PARTON OAM: EXPERIMENTAL LEADS

Transcription

1 PARTON OAM: EXPERIMENTAL LEADS 3D STRUCTURE OF THE NUCLEON AUGUST 29, 2017 INT UNIVERSITY OF WASHINGTON Simonetta Liuti University of Virginia

2 8/27/17 2

3 8/27/17 3 arxiv:1709.xxxx [hep-ph]

4 8/29/17 4 Outline 1. Introduction 2. Definitions ~ 3. Lorentz Invariant Relations! twist three GPD E 2T " 4. Equations of Motion Relations 5. A probe of QCD at the amplitude level: FSI and genuine tw 3 6. Nuclei 7. A glimpse on Gluon Angular Momