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Arie Bodek Univ. of Rochester and Un-Ki Yang Univ. of Chicago July 1-6, 2002

Modeling of (e/ m / n -(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.

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Arie Bodek Univ. of Rochester and Un-Ki Yang Univ. of Chicago July 1-6, 2002

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  1. Modeling of (e/m/n-(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 Arie Bodek, Univ. of Rochester

  2. Arie Bodek- Univ. of RochesterNuFact02 - 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/ m/n 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/ m /n Cross Sections from DIS to Resonance Region 15 min: Short Talk - summary Arie Bodek, Univ. of Rochester

  3. 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 n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/m/n DIS experiments) • However, neutrino differential cross sections and final states in the few GeV region are poorly understood. ( especially, resonance and low Q2 DIS contributions). In contrast, there is enormous amount of e-N data from SLAC and Jlab in this region. Arie Bodek, Univ. of Rochester

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

  5. Neutrino interactions Quasi-Elastic / Elastic(W=Mp) nm + n --> m- + 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 Q2, W< 2) nm + p-->m- + p + p Poorly measured and only 1st resonance described by Rein and Seghal Deep Inelastic nm + p --> m- + X(high Q2, 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 Q2 region. (e.g. JLAB data at Q2=0.22) Issues at few GeV : Resonance production and low Q2 DIS contribution meet. The challenge is to describe both processes at a given neutrino (or electron) energy. Neutrino cross sections GRV94 LO 1st resonance Arie Bodek, Univ. of Rochester

  6. Can we build up a model to describe all Q2 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 Q2? - 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) Building up a model for all Q2 region.. Challenges Arie Bodek, Univ. of Rochester

  7. What are Higher Twist Effects- page 1 • Higher Twist Effects are terms in the structure functions that behave like a power series in (1/Q2 ) or [Q2/(Q4+A)],… (1/Q4 ) etc…. Higher Twist: Interaction between interacting and Spectator quarks via gluon exchange at Low Q2 affects the Cross Section. Terms are like (1/Q2 ) or [Q2/(Q4+A)],… (1/Q4 ) • While pQCD predicts terms in as2 ( ~1/[ln(Q2/ L2 )] )… as4 etc…(i.e. LO, NLO, NNLO etc.) • In the few GeV region, the two power series cannot be distinguished In pQCD high Q2 impulse approximation, the interacting quark and the spectator quarks are resolved and do not affect each other. In NNLO p-QCD more gluons emission affects the cross section like as2 ( ~1/[ln(Q2/ L2 )] )… as4 Spectator quarks are not Involved. Arie Bodek, Univ. of Rochester

  8. What are Higher Twist Effects - Page 2 • Higher Twist Effects are terms in the structure functions that behave like a power series in (1/Q2 ) or [Q2/(Q4+A)],… (1/Q4 ) etc…. • While pQCD predicts terms in as2 ( ~1/[ln(Q2/ L2 )] )… as4 (i.e. LO, NLO, NNLO etc.) -->In the few GeV region, the two power series cannot be distinguished • Nature has “evolved” the high Q2 PDF from the low Q2 PDF, therefore, the high Q2 PDF include the information about the higher twists . • High Q2 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 Q2 PDFs “remember” the higher twists, which originate from the non-perturbative QCD terms. • Evolving back the high Q2 PDFs to low Q2 (e.g. NLO-QCD) and comparing to low Q2 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. F2(Q2 =0) = 0]] Arie Bodek, Univ. of Rochester

  9. Duality between fixed W and DIS • OLD Picture fixed W: Elastic Scattering, Resonance Production Electric and Magnetic Form Factors (GE and GM) versus Q2 measure size of object (the electric charge and magnetization distributions). • Elastic scattering W = Mp = M, single nucleon in final state: Form factor measures size of nucleon. Matrix element squared | <p f | V(r) | p i > |2 between initial and final state lepton plane waves. Which becomes: | < e -i k2. r | V(r) | e +i k1 . r > | 2 q = k1 - k2 = momentum transfer GE (q) = INT{e i q . r r(r) d3r} = Electric form factor is the Fourier transform of the charge distribution. In the language of structure functions for example: 2xF1(x ,Q2)elastic = x2 GM2 elasticd (x-1) • Resonance Production, W=MR, 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 d (x-1). 2xF1(x ,Q2) resonance ~ x2 GM2 Resonance transitionBW (W-1.238) e +i k2 . r e +i k1.r rMp Mp e +i k2 . r e +i k1 . r q MR Mp Arie Bodek, Univ. of Rochester

