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Global QCD Analysis and Hadron Collider Physics

The role of global QCD analysis of parton distribution functions (PDFs) in collider physics at the Tevatron and LHC is surveyed. Current status of PDF analyses are reviewed, emphasizing the uncertainties and the open issues. The stability of NLO QCD global analysis and its prediction on"standard candle"W/Z cross sections at hadron colliders are investigated. The importance of the precise measurement of various W/Z cross sections at the Tevatron in advancing our knowledge of PDFs, hence in enhancing the capabilities of making significant progress in W mass and top quark parameter measurements, as well as the discovery potentials of Higgs and New Physics at the Tevatron and LHC, is emphasized.


Introduction
As the physics program at the Tevatron advances from Run I to Run II, and as the planning of LHC physics shifts into high gear, it is important to assess the foundation on which most of the relevant physics calculations are based. All experimental measurements are made on lepton-and hadron-initial (and final) states. On the other hand, the fundamental physics we are trying to unravel is formulated in terms of interactions between leptons, vector bosons, quarks, gluons, Higgs and other new particles. To make any progress, we need to ask: How well do we know the parton (quark and gluon) distribution functions (PDFs) of the nucleon? What important pieces of information on PDFs are still missing? What are the uncertainties of the "known" pieces? What information gained at the Tevatron Run II and HERA II will be important to predict, and help sort out, the physics at LHC? Some, but not all, of these questions are addressed in this talk. Due to space limitations, only representative results from the talk can be included in this written report. For the same reason, references will be limited to the essential ones. Details on many topics can be found in recent literature and in related articles in these proceedings.

Status and Open Issues in Global QCD Analysis
A great deal of progress has been made in global QCD analysis over the last 20 years, due to new experimental input from a variety of hard processes (DIS, DY, W/Z and jet production in colliders), advances in theoretical calculations (NLO, NNLO and resummation), and more powerful phenomenological analysis tools (improved Hessian methods, Lagrange Multiplier technique, and functional approaches). Nonetheless, many features of the PDFs are still uncertain, or unknown. These open issues are summarized here, with some brief background information in each case.
The u and d distributions Improved data have caused considerable changes in the PDFs u(x, Q) and d(x, Q) over the years, particularly at small x (as a direct consequence of the advent of the HERA data since the mid-1990's). Combined high precision fixed-target and HERA data, supplemented by constraints from DY and W production data, now cover roughly the x range of (10 −5 , 0.75). These data result in rather well-determined u and d distributions in most recent global analyses. [1,2] The remaining uncertainties concern mainly the large x behavior, beyond the measured range, particularly the d/u ratio. The discrimination between these two flavors is currently hampered by unknown nuclear corrections associated with the use of DIS and DY data on deuteron targets. Recent preliminary DY cross section data from NuSea [3] have stimulated renewed interest in this problem. However, no conclusion can be reached until the data is finalized. In principle, more precise e ± p charged-current scattering data at HERA (W + /W − exchanges) would be able to determine the d/u ratio free of nuclear effects. So will data, eventually, on the rapidity distributions of W + /W − production at the LHC.

The gluon distribution
It is well known that the constraints on g(x, Q) are much looser than for u and d. Hence, the first determinations of the gluon distribution varied over a wide range. The initial HERA data forced a much steeper rise of g(x, Q) toward small x. However, more recent global analyses all have a much more moderate rise, or, in some cases, even a fall, in the small-x behavior of the gluon at low Q. An important contributing factor for this turn-around is the indirect effect of including single-jet inclusive production data from the Tevatron. These data favor a larger g(x, Q) at high x, which takes away gluons at small x because of the overall momentum sum rule. The range over which the gluon distribution has developed over these years shows vividly both how global QCD analysis has been evolving, and how much further we need to go to determine the parton structure of the nucleon with confidence. The fractional uncertainty on g(x, Q) is largest at high x, where experimental constraints are scarce. At small x, there is a whole range of open theoretical and phenomenological issues. In fact, in the MRST analyses, the necessity of introducing a negative gluon distribution at low x and small Q has been proposed [1]. These issues will be discussed in a later section.
The strangeness sector An important, but so far poorly determined, frontier of PDF study is the strangeness sector. This can be separated into two parts: the total strangeness sea, s + (x) = s(x) +s(x), and the strangeness charge asymmetry s − (x) = s(x) −s(x). Advance on both fronts is expected by detailed study of the recent CCFR-NuTeV dimuon production data in neutrino and anti-neutrino scattering [4] in global analysis. 1

Flavor SU(3) breaking:
The naive expectation of flavor SU(3) symmetry, s =s =d =ū, assumed in some early PDF studies, is clearly unrealistic: the strange quark mass alone would induce a difference between (s,s) and, say, (ū +d)/2. Experimental evidence suggests that the ratio R s+ ≡ s+s u+d , is of the order 0.5 at a scale of 1-2 GeV. Up to now, this ratio has mainly been enforced as a (constant) factor in global analyses, rather than as the result of actual fitting to data, because the relevant data have not been presented in a form suitable for global analysis.
Strangeness Charge Asymmetry: In order to separate the s ands distributions, and hence allow the measurement of the strangeness charge asymmetry, s − (x), the separate measurement of ν andν cross sections is required. Although the strangeness number sum rule requires 1 0 s − (x) dx = 0, a charge asymmetric strange sea, manifested by, e.g. a non-zero first moment [S − ] = 1 0 x s − (x) dx, has important physical consequences such as the NuTeV anomaly [5].
Quantitative study of this issue is currently limited by the fact that CCFR-NuTeV measures ν andν cross sections with specific kinematic cuts [4], rather than inclusive cross sections that theory can calculate. In order to perform a global fit incorporating these new data, a LO model bridging the inclusive and the measured cross sections has to be invoked. With this caveat, CTEQ has found that (i) a positive value for [S − ] on the order of 0.0017 is favored; and (ii) the range of uncertainty, both theoretical and experimental, is rather large: −0.001 < [S − ] < 0.004. [6] The analysis by CCFR-NuTeV, using their own data only, reached a somewhat different conclusion [4]. The two groups are working together to understand the differences, and to extend both analyses to a full NLO one.
Possible Isospin Violation in the Parton Structure of the Nucleon Interest in possible explanations of the NuTeV anomaly has also motivated the study of the possibility that u proton (x, Q) = d neutron (x, Q). Experimental constraints on this effect are very weak, even without taking into account the large uncertainties about nuclear corrections that are needed to measure neutron structure functions [1]. Theoretically, it has been pointed out that isospin violation in PDFs arises naturally when one tries to include electroweak corrections in the global QCD analysis: the evolution equations of PDFs will then include photon distributions of the nucleon; the u proton (x, Q) and d neutron (x, Q) distributions will evolve differently due to their different electric charge. This effect has recently been studied [7]; it is small, but still can be physically relevant.

Heavy Quark distributions-Charm and Bottom
Although there has been much discussion about physical processes involving heavy quark production in the literature, there is as yet not very much reliable information on the heavy flavor parton distributions. Of the heavy quarks c, b, and t, only the c and b quark-partons actively participate in PQCD calculations of high energy physical processes at energy scales even up to those of LHC. The definition of heavy quark partons is even more schemedependent than that of light quark flavors. In the so-called (fixed) 3-flavor scheme, there are, by definition, no heavy quark partons at all; whereas in the (fixed) 4-flavor scheme, there is a charm distribution, but no bottom distribution. These schemes are useful only for limited energy ranges. In recent years, a consensus has emerged that the variable-flavor number scheme, which is a generalization of the conventional M S zero-mass parton scheme [8], is the appropriate one to use for calculations that cover a wide range of Q. If one assumes that there is no non-perturbative heavy flavor content in the nucleon, then the heavy flavor parton distribution functions c and b are "radiatively generated" by QCD evolution from their respective thresholds. This is the assumption used in all existing global analyses of PDFs. Unfortunately, the concept of radiatively generated heavy quark partons is not fully welldefined, since the location of "heavy quark threshold" for a given flavor is itself ambiguous: it can be any value of the same order of magnitude as the heavy quark mass or the physical heavy flavor particles. Ways to actually measure c and b quark distributions at the Tevatron and LHC will be discussed in a later section.

