1. Parton Structure of the Nucleon: Progress & Challenges Wu-Ki Tung Michigan State Univ. & Univ. of Washington

2. Global QCD Analysis of the Parton Structure of the Nucleon (unpolarized) Current Status and Challenges Recent Developments and Advances  Experimental HERA Fixed Target (NuTeV, E866, Jlab)  Theoretical: NNLO and Resummation Consistent Treatment of Quark Mass Effects “New” Approaches to Global Analysis Outlook

3. universal, extracted by applying global analysis methods Theoretical Calculations Recent Progress on all three fronts

4. Challenges for Global QCD Analysis of Nucleon Parton Distributions—from here to LHC Gluon Distribution; Small-x and Large-x behavior of all distributions; Strange distribution; Charm and bottom distributions; Quantifying uncertainties of all PDFs. In spite of steady progress in over 20 years of global analysis of PDFs, it is surprising how much knowledge is still missing on the parton structure of the nucleon ! The future of HEP, particularly the success of the Tevatron Run II and the LHC physics programs, depends on making substantial improvements on most of these fronts.

5. Estimated Uncertainties of PDFs: CTEQ6 by an iterative Hessian method, using orthonormal eigenvector sets Q 2 = 10 GeV 2 Theory uncertainties not explicitly included; but some allowance is made in choosing the tolerance criteria. u-quark d-quark gluon

6. New Experimental Input to Current Global Analysis Extensive HERA I data sets (complete?) on  total inclusive NC and CC cross sections, covering a wide range of kinematic phase space;  semi-inclusive (tagged heavy flavor) cross sections: charm and bottom;  semi-inclusive jet cross sections. (Note: out go the SFs, F 1,2,3 ; in come the xSec’s!) Fixed-target Experiments (Last of the kind?)  NuTeV DIS S.F.s and cross sections;  E866 DY pp and pd cross sections (finally?). New Tevatron Data on W/Z production, jet production, … etc.

7. Available HERA Data Sets for Current Global Analysis H1  CCe+9497X  CCe+9900X  CCe-9899X  NCe+9497X  NCe+9697X  NCe+9900X  NCe-9899X  NCe-9900X  NCe+9697F 2 c  NCe+9900Xc  NCe+9900Xb ZEUS  CC+9497X  CC+9900X  CC-9899X  NC+9697X  NC+9900X  NC-9899X  NC+9697F 2 c  NC+9890F 2 c HERA II data are beginning to emerge (DIS2006): (i) Polarized beam; (ii) Precision will improve considerably more. ep  p w CC ep  ep ,z NC HERA I

8. Progress on the Theoretical Foundation of Global QCD Analysis NNLO: Evolution kernels + Hard Scattering cross section calculations + Global Analysis; (Much publicized; but how important?) Resummations in multi-scale problems (small-x, large-x, p t, …); (Their days are finally arriving?) Consistent and efficient implementation of the general PQCD formalism with Heavy Quark masses** (  Precision Global Analysis with new Hera data.) Power-law Corrections (aka Higher-twist, Renormalons, … etc.; still too much unknowns.)

9. E866 “ The results imply that the u quark distributions in CTEQ6 and MRST2001 are overestimated as x  1. —hep-ex/0302019 These.. discrepancies.. imply that future PDF fits will see a substantial correction to the u and d quark distributions at large x.” Reports by E866 in subsequent conferences indicate that improved radiative corrections reduce the discrepancy, but it has not completely gone away.Cf. talk by Reimer at APS06, Dallas CTEQ studied possible sources of this discrepancy, particularly deuteron target corrections. Results were inconclusive.

10. F 2 Measurement Isoscalar ν-Fe F 2 NuTeV F 2 is compared with CCFR and CDHSW results - the line is a fit to NuTeV data All systematic uncertainties are included All data sets agree for x<0.4. At x>0.4 NuTeV agrees with CDHSW At x>0.4 NuTeV is systematically above CCFR Tzanov, DIS2005 Notable NuTeV Results

11. Attempts to Incorporate the NuTeV data in Global QCD Analysis Both CTEQ and MRST find this data set “incompatible” with the other data sets in the global analysis—with our usual assumptions; CTEQ tried to: (i) vary nuclear target corrections; (ii) include target and heavy quark mass effects, … None of these help; some make things worse; MRST find similar problems: their “best fit” with free parametrization of “nuclear correction factor” yields unrealistic shape of this factor; New CHORUS data agree more with CCFR than with NuTeV; Clearly this problem is very much an open one at present.

12. Can Nuclear Corrections Help? (CTEQ) For each x, data are combined and errors are weighted; See a systematic x- dependent deviation that cannot be reduced substantially by any reasonable nuclear correction models.

13. NNLO Calculations and Global Analysis ( Cf. talks & reviews at DIS06 ) Recent Milestone calculations ( many hundreds of papers. ):  3-loop evolution kernel;  Multi-jet final states;  2-loop hard cross sections (Wilson coefficients) for Rapidity differential cross section of DY process Higgs cross sections …  Many significant advances in calculational techniques and insight into new structures of perturbative amplitudes. NNLO Global Analysis is call of the day ( Alekhin, MRST )  How important, and necessary, is NNLO global analysis, given the uncertainties of current experiments and the limitations of current theory? Needs a critical look …

14. A close examination of NNLO Global Analysis of PDF’s Experimental considerations:  The most accurate DIS data have errors > 2-3%;  DY, W-, jet prod. data: stat. errors 5-10%, with even larger sys. errors in certain kinematic regions. Theoretical considerations:  Naïve error estimates (power-counting): LO ~ 10% ( µ  s ); NLO ~ 1% ( µ  s 2 ); NNLO ~ 0.1% ( µ  s 3 ) — too optimistic?;  Estimates based on scale-dependence are very much process-dependent, and are generally larger than the naïve estimate above;  Actual NNLO corrections are now known for DIS: MVV (Moch, Vermaseren and Vogt) How big are they?

15. Typical NNLO Results from Vogt, Moch and Vermaseren: divergence spurious: (0/0) 2% NNLO corrections 10 -3. Large corrections for x < 10 -3 due to x -1 ln n (x) terms in expansion.

16. Experimental considerations:  Most accurate DIS data have errors > 2-3%  DY, W-, jet prod. data: stat. errors 5-10%, with even larger sys. errors in certain kinematic regions. Theoretical considerations:  Naïve error estimates (power-counting): LO ~ 10% ( µ  s ); NLO ~ 1% ( µ  s 2 ); NNLO ~ 0.1% ( µ  s 3 ) — too optimistic;  Estimates based on scale-dependence are very much process-dependent, and generally larger than the above;  Actual NNLO corrections are now known for DIS: In light of these facts, what do we make of the necessity for NNLO Global Analysis? They are 10 -3 ; Corrections are significant (vs. exptl errors) only for x < 10 -3 —due to large x -1 ln n (x) terms, which would require small-x resummation to render results stable.

17. Typical motivation for NNLO Global Analysis MST 2006 Large NNLO corrections for x < 10 -3 mandate inclusion? Negative physical predictions, e.g. NLO F L for x < 10 -4, suggests trouble at this order; needs be cured by NNLO? Not necessarily. (per previous slide) NLO F L (x < 10 -4 ) < 0 more tied to g MST (x < 10 -4 ) < 0, not directly to NLO vs. NNLO. (CTEQ F L (x) stays > 0 for all x.) (Alekhin, MRST)

18. In summary, for global analysis … The NNLO results, where they are “large” compared to NLO ones (x < 10 -3 ), could either stabilize or de-stabilize global QCD analysis of PDFs, hence are not necessarily more reliable than the NLO results. When not large compared to NLO ones (x > 10 -3 ), they are also much smaller then the experimental uncertainties of the available data sets currently used for global analysis. In addition, NNLO Wilson coefficients for some processes used in global analysis (eg. inclusive jet cross section) don’t yet exist. Þ NNLO global analysis is a luxury—very expensive one computationally, at that—but not really a necessity. A rhetoric question for the spin community: now that NNLO polarized DIS coefficients are available (MVV), shouldn’t global analysis of spin-dependent PDFs be made at NNLO?

