1. Arie Bodek, Univ. of Rochester1 Modeling -(Nucleon/Nucleus) Cross Sections at all Energies - from the Few GeV to the Multi GeV Region Modeling of (e/  /  -(Nucleon/Nucleus) Cross Sections at all Energies - from the Few GeV to the Multi GeV Region Arie Bodek Univ. of Rochester and Un-Ki Yang Univ. of Chicago July 1-6, 2002 NuFact02 - 4th International Workshop Imperial College, London

2. Arie Bodek, Univ. of Rochester2 Arie Bodek- Univ. of Rochester NuFact02 - 4th International Workshop - Imperial College, London This presentation contains slides for two talks: (1)Working Group 3 - Non-Oscillation Neutrino Physics: Physics and Detector: Wednesday (3rd July, 2002) Session: Correlations Between HT and Higher-Order Perturbative QCD Corrections(Wed. 11:30-13:00) - Chairman: S. Kumano Talk 1- Wed. 11:30 AM : A. Bodek - Next-to-next-to-leading Order Fits with HT Corrections and Modeling (e/  /  Cross Sections from DIS to Resonance - (25 min + 5 min discussion)- Long talk (2) Working Group 2 - Neutrino Oscillation: Physics and Detector Thursday (4th July, 2002) - (Joint Session with WG3) Cross Sections, Detector Issues and Beam Systematics Issues Chairs: Kevin McFarland and Debbie Harris Session: Cross Sections: from Quasielastics to DIS (Thursday 10-11, 11:30-1) Talk 2 - Thu. 10:15 AM : A. Bodek- Modeling(e/  /  Cross Sections from DIS to Resonance Region 15 min: Short Talk - summary

3. Arie Bodek, Univ. of Rochester3 Neutrino cross sections at low energy?  Neutrino oscillation experiments (K2K, MINOS, CNGS, MiniBooNE, and future experiments with Superbeams at JHF,NUMI, CERN) are in the few GeV region Important to correctly model neutrino-nucleon and neutrino-nucleus reactions at 0.5 to 4 GeV region (essential for precise next generation neutrino oscillation experiments with super neutrino beams ) as well as at the 15-30 GeV region (for future  factories)  The very high energy region in neutrino-nucleon scatterings (50-300 GeV) is well understood at the few percent level in terms QCD and Parton Distributions Functions (PDFs) within the framework of the quark-parton model (data from a series of e/  / DIS experiments)  However, neutrino differential cross sections and final states in the few GeV region are poorly understood. ( especially, resonance and low Q 2 DIS contributions). In contrast, there is enormous amount of e-N data from SLAC and Jlab in this region.

4. Arie Bodek, Univ. of Rochester4 How are PDFs Extracted from High Q2 Deep Inelastic e/  / Data At high x, deuteron binding effects introduce an uncertainty in the d distribution extracted from F2d data (but not from the W asymmetry data). MRSR2 PDFs

5. Arie Bodek, Univ. of Rochester5 Neutrino cross sections Neutrino interactions  Quasi-Elastic / Elastic ( W=Mp) –  + n -->  - + p (x =1, W=Mp)  well measured and described by form factors (but need to account for Fermi Motion/binding effects in nucleus)  Bodek and Ritchie (Phys. Rev. D23, 1070 (1981 )  Resonance (low Q 2, W< 2)   + p -->  - + p +  Poorly measured and only 1st resonance described by Rein and Seghal  Deep Inelastic –  + p -->  - + X (high Q 2, W> 2) –well measured by high energy experiments and well described by quark-parton model (pQCD with NLO PDFs), but doesn’t work well at low Q 2 region. (e.g. JLAB data at Q 2 =0.22)  Issues at few GeV :  Resonance production and low Q 2 DIS contribution meet.  The challenge is to describe both processes at a given neutrino (or electron) energy. GRV94 LO 1 st resonance

6. Arie Bodek, Univ. of Rochester6 Building up a model for all Q 2 region..  Can we build up a model to describe all Q 2 region from high down to very low energies ?  [resonance, DIS, even photo production]  Advantage if we describe it in terms of the quark-parton model.  - then it is straightforward to convert charged-lepton scattering cross sections into neutrino cross section. (just matter of different couplings )  Understanding of high x PDFs at very low Q 2 ? - There is a of wealth SLAC, JLAB data, but it requires understanding of non- perturbative QCD effects. - Need better understanding of resonance scattering in terms of the quark-parton model? (duality works, many studies by JLAB) Challenges

7. Arie Bodek, Univ. of Rochester7 What are Higher Twist Effects- page 1  Higher Twist Effects are terms in the structure functions that behave like a power series in (1/Q 2 ) or [Q 2 /(Q 4 +A)],… (1/Q 4 ) etc….  While pQCD predicts terms in  s 2 ( ~1/[ln(Q 2 /  2 )] )…  s 4 etc…(i.e. LO, NLO, NNLO etc.)  In the few GeV region, the two power series cannot be distinguished Higher Twist: Interaction between interacting and Spectator quarks via gluon exchange at Low Q2 affects the Cross Section. Terms are like (1/Q 2 ) or [Q 2 /(Q 4 +A)],… (1/Q 4 ) In NNLO p-QCD more gluons emission affects the cross section like  s 2 ( ~1/[ln(Q 2 /  2 )] )…  s 4 Spectator quarks are not Involved.

8. Arie Bodek, Univ. of Rochester8 What are Higher Twist Effects - Page 2  Higher Twist Effects are terms in the structure functions that behave like a power series in (1/Q 2 ) or [Q 2 /(Q 4 +A)],… (1/Q 4 ) etc….  While pQCD predicts terms in  s 2 ( ~1/[ln(Q 2 /  2 )] )…  s 4 (i.e. LO, NLO, NNLO etc.) -- >In the few GeV region, the two power series cannot be distinguished  Nature has “evolved” the high Q 2 PDF from the low Q 2 PDF, therefore, the high Q 2 PDF include the information about the higher twists.  High Q 2 manifestations of higher twist/non perturbative effects include: difference between u and d, the difference between d-bar, u-bar and s-bar etc. High Q 2 PDFs “remember” the higher twists, which originate from the non-perturbative QCD terms.  Evolving back the high Q 2 PDFs to low Q 2 (e.g. NLO-QCD) and comparing to low Q 2 data is one way to check for the effects of higher order terms.  What do these higher twists come from?  Kinematic higher twist – initial state target mass binding (Mp), initial state and final state quark masses (e.g. charm production)  Dynamic higher twist – correlations between quarks in initial or final state.==> Examples : Initial or final state multiquark correlations: diquarks, elastic scattering, excitation of quarks to higher bound states e.g. resonance production, exchange of many gluons  Non-perturbative effects to satisfy gauge invariance and connection to photo- production [e.g. F 2 (Q 2 = 0) = 0]]