Uncertainties of Parton Distributions
In parallel with the determination of ever improving "best-fit" PDFs, an equally important front in global analysis has been opened in recent years-the development of quantifiable uncertainties on the PDFs and their physical predictions. Several groups have carried out extensive studies with different techniques and emphases. Much progress has been made; many useful results have been obtained; but there are, as yet, no unambiguous conclusions. The basic problem lies with the complexity of the global analysis that utilizes results from many experiments on a variety of physical processes, with diverse characteristics and errors, and which are often not mutually compatible according to textbook statistics. The analyses are also sensitive to many theoretical uncertainties that cannot be readily quantified; and they can depend on the choice of parametrization of the non-perturbative functions used in the analysis. Individually and collectively, these factors render a rigorous approach to error analysis untenable [9,10].
As an illustration of the basic problem, we briefly describe results on a study of the uncertainty of the W production cross section at the Tevatron due to known experimental errors on the input data sets, conducted by the CTEQ group [2] using the Lagrange Multiplier method [9]. First, we obtain a series of PDFs that provide best fits to the global data, but constrained to yield a series of possible values of σ W at the Tevatron around the CTEQ6M value (which corresponds to the least overall χ 2 by definition). Then, we evaluate the individual χ 2 of each experimental data set to gauge the consistency between the data sets, as well as to assess in a sensible way to quantify the overall uncertainty of the prediction on σ W due to the input experimental uncertainties. The results are shown in the two plots of Fig. 1. The horizontal axes correspond to the values of σ W . For each of the 15 input experimental data sets, a best-fit value and a range is shown. These are arranged vertically, in no particular order. On the left plot, each range corresponds to a ∆χ 2 = 1 error due to that experiment; while on the right plot, it corresponds to a "90% confidence level" (cumulative distribution function of the χ 2 normalized to the best fit). We see that, if a ∆χ 2 = 1 error criterion is enforced, there is no common value for the predicted σ W (or, equivalently, some of the data sets must be deemed mutually incompatible). But within the 90% confidence level range, there is a common range for σ W that spans the values indicated by the dashed Figure 1: Predicted value of σ W at the Tevatron: (a) ∆χ 2 = 1 error ranges for individual experimental data sets evaluated from PDF sets obtained by the Lagrange Multiplier method in constrained global fits; and (b) 90% confidence level ranges for the same data sets and PDF sets. vertical lines.
Faced with the problem of nominally incompatible data sets (which is common in combined analysis of data from diverse experiments, e.g. PDG work), subjective assumptions and compromise measures are necessary to obtain sensible results. Several approaches have been followed by the different global analysis groups. CTEQ uses the ansatz that the range of uncertainty indicated in Fig. 1b represents a 90% C.L. uncertainty on σ W ; and, in general, characterizes the PDF uncertainties by using similar criteria along 20 orthonormal eigenvector directions in the PDF parameter space, using an improved Hessian method. 2 MRST has adopted the same approach [1], albeit choosing a slightly narrower range of the uncertainty. The H1 and ZEUS PDF analysis groups also adopt similar methods, but, by restricting the input data sets to DIS experiments only, apply their own definition of the range. [10] The important fact is that these different groups (all using the leading twist PQCD formalism) arrive at quite comparable results, both for the PDFs and for the magnitude of the error bands, even if some details are different because of the variations in experimental and theoretical inputs.
With this approach, both CTEQ and MRST have been able to make estimates on PDFs as well as their predictions on future measurements. Two examples in the latter category are given in Fig. 2. Fig. 2a shows fractional uncertainties in the predicted qq and GG parton luminosity functions as a function of √ŝ at the Tevatron energy obtained by CTEQ, from which the values and the uncertainty ranges of a variety of physical processes, both in the SM and beyond, can be estimated. We see a considerable uncertainty associated with the gluon-gluon luminosity at large x. Fig. 2b shows contours of increasing χ 2 in the σ W -σ H Figure 2: Examples of ranges of predictions at Tevatron Run II and LHC associated with uncertainties of PDFs due to experimental input. See text.
plane, due to PDF uncertainties for the LHC, obtained by MRST. Theoretical uncertainties are not included in either plot.
A different approach is followed by Alekhin [11]. The experimental input is restricted to DIS experiments only, and the theoretical framework is broadened to include higher-twist effects, among others, in order to better accommodate the different data sets. A consistent fit is then achieved in the strict statistical sense; and the uncertainty range is defined according to the classic ∆χ 2 = 1 criterion. However, by forgoing the critical experimental constraints provided by Drell-Yan and inclusive jet production data, the determination of the PDFs cannot be complete. Applying the Alekhin PDFs to the available DY data sets (E605, CDF W-asymmetry, E866), one obtains a χ 2 of 892 for 145 data points-a clear indication that vital information is missing on certain aspects of PDFs. This can be seen in a plot ofd −ū d+ū where the Alekhin prediction is completely different from the experimentally determined ratio obtained from σ pd /σ pp DY data [12]. Under these circumstances, one might ask, whether these PDF uncertainties defined by the textbook ∆χ 2 = 1 rule are of any practical use? Giele et al. [13] also emphasize a rigorous statistical approach, using the more general likelihood method. Within the leading twist PQCD approach, this leads to acceptable results only if one restricts the input experimental data sets to one or a few DIS sets. Thus, depending on which subsets of data are used, one gets many predictions on physical quantities (such as σ W ) with "1σ-error" ranges, which do not overlap with each other. This approach leaves unanswered the important question: "What is the best estimate of current uncertainty, given all available experiments?".
Thus, the underlying facts seen by all groups are consistent with one another. The differences in interpretation lie in the emphases placed to cope with these facts. In principle, all methods are valid and equivalent: in an ideal world where all experiments came up with textbook-like errors, they would all yield the same results. In reality, in the complex world of global analysis, the results appear different or inconsistent (if strict criteria are applied), depending on subjective judgements made in placing the emphases. This state of affairs requires that users of PDFs must be well-informed about the nature of the "uncertainties" provided by the various global analysis groups and to use these results judiciously according to their own (subjective) judgement and taste.
Unfortunate as it may be, but there is no "1-σ PDF error" that can be defended scientifically on all accounts. This fact points to the need for continued hard work, both on the experimental and theoretical fronts, in order to improve the situation, and to reduce the ambiguities described above. To this end, the physics programs of HERA II, Tevatron Run II, as well as several fixed-target experiments, can make important contributions in the immediate future. Through these efforts, the PDFs and their uncertainties will certainly be better known when the LHC comes on line. This will lead to better predictions on both SM and new physics processes, and hence improve the potential for all discoveries. In addition, the high reaches of the LHC, both in energy range and in statistics, will provide additional constraints on PDFs, and thus allow even better determination of the parton structure of the nucleon.

