1. Arie Bodek, Univ. of Rochester1 Quarks for Dummies: Quarks for Dummies: Modeling (e/  /  -N Cross Sections from Low to High Energies: from DIS to Resonance, to Quasielastic Scattering Arie Bodek, Univ. of Rochester Un-Ki Yang, Univ of Chicago Work in progress: most recently presented in invited talks at 1. NuFact02 -Imperial College, London, July, 2002 (2 talks) 2. DPF Meeting, Virginia May 2001 3. APS Meeting, New Mexico April, 2001 4. NuInt01, KEK Japan Dec. 2001 Final studies with  ’ w & A(W,Q2) in preparation for NuInt02, Irvine, Dec. 2002. oStudies in LO (  w HT scaling) - Being written for proceedings of NuFact 02 oStudies in LO (X w HT scaling) - Bodek and Yang: hep-ex/0203009 (2002) to appear in proceedings of NuInt 01 (Nuclear Physics B) o Studies in NNLO+HT - Yang and Bodek: Eur. Phys. J. C13, 241 (2000) oStudies in NLO+HT - Yang and Bodek: Phys. Rev. Lett 82, 2467 (1999) oStudies in 0th ORDER (QPM + X w HT scaling) - Bodek, el al PRD 20, 1471 (1979

2. Arie Bodek, Univ. of Rochester2 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 (essential for precise next generation neutrino oscillation experiments with super neutrino beams ) as well as at the 15-30 GeV (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.

3. Arie Bodek, Univ. of Rochester3 Examples of Current Low Energy Neutrino Data: Quasi-elastic cross section  tot /E

4. Arie Bodek, Univ. of Rochester4 How are PDFs Extracted from global fits to 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 Note: additional information on Antiquarks from Drell-Yan and on Gluons from p-pbar jets also used.

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) e.g. Bodek and Ritchie (Phys. Rev. D23, 1070 (1981) Resonance (low Q 2, W  - + 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. othen 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 terms of the two power series cannot be distinguished (a)Higher Twist: Interaction between Interacting and Spectator quarks via gluon exchange at Low Q2-mostly at low W (b) Interacting quark TM binding, initial Pt and Missing Higher Order QCD terms -In DIS region. -> (1/Q 2 ) or [Q 2 /(Q 4 +A)],… (1/Q 4 ). In NNLO p-QCD additional gluons emission: terms like  s 2 ( ~1/[ln(Q 2 /  2 )] )…  s 4 Spectator quarks are not Involved.  Pt  

8. Arie Bodek, Univ. of Rochester8 What are Higher Twist Effects - Page 2 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,  TM) initial state and final state quark masses (e.g. charm production)-  TM important at high x 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: important at low W Non-perturbative effects to satisfy gauge invariance and connection to photo- production [e.g. F 2 (, Q 2 = 0) = Q 2 / [Q 2 +C]= 0]. important at very low Q2. Higher Order QCD effects - to e.g. NNLO+ multi-gluon emission”looks like” Power higher twist corrections since a LO or NLO calculation do not take these into account, also quark intrinsic P T (terms like P T 2 /Q 2 ). Important at all x (look like Dynamic Higher Twist)

9. Arie Bodek, Univ. of Rochester9 Old Picture of fixed W scattering - form factors 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 final state nucleon: 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) =  {e i q. r  (r) d 3 r } = Electric form factor is the Fourier transform of the charge distribution. Similarly for the magnetization distribution for G M Form factors are relates to structure function by: 2xF 1 (x, Q 2 ) elastic = x 2 G M 2 elastic ( Q 2 )  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 Res. transition ( Q 2 ) 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 at Low W 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 A w (w, Q 2 ) 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. A w (w, Q 2 ) =  x-1) (times) {integral over , from pion threshold to  =1 } : local duality (This is just a check of local duality, better to use Ge,Gm) Resonance Production, W=M R, e.g. delta 1.238 resonance. The scattering is from a quark with a high value of , is such that that the final state interaction makes a low mass resonance. A w (w, Q 2 ) includes Breit-Wigners. Local duality 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 resonance A (w, Q 2 ) we could also reproduce the various Breit-Wigners + continuum. 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. Non-perturbative effect. There are no longitudinally polarized photons at Q 2 =0  L (, Q 2 ) = 0 limit as Q 2 -->0 Implies R (, Q 2 ) =  L /  T ~ Q 2 / [Q 2 +const] --> 0 limit as Q 2 -->0   -proton, ) =  T (, Q 2 ) limit as Q 2 -->0  implies   -proton, ) = 0.112 mb 2xF 1 (, Q 2 ) / (KQ 2 ) limit as Q 2 -->0   -proton, ) = 0.112 mb F 2 (, Q 2 ) D / KQ 2 limit as Q 2 -->0  or F 2 (, Q 2 ) ~ Q 2 / [Q 2 +C] --> 0 limit as Q 2 -->0 K = [1 - Q 2 / 2M  D = (1+ Q 2 / 2 )/(1+R) If we want PDFs to work down to Q 2 =0 where pQCD freezes The PDFs must be multiplied by a factor Q 2 / [Q 2 +C] (where C is a small number). The scaling variable x does not work since   -proton, ) =  T (, Q 2 ) 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 F 2 (, Q 2 ) = F 2 (  c, Q 2 ) = F 2 [B/ (2M , 0] limit as Q 2 -->0

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. oStudies in NLO - Yang and Bodek: Phys. Rev. Lett 82, 2467 (1999) ;ibid 84, 3456 (2000) oStudies in NNLO - Yang and Bodek: Eur. Phys. J. C13, 241 (2000) oStudies in LO - Bodek and Yang: hep-ex/0203009 (2002 ) oStudies in QPM 0th order - Bodek, el al PRD 20, 1471 (1979)

13. Arie Bodek, Univ. of Rochester13 Lessons from previous “NLO QCD” study Our NLO study comparing NLO PDFs to DIS SLAC, NMC, and BCDMS e/  scattering data on H and D targets shows (for Q 2 > 1 GeV 2 ) [ref:Yang and Bodek: Phys. Rev. Lett 82, 2467 (1999) ] oKinematic Higher Twist (target mass ) effects are large and important at large x, and must be included in the form of Georgi & Politzer  TM scaling. oDynamic Higher Twist effects are smaller, but need to be included. (A second NNLO study established their origin) oThe ratio of d/u at high x must be increased if nuclear binding effects in the deuteron are taken into account. oThe Very high x (=0.9) region - is described by NLO QCD (if target mass and renormalon higher twist effects are included) to better than 10%. SPECTATOR QUARKS modulate A(W,Q 2 ) ONLY. oResonance region: NLO pQCD + Target mass + Higher Twist describes average F 2 in the resonance region (duality works). Include A w (w, Q 2 ) resonance modulating function from spectator quarks later. A similar NNLO study using NNLO QCD we find that the “empirically measured “effective” Dynamic Higher Twist Effects in the NLO study come from the missing NNLO higher order QCD terms. [ref: Yang and Bodek Eur. Phys. J. C13, 241 (2000) ]

14. Arie Bodek, Univ. of Rochester14 F 2, R comparison of QCD+TM plot vs. NLO QCD+TM+HT (use QCD Renormalon Model for HT) PDFs and QCD in NLO + TM + QCD Renormalon Model for Dynamic HTdescribe the F2 and R data re well, with only 2 parameters. Dynamic HT effects are there but small

15. Arie Bodek, Univ. of Rochester15 Same study showing the QCD-only Plot 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 A simlar study of NLO QCD+TM vs. QCD+TM+HT (use here Empirical Model for Dynamic HT) - backup slide ** PDFs and QCD in NLO + TM + Empirical Model for Dynamic HT describe the data for F2 (only) reasonably well with 3 parameters. Dynamic HT effects are there but small 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) Georgi and Politzer Phys. Rev. D14, 1829 (1976): Well known  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 (Derivation of  TM in Appendix) 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 from TM

18. Arie Bodek, Univ. of Rochester18 Kinematic Higher-Twist (target mass:TM)  TM = Q 2 / [M (1+ (1+Q 2 / 2 ) 1/2 ) ] 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- Renormalon Model Use: Renormalon QCD model of Webber&Dasgupta- Phys. Lett. B382, 272 (1996), Two parameters a 2 and a 4. This model includes the (1/ Q 2 ) and (1/ Q 4 ) terms from gluon radiation turning into virtual quark antiquark fermion loops (from the interacting quark only, the spectator quarks are not involved). F 2 theory (x,Q 2 ) = F 2 PQCD+TM [1+ D 2 (x,Q 2 ) + D 4 (x,Q 2 ) ] D 2 (x,Q 2 ) = (1/ Q 2 ) [ a 2 / q (x,Q 2 ) ] ∫ (dz/z) c 2 (z) q(x/z, Q 2 ) D 4 (x,Q 2 ) = (1/ Q 4 ) [ 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. (the D 2 (x,Q 2 ) term is the same for 2xF 1 and, xF 3 ) 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). q-qbar loops

20. Arie Bodek, Univ. of Rochester20 QCD Power Law Corrections Renormalon QCD model of Webber&Dasgupta- Phys. Lett. B382, 272 (1996), includes only two parameters a 2 and a 4. This infrared renormalon model (only one renormalon chain of bubble graphs) leads the (1/ Q 2 ) and (1/ Q 4 ) terms from gluon radiation turning into virtual quark antiquark fermion loops (from the interacting quark), the spectator quarks are not involved). QCD is an asymptotic series which gets closer to the correct answer if taken up up to a certain N and then individual terms begin to factorially increase at any fixed x. This introduces an ambiguity into the theory (trying to estimate the size of the infinite number of remaining terms in the series). This ambiguity is referred to as the power corrections, or infrared renomalons (while ultraviolet renormalons are a power series in  s). Calculations show that the infrared renormalons (to estimate the remaining terms in the series) have a power law dependence, and therefore look like higher twist. Note that these power law corrections are from interactions of the Interacting Quark only, and have to do with the corrections for the missing higher order terms in QCD. These terms lead to multiple final state gluons (I.e. final state quark effective mass), and initial state Pt and effective mass from multiple gluon emission. What we have shown, is that NNLO QCD (with the world average of  s ) is already very good, since the “extracted” power law corrections within the renomalon model are very small. We will go back later and focus on a LEADING ORDER analysis. We can now correct for the missing higher order NLO, NNLO etc. within our new physical model for the renormalon Higher Twist (for LO) in terms of effective initial quark Pt and mass, and effective final quark mass, and enhanced Target Mass corrections. q-qbar loops Renormalon Power Corr. Higher Order QCD Corr.

21. Arie Bodek, Univ. of Rochester21 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 x  x  A w (w, Q 2 ) will account for interactions with spectator quarks

22. Arie Bodek, Univ. of Rochester22 Look at Q 2 = 8, 15, 25 GeV 2 very high x data- backup slide* Pion production threshold A w (w, Q 2 ) 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. oThe 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

23. Arie Bodek, Univ. of Rochester23 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 (but not 0) a2= -0.009 (in NNLO) versus –0.1( in NLO ) - > factor of 10 smaller, a4 nonzero

24. Arie Bodek, Univ. of Rochester24 “Converting” NLO PDFs to NNLO PDFs backup slide ** 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. True NNLO PDFs in a year or two Floating factors NNLO NLO

25. Arie Bodek, Univ. of Rochester25 Lessons from the NNLO pQCD analysis For Q 2 >1 GeV 2 The origin of the “empirically measured small dynamic higher twist effects” in NLO 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 (e.g. QCD renormalon) effects in NLO both have the same Q2 dependence at fixed x. Both involve only gluon and fermion loops off the legs of the interacting quark (not spectator quarks). (F 2NNL0 /F 2NLO )-1

26. Arie Bodek, Univ. of Rochester26 Derivation: for initial quark mass m I and final mass m * bound in a proton of mass M - INCLUDING Quark INITIAL P 2 t Georgi and Politzer Phys. Rev. D14, 1829 (1976)]- GP did not include Pt (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 +P 2 t) /[  M] or [Pi 0 -Pi 3 ] = (m I 2 +P 2 t) /[  M] --- in lab Get 2 Pi 0 =  M + (m I 2 +P 2 t) /[  M] Plug into (eq. 1) and 2 Pi 3 =  M - (m I 2 +P 2 t) /[  M] {  M + (m I 2 +P 2 t) /[  M] } + {q3  M - q3 (m I 2 +P 2 t) 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 +P 2 t) (  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  Add B and A account for effects of additional  m 2 from NLO and NNLO effects. (at high Q 2 these are current quark masses, but at low Q 2 maybe constituent masses?) Get  w= [Q’ 2 +B ] / [ M (1+(1+Q 2 / 2 ) 1/2 ) +A ] where2Q’ 2 = [Q 2 + m * 2 - m I 2 ] + [ ( Q 2 +m * 2 - m I 2 ) 2 + 4Q 2 (m I 2 +P 2 t ) ] 1/2 or 2Q’ 2 = [Q 2 + m F 2 - m I 2 ] + [ Q 4 +2 Q 2 (m F 2 + m I 2 +2P 2 t ) + (m F 2 - m I 2 ) 2 ] 1/2 If m i =0 2Q’ 2 = [Q 2 + m * 2 ] + [ ( Q 2 +m * 2 ) 2 + 4Q 2 ( P 2 t ) ] 1/2 P= P 0 + P 3,M Pf, m* Pi= Pi 0,Pi 3,m I q

27. Arie Bodek, Univ. of Rochester27 Pseudo Next to Leading Order Calculations  Add B and A account for effects of additional  m 2 from NLO and NNLO effects. There are many examples of taking Leading Order Calculations and correcting them for NLO and NNLO effects using external inputs from measurements or additional calculations: e.g. 2.Direct Photon Production - account for initial quark intrinsic Pt and Pt due to initial state gluon emission in NLO and NNLO processes by smearing the calculation with the MEASURED Pt extracted from the Pt spectrum of Drell Yan dileptons as a function of Q2 (mass). 3.W and Z production in hadron colliders. Calculate from LO, multiply by K factor to get NLO, smear the final state W Pt from fits to Z Pt data (within gluon resummation model parameters) to account for initial state multi-gluon emission. 4.K factors to convert Drell-Yan LO calculations to NLO cross sections. Measure final state Pt. 3.K factors to convert NLO PDFs to NNLO PDFs 4.Prediction of 2xF1 from leading order fits to F2 data, and imputing an empirical parametrization of R (since R=0 in QCD leading order). 5.THIS IS THE APPROACH TAKEN HERE. i.e. a Leading Order Calculation with input of effective initial quark masses and Pt and final quark masses, all from gluon emission. where2Q’ 2 = [Q 2 + m * 2 - m I 2 ] + [ ( Q 2 +m * 2 - m I 2 ) 2 + 4Q 2 (m I 2 +P 2 t ) ] 1/2 or 2Q’ 2 = [Q 2 + m F 2 - m I 2 ] + [ Q 4 +2 Q 2 (m F 2 + m I 2 +2P 2 t ) + (m F 2 - m I 2 ) 2 ] 1/2 P= P 0 + P 3,M Pf, m* Pi= Pi 0,Pi 3,m I q Xw= [Q 2 +B ] / [ 2M +A ]  w= [Q’ 2 +B ] / [ M (1+(1+Q 2 / 2 ) 1/2 ) +A ] Use LO PDFs (Xw) times (Q 2 /Q 2 +C) And PDFs (  w) times (Q 2 /Q 2 +C)

28. Arie Bodek, Univ. of Rochester28 At low x, Q 2 “NNLO terms” look similar to “kinematic final state mass higher twist” or “effective final state quark mass -> “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 pions etc Or gluon emission from the Interacting quark   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

29. Arie Bodek, Univ. of Rochester29 At high x, “NNLO QCD terms” have a similar form to the “kinematic -Georgi-Politzer  TM TM effects” -> look like “enhanced” QCD evolution at low Q Target Mass (G-P):  tgt mass  2 TM M + 2  TM q.P - Q 2 = 0 (Q 2 = -q 2 )  mnemonic- 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 +B] / [ M (1+(1+Q 2 / 2 ) 1/2 ) +A ] - includes both initial state target proton mass and final state M* mass effect) - Exact derivation in Appendix. Add B and A account for additional  m 2 from NLO and NNLO effects. (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:Georgi and Politzer Phys. Rev. D14, 1829 (1976)]]

30. Arie Bodek, Univ. of Rochester30 Towards a unified model oWe learned that the NNLO+ TM describes the DIS and resonance data very well. Theoretically, this breaks down at low Q 2 <1 Practically, no way to implement it in MC oHT 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. oWell, can we do something with LO QCD and LO PDFs ? YES Different scaling variables Resonance, higher twist, and TM  =0.9 X=-0.95 M* (final state interaction) Or multigluon emission  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 pre-QCD early fits to SLAC data in 1972) And then follow up by using the above  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

31. Arie Bodek, Univ. of Rochester31 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. i.e. SPECTATOR QUARKS ONLY MODULATE THE CROSS SECTION AT LOW W. THEY DO NOT CONTRIBUTE TO DIS HT. 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 Pt, 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, Pt 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. I.e. Do a fit with: 4. F 2 (x, Q 2 ) = Q 2 / [Q 2 +C] F 2QCD (  w, Q 2 ) A (w, Q 2 ) (set A w (w, Q 2 ) =1 for now - spectator quarks)  w= [Q 2 +B ] / [ M (1+(1+Q 2 / 2 ) 1/2 ) + A] or Xw = [Q 2 +B] /[2M + A] 6. B=effective final state quark mass. A=enhanced TM term, [Ref:Bodek and Yang hep-ex/0203009] Philosophy

32. Arie Bodek, Univ. of Rochester32 Modified LO PDFs for all Q 2 (including 0) 1. Start with GRV94 LO (Q 2 min =0.23 GeV 2 ) - describe F2 data at high Q 2 2A. Replace X with a new scaling, Xw  x= [Q 2 ] / [2M   Xw= [Q 2 +B] / [2M +A]  A: initial binding/target mass effect plus NLO +NNLO terms )  B: final state mass effect(but also photo production limit) 2B. Or Replace X with a new scaling,  w  w= [Q 2 +B ] / [ M (1+(1+Q 2 / 2 ) 1/2 ) + A] 3. Multiply all PDFs by a factor of Q 2 /[Q 2 +c] for photo prod. Limit+non-perturbative F 2 (x, Q 2 ) = Q 2 /[Q 2 +C] F 2QCD (  w, Q 2 ) A (w, Q 2 ) 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) A. 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. B. HERE INCLUDE DATA WITH Q2<1 if it is not in the resonance region Modified LO GRV94 PDFs with three parameters (a new scaling variable, Xw,  w) describe DIS F2 H, D data (SLAC/BCDMS/NMC) well. A=1.735, B=0.624, and C=0.188 Xw (note for Xw, A includes the Proton M) A=0.700, B=0.327, and C=0.197  w works better as expected Keep final state interaction resonance modulating function A (w, Q 2 )=1 for now (will be included in the future). Fit DIS Only Compare with SLAC/Jlab resonance data (not used in our fit) -> A (w, Q 2 ) Compare with photo production data (not used in our fit)-> check on C Compare with medium energy 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]

33. Arie Bodek, Univ. of Rochester33 Comparison of Xw Fit and  w Fit backup slide * Xw = [Q 2 +B] / [2M +A] used in 1972  w = [Q 2 +B ] / [ M (1+(1+Q 2 / 2 ) 1/2 ) + A] (theoretically derived) Multiply all PDFs by a factor of Q 2 /[Q 2 +C] 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 and the scaling variable, Xw, describe DIS F2 H, D data (SLAC/BCDMS/NMC) reasonably well. A=1.735, B=0.624, and C=0.188 (+- 0.022) (+-0.014) ( +-0.004)  2 = 1555 /958 DOF With  w A and B are smaller Modified LO GRV94 PDFs with three parameters and the 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. GRV94 Assumed to be PEFECT (no f(x) floating factors). Better fits expected with GRV98 and floating factors f(x) Same construction for Xw and  w fits Comparison

34. Arie Bodek, Univ. of Rochester34 LO+HT fit Comparison with DIS F 2 (H, D) data [These SLAC/BCDMS/NMC are used in this Xw fit  2 = 1555 /958 DOF ] Proton Deuteron

35. Arie Bodek, Univ. of Rochester35 LO+HT fit Comparison with DIS F 2 (H, D) data SLAC/BCDMS/NMC  w works better  2 = 1351 /958 DOF Proton Deuteron

36. Arie Bodek, Univ. of Rochester36 Comparison with F2 resonance data [ SLAC/ Jlab] (These data were not included in this Xw 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 <---Xw fit. For now, lets compare to neutrino data and photoproduction Next repeat with  w Next repeat with GRV98 and f(x) Note QCD evolution between Q2=0.85 qnd Q2=0.25 small. Can use GRV98 then: add the A w (w, Q 2 ) modulating function (to account for interaction with spectator quarks at low W) Also, check the x=1 Elasic Scattering Limit. Q 2 = 0.07 GeV 2 Q 2 = 1 5 GeV 2 Q 2 = 2 5 GeV 2 Q 2 = 3 GeV 2 Q 2 = 9 GeV 2 Q 2 = 1. 4 GeV 2 Q 2 = 0.8 5 GeV 2 Q 2 = 0.25 GeV 2

37. Arie Bodek, Univ. of Rochester37 Comparison with F2 resonance data [ SLAC/ Jlab] (These data were not included in this  w fit) The modified LO GRV94 PDFs with a new scaling variable,  w 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 <---  w fit. For now, lets compare to neutrino data and photoproduction Next repeat with GRV98 and f(x) Note QCD evolution between Q2=0.85 qnd Q2=0.25 small. Can use GRV98 then: add the A w (w, Q 2 ) modulating function. (to account for interaction with spectator quarks at low W) Also, check the x=1 Elasic Scattering Limit. Q 2 = 0.07 GeV 2 Q 2 = 1 5 GeV 2 Q 2 = 2 5 GeV 2 Q 2 = 3 GeV 2 Q 2 = 9 GeV 2 Q 2 = 1. 4 GeV 2 Q 2 = 0.8 5 GeV 2 Q 2 = 0.25 GeV 2

38. Arie Bodek, Univ. of Rochester38 Comparison of LO+HT to neutrino data on Iron [CCFR] (not used in this Xw 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. Will repeat with  w Construction Xw fit

39. Arie Bodek, Univ. of Rochester39 Comparison with photo production data (not included in this 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(see appendix R data figure) 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. So will switch to GRV98 or 2002 LO PDFs. 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 Xw

40. Arie Bodek, Univ. of Rochester40 Comparison with photo production data (not included in this  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 is ver small (see appendix R data figure) 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  w ), the GRV94 need to be updated to fit latest HERA data at very low x and low Q 2. So will switch to GRV98 or 2002 LO PDFs. 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  w w

41. Arie Bodek, Univ. of Rochester41 The GRV LO need to be updated to fit latest HERA data at very low x and low Q 2. We used 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. Now it looks like we can freeze at Q2=0.8 and have no problems. So switch to modern PDFs. 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

42. Arie Bodek, Univ. of Rochester42 Summary Our modified GRV94 LO PDFs with a modified scaling variables, Xw and  w describe all SLAC/BCDMS/NMC DIS data. (We will investigate further refinements to  w, and move to more modern PDFs GRV98, 2002 MRST and CTTEQ LO PDFs. ) 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 This work is continuing… focus on further improvement to  w (although very good already) and A(W, Q 2) (low W spectator quark modulating function). What are the further improvement in  w - Mostly to reduce the size of the three free parameters a, b, c as more theoretically motivated terms are added into the formalism (mostly intellectual curiosity, since the model is already good enough).

43. Arie Bodek, Univ. of Rochester43 Future Work - part 1 Implement A e/  (W,Q 2 ) resonances into the model for F 2 with  w scaling. For this need to fit all DIS and SLAC and JLAB resonance date and Photo- production H and D data and CCFR neutrino data. Check for local duality between  w scaling curve and elastic form factors Ge, Gm in electron scattering. - Check method where its applicability will break down. Check for local duality of  w scaling curve and quasielastic form factors G A, G V in quasielastic neutrino and antineutrino scattering.- Good check on the applicability of the method in predicting exclusive production of strange and charm hyperons Compare our model prediction with the Rein and Seghal model for the 1 st resonance (in neutrino scattering). Implement differences between  and e/  final state resonance masses in terms of A,  bar (W,Q 2 ) {See Appendix) Look at Jlab and SLAC heavy target data for possible Q 2 dependence of nuclear dependence on Iron. 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 update Rw to include Jlab R data in resonance region. Compare to low-energy neutrino data (only low statistics data, thus new measurements of neutrino differential cross sections at low energy are important). Check other forms of scaling e.g. F 2 =(1+ Q 2 /    W 2 (for low energies)

44. Arie Bodek, Univ. of Rochester44 Future Work - part 2 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., Note:  w = [Q 2 +B ] / [ M (1+ (1+Q 2 / 2 ) 1/2 ) +A ] assumes m F = m i =0, P 2 t=0  More sophisticated General expression (see derivation in Appendix):  w ’ =[ Q’ 2 +B ] / [M (1+ (1+Q 2 / 2 ) 1/2 ) +A] with 2Q’ 2 = [Q 2 + m F 2 - m I 2 ] + [ ( Q 2 + m F 2 - m I 2 ) 2 + 4Q 2 (m I 2 +P 2 t) ] 1/2 or 2Q’ 2 = [Q 2 + m F 2 - m I 2 ] + [ Q 4 +2 Q 2 (m F 2 + m I 2 +2P 2 t ) + (m F 2 - m I 2 ) 2 ] 1/2 Here B and A account for effects of additional  m 2 from NLO and NNLO effects. However, one can include P 2 t, as well as m F, m i as the current quark masses (e.g. Charm, production in neutrino scattering, strange particle production etc.). In  w, B and A account for effective masses+initial Pt. When including Pt in the fits, constrain the Q2 dependence of Pt to agree with the measured mean Pt of Drell Yan data versus Q2. Include a floating factor f(x) to change the x dependence of the GRV94 PDFs such that they provide a good fit all high energy DIS, HERA, Drell-Yan, W-asymmetry, CDF Jets etc, for a global PDF QCD LO fit to include Pt, quark masses A, B for  w scaling and the Q 2 /(Q 2 +C) factor, and A e/  (W,Q 2 ) as a first step towards modern PDFs Later work with PDF fitters to produce PDFs: GRV-LO-02-Xsiw, MRST-LO-00- xsiw, CTEQ-LO-02-Xsiw, we should good down to Q 2 =0, including A (W,Q 2 ), A, B, C, Pt, quark masses etc. I.e. fit everything all at once. Put in fragmentation functions versus W, Q2, quark type and nuclear target

45. Arie Bodek, Univ. of Rochester45 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.

46. Arie Bodek, Univ. of Rochester46 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).

47. Arie Bodek, Univ. of Rochester47 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.

48. Arie Bodek, Univ. of Rochester48 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; Georgi 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

49. Arie Bodek, Univ. of Rochester49 One Page Derivation: In general GP derive for initial quark mass m I and final mass m * bound in a proton of mass M (neglect quark initial Pt) Georgi 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 +B ] / [ M (1+(1+Q 2 / 2 ) 1/2 ) +A ]  = [Q’ 2 +B] / [M  [  +  4M 2 x 2 / Q 2 ] 1/2 ) +A ] (equivalent form) Add B and A account for effects of additional  m 2 from NLO and NNLO effects. (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 +2B ] / [Q 2 + (Q 4 +4x 2 M 2 Q 2 ) 1/2 +2Ax ] (equivalent form) P= P 0 + P 3,M Pf, m* Pi= Pi 0,Pi 3,m I q

50. Arie Bodek, Univ. of Rochester50 Derivation: for initial quark mass m I and final mass m * bound in a proton of mass M - INCLUDING Quark INITIAL P 2 t Georgi and Politzer Phys. Rev. D14, 1829 (1976)]- GP did not include Pt (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 +P 2 t) /[  M] or [Pi 0 -Pi 3 ] = (m I 2 +P 2 t) /[  M] --- in lab Get 2 Pi 0 =  M + (m I 2 +P 2 t) /[  M] Plug into (eq. 1) and 2 Pi 3 =  M - (m I 2 +P 2 t) /[  M] {  M + (m I 2 +P 2 t) /[  M] } + {q3  M - q3 (m I 2 +P 2 t) 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 +P 2 t) (  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 +B ] / [ M (1+(1+Q 2 / 2 ) 1/2 ) +A ]  Add B and A account for effects of additional  m 2 from NLO and NNLO effects. (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 2 +m * 2 - m I 2 ) 2 + 4Q 2 (m I 2 +P 2 t ) ] 1/2 or 2Q’ 2 = [Q 2 + m F 2 - m I 2 ] + [ Q 4 +2 Q 2 (m F 2 + m I 2 +2P 2 t ) + (m F 2 - m I 2 ) 2 ] 1/2 If m i =0 2Q’ 2 = [Q 2 + m * 2 ] + [ ( Q 2 +m * 2 ) 2 + 4Q 2 ( P 2 t ) ] 1/2 P= P 0 + P 3,M Pf, m* Pi= Pi 0,Pi 3,m I q

51. Arie Bodek, Univ. of Rochester51 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 (neglect quark initial Pt)  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

52. Arie Bodek, Univ. of Rochester52 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 (neglect quark initial Pt)  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 +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 (at high Q 2 these are current quark masses, but at low Q 2 maybe constituent masses?) Add B and A account for effects of additional  m 2 from NLO and NNLO effects. q=q3,q0

53. Arie Bodek, Univ. of Rochester53 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 - (neglect quark initial Pt) why did Xw work in 1970 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 +2B = 2 Q 2 +2B : current quarks ( m i = m F =0) m * =m I : 2Q’ 2 +2B = 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 +2B = 2Q 2 + 2 m * 2 +2B : final state mass ( m i =0, m F =charm) m * =0 : 2Q’ 2 +2B = 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 approximately 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

54. Arie Bodek, Univ. of Rochester54 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/final quark masses for u-I, u-F, d-I,d-F, s-I,s-F,c-F. and A and B

55. Arie Bodek, Univ. of Rochester55 Initial quark mass m I and final mass m F =m * bound in a proton of mass M Page 5 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

56. Arie Bodek, Univ. of Rochester56 Examples of Low Energy Neutrino Data: Total (inelastic and quasielastic) cross section E GeV

57. Arie Bodek, Univ. of Rochester57 Examples of Current Low Energy Neutrino Data: Single charged and neutral pion production

58. Arie Bodek, Univ. of Rochester58 Examples of Current Low Energy Neutrino Data: Quasi-elastic cross section

59. Arie Bodek, Univ. of Rochester59 Appendix B - Studies of d/u Comparison of electron, muon, neutrino and e-p, and hadron collider data.

60. Arie Bodek, Univ. of Rochester60 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

61. Arie Bodek, Univ. of Rochester61 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).

62. Arie Bodek, Univ. of Rochester62 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)

63. Arie Bodek, Univ. of Rochester63 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)

64. Arie Bodek, Univ. of Rochester64 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.

65. Arie Bodek, Univ. of Rochester65 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

66. Arie Bodek, Univ. of Rochester66 Iinitial quark mass m I and final mass,m F =m * bound in a proton of mass M -- Page 1 INCLUDE quark initial Pt)  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

67. Arie Bodek, Univ. of Rochester67 initial quark mass m I and final mass m F =m * bound in a proton of mass M -- Page 2 INCLUDE quark initial Pt)  For the case of non zero m I,P t (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 all terms here and : multiply by  M and group terms in  qnd  2  2 M 2 (  q3) -  M [Q 2 + m F 2 - m I 2 ] + [m I 2 +Pt 2 (  q3) 2 ] = 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 q=q3,q0 or 2Q’ 2 = [Q 2 + m F 2 - m I 2 ] + [ Q 4 +2 Q 2 (m F 2 + m I 2 +2P 2 t ) + (m F 2 - m I 2 ) 2 ] 1/2   w = [Q’ 2 +B] / [M  [  +  4M 2 x 2 / Q 2 ] 1/2 ) +A] (equivalent form)  w = x [2Q’ 2 + 2B] / [Q 2 + (Q 4 +4x 2 M 2 Q 2 ) 1/2 +2Ax ] (equivalent form)  w = [Q’ 2 +B] / [ M (1+(1+Q 2 / 2 ) ) 1/2 +A]  where 2Q’ 2 = [Q 2 + m F 2 - m I 2 ] + [ ( Q 2 +m F 2 - m I 2 ) 2 + 4Q 2 (m I 2 +P 2 t) ] 1/2  Add B and A account for effects of additional  m 2 from NLO and NNLO effects.

68. Arie Bodek, Univ. of Rochester68 Initial quark mass m I and final mass m F =m * bound in a proton of mass M --- Page 3 - (INCLUDE initial Pt) why did Xw work in 1970 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 2 +m * 2 - m I 2 ) 2 + 4Q 2 (m I 2 +P 2 t ) ] 1/2 or 2Q’ 2 = [Q 2 + m F 2 - m I 2 ] + [ Q 4 +2 Q 2 (m F 2 + m I 2 +2P 2 t ) + (m F 2 - m I 2 ) 2 ] 1/2 Numerator: = x [2Q’ 2 ]: Special cases for 2Q’ 2 +2B m * =m I =0: 2Q’ +2B = Q 2 +2B+ [ Q 4 + 4 Q 2 P 2 t ] 1/2 : current quarks ( m i = m F =0) m * =m I : 2Q’ 2 +2B = Q 2 + [ Q 4 +4 Q 2 m * 2 + 4 Q 2 P 2 t ] 1/2 +2B: constituent mass ( m i = m F =0.3 GeV) m I =0 :2Q’ 2 +2B = Q 2 + m F 2 + [ ( Q 2 +m * 2 ) 2 + 4Q 2 ( P 2 t ) ] 1/2 2 +2B : final state mass ( m i =0, m F =charm) m * =0 : 2Q’ 2 +2B = Q 2 - m I 2 + [ ( Q 2 +- m I 2 ) 2 + 4Q 2 (m I 2 +P 2 t ) ] 1/2 -----> : 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 approximately 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

69. Arie Bodek, Univ. of Rochester69 GP target mass:TM backup slide ** TM effects yield a non-xero R at large x  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 (Derivation of  TM in Appendix) 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 ]) R = k 2 (F 2 /2x F 1 ) -1 In leading order R = {F 2 pQCD ( , Q 2 ) x 2 k 2 / [k 3  2 ] /2x 2 F 1 pQCD ( , Q 2 )/ [k  ]} -1 R ={F 2 pQCD ( , Q 2 ) / 2  F 1 pQCD ( , Q 2 ) } -1

70. Arie Bodek, Univ. of Rochester70 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

71. Arie Bodek, Univ. of Rochester71 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 - University of Rochester Next-to-next-to-leading Order Fits with HT Corrections and Modeling (e/  /  Cross Sections from DIS to Resonance - (25 min + 15 min)- review 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 Summary Talk