  10. Duality: Parton Model Pictures of Elastic and Resonance Production • Elastic Scattering, Resonance Production • Scatter from one quark with the correct parton momentum x, and the two spectator are just right such that a final state interaction makes up a proton, or a resonance. • Elastic scattering W = Mp = M, single nucleon in final state. • The scattering is from a quark with a very high value of x, is such that one cannot produce a single pion in the final state and the final state interaction makes a proton. • Resonance Production, W=MR, e.g. delta 1.238 resonance The scattering is from a quark with a high value of x, 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 usingQCD PDFs q (x, Q2) 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, Q2 Q2) we could also reproduce the various Breit-Wigners. q X= 1.0 x=0.95 Mp Mp X= 0.95 x=0.90 MR Mp Arie Bodek, Univ. of Rochester

  11. Photo-production Limit Q2=0Non-Perturbative - QCD evolution freezes • Photo-production Limit: Transverse Virtual and Real Photo-production cross sections must be equal at Q2=0. • There are no longitudinally polarized photons at Q2=0 • s(g-proton, n) = 0.112 mb F2(n, Q2) / Q2 limit as Q2 -->0 s(g-proton, n) = 0.112 mb 2xF1(n, Q2) / Q2 limit as Q2 -->0 s(g-proton, n) = T(n, Q2) limit as Q2 -->0. • F2(n, Q2) ~ Q2/ [Q2+C] --> 0 limit as Q2 -->0 • R(n, Q2) = L/ T ~ Q2/ [Q2+const] --> 0 limit as Q2 -->0 • If we want PDFs to work down to Q2 = 0 they must be multiplied • by a factor Q2/ [Q2+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 n. Since at Q2 = 0 • F2(n, Q2) = F2(x , Q2) with x = Q2 /( 2Mn) reduces to one point x=0 • However, a scaling variable xc= (Q2 +B) /( 2Mn) works at Q2 = 0 since • F2(n, Q2) = F2(x, Q2) becomes a a function of B/ (2Mn). Arie Bodek, Univ. of Rochester

  12. How do we “measure” higher twist (HT) • Take a set of QCD PDF which were fit to high Q2 (e/m/n data (in Leading Order-LO, or NLO, or NNLO) • Evolve to low Q2 (NNLO, NLO to Q2=1 GeV2) (LO to Q2=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 Q2data 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/Q2and 1/LnQ2 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) Arie Bodek, Univ. of Rochester

  13. Lessons from previous “NLO QCD” study • Our previous studies of comparing NLO PDFs to DIS data: SLAC, NMC, and BCDMS e/m 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 xTM 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) ] Arie Bodek, Univ. of Rochester

  14. F2, R comparison with QCD+TMvs. 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 Arie Bodek, Univ. of Rochester

  15. F2, R comparison with QCD-onlyvs. 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 Arie Bodek, Univ. of Rochester

  16. F2 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. F2 theory (x,Q2) = F2PQCD+TM [1+ h(x)/ Q2] f(x) f(x) = floating factor, should be 1.0 if PDFs have the correct x dependence. h(x) = a (xb/(1-x) -c) Arie Bodek, Univ. of Rochester

  17. Kinematic Higher-Twist (GP target mass:TM) Goergi and Politzer Phys. Rev. D14, 1829 (1976) xTM= { 2x / [1 + k ] } [1+ Mc2 /Q2 ] (last term only for heavy charm product) k= ( 1 +4x2M2 /Q2) 1/2 For Q2 large (valence) F2=2 xF1= xF3 F2 pQCD+TM(x,Q2) =F2pQCD (x, Q2) x2 / [k3x2] +J1* (6M2x3 / [Q2k4]) + J2*(12M4x4 / [Q4k5]) 2F1 pQCD+TM(x,Q2) =2F1pQCD (x, Q2) x/ [kx ] +J1 * (2M2x2 / [Q2k2 ] ) + J2*(4M4x4 / [Q4k5]) F3 pQCD+TM(x,Q2) =F3pQCD(x, Q2) x / [k2x ] +J1F3 * (4M2x2 / [Q2k3 ]) Ratio F2 (pQCD+TM)/F2pQCD At very large x, factors of 2-50 increase at Q2=15 GeV2 Arie Bodek, Univ. of Rochester

  18. Kinematic Higher-Twist (target mass:TM) Compare complete Target-Mass calculation to simple rescaling in xTM • The Target Mass Kinematic Higher Twist effects comes from the fact that the quarks are bound in the nucleon. They are important at low Q2 and high x. They involve change in the scaling variable from x to xTM and various kinematic factors and convolution integrals in terms of the PDFs for xF1, F2 and xF3 • Above x=0.9, this effect is mostly explained by a simple rescaling in xTM. F2pQCD+TM(x,Q2) =F2pQCD(xTMQ2) Ratio F2 (pQCD+TM)/F2pQCD Q2=15 GeV2 Arie Bodek, Univ. of Rochester

  19. Dynamic Higher Twist • Use: Renormalon QCD model of Webber&Dasgupta- Phys. Lett. B382, 272 (1996), Two parameters a2 and a4. • F2 theory (x,Q2) = F2PQCD+TM [1+ D2 (x,Q2)/ Q2 + D4 (x,Q2)/ Q4 ] D2 (x,Q2) = [ a2/ q (x,Q2) ] ∫ (dz/z) c2(z) q(x/z, Q2) D4 (x,Q2) = [ a4 times function of x) In this model, the higher twist effects are different for 2xF1, xF3 ,F2. With complicated x dependences which are defined by only two parameters a2 and a4 . Fit a2 and a4to experimental data for F2 and R=FL/2xF1. F2 data (x,Q2) = [ F2measured + l dF2 syst ] ( 1+ N ) : c2weighted by errors where N is the fitted normalization (within errors) and dF2 syst is the is the fitted correlated systematic error BCDMS (within errors). 1 Arie Bodek, Univ. of Rochester

  20. Very high x F2 proton data (DIS + resonance)(not included in the original fits Q2=1. 5 to 25 GeV2) Q2= 25 GeV2 Ratio F2data/F2pQCD F2 resonance Data versus F2pQCD+TM+HT NLO pQCD + x TM + higher twist describes very high x DIS F2 and resonance F2 data well. (duality works) Q2=1. 5 to 25 GeV2 Q2= 1. 5 GeV2 pQCD ONLY Q2= 3 GeV2 Q2= 25 GeV2 Ratio F2data/ F2pQCD+TM pQCD+TM Q2= 9 GeV2 Q2= 15 GeV2 Q2= 25 GeV2 Ratio F2data/F2pQCD+TM+HT pQCD+TM+HT Q2= 25 GeV2 pQCD+TM+HT Arie Bodek, Univ. of Rochester

  21. Pion production threshold Now Look at lower Q2 (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% Look at Q2= 8, 15, 25 GeV2 very high x data Ratio F2data/F2pQCD+TM+HT Q2= 9 GeV2 Q2= 15 GeV2 Q2= 25 GeV2 Arie Bodek, Univ. of Rochester

  22. F2, 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) Arie Bodek, Univ. of Rochester

  23. 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. fNLO: Here the theory is pQCD(NLO)+TM+HT using NLO PDFs. fNNLO: Here the theory is pQCD(NNLO)+TM+HT alsousing NLO PDFs Therefore fNNLO / fNLOis 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 “Converting” NLO PDFs to NNLO PDFs NLO NNLO Arie Bodek, Univ. of Rochester

  24. 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) (F2NNL0/F2NLO)-1 The NNLO pQCD corrections and the Dynamic Higher Twist effects in NLO both have the same Q2 dependence at fixed x. Arie Bodek, Univ. of Rochester

  25. 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 Charm production s to c quarks in neutrino scattering-slow rescaling u M* (final state interaction) Production of pons etc c u s Mc (final state quark mass • 2 x C q.P = Q2 +Mc2 (Q2 = -q2 ) • 2 xC Mn = Q2 +Mc2 x C - slow re-scaling • x C= [Q2+Mc2 ] / [ 2Mn] (final state charm mass • (Pi + q)2 = Pi2 + 2q.Pi + q2 = Pf2 = Mc2 • x C= [Q2+M*2 ] / [ 2Mn] (final state M* mass)) • versus for mass-less quarks 2x q.P= Q2 • x = [Q2] / [2Mn] (compared to x] • (Pi + q)2 = Pi2 + 2q.Pi + q2 = Pf2 = M*2 F2 x Low x QCD evolution x C slow rescaling looks like faster evolving QCD Since QCD and slow rescaling are both present at the same Q2 At Low x, low Q2 x C > x (slow rescaling x C) (and the PDF is smaller at high x, so the low Q2 cross section is suppressed - threshold effect. Final state mass effect Lambda QCD Ln Q2 Arie Bodek, Univ. of Rochester

  26. 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): x - tgt mass Final state mass • x2 TMM + 2 x TM q.P - Q2 = 0(Q2 = -q2 ) • solve quadratic equation • x TM = Q2/[Mn (1+(1+Q2/n2) 1/2 ] proton target mass effect in Denominator) • Versus : Numerator in • x C= [Q2+M*2] / [ 2Mn] (final state M* mass) • Combine both target mass and final state mass: • xC+TM = [Q2+M*2] / [ Mn (1+(1+Q2/n2) 1/2 ] - includes both initial state target proton mass and final state M* mass effect) Initial state target mass • (Pi + q)2 = Pi2 + 2qPi + Q2 = Pf2 F2 fixed Q2 x < x C xTM< x X=0 X=1 F2 At high x, low Q2 x TM < x (tgt mass x) (and the PDF is higher at lower x, so the low Q2 cross section is enhanced . x Target mass effects [Ref:Goergi and Politzer GP Phys. Rev. D14, 1829 (1976)]] QCD evolution High x Mproton Ln Q2 Arie Bodek, Univ. of Rochester

  27. Towards a unified model Different scaling variables • We learned that the NNLO+ TM describes the DIS and resonance data very well. • Theoretically, this breaks down at low Q2 • 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 Q2 • No way to implement photo-production limit. • Well, can we do something with the LO QCD and PDFs ? YES Resonance, higher twist, and TM q X=-0.95 x=0.9 M* (final state interaction) M • (Pi + q)2 = Pi2 + 2qPi + Q2 = M*2 • x C=[Q2+M *2] / ( 2Mn) (quark final state M* mass) • xTM= Q2/[Mn (1+ (1+Q2/n2) 1/2 ] (initial proton mass) • x = [Q2+M *2] / [ Mn (1+(1+Q2/n2) 1/2 ] combined • x = x [2Q2+2M *2] / [Q2 + (Q4 +4x2M2 Q2) 1/2 ] F2 B term (M *) TRY: Xw = [Q2+B] /[2Mn + A] = x [Q2+B] / [Q2 + Ax] (used in early fits to SLAC data in 1972) And then follow up by trying x w= [Q2+B] / [ Mn (1+(1+Q2/n2) 1/2 + A] (xw works better and is theoretically motivated) Xw worked in 1972 because it approximates xw Low x Xw Photoproduction limit- Need to multiply by Q2/[Q2+C] A term (tgt mass) High x Lambda QCD Ln Q2 Arie Bodek, Univ. of Rochester

  28. 1. We find that NNLO QCD+tgt mass works very well for Q2 > 1 GeV2. 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 Q2=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 Q2=1, and mimic the higher order QCD terms for Q2>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 Q2 (and extract NNLO PDFs), we can reverse the approach and have “higher twist” model non-perturbative QCD, down to Q2=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] Modified LO PDFs for all Q2 region? Philosophy Arie Bodek, Univ. of Rochester

  29. 1. Start with GRV94 LO (Q2min=0.23 GeV2 ) - describe F2 data at high Q2 2. Replace X with a new scaling, Xw x= [Q2] / [2Mn] Xw=[Q2+B] / [2Mn+A]= x[Q2+B]/[Q2+ 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 Q2/[Q2+C]for photo prod. limit and higher twist 4.Freeze the evolution at Q2 = 0.24 GeV2 - F2(x, Q2 < 0.24) = Q2/[Q2+C] F2(Xw, Q2=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 xF3, Modified LO PDFs for all Q2 region? Construction Results [Ref:Bodek and Yang hep-ex/0203009] Arie Bodek, Univ. of Rochester

