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Gary R. Goldstein Tufts University PowerPoint Presentation
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Gary R. Goldstein Tufts University

Gary R. Goldstein Tufts University

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Gary R. Goldstein Tufts University

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  1. Azimuthal Asymmetries in Unpolarized N+anti-N Drell-Yan Processes and Transversity-odd Distributions Gary R. Goldstein Tufts University With L. Gamberg (Penn.State -Berks) & K. Oganessyan (formerly Frascati) QCD-N2006 Frascati G.R.Goldstein

  2. Abstract Drell-Yan unpolarized processes display largeazimuthal asymmetries. One such asymmetry, cos(2), is directly related to the leading twist “transversity-odd” distribution h1(x,kT). We use a model developed for semi-inclusive deep inelastic scattering that determines the “Sivers function” f1T (x,kT) to predict the Drell-Yan asymmetry  as a function of Q2, QT and either x or xFor a new variable, . The resulting predictions include a non-leading twist contribution from spin-averaged distributions that measurably affect lower energy results. QCD-N2006 Frascati G.R.Goldstein

  3. Outline • Transversity • Short history • Helicity flip, chirality, phases & k • SIDIS • Asymmetries: SSA & azimuthal • Rescattering & leading twist contributions • Drell-Yan • Transversity-odd distribution functions • N (& ) distributions • cos2 asymmetry • Conclusions QCD-N2006 Frascati G.R.Goldstein

  4. From Frank Close at Dalitz Memorial Meeting Oxford, June 3, 2006 Email from G Rajasekran on origin of quark model: physics/0602131 During one of the lectures, Dalitz questioned him about the triplets. Why is he ignoring them? Gell-Mann managed to evade it, inspite of Dalitz's repeated questioning. If Gell-Mann had answered the question directly, quarks would have been born in Bangalore in 1961 instead of having to wait for another three years...." QCD-N2006 Frascati G.R.Goldstein

  5. Les Houchcs 1965 QCD-N2006 Frascati G.R.Goldstein

  6. O F.Close: What Dalitz had proposed Les Houches 1965 L=1 O O O L=2 O O QCD-N2006 Frascati G.R.Goldstein

  7. PDG 2006 !!! 1720 1710 1690 1700 1690 1550 1520 1520 1405 O L=1 O O QCD-N2006 Frascati G.R.Goldstein

  8. Transversity - some history • 2-body scattering amps - Exclusive hadronic • fa,b;c,d(s,t) with spin projections a,b;c,d • GPD’s bring back 2-body picture • What spin frame leads to simplest description of theory or data? Amps to observables? • helicity has easy relativistic covariance - theory • states of S·p, e.g. |+1/2 ,|-1/2 , etc. • transversity: eigenstates of S·(p1p2) | 1/2 )T = {|+1/2 (i)|-1/2}/√2 for spin 1/2, etc. Especially for relating to single spin asymmetries - only S·n where interference or imaginary parts are relevant QCD-N2006 Frascati G.R.Goldstein

  9. Transversity & simplicity • states of {S·(p1p2)} or {S·(p1 p2)} are transversity normal to or parallel to scattering plane non-diagonal in helicity basis • Spin 1: | 1)T = {|+1> 2 |0> + |-1>}/2 | 0)T = {|+1> - |-1>}/ 2 • photon: | 1)T = {|+1> + |-1>}/2 linear polzn normal to plane | 0)T = {|+1> - |-1>}/2 linear polzn parallel to plane useful in on-shell photoproduction dynamics GRG & M.J.Moravcsik, Ann.Phys.98 (1976) 128. also in other context: Gault, GRG, Jones, NC27 (1975) 174. early: Cohen-Tannoudji, Morel, Navelet, Ann.Phys.46 (1968) 239.GRG & Moravcsik & Arash 1980’s QCD-N2006 Frascati G.R.Goldstein

  10. Phases & SSA • Single Spin Asymmetries (SSA) in 2-body • Parity allows only <S·n> non-zero for any single spinning particle. Requires some helicity flip or chirality flip for m=0 quarks & phase. <S·n>f*ab,cd[·n]dd’fab,cd’  Im[f*ab,c+ fab,c-] for particle D’s SSA n requires some p2transverse to p1 (at quark level? m=0 & PQCD conserve helicity no SSA) • Inclusive A+B->X+D: sum&∫ over all C particles & relate to A+B+anti-D forward elastic. GRG & J.F.Owens (76) np1 p2 p2  p1 QCD-N2006 Frascati G.R.Goldstein

  11. Azimuthal asymmetries - kinematics • Why similar to spin asymmetries? • Need plane established (P1P2)  transverse P • Need azimuthal angle relative to 1st plane, i.e. 2nd plane • via fragmentation or decay or pair production • How does orientation information get transferred from 1st plane to 2nd plane? Dynamical question. Polarized intermediate particles &/or plane dependent observables - leading twist (&kinematic mass/Q) or non-leading • SIDIS & Drell-Yan involve off-shell photons - like massive vector particles with  (& longitudinal) polarization via QED QCD-N2006 Frascati G.R.Goldstein

  12. SSA & AzAs dynamics:require loops & k • <S·n> f*AB,CD[·n]DD’fAB,CD’ helicity basis or in transversity basis: {|fAB,C(+)|2 - |fAB,C(-)|2} • Imaginary part or phase requires beyond tree level in field theory • PQCD efforts to explain polarized hyperons via s-quark polzn • What is tree level in soft physics &/or “effective” field theory? • Mixing PQCD & soft physics • Helicity or chirality flip requires a flipping interaction (m≠0,…) & non-zero transverse momentum of participants or k’s QCD-N2006 Frascati G.R.Goldstein

  13. Brodsky, Hwang & Schmidt provided non-trivial model calculation Final state corrections to tree-level DISf1T(x,p2) Sivers & SSA LG&GG: same FSI contribute to other SSA’s & f1T(x,p2) = + h1(x,p2) Boer-Mulders Need SIDIS or D-Y to make functions experimentally accessible in asymmetry or polarization Brodsky, Hwang, Schmidt PLB 2002 Collins PLB 2002; Ji & Yuan PLB 2002 Goldstein & Gamberg ICHEP 2002 Gamberg, Goldstein, Oganessyan PRD 2003-5 &hep-ph QCD-N2006 Frascati G.R.Goldstein

  14. h1 • h1(x,p2) is “Transversity-odd” distribution -(Boer-Mulders) probability of finding quark with non-zero transversity in unpolarized hadron (it is P-even) • Vanishes at tree level in T-conserving models, as in spectator diquark model e.g. N quark+diquark where q is struck quark (like ordinary decay amps - final state interactions are essential - no T violation) • Simple quark-diquark or spectator picture is starting point for getting at properties to expect & observable consequences QCD-N2006 Frascati G.R.Goldstein

  15. SIDIS asymmetries In model yellow inclusive blob becomes diquark - scalar for simplicity (ud flavor) leaving u-quark being struck by q 1+ diquarks include uu (& dd) allowing for d-quark struck - 2 couplings Note: diquark actually has structure also - diq-g-diq form factors alternate method *+Nq+diq SSA CM helicity amps then light-cone limit (Dharmaratna&GG (92)) QCD-N2006 Frascati G.R.Goldstein

  16. Brodsky, Hwang, Schmidt rescattering Sivers fn, Collins =1 QCD-N2006 Frascati G.R.Goldstein

  17. Model calculation QCD-N2006 Frascati G.R.Goldstein

  18. Calculating h1(x,kT) QCD-N2006 Frascati G.R.Goldstein

  19. Integration results Spin independent tree level: Transversity T-odd TMD: ( .. k.jfactoron both sides) QCD-N2006 Frascati G.R.Goldstein

  20. Regularization • SSA’s & asymmetries involve moments of distribution & fragmentation functions e.g. h1(1)(x) = ∫ d2k k2 h1(x, k2) which would diverge without k2 damping QCD-N2006 Frascati G.R.Goldstein

  21. Transverse momentum hadronic tensor QCD-N2006 Frascati G.R.Goldstein

  22. h1(x,k) calculation with Gaussian h 1/(m2-k2)=(1-x)/(kT2) result of p->q+diq kinematics h1(x,k)= f1T (x,k) in diquark model Gamberg, Goldstein, Oganessyan PRD 2003 QCD-N2006 Frascati G.R.Goldstein

  23. Drell-Yan coordinates lepton CM frame defines plane tilted at  rel.t. hadron plane of P1 &P2 Coordinates? z is direction of q in initial CM frame or x is direction along qT from initial cm boost (Collins-Soper frame) or … y x l’ P2  P1 z  l QCD-N2006 Frascati G.R.Goldstein

  24. Drell-Yan Cross Sections see early papers ‘70’s Collins&Soper (1977) effects of transverse momenta -> ,, non-zero Boer, Mulders, Teryaev (1998), Boer (1999) & D.Boer, S.J. Brodsky & D.S. Hwang (2002) Unpolarized pair of hadrons  l + l’ + X  involves transversity at leading twist QCD-N2006 Frascati G.R.Goldstein

  25. D-Y angular dependent  How are angular asymmetries calculated? q+anti-q annihilation (& q+anti-q + gluons). Cross sections require convoluting hadron->u with hadron->anti-u distributions.  is related to T-odd distributions at leading twist (D. Boer). QCD-N2006 Frascati G.R.Goldstein

  26. Failure of Lam-Tung relation QCD-N2006 Frascati G.R.Goldstein

  27. AzAs: h1(1) •H1(1) cos2 Both distribution & fragmentation calculated in spectator models with gaussian k π π +h.c. QCD-N2006 Frascati G.R.Goldstein

  28. From SIDIS to Drell-Yan - analogous calculations Beam (π, p, p, …) QCD-N2006 Frascati G.R.Goldstein

  29. Azimuthal asymmetry Integrate over all quark transverse momenta. (p)+pl+l- X is in process; p+anti-p is calculated for various s. x direction is QT direction Notation of Boer, Mulders, Teryaev & Boer, Brodsky, Hwang QCD-N2006 Frascati G.R.Goldstein

  30. QT dependences General form: expectation of a hadronic tensor with distributions from quark model of incoming particles • Asymmetry must vanish as QT0 ; no 1st plane orientation in forward limit of initial state. • What is role of quark spin? • In lepton rest frame or q+q (CM) “fat” photon produced. • Whether q & q polarized or not, photon’s spin tensor (T & L) is fixed by QED. • Unpolarized q+q defines a plane via QT & tensor behaves ~ (QT2 / Q2)2 . • Transversely polarized q+q have ST1 ST2 tensor structure to combine with kT & pT (2 planes) (or 2 planes w/o spins, but higher twist) QCD-N2006 Frascati G.R.Goldstein

  31. Non-leading contributions Spin dependent leading part ~ QT2 for small QT2 / Q2 Non-leading, spin independent part ~ extra QT2 Collins & Soper ‘77 defined tensor A2 = B = (2kTx pTx  kTpT) / M2 QCD-N2006 Frascati G.R.Goldstein

  32. convolutions There will be the tensor B and azimuthal dependence, crucial for transmitting plane orientation information. Integrate numerically to obtain convoluted functions depending on x, mee, QT (and s). Note x ≈ mee2/xs for s >> mee2 >>QT2. Convolutions of h’s have extra factors of S at appropriate scale compared to f’s. But f’s in numerator have QT2relative to h’s. QCD-N2006 Frascati G.R.Goldstein

  33. Drell-Yan kinematics Asymmetry  is function of 3 variables: x, √Q2 =m ,qT (after separating sin2 cos2 dependence) Want to obtain  integrated over 2 variables to c.f. experiments. How to do this while keeping “symmetry” x1, x2 Using xFtreats x1, x2 symmetrically, but different range vs. q. Use = xF/2(1-) from -1/2 to +1/2 QCD-N2006 Frascati G.R.Goldstein

  34. AzAs function Insert convolutions into asymmetry expression: Obtained for range of x, mee, QT (and s). Choose kinematic ranges of Conway, et al. (FNAL fixed target π p) applied to p p . Sum over their (limited) ranges to obtain (x), (mee2), (QT) . QCD-N2006 Frascati G.R.Goldstein

  35. (QT2) leading h1 contribution Calculated s50 Gev2 - lower kinematic range than Conway, et al. E615. Antiproton beam vs. π Very similar to Boer, Brodsky, Hwang But gaussian supressed Data for π-p at s=500 GeV2 E615 f f part is at most 10% At higher s 500 Gev2 with comparable range curve decreases a bit f f part can be 10 - 15% of this for some values of 3 variables. QCD-N2006 Frascati G.R.Goldstein

  36. Including qT2 kinematic corrections Values lower ~10% QCD-N2006 Frascati G.R.Goldstein

  37. (m) leading h1 contribution (s=50 GeV2) Data for π-p at s=500 GeV2 E615 QCD-N2006 Frascati G.R.Goldstein

  38. (Q2+QT2) vs.  vs. x x = 0.9 x = 0.8 x = 0.7 x = 0.6 x = 0.5 x = 0.4 x = 0.3 x = 0.2 x = 0.1 QCD-N2006 Frascati G.R.Goldstein

  39. () s=50 GeV2 () Blue -leading Red - with non-leading  QCD-N2006 Frascati G.R.Goldstein

  40.  versus x1s=50 GeV2 Leading twist only Including non-leading QCD-N2006 Frascati G.R.Goldstein

  41.  vs. xF s=50 GeV2 QCD-N2006 Frascati G.R.Goldstein

  42. p+p Drell-Yan model Need anti-quark from sea to annihilate valence quark pganti-q + q Fermilab E866 ppl+l- X asymmetry QCD-N2006 Frascati G.R.Goldstein

  43. Summary& Conclusions • Transversity is important for full understanding of hadron spin composition. Accessed experimentally via SIDIS & Drell-Yan with SSA’s & azimuthal asymmetries. • Require helicity flips & loops; combinations of factorized soft-hadronic & PQCD. • BHS rescattering is mechanism for generating TMD’s at leading twist that can be measured via SSA’s & AzAs’s. • quark-diquark (S=0) model with gaussian regulators allows simple calculations to demonstrate existence & size of interesting TMD’s and thus SSA’s & AzAs’s. QCD-N2006 Frascati G.R.Goldstein

  44. Summary (cont’d) • Example considered: “Transversity-odd” contributions to cos2 in D-Y compared to non-leading spin independent piece (Collins&Soper). • Does data support “Transversity-odd” TMD? Large effect in π+p at hi s makes this very plausible. Cleanest determination would be AzAs data on anti-p+p (GSI - PAX?). p+p data also. • Improvements: • S=1 diquark is I=1 uu flavor  p->d+diq • better starting model (2-body constraints are limiting) • Questions: • How do Transversity-odd TMD’s evolve? • Are Sudakov effects important for low qT in AzAs’s? • Gluon bremsstrahlung, Qui-Sterman mechanism at other kinematics • π and p to anti-quark - analog of h1T? QCD-N2006 Frascati G.R.Goldstein