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					Spin Structure of the Nucleon
              Krzysztof Kurek
 The Andrzej Sołtan Institute for Nuclear Studies,
              Warsaw, Poland
                     and
          COMPASS Collaboration




        XXVIII PHYSICS IN COLLISION
        Perugia, Italy, June 25-28, 2008
                                    Contents
             Introduction and problems
             Longitudinal spin structure
             Gluon polarization
             Transversity
             Orbital angular momentum
             Summary and the future

          Topics not covered: GDH sum rule, g2 (mainly JLAB),
                              GPD’s meas., Belle Collins FF


XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                  Krzysztof Kurek   Spin Structure of the Nucleon   2

   Spin Structure of the Nucleon
                                                                                   Introduction




XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                  Krzysztof Kurek   Spin Structure of the Nucleon      3

   Spin Structure of the Nucleon
 Data selection          Introduction   Longitudinal spin structure
                   Gluon polarization   Transversity                        Spin degree of freedom
          Orbital angular momentum      Summary

      Spin is a fundamental degree of freedom originated from the space-
       Spin is a fundamental degree of freedom originated from the space-
      time symmetry.
      time symmetry.
      Spin plays a critical role in determining the basic structure of
       Spin plays a critical role in determining the basic structure of
      fundamental interactions.
      fundamental interactions.
      Test of a theory is not complete without a full test of spin-dependent
       Test of a theory is not complete without a full test of spin-dependent
      decays and scattering.
      decays and scattering.

  Spin provides a unique opportunity to probe the inner structure of a composite
  Spin provides a unique opportunity to probe the inner structure of a composite
  system (such as the proton) and hence testing our ability to understand the working
  system (such as the proton) and hence testing our ability to understand the working
  of non-perturbative QCD
  of non-perturbative QCD

       The driving question for
        The driving question for
                                                                             Spin budget spin =
                                                                           Total protonof the proton 1/2
                                                                                                  Quark spin measured
       QCD spin physics is where
       QCD spin physics is where
                                                                                                  in DIS (~1/3 if sum rule)
       the nucleon spin comes from?
       the nucleon spin comes from?                                                                   25%




                                                                                 75%

         X.Ji, DIS2008
         X.Ji, DIS2008                                                 “Dark” angular momentum?

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon                           4

   Spin Structure of the Nucleon
                        Introduction       Longitudinal spin structure
                  Gluon polarization       Transversity                        Structure functions and anomaly
         Orbital angular momentum          Summary



  DIS:
             σ ~ F1 ( x) = 1 ∑ eq qi ( x)
                           2
                                2
                                                              and F2 ( x ) ≈ 2 xF1
                                       i

             ∆σ = σ − σ ~ g1 ( x) =                   1
                                                      2   ∑ qi ( ) and g 2
                                                           eq ∆Notxlonger true(?)
                                                            2
                           The problem was very recently announced
                                   i
  where: Dq(x)=q+(x) - q--(x)
                 +           to be solved: PRL 100, 232002-1(2008)

  „Switching on” spin lead us to two complications:

  q+ ~ Y (1+g5) gmY, q-- ~ Y (1-g5)gmY fl Dq(x) ~ Yg5gmY
   +         5
                 m
                                 5
                                    m
                                                   5
                                                     m


  1. Axial vector current is not conserved – triangle anomaly (Adler-Bell-Jackiw)
  1. Axial vector current is not conserved – triangle anomaly (Adler-Bell-Jackiw)
  2. There is no local and gauge invariant, dimension-3 axial vector operator for
  2. There is no local and gauge invariant, dimension-3 axial vector operator for
     gluons in QCD; (local in axial gauge). solved? See PRL100 (2008),232002-1
     gluons in QCD; (local in axial gauge).

  However there is a good helicity gluon distribution function defined in axial
  However there is a good helicity
  gauge where QPM is formulated! (infinite momentum frame).

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                       Krzysztof Kurek        Spin Structure of the Nucleon           5

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                         Sum rules
         Orbital angular momentum       Summary



            Γ1 = ∫ g1 ( x)dx
                             g A NS
            Γ1p − Γ1n =          C1                                              (Bjorken sum rule)
                            6 gV
                           a8  C1NS      C1S
           Γ =  ± a3 + 
              1
               p ,n
                                      + a0           (Ellis-Jaffe sum rule)
                           3  12          9
          a3 , a8 , g A,V - weak b hyperon decays + SUf(3);
          C1S , NS     - Calculable in pQCD


                                                                                       a0= DS - (3aS/2p) DG
                                                                                        0          S
                          1  4 (1)      1(4)     1 
                 Γ p(n) =          ∆u +      ∆d + ∆s 
                  1       2 9
                                         9       9  
                                                                 Invariant triangle anomaly term
                                1              1              1
                        = ( −) (∆u − ∆d ) + (∆u + ∆d − 2∆s ) + (∆u + ∆d + ∆s )
                               12             36              9
                                             a3                       a8                      a0

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek         Spin Structure of the Nucleon          6

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                        Decomposition of the nucleon spin
         Orbital angular momentum       Summary




                                   ½= ½∆Σ + ∆G + L
     Naive scenario: DS=1 and the rest 0.(e.g. SU(6) static model)
     Relativistic corrections change this requirement to ~ 0.6!

