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Arie Bodek, Univ. of Rochester Un-Ki Yang, Univ of Chicago

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  1. Quarks for Dummies:Modeling (e/ m /n-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 • NuFact02 -Imperial College, London , July, 2002 (2 talks) • DPF Meeting, Virginia May 2001 • APS Meeting, New Mexico April, 2001 • NuInt01, KEK Japan Dec. 2001 Final studies with x’w & A(W,Q2) in preparation for NuInt02, Irvine, Dec. 2002. • Studies in LO (xw HT scaling) - Being written for proceedings of NuFact 02 • Studies in LO (Xw HT scaling) - Bodek and Yang: hep-ex/0203009 (2002) to appear in proceedings of NuInt 01 (Nuclear Physics B) • Studies in NNLO+HT - Yang and Bodek: Eur. Phys. J. C13, 241 (2000) • Studies in NLO+HT - Yang and Bodek: Phys. Rev. Lett 82, 2467 (1999) • Studies in 0th ORDER (QPM + Xw HT scaling) - Bodek, el al PRD 20, 1471 (1979 Arie Bodek, Univ. of Rochester

  2. 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 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

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

  4. How are PDFs Extracted from global fits to High Q2 Deep Inelastic e/m/n Data Note: additional information on Antiquarks from Drell-Yan and on Gluons from p-pbar jets also used. 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) e.g. 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…. (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/Q2 ) or [Q2/(Q4+A)],… (1/Q4 ).  Pt • While pQCD predicts terms in as2 ( ~1/[ln(Q2/ L2 )] )… as4 etc… • (i.e. LO, NLO, NNLO etc.) In the few GeV region, the terms of the two power series cannot be distinguished In NNLO p-QCD additional gluons emission: terms like as2 ( ~1/[ln(Q2/ L2 )] )… as4 Spectator quarks are not Involved. In pQCD high Q2 impulse approximation, the interacting quark and the spectator quarks are resolved and do not affect each other.  Arie Bodek, Univ. of Rochester

  8. What are Higher Twist Effects - Page 2 • 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, xTM) initial state and final state quark masses (e.g. charm production)- xTMimportant 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. F2(n ,Q2 =0) = Q2/ [Q2+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 PT (terms like PT2/Q2). Important at all x (look like Dynamic Higher Twist) Arie Bodek, Univ. of Rochester

  9. Old Picture of fixed W scattering - form factors • 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 final state nucleon: 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) =  {e i q . r r(r) d3r} = Electric form factor is the Fourier transform of the charge distribution. Similarly for the magnetization distribution for GM Form factors are relates to structure function by: • 2xF1(x ,Q2)elastic = x2 GM2 elastic (Q2)d (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 Res. transition(Q2)BW (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 at Low W 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 Aw(w, Q2 ) 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. Aw(w, Q2 ) = d (x-1) (times) {integral over x, from pion threshold to x =1 } : local duality (This is just a check of local duality, better to use Ge,Gm) 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. Aw(w, Q2 ) 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 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 resonanceA (w, Q2 ) we could also reproduce the various Breit-Wigners + continuum. 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. Non-perturbative effect. • There are no longitudinally polarized photons at Q2=0 • L(n, Q2) = 0 limit as Q2 -->0 • Implies R(n, Q2) = L/ T ~ Q2/ [Q2+const] --> 0 limit as Q2 -->0 • s(g-proton, n) = T(n, Q2) limit as Q2 -->0 • implies s(g-proton, n) = 0.112 mb 2xF1(n, Q2) / (KQ2)limit as Q2 -->0 • s(g-proton, n) = 0.112 mb F2(n, Q2) D / KQ2limit as Q2 -->0 • or F2(n, Q2) ~ Q2/ [Q2+C] --> 0 limit as Q2 -->0 • K= [1 - Q2/ 2M D = (1+ Q2/ 2 )/(1+R) • If we want PDFs to work down to Q2=0 where pQCD freezes • The PDFs must be multiplied by a factor Q2/ [Q2+C] (where C is a small number). • The scaling variable x does not work since s(g-proton, n) = T(n, Q2) • 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 • F2(n, Q2) = F2(xc, Q2) = F2 [B/ (2Mn), 0] limit as Q2 -->0 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) • Studies in QPM 0th order - Bodek, el al PRD 20, 1471 (1979) Arie Bodek, Univ. of Rochester

  13. Lessons from previous “NLO QCD” study • Our NLO study comparing NLO PDFs to DIS SLAC, NMC, and BCDMS e/m scattering data on H and D targets shows (for Q2 > 1 GeV2) [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. (A second NNLO study established their origin) • 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 renormalon higher twist effects are included) to better than 10%. SPECTATOR QUARKS modulate A(W,Q2) ONLY. • Resonance region: NLO pQCD + Target mass + Higher Twist describes average F2 in the resonance region (duality works). Include Aw(w, Q2 ) 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) ] Arie Bodek, Univ. of Rochester

  14. F2, R comparison of QCD+TM plotvs. 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 Arie Bodek, Univ. of Rochester

  15. Same study showing the QCD-only Plotvs. 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. 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. 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) Georgi and Politzer Phys. Rev. D14, 1829 (1976): Well known xTM= { 2x / [1 + k ] } [1+ Mc2 /Q2 ] (last term only for heavy charm product) k= ( 1 +4x2M2 /Q2) 1/2 (Derivation of xTM in Appendix) 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 from TM Arie Bodek, Univ. of Rochester

  18. Kinematic Higher-Twist (target mass:TM)x TM = Q2/ [Mn(1+(1+Q2/n2) 1/2 )] 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- Renormalon Model • Use: Renormalon QCD model of Webber&Dasgupta- Phys. Lett. B382, 272 (1996), Two parameters a2 and a4. This model includes the (1/ Q2) and (1/ Q4) terms from gluon radiation turning into virtual quark antiquark fermion loops (from the interacting quark only,the spectator quarks are not involved). • F2 theory (x,Q2) = F2PQCD+TM [1+ D2 (x,Q2) + D4 (x,Q2) ] D2 (x,Q2) = (1/ Q2) [ a2/ q (x,Q2) ] ∫ (dz/z) c2(z) q(x/z, Q2) D4 (x,Q2) = (1/ Q4) [ 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 . (the D2 (x,Q2) term is the same for 2xF1 and , xF3 ) Fit a2 and a4to experimental data for F2 and R=FL/2xF1. F2 data (x,Q2) = [ F2measured + l dF2 syst ] ( 1+ N ) : c2 weighted by errors where N is the fitted normalization (within errors) and dF2 syst is the is the fitted correlated systematic error BCDMS (within errors). q-qbar loops Arie Bodek, Univ. of Rochester

  20. QCD Power Law Corrections Renormalon QCD model of Webber&Dasgupta- Phys. Lett. B382, 272 (1996), includes only two parameters a2 and a4. This infrared renormalon model (only one renormalon chain of bubble graphs) leads the (1/ Q2) and (1/ Q4) 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. Higher Order QCD Corr. q-qbar loops Renormalon Power Corr. Arie Bodek, Univ. of Rochester

  21. Very high x F2 proton data (DIS + resonance)(not included in the original fits Q2=1. 5 to 25 GeV2) Q2= 25 GeV2Ratio 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 x  x  Aw(w, Q2 ) will account for interactions with spectator quarks Arie Bodek, Univ. of Rochester

  22. Pion production threshold Aw(w, Q2 ) 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-backup slide* Ratio F2data/F2pQCD+TM+HT Q2= 9 GeV2 Q2= 15 GeV2 Q2= 25 GeV2 Arie Bodek, Univ. of Rochester

  23. F2, 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 Arie Bodek, Univ. of Rochester

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

  25. Lessons from the NNLO pQCD analysis • For Q2>1 GeV2 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) (F2NNL0/F2NLO)-1 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). Arie Bodek, Univ. of Rochester

  26. Georgi and Politzer Phys. Rev. D14, 1829 (1976)]- GP did not include Pt (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 +P2t) /[xM] or [Pi0 -Pi3] = (m I 2 +P2t) /[xM] --- in lab Get 2 Pi0 = xM + (m I 2 +P2t) /[xM] Plug into (eq. 1) and 2 Pi3 = xM - (m I 2 +P2t) /[xM] { nxM + n(m I 2 +P2t) /[xM]} + {q3 xM - q3 (m I 2 +P2t)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 +P2t)(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 Derivation: for initial quark mass m I and final massm * bound in a proton of mass M - INCLUDING Quark INITIAL P2t q Pi= Pi0,Pi3,mI Pf, m* P= P0 + P3,M Get xw= [Q’2 +B] / [ Mn (1+(1+Q2/n2) 1/2 ) +A ] where 2Q’2 =[Q2+ m *2 -m I 2] + [ (Q2+m *2 -m I 2 ) 2 + 4Q2 (m I 2+P2t) ]1/2 or 2Q’2 =[Q2+ m F2 -m I 2] + [ Q4 +2Q2(m F2 +m I 2 +2P2t ) + (m F2 -m I 2 ) 2 ] 1/2 If mi=0 2Q’2 =[Q2+ m *2 ] + [ (Q2+m *2 ) 2 + 4Q2 (P2t) ]1/2 AddB and Aaccount for effects of additional  m2 from NLO and NNLO effects. (at high Q2 these are current quark masses, but at low Q2 maybe constituent masses?) Arie Bodek, Univ. of Rochester

  27. Pseudo Next to Leading Order Calculations Use LO PDFs (Xw) times (Q2/Q2+C) And PDFs (xw) times (Q2/Q2+C) q Pi= Pi0,Pi3,mI Xw= [Q2 +B] / [ 2Mn+A ] xw= [Q’2 +B] / [ Mn (1+(1+Q2/n2) 1/2 ) +A ] Pf, m* P= P0 + P3,M where 2Q’2 =[Q2+ m *2 -m I 2] + [ (Q2+m *2 -m I 2 ) 2 + 4Q2 (m I 2+P2t) ]1/2 or 2Q’2 =[Q2+ m F2 -m I 2] + [ Q4 +2Q2(m F2 +m I 2 +2P2t ) + (m F2 -m I 2 ) 2 ] 1/2 AddB and Aaccount for effects of additional  m2 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. 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). 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. K factors to convert Drell-Yan LO calculations to NLO cross sections. Measure final state Pt. K factors to convert NLO PDFs to NNLO PDFs Prediction of 2xF1 from leading order fits to F2 data , and imputing an empirical parametrization of R (since R=0 in QCD leading order). 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. Arie Bodek, Univ. of Rochester

  28. At low x, Q2 “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 Charm production s to c quarks in neutrino scattering-slow rescaling u M* (final state interaction) Production of pions etc Or gluon emission from the Interacting quark 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

  29. At high x, “NNLO QCD terms” have a similar form to the “kinematic -Georgi-Politzer x TMTM effects” -> look like “enhanced” QCD evolution at low Q Target Mass (G-P): x - tgt mass Final state mass • x2 TMM + 2 x TM q.P - Q2 = 0(Q2 = -q2 ) • mnemonic- 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+B] / [ Mn (1+(1+Q2/n2) 1/2 ) +A ] - includes both initial state target proton mass and final state M* mass effect) - Exact derivation in Appendix. Add B and Aaccount for additional  m2 from NLO and NNLO effects. 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:Georgi and Politzer Phys. Rev. D14, 1829 (1976)]] QCD evolution High x Mproton Ln Q2 Arie Bodek, Univ. of Rochester

  30. 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<1 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 LO QCD and LO PDFs ? YES Resonance, higher twist, and TM q X=-0.95 x=0.9 M* (final state interaction) Or multigluon emission 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 pre-QCD early fits to SLAC data in 1972) And then follow up by using the above 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

  31. 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. 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 Q2=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 Q2=1, and mimic the higher order QCD terms for Q2>1 (in terms of effective masses, Pt due to gluon emission). 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. I.e. Do a fit with: F2(x, Q2 ) = Q2/ [Q2+C] F2QCD(x w, Q2) A (w, Q2 ) (set Aw(w, Q2 ) =1 for now - spectator quarks) x w= [Q2+B] / [ Mn (1+(1+Q2/n2) 1/2 ) + A] or Xw = [Q2+B] /[2Mn + A] B=effective final state quark mass. A=enhanced TM term,[Ref:Bodek and Yang hep-ex/0203009] Modified LO PDFs for all Q2 region? Philosophy Arie Bodek, Univ. of Rochester

  32. 1. Start with GRV94 LO (Q2min=0.23 GeV2 ) - describe F2 data at high Q2 2A. Replace X with a new scaling, Xw x= [Q2] / [2Mn] Xw= [Q2+B] / [2Mn+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, x w x w= [Q2+B] / [ Mn (1+(1+Q2/n2) 1/2 ) + A] 3. Multiply all PDFs by a factor of Q2/[Q2+c] for photo prod. Limit+non-perturbative F2(x, Q2 ) = Q2/[Q2+C] F2QCD(x w, Q2) A (w, Q2 ) 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 if it is not in the resonance region Modified LO GRV94 PDFs with three parameters (a new scaling variable, Xw, x 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 x w works better as expected Keep final state interaction resonance modulating function A (w, Q2 )=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, Q2 ) 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 xF3 Modified LO PDFs for all Q2 (including 0) Construction Results [Ref:Bodek and Yang hep-ex/0203009] Arie Bodek, Univ. of Rochester

  33. Xw = [Q2+B] / [2Mn+A] used in 1972 x w = [Q2+B] / [ Mn (1+(1+Q2/n2) 1/2 ) + A] (theoretically derived) Multiply all PDFs by a factor of Q2/[Q2+C] 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 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 Withx w A and B are smallerModified LO GRV94 PDFs with three parameters and the 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. GRV94 Assumed to be PEFECT (no f(x) floating factors). Better fits expected with GRV98 and floating factors f(x) Comparison of Xw Fit and xw Fit backup slide * Same construction for Xw and x w fits Comparison Arie Bodek, Univ. of Rochester

  34. Proton LO+HT fit Comparison with DIS F2 (H, D) data[These SLAC/BCDMS/NMC are used in this Xw fit 2 = 1555 /958 DOF ] Deuteron Arie Bodek, Univ. of Rochester

  35. Proton LO+HT fit Comparison with DIS F2 (H, D) dataSLAC/BCDMS/NMC wworks better 2 = 1351 /958 DOF Deuteron Arie Bodek, Univ. of Rochester

  36. 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 <---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 Aw(w, Q2 ) modulating function (to account for interaction with spectator quarks at low W) Also, check the x=1 Elasic Scattering Limit. Comparison with F2 resonance data [ SLAC/ Jlab] (These data were not included in this Xw fit) Q2= 0.07 GeV2 Q2= 0.25 GeV2 Q2= 0.8 5 GeV2 Q2= 1. 4 GeV2 Q2= 9 GeV2 Q2= 3 GeV2 Q2= 1 5 GeV2 Q2= 2 5 GeV2 Arie Bodek, Univ. of Rochester

  37. The modified LO GRV94 PDFs with a new scaling variable, w 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 <--- 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 Aw(w, Q2 ) modulating function. (to account for interaction with spectator quarks at low W) Also, check the x=1 Elasic Scattering Limit. Comparison with F2 resonance data [ SLAC/ Jlab] (These data were not included in this w fit) Q2= 0.07 GeV2 Q2= 0.25 GeV2 Q2= 0.8 5 GeV2 Q2= 1. 4 GeV2 Q2= 9 GeV2 Q2= 3 GeV2 Q2= 1 5 GeV2 Q2= 2 5 GeV2 Arie Bodek, Univ. of Rochester

  38. 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 this Xwfit) Construction Xw fit The modified GRV94 LO PDFs with a new scaling variable, Xw describe the CCFR diff. cross section data (En=30–300 GeV) well. Will repeat with w Arie Bodek, Univ. of Rochester

  39. 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(see appendix R data figure) Comparison with photo production data(not included in this Xw fit) mb Xw 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. 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+ 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

  40. 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 is ver small (see appendix R data figure) Comparison with photo production data(not included in this w fit) mb w 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 w), the GRV94 need to be updated to fit latest HERA data at very low x and low Q2. 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+ 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

  41. 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 used 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. Now it looks like we can freeze at Q2=0.8 and have no problems. So switch to modern PDFs. Arie Bodek, Univ. of Rochester

  42. 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 Q2 = 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, Q2) (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). Arie Bodek, Univ. of Rochester

  43. Future Work - part 1 • Implement Ae/(W,Q2) resonances into the model for F2 with x 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 x 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 x w scaling curve and quasielastic form factors GA, GV 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 1st resonance (in neutrino scattering). • Implement differences between n and e/ final state resonance masses in terms of An,n bar(W,Q2){See Appendix) • Look at Jlab and SLAC heavy target data for possible Q2 dependence of nuclear dependence on Iron. • Implementation for R (and 2xF1) is done exactly - use empirical fits to R (agrees with NNLO+GP tgt mass for Q2>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. F2=(1+ Q2/ n2 )n W2 (for low energies) Arie Bodek, Univ. of Rochester

  44. Future Work - part 2 • Investigate different scaling variables for different flavor quark masses (u, d, s, uv, dv, usea, dsea in initial and final state) for F2. , • Note: x w = [Q2+B] / [ Mn (1+ (1+Q2/n2) 1/2 ) +A ] assumes m F =m i=0, P2t=0 • More sophisticated General expression (see derivation in Appendix): • x w’ =[ Q’ 2+B] / [Mn (1+ (1+Q2/n2) 1/2 ) +A] with • 2Q’2 =[Q2+ m F2 -m I 2] + [ (Q2+ m F2 -m I 2 ) 2 + 4Q2 (m I 2+P2t) ]1/2 • or 2Q’2 =[Q2+ m F2 -m I 2] + [ Q4 +2Q2(m F2 +m I 2 +2P2t ) + (m F2 -m I 2 ) 2 ] 1/2 Here B and Aaccount for effects of additional  m2 from NLO and NNLO effects. However, one can include P2t, as well as m F ,m ias the current quark masses (e.g. Charm, production in neutrino scattering, strange particle production etc.). In x 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 forx w scaling and the Q2/(Q2+C) factor, and Ae/(W,Q2) 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 Q2=0, including A (W,Q2), 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 Arie Bodek, Univ. of Rochester

  45. 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

  46. 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

  47. 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

  48. Bodek and Yang hep-ex/0203009; Georgi 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

  49. Georgi 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 +B] / [ Mn (1+(1+Q2/n2) 1/2 ) +A ] x = [Q’2 +B] / [M n (1 + [ 1 +4M2 x2/Q 2 ] 1/2 )+A] (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 (neglect quark initial Pt) 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 +2B] / [Q2 + (Q4 +4x2M2 Q2) 1/2+2Ax] (equivalent form) AddB and Aaccount for effects of additional  m2 from NLO and NNLO effects. (at high Q2 these are current quark masses, but at low Q2 maybe constituent masses?) Arie Bodek, Univ. of Rochester

  50. Georgi and Politzer Phys. Rev. D14, 1829 (1976)]- GP did not include Pt (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 +P2t) /[xM] or [Pi0 -Pi3] = (m I 2 +P2t) /[xM] --- in lab Get 2 Pi0 = xM + (m I 2 +P2t) /[xM] Plug into (eq. 1) and 2 Pi3 = xM - (m I 2 +P2t) /[xM] { nxM + n(m I 2 +P2t) /[xM]} + {q3 xM - q3 (m I 2 +P2t)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 +P2t)(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 +B] / [ Mn (1+(1+Q2/n2) 1/2 ) +A ] Derivation: for initial quark mass m I and final massm * bound in a proton of mass M - INCLUDING Quark INITIAL P2t q Pi= Pi0,Pi3,mI Pf, m* P= P0 + P3,M where 2Q’2 =[Q2+ m *2 -m I 2] + [ (Q2+m *2 -m I 2 ) 2 + 4Q2 (m I 2+P2t) ]1/2 or 2Q’2 =[Q2+ m F2 -m I 2] + [ Q4 +2Q2(m F2 +m I 2 +2P2t ) + (m F2 -m I 2 ) 2 ] 1/2 If mi=0 2Q’2 =[Q2+ m *2 ] + [ (Q2+m *2 ) 2 + 4Q2 (P2t) ]1/2 AddB and Aaccount for effects of additional  m2 from NLO and NNLO effects. (at high Q2 these are current quark masses, but at low Q2 maybe constituent masses?) Arie Bodek, Univ. of Rochester