9. Arie Bodek, Univ. of Rochester9 Duality between fixed W and DIS  OLD Picture fixed W: Elastic Scattering, Resonance Production Electric and Magnetic Form Factors (G E and G M ) versus Q 2 measure size of object (the electric charge and magnetization distributions).  Elastic scattering W = M p = M, single nucleon in final state: Form factor measures size of nucleon. Matrix element squared | | 2 between initial and final state lepton plane waves. Which becomes: | | 2 q = k1 - k2 = momentum transfer G E (q) = INT{e i q. r  (r) d 3 r } = Electric form factor is the Fourier transform of the charge distribution. In the language of structure functions for example: 2xF 1 (x, Q 2 ) elastic = x 2 G M 2 elastic  x-1)  Resonance Production, W=M R, Measure transition form factor between a quark in the ground state and a quark in the first excited state. For the Delta 1.238 GeV first resonance, we have a Breit- Wigner instead of  x-1). 2xF 1 (x,Q 2 ) resonance ~ x 2 G M 2 Resonance transition BW  W-1.238) e +i k2. r e +i k1.r r Mp Mp q Mp MRMR e +i k1. r e +i k2. r

10. Arie Bodek, Univ. of Rochester10 Duality: Parton Model Pictures of Elastic and Resonance Production  Elastic Scattering, Resonance Production  Scatter from one quark with the correct parton momentum , and the two spectator are just right such that a final state interaction makes up a proton, or a resonance.  Elastic scattering W = M p = M, single nucleon in final state.  The scattering is from a quark with a very high value of , is such that one cannot produce a single pion in the final state and the final state interaction makes a proton.  Resonance Production, W=M R, e.g. delta 1.238 resonanceThe scattering is from a quark with a high value of , is such that that the final state interaction makes a low mass resonance.  Therefore, with the correct scaling variable, and if we account for low W and low Q2 higher twist effects, the prediction using QCD PDFs q ( , Q 2 ) should give an average of F2 in the elastic scattering and in the resonance region. (including both resonance and continuum contributions). If we modulate the PDFs with a final state interaction A(W, Q 2 Q 2 ) we could also reproduce the various Breit-Wigners. X= 1.0  =0.95 Mp Mp Mp q X= 0.95  =0.90 Mp MR MR

11. Arie Bodek, Univ. of Rochester11 Photo-production Limit Q 2 =0 Non-Perturbative - QCD evolution freezes  Photo-production Limit: Transverse Virtual and Real Photo- production cross sections must be equal at Q 2 =0.  There are no longitudinally polarized photons at Q 2 =0   -proton, ) = 0.112 mb F 2 (, Q 2 ) / Q 2 limit as Q 2 -->0  -proton, ) = 0.112 mb 2xF 1 (, Q 2 ) / Q 2 limit as Q 2 -->0  -proton, ) =  T (, Q 2 ) limit as Q 2 -->0. F 2 (, Q 2 ) ~ Q 2 / [Q 2 +C] --> 0 limit as Q 2 -->0  R (, Q 2 ) =  L /  T ~ Q 2 / [Q 2 +const] --> 0 limit as Q 2 -->0   If we want PDFs to work down to Q 2 = 0 they must be multiplied  by a factor Q 2 / [Q 2 +C] (where C is a small number).  In addition, the scaling variable x does not work since the photo- production cross section is a function of  Since at Q 2 = 0  F 2 (, Q 2 ) = F 2 (x, Q 2 ) with x = Q 2 /( 2M  reduces to one point x=0  However, a scaling variable  c  = (Q 2 +B) /( 2M  works at Q 2 = 0 since  F 2 (, Q 2 ) = F 2 ( , Q 2 )  becomes a a function of B/ (2M .

12. Arie Bodek, Univ. of Rochester12 How do we “measure” higher twist (HT)  Take a set of QCD PDF which were fit to high Q 2 (e/  /  data (in Leading Order-LO, or NLO, or NNLO)  Evolve to low Q2 (NNLO, NLO to Q 2 =1 GeV 2 ) (LO to Q 2 =0.24)  Include the “known” kinematic higher twist from initial target mass (proton mass) and final heavy quark masses (e.g. charm production).  Compare to low Q 2 data in the DIS region (e.g. SLAC)  The difference between data and QCD+target mass predictions is the extracted “effective” dynamic higher twists.  Describe the extracted “effective” dynamic higher twist within a specific HT model (e.g. QCD renormalons, or a purely empirical model).  Obviously - results will depend on the QCD order LO, NLO, NNLO (since in the 1 GeV region 1/Q 2 and 1/LnQ 2 are similar). In lower orders, the “effective higher twist” will also account for missing QCD higher order terms. The question is the relative size of the terms.  Studies in NLO  Yang and Bodek: Phys. Rev. Lett 82, 2467 (1999) ;ibid 84, 3456 (2000)  Studies in NNLO - Yang and Bodek: Eur. Phys. J. C13, 241 (2000)  Studies in LO - Bodek and Yang: hep-ex/0203009 (2002 )

13. Arie Bodek, Univ. of Rochester13 Lessons from previous “NLO QCD” study  Our previous studies of comparing NLO PDFs to DIS data: SLAC, NMC, and BCDMS e/  scattering data on H and D targets shows that.. [ref:Yang and Bodek: Phys. Rev. Lett 82, 2467 (1999) ]  Kinematic Higher Twist (target mass ) effects are large and important at large x, and must be included in the form of Georgi & Politzer  TM scaling.  Dynamic Higher Twist effects are smaller, but need to be included.  The ratio of d/u at high x must be increased if nuclear binding effects in the deuteron are taken into account.  The Very high x (=0.9) region - is described by NLO QCD  (if target mass and higher twist effects are included) to better than 10%  Resonance region: NLO pQCD + Target mass + Higher Twist describes average F2 in the resonance region (duality works)  Also, in a subsequent study in NNLO QCD we find that the “empirically measured Dynamic Higher Twist effects in the NLO study come from the missing NNLO QCD terms.  [ref: Yang and Bodek Eur. Phys. J. C13, 241 (2000) ]

14. Arie Bodek, Univ. of Rochester14 F 2, R comparison with QCD+TM vs. NLO QCD+TM+HT (use QCD Renormalon Model for HT) PDFs and QCD in NLO + TM + QCD Renormalon Model for Dynamic Higher Twist describe the F2 and R data reasonably well, with only 2 parameters