  Simple example - MIT Bag model: „bag” with relativistic quarks
                    confined in sphere with radius R.
  Lower spinor („p” state) – effectively transfers spin to orbital angular
  momentum and the contribution to the nucleon „spin” is smaller: ~0.65
                f  ↑
            Ψ =       →   ⇒ ( f 2 − 1 g 2 ) ~0.65
                iσrg   ↓            3
                  ˆˆ   


XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon             7

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                        Transversity
         Orbital angular momentum       Summary




            q(x) : unpolarized

            ∆q(x) = q                  - q = q+- q- : helicity

            ∆Tq(x) = q - q : transversity

             ∆Tq(x) is C-odd and chiral odd →not in inclusive DIS

         In Drell-Yan: ∆Tq(x) ⊗ ∆Tq(x)

         SIDIS (semi-inclusive…): ∆Tq(x) ⊗ DTDqh(z)

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon   8

   Spin Structure of the Nucleon
                        Introduction       Longitudinal spin structure
                  Gluon polarization       Transversity                        Transversity
         Orbital angular momentum          Summary


                       Key features of transversity:
                        probes relativistic nature of quarks
                        no gluon analog for spin-1/2 nucleon
                        different Q2 evolution and sum rule than ∆q(x)
                        sensitive to valence quark polarization


     Nonrelativistic quarks – no differences
     Nonrelativistic quarks – no differences
     between transversity and helicity –
     between transversity and helicity –                                          Leader sum rule (04):
     boosts and rotations commute.
     boosts and rotations commute.
                                                                                 1 1
     Relativistic quarks gives the difference
     Relativistic quarks gives the difference                                     = ∑ ∫ dx ⋅ ∆ T q ( x) + ∑ Lz
     example - again MIT Bag model:
     example - again MIT Bag model:                                              2 2 q ,q                 q ,q , g

                                       „renormalization” factor:
                                                                                    in analogy with:
      f  ↑                                                                            1
      iσrg  →  ↓  ⇒ ( f − 3 g )
  Ψ =                                        helicity – 0.65
                           2     2
                                                                                     S z = ∆Σ + ∆G + Lz
                              1
              
      ˆˆ   
                                                                                          2
                           ( f 2 + 1 g2)
                                   3
                                             transversity – 0.83


XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                       Krzysztof Kurek        Spin Structure of the Nucleon               9

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                        Collins and Sivers effects
         Orbital angular momentum       Summary


              Collins effects                                                      Sivers effects
       Nh = Nh ⋅ [1 ± A1 ⋅ sinΦColl ]
        ±    0
                                                                            Nh = Nh ⋅ [1 ± A1 ⋅ sinΦSiv ]
                                                                             ±    0

         A1 = f ⋅ PT ⋅ D ⋅ A Coll                                                A1 = f ⋅ PT ⋅ D ⋅ A Siv
                            2         0 h
                        ∑q e ⋅ ∆ q ⋅ ∆ D                                           2 T         h
                            q   T     T q                                      ∑q eq ⋅ ∆ q ⋅ Dq
           A     =                                                                      0
                                                                        ASiv =
            Coll                 2       h                                          2        h
Polarimeter!
                             ∑q e ⋅ q ⋅ D                                       ∑q eq ⋅ q ⋅ Dq
                                 q       q

 0 h describes the spin-dependent part                                    probability of finding unpolarized
∆ D                                                                       quarks inside transversally polarized
 T q of the hadronisation of a transversally                              nucleon
             polarised quark q into a hadron h
                                                       v          v 2    Tq (x, k 2).sinΦ
                                                                                v
Intrinsic kT dependence of the quark distribution q (x,kT ) = q(x,k ) + ∆
                                                   T               T     0        T      S
   Measurements: single-spin asymmetry on transversely polarized target
XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon               10

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                        Sivers effect
         Orbital angular momentum       Summary

     Final state interactions generate asymmetry before the active quark
    fragments – condition: kT of quark in tranversely polarized nucleon
                              T
     Non-integrated distributions (in kT)
                                       T
     Chiral-even, T-odd (Collins: chiral-odd, T-even)

    Transverse structure (kT) – beyond the simple QPM model of paralell
                           T
    stream of partons – connection with orbital angular momentum?




 Sivers effect requires:
    Correlation of the two QCD amplitudes
       g pÆö F i g p∞ö F – where F – same final state
    Boths amplitudes should have different phases (T-odd)
    Amplitudes with two different spin projections ï DL=1!
    FSI (dla DIS) i ISI (dla DY) – reason why Sivers is not
    universal distribution

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon   11

   Spin Structure of the Nucleon
                                              Longitudinal spin structure




XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                  Krzysztof Kurek   Spin Structure of the Nucleon   12

   Spin Structure of the Nucleon
                                 Introduction                Longitudinal spin structure
                           Gluon polarization                Transversity                        Experiments
                  Orbital angular momentum                   Summary

            Experiments:
             Inclusive spin-dependent DIS
                  EMC, SMC, COMPASS (CERN),E142,E143,E154,E156,
                  HERMES (DESY) ,JLAB-Hall,A,B(CLAS)
             Semi-inclusive DIS: SMC, COMPASS,HERMES
             Semi-inclusive DIS:
             Polarized pp collision RHIC,PHENIX & STAR, BRAHMS (Brookhaven)
             Polarized pp collision
             ee: BELLE (KEK) (Fragmentation functions)
             ee: BELLE (KEK) (Fragmentation
                                                                                           Beam: 27.5 GeV e± Pbeam~(53±1.0)%
                        Beam: 160 GeV ,Pbeam:~75%
                        Target: 6LiD; 50% pol, NH3 (2007)                                  Internal Gas Target: pol.: He , H , D
                                                        Trigger Hodoscopes                 unpol: H2, D2, He, N, Ne, Kr, Xe
                        µ −Filter
                                    ECal1/2 & HCal1/2