  30. Try Also x w= [Q2+B] / [ Mn (1+(1+Q2/n2) 1/2 + A] Fitted normalizations HT fitting with Xw p d SLAC 0.979 +-0.0024 0.967 +- 0.0025 NMC 0.993 +-0.0032 0.990 +- 0.0028 BCDMS 0.956 +-0.0015 0.974 +- 0.0020 BCDMS Lambda = 1.01 +-0.156 HT fitting with xw p d SLAC 0.982 +-0.0024 0.973 +- 0.0025 NMC 0.995 +-0.0032 0.994 +- 0.0028 BCDMS 0.958 +-0.0015 0.975 +- 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, x 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) Comparison of Xw Fit and xw Fit Same construction for Xw and x w fits Comparison Arie Bodek, Univ. of Rochester

  31. Proton LO+HT fit Comparison with DIS F2 (H, D) data[These SLAC/BCDMS/NMC are used in the Xw fit ] Deuteron Arie Bodek, Univ. of Rochester

  32. Proton LO+HT fit Comparison with DIS F2 (H, D) data[Try again with w: These SLAC/BCDMS/NMC are used in the w fit ] Deuteron Arie Bodek, Univ. of Rochester

  33. Apply nuclear corrections using e/m scattering data. Calculate F2 and xF3 from the modified PDFs with Xw Use R=Rworld fit to get 2xF1 from F2 Implement charm mass effect through a slow rescaling algorithm, for F2 2xF1, and XF3 Comparison of LO+HT to neutrino data on Iron [CCFR] (not used in the fit) Construction The modified GRV94 LO PDFs with a new scaling variable, Xw describe the CCFR diff. cross section data (En=30–300 GeV) well. Arie Bodek, Univ. of Rochester

  34. The modified LO GRV94 PDFs with a new scaling variable, Xw describe the SLAC/Jlab resonance data very well (on average). Even down to Q2 = 0.07 GeV2 Duality works: The DIS curve describes the average over resonance region Comparison with F2 resonance data [ SLAC/ Jlab] (These data were not included in the fit) Arie Bodek, Univ. of Rochester

  35. sg-proton) = s Q2=0, Xw) s = 0.112 mb 2xF1/( KQ2 ) K depends on definition of virtual photon flux for usual definition K= [1 - Q2/ 2M s = 0.112 mb F2(x, Q2) D(, Q2)/( KQ2 ) D = (1+ Q2/ 2 )/(1+R) F2(x, Q2 ) limit as Q2 -->0 = Q2/(Q2+0.188) * F2-GRV94 (Xw, Q2 =0.24) Try: R = 0 R= Q2/ 2 ( evaluated at Q2 =0.24) R = Rw (evaluated at Q2 =0.24) Note: Rw=0.034 at Q2 =0.24 Comparison with photo production data(not included in the Xw fit) mb The modified LO GRV94 PDFs with a new scaling variable, Xw also describe photo production data (Q2=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 Q2. 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+ Q2/ 2 )/(1+R) more precisely, we also need to compare measured Jlab R data in the Resonance Region at Q2 =0.24 to the Rw parametrization. Arie Bodek, Univ. of Rochester

  36. sg-proton) = s Q2=0, Xw) s = 0.112 mb 2xF1/( KQ2 ) K depends on definition of virtual photon flux for usual definition K= [1 - Q2/ 2M s = 0.112 mb F2(x, Q2) D(, Q2)/( KQ2 ) D = (1+ Q2/ 2 )/(1+R) F2(x, Q2 ) limit as Q2 -->0 = Q2/(Q2+0.188) * F2-GRV94 ( w, Q2 =0.24) Try: R = 0 R= Q2/ 2 ( evaluated at Q2 =0.24) R = Rw (evaluated at Q2 =0.24) Note: Rw=0.034 at Q2 =0.24 Comparison with photo production data(not included in the w fit) mb The modified LO GRV94 PDFs with a new scaling variable, w also describe photo production data (Q2=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 Q2. 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+ Q2/ 2 )/(1+R) more precisely, we also need to compare measured Jlab R data in the Resonance Region at Q2 =0.24 to the Rw parametrization. Arie Bodek, Univ. of Rochester

  37. 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) Q2=10 GeV2 Q2=0.5 GeV2 Q2=1 GeV2 CTEQ6L CTEQ4L GRV94 CTEQ4L CTEQ6L CTEQ6L CTEQ4L GRV94 GRV94 X=0.01 X=0.0001 The GRV LO need to be updated to fit latest HERA data at very low x and low Q2. We use GRV94 since they are the only PDFs to evolve down to Q2=0.24 GeV2 . All other PDFs (LO) e.g. GRV98 stop at 1 GeV2 or 0.5 GeV2. Arie Bodek, Univ. of Rochester

  38. 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 Q2 = 0 • Also good agreement with high energy neutrino data. • Therefore, this model should also describe a low energy neutrino cross sections reasonably well. Arie Bodek, Univ. of Rochester

  39. Future Work • Implement A e/(W,Q2) resonances into the model for F2 . • 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 n and e/ final state resonance masses in terms of A n,n bar(w){See Appendix) • Here Include Jlab and SLAC heavy target data for possible Q2 dependence of nuclear dependence on Iron. • Investigate different scaling variables for different flavor quark masses (u, d, s, uv, dv, usea, dsea in initial and final state) for F2. , Xw=[Q2+B] / [2Mn+A] versus: • x w = [Q2+B] / [ Mn (1+ (1+Q2/n2) 1/2 +A ] (see Appendix) •  or more sophisticated Genral expression: x w’ =[ Q’ 2+B] / [Mn (1+ (1+Q2/n2) 1/2 +A] with • 2Q’2 =[Q2+ m *2 -m I 2] + [ Q4 +2Q2(m *2 +m I 2 ) + (m *2 -m I 2 ) 2 ] 1/2 • Check other forms of scaling e.g. F2=(1+ Q2/ n2 )n W2 • Implementation for R (and 2xF1) is done exactly - use empirical fits to R (agrees with NNLO+GP tgt mass for Q2>1); Need to compare to Jlab R data in resonance region. • Compare our model prediction with the Rein and Seghal model for the 1st 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) Arie Bodek, Univ. of Rochester

  40. 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. Arie Bodek, Univ. of Rochester

  41. F2(iron)/(deuteron) Nuclear effects on heavy targets • F2(deuteron)/(free N+P) What are nuclear effects for F2 versus XF3; what are they at low Q2; possible differences between Electron, Neutrino CC and Neutrino NC at low Q2 (Vector dominance effects). Arie Bodek, Univ. of Rochester

  42. Current understanding of R We find for R (Q2>1 GeV2) 1. Rw empirical fit works well (down to Q2=0.35 GeV2) 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 Q2=0. >> Use Rw for Q2 > 0.35 GeV2 For Q2 < 0.35 use: R(x, Q2 ) = = 3.207 {Q2 / [Q4 +1) } R(x, Q2=0.35 GeV2 ) * Plan to compare to recent Jlab Data for R in the Resonance region at low Q2. Arie Bodek, Univ. of Rochester

  43. Bodek and Yang hep-ex/0203009; Goergi and Politzer Phys. Rev. D14, 1829 (1976) Different Scaling variables for u,d,s,c in initial and u, d, s,c in final states and valence vs. sea For further study We Use: Xw = [Q2+B] / [2Mn + A] = x [Q2+B] / [Q2 + Ax] Could also try: x w = [Q2+B] / [ Mn (1+ (1+Q2/n2) 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 ?? x w’ =[ Q’ 2+B] / [Mn (1+ (1+Q2/n2) 1/2 +A] q x m I 2 m * 2 M • x =[Q2+m *2] / ( 2Mn) (quark final state m* mass) • x = Q2/[Mn (1+ (1+Q2/n2) 1/2 ] (initial proton mass) • x = [Q2+m *2] / [ Mn (1+ (1+Q2/n2) 1/2 ] combined • x = x [Q2+m *2] 2 / [Q2 + (Q4 +4x2M2 Q2) 1/2 ] • (Pi + q)2 = Pi2 + 2qPi + q2 = m*2 In general GP derive for initial quark mass m I and final massm * bound in a proton of mass M (at high Q2 these are current quark masses, but at low Q2maybe constituent masses? ) GP get 2Q’2 =[Q2+ m *2 -m I 2] + [ Q4 +2Q2(m *2 +m I 2 ) + (m *2 -m I 2 ) 2 ] 1/2 x = x [2Q’2] / [Q2 + (Q4 +4x2M2 Q2) 1/2 ] - note masses may depend on Q2 x = [Q’2] / [ Mn (1+(1+Q2/n2) 1/2 ] (equivalent form) 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. Arie Bodek, Univ. of Rochester

  44. Goergi and Politzer Phys. Rev. D14, 1829 (1976)]- GP (Pi + q)2 = m I 2 + 2qPi + q2 = m *2 and q = (n, q3) in lab 2qPi = 2 [nPi0 + q3 Pi3] = [Q2+ m *2 -m I 2] (eq. 1) ( note + sign sinceq3andPi3are pointing at each other) x = [Pi0 +Pi3] /[P0 + P3] --- frame invariant definition x = [Pi0 +Pi3] /M ---- In lab or [Pi0 +Pi3] = xM --- in lab x [Pi0 -Pi3] = [Pi0 +Pi3][Pi0 -Pi3] / M -- multiplied by [Pi0 -Pi3] Pi0 -Pi3= [ (Pi0 ) 2 - ( Pi3 ) 2 ]/ M = m I 2 /[xM] or [Pi0 -Pi3] = m I 2 /[xM] --- in lab Get 2 Pi0 = xM + m I 2 /[xM] Plug into (eq. 1) and 2 Pi3 = xM - m I 2 /[xM] { nxM + nm I 2 /[xM]} + {q3 xM - q3 m I 2 /[xM]} - [Q2+ m *2 -m I 2] = 0 a b c x2M 2 (n+ q3) -xM [Q2+ m *2 -m I 2] + m I 2(n- q3) = 0 >>> x = [-b +(b 2 - 4ac) 1/2 ] / 2a => solution use (n 2- q3 2) = q 2 = -Q 2 and (n+ q3) = n + n [ 1 +Q 2/n 2 ] 1/2 = n + n [ 1 +4M2 x2/Q 2 ] 1/2 Get x = [Q’2] / [ Mn (1+(1+Q2/n2) 1/2 ] x = [Q’2] / [M n (1 + [ 1 +4M2 x2/Q 2 ] 1/2)] (equivalent form) One Page Derivation: In general GP derive for initial quark mass m I and final massm * bound in a proton of mass M q Pi= Pi0,Pi3,mI Pf, m* P= P0 + P3,M where 2Q’2 =[Q2+ m *2 -m I 2] + [ Q4 +2Q2(m *2 +m I 2 ) + (m *2 -m I 2 ) 2]1/2 or x = x [2Q’2] / [Q2 + (Q4 +4x2M2 Q2) ] 1/2 (equivalent form) (at high Q2 these are current quark masses, but at low Q2 maybe constituent masses?) Arie Bodek, Univ. of Rochester

  45. xIs the correct variable which is Invariant in any frame : q3 and P in opposite directions. x In general GP derive for initial quark mass m I and final mass,mF=m *bound in a proton of mass M -- Page 1 q=q3,q0 PF= PF0,PF3,mF=m* PF= PI0,PI3,mI P= P0 + P3,M Arie Bodek, Univ. of Rochester

  46. xFor the case of non zero mI (note P and q3 are opposite) x In general GP derive for initial quark mass m I and final massmF=m*bound in a proton of mass M -- Page 2 q=q3,q0 PF= PF0,PF3,mF=m* PF= PI0,PI3,mI P= P0 + P3,M ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Keep terms with mI : multiply by xM and group terms inx qnd x2 x2M 2 (n+ q3) -xM [Q2+ m *2 -m I 2] + m I 2(n- q3) = 0 General Equation a b c => solution of quadratic equation x = [-b +(b 2 - 4ac) 1/2 ] / 2a use (n 2- q3 2) = q 2 = -Q 2 and (n+ q3) = n + n [ 1 +Q 2/n 2 ] 1/2 = n + n [ 1 +4M2 x2/Q 2 ] 1/2 Get x = [Q’2] / [ Mn (1+(1+Q2/n2) 1/2 ] x = [Q’2] / [M n (1 + [ 1 +4M2 x2/Q 2 ] 1/2)] or x = x [2Q’2] / [Q2 + √ (Q4 +4x2M2 Q2) ] (equivalent form) where 2Q’2 =[Q2+ m *2 -m I 2] + [ Q4 +2Q2(m *2 +m I 2 ) + (m *2 -m I 2 ) 2 ] 1/2 (at high Q2 these are current quark masses, but at low Q2 maybe constituent masses?) Arie Bodek, Univ. of Rochester

  47. Get x = [Q’2 +B ] / [ Mn (1+(1+Q2/n2) 1/2 +A ] x = [Q’2 +B ] / [M n (1 + [ 1 +4M2 x2/Q 2 ] 1/2) +A] or x = x [2Q’2 +2B ] / [Q2 + (Q4 +4x2M2 Q2) 1/2 +2Ax ] (equivalent form) where 2Q’2 =[Q2+ m *2 -m I 2] + [ Q4 +2Q2(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 Q2 +2B : current quarks (mi =mF =0) m *=m I : 2Q’2 =Q2+ [ Q4 +4Q2 m *2 ] 1/2 +2B: constituent mass (mi =mF =0.3 GeV) m I =0 : 2Q’2 =2Q2 + 2m *2 +2B : final state mass (mi =0,mF =charm) m *=0 :2Q’2 =2Q2 +2B -----> : initial constituent(mi=0.3); final state current (mF =0) denominator : [Q2 + (Q4 +4x2M2 Q2) 1/2 +2Ax ] at large Q2 ---> 2 Q2 + 2 M2x2 +2Ax Or [ Mn (1+(1+Q2/n2) 1/2 +A ]which at small Q2 ---> 2Mn + M2/x +A Therefore A = M2 x at large Q2 and M2/x at small Q2 ->A=constant is approximate;y OK versus Xw = [Q2+B] / [2Mn + A] = x [2Q2+2B] / [2Q2 + 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. x In general GP derive for initial quark mass m I and final massmF=m*bound in a proton of mass M --- Page 3 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 PF= PF0,PF3,mF=m* PF= PI0,PI3,mI P= P0 + P3,M Arie Bodek, Univ. of Rochester

  48. 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= (Wthreshold) 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 Wthreshold m*2 m* mF1 +mF2 quark quark threshold GeV GeV 2 GeV quarks e-P u or d u or d Mp+Mpion 1.12 0.29 0.53 0.01 e-P s s+sbar MLambdS+MK 1.61 1.63 1.28 0.34  -N d c+u MLamdaC+Mpion 2.42 4.89 2.21 1.36  -N s c+sbar MLamdaC+MK 2.78 6.76 2.60 1.52 e-P c c+cbar MLamdaC+MD 4.15 16.29 4.04 2.70 e-P c c+cbar Mp +MD+MD 4.68 20.90 4.57 2.70 x Initial quark mass m I and final massmF=m*bound in a proton of mass MPage 4 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 PF= PF0,PF3,mF=m* PF= PI0,PI3,mI P= P0 + P3,M Arie Bodek, Univ. of Rochester

  49. 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 - mud*2 is more relevant since mud*2 is already included in the scaling violation fits for the u and d PDFs Reaction initial state final state final state Wthreshold (m)*2  m* mF1 +mF2 quark quark threshold GeV GeV 2 GeV quarks e-P u or d u or d Mp+Mpion 1.12 0.00 0.00 0.00 e-P s s+sbar MLambda+MK 1.61 1.34 1.16 0.34 -N d c+u MLamdaC+Mpion 2.42 4.61 2.15 1.36  N s c+sbar MLamdaC+MK 2.78 6.47 2.54 1.52 e-P c c+cbar MLamdaC+MD 4.15 16.00 4.00 2.70 e-P c c+cbar Mp +MD+MD 4.68 20.62 4.54 2.70 FAST re-scaling (function ) is a crude implementation of A(W,Q2). x Initial quark mass m I and final massmF=m*bound in a proton of mass MPage 4 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 PF= PF0,PF3,mF=m* PF= PI0,PI3,mI P= P0 + P3,M Arie Bodek, Univ. of Rochester

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

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