Global QCD Analysis and Collider Physics Standard Candle Processes and Stability of NLO QCD Predictions
Because W and Z bosons are copiously produced at the Tevatron and LHC, and because the PQCD theory for this process is well established, the W/Z cross sections have been widely considered as "standard candle" measurements. It is therefore crucial to understand the uncertainty and the stability of PQCD analyses of this process. The former has been discussed in the previous section. We now look at the stability problem.

Stability of NLO global analysis and the total W cross section
The vast majority of work on global QCD analysis of parton distribution functions (PDFs) and the application of PDFs to the calculation of high energy processes has been performed at NLO (1-loop hard cross sections and 2-loop evolution kernel). In recent years, some preliminary NNLO analyses have been carried out, even if not all necessary hard cross sections are yet available, and the evolution kernel was only known approximately (until very recently). Since errors on most of the experimental data used in global analysis are generally larger than the known NNLO corrections, the necessity to extend the analysis to NNLO, at the expense of vastly more complicated calculations, is not obvious.
A strong motivation to go to NNLO would exist, however, if the conventional NLO analysis does not yield stable PDFs and physical predictions. This possibility was indeed raised by a recent MRST study [1]. In particular, they found a 20% variation in the predicted cross section for W production at the LHC-a very important "standard candle" process for hadron colliders-in their NLO analysis, depending on kinematic cuts placed on input data. 3 Their results on the total cross section and on the rapidity distribution of the W boson are shown in Fig. . For both the Tevatron Run II and LHC physics programs this is clearly a critical issue; it is important to investigate whether this result is confirmed by an independent study. We have investigated this problem in considerable detail within the CTEQ global analysis framework [2]. Applying the same theoretical and experimental inputs to the global analysis, we systematically varied the kinematic cuts in x and Q, and generated new sets of best-fit PDFs. To explore the dependence of the results on assumptions made about the parametrization of PDFs at the starting scale Q 0 (1.3 GeV), we also studied cases where the gluon distribution function is allow to go negative at small x, a possibility favored by the MRST NLO analysis in the past few years. The main results of this study [14] are discussed below. Fig. 4a shows the variation of the W total (W + +W − ) cross section at LHC with respect to a series of best fit PDFs obtained with increasing x-cuts comparable to those of the MRST study; the results are super-imposed on Fig. a. Also included is a point corresponding to a higher Q-cut of Q 2 > 10 GeV 2 . These results show full stability of the predicted W cross section versus the kinematic cuts used in the global analysis of PDFs, in marked contrast to the results of [1]. (Results at Tevatron energies, not shown due to lack of space, are even more stable, as one would expect.) Fig. 4b shows a comparison of the predicted rapidity (y) distribution for (W + +W − ) at LHC. Instead of showing the predicted y-distribution from the fits with different kinematic cuts (which are all similar to one another), we show the range of variation of the y-distribution when the 2nd moment of the y-distribution is extremized using the Lagrange multiplier method of [9]. These "extreme" cases allowed by our global analysis are compared to the MRST "default" and "conservative" predictions. The MRST conservative prediction, which corresponds to strong kinematic cuts, clearly stands out.
The stability of the prediction with respect to variations of kinematic cuts seen in analysis is reassuring. It indicates that NLO QCD should provide an adequate framework for studying high energy phenomenology at the Tevatron and LHC, except perhaps for processes that are known to have large corrections to the hard cross section beyond NLO. On the other hand, the difference between our results and those of MRST needs to be understood. The two global analysis efforts share many common theoretical and experimental inputs. Most of their predictions have been found to be compatible with each other. However, historically, Figure 4: (a) Dependence of CTEQ predictions (light dots) on W total cross section at LHC on kinematic cuts of input data, super-imposed on Fig. a; (b) W rapidity distribution at LHC, CTEQ6 prediction plus those extremized w.r.t. the 2nd moment of y in solid color lines, compared to that of the MRST default and conservative PDF predictions (in black solid and dotted lines). they also have arrived at different conclusions on a number of specific issues, due to subtle (or not so subtle) differences in methodology and/or input. These were eventually resolved when the causes for the difference were identified. The case on the stability of NLO analysis may represent such a situation. In particular, the instability of the MRST NLO analyses appears to be associated with two other unique features of their recent analyses: (i) an apparent "tension" between inclusive jet production data at relatively large x and DIS data at small x from HERA; and (ii) a relatively strong preference for a negative gluon distribution g(x, Q) at small x and small Q.
