1. Heavy Quark Mass Effects and Heavy Flavor Parton Distributions DIS06TsukubaTung

2. Outline The importance of studying the Heavy Flavor Sector of the Parton Structure of the Nucleon PQCD calculation including quark mass effects:  General-Mass (GM) formalism (Collins);  A coherent and efficient implementation of the GM formalism, emphasizing the simplicity of the physical effects of non-zero quark masses;  Numerical results and comparisons, including F L. Applications: (i) precision global analysis (cf. talk in SF group); (ii) how well do current experimental data constrain the charm distribution of the proton? Outlook Collaborators: Belyaev, Lai, Pumplin, Stump, Yuan Thanks to: Collins, Kretzer, Olness, Schmidt

3. The General Mass (GM) PQCD Formalism Factorization Thm (valid order-by-order, to all orders of PQCD) Conventional proofs: assume Zero Mass (ZM) partons; Collins (85-97): FacThm proof is independent of the mass parameters in PQCD—General Mass (GM) case. aa  H.Order The Parton Picture: is based on:

4. Total Inclusive Structure Functions F tot — Forward Compton Amp.:  final State parton flavors * Solid lines: heavy quark; * dashed line: light quark; * subtraction terms to remove overlap are not explicitly shown Only diagrams with HQ lines are shown.  (Q) mCmC 3-flavor scheme 4-flavor scheme LONLO NNLO O(  s 0 ) terms with light quarks only O(  s 1 ) terms with light quarks only F tot O(  s 2 ) terms with light quarks only + …

5. New Implementation of the GM Formalism: F tot Some fine prints: Subtraction terms to remove overlap are not explicitly shown; Slightly different treatment of mass dependence in NC and CC cases. SACOT   mCmC 3-flavor scheme 4-flavor scheme O(  s 0 ) O(  s 0 ) terms with light quarks only O(  s 1 ) terms with light quarks only Summing Initial S. partons  Variable-flavor # schemes (3,4,5: depends on Q) Summing Final S. partons  All flavors allowed by P.S. Kinematic Constraints:  Phase space integration limits;  Rescaling—smooth and physical threshold behavior CC (Barnett) NC (Acot  ) Wilson Coefficients:  Simplified ACOT (initial state parton mass  0)— more natural parton kinematics and greatly simplified W.C. O(  s 1 )

6. When and where do mass effects matter? In the kinematic phase space:  When  is different from x, and where f(x,Q) is steep in x  (slides) For Physics quantities that vanish in the zero-mass limit, such as LO F longitudinal.  (slides) In real-life precision phenomenology:  Certain HERA data sets—in the low Q 2 region (H1NCe+9697X and ZeusNCe+9697X) (slides) NC: Kretzer, Schmidt, wkt (cf. CC: Barnett)

7. Comparison of GM and ZM Calculations: where in the (x,Q) plane do the differences matter? F 2 (x,Q) GM ZM low Q 2 mostly

8. Comparison of GM and ZM Calculations: where in the (x,Q) plane do the differences matter? F L (x,Q) GM ZM low Q 2 mostly Ignore: top mass effect

9. The Longitudinal Structure Function For the zero mass case, F L = 0 at LO.  This is a good place to look for and test mass effects (and NLO effect—e.g. measure the gluon). Results:  F L (x,Q)/F L (x,Q) | M/=0 vs. M=0 (previous slide); F L (x,Q) vs. x for fixed Q; F L (x,Q) vs. Q for fixed Q; Comparison with data on reduced cross sections;

10. x-dependence of F long at fixed Q Positive definite, as physical observables should be; Smooth dependence; Increase at small x due to growth of NLO contribution from the gluon distribution

11. Q-dependence of F long for several x values Continuity across both the charm and the bottom quark thresholds (where the flavor # increase by 1). This is guaranteed by the ACOT formalism + the correct treatment of kinematics à la the Acot  rescaling.