19. For global analysis of PDFs it makes more physics, and practical, sense to combine NLO PQCD with appropriate small-x resummation, rather than insist on NNLO for all x. For processes that have large NNLO Wilson coefficients (e.g. Higgs cross section), they should definitely be included—paired with the most precisely determined PDFs available (cf. above). Potential “mis-matches”? Due to the presence of multiple large- scales (, Q, M H, p T, …), PQCD is a many-facet formalism. Mixing orders are permissible if errors are smaller than that due to more significant physical effects included—strict rules (eg. order-matching) is too simplistic for multi-scale problems. Sensible approaches require flexibility based on understanding of the underlying physics issues. Witness the usefulness and successes of MC event generators that mix fixed order matrix elements with multiple radiation effects! NNLO calculations are theoretically important, but …

20. Progress on the theoretical treatment of Heavy Quark mass effects in global analysis

21. The Heavy Quark Parton Dichotomy Textbook PQCD: Heavy quarks (c,b,t) as zero-mass partons:  Parton Distributions (D-O/EHLQ/… /MRST/CTEQ)  Event Generators (Pythia/Herwig/ … )  Most calculations on high energy SM and New Physics signals and backgrounds in the literature. --- (ZM VFNS) Clearly not an appropriate description when Q ~ M. Theoretical Calculations of Heavy quark Production:  Assume M >> Q; treat HQ particle just like W/Z/Higgs;  E.g. for bottom prod.,  g  b b,  g  b b g, … (Fixed-flavor-number scheme, FFNS) Becomes singular (IR unsafe) as Q >> M (~  s n ln n (Q/M) ). - -

22. PQCD with HQ mass—Brief background Two key papers in Ancient History (1970’s): Heavy Quark parton: Witten; Composite Renormalization Scheme for heavy particles: Collins, Wilczek, Zee (CWZ) Heavy Quarks in Modern PQCD Applications: Basic Idea: Collins-wkt ’85 ; NLO realization of formalism: ACOT ’94 ; General Proof of factorization: Collins ’97. Partial implementation in Global Analyses CTEQ and MRST: 95 – 05 (two different variants) Full implementation in Global Analyses MST: Thorne: DIS05-06 ; CTEQ: WKT: DIS2006  

23. Physics of Heavy Quark Partons in General PQCD (ACOT) E.g. charm prod. in DIS gluon fusion Start with Q ~ M: basis mechanism is, 3-flavor scheme (no c parton)

24. At high energies Q >> M H,  s ln(Q/M) ~ 1 because of the colinear region of final state phase space: Physics of Heavy Quark Partons in General PQCD E.g. charm prod. in DIS gluon fusion colinear, infra-red unsafe start with Q ~ M PQCD breaks down! Must subtract these “singular” terms to restore IRS.

25. Q >> M H  s ln(Q/M)~1 Physics of Heavy Quark Partons in General PQCD E.g. charm prod. in DIS gluon fusion colinear, infra-red unsafe start with Q ~ M ^ IRS “NLO corr.” Must put it back in, and systematically control these singular terms: resum it together with similar unsafe terms to all orders, yielding the universal charm PDF! Can’t just arbitrarily throw away a term that one does not like ! Finite mass-dependence; no more singularities.

26. Q >> M H  s ln(Q/M)~1 Physics of Heavy Quark Partons in General PQCD E.g. charm prod. in DIS start with Q ~ M ^ NLO corr. resum to all orders:  c PDF LO c prod. at high E 4-flavor scheme (with c parton, and c mass in final state for correct kinmatics) How does the resummed calculation do back in the threshold region (where we started)?

27. 0 + O(  s 2 ) Q ~ M H ln(Q/M)~1 not large “LO” c prod. ~ threshold Physics of Heavy Quark Partons in General PQCD E.g. charm prod. in DIS Return to 3-flavor scheme result, with some higher order corrections. We arrive at a general formalism that is valid over the entire range of Q, from threshold to high energies!

28. Implementation of the General Formalism Needs more than just the formalism to make reliable calculations:  Choices to be made within PQCD: Renorm. & fact. scales; matching/transition points for the composite scheme calculation;  Proper (and consistent) treatment of the kinematics of light/heavy partons, as well as produced hadrons. Incremental improvements make, ’94 – now. New elements of our 2006 implementation:  Proper differentiation between incoming parton (scheme-dep) and final-state HQ (on mass shell);  “Physical” treatment of kinematics—important to introduce rescaling variable. The new package is comprehensive, consistent, and surprisingly simple (hence efficient).

