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Overview on Transverse Momentum Dependent Distribution and Fragmentation F unctions

Overview on Transverse Momentum Dependent Distribution and Fragmentation F unctions. M. Boglione. xP. P. Deep Inelastic Scattering. (E,k’). (E,k). q. q. The nucleon has an internal structure x is the fraction of proton momentum carried by the parton

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Overview on Transverse Momentum Dependent Distribution and Fragmentation F unctions

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  1. Overview on Transverse Momentum Dependent Distribution and Fragmentation Functions M. Boglione

  2. xP P Deep Inelastic Scattering (E,k’) (E,k) q q • The nucleon has an internal structure • x is the fraction of proton momentum carried by the parton • The cross section is the incoherent sum of all partonic contributions convoluted • with the parton distribution function, which only depends on x at LO M. Boglione

  3. Q2 Evolution • QCD corrections induce Q2dependence • f(x) → f(x,Q2) • DGLAP evolution equations exactly • predict this Q2 dependence NOTE : As we will see later, the evolution of some TMD distribution and fragmentation functions is yet unknown M. Boglione

  4. - Parton distribution functions Unpolarized distribution functions fq/p(x) Δfsz/+(x) Δfq↑/p↑(x) - Transversity distribution functions Helicity distribution functions M. Boglione

  5. Correlator D.E. Soper, Phys. Rev. D 15 (1977) 1141; Phys. Rev. Lett. 43 (1979) 1847; J.C. Collins and D.E. Soper, Nucl. Phys. B194 (1982) 445; R.L. Jaffe, Nucl. Phys. B 229 (1983) 205. M. Boglione ΔTq q Δq

  6. Parton distribution functions M. Boglione • Verygoodknowledgeofunpolarizeddistributionfunctions, q(x,Q2) andg(x,Q2) • Fairlygoodknowledgeoflongitudinallypolarized, partonicdistributions, Δq(x,Q2); poorknowledgeoflongitudinallypolarizedgluonsΔg(x,Q2) • NO direct information on transverselypolarizedpartonicdistributions,ΔTq(x,Q2), from DIS

  7. Parton distribution functions M. Boglione

  8. + - + – Transversity Drell-Yan SIDIS + – – + – + + + – – – + M. Boglione • There is no gluontransversity distribution function • Transversity cannot be studied in deep inelastic scattering because it is chirally odd • Transversitycan only appear in a cross-section convoluted to another chirally odd function

  9. Semi Inclusive Deep Inelastic Scattering ℓ ℓ’ g* fragmentation function elementary cross- section k’ h Dh/q (z) final detected hadron k Incoming proton fq(x) X P distribution function M. Boglione

  10. IntrinsicTransverse Momentum M. Boglione

  11. Intrinsic TransverseMomentum P k xP Plenty of theoretical and experimental evidence for transverse motion of partons within nucleons, and of hadrons within fragmentation jets M. Boglione

  12. Intrinsic TransverseMomentum l+ Q2 = M2 l– qT p p qL qTdistributionofleptonpairs in D-Y processes pTdistributionof hadrons in SIDIS M. Boglione

  13. Intrinsic TransverseMomentum • Distribution and fragmentation functions now depend • on the lightcone momentum fraction • (xfor the distributions and zfor the fragmentations) • on Q2 (→pQCD evolution), • on the intrinsic transverse momentum of the partons, • (kfor the distributions and pfor the fragmentations) • OPEN QUESTIONS: • How do TMD’s depend on the intrinsic transverse momentum ? • Gaussian behaviour in the central region … • Power law decrease at large transverse momentum… • Does the partonic intrinsic transverse momentum k (p) depend on x (z) ? M. Boglione

  14. Leading twist TMD Correlator Mulders and Tangermann, NP B461 (1996) 197, Boer and Mulders, PR D57 (1998) 5780 M. Boglione

  15. TransverseMometum Dependent Distribution Functions Correlationsbetweenspin and transversemomentum Courtesy of Aram Kotzinian M. Boglione

  16. TransverseMomentum Dependent Distribution Functions QUARK POLARIZATION U L T • f1(x,k) • Unpolarized • h1 (x,k) • Boer-Mulders U NUCLEON POLARIZATION • g1(x,k) • Helicity L • h1L(x,k) • f1T(x,k) • Sivers • h1T(x,k) • h1T (x,k) • g1T(x,k) T Transversity Courtesy of A. Bacchetta • Functions in bold face survive k integration • Functions in shaded cells are naïve T-odd • Functions in red box are chirally odd M. Boglione

  17. General Formalism with Helicity Amplitudes M. Anselmino, M. Boglione, U. D-Alesio, E. Leader, S. Melis, F. Murgia, PR D71, 014002 (2005), PR D73, 014020 (2006) M. Anselmino, M. Boglione, U. D-Alesio, S. Melis, F. Murgia, A. Prokudin (2010) in preparation The distribution function fa/A is the number density of partons of type a inside hadronA a A X M. Boglione