In order to look a little closer into these issues, we have investigated the χ 2 values of the jet data sets and the HERA data sets separately as we vary the kinematic cuts. No discernable improvement in the fit to the jet data sets is seen as the x-cut on input DIS data is raised. 4 The χ 2 /N for the CDF and D0 experiments are (1.47, 1.48, 1.47, 1.47, 1.48) and (0.718, 0.715, 0.723, 0.726, 0.697) for x-cuts of (0, .001, .0025, .005, .01) respectively. The quality of fit to the HERA data (as measured by χ 2 /N) also remains stable with respect to the change in the kinematic cuts.
An important effect of raising the kinematic cuts in x and Q is that constraints provided by the precision HERA data in the small x and low Q region are removed from the global fit. Thus, the uncertainty of the resulting PDFs and their physical predictions will increase. We quantified this effect for the specific case of W production at LHC. Using the Lagrange Multiplier method [9], we examine the best χ 2 values obtained in constrained global fits, as a function of the W cross section, for several choices of kinematic cuts: (i) the normal CTEQ cuts (Q 2 > 4 GeV 2 ); (ii) medium cuts (Q 2 > 6.25 GeV 2 , x > 0.001); (iii) strong cuts (Q 2 > 10 GeV 2 , x > 0.005); and (iv) very strong cuts (Q 2 > 100 GeV 2 ). 5 Fig. 5a shows the results when the normal CTEQ parametrization, with positive PDFs at the input scale of Q = 1.3 GeV, is used. We see that, as stronger kinematic cuts are imposed, the range of uncertainty on the prediction for the W production cross section becomes progressively larger. Whereas the change from the normal to medium cuts is small, the range of uncertainty almost doubles from normal to strong cuts at any tolerance level of χ 2 . The very-strong-cuts case removes most DIS data from the fit, thus heavily emphasizes collider data-it represents a step toward a hypothetical "collider-data-only prediction" from Tevatron to LHC. It results in a very large range of uncertainty for the LHC cross section, as one would have expected. 16 17 We have also performed similar fits, but removing our usual constraint that parton distributions are positive definite at the scale Q 0 . This affects mostly the gluon, as quark distributions are more directly related to (positive) input structure functions. 6 As a rule, our best fits for any given sets of kinematic cuts do not require negative gluons at our input scale of Q 0 = 1.3 GeV. However, when σ LHC W is pulled away from the best-fit value by the constrained fits in the Lagrange Multiplier study, the additional freedom provided by a negative gluon widens the allowed range of variation of σ LHC W . The results of this analysis (except that of the very strong cuts) are shown in Fig. 5b. Compared to the corresponding cases in Fig. 5a, we see that the medium-cuts curve has opened up noticeably at the lower end, and the strong-cuts case has widened at both ends.
Should the fits with negative gluons be taken seriously as candidate PDF sets, hence providing better determination of the true range of uncertainty? We think not. The reason is that whereas PDFs are not strictly forbidden to become negative, all physical quantities calculated from them, in all parts of physical phase space, must remain positive definite. This is a very strong condition that is extremely difficult to satisfy if some PDFs become negative in some region of the (x, Q) plane. For all the solutions involving negative gluons at small x and low Q (including the MRST ones), we found it is possible to identify some physical cross sections, at some high energies, near some boundary of phase space, that MRST conservative fit. 6 A strong enough negative initial gluon at small x can induce negative quark distributions at small x and larger Q through QCD evolution. This happens to the MRST conservative PDF set, for instance, at Q 2 = 5 GeV 2 for x < 0.00015. become negative. These negative gluon solutions must therefore be considered as unphysical (at this order of α s ).
Another possible source of discrepancy between the CTEQ and MRST results is the treatment of heavy quark mass effects in DIS. We primarily use the widely used zero-mass M S formalism, whereas MRST uses the Thorne-Roberts prescription for the non-zero-charmmass variable flavor number scheme. In general, the difference is nominally small. However, since charm production accounts for nearly 25% of the DIS cross section at small x, and since the HERA data at small x have very small errors, the difference could have an effect on the issues discussed above, because the R-T prescription results in an unusual behavior of the charm structure function just above the threshold. Further study is clearly needed to settle fully the stability issue.
We should add that, independent of the stability of NLO global analysis, NNLO calculations are needed for processes that require a high level of accuracy, or that are known to have large corrections.

Precision W/Z Differential Distributions as Input to PDF Analysis
The differential cross section for W/Z production d 2 σ / dy dp T (or, more practically, the cross section d 2 σ / dy dp T for one of the decay leptons in the semi-leptonic decay channel) is sensitive to details of PDFs. Precise data on these cross sections can play a decisive role in narrowing the uncertainties and clarifying many of the open issues on PDFs described in the first part of this review. This is because: (i) they measure completely different combinations of PDFs, thus provide constraints on many independent quantities not accessible in DIS experiments; and (ii) the kinematic coverage of the collider cross section data will greatly expand that of available DIS data. It is particularly important that the W/Z cross sections be measured as precisely, and in as wide a kinematic range, as is possible at the Tevatron, in order to determine the PDFs well enough to enable better predictions, hence improved discovery potentials, at the LHC.
The Tevatron and LHC are W/Z factories. The reason that their potential for contributing to the next generation of global QCD analysis has so far not attracted much attention has perhaps to do with the fact that the measured cross sections, involving convolutions of the products of two PDFs, do not depend on the PDFs in as direct a way as the structure functions of DIS scattering. Thus, it is difficult to highlight which measurement will determine what particular features of PDFs. But, since most of the open issues in current PDF analysis concern subdominant effects, the more subtle role to be played by precision W/Z data will be both natural and vital. Instead of looking at LO parton formulas for motivation to focus on certain measurements, we now need detailed phenomenological studies of the effects of various measurements on the remaining uncertainties of PDFs in the global analysis context, utilizing the new tools developed in recent years, such as the Lagrange multiplier method. Efforts of this kind have not yet been systematically carried out; but are crucial for the success of the Tevatron and LHC physics programs.
Many of the same comments apply to cross sections of W/Z plus jets, for which there will also be abundant data. A great deal of theoretical work has been done recently on this subject. Since the definition of jets, and related issues also come into play, the detailed study of this process will probably concern less about PDF analysis, and a lot more with new physics discovery backgrounds.

