1. Simona Malace University of South Carolina

2. Overview  Standard pQCD fits and their limitations (example => CTEQ6)  Another kind of QCD fits: extension of fits at larger x in the nonperturbative region (example => Alekhin)  Can we go even further? Quark-hadron duality: => experimental observation & working hypothesis for PDFs extension at large x => recent results from Jlab on quark-hadron duality in the F 2 p,d structure function => Quark-hadron duality in F 2 n  Plans for future

3. Complete picture (or closer to …) Naive picture Operator Product Expansion in pQCD: leading-twist higher-twist The quark and gluon structure of the Proton in QCD perturbative ln(Q 2 ) corrections nonperturbative corrections

4. How does it compare to data?  Very good, where only the leading twist is expected to contribute  Most cases, parton distribution functions (PDFs) are extracted from data from “safe kinematic regions”, only (no nonperturbative effects)  What is the price to pay? A: Unconstrained PDFs outside the “safe kinematic regions” Let’s see why….

5. 234234 PDFs Extraction in pQCD  Two basic ideas of QCD: Factorization: separate the long-distance from short-distance dependence perturbative nonperturbative input (PDF) Evolution: knowledge of implies knowledge of ……………at all Q 2 > ( DGLAP equations splitting functions

6. 234234 PDFs Extraction in pQCD: Recap  Three basic quantities needed for pQCD calculation of F 2 : Computed perturbatively as power series in  s  Examples of parameterizations for nonperturbative input: Only requirements: flexible enough to accommodate small/large x behavior + obey the sum rules Q 2 evolution of PDF calculated via DGLAP equations x dependence of PDF assumed and constrained by data CTEQ6: MSTW: Alekhin:  To constrain the x dependence is evolved to all Q 2 where data exist in the “safe kinematic regions”

7. 234234 Standard pQCD fits: PDFs from CTEQ6  CTEQ6: pQCD fit to hard scattering and DIS data with Q 2 > 4 GeV 2 and W 2 > 12.25 GeV 2 ; the x dependence of PDFs parameterized at Q 2 = 1.3 GeV 2 ; evolution up to NLO JHEP 0207:012, 2002

8. 234234 CTEQ6: Comparison to Data  Good fit to data in the “safe kinematic regions” but beyond …  /ndf = 1.1  /ndf = 1.52

9. 234234 CTEQ6: Large Uncertainties at Large x  Large x region important for (see Alberto’s talk): - study the mechanism of spin-flavor symmetry breaking in valence …..quark distributions - determining high-energy cross sections at collider energies - quantification of quark-hadron duality, etc. but what’s involved in extending PDFs validity to larger x?  Large uncertainties where there are no constraints from data

10. 234234 Complete picture  Corrections beyond leading twist PDFs at Large x and low Q 2 1)Higher-Twists: kinematic and dynamical Kinematic HT – associated to twist-2 operator => no additional information on the quark dynamics Dynamical HT – contains information about the valence quarks dynamics (confinament) 2) Large-x resummation 3) Nuclear Corrections – for the neutron  Messy but needs to be done to achieve exhaustive knowledge of the dynamic of the nucleon! 4) …

11. 234234  Stepping out of the “safe kinematic region” => inclusion of nonperturbative effects (TMC, HT) (and nuclear effects for nuclear targets) Example: PDFs from ALEKHIN Phys. Rev. D 68, 014002 (2003); JETP Lett. 82, 628 (2005)  Extension of PDF fits to larger x: kinematic cuts (W 2,Q 2,x,) are relaxed to provide more constraints from data ALEKHIN CTEQ6  The x dependence of PDFs parameterized at Q 2 = 9 GeV 2 ; evolution up to NNLO

12. 234234 Uncertainties: Alekhin vs CTEQ6  Result: smaller uncertainties at large x Relative experimental uncertainties of PDFs at a Q 2 of 9 GeV 2 : full = Alekhin; dotted = CTEQ6 Reduction by ~ 10 of d uncertainty at large x Reduction by ~ 4 of u uncertainty at large x Phys. Rev. D 68, 014002 (2003)