15. Arie Bodek, Univ. of Rochester15 F 2, R comparison with QCD-only vs. NLO QCD+TM+HT (use QCD Renormalon Model for HT) PDFs and QCD in NLO + TM + QCD Renormalon Model for Dynamic Higher Twist describe the F2 and R data reasonably well. TM Effects are LARGE

16. Arie Bodek, Univ. of Rochester16 F 2 comparison with QCD+TM vs. NLO QCD+TM+HT (use Empirical Model for Dynamic HT) PDFs and QCD in NLO + TM + Empirical Model for Dynamic Higher Twist describe the data for F2 (only) reasonably well with 3 paremeters Here we used an Empirical form for Dynamic HT. Three parameters a, b, c. F 2 theory (x,Q 2 ) = F 2 PQCD+TM [1+ h(x)/ Q 2 ] f(x) f(x) = floating factor, should be 1.0 if PDFs have the correct x dependence. h(x) = a (x b /(1-x) -c )

17. Arie Bodek, Univ. of Rochester17 Kinematic Higher-Twist (GP target mass:TM) Goergi and Politzer Phys. Rev. D14, 1829 (1976)  TM = { 2x / [1 + k ] } [1+ Mc 2 / Q 2 ] (last term only for heavy charm product) k= ( 1 +4x 2 M 2 / Q 2 ) 1/2 For Q 2 large (valence) F 2 =2  F 1 =  F 3 F 2 pQCD+TM (x,Q 2 ) =F 2 pQCD ( , Q 2 ) x 2 / [k 3  2 ] +J 1* (6M 2 x 3 / [Q 2 k 4 ] ) + J 2* (12M 4 x 4 / [Q 4 k 5 ] ) 2F 1 pQCD+TM (x,Q 2 ) =2F 1 pQCD ( , Q 2 ) x / [k  ] +J 1 * (2M 2 x 2 / [Q 2 k 2 ] ) + J 2* (4M 4 x 4 / [Q 4 k 5 ] ) F 3 pQCD+TM (x,Q 2 ) =F 3 pQCD ( , Q 2 ) x / [k 2  ] +J 1F3 * (4M 2 x 2 / [Q 2 k 3 ]) Ratio F 2 (pQCD+TM)/F 2 pQCD At very large x, factors of 2-50 increase at Q 2 =15 GeV 2

18. Arie Bodek, Univ. of Rochester18 Kinematic Higher-Twist (target mass:TM) The Target Mass Kinematic Higher Twist effects comes from the fact that the quarks are bound in the nucleon. They are important at low Q 2 and high x. They involve change in the scaling variable from x to  TM  and various kinematic factors and convolution integrals in terms of the PDFs for xF1, F 2 and xF 3 Above x=0.9, this effect is mostly explained by a simple rescaling in  TM  F 2 pQCD+TM (x, Q 2 ) =F 2 pQCD (  TM  Q 2  Compare complete Target-Mass calculation to simple rescaling in  TM Ratio F 2 (pQCD+TM)/F 2 pQCD Q 2 =15 GeV 2

19. Arie Bodek, Univ. of Rochester19 Dynamic Higher Twist Use: Renormalon QCD model of Webber&Dasgupta- Phys. Lett. B382, 272 (1996), Two parameters a 2 and a 4. F 2 theory (x,Q 2 ) = F 2 PQCD+TM [1+ D 2 (x,Q 2 )/ Q 2 + D 4 (x,Q 2 )/ Q 4 ] D 2 (x,Q 2 ) = [ a 2 / q (x,Q 2 ) ] ∫ (dz/z) c 2 (z) q(x/z, Q 2 ) D 4 (x,Q 2 ) = [ a 4 times function of x) In this model, the higher twist effects are different for 2xF 1, xF 3,F 2. With complicated x dependences which are defined by only two parameters a 2 and a 4. Fit a 2 and a 4 to experimental data for F 2 and R=F L /2xF 1. F 2 data (x,Q 2 ) = [ F 2 measured +  F 2 syst ] ( 1+ N ) :  2 weighted by errors where N is the fitted normalization (within errors) and  F 2 syst is the is the fitted correlated systematic error BCDMS (within errors). 1

20. Arie Bodek, Univ. of Rochester20 Very high x F2 proton data (DIS + resonance) (not included in the original fits Q 2 =1. 5 to 25 GeV 2 ) NLO pQCD +  TM + higher twist describes very high x DIS F 2 and resonance F 2 data well. (duality works) Q 2 =1. 5 to 25 GeV 2 Q 2 = 25 GeV 2 Ratio F 2 data/F 2 pQCD Q 2 = 25 GeV 2 Ratio F 2 data/ F 2 pQCD+TM Q 2 = 25 GeV 2 Ratio F 2 data/F 2 pQCD+TM+HT F2 resonance Data versus F 2 pQCD+TM+HT pQCD ONLY pQCD+TM pQCD+TM+HT Q 2 = 25 GeV 2 Q 2 = 15 GeV 2 Q 2 = 9 GeV 2 Q 2 = 3 GeV 2 Q 2 = 1. 5 GeV 2

21. Arie Bodek, Univ. of Rochester21 Look at Q 2 = 8, 15, 25 GeV 2 very high x data Pion production threshold Now Look at lower Q 2 (8,15 vs 25) DIS and resonance data for the ratio of F2 data/( NLO pQCD +TM +HT} High x ratio of F2 data to NLO pQCD +TM +HT parameters extracted from lower x data. These high x data were not included in the fit.  The Very high x(=0.9) region: It is described by NLO pQCD (if target mass and higher twist effects are included) to better than 10% Ratio F 2 data/F 2 pQCD+TM+HT Q 2 = 25 GeV 2 Q 2 = 15 GeV 2 Q 2 = 9 GeV 2

22. Arie Bodek, Univ. of Rochester22 F 2, R comparison with NNLO QCD => NLO HT mostly missing NNLO terms Size of the higher twist effect with NNLO analysis is really small (a2=-0.009(NNLO) vs –0.1(NLO)

23. Arie Bodek, Univ. of Rochester23 “Converting” NLO PDFs to NNLO PDFs f(x) = the fitted floating factor, which is the fitted ratio of the data to theory. Note f(x) =1.00 if pQCD PDFs describe the data. f NLO : Here the theory is pQCD(NLO)+TM+HT using NLO PDFs. f NNLO : Here the theory is pQCD(NNLO)+TM+HT also using NLO PDFs Therefore f NNLO / f NLO is the factor to “convert” NLO PDFs to NNLO PDFs (NNLO PDFs are not yet available. NNLO PDFs are lower at high x and higher at low x. Use f(x) in NNLO calculation of QCD processes (e.g. hardon colliders) Recently MRST did a similar analysis including NNLO gluons. Floating factors NNLO NLO