W Mass Measurement
The precise determination of the W mass is one of the key measurements at hadron colliders. One of the main uncertainties of this measurement is the error attributable to PDFs. In spite of a great of deal of effort, current estimates are still not well founded. Detailed studies of the kind described in the previous section can help expose the effects due to uncertainties in regions of the PDF parameter space that have not been included in previous estimates, as well as identify new measurements, prior to the full mass measurement analysis, that can reduce the uncertainties. A concerted effort by theorists and experimentalists working together is essential in this endeavor, because the task involves a strong interplay between theoretical and experimental considerations.

W/Z Plus Tagged Heavy Flavor Production and Heavy Quark Distributions
One area where hadron collider measurements can, in principle, make clearly defined contributions to the parton structure of the nucleon is heavy quark distributions, c(x), b(x). The relevant processes involve the production of a W/Z boson with an associated heavy flavor meson, as illustrated here: These final states are very difficult to measure reliably and precisely. But they do provide direct handles on the heavy quark distributions of the nucleon that are not available otherwise. Hence, it is a very important challenge that needs to be met.

Predictions for Top Production at Hadron Colliders
The quantitative study of the top quark properties and its production cross sections is a high priority both at the Tevatron and at LHC. The uncertainties of PQCD predictions on the tt production cross section have been studied extensively. As an example, some of the results of the detailed study of [15] are captured in the following table.
CTEQ PDF uncertainties MRST PDF uncertainties MRST PDF uncertainties + scale dependence The first row shows the range of predicted cross sections associated with uncertainties of PDFs due to experimental inputs, as given by CTEQ and MRST. The second row shows the range due to the variation in α s , with associated change in PDFs, as given by MRST.
The anticipated measurement of single top production will reveal valuable information on the top coupling to gauge bosons, among other things. The production mechanism, involving an intricate interplay between gW →bt and bW → t subprocesses, provides an important forum to study the PQCD physics of heavy quark partons. Although the basic theory behind this physics has, by now, been well-established [8,16], there are many implementation issues, as well as the general lack of information on the b distribution (cf. previous section on heavy quark PDFs), that make this process an extremely interesting one to study at both hadron colliders.

Predictions for Higgs Production at Hadron Colliders
The intense interest in discovering the Higgs particle, whether within the SM or beyond, also underlines the importance of having precise knowledge on PDFs. The many relevant issues will be explored in detail in other sessions of this workshop. We will only quote one example: the uncertainty of the predicted cross section due to the subprocesses gg → HX is clearly tied to the (rather large) uncertainty on the gluon distribution, which we discussed earlier. This uncertainty is larger at the Tevatron than at LHC, mainly because of the different x-range involved. The results of [17], comparing the predictions of three recent PDFs, are shown below.

Conclusions
The importance of precise knowledge of PDFs for calculating SM processes and making predictions of New Physics discoveries at the hadron colliders has been well recognized. In reviewing this subject, we emphasize that, in spite of the steady progress made in global QCD analysis, there are still many gaps in our understanding of the parton structure of the nucleon. Continued advances to fill these gaps will require a sustained effort by the whole HEP community.
There are many measurements at the Tevatron and LHC that can help. For instance: • many processes at the high energy colliders are dominated by gluon initiated subprocesses.
Better data on these processes (such as jet cross sections, heavy quark production, ... etc.) will provide much needed constraints on the not-so-well-determined gluon distribution; • since the colliders are W/Z factories, precisely measured rapidity and transverse momentum distributions can provide constraints on the quark distributions that are complementary to those afforded by DIS data. For instance, the W ± rapidity distributions at the LHC can differentiate between u(x) and d(x) without the uncertainties of nuclear corrections that beset current analyses that use deuteron targets; • W/Z plus heavy flavor channels provide unique handles on heavy quark distributions, which are, so far, virtually untested. All these can have significant impact on the precision measurement of the W mass and top quark parameters, as well as on the discovery potentials for Higgs and New Physics signals.