We find the CTEQ5HQ set gives a slightly better overall fit to the full data sets than CTEQ5M; the difference in χ 2 being noticeable only in the HERA experiments, as expected. This difference is, however, not particularly significant since both are within experimental errors. To show the effect of the scheme choice, Fig.  15 compares the parton distributions from CTEQ5M and CTEQ5HQ at Q = 5 GeV. The differences for c(x, Q) and G(x, Q) are surprisingly small; but for u(x, Q) and d(x, Q) they are quite noticeable in the small x region. 10 This underlines the non-trivial coupling between the various flavors when theoretical or experimental input to the global analysis is varied. Expectations based on direct correlations do not always hold. Given the different treatment of charm quark mass in extracting parton distributions in different analyses, and by various groups, an important practical question is: how much error is incurred if these parton distributions are applied "incorrectly", by convoluting them with hard cross-sections calculated in a different scheme? To answer this question quantitatively, we evaluated the nominal χ 2 value of the full data sets used in our global analysis by mixing CTEQ5M parton distributions with hard cross-sections used in CTEQ5HQ extraction and vice versa. The χ 2 increased by 600 for around 1000 DIS data points (but changed little for other processes). This big increase in χ 2 highlights both the accuracy of current DIS data and the importance of maintaining consistency in applying parton distributions in quantitative QCD calculations. Mixed use of parton distributions defined in different schemes is clearly unacceptable, even if the distributions may look rather similar in graphical displays.
Although we did not use data from charm production in DIS for the CTEQ5HQ analysis, because of the theoretical and experimental problems mentioned in Sec. 3, it is interesting for comparison purposes, to evaluate an effective "F c 2 " based on the order α s formulas in the ACOT scheme [28,38] and check it against the existing measurement. Fig. 16 shows the result: the agreement is good.
In some heavy quark applications, various authors prefer to use fixed-flavornumber schemes, with the number of quark-partons fixed at either 3 or 4, for all momentum scales. For these application, we present the CTEQ5F3(4) sets in the fixed-3(4)-flavor scheme which treat charm (bottom) quarks as heavy particles, not partons. The CTEQ5F3(4) sets are updated versions of CTEQ4F3(4) [38]. They are similar in spirit to the GRV parton distributions, in particular the GRV98 set [40]. Whereas the fixed-flavor-number scheme is appropriate for certain applications, such as charm/bottom production at energy scales not far above the threshold; it is obviously inappropriate for processes in which charm/bottom quarks play a similar role as the light quarks, such as inclusive jet production at hadron colliders. Thus, the range of applicability of these distributions is much more restricted.

Comparison with Other Parton Distributions
To compare the CTEQ5 parton distributions with other recent parton distribution sets, it is important to take into account not only possible differences in input data sets and analysis procedures, but also the choices of renormalization and factorization schemes. As already mentioned in the description of the CTEQ5 global analysis, in addition to the familiar differences between MS and DIS schemes, the more refined recent parton distribution sets are also distinguished by their choice of the scheme for treating heavy quarks.
The MRS group adopted a new procedure for treating charm quark mass effects in DIS processes in their MRST (MRS98) analysis [20] by applying the method of Ref. [41]. The conventional zero-mass formalism is used for the other processes. This procedure is similar to that used for CTEQ4HQ and CTEQ5HQ, although the method of [41] does differ from that of [28,29] in the specifics of treating the mass effects (see below). With this in mind, we first show in Fig. 17 an overview of the comparison between the CTEQ5HQ parton distributions and those of MRST. The most striking difference is in the charm distribution; although less obvious differences in the other flavors are also present. Fig. 18 makes clear that the difference in c (x, Q) spans the entire x range. This difference can be attributed to: (i) the different choice of m c -1.30 GeV for CTEQ5 versus 1.35 GeV for MRST; and (ii) the residual difference in the procedure of treating charm mass effects in the Wilson coefficients. This does not directly affect the phenomenology of charm production, since, when used in conjunction with the appropriate Wilson coefficients, both reproduce well the measured physical DIS structure functions, including "F c 2 ". Cf. Fig. 16 above and Ref. [20] respectively. Of more phenomenological interest is the comparison of the gluon distribution in the CTEQ and MRST analyses, because of its implications for future high energy processes. On this issue, the difference due to the choice of scheme is completely overshadowed by that due to the choice of experimental input: to complement the DIS constraints in determining G(x, Q), we used the inclusive jet data of CDF and D0, as discussed above; whereas MRST relied on direct photon production results of WA70, applying a range of k T broadening corrections using the E706 data as a constraint. These experiments affect directly the determination of G(x, Q) in the medium to large x region. Fig. 19 shows the comparison of G(x, Q) from CTEQ5M and CTEQ5HJ with those of MRST at Q = 5 GeV. The significant difference observed can be readily understood in terms of the inputs.
The large range of variation between the MRST sets in the region around x ∼ 0.25 reflects the freedom of choice of the k T -broadening parameter k T which produces a very significant correction factor to the theoretical cross-section (recall this factor needs to be of the order of 2 ∼ 3 for E706 to agree with data), in addition to the well-known large scale dependence for NLO QCD predictions [42,18,21]. For a detailed discussion of the choices made to obtain this range, see Ref. [20]. The much narrower apparent range seen between the two CTEQ5 sets in this x span is due to the constraints on the shape of G(x, Q) imposed by the inclusive jet crosssection (which has rather stable NLO QCD theory predictions) and the requirement of best fit for the CTEQ5M and CTEQ5HJ conditions (with no attempt being made to explore the possible range as did in Ref. [36]). The MRST-G↑ (MRS98-2 in the figure) set uses WA70 data with zero k T broadening. Its G(x, Q) is closest to that of CTEQ5M, as can be seen in Fig. 19.
For the x > 0.5 region, the wide range of variation of the CTEQ5 sets reflects the lack of experimental constraints on G(x, Q) at large x. The convergence of the MRST gluons in this region appears to be due to choosing the same parametrization at large x for all these sets. Finally, the differences between the two series in the range 0.01 < x < 0.1 is most likely correlated to the differences in 0.1 < x < 0.6 as the result of the momentum sum rule constraint.
The other relevant process for this discussion is inclusive jet production. In Fig. 20, we show the comparison of the D0 data with NLO QCD calculations using the two CTEQ5 and MRST series of parton distribution sets. The calculation is performed using the Ellis-Kunzst-Soper program [24] with the scale parameter µ = E T /2 and the jet-separation parameter R sep = 1.3 (which is the current value favored by both CDF and D0). For this comparison, the experimental normalization is not floated, as done in fitting the parton distributions, for the obvious reason that the same experimental data points cannot have many different normalizations. The MRST curves lie considerably lower than the CTEQ5 ones, because their G(x, Q) is much lower in the relevant x range, as already seen in Fig. 19. The corresponding comparison to the CDF data is shown in Fig. 21. The significance of the observed differences must be assessed within the context of relevant theoretical and experimental considerations, some of which have been discussed above.

Conclusions and Comments on Uncertainties of Parton Distributions
As both theory and experiment improve steadily, global QCD analyses continue to show a remarkable agreement of perturbative QCD with available data on the wide range of hard-scattering processes and allow us to extract the non-perturbative parton distributions with increasing accuracy. There are, however, still many areas where more detailed theoretical and experimental work will help to clear up current uncertainties, and allow more precise determination of the parton structure of the nucleon. We devote this concluding section to discussions of these areas of uncertainty.
On the theory side, the most desirable advance would be a reliable calculation of direct photon production (especially in the p T range of fixed-target experiments), which could elevate the phenomenology of this process to the same level of confidence as for DIS, DY, and jet processes, and thereby lead to a definitive determination of the gluon distribution. Many theorists are working on the soft-gluon resummation corrections to the NLO QCD calculation to see if this can lead to a quantitative theory [22], accounting for the factor of 3 or more difference between the NLO theory and experiment beyond E706 energies. However, this explanation of the discrepancy is not yet universally accepted [21].
Considerable progress has been made on the differentiation between u and d quarks in the last year, as the result of complementary information provided by several different DIS and DY measurements, as discussed in Sec. 2 and 3. However, this analysis relies heavily on: (i) the assumption of charge symmetry ) (which has been questioned in recent literature [17]; and (ii) the extraction of neutron cross-sections from actually measured deuteron crosssections. The size of nuclear corrections needed to extract the neutron crosssection is still a subject of some controversy. These corrections could affect the determination of d/u, especially at large x [16]. We found that, in the global analysis context, all current data can be consistently described within the PQCD formalism with or without applying a deuteron correction; and chose to take the simple option of not applying any such correction. A detailed study is underway to probe this issue more thoroughly. Such studies will clearly benefit from a better theory for nuclear corrections. Conversely, better phenomenological analyses of the existing abundant data could provide useful input to the study of the nuclear effects.
There has been little advance in the unambiguous determination of the strange quark distribution. The long-standing dilemma associated with the discrepency of the strange quark distribution inferred from the di-muon neutrino data and that from the difference of neutral and charged current structure functions [1] remains unresolved. This problem may be related to that of charge symmetry [17]. To make real progress, the most useful development would be measurements of physical cross-sections (or structure functions) for charm production in neutrinonucleon scattering, which can then be incorporated in the global analysis. If this cannot be done for existing measurements, one hopes it will be achievable in the analysis of the NuTeV experiment.
The charm quark distribution has entered the arena of global QCD analysis with the availability of charm production data in neutral current interactions, particularly at HERA. This has directed attention to more precise formulations of QCD theory including massive quarks, which have been actively pursued over the last ten years. Unfortunately, more precise formulations necessarily lead to additional scheme dependence of the PQCD calculations, thereby complicate the application of the parton formalism for users of parton distributions. We briefly described some of the pertinent issues in Sec. 3 and in the Appendix. An interesting related question is: is there a non-perturbative component of charm inside the nucleon? [43,44] This question has not yet been addressed by any of the existing global analysis efforts -all assume a purely radiatively generated charm distribution which vanishes at the threshold scale. Since the charm mass is only slightly above the nucleon mass, there is no strong argument against the existence of an additional non-perturbative component of charm. This issue can be studied once more abundant precision data become available.
It is universally recognized that for a wide range of theoretical and experimental applications, it is extremely important to know the range of uncertainties of the parton distributions. The ultimate goal would be to have parton distribution sets with a well-defined correlation matrix for their parameters [45]. To see what needs to be done toward achieving this goal, it is first necessary to recognize the major sources of uncertainties in global QCD analysis and address them systematically.
The most obvious uncertainties are the reported experimental errors. The nontrivial aspect of these are the correlated systematic errors. In principle, there are standard methods to incorporate these errors, often represented as covariance matrices, in data-fitting. Several recent attempts and proposals have been made to pursue this approach [46]. In practice, since only a limited number of experiments present information on correlated errors, the input data sets for the global analysis are much more restricted than required to determine the different parton flavors. In addition, this task is much more complex than appears on the surface, because: (i) it is known that the standard covariance matrix method is not robust under certain conditions [47] and can lead to pathological results 11 , and (ii) the diversity of experiments involved in a global analysis, and the non-uniform information they provide, can easily vitiate some of the essential assumptions underlying the statistical analysis method.
Theoretical uncertainties that affect the global analysis are much less obvious and much harder to quantify than the experimental errors. The magnitudes of the uncertainties due to higher-order effects, scale-dependence, soft-gluon resummation, higher-twist effects, nuclear (deuteron) corrections, etc., vary widely from process to process, and from one kinematic region to another. Thus, while the uncertainties of NLO calculations of DIS and DY processes are known to be under control (except near the boundaries of the kinematic region), and those of inclusive jet cross-section are also stable, the same is far from true for direct photon production (at p T values of most available data) and for heavy quark production in hadron collisions. These uncertainties have to be dealt with on a case by case basis, using the most up-to-date knowledge of the specific process.
Last, but by no means least, there are hidden uncertainties associated with the choice of functional forms for the non-perturbative initial parton distributions. Although the parameters in these functions are determined by comparison with experiment, the choice of functional form introduces implicit correlations between the parton distributions at different x ranges. We have encountered this hidden correlation often in our investigation of the range of variations of the gluon distribution in previous and current CTEQ analyses. The simpler the functional form (or the more economical the parametrization),the more rigid is the implied correlation. 12 To reduce this undesirable correlation, one cannot, however, indis-criminately increase the degrees of freedom of the parametrization. If there are not enough experimental constraints to determine the parameters, one will get unpredictable artificial behavior of the parton distributions that is not related to the experimental input. We have also encountered examples of this kind in the course of our analyses. Only as more precise experimental data become available for more processes, does it become possible to refine the parametrization in a progressive manner.
The presence of uncertainties of the second and third kind has important implications on efforts to quantify the implications of experimental systematic error on parton distribution analysis, because both uncertainties are of a highly correlated nature and all three are inextricably intertwined.
In the CTEQ series of global QCD analyses [1,2,38,36,48], we try to assess the current knowledge of the parton distributions keeping all the above sources of uncertainties in perspective, and make the best educated estimates on the uncertainties as possible. The global analysis of parton distributions is yet far from being an exact science, due to its complexity and comprehensive scope. However, the steady progress that has been achieved clearly demonstrates that vigorous pursuit of the open problems summarized above will continue to improve our knowledge of the parton structure of hadrons, and pave the way for advances in all fronts in elementary particle physics.
Note added in proof: Until recently, CTEQ global analyses used QCD evolution codes which yield slightly different results compared to those eventually arrived at by a 1995/96 HERA working group. 13 This was due to certain numerical approximations adopted for the NLO evolution. The approximations have since been eliminated, so that the differences between our results (based on an x-space method) and that of the HERA group based on the same method are now much smaller than the differences between their x-space and moment-space methods. As pointed out by the HERA working group, the latter differences represent a measure of the intrinsic uncertainty of these perturbative calculations. We have repeated the initial CTEQ5 global fits using the improved code. Although there are slight shifts in some parton distributions in some kinematic regions, all results described in this paper are independent of such small shifts. In particular, because the fits to data are of identical quality, comparisons of theory and experimental data are indistinguishable. For the same reason, the differences in current physical calculations due to the two versions of PDF's are small -certainly insignificant at x=0.18 from the NA51 measurement. The dramatic turn around, seen in the CTEQ5M curve, is brought about by the more extensive data set of E866 which forced a change of the function form. 13  compared to those due to the many, much larger, uncertainties in the global analysis, as discussed above. The relative sizes of these differences will be described in forthcoming studies on the effect of parton distribution uncertainties on precision electro-weak and QCD phenomenology at the Tevatron and LHC.

Appendix: Renormalization and Factorization Schemes
In the presence of heavy flavors, perturbative QCD becomes more involved than commonly formulated because the heavy quark masses, m i , appear as extra scales in the problem. The magnitude of the hard scale Q of the physical process relative to m i is an important determining factor in what is the appropriate renormalization and factorization scheme to adopt. For a concise review of the relevant issues, see Ref. [49]; for a rigorous theoretical treatment of the problem, see Ref. [29]. The following discussion applies to both charm and bottom quarks, but for definiteness we shall focus on charm.
Most universally used parton distribution functions in the past have been generated using the conventional 4/5-quark-flavor scheme, using zero-mass hard crosssections for all active flavors. The same is true for practically all the popular Monte Carlo programs used in data analyses. The implicit assumption in this practice is that Q ≫ m c , which is obviously not a good approximation if the application includes the region Q m c . In contrast, the fixed "three-flavor scheme" (in which the charm quark never enters as a parton) has been used in many LO [26] and NLO [27] heavy-quark production calculations, as well as in the GRV parton distribution determination [50,40]. The implicit assumption here is that the charm quark always behaves like a heavy particle, irrespective of the physical energy scale Q. Although this represents the correct physical picture in the threshold region, it is clearly inappropriate in the asymptotic region Q ≫ m c . In recent years, a unified approach (the on-mass-shell, or ACOT scheme [28,29,49,51,52]) which incorporates non-zero charm-parton mass effects near threshold, and which contains the above two cases as distinct limits has been formulated. Stimulated by HERA data on charm production [7,8], variants of this approach have also been proposed and used in the recent literature [20,41,53]. Recently, Collins has provided a rigorous basis for the ACOT scheme by establishing a generalized factorization theorem including the heavy quark masses, to all orders of PQCD [29].
The basic ideas of the on-mass-shell scheme are relatively simple: (i) in the region below and just above the charm threshold (Q m c ), one adopts the natural three-flavor scheme; (ii) somewhere past the charm threshold region (i.e. for Q > m c ), one switches to a four-flavor scheme in which the infrared unsafe factors involving ln(Q/m c ) are resummed into charm parton distribution (or fragmenta-tion) functions, while keeping the remaining infrared safe m c dependence; 14 and (iii) the transition from the 3-flavor to the 4-flavor scheme can be carried out anywhere over a fairly large region in which both schemes apply, provided appropriate matching conditions between the two schemes are implemented. The resummation of the infrared unsafe factors enables this approach to reproduce the conventional zero-mass parton formalism asymptotically (i.e. for Q ≫ m c ), while keeping the infrared safe m c effects ensures that it reproduces the fixedthree-flavor scheme results near threshold. If the four-flavor scheme part is defined with MS subtraction, the parton distributions satisfy evolution equations with the same mass-independent evolution kernel as in the conventional picture. The hard scattering cross-sections, however, will be different from the conventional ones in the literature if infrared safe m c effects are retained in the transition region for accuracy. Furthermore, because of inherent approximations in the perturbative approach, there are different, but equally valid, ways to include the quark mass effects, as exemplified by the differing results on DIS used in Refs. [28,29,49,52] on the one hand, and Refs. [20,41] on the other.
In principle, the new on-mass-shell scheme is the natural one to use for global analysis, because of its relative generality and its special relevance to the charm production process. However, aside from the DIS process, no NLO hard crosssection calculation has yet been carried out in the more general scheme. Thus, both in the previous CTEQ study, resulting in the CTEQ4HQ distributions [38], and in the recent MRST analysis [20], non-zero-mass hard cross-sections for the DIS structure functions are employed along with zero-mass hard cross-sections for the other processes. This procedure appears to mix two different ways for calculating hard cross-sections. It can however be justified in practice, if the errors due to the zero-mass approximation are negligible over the kinematic ranges of the relevant processes included in the analysis. This is clearly the case for hadron collider processes (W-, Z-, and jet-production) where the relevant momentum scale is always far greater than m c . For processes involving momentum scales comparable to, or not far greater than m c , the zero-mass parton approximation can still be acceptable if the contribution from the so-called (heavy) flavor-creation (FC) subprocesses is sub-leading, because mass effects are most significant in these subprocesses [29]. Thus it is more important to keep the mass in the DIS calculation (where FC comes in order α s ), especially for observables involving charm particles in the final state (for which order α s is in fact the leading order at a scale Q m c ), than in the DY calculation (where FC only comes in at order α 2 s ). We shall not be concerned about charm particle production in DIS in this study, since both experimentally and theoretically, the only well-defined physical quantities are F D,D * 2 ; but it is not possible at present to use F D,D * 2 to place useful constraints on parton distributions because they also depend on the poorly known fragmentation functions of D, D * . 15 (cf. Sec. 3) The remaining issue is then, should one always adopt the on-mass-shell formalism for treating the total inclusive DIS structure functions F total 2,3 in the global analysis, knowing that leading order contributions to F total 2,3 is of order α 0 s and the FC contribution is of order α 1 s near the threshold region where mass effects are important. From the theoretical point of view, the answer appears to be yes -since the formalism already exists, and it is desirable to be as precise as possible. However, switching to the on-shell formalism does bring about practical complications: (i) to use the resulting parton distributions, one needs to convolute them with the matching Wilson coefficients (at least for DIS processes) which are not incorporated in most applications; and (ii) as we show in Sec. 4, different implementations of the (in principle equivalent) on-massshell scheme actually result in quite non-negligible differences in DIS structure function calculations at the NLO accuracy, thus users of these distributions need to have multiple versions of hard-cross-sections to match the corresponding parton distributions from different groups.
These considerations underlie our decision to update the conventional zeroparton-mass distributions in the form of CTEQ5M,D,L,HJ sets, which can be used with standard hard cross-section formulas to produce accurate PQCD calculations of all processes, along with the on-mass-shell CTEQ5HQ set which is needed for applications involving heavy quark final states. For these latter class of applications, we also include CTEQ5F3/4 sets in the 3/4 fixed-flavor-number schemes which are the appropriate ones to use with existing calculations of heavy quark production processes done in these schemes. From the discussions above, it should be clear that, in fact, CTEQ5HQ can be regarded as the most general among all these sets, since the on-shell scheme contains the other schemes as limiting cases. However, in order to produce precise calculations for DIS structure functions, CTEQ5HQ distributions must be matched with Wilson coefficients calculated in the ACOT scheme.    Incl. Jet : p t 7 * dσ/dp t  Figure 9 Comparison of D0 inclusive jet production data to the CTEQ5 fits. The bottom plot shows the measured cross-section dσ/dp t , multiplied by p 7 t in order to allow a linear display. The top plot shows the ratio of the measured cross-section to that calculated with CTEQ5M, as well as the ratios of CTEQ5HJ to CTEQ5M.  Figure 11 Comparison of the WA70 direct photon data with NLO QCD calculations using CTEQ5M. A normalization factor of 1.08 has been applied. E706 Direct Photon p t 7 * E d 3 σ/d 3 p t E706-530GeV E706-800GeV NLO QCD (CTEQ5M) Figure 12 Comparison of the E706 direct photon data with NLO QCD calculations using CTEQ5M. An initial state parton k T broadening effect has been applied, as described in the text.  Figure 13 Comparison of the gluon distributions from CTEQ5M and CTEQ5HJ at three energy scales: 2, 5, and 80 GeV. Note the effect of QCD evolution. The dip at x ∼ 0.5 of the dashed curve is not physically significant: it is the result of the parametrization (being the product of two factors, one rising and one falling).
Only the general magnitude of the curve in this region is meaningful.  Figure 15 Comparison of the parton distributions of CTEQ5HQ, defined in the on-mass-shell ACOT scheme for heavy quarks, with those of CTEQ5M, which uses the conventional zero-mass approximation.  Figure 16 Comparison of preliminary HERA measurement of "F c 2 " with order α s ACOT scheme calculation using CTEQ5HQ.  Figure 17 Comparison of parton distributions from MRST, in the on-mass-shell charm scheme of Ref. [41], with those from CTEQ5HQ, at 5 GeV.