24. Arie Bodek, Univ. of Rochester24 Lessons from the NNLO pQCD analysis  The origin of the “empirically measured dynamic higher twist effects” is from the missing NNLO QCD terms.  Both TM and Dynamic higher twists effects should be similar in electron and neutrino reactions (aside from known mass differences, e.g. charm production) The NNLO pQCD corrections and the Dynamic Higher Twist effects in NLO both have the same Q2 dependence at fixed x. (F 2NNL0 /F 2NLO )-1

25. Arie Bodek, Univ. of Rochester25 In the few GeV region, At low x, “dynamic higher twist” look similar to “kinematic final state mass higher twist” -->both look like “enhanced” QCD At low Q2, the final state u and d quark effective mass is not zero u u M* (final state interaction) Production of pons etc   C  = [Q 2 +M* 2 ] / [ 2M ] (final state M* mass))  versus for mass-less quarks 2x q.P= Q 2  x = [Q 2 ] / [2M ] (compared to x] (Pi + q) 2 = Pi 2 + 2q.Pi + q 2 = Pf 2 = M* 2 Ln Q2 F2 Lambda QCD Low x QCD evolution  C slow rescaling looks like faster evolving QCD Since QCD and slow rescaling are both present at the same Q2  Charm production s to c quarks in neutrino scattering-slow rescaling s c Mc (final state quark mass  2  C q.P = Q 2 + Mc 2 (Q 2 = -q 2 )  2  C M  = Q 2 + Mc 2  C  slow re-scaling   C = [Q 2 +Mc 2 ] / [ 2M ] (final state charm mass (Pi + q) 2 = Pi 2 + 2q.Pi + q 2 = Pf 2 = Mc 2 At Low x, low Q 2  C  x (slow rescaling  C  (and the PDF is smaller at high x, so the low Q 2 cross section is suppressed - threshold effect. Final state mass effect

26. Arie Bodek, Univ. of Rochester26 At high x, “dynamic higher twists” have a similar form to the “kinematic Goergi-Politzer proton target mass effects” --> both look like “enhanced” QCD Target Mass (G-P):  tgt mass   2 TM M + 2  TM q.P - Q 2 = 0 (Q 2 = -q 2 )  solve quadratic equation  TM = Q 2 /[M (1+ (1+Q 2 / 2 ) 1/2 ] proton target mass effect in Denominator)  Versus : Numerator in  C  = [Q 2 +M* 2 ] / [ 2M ] (final state M* mass)  Combine both target mass and final state mass:  C+TM = [Q 2 +M* 2 ] / [ M (1+(1+Q 2 / 2 ) 1/2 ] - includes both initial state target proton mass and final state M* mass effect) (Pi + q) 2 = Pi 2 + 2qPi + Q 2 = Pf 2 Ln Q 2 F2 Mproton High x  At high x, low Q 2  TM  x (tgt mass  (and the PDF is higher at lower x, so the low Q 2 cross section is enhanced. F2 fixed Q 2 X=1X=0  TM < x x <  C Final state mass Initial state target mass QCD evolution Target mass effects [Ref:Goergi and Politzer GP Phys. Rev. D14, 1829 (1976)]]

27. Arie Bodek, Univ. of Rochester27 Towards a unified model  We learned that the NNLO+ TM describes the DIS and resonance data very well. Theoretically, this breaks down at low Q 2 Practically, no way to implement it in MC  HT takes care of the NNLO term. So what about NLO + TM + HT? Still, it break down at very low Q 2 No way to implement photo-production limit.  Well, can we do something with the LO QCD and PDFs ? YES Different scaling variables Resonance, higher twist, and TM  =0.9 X=-0.95 M* (final state interaction)  C  =[Q 2 +M * 2 ] / ( 2M ) (quark final state M* mass)  TM = Q 2 /[M (1+ (1+Q 2 / 2 ) 1/2 ] (initial proton mass)   = [Q 2 +M * 2 ] / [ M (1+(1+Q 2 / 2 ) 1/2 ] combined   = x [2Q 2 +2M * 2 ] / [Q 2 + (Q 4 +4x 2 M 2 Q 2 ) 1/2 ] (Pi + q) 2 = Pi 2 + 2qPi + Q 2 = M* 2 Ln Q 2 F2 Lambda QCD Low x High x Xw Photoproduction limit- Need to multiply by Q 2 /[Q 2 +C] TRY: Xw = [Q 2 +B] /[2M + A] = x [Q 2 +B] / [Q 2 + Ax] (used in early fits to SLAC data in 1972) And then follow up by trying  w= [Q 2 +B ] / [ M (1+(1+Q 2 / 2 ) 1/2 + A] (  w works better and is theoretically motivated) Xw worked in 1972 because it approximates  w M B term (M *) A term (tgt mass) q

28. Arie Bodek, Univ. of Rochester28 Modified LO PDFs for all Q 2 region? 1. We find that NNLO QCD+tgt mass works very well for Q 2 > 1 GeV 2. 2. That target mass and missing NNLO terms “explain” what we extract as higher twists in a NLO analysis. 2. However, we want to go down all the way to Q 2 =0. All NNLO and NLO terms blow up. However, higher twist formalism in terms of initial state target mass binding and final state mass are valid below Q 2 =1, and mimic the higher order QCD terms for Q 2 >1 (in terms of effective masses due to gluon emission). 3. While the original approach was to explain the “empirical higher twists” in terms of NNLO QCD at low Q 2 (and extract NNLO PDFs), we can reverse the approach and have “higher twist” model non-perturbative QCD, down to Q 2 =0, by using LO PDFs and “effective target mass and final state masses” to account for initial target mass, final target mass, and missing NLO and NNLO terms. [Ref:Bodek and Yang hep-ex/0203009] Philosophy