                                                                                 MWPC
                                      SM2


                     RICH

                  SM1
                                                      Scifi
  Target                                           GEM & MWPC


                                         Straws
    a   m
 be                            Drift chamber
         Scifi          MicroMegas
        Silicon


XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                                           Krzysztof Kurek      Spin Structure of the Nucleon           13

    Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                        Experiments
         Orbital angular momentum       Summary


       STAR



                                                             RHIC

                                                                                   PHENIX detector

                                               Beam: 200 GeV pp
                                               Pbeam:~60 %




                       Hall B      RHIC
    JLAB
                                                                      Beam: ≤6 GeV e-; 85% polarization
                         Large acceptance spect.                      Target: polarized targets 3He, 6LiD, NH3
                          electron/photon beams

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon              14

   Spin Structure of the Nucleon
                        Introduction       Longitudinal spin structure
                  Gluon polarization       Transversity                         A1 and g1
         Orbital angular momentum          Summary


    How to translate measured longitudinal asymmetry to g1 structure function?

                           sÆ∞ - sÆÆ
             A|| =
              ||           sÆ∞ + sÆÆ
                                           = D (A1 + h A2 )
                                                 1      2
                   d,p,n           d,p,n
          |h A2
              2
                           |á |A1
                                1
                                           |,
             p,n       gN       s1/2 - s3/2
          A1 = A =
           1
                                                        for nucleon
                                s1/2 +       s3/2

             d        gd        s0 - s2
          A1 = A =
           1
                                                        for deuteron
                                s0 + s2

          Ameas ~ A|| ~ A1
           meas    ||    1

          For deuteron:                                                                         d
          gd =
           g1 = ½ (g1 + g1 ) (1-
            1       1    1 (g p +   gn)               3w )>
                                                        d
                                                        d
                                                                     Ad     Fd =
                                                                         A1 F1 =
                                                                          1 1        A1 F2
                                                                                     Ad
                                                                                      1
                                                      2                                 2x(1+R)

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                       Krzysztof Kurek         Spin Structure of the Nucleon   15

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                        New results: COMPASS
         Orbital angular momentum       Summary


             COMPASS 2002-2004 data (Q2>1(GeV/c)2)
                                      2         2                                               PLB 647 (2007)8




                                                                                           Good agreement
                                                                                          between experiments,
                                                                                          improved significantly
                                                                                          statistics at low x,
                                                                                          no tendency towards
            Q2 > 1 (GeV/c)2
                                                                                          negative values at
                                                                                          x < 0.03




               COMPASS 2002-2003 data (Q2<1(GeV/c)2)
                                        2         2                                          PLB 647 (2007)330

     10-20 times lower statistical errors compared to SMC- very precise!
     The results compatible with zero for small x
XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon                16

   Spin Structure of the Nucleon
                         Introduction   Longitudinal spin structure
                   Gluon polarization   Transversity                        New results: COMPASS
          Orbital angular momentum      Summary


                          COMPASS 2002-2004 data (Q2>1(GeV/c)2)
                                                   2         2



                                                       F2d
   g = g (1 − ωd ) =
      d
      1            1
                    N              3
                                   2               2 x (1+ R )    A1d



                      Q2 > 1 (GeV/c)2




XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon   17

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                        New results: HERMES
         Orbital angular momentum       Summary

                                                                                PRD 75:012007(2007).




                                                                                       proton




                                                                                       deuteron




XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon    18

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                        New results: HERMES
         Orbital angular momentum       Summary



                                                                               Neutron results

                                                                              2
                                                                        g =
                                                                          n
                                                                          1       ⋅ g 1d − g 1p
                                                                              3
                                                                            1− ωD
                                                                              2


                                                                 g1n negative everywhere except at very
                                                                 high-x
                                                                 Low-Q2 data tends to zero at low-x
                                                                  – Does not support earlier conjecture
                                                                     of strong decrease for x → 0




XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon   19

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                         New results: JLAB
         Orbital angular momentum       Summary



       JLAB Hall B: 3He structure functions („effective neutron”)




   Small Q2 !                                                         JLAB – g1 for large x – important (CLAS)
   Note: precision of g2                                              PRL92 (2004)012004
XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek         Spin Structure of the Nucleon             20

   Spin Structure of the Nucleon
                            Introduction   Longitudinal spin structure
                      Gluon polarization   Transversity                        Monent of g1 (COMPASS)
             Orbital angular momentum      Summary


                       Compass data only                                                     Phys.Lett.B 647(2007)8


    1
     N
         (
  Γ Q = 3GeV   2
               0
                               2
                                   ) = ∫ g (x )dx = 0.050 ± 0.003(stat ) ± 0.003(evol ) ± 0.005(syst )
                                       1

                                       0
                                           1
                                            N



           1  α s (Q 2 )             
         1
          N

           9
              ( )
      Γ Q = 1 −   2

                   π
                                                    1 
                          + O (α s2 )  a0 (Q 2 ) + a8 
                                                   4 
                                                                                        (NLO QCD)
                                     

                                                      a8 = 0.585 ± 0.025
                   (Goto et al., PRD62 (2000) 034017: SU(3)f assumed for weak decays)

                           a0|Q 2 =3( GeV / c ) 2 = 0.35 ± 0.03( stat ) ± 0.05( syst )
                                   0


                                   Contribution from unmeasured x range º 4%

       HERMES: The deuteron integral is observed to saturate
       a0 = 0.330 ± 0.011(theor) ± 0.025 (exp) ± 0.028 (evol) at 5 (GeV/c)2

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                       Krzysztof Kurek        Spin Structure of the Nucleon                21

   Spin Structure of the Nucleon
                          Introduction   Longitudinal spin structure
                    Gluon polarization   Transversity                        Strange sea polarization (HERMES)
           Orbital angular momentum      Summary

       Seminclusive DIS – SIDIS: separation of the flavours. r u r
                                                             e + d → e'K ± X
       Example: strange quark sea from HERMES
       Example:
                         All needed information can be extracted from HERMES data alone!
                           - inclusive A1,d(x,Q2) and kaon AK1,d (x,Q2) asym.
                           - Kaon multiplicities    DQK and DSK
                           - Isoscalar target (deuteron)    fragmentation simplifies
                                                                                             Leader et al., PRD73, 034023 (2006)
                                                                               0.2
                                                                                      x∆Q(x)
                                         Fit to the x-dependence               0.1

                                         of multiplicities to get                0
                                         S(x) PDF and Kaon FF                  0.2
                                                                                      x∆S(x)
                                                                               0.1

                                                                                0
           0.6
       ∫   0.02
                  ∆Q = 0.359 ± 0.026 ± 0.018                                  -0.1
                                                                                      0.02                     0.1                 0.6
                                                                                                                                     X
           0.6
       ∫   0.02
                  ∆S = 0.037 ± 0.019 ± 0.027
                                               1
                          COMPASS: (∆s + ∆s ) = (a0 − a8 ) = −0.08 ± 0.01( stat ) ± 0.02( syst )
                                                 ˆ
                                               3
XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                     Krzysztof Kurek        Spin Structure of the Nucleon                               22

   Spin Structure of the Nucleon
                        Introduction        Longitudinal spin structure
                  Gluon polarization        Transversity                        Difference asymmetries
         Orbital angular momentum           Summary


        Semi-inclusive hadron asymmetries                                               Idea: PLB 230(1989)141,
                                                                                        SMC:PLB 369(1996)93,
                    h+         h+                     h−       h−
                σ −σ                               σ −σ
        A+ =        ↑↓
                    h+
                               ↑↑
                               h+
                                            A− =      ↑↓
                                                      h−
                                                               ↑↑
                                                               h−
                σ +σ↑↓         ↑↑
                                                   σ +σ
                                                      ↑↓       ↑↑

                         Difference asymmetry                                                  ∑ e (∆qD
                                                                                                q
                                                                                                        2
                                                                                                        q
                                                                                                                h
                                                                                                                q   + ∆q Dq )
                                                                                                                          h


                                                                                       A1h   =
                                                                                                ∑ e (qD
                           +            −              +            −
                    (σ ↑↓ − σ ↑↓ ) − (σ ↑↑ − σ ↑↑ )
                       h      h         h      h                                                            2
                                                                                                            q
                                                                                                                h
                                                                                                                q   + q Dq )
                                                                                                                         h

        A+ − =            h+          h−             h+         h−
                                                                                                    q

                    (σ    ↑↓
                                − σ ) + (σ
                                      ↑↓             ↑↑
                                                           −σ ) ↑↑

     Fragmentation functions cancel out in LO and
     under the assuption of independent fragmentation.


         π + −π −                   K + −K −          ∆uv ( x) + ∆d v ( x)                     Only valence quarks!
      Ad            ( x) = A        d          ( x) =
                                                       uv ( x ) + d v ( x )

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                        Krzysztof Kurek        Spin Structure of the Nucleon                         23

   Spin Structure of the Nucleon
                          Introduction      Longitudinal spin structure
                    Gluon polarization      Transversity                                  Valence quarks (COMPASS)
           Orbital angular momentum         Summary

             1
     Γv = ∫ (∆uv ( x) + ∆d v ( x))dx
             0

     ∆u + ∆d = 3Γ1N − 1 Γv + 12 a8 = (∆s + ∆s ) + 1 (a8 − Γv )
                      2
                              1
                                                  2

                                                                                         x(∆ uv+∆ dv)                 SIDIS+DIS, Q2=10 GeV2
      ∆u = ∆d = ∆s = ∆s                     symmetric
                                                                              scenario     0.5          From diff asymmetry
      ∆u = − ∆d                          asymmetric
                                                                                           0.4
                                                                                                        From Inclusive g
                                                                                                                          d
                                                                                                                          1
                                                                                                        DNS fit (prediction)

∫x
 0.7
  (∆ uv+∆ dv)dx               SIDIS+DIS, Q2=10 GeV2                                        0.3
                                                                                                                             SS
                                                                                                                          PA
     0.6    ∆ u=∆ d=∆ s=∆ s              COMPASS                                           0.2                       CO
                                                                                                                        M

     0.5                                                                                   0.1

                                                                                             0
     0.4
                                                             unmeasured



                                                                                          -0.1
     0.3      ∆ u=-∆ d
                                                                                          -0.2
                                                                                                    10-2                       10-1           1
     0.2
            0.7
                                                                                                                                        x
     0.1
            ∫ (∆u +∆d )dx=0.40±0.07±0.06
           0.006
                    v     v                                                        Data: asymmetric scenario!
       0
                   10-2              10-1                                 1               PLB 660(2008)458
                                                           x
XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                        Krzysztof Kurek                 Spin Structure of the Nucleon                          24

     Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                          QCD analysis (fits)
         Orbital angular momentum       Summary


        Idea:
                                                              p,n,d
            Measured structure function g1                             (different x and Q2)
                       2        1
                                2
                                       2
                                             [
         g1 ( x, Q ) = 〈e 〉 C ⊗ ∆Σ + C ⊗ ∆q + 2n f CG ⊗ ∆G
                                                 s
                                                 q
                                                                        NS
                                                                        q
                                                                                        NS
                                                                                                                             ]
                                                             d ∆q NS    αs (t ) NS
                                                             dt
                                                                    = 2π Pqq ⊗ ∆q NS
            DGLAP equations:
                                                          ∆Σ  αs (t )  Pqq 2n f PqG   ∆Σ 
                                                                           S        S

                 t = log     ( )
                              Q2
                              Λ2
                                                     d   = 2π  S
                                                     dt  ∆G          P
                                                                                       ⊗ 
                                                                                PGG   ∆G 
                                                                                  S
                                                                      Gq             


            Initial parametrization:
                                                         (∆Σ, ∆qs , ∆q8 , ∆G ) = η                       1
                                                                                                             xα (1− x ) β (1+γx )


             x dependence at fixed Q2                                                                    ∫   xα (1− x ) β (1+γx ) dx
                                                                                                         0
         (g ≠0 for singlet only for DG >0)
                                                              χ =∑ 2
                                                                            N
                                                                                  [g
                                                                                   calc
                                                                                   1
                                                                                                      exp
                                                                                        ( x ,Q 2 ) − g1 ( x ,Q 2 )]2


            Minimization routine
                                                                           i =1
                                                                                         [   exp
                                                                                                         ]
                                                                                          σ stat ( x ,Q 2 )
                                                                                                            2



XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                      Krzysztof Kurek       Spin Structure of the Nucleon                                25

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                         QCD analysis (fits)
         Orbital angular momentum       Summary



       Make some generic assumptions about the functional form with a few
       parameters fixing by fitting the data
       Many efforts in the past have been made
        – Gluck, Reya, Stratmann, Vogelsang (2001)
        – Blumlein and Bottcher (2003)
        – Leader, Sidorov, Stamenov (2006)
        – Hirai, Kumano, Saito (2006)
        – …..
       One of the most recent is the NLO fit by de Florian, Sassot,
       Stratmann and Vogelsang (hep-ph/0804.0422) in which pp
       collision jet data are first included. (Technically challenging!)




XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon   26

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                         QCD analysis – DSSV PDF
         Orbital angular momentum       Summary




   DSSV PDF
                  Polarized sea distributions
                                                                                   RHIC spin asymmetries




XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon        27

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                         QCD analysis – DSSV PDF
         Orbital angular momentum       Summary




   DSSV PDF
                  Polarized sea distributions
                                                                                   RHIC spin asymmetries




                                                           The gluon polarisation is small but
DS ~from ¼ up to 1/3                                       still with large errors!

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon        28

   Spin Structure of the Nucleon
                                                                   Gluon polarization




XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                  Krzysztof Kurek   Spin Structure of the Nucleon   29

   Spin Structure of the Nucleon
                        Introduction              Longitudinal spin structure
                  Gluon polarization              Transversity                          Photon-Gluon Fusion
         Orbital angular momentum                 Summary



                                                                                                 Open Charm
                                                                                 cross section difference in charmed
                                                                                 meson production
                                                                                      γ *g → c c → D 0 X
                                                                                 → clean channel
                                                                                 → but experimentally difficult
                                                                                  and NLO corr. can be important
         N
                                π, p, K




                                          X




                                                                                   High-pT Hadron Pairs
                                              p
                                          t




                                                                                         T
                                                                                   cross section difference in 2
                                                                                   hadrons production
                                                                                        *
                                                                                         γ g → q q → hh
                                                                                   → large gain in statistics
                                                                                   → but physical background
 Analysing power Gluon polarization
                                                                                     and analysis MC dependent
XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                              Krzysztof Kurek        Spin Structure of the Nucleon      30

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure      New results: COMPASS
                  Gluon polarization    Transversity
         Orbital angular momentum       Summary                          DG/G measurements summary

                                                                      ∆G/G measurements
                                                                          new high pT
                                                                             point
                                                                                         New high-pT COMPASS result:
                                                                                         Data: 2002-2004 Q2>1(GeV/c)2
                                                                                         DG/G=0.08±0.1(stat.)±0.05(sys.)
                                                                             new charm
                                                                               point



                                                                                    New charm COMPASS result:
                                                                                    Data: 2002-2006
                                                                                    DG/G=-0.49±0.27(stat.)±0.11(sys.)



          QCD fits   ∆G ≈| 0.2−0.3| as direct measurements point to a small
         value of ∆G ….. axial anomaly contribution is small       a0 ≈ ∆Σ
XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon                     31

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                         New results: RHIC-STAR p
         Orbital angular momentum       Summary




                                                                      STAR/PHENIX p0 asymmetries
                                                                      QCD NLO analysis




XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek         Spin Structure of the Nucleon    32

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                        New results: STAR inclusive jets
         Orbital angular momentum       Summary




                                   2005
                                                                                                      2006




      PRL100 (2008)28
                                                   GRSV-stnd. excluded 99% CL
     Statistical uncertainty are 3-4 times smaller DG<-0.7: 90%
     than in 2005 for pT>13 GeV/c
XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon            33

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                        New results: PHENIX, p
         Orbital angular momentum       Summary




                     Summary: Gluon polarization rather close to 0 however
                          still DG on the level ~0.2-0.3 not excluded!
                       More precise measurements (large x) are needed!

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon   34

   Spin Structure of the Nucleon
                                                                                 Transversity




XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                  Krzysztof Kurek   Spin Structure of the Nucleon    35

   Spin Structure of the Nucleon
                         Introduction
                        Introduction     Longitudinal spin structure
                                        Longitudinal spin structure
                   Gluon polarization
                  Gluon polarization     Transversity
                                        Transversity                         Collins asymmetries (HERMES)
          Orbital angular momentum
         Orbital angular momentum        Summary
                                        Summary

                  ep         πX
                                                                                δq ( x) ⊗ H 1⊥ q ( z )

                π+                                                                first time: transversity &
                                                                                  Collins FF are non-zero!

                                                                                • π+ asymmetries positive – no
                                                                                surprise: u-quark dominance and
                                                                                expect δq>0 since ∆q>0

                                                                                • large negative π− asymmetries
                                                                                – ARE a surprise: suggests the
                                                                                disfavoured CollinsFF being large

               π−                                                               and with oposite sign:

                                                                                     H 1⊥ ,disfav ( z ) ≈ − H 1⊥ ,fav ( z )

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek         Spin Structure of the Nucleon                          36

   Spin Structure of the Nucleon
                         Introduction
                        Introduction     Longitudinal spin structure
                                        Longitudinal spin structure
                   Gluon polarization
                  Gluon polarization     Transversity
                                        Transversity                         Sivers asymmetries (HERMES)
          Orbital angular momentum
         Orbital angular momentum        Summary
                                        Summary

                  ep         πX
                                                                                 f1⊥ q ( x) ⊗ D1q ( z )
                                                                                   T
                                                                                π+ are subtantial and positive:
           π+                                                                    • first unambiguous evidence for
                                                                                 a non-zero T-odd distribution
                                                                                 function in DIS
                                                                                  • a signature for quark orbital
                                                                                  angular momentum !
                                                                                   • SURPRISE:
                                                                                   K+ amplitude 2.3±0.3 times
                                                                                   larger than for π+
                                                                                      conflicts with usual
                                                                                    expectations based on u-quark
                                                                                    dominance
              π−                                                                       suggests substantial
                                                                                    magnitude of the Sivers fct. for
                                                                                    sea quarks
XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek         Spin Structure of the Nucleon                   37

   Spin Structure of the Nucleon
                            Introduction       Longitudinal spin structure          Collins and Sivers asymmetries
                      Gluon polarization       Transversity
             Orbital angular momentum          Summary                              new results from COMPASS
         Sivers and Collins asymmetries measured by COMPASS on deuteron target
         Sivers and Collins asymmetries measured by COMPASS on deuteron target
         are compatible with zero (cancellation?, subm. PLB ,hep-ex/0802.2160)
         are compatible with zero (cancellation?, subm. PLB hep-ex/0802.2160)
     0.2
 AColl




                positive hadrons
                                         Collins COMPASS              2007 proton data (part.)                Proton data:
p




                negative hadrons
     0.1
                                                                                                              Non-zero
         0
                                                                                                              as in HERMES
    -0.1

                                                preliminary
              10-2        10-1           0.2    0.4    0.6     0.8            0.5     1      1.5
                                   xBj                               z                      p [GeV/c]
                                                                                             t
 ASiv




                positive hadrons
                                           Sivers COMPASS
p




                negative hadrons                                         2007 proton data (part.)
                                                                                                               Zero!??
     0.1


         0


    -0.1                                                                                                      Subm. to PRL
                                                preliminary                                                   arXiv:0801.1078
    -0.2
              10-2        10-1           0.2    0.4    0.6     0.8            0.5     1      1.5
                                   xBj                               z                      p [GeV/c]
                                                                                             t


             Star: Di-jet Sivers measurement: 2006 p+p, 11pb-1
             A~0 ö ISI and FSI cancel?               PRL 99(2007)142003,
XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                           Krzysztof Kurek         Spin Structure of the Nucleon                     38

    Spin Structure of the Nucleon
                                                   Orbital angular momentum




XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                  Krzysztof Kurek   Spin Structure of the Nucleon   39

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                        Spin in the asymptotic limit
         Orbital angular momentum       Summary


         Scale evolution equation
                             d  Jq µ 
                                   
                                        ( )
                                          2
                                              = αs 
                                                       − 16 ,3n f     J q µ 
                                                                      
                                                                            ( )
                                                                              2
                                                                                
                                       ( )
                          d ln µ 2  J g µ 2  2π ⋅ 9 16,−3n f
                                                     
                                                                       J µ 2 
                                                                       g  ( ) 
         Asymptotic solution
                                       1 3n f                   1 16
                            J q (∞ ) =             , J g (∞ ) =
                                       2 16 + 3n f              2 16 + 3n f

    Roughly half of the angular momentum is carried by gluons!
    L must be important! (X.Ji)

      MIT Bag model – simple but not agrees with DIS measurements
      Nonperturbative QCD on Lattice
      GPD’s – Generalized Parton Distributions
      „room” for L – non-zero internal kT
                                        T
      Cahn asymmetry - seen in SIDIS; new COMPASS result : ~10%

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon        40

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                        New results: Lattice QCD
         Orbital angular momentum       Summary

                      Quark spin, OAM and total angular momentum
                                   LHPC                                                       QCDSF/UKQCD




 Agrees with GPD’s measurements
 Agrees with GPD’s measurements
 HERMES & JLAB!                                      overall agreement within errors
 HERMES & JLAB!

         J µ ≈ 0.22 ± 0.02 ≅ 44% of 1 / 2                             Ld ≈ − Lu ≈ 0.20 ± 0.04 ≅ 40% of 1 / 2
         J d ≈ 0.00 ± 0.02                                            Lu +d ≈ 0.00 ± 0.04

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon            41

   Spin Structure of the Nucleon
                                                                                         Summary




XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                  Krzysztof Kurek   Spin Structure of the Nucleon       42

   Spin Structure of the Nucleon
                  Summary
      Precise mesurements of the spin structure function g1 show that quarks
      contribute only in ~1/3 to the spin of the nucleon. This result is also
      confirmed by QCD fits and by independent measurement of the
      valence quark polarization (difference asymmetries) .
      Direct measurements point to a small value of gluon polarization however
      still 0.2-0.3 is not excluded (QCD fits, DG/G at large x).
      Different types of asymmetries precise measured at RHIC also indicate
      that large gluon polarization is rather excluded.
      New results from difference asymmetries show that asymmetric scenario
       is observed: Du = - Dd
      Small positive polarization for strange quark polarization have been
      measuered by HERMES while COMPASS results based on sum rules
      give small negative polarization of strange sea.



XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                  Krzysztof Kurek   Spin Structure of the Nucleon   43

   Spin Structure of the Nucleon
                  Summary
      Collins asymmetry measured on proton target is non-zero for larger xBjk.
      Collins and Sivers asymmetries measured by COMPASS on deuteron
      target are compatible with zero (cancellation in deuteron?).
      Non-zero Sivers asymmetry is observed by HERMES on proton target.
      Compass results from proton data are compatible with zero!
      Non-zero Sivers asymmetry indicates non-zero orbital angular momentum.
      The cancellation of the L for quarks is predicted by non-perturbative
      QCD calculations on lattices (contribution from quark helicity ~40%)
      New results from DVCS measurements (Hermes, JLab) agree with
      the Lattice QCD results.

  Possible scenarios (quarks ~1/3) :
  1. Gluon polarization ~0.3 (mainly large x effect) - no needs of L (L=0)
  2. Gluon polarization ~0 - significant contribution from L (quarks and gluons)
  3. Gluon polarization ~0 and Lattice calcul. – only gluons give L contribution

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                  Krzysztof Kurek   Spin Structure of the Nucleon   44

   Spin Structure of the Nucleon
                  Future
     New data from RHIC (Star jets, run 2009, beyond 2009 – 500 GeV)
     New results from HERMES and COMPASS (proton data, flavour separation)
     New results from JLAB (g1 for large x, g2)
     DVSC + GPD’s – future plans for COMPASS, JLAB upgrade
     Lattice QCD – probably the most promissing tool to study L


   Still a lot of things to do to understand the spin structure of the nucleon –
   particle from which we are made of                      ☺




XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                  Krzysztof Kurek   Spin Structure of the Nucleon   45

   Spin Structure of the Nucleon
                                                                     Backup slides




XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                  Krzysztof Kurek   Spin Structure of the Nucleon   46

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                        Collins and Sivers effects
         Orbital angular momentum       Summary



                         Collins and Sivers angles

       ΦC = φh - φS’


      Φ S = φh - φS




      φS ’     azimuthal angle of spin vector of fragmenting quark (φS’’ = π - φS)

      φh       azimuthal angle of hadron momentum

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon      47

   Spin Structure of the Nucleon
                         Introduction    Longitudinal spin structure
                   Gluon polarization    Transversity                           QCD analysis (fits) COMPASS
          Orbital angular momentum       Summary


                                             Compass fit (world data)
                                                                                           Phys.Lett.B 647(2007)8
              Quark polarisation:
                 ηG > 0                 ηG < 0                              ηΣ = 0.30 ± 0.01(stat ) ± 0.02(evol)
     ηΣ       0.27 ± 0.01          0.32 ± 0.01
                                                                   1
                                                                           
                                                                ∫
                                                       ηK =                         (error ≈ factor 2 larger without
                                                                   ∆ k dx 
                                                                                    COMPASS data)
                                                                 0        


             Gluon polarisation (indirect determination via DGLAP):
                                                            prog1                              prog2
      • Solutions with ηG > 0:                           ηG                   +0 05
                                                                        = 0.34−0..07 , ηG                      +0 04
                                                                                                         = 0.23−0..05
                                                            prog1                              prog2
      • Solutions with ηG < 0:                           ηG                    + 0 10
                                                                        = −0.31−0..14 , ηG                      + 0 06
                                                                                                         = −0.19−0..11


                                              |ηG| ≈ 0.2 - 0.3
XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                     Krzysztof Kurek          Spin Structure of the Nucleon                   48

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                        Cahn’s asymmetry (COMPASS)
         Orbital angular momentum       Summary

          Unpolarized: „pure” kinematical effect which reflects
          the fact of existence of internal kT in the nucleon – potential „room” for L
                                             T



                                                                                                      Cahn - 10% effect
        Cos(fh)



                                                                                                        Cahn+
       Cos(2fh)                                                                                         Boer-Mulders



          Sin(fh)                                                                                     Beam polarization


              Similar for h-
XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon                    49

   Spin Structure of the Nucleon
                          Introduction   Longitudinal spin structure
                    Gluon polarization   Transversity                        BRAHMS
           Orbital angular momentum      Summary
                                                                                              Submitted to Physical Review Letters
                                                                                              arXiv:0801.1078




                                                                                                       Twist 3



                                                                                                              sivers




       •      Large AN(π): 0.3-0.4 at xF~0.6 pT~1.3 GeV
       •      Strong xF -pT dependence. Though |AN(π+)| ~ !AN(π−)| |AN(π+)/AN(π−)|
              decreases with xF-pT

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                     Krzysztof Kurek        Spin Structure of the Nucleon                          50

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                        BRAHMS
         Orbital angular momentum       Summary




                    Unifying 62 and 200 GeV BRAHMS + E704




                                   E704 data – all pt (small star) pt>0.7 red star.

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon   51

   Spin Structure of the Nucleon
                                            Longitudinal spin structure
                            Introduction Longitudinal spin structure
                        Introduction
                     Gluon polarization Transversity
                  Gluon polarization        Transversity                    Lattice QCD
            Orbital angular momentum Summary
         Orbital angular momentum           Summary



    X.Ji spin sum rule
                                             1
                                             2   = S z = 1 ∆Σ q + Lq + ∆G + Lg
                                                         2

                                                 J q = 1 ( A20 (0) + B20 (0))
                                                       2
                                                            q         q
                                                                                          Jg          lattice




            „graviton-like coupling“
                in lattice QCD


                                                                                                                exp/
                                                                                                                pheno
   J u ≈ 40% of 1 / 2              Jd ≈0


          be aware of systematic
              uncertainties of                                                                           Jlab Hall A
                                                                                                         PRL 2007
            lattice simulations
XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                      Krzysztof Kurek      Spin Structure of the Nucleon                    52

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                                Lattice QCD
         Orbital angular momentum       Summary



                        e − E (T −τ ) − Eτ P' , Λ ' q [ΓDirac D µ1 D µ 2 ...]q P, Λ

                        ~ ∆Σ, δq ( x), F1 (t ), A20 (t ), B20 (t )...                           t = ∆2
       = vector-, axialvector-, graviton-, quark spin flip-, „spin-n“ coupling

                                                                                                            gauge fields/links U
                                                                                                                        quarks

                                                                                                       quark propagators
                       L      N(P’)                                                                                        u
                                                                                    u
                                                                                          N(P)                      u
                                                                                   u
 Philipp Hägler                                                                      d     z                         d
 talk on DIS2008                                                                                                proton
                                        t

                                                              [∫ DUe                        ]
                                                                                                                    N
      q2 q1 ~ ∫ DADqDqe                iS [ q , q , A ]
                                                          →            − S [U ]
                                                                                  det D[U ] D    −1
                                                                                                1→ 2   [U ] ≈   1
                                                                                                                N   ∑ →
                                                                                                                     D1−1 2 [U i ]
                                                                                                                    i =1

XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                         Krzysztof Kurek
                                                                            compute the path-integral numerically
                                                                            Spin Structure of the Nucleon                            53

   Spin Structure of the Nucleon
    Towards the decomposition of the nucleon spin



        substantial uncertainties
       due to model dependence


           uncertainties will be
           reduced in future →
          more HERMES data,
          JLab 12 GeV upgrade




       be aware of systematic                                                                  HERMES 0802.2499
           uncertainties of
         lattice simulations
XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                  Krzysztof Kurek   Spin Structure of the Nucleon                 54

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                         GPD’s ideology
         Orbital angular momentum       Summary

   GPD’s: distributions in quantum phase space
        In the past, we only knows how
        to imagine quarks either in
         – Coordinate space
             (form factors)                                           DVCS:
         – Momentum space
             (parton distributions)                                                   ~       ~
        GPDs provide correlated                                       H ( x, ξ , t ), H , E , E        t
        distributions of quarks and                                   H ( x,0,0) = q ( x)
        partons in combined coordinate                                ~
        and momentum (phase) space                                    H ( x,0,0) = ∆q ( x)
         – Wigner distribution in
            Quantum Mechanics(1932)
                                                                      ∫ H ( x,ξ , t )dx = F (t )
         – GPD: Wigner-type quark                                        H(x,0,t) → 3D view of nucleon (x,d⊥)
                                                                                  related to Lz (Ji sum rule)
                     distributions
                                                                         2Jq = ∫ x (H +E )( x, ξ , 0 ) dx
        Measurements: HERMES, JLAB
XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek         Spin Structure of the Nucleon            55

   Spin Structure of the Nucleon
XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                  Krzysztof Kurek   Spin Structure of the Nucleon   56

   Spin Structure of the Nucleon
                        Introduction    Longitudinal spin structure
                  Gluon polarization    Transversity                        GPD’s
         Orbital angular momentum       Summary


                               Generalized Parton Distributions
                                             δ z   ⊥

                                       xp
                                                       b   ⊥

                                        z                                                 x


  The GPDs contain the information on the longitudinal momentum AND
  the transverse spatial distributions of the partons in the nucleon

                                                                           f ( x ,b   ⊥   )




                                                                      0
                                                                                              b   ⊥


                      x
                                   1


XXVIII Physics in Collision 2008 , Perugia
  6/28/2008                                    Krzysztof Kurek        Spin Structure of the Nucleon   57

   Spin Structure of the Nucleon

				
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