13. 234234 Dynamical Higher Twist  Interplay of Higher-Order QCD corrections and dynamical Higher Twists  Decrease of magnitude of HT with increase of pQCD order but HT don’t vanish in NNLO  HT contribution to F 2 : at most ~10% of Leading Twist (maximal at x~0.6 and Q 2 = 5 GeV 2 ) order  S LO0.1301 +/- 0.0026 NLO0.1171 +/- 0.0015 NNLO0.1143 +/- 0.0014  From extrapolation: HT not expected to vanish in NNNLO either Phys. Rev. D 68, 014002 (2003)

14. 234234 How about extending PDFs to even large x? 2 nd resonance region at Q 2 = 2 GeV 2 2 nd resonance region at Q 2 = 5 GeV 2 Q 2 = 2 GeV 2 Q 2 = 5 GeV 2  Extending to larger x at finite Q 2 => encounter the resonance region  Resonances are basically “made” of higher twists  The contribution of higher-twist terms in the resonance region would be expected to be large… Or is it?

15. 234234 Bloom-Gilman Duality  The resonance region data: - oscillate around the scaling curve - are on average equivalent to the scaling curve. - “slide” along the deep inelastic curve with increasing Q 2  Quantitatively: comparing the lhs to the rhs, relative difference 10% for Q 2 =1 GeV 2 to <2% for Q 2 =2 GeV 2. Phys. Rev. Lett. 25, 1140 (1970) “… resonances are not a separate entity but are an intrinsic part of the scaling behavior of W 2 …” Yes, but not on average

16. 234234 Duality in QCD W 2 Q 2 = 1 GeV 2 Q 2 = 3 GeV 2 Q 2 = 5 GeV 2 data pQCD  De Rujula, Georgi, Politzer: “ The most intriguing aspects of SLAC data on inclusive electroproduction are precocious scaling and local duality ” Phys. Lett. B 64, 428 (1976)  Duality = higher-twists are either small or cancel on average (on average, the interactions between the valence quarks are suppressed) Operator product expansion: Mellin transform of twist-2 pQCD calculation of W 2 twist-2  On average, the resonance region data mimic the twist-2 pQCD calculation

17. 234234 Quark-Hadron Duality in F 2 : Recent Experiments at JLab Jefferson Lab Electron-beam accelerator As of now, beam energies up to 6 GeV As of now, three experimental halls: A, B, C Two spectrometers: HMS & SOS

18. 234234  1996 JLab-96 (I. Niculescu): duality dedicated experiment; measures H(e,e’) & D(e,e’) cross sections  1998 E94-110 (Y. Liang): performs Rosenbluth separation (measures R =  L /  T ); measures H(e,e’) cross sections  2003 E00-116 (S. Malace): duality dedicated experiment, push to larger x and Q 2; measures H(e,e’) & D(e,e’) cross sections Inclusive Resonance Region Measurements in Hall C Among other, three experiments: JLab-96, E94-110, E00-116 Kinematics covered: x between ~0.3 and 0.9, Q 2 up to 7 GeV 2, in the resonance region (mainly) JLab-96 E94-110 E00-116 CTEQ6 ALEKHIN

19. 234234 Procedure for F 2 extraction  Differential one-photon exchange (Born) cross section  F 2 extraction requires the knowledge of cross section and R E94-110: measured R JLab-96: used R from E94-110 E00-116: used R from R1998 (R < 0.2 @ E00-116 kinematics) Experimental natural variables: momentum and angle of scattered electron + energy of incoming electron x, Q 2, W 2 crap

20. 234234 Physics Results from JLab-96  Verifying quark-hadron duality “a la Bloom-Gilman” NMC fit to DIS data at the same  but higher W 2, Q 2 than RES data  The new precision data display the signature oscillation around the DIS curve (the agreement, on average, better than 10%)  JLab-96 conclusively verifies the observations of Bloom and Gilman

21. 234234 Physics Results from JLab-96  Verifying quark-hadron duality in a pQCD framework: analysis in fixed W 2 bins Averaged RES data pQCD(NLO) pQCD(NLO)+TMC large-x resummation: brings pQCD calculation in better agreement with data  TMC significant effect: pQCD calculation in better agreement with data  LxR: resummation on ln(1-z) in x space => Q 2 scale replaced by Q 2 (1-z)/z  HT: in RES region similar to those for W 2 > 10, with exception of 

22. 234234 Physics Results from E94-110  More precise data from JLab: the resonances average to pQCD+TMC calculations from CTEQ and MRST  The resonance data slide with increasing Q 2 to higher x always following the pQCD curves  The ratio of F 2 integrals data to pQCD better than 5% at Q 2 = 0.5 GeV 2 but ~ 18% at Q 2 = 3.5 GeV 2 ?!?  Violation of duality, unconstrained PDFs at large x, something else ? inegrals over entire RES region

23. 234234 Physics Results from E00-116 1 st 2 st 2 nd 2 st 3 rd 4 th DIS  Verify quark-hadron duality at higher Q 2 Region W min W max 1 st 1.3 1.9 2 nd 1.9 2.5 3 rd 2.5 3.1 4 th 3.1 3.9 DIS 3.9 4.5 Calculate: Define: Data from E00-116, E94-110, JLab-96 and SLAC; parametrizations from CTEQ6, MRST, ALEKHIN Compare: Large discrepancies in the description of F 2 at large x

24. 234234 Physics Results from E00-116 Comparison: data [H(e,e’)] to CTEQ6M (NLO) + TM  I ~ 1 at Q 2 ~ 1.5 GeV 2 then rises with increasing Q 2 and reaches a plateau at ~ 4 GeV 2 ; above this value Q 2 dependence saturates  This behavior displayed when integrating globally and locally except for first resonance. Not a failure of pQCD in describing the Q 2 evolution but a paucity in the strength of PDFs at large x  I becomes constant at different value for each RES region Related to growing uncertainty of PDFs strength at large x Phys. Rev. C 80, 035207 2009

25. 234234 Physics Results from E00-116 Comparison: data [H(e,e’)] to MRST04 (NNLO) + TM  The observed Q 2 dependence of I yields similar conclusions as drawn from the CTEQ6 Not surprising: the extraction procedure (and kinematic cuts) of PDFs similar for MRST04 and CTEQ6 Differences:  MRST04 undershoots the data by an even larger amount and I saturates at a larger value of Q 2 than for CTEQ6 Possibly results from the difference in modeling the x dependence of PDFs (?) Phys. Rev. C 80, 035207 2009

26. 234234 Physics Results from E00-116 Comparison: data [H(e,e’)] to ALEKHIN (NNLO) + HT + TM  Due to cuts employed for data selection, Alekhin’s fits far better constrained at large x  For the 4 th RES region and DIS, I very close to 1 for entire Q 2 range analyzed  Good agreement for 3 rd and 2 nd RES regions: I deviates from 1 by about 5% HT in RES region, on average, differ by at most 5% from those extracted by Alekhin  1 st resonance in disagreement with Alekhin’s fit: the validity of the fit questionable at these kinematics  Averaged RES data could be used to constrain PDF fits Phys. Rev. C 80, 035207 2009

27. 234234 Physics Results from E00-116  Good description at Q 2 = 3,5 GeV 2 (except for largest x regime: 1 st RES)  Q 2 = 7 GeV 2 : probing the largest x regime (ALEKHIN least constrained) => growing discrepancy  Q 2 = 1 GeV 2 : discrepancy as x grows  reached limits of applicability ALEKHIN CTEQ6  Fails to describe x dependence of data  Better data description by ALEKHIN than CTEQ6

28. 234234 Physics Results from E00-116 Comparison: data [D(e,e’)] to CTEQ6 and ALEKHIN  F 2 d (ALEKHIN,CTEQ6) = F 2 p (ALEKHIN,CTEQ6) * d/p (from empirical fit)  The Q 2 dependence of I: similar characteristics as in the study of H(e,e’)  ALEKHIN offers better description of averaged RES data than CTEQ6

29. 234234 Is Quark-Hadron Duality Verified in the Proton?  Duality is an experimental observation and could be a working hypothesis for extending PDFs at large x => needs to be verified and quantified  It has been observed to work better 5% down to a Q 2 as low as 1 GeV 2 when compared to pQCD fits from MRST: E94-110  Surprisingly, it has been observed that the violation of duality becomes more pronounced as x and Q 2 increase  Our studies indicate that this increasing violation of duality with Q 2 is very likely only APPARENT: duality studies involve extrapolations of pQCD fits (unconstrained at large x)  The unconstrained PDFs at large x pose problems for quantifying how well duality holds in this kinematic regime (and that’s not good)

30. 234234 Extraction of the Neutron Structure Function F 2 n smearing functions (can be calculated from nuclear wave function)  Impulse Approximation (IA) – virtual photon scatters incoherently from individual nucleons  Beyond IA: nuclear shadowing, MEC, FSI, relativistic effects, off-shell corrections (most not addressed in present analysis) New method of extracting neutron SF from inclusive SFs of nuclei New method of extracting neutron SF from inclusive SFs of nuclei: employs iterative procedure of solving integral convolution equations ( Phys. Rev. C 79, 035205 2009 )  Can write the nuclear structure functions as convolutions of nucleon structure functions Present analysis does not attempt to provide a complete description of nuclear SFs (yet)

31. 234234 Extraction of F 2 n  In the Weak Bound Approximation (WKA): the deuteron SF is sum of smeared proton and neutron SF and an additive term to account for modifications of SF off-shell  The effective smeared neutron SF:  Need to solve equation: Method - Parameterize the nuclear corrections by an additive term  - F 2 n extracted using an iterative procedure which gives after first iteration assumed  Study sensitivity of extraction to: number of iterations, first guess for neutron SF etc.

32. 234234 Results from E00-116: Extraction of F 2 n  The resonances are obvious in the extracted F 2 n  After only two iterations F 2 d reconstructed from F 2 p data and extracted neutron F 2 n agrees well with the F 2 d data  The extracted F 2 n yields similar results after two iterations when different inputs are used [F 2 n(0) = F 2 p & F 2 n(0) = F 2 p /2]  Both F 2 p and F 2 n average to the QCD fit from Alekhin suggesting the onset of duality How well? Application of method to data ( Phys. Rev. Lett., xx, to be submitted )

33. 234234 Quark-Hadron Duality in F 2 n Compare integrals of neutron “data” to integrals of Alekhin’s newest fit ( arXiv:0908.2766, August 2009 )  Without HT: agreement at the level of 10-15% for Q 2 < 3 GeV 2 (except for  )   covers the highest x regime (the fit least constrained)  The discrepancy increases with increasing Q 2 (unconstrained PDFs at larger x?, …) … sounds familiar?  With HT: good agreement; deviation less than 10% in most cases

34. 234234 Neutron/Proton vs Q 2 & x  Good agreement between data and pQCD fits, except for  region which is somehow underestimated  The agreement slightly worsens as we go to larger Q 2 and x

35. 234234 Need more data at large x and “low” Q 2 ? We can help … To be proposed at the next PAC in January 2010: CTEQ6 ALEKHIN Measurements at 11 GeV @ JLab  Extend RES region and low W 2 DIS region measurements at even higher x and Q 2 at JLab  Systematic study of quark-hadron duality; extraction of dynamical HT (interesting in their own right); additional constraints for PDFs at large x; extract the neutron SF at even larger x (and maybe constrain the d quarks distribution better) …