29. Arie Bodek, Univ. of Rochester29 Modified LO PDFs for all Q2 region? 1. Start with GRV94 LO (Q 2 min =0.23 GeV 2 ) - describe F2 data at high Q 2 2. Replace X with a new scaling, Xw  x= [Q 2 ] / [2M   Xw=[Q 2 +B] / [2M +A]= x[Q 2 +B]/[Q 2 + Ax]  A: initial binding/target mass effect ( but also higher twist and NLO effect)  B: final state mass effect (but also photo production limit) 3. Multiply all PDFs by a factor of Q 2 /[Q 2 +C] for photo prod. limit and higher twist 4. Freeze the evolution at Q 2 = 0.24 GeV 2 - F 2 (x, Q 2 < 0.24) = Q 2 /[Q 2 +C] F 2 (Xw, Q 2 =0.24)  Do a fit to SLAC/NMC/BCDMS H, D data, allow the normalization of the experiments and the BCDMS major systematic error to float within errors.  HERE INCLUDE DATA WITH Q2<1 which is not in the resonance region  Modified LO GRV94 PDFs with three parameters (a new scaling variable, Xw) describe DIS F2 H, D data (SLAC/BCDMS/NMC) very well. A=1.735, B=0.624, and C=0.188 Compare with resonance data (not used in our fit) Compare with photo production data (not used in our fit) Compare with medium neutrino data (not used in our fit)- except to the extent that GRV94 originally included very high energy data on xF 3, Construction Results [Ref:Bodek and Yang hep-ex/0203009]

30. Arie Bodek, Univ. of Rochester30 Comparison of Xw Fit and  w Fit Try Also  w= [Q 2 +B ] / [ M (1+(1+Q 2 / 2 ) 1/2 + A] Fitted normalizations HT fitting with Xw pd SLAC 0.979 +-0.00240.967 +- 0.0025 NMC 0.993 +-0.00320.990 +- 0.0028 BCDMS 0.956 +-0.0015 0.974 +- 0.0020 BCDMS Lambda = 1.01 +-0.156 HT fitting with  w pd SLAC 0.982 +-0.00240.973 +- 0.0025 NMC 0.995 +-0.00320.994 +- 0.0028 BCDMS 0.958 +-0.00150.975 +- 0.0020 BCDMS Lambda = 0.976 +- 0.156.  Modified LO GRV94 PDFs with three parameters (a new scaling variable, Xw) describe DIS F2 H, D data (SLAC/BCDMS/NMC) very well.  A=1.735, B=0.624, and C=0.188  (+-0.022) (+-0.014) ( +-0.004)   2 = 1555 /958 DOF  Modified LO GRV94 PDFs with three parameters (a new scaling variable,  w ) describe DIS F2 H, D data (SLAC/BCDMS/NMC) EVEN BETTER  A=0.700, B=0.327, and C=0.197  (+-0.020) (+-0.012) ( +-0.004)   2 = 1351 /958 DOF  Note: No systematic errors (except for normalization and BCDMS B field error were included) Same construction for Xw and  w fits Comparison

31. Arie Bodek, Univ. of Rochester31 LO+HT fit Comparison with DIS F 2 (H, D) data [These SLAC/BCDMS/NMC are used in the Xw fit ] Proton Deuteron

32. Arie Bodek, Univ. of Rochester32 LO+HT fit Comparison with DIS F 2 (H, D) data [Try again with  w: These SLAC/BCDMS/NMC are used in the  w fit ] Proton Deuteron

33. Arie Bodek, Univ. of Rochester33 Comparison of LO+HT to neutrino data on Iron [CCFR] (not used in the fit)  Apply nuclear corrections using e/  scattering data.  Calculate F 2 and xF 3 from the modified PDFs with Xw  Use R=Rworld fit to get 2xF 1 from F 2  Implement charm mass effect through a slow rescaling algorithm, for F 2 2xF 1, and XF 3 The modified GRV94 LO PDFs with a new scaling variable, Xw describe the CCFR diff. cross section data (E =30–300 GeV) well. Construction

34. Arie Bodek, Univ. of Rochester34 Comparison with F2 resonance data [ SLAC/ Jlab] (These data were not included in the fit)  The modified LO GRV94 PDFs with a new scaling variable, Xw describe the SLAC/Jlab resonance data very well (on average).  Even down to Q 2 = 0.07 GeV 2  Duality works: The DIS curve describes the average over resonance region

35. Arie Bodek, Univ. of Rochester35 Comparison with photo production data (not included in the Xw fit)   -proton) =  Q 2 =0, Xw)   = 0.112 mb 2xF 1 /( KQ 2 )  K depends on definition of virtual photon flux for usual definition K= [1 - Q 2 / 2M    = 0.112 mb F 2 (x, Q 2 ) D(, Q 2 ) /( KQ 2 )  D = (1+ Q 2 / 2 )/(1+R) F 2 (x, Q 2 ) limit as Q 2 -->0 = Q 2 /(Q 2 +0.188) * F 2-GRV94 (Xw, Q 2 =0.24) Try: R = 0 R= Q 2 / 2 ( evaluated at Q 2 =0.24) R = Rw (evaluated at Q 2 =0.24) Note: Rw=0.034 at Q 2 =0.24 The modified LO GRV94 PDFs with a new scaling variable, Xw also describe photo production data (Q 2 =0) to within 25%: To get better agreement at high  100 GeV (very low Xw), the GRV94 need to be updated to fit latest HERA data at very low x and low Q 2. If we include these photoproduction data in the fit, we will get C of about 0.22, and agreement at the few percent level. To evaluate D = (1+ Q 2 / 2 )/(1+R) more precisely, we also need to compare measured Jlab R data in the Resonance Region at Q 2 =0.24 to the Rw parametrization. mb

36. Arie Bodek, Univ. of Rochester36 Comparison with photo production data (not included in the  w fit)   -proton) =  Q 2 =0, Xw)   = 0.112 mb 2xF 1 /( KQ 2 )  K depends on definition of virtual photon flux for usual definition K= [1 - Q 2 / 2M    = 0.112 mb F 2 (x, Q 2 ) D(, Q 2 ) /( KQ 2 )  D = (1+ Q 2 / 2 )/(1+R) F 2 (x, Q 2 ) limit as Q 2 -->0 = Q 2 /(Q 2 +0.188) * F 2-GRV94 (  w, Q 2 =0.24) Try: R = 0 R= Q 2 / 2 ( evaluated at Q 2 =0.24) R = Rw (evaluated at Q 2 =0.24) Note: Rw=0.034 at Q 2 =0.24 The modified LO GRV94 PDFs with a new scaling variable,  w also describe photo production data (Q 2 =0) to within 15%: To get better agreement at high  100 GeV (very low Xw), the GRV94 need to be updated to fit latest HERA data at very low x and low Q 2. If we include these photoproduction data in the fit, we will get C of about 0.22, and agreement at the few percent level. To evaluate D = (1+ Q 2 / 2 )/(1+R) more precisely, we also need to compare measured Jlab R data in the Resonance Region at Q 2 =0.24 to the Rw parametrization. mb