12. F 2 —Heavy Flavor Threshold Behavior * Smooth behavior across both charm and bottom thresholds

13. 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

14. Remarks on the new implementation of the GM formalism Physical quantities, such as F 2, F L, are positive definite and smooth across heavy flavor thresholds. Simplicity of the general formalism + the new implementation (including proper kinematics) underpin these results. This combination has proven to be very successful as the basis of a comprehensive new Global Analysis, including all relevant HERA I data. (Cf. talk in Session 1 of the SF group.) F L contributes to the xSec formula, especially at large y. The success of the precision Global Analysis (including the large y data) implies a good fit of theory to F L ; Dedicated fits to F L can be done, but, in principle, does not add anything, since the SFs are extracted from the xSec data in the first place (often with additional assumptions or approximations). This will be looked into in much greater detail. “Less is More?”

15. Application of the New Implementation of the GM calculation, (in addition to global analysis (cf. session 1 of SF group) First phenomenological study of the heavy flavor parton distributions:  Is there room for intrinsic charm in the nucleon?  If yes, how much?

16. 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 still 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) …

17. 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 you-know-who!). (A hybrid model is also possible, but clearly there is not enough experimental constraints yet to warrant a separate study.) 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.

18. A little bit of detail (that is going to be asked anyway) Since current experimental constraints are rather loose, we must limit the new degrees of freedom: For the sea-like scenario, assume the shape of c(x,Q 0 ) is the same as s(x,Q 0 ) and only vary the normalization; For the light-cone model scenario, take the shape of c(x,Q 0 ) to be that of Brodsky etal, and only vary the normalization. Preliminary results on the non- perturbative charm content of the nucleon …

19. Goodness-of-fit vs. input non-perturbative Charm momentum fraction  (charm mom. frac.) The appropriate value for   in the current global analysis environment has not yet been investigated. Hence, the value for the allowed charm mom. frac. should be taken as indicative only.  

20. Parton Distributions in the presence of a non- zero component of charm Charm Distribution  @ Q 0, Q 2 = 10 GeV, Q = 85 GeV & for Scenarios B— light-cone like charm component. Gluon Distribution  (same as above) Not shown due to lack of space:  Strange Distribution;  Ubar+dbar Distribution. (these can be affected by the charm content in scenario A—the sea-like input charm tied to the light flavors)

21. 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.

22. 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.

23. 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.

24. Outlook This is just the beginning. Looking forward to more comprehensive and accurate data from HERA II With W/Z/  + tagged heavy flavor events at the hadron colliders, we can get direct information on s/c/b quark distributions; c-quark and b-quark are important phenomenologically in the physics program at LHC for exploring beyond the SM scenarios.

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26. Outline The importance of studying the Heavy Flavor Sector of the Parton Structure of the Nucleon PQCD calculation including quark mass effects:  General-Mass (GM) formalism (Collins);  A coherent and efficient implementation of the GM formalism, emphasizing the simplicity of the physical effects of non-zero quark masses;  Numerical results and comparisons, including F L. Applications: (i) precision global analysis (cf. talk in SF group); (ii) how well do current experimental data constrain the charm distribution of the proton? It is interesting that a reason able limit can be estimated already: < 0.015

27. Extras

28. Comments on NNLO In the perturbative approach, for the total inclusive S.F.s and cross sections, once a comprehensive NLO calculation is in place, it is straightforward to include known NNLO corrections additively. However, one needs to realize that, unlike total inclusive F 2,L, quantities such as ”F 2 c ” is not well defined theoretically at NNLO and beyond. (It is not infra-red safe!) It is rather misleading to talk about a true “NNLO theory” of F 2 c (except within the 3-flv scheme, which has only a limited range of applicability). Extending global analysis to NNLO is certainly desirable, but not necessarily urgent for current applications (cf. excellent global fits), since experimental errors for most measured quantities, as well as other sources of uncertainties (such as parametrization, power-law corrections …), largely outweigh the NNLO corrections.