1. 1 Transverse spin physics Piet Mulders ‘Transverse spin physics’ RIKEN Spinfest, June 28 & 29, 2007 mulders@few.vu.nl

2. 2 Abstract QCD is the theory underlying the strong interactions and the structure of hadrons. The properties of hadrons and their response in scattering processes provide in principle a large number of observables. For comparison with theory (lattice calculations or models), it is convenient if these observables can be identified with well-defined correlators, hadronic matrix elements that involve only one hadron and known local or nonlocal combinations of quark and gluon operators. Well- known examples are static properties, such as mass or charge, form factors and parton distribution and fragmentation functions. For the partonic structure, accessible in high-energy (hard) scattering processes, a lot of information can be obtained, in particular if one finds ways to probe the ‘transverse structure’ (momenta and spins) of partons. Relevant scattering experiments to extract such correlations usually require polarized beams and targets and measurements of azimuthal asymmetries. Among these, single spin asymmetries are special because of their particular time-reversal behavior. The strength of single spin asymmetries depends on the flow of color in the hard scattering process, which affects the nonlocal structure of quark and gluon field operators in the correlators.

3. 3 Content Lecture 1: –Partonic structure of hadrons –correlators: distribution/fragmentation Lecture 2: –Correlators: parameterization, interpretation, sum rules –Orbital angular momentum? Lecture 3: –Including transverse momentum dependence –Single spin asymmetries Lecture 4: –Hadronic scattering processes –Theoretical issues on universality and factorization

4. 4 3 colors Valence structure of hadrons: global properties of nucleons d uu proton mass charge spin magnetic moment isospin, strangeness baryon number M p  M n  940 MeV Q p = 1, Q n = 0 s = ½ g p  5.59, g n  -3.83 I = ½: (p,n) S = 0 B = 1 Quarks as constituents

5. 5 A real look at the proton  + N  …. Nucleon excitation spectrum E ~ 1/R ~ 200 MeV R ~ 1 fm

6. 6 A (weak) look at the nucleon n  p + e  +  = 900 s  Axial charge G A (0) = 1.26

7. 7 A virtual look at the proton    N N   + N  N _

8. 8 Local – forward and off-forward m.e. Local operators (coordinate space densities): PP’  Static properties: Form factors Examples: (axial) charge mass spin magnetic moment angular momentum

9. 9 Nucleon densities from virtual look proton neutron charge density  0 u more central than d? role of antiquarks? n = n 0 + p   + … ?

10. 10 Quark and gluon operators Given the QCD framework, the operators are known quarkic or gluonic currents such as probed in specific combinations by photons, Z- or W-bosons (axial) vector currents energy-momentum currents probed by gravitons

11. 11 Towards the quarks themselves The current provides the densities but only in specific combinations, e.g. quarks minus antiquarks and only flavor weighted No information about their correlations, (effectively) pions, or … Can we go beyond these global observables (which correspond to local operators)? Yes, in high energy (semi-)inclusive measurements we will have access to non-local operators! L QCD (quarks, gluons) known!

12. 12 Deep inelastic experiments fragmenting quark scattered electron proton remnants Results directly reflect quark, antiquark and gluon distributions in the proton xBxB

13. 13 QCD & Standard Model QCD framework (including electroweak theory) provides the machinery to calculate cross sections, e.g.  *q  q, qq   *,  *  qq, qq  qq, qg  qg, etc. E.g. qg  qg Calculations work for plane waves __

14. 14 Confinement in QCD Confinement limits us to hadrons as ‘quark sources’ or ‘targets’ These involve nucleon states At high energies interference terms between different hadrons disappear as 1/P 1.P 2 Thus, the theoretical description/calculation involves for hard processes, a forward matrix element of the form quark momentum

15. 15 Partonic structure of hadrons Hard (high energy) processes –Inclusive leptoproduction –1-particle inclusive leptoproduction –Drell-Yan –1-particle inclusive hadroproduction Elementary hard processes Universal (?) soft parts - distribution functions f - fragmentation functions D

16. 16 Partonic structure of hadrons PHPH HPhPh pk h distribution correlator fragmentation correlator hard process Need P H.P h ~ s (large) to get separation of soft and hard parts Allows  d  … =  d(p.P)… PHPH PHPH  (x, p T ) PhPh PhPh  (z, k T )

17. 17 Intrinsic transverse momenta Hard processes: Sudakov decomposition for momenta: p = xP H + p T +  n zero: p T.P H = n 2 = p T.n large: P H.n ~  s hadronic: p T 2 ~ P H 2 = M H 2 small:  ~ (p.P H,p 2,M H 2 )/  s Parton virtuality enters in  and is integrated out   H  q (x,p T ) describing quark distributions Integrating p T  collinear  H  q (x) Lightlike vector n enters in  (x,p T ), but is irrelevant in cross sections Similarly for quark fragmentation: k = z  K h + k T +  ’ n’  correlator  q  h (z,k T ) PHPH H KhKh p k h

18. 18 (calculation of) cross section in DIS Full calculation + … + + + LEADING (in 1/Q) x = x B =  q 2 /P.q  (x)

19. Lightcone dominance in DIS

20. 20 Parametrization of lightcone correlator Jaffe & Ji NP B 375 (1992) 527 Jaffe & Ji PRL 71 (1993) 2547 leading part M/P + parts appear as M/Q terms in cross section T-reversal applies to  (x)  no T-odd functions

21. 21 Basis of partons  ‘Good part’ of Dirac space is 2-dimensional  Interpretation of DF’s unpolarized quark distribution helicity or chirality distribution transverse spin distr. or transversity

22. 22  Off-diagonal elements (RL or LR) are chiral-odd functions  Chiral-odd soft parts must appear with partner in e.g. SIDIS, DY Matrix representation for M = [  (x)  + ] T Quark production matrix, directly related to the helicity formalism Anselmino et al. Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712

23. 23 Results for deep inelastic processes  This has resulted in a good knowledge of u(x) = f 1 p  u (x), d(x), u(x), d(x) and (through evolution equations) also G(x)  For example in proton d > u, in neutron u > d (naturally explained by a   p component in neutron, providing additional insight beyond G E )  Polarized experiments (double spin asymmetries) have provided spin densities  u(x) = g 1 p  u (x), etc. _ _ _ _ __

24. 24 End lecture 1

25. 25 Lecture 2

26. 26 Local – forward and off-forward Local operators (coordinate space densities): PP’  Static properties: Examples: (axial) charge mass spin magnetic moment angular momentum Form factors

27. 27 Nonlocal - forward Nonlocal forward operators (correlators): Specifically useful: ‘squares’ Momentum space densities of  -ons: Sum rules  form factors Selectivity at high energies: q = p

28. 28 Quark number Quark distribution and quark number Sum rule: Next higher moment gives ‘momentum sum rule’

29. 29 Quark axial charge/spin sum rule Quark chirality distribution and quark spin/axial charge Sum rule: This is one part of the spin sum rule

30. 30 Full spin sum rule The angular momentum operators in this spin sum rule The off-forward matrix elements of the (symmetric) energy momentum tensor give access to J Q and J G

31. 31 Local – forward and off-forward Local operators (coordinate space densities): PP’  Static properties: Form factors

32. 32 Nonlocal - forward Nonlocal forward operators (correlators): Specifically useful: ‘squares’ Momentum space densities of  -ons: Sum rules  form factors Selectivity at high energies: q = p

33. 33 Nonlocal – off-forward Nonlocal off-forward operators (correlators AND densities): Sum rules  form factors Forward limit  correlators GPD’s b Selectivity q = p

34. 34 Quark tensor charge Quark chirality distribution and quark spin/axial charge Sum rule: Note that this is not a ‘spin’ measure, even if h 1 (x) is the distribution of transversely polarized quarks in a transversely polarized nucleon! ‘Transverse spin’ ~ (no decent operator!)

35. 35 A transverse spin rule One can write down a ‘transverse spin’ sum rule It was first discussed by Teryaev and Ratcliffe, but it involves the twist-3 funtion g T =g 1 +g 2 (Burkhardt-Cottingham sumrule) … and a similar gluon sumrule It does not involve the ‘transverse spin’. This appears in the Bakker-Leader-Trueman sumrule (which involves the assumption of having ‘free’ quarks). (my version of Trieste meeting)

36. 36 End lecture 2

37. 37 Lecture 3

38. 38 Issues Knowledge of partonic structure can be extended by looking at the ‘transverse structure’ Time reversal invariance provides a nice discriminator for ‘special effects’ Example is the color flow in hard processes, which is reflected in the nonlocal structure of matrix elements and shows up in single spin asymmetries Single spin asymmetries are being measured (HERMES@DESY, JLAB, COMPASS@CERN, KEK, RHIC@Brookhaven)

39. 39 The partonic structure of hadrons The cross section can be expressed in hard squared QCD-amplitudes and distribution and fragmentation functions entering in forward matrix elements of nonlocal combinations of quark and gluon field operators (    or G) lightcone lightfront:   = 0 TMD FF

40. 40 Partonic structure of hadrons PHPH HPhPh pk h distribution correlator fragmentation correlator hard process Need P H.P h ~ s (large) to get separation of soft and hard parts Allows  d  … =  d(p.P)… PHPH PHPH  (x, p T ) PhPh PhPh  (z, k T )

41. 41 (calculation of) cross section in SIDIS Full calculation + + … + + LEADING (in 1/Q)

42. Lightfront dominance in SIDIS Three external momenta P P h q transverse directions relevant q T = q + x B P – P h /z h or q T = -P h  /z h

43. 43 Gauge link in DIS In limit of large Q 2 the result of ‘handbag diagram’ survives … + contributions from A + gluons ensuring color gauge invariance A + gluons  gauge link Ellis, Furmanski, Petronzio Efremov, Radyushkin A+A+

44. Distribution From  A T (  )  m.e. including the gauge link (in SIDIS) A+A+ One needs also A T G +  =  + A T  A T  (  )= A T  ( ∞ ) +  d  G +  Belitsky, Ji, Yuan, hep-ph/0208038 Boer, M, Pijlman, hep-ph/0303034

45. 45 Parametrization of  (x,p T ) Link dependence allows also T-odd distribution functions since T U[0,  ] T † = U[0,-  ] Functions h 1  and f 1T  (Sivers) nonzero! Similar functions (of course) exist as fragmentation functions (no T-constraints) H 1  (Collins) and D 1T 

46. 46 Interpretation unpolarized quark distribution helicity or chirality distribution transverse spin distr. or transversity need p T T-odd

47. 47  p T -dependent functions T-odd: g 1T  g 1T – i f 1T  and h 1L   h 1L  + i h 1  (imaginary parts) Matrix representation for M = [   ±] (x,p T )  + ] T Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712

48. 48 T-odd  single spin asymmetry with time reversal constraint only even-spin asymmetries the time reversal constraint cannot be applied in DY or in  1-particle inclusive DIS or e  e  In those cases single spin asymmetries can be used to measure T-odd quantities (such as T-odd distribution or fragmentation functions) * *  W  (q;P,S;P h,S h ) =  W  (  q;P,S;P h,S h )  W  (q;P,S;P h,S h ) = W  (q;P,S;P h,S h )  W  (q;P,S;P h,S h ) = W  (q;P,  S;P h,  S h )  W  (q;P,S;P h,S h ) = W  (q;P,S;P h,S h ) _ __ _ _ ___ _ _ _ _ time reversal symmetry structure parity hermiticity * *

49. 49 Lepto-production of pions H 1  is T-odd and chiral-odd

50. 50 End lecture 3

51. 51 Lecture 4

52. 52 Quarks Integration over   =  P  allows ‘twist’ expansion Gauge link essential for color gauge invariance Arises from all ‘leading’ matrix elements containing     

53. 53 Sensitivity to intrinsic transverse momenta In a hard process one probes partons (quarks and gluons) Momenta fixed by kinematics (external momenta) DISx = x B = Q 2 /2P.q SIDIS z = z h = P.K h /P.q Also possible for transverse momenta SIDIS q T = q + x B P – K h /z h   K h  /z h = k T – p T 2-particle inclusive hadron-hadron scattering q T = K 1 /z 1 + K 2 /z 2  x 1 P 1  x 2 P 2  K 1  /z 1 + K 2  /z 2 = p 1T + p 2T – k 1T – k 2T Sensitivity for transverse momenta requires  3 momenta SIDIS:  * + H  h + X DY: H 1 + H 2   * + X e+e-:  *  h 1 + h 2 + X hadronproduction: H 1 + H 2  h 1 + h 2 + X  h + X (?) p  x P + p T k  z -1 K + k T K2K2 K1K1      pp-scattering Knowledge of hard process(es)!

54. 54 Generic hard processes C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596 (2004) 277 [hep-ph/0406099]; EPJ C 47 (2006) 147 [hep-ph/0601171] Link structure for fields in correlator 1 E.g. qq-scattering as hard subprocess Matrix elements involving parton 1 and additional gluon(s) A + = A.n appear at same (leading) order in ‘twist’ expansion insertions of gluons collinear with parton 1 are possible at many places this leads for correlator involving parton 1 to gauge links to lightcone infinity

55. 55 SIDIS SIDIS   [U + ] =  [+] DY   [U - ] =  [  ]

56. 56 A 2  2 hard processes: qq  qq E.g. qq-scattering as hard subprocess The correlator  (x,p T ) enters for each contributing term in squared amplitude with specific link  [Tr(U □ )U + ] (x,p T ) U □ = U + U  †  [U □ U + ] (x,p T )

57. 57 Gluons Using 3x3 matrix representation for U, one finds in gluon correlator appearance of two links, possibly with different paths. Note that standard field displacement involves C = C’

58. 58 Integrating  [±] (x,p T )   [±] (x) collinear correlator 

59. 59 Integrating  [±] (x,p T )     [±] (x) transverse moment  G (p,p  p 1 ) T-evenT-odd

60. 60 Gluonic poles Thus:  [U] (x) =  (x)   [U]  (x) =    (x) + C G [U]  G  (x,x) Universal gluonic pole m.e. (T-odd for distributions)  G (x) contains the weighted T-odd functions h 1  (1) (x) [Boer-Mulders] and (for transversely polarized hadrons) the function f 1T  (1) (x) [Sivers]   (x) contains the T-even functions h 1L  (1) (x) and g 1T  (1) (x) For SIDIS/DY links: C G [U ± ] = ±1 In other hard processes one encounters different factors: C G [U □ U + ] = 3, C G [Tr(U □ )U + ] = N c Efremov and Teryaev 1982; Qiu and Sterman 1991 Boer, Mulders, Pijlman, NPB 667 (2003) 201 ~ ~

61. 61 A 2  2 hard processes: qq  qq E.g. qq-scattering as hard subprocess The correlator  (x,p T ) enters for each contributing term in squared amplitude with specific link  [Tr(U □ )U + ] (x,p T ) U □ = U + U  †  [U □ U + ] (x,p T )

62. 62 examples: qq  qq in pp C G [D 1 ] D1D1 = C G [D 2 ] = C G [D 4 ] C G [D 3 ] D2D2 D3D3 D4D4 Bacchetta, Bomhof, Pijlman, Mulders, PRD 72 (2005) 034030; hep-ph/0505268

63. 63 examples: qq  qq in pp D1D1 For N c  : C G [D 1 ]   1 (color flow as DY) Bacchetta, Bomhof, D’Alesio,Bomhof, Mulders, Murgia, hep-ph/0703153

64. 64 Gluonic pole cross sections In order to absorb the factors C G [U], one can define specific hard cross sections for gluonic poles (which will appear with the functions in transverse moments) for pp: etc. for SIDIS: for DY: Bomhof, Mulders, JHEP 0702 (2007) 029 [hep-ph/0609206] (gluonic pole cross section) y

65. 65 examples: qg  q  in pp D1D1 D2D2 D3D3 D4D4 Only one factor, but more DY-like than SIDIS Note: also etc. Transverse momentum dependent collinear

66. 66 examples: qg  qg D1D1 D2D2 D3D3 D4D4 D5D5 e.g. relevant in Bomhof, Mulders, Vogelsang, Yuan, PRD 75 (2007) 074019 Transverse momentum dependent collinear

67. 67 examples: qg  qg Transverse momentum dependent collinear

68. 68 examples: qg  qg Transverse momentum dependent collinear

69. 69 examples: qg  qg It is also possible to group the TMD functions in a smart way into two! (nontrivial for nine diagrams/four color-flow possibilities) But still no factorization! Transverse momentum dependent collinear

70. 70 ‘Residual’ TMDs We find that we can work with basic TMD functions  [±] (x,p T ) + ‘junk’ The ‘junk’ constitutes process-dependent residual TMDs The residuals satisfies  int  (x) = 0 and  int G (x,x) = 0, i.e. cancelling k T contributions; moreover they most likely disappear for large k T definite T-behavior no definite T-behavior

71. 71 Conclusions Appearance of single spin asymmetries in hard processes is calculable For integrated and weighted functions factorization is possible For TMDs one cannot factorize cross sections, introducing besides the normal ‘partonic cross sections’ some ‘gluonic pole cross sections’ Opportunities: the breaking of universality can be made explicit and be attributed to specific matrix elements Related: Qiu, Vogelsang, Yuan, hep-ph/0704.1153 Collins, Qiu, hep-ph/0705.2141 Qiu, Vogelsang, Yan, hep-ph/0706.1196 Meissner, Metz, Goeke, hep-ph/0703176