37. Arie Bodek, Univ. of Rochester37 The GRV LO need to be updated to fit latest HERA data at very low x and low Q 2. We use GRV94 since they are the only PDFs to evolve down to Q2=0.24 GeV 2. All other PDFs (LO) e.g. GRV98 stop at 1 GeV 2 or 0.5 GeV 2. GRV94 LO PDFs need to be updated.at very low x, but this is not important in the few GeV region Comparison of u quark PDF for GRV94 and CTEQ4L and CTEQ6L (more modern PDFs) Q 2 =10 GeV 2 Q 2 =1 GeV 2 Q 2 =0.5 GeV 2 GRV94 CTEQ6L CTEQ4L CTEQ6L GRV94 CTEQ4L GRV94 CTEQ6L CTEQ4L X=0.01 X=0.0001

38. Arie Bodek, Univ. of Rochester38 Summary  Our modified GRV94 LO PDFs with a modified scaling variable, Xw describe all SLAC/BCDMS/NMC DIS data. (We will investigate other variables also)  The modified PDFs also yields the average value over the resonance region as expected from duality argument, ALL THE WAY TO Q 2 = 0  Also good agreement with high energy neutrino data.  Therefore, this model should also describe a low energy neutrino cross sections reasonably well.

39. Arie Bodek, Univ. of Rochester39 Future Work  Implement A e/  (W,Q 2 ) resonances into the model for F 2.  For this need to fit all DIS and SLAC and JLAB resonance date and Photo- production H and D data and CCFR neutrino data.  Implement differences between  and e/  final state resonance masses in terms of A,  bar (w) {See Appendix)  Here Include Jlab and SLAC heavy target data for possible Q 2 dependence of nuclear dependence on Iron.  Investigate different scaling variables for different flavor quark masses (u, d, s, u v, d v, u sea, d sea in initial and final state) for F 2., Xw=[ Q 2 +B] / [ 2M +A] versus:   w = [Q 2 +B ] / [ M (1+ (1+Q 2 / 2 ) 1/2 +A ] (see Appendix)   or more sophisticated Genral expression:  w ’ =[ Q’ 2 +B ] / [M (1+ (1+Q 2 / 2 ) 1/2 +A] with  2Q’ 2 = [Q 2 + m * 2 - m I 2 ] + [ Q 4 +2 Q 2 (m * 2 + m I 2 ) + (m * 2 - m I 2 ) 2 ] 1/2  Check other forms of scaling e.g. F 2 =(1+ Q 2 /    W 2  Implementation for R (and 2xF 1 ) is done exactly - use empirical fits to R (agrees with NNLO+GP tgt mass for Q 2 >1); Need to compare to Jlab R data in resonance region.  Compare our model prediction with the Rein and Seghal model for the 1 st resonance (in neutrino scattering)  Compare to low-energy neutrino data (only low statistics data, thus new measurements of neutrino differential cross sections at low energy are important).  Do analysis with Xw with other LO PDFs like NuTeV Buras-Gamers LO PDFs etc. (as an alternative LO model) - Would like to have GRV02 LO (now using GRV94)

40. Arie Bodek, Univ. of Rochester40 Future Neutrino Experiments -JHF,NUMI Need to know the properties of neutrino interactions (both structure functions AND detailed final states on nuclear targets (e.g. Carbon, Oxygen (Water), Iron). Need to understand differences between neutrino and electron data for H, D and nuclear effects for the structure functions and the final states. Need to understand neutral current structure functions and final states. Need to understand implementation of Fermi motion for quasielastic scattering and the identification of Quasielastic and Inelastic processes in neutrino detectors (subject of another talk). A combined effort in understanding electron, muon, photoproduction and neutrino data of all these processes within a theoretical framework is needed for future precision neutrino oscillations experiments in the next decade.

41. Arie Bodek, Univ. of Rochester41 Nuclear effects on heavy targets F2(iron)/(deuteron) F2(deuteron)/(free N+P) What are nuclear effects for F 2 versus XF 3 ; what are they at low Q 2 ; possible differences between Electron, Neutrino CC and Neutrino NC at low Q 2 (Vector dominance effects).

42. Arie Bodek, Univ. of Rochester42 Current understanding of R We find for R (Q 2 >1 GeV 2 ) 1. Rw empirical fit works well (down to Q 2 =0.35 GeV 2) 2. Rqcd (NNLO) + tg-tmass also works well (HT are small in NNLO) 3. Rqcd(NLO) +tgt mass +HT works well (since HT in NLO mimic missing NNLO terms) 4. Need to constrain R to zero at Q 2 =0. >> Use Rw for Q 2 > 0.35 GeV 2 For Q 2 < 0.35 use: R(x, Q 2 ) = = 3.207 {Q 2 / [Q 4 +1) } R(x, Q 2 =0.35 GeV 2 ) * Plan to compare to recent Jlab Data for R in the Resonance region at low Q2.

43. Arie Bodek, Univ. of Rochester43 Different Scaling variables for u,d,s,c in initial and u, d, s,c in final states and valence vs. sea Bodek and Yang hep-ex/0203009; Goergi and Politzer Phys. Rev. D14, 1829 (1976) For further study   =[Q 2 +m * 2 ] / ( 2M ) (quark final state m* mass)  = Q 2 /[M (1+ (1+Q 2 / 2 ) 1/2 ] (initial proton mass)   = [Q 2 +m * 2 ] / [ M (1+ (1+Q 2 / 2 ) 1/2 ] combined   = x [Q 2 +m * 2 ] 2 / [Q 2 + (Q 4 +4x 2 M 2 Q 2 ) 1/2 ] (Pi + q) 2 = Pi 2 + 2qPi + q 2 = m* 2 M  m * 2 We Use: Xw = [Q 2 +B] / [2M + A] = x [Q 2 +B] / [Q 2 + Ax] Could also try:  w = [Q 2 +B ] / [ M (1+ (1+Q 2 / 2 ) 1/2 +A ] Or fitted effective initial and final state quark masses that mimic higher twist (NLO+NNLO QCD), binding effects, +final state intractions could be different for initial and final state u,d,s, and valence vs sea? Can try ??  w ’ =[ Q’ 2 +B ] / [M (1+ (1+Q 2 / 2 ) 1/2 +A] In general GP derive for initial quark mass m I and final mass m * bound in a proton of mass M (at high Q 2 these are current quark masses, but at low Q 2 maybe constituent masses? ) GP get2Q’ 2 = [Q 2 + m * 2 - m I 2 ] + [ Q 4 +2 Q 2 (m * 2 + m I 2 ) + (m * 2 - m I 2 ) 2 ] 1/2  = x [2Q’ 2 ] / [Q 2 + (Q 4 +4x 2 M 2 Q 2 ) 1/2 ] - note masses may depend on Q2  = [Q’ 2 ] / [ M (1+(1+Q 2 / 2 ) 1/2 ] (equivalent form) m I 2 Can try to models quark masses, binding effects Higher twist, NLO and NNLO terms- All in terms of effective initial and final state quark masses and different target mass M with more complex form. q

44. Arie Bodek, Univ. of Rochester44 One Page Derivation: In general GP derive for initial quark mass m I and final mass m * bound in a proton of mass M Goergi and Politzer Phys. Rev. D14, 1829 (1976)]- GP (Pi + q) 2 = m I 2 + 2qPi + q 2 = m * 2 and q = (  q3) in lab 2qPi = 2 [ Pi 0 + q3 Pi 3 ] = [Q 2 + m * 2 - m I 2 ] (eq. 1) ( note + sign since q3 and Pi 3 are pointing at each other)  = [Pi 0 +Pi 3 ] /[P 0 + P 3 ] --- frame invariant definition  = [Pi 0 +Pi 3 ] /M ---- In lab or [Pi 0 +Pi 3 ] =  M --- in lab  [Pi 0 -Pi 3 ] = [Pi 0 +Pi 3 ] [Pi 0 -Pi 3 ] / M -- multiplied by [Pi 0 -Pi 3 ] Pi 0 -Pi 3 = [ (Pi 0 ) 2 - ( Pi 3 ) 2 ]/ M = m I 2 /[  M] or [Pi 0 -Pi 3 ] = m I 2 /[  M] --- in lab Get 2 Pi 0 =  M + m I 2 /[  M] Plug into (eq. 1) and 2 Pi 3 =  M - m I 2 /[  M] {  M + m I 2 /[  M] } + {q3  M - q3 m I 2 /[  M] } - [Q 2 + m * 2 - m I 2 ] = 0 abc  2 M 2 (  q3) -  M [Q 2 + m * 2 - m I 2 ] + m I 2 (  q3) = 0 >>>  = [-b +(b 2 - 4ac) 1/2 ] / 2a => solution use (  2  q3 2 ) = q 2 = -Q 2 and (  q3) =  [  +  Q 2 /  2 ] 1/2 =  [  +  4M 2 x 2 / Q 2 ] 1/2 Get  = [Q’ 2 ] / [ M (1+(1+Q 2 / 2 ) 1/2 ]  = [Q’ 2 ] / [M  [  +  4M 2 x 2 / Q 2 ] 1/2) ] (equivalent form) (at high Q 2 these are current quark masses, but at low Q 2 maybe constituent masses?) where2Q’ 2 = [Q 2 + m * 2 - m I 2 ] + [ Q 4 +2 Q 2 (m * 2 + m I 2 ) + (m * 2 - m I 2 ) 2 ] 1/2 or  = x [2Q’ 2 ] / [Q 2 + (Q 4 +4x 2 M 2 Q 2 ) ] 1/2 (equivalent form) P= P 0 + P 3,M Pf, m* Pi= Pi 0,Pi 3,m I q

45. Arie Bodek, Univ. of Rochester45 In general GP derive for initial quark mass m I and final mass,m F =m * bound in a proton of mass M -- Page 1  Is the correct variable which is Invariant in any frame : q3 and P in opposite directions.  P= P 0 + P 3,M P F = P I 0,P I 3,m I P F = P F 0,P F 3,m F =m * q=q3,q0

46. Arie Bodek, Univ. of Rochester46 In general GP derive for initial quark mass m I and final mass m F =m * bound in a proton of mass M -- Page 2  For the case of non zero m I (note P and q3 are opposite)  P= P 0 + P 3,M P F = P I 0,P I 3,m I P F = P F 0,P F 3,m F =m * ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Keep terms with m I : multiply by  M and group terms in  qnd  2  2 M 2 (  q3) -  M [Q 2 + m * 2 - m I 2 ] + m I 2 (  q3) = 0 General Equation abc => solution of quadratic equation  = [-b +(b 2 - 4ac) 1/2 ] / 2a use (  2  q3 2 ) = q 2 = -Q 2 and (  q3) =  [  +  Q 2 /  2 ] 1/2 =  [  +  4M 2 x 2 / Q 2 ] 1/2 Get  = [Q’ 2 ] / [ M (1+(1+Q 2 / 2 ) 1/2 ]  = [Q’ 2 ] / [M  [  +  4M 2 x 2 / Q 2 ] 1/2 ) ] or  = x [2Q’ 2 ] / [Q 2 + √ (Q 4 +4x 2 M 2 Q 2 ) ] (equivalent form) where2Q’ 2 = [Q 2 + m * 2 - m I 2 ] + [ Q 4 +2 Q 2 (m * 2 + m I 2 ) + (m * 2 - m I 2 ) 2 ] 1/2 (at high Q 2 these are current quark masses, but at low Q 2 maybe constituent masses?) q=q3,q0

47. Arie Bodek, Univ. of Rochester47 In general GP derive for initial quark mass m I and final mass m F =m * bound in a proton of mass M --- Page 3 Get  = [Q’ 2 +B ] / [ M (1+(1+Q 2 / 2 ) 1/2 +A ]  = [Q’ 2 +B ] / [M  [  +  4M 2 x 2 / Q 2 ] 1/2) +A] or  = x [2Q’ 2 +2B ] / [Q 2 + (Q 4 +4x 2 M 2 Q 2 ) 1/2 +2Ax ] (equivalent form) where2Q’ 2 = [Q 2 + m * 2 - m I 2 ] + [ Q 4 +2 Q 2 (m * 2 + m I 2 ) + (m * 2 - m I 2 ) 2 ] 1/2 Numerator: = x [2Q’ 2 ]: Special cases for 2Q’ 2 +2B m * =m I =0: 2Q’ 2 = 2 Q 2 +2B : current quarks ( m i = m F =0) m * =m I : 2Q’ 2 = Q 2 + [ Q 4 +4 Q 2 m * 2 ] 1/2 +2B: constituent mass ( m i = m F =0.3 GeV) m I =0 : 2Q’ 2 = 2Q 2 + 2 m * 2 +2B : final state mass ( m i =0, m F =charm) m * =0 : 2Q’ 2 = 2Q 2 +2B -----> : initial constituent( m i =0.3); final state current ( m F =0) denominator : [Q 2 + (Q 4 +4x 2 M 2 Q 2 ) 1/2 +2Ax ] at large Q 2 ---> 2 Q 2 + 2 M 2 x 2 +2Ax Or [ M (1+(1+Q 2 / 2 ) 1/2 +A ] which at small Q 2 ---> 2M  M 2 /x +A Therefore A = M 2 x at large Q 2 and M 2 /x at small Q 2 ->A=constant is approximate;y OK versus Xw = [Q 2 +B] / [2M + A] = x [2Q 2 +2B] / [2Q 2 + 2Ax] In future try to include fit using above form with floating masses and B and A. Expect A,B to be much smaller than for Xw. C is well determined if we include photo-production in the fit.  P= P 0 + P 3,M P F = P I 0,P I 3,m I P F = P F 0,P F 3,m F =m * q=q3,q0 Plan to try to fit for different initial/final quark masses for u-I, u-F, d-I,d-F, s-I,s-F,c-F. and A and B for HT

48. Arie Bodek, Univ. of Rochester48 Initial quark mass m I and final mass m F =m * bound in a proton of mass M Page 4 At High Q 2, we expect that the initial and final state quark masses are the current quark masses: (e.g. u=5 MeV, d=9 MeV, s=170 MeV, c=1.35 GeV, b=4.4 GeV. For massive final state quarks, this is known as slow-rescaling : At low Q 2, Donnachie and Landshoff (Z. Phys. C. 61, 145 (1994)] say that The effective final state mass should reflect the true threshold conditions as follows: m* 2 = (W threshold ) 2 -Mp 2. Probably not exactly true since A(w) should take care of it if target mass effects are included. Nonetheless it is indicative of the order of the final state interaction. (This is known as fast rescaling - I.e. introducing a   function to account for threshold) Reaction initial state final state final state W threshold m* 2 m* m F1 + m F2 quark quark threshold GeV GeV 2 GeV quarks e-P u or d u or d M p +M pion 1.12 0.29 0.53 0.01 e-P s s+s bar M LambdS +M K 1.61 1.63 1.28 0.34 -N d c+u M LamdaC +M pion 2.42 4.89 2.21 1.36 -N s c+s bar M LamdaC +M K 2.78 6.76 2.60 1.52 e-P c c+c bar M LamdaC +M D 4.15 16.29 4.04 2.70 e-P c c+c bar M p +M D +M D 4.68 20.90 4.57 2.70  P= P 0 + P 3,M P F = P I 0,P I 3,m I P F = P F 0,P F 3,m F =m * q=q3,q0 Plan to try to fit for different intial/fnal quark masses for u-I, u-F, d-I,d-F, s-I,s-F,c-F. and A and B

49. Arie Bodek, Univ. of Rochester49 Initial quark mass m I and final mass m F =m * bound in a proton of mass M Page 4 At High Q 2, we expect that the initial and final state quark masses are the current quark masses: (e.g. u=5 MeV, d=9 MeV, s=170 Mev, c=1.35 GeV, b=4.4 GeV. However, (  m)* 2 = m* 2 - m ud * 2 is more relevant since m ud * 2 is already included in the scaling violation fits for the u and d PDFs Reaction initial state final state final state W threshold (  m)* 2  m* m F1 + m F2 quark quark threshold GeV GeV 2 GeV quarks e-P u or d u or d M p +M pion 1.12 0.00 0.00 0.00 e-P s s+s bar M Lambda +M K 1.61 1.34 1.16 0.34 -N d c+u M LamdaC +M pion 2.42 4.61 2.15 1.36 N s c+s bar M LamdaC +M K 2.78 6.47 2.54 1.52 e-P c c+c bar M LamdaC +M D 4.15 16.00 4.00 2.70 e-P c c+c bar M p +M D +M D 4.68 20.62 4.54 2.70 FAST re-scaling (   function ) is a crude implementation of A(W,Q2).  P= P 0 + P 3,M P F = P I 0,P I 3,m I P F = P F 0,P F 3,m F =m * q=q3,q0 Plan to try to fit for different intial/fnal quark masses for u-I, u-F, d-I,d-F, s-I,s-F,c-F. and A and B

50. Arie Bodek, Univ. of Rochester50 Examples of Low Energy Neutrino Data: Total (inelastic and quasielastic) cross section

51. Arie Bodek, Univ. of Rochester51 Examples of Current Low Energy Neutrino Data: Single charged and neutral pion production

52. Arie Bodek, Univ. of Rochester52 Examples of Current Low Energy Neutrino Data: Quasi-elastic cross section

53. Arie Bodek, Univ. of Rochester53 Appendix B - Studies of d/u Comparison of electron, muon, neutrino and e-p, and hadron collider data.

54. Arie Bodek, Univ. of Rochester54 Status of d/u a few years ago with CTEQ3 and MRS A, G, R PDFs In 1997 PDFs did not describe the new CDF W Asymmetry Data. We decided to look at F2N/F2P Correcting for Nuclear Effects in the deuteron resulted in even poorer agreement of PDFs with the corrected F2n/F2p data

55. Arie Bodek, Univ. of Rochester55 The d/u correction to MRSR2 PDFs (which were originally constrained to have d/u --> 0 at x=1) We parametrized a correction to the d/u ratio in the MRSR2 PDFs such that they would agree with the NMC experimentally measured F 2N /F 2P ratio (corrected for nuclear binding effects in the deuteron).

56. Arie Bodek, Univ. of Rochester56 Comparison to CDF Pbar-PTevatron W asymmetry The modified PDFs (After d/u correction) are in better agreement with the CDF Tevatron very high energy data on unbound protons. Yang and Bodek PRL 82, 2467 (1999)

57. Arie Bodek, Univ. of Rochester57 Comparison of modified PDFs to HERA charged current data for e+P and e-P After d/u correction Comparison of modified PDFs to HERA data for the ratio of e+P and e-P charged current data. Modified PDFs are in better agreement with very high energy data on unbound protons. Yang and Bodek, Phys. Rev. Lett. 82, 2467 (1999)

58. Arie Bodek, Univ. of Rochester58 Comparison to CHDS Neutrino/Antineutrino Ratio on H2 and to QCD predictions The d/u from CDHS neutrino and antineutrino data on hydrogen versus MRSR2 PDFs with and without our d/u corrections. Note, QCD predicts d/u=0.2 as x-> 1. The modified PDFs agree with CDHS data on unbound protons and with the QCD prediction.

59. Arie Bodek, Univ. of Rochester59 Comparison to CDF and Dzero High Pt Jet Data at Higher Pt (i.e higher x) A higher d quark density at high x yields better agreement with the rate of high Pt jets in CDF and Dzero on unbound protons and antiprotons