29. Applications of the New Implementation of the GM calculation In conjunction with the comprehensive HERA I data (+ Fixed Target and Hadron Collider data), the new GM calculation  Precision global QCD analysis of PDFs First phenomenological study of the heavy flavor parton distributions:  Is there room for intrinsic charm in the nucleon?  If yes, how much? Collaborators: Belyaev, Lai, Pumplin, Stump, Yuan (MSU)

30. New Precision Global Analysis Excellent fit to 32 sets of data—CTEQ6C0; (representative plots to follow.) Do mass effects matter in the global analysis? Yes:  H1NCe+9697X, ZeusNCe+9697X — low Q 2 data.  For the longitudinal structure function F L (x,Q), it matters a lot, since F L (x,Q)| LO QCD = 0.

31. NC e+ 96-97 X (  -exchange) H1 ZEUS

32. H1 Zeus NC e+ 99-00 X (  *-exchange and  *-Z interference)

33. Pull Plots: NC e+ 99-00 X H1 Zeus Close to normal distribution.

34. Lines: theory (fits); Red points: raw data points; Blue points: data points shifted by optimal correlated SysErr. (usually within 1  ) H1 CC e+ 99-00 X Comparison to CC data (W-exchange)

35. CC e+ 99-00X ZEUS

36. GM global analysis and HERA I Charm Production data H1 NC e + p 96-97 F 2 c Zeus NC e + p 98-00 F 2 c Zeus NC e + p 96-97 F 2 c H1 NC e + p 99-00 X

37. The Charm Content of the Nucleon Conventional global analysis assume that heavy flavor partons are exclusively generated “radiatively”, i.e. by gluon splitting. This assumption/ansatz more or less agrees with existing data on production of charm. “More or less” since: (i) experimentally, errors on data are large; and (ii) theoretically, the ansatz is ambiguous: at what scale does the radiation start? Why should we care about c(x,Q)?  Intrinsic interest: the structure of the nucleon;  Practical interests: collider phenomenology, especially beyond the SM, e.g. Charged Higgs production, c + s-bar --> H + ; Single top production in DIS (flavor-changing NC) …

38. Is there a non-perturbative charm component in the nucleon; and if so, how big can it be? Theoretical preconceptions aside, let nature speaks for herself: Perform unbiased global analysis, allowing charm to have its own degrees of freedom, in two scenarios:  A sea-like component at some initial scale Q 0 ;  A light-cone model component (centered at moderate x)—aka “intrinsic charm” (championed by Brodsky.). Method: (i) For various assumed input charm c(x,Q 0 ), do independent global fits, and compare the resulting goodness-of-fit,  2 global ; (ii) Define the range of allowed c(x,Q 0 ) by the currently used  2 global for defining PDF uncertainties.

39. Goodness-of-fit (  2 ) vs. input non-perturbative Charm momentum fraction  (charm mom. frac.) The value for the allowed charm momentum fraction is found to be ~ 0 – 0.018.  

40. Charm and Gluon Distributions at Q = 1.3 GeV Horizontal axis is scaled in x 1/3 —inbetween linear and log— in order to exhibit the behavior at both large and small x. Varying amounts of input lightcone charm components (à la Brodsky etal.) : Momentum frac. at Q 0 = 0 — 0.02.

41. Charm and Gluon Distributions at Q 2 = 10 GeV 2 * Two-component charm distr. is apparent! (The radiatively generated component is represented by C6C0l (black) curve. Varying amounts of input lightcone charm components (à la Brodsky etal.) : Momentum frac. at Q0 = 0 — 0.02.

42. Charm and Gluon Distributions at Q 2 = (85 GeV) 2 * Very substantial amount of charm, over the radiatively generated component (C6C0l), still persists at this very large scale  there can be interesting phenomenological consequences even at LHC. Varying amounts of input lightcone charm components (à la Brodsky etal.) : Momentum frac. at Q0 = 0 — 0.02.

43. Summary and Outlook The impressive consistency between the improved theoretical calculation and much improved experimental input on DIS NC, CC & heavy flavor production (and other F.T. and hadron collider processes) provides a new basis for performing precision phenomenology within and beyond the SM. A lot of work remains to be done to pin down the full parton structure of the nucleon (particularly gluon, s, c, b); HERA II and Tevatron Run II data can contribute substantially to fill the gaps. More specifically, With more accurate data on CC cross sections, we gain additional (clean) handles for differentiating up and down types of quarks; Direct F Long measurement in the cards? With W/Z/  + tagged heavy flavor events at the hadron colliders, we can get direct information on s/c/b quark distributions; …. LHC physics is waiting for these advances …