  18. General Formalism with Helicity Amplitudes M. Boglione

  19. General Formalism with Helicity Amplitudes M. Boglione

  20. TransverseMometum Dependent Distribution Functions ΔNfq/p↑(x,k) Sivers function Δfsz/p↑(x,k) Δfsy/p(x,k) Boer-Mulders function fq/p(x,k) Δfsz/p+(x,k) Helicity fn. Δfsx/p↑(x,k) Transversity function Δfsx/p+(x,k) Δfsy/p↑(x,k) Transversity function Correlationsbetweenspin and transversemomentum M. Boglione

  21. TransverseMometum Dependent Distribution Functions QUARK POLARIZATION U L T • fq/p(x,k) • Unpolarized • Δfsy/p(x,k) • Boer-Mulders U NUCLEON POLARIZATION • Δfsz/p+(x,k) • Helicity L • Δfsx/p+(x,k) • Δfsx/p↑(x,k) • Δfsy/p↑(x,k) • ΔNfq/p↑(x,k) • Sivers • Δfsz/p↑(x,k) T Transversity • Functions in bold face survive k integration • Functions in shaded cells are naïve T-odd • Functions in red box are chirally odd M. Boglione

  22. General SIDIS Cross Section M. Boglione

  23. SIDIS FACTORIZATION All three fundamental blocks contain phases. The most general expression for the cross-section is obtained when all of them are kept into account. hard scattering TMD-PDF TMD-FF M. Boglione

  24. TMD’s in SIDIS Trento conventions SIDIS in partonmodelwithintrinsick┴ Factorizationholds at largeQ2, and (Ji, Ma, Yuan) Unpolarized Cross Section M. Boglione

  25. TMD in unpolarized SIDIS → Cahn Effect Azimuthal dependence induced by quark intrinsic motion EMC data,µp and µd, E between 100 and 280 GeV W. Käfer, COMPASS collaboration, talk at Transversity 2008, Ferrara CLAS data, arXiv:0809.1153 [hep-ex] F. Giordano and R. Lamb, AIP Conf.Proc.1149:423-426,2009. M. Boglione

  26. TMD in unpolarized SIDIS → Cahn Effect • Assume a simple, factorized form for the TMD distribution and fragmentation functions, • with a gaussian dependence on the intrinsic transverse momentum • Determine the free parameters by fitting experimental data A cosφ dependence is also generated by Boer-Mulders⊗Collins term, via a kinematical effect in dΔσ̂ , not included in this fit. At further dependence on is generated EMC data,µp and µd, E between 100 and 280 GeV M.Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin, Phys. Rev. D71:074006,2005. M. Boglione

  27. TMD in unpolarized SIDIS → Cahn Effect W. Käfer, COMPASS collaboration, talk at Transversity 2008, Ferrara Comparison with M. Anselmino, M. Boglione, A. Prokudin, C. Türk Eur. Phys. J. A 31, 373-381 (2007) does not include the Boer – Mulders contribution M. Boglione

  28. PT dependence of data in agreement with a Gaussian k⊥ dependence of unpolarized TMDs CLAS data, arXiv:0809.1153 [hep-ex] CLAS data, arXiv:0809.1153 [hep-ex] No hint of x dependence in the explored region Hint of a z-dependence at small z values Gaussian TMD’s with solid line → M. Boglione

  29. Polarized SIDIS cross section M. Boglione

  30. Polarized SIDIS cross section Unpolarized proton and lepton The F structure functions contain all the TMD’s Longitudinally polarized proton, polarized lepton Transversely polarized proton, unpolarized lepton Transversely polarized proton, polarized lepton • Studying Sivers, Collins and other mechanisms is complicated by the fact that all these effects mix and overlap in the polarized SIDIS cross section and azimuthal asymmetries • Way out : build appropriately “weighted”azimuthal asymmetries ! Kotzinian, NP B441 (1995) 234, Muldersand Tangerman, NP B461 (1996) 197; Boer and Mulders, PR D57 (1998) 5780, Bacchettaet al., PL B595 (2004) 309, Bacchettaet al., JHEP 0702 (2007) 093, M. Anselmino, M. Boglione, U.D’Alesio, S. Melis, F. Murgia, A. Prokudin, (in preparation).

  31. TMD’s in polarized SIDIS chiral-even TMDs chiral-odd TMDs Cahn kinematical effects (Avakian, Efremov, Schweitzer, Metz, Teckentrup, arXiv:0902.0689) M. Boglione

  32. TMD’s in SIDIS one by one … M. Boglione

  33. X The SiversDistribution Function The Sivers function is related to the probability of finding an unpolarized quark inside a transversely polarized proton The Sivers function is T-odd The Sivers function inbeds the correlation between the proton spin and the quark transverse momentum M. Boglione

  34. The SiversDistribution Function HERMES Collaboration, Phys.Rev.Lett.103:152002,2009. HERMES Collaboration, Phys.Rev.Lett.103:152002,2009. M. Boglione

  35. The SiversDistribution Function COMPASS proton data COMPASS deuteron data S. Levorato for the COMPASS Collaboration, Transversity 2008 COMPASS vs HERMES, problems? COMPASS Collaboration, Phys. Lett. B673: 127-135, 2009 Compass finds Sivers = 0. Up to now, only HERMES data reveal non-zero Sivers effect. We need further check s! Courtesy of F. Bradamante M. Boglione

  36. The SiversDistribution Function Bacchetta, Gamberg, Goldstain, Mukherjee, Metz, Amrath, Shaefer, Yang, Brodsky, Schmidt, Hwang, Scopetta, Courtoy, Frattini, Vento, Radici, Pasquini, Yuan … Models → Determining the Sivers function M. Anselmino, M. Boglione, J.C. Collins, U. D’Alesio, A.V. Efremov, K. Goeke, A. Kotzinian, S. Menze, A. Metz, F. Murgia, A. Prokudin, P. Schweitzer, W. Vogelsang, F. Yuan Fits → The first and 1/2-transverse moments of the Sivers quark distribution functions. The fits were constrained mainly (or solely) by the preliminary HERMES data in the indicated x-range. The curves indicate the 1-σ regions of the various parameterizations. M. Boglione

  37. Extraction ofSivers Function from SIDIS experimental data New data, old fit ! M.Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, S. Melis, F. Murgia, A. Prokudin, C. Türk, Eur.Phys.J.A39:89-100,2009. M. Boglione

  38. Sivers Function … … to do list • Update the fit using new analysis from HERMES and COMPASS and, • possibly, more modern fragmentation functions • Study evolution equations • Test new parametrizations with non-gaussian intrinsic transverse • momentum dependence • Comparison SIDIS vs. Drell-Yan M. Boglione

  39. The SiversDistribution Function Spin-orbit correlations A non-zero Siversfunction requires non-zero orbital angular momentum ! [Matthias Burkardt] Lu>0 Ld<0 Lattice [P. Haegleret al.] M. Boglione

  40. 3D view of the proton Courtesy of Alexei Prokudin M. Boglione

  41. 3D view of the proton Courtesy of Alexei Prokudin M. Boglione

  42. X The Boer-MuldersDistribution Function The Boer-Mulders function isrelated to the probability of finding a polarized quark inside an unpolarized proton The Boer-Mulders function is chirally odd and T-odd The Boer-Mulders function inbeds the correlation between the quark spin and its transverse momentum M. Boglione

  43. The Boer-MuldersDistribution Function F. Giordano and R. Lamb, AIP Conf.Proc.1149:423-426,2009. W. Käfer, COMPASS collaboration, talk at Transversity 2008, Ferrara Diquark spectator model L. P. Gamberg and G. R. Goldstein,Phys.Rev.D77:094016,2008. M. Boglione

  44. The Boer-MuldersDistribution Function A. Prokudin, talk at Workshop on Transverse Spin Physics, Beijing (2008) V. Barone, S. Melis, A. ProkudinArXiv: 0912.5194 • There are two contributions to <cos 2φ>: • Boer-Mulders + Collins (no suppression in 1/Q) • Cahn effect at O(k2/Q2), which turns out to be large M. Boglione

  45. The Boer-MuldersDistribution Function ? M. Boglione

  46. X The Collins Fragmentation Function The Collins function is related to the probability that a transversely polarized struck quark will fragment into a spinlesshadron The Collins function is chirally odd The Collins function inbeds the correlation between the fragmenting quark spin and the transverse momentum of the produced hadron M. Boglione

  47. The Collins Effect in SIDIS The Collins effectin SIDIS couples totransversity COMPASS 2002-2004 COMPASS proton data S. Levorato for the COMPASS Collaboration, Transversity 2008 M. Boglione

  48. Simultaneous determination of Transversity and Collins functions • We need to determine two convoluted unknown functions • Fix one of the two functions according to some theoretical model and use SIDIS data todetermine the other (seeforexampleEfremov, Goeke,Schweitzer) • Perform a simultaneous fit of SIDIS and e+e-→h1h2X BELLE data. BELLE Hadronic plane method BELLE Thrust axis method M. Boglione

  49. Simultaneous determination of Transversity and Collins functions M.Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, S. Melis, F. Murgia, A. Prokudin, C. Türk Models for the Collins functions M. Boglione

  50. The last three TMD’s … what about the last 3 TMDs? any relation with the others? neglecting twist-3 contributions similar to the Wandzura-Wilczek relation supported by experiment for a recent model of all twist-2 TMDs see Bacchettaet al., arXiv:0807.0323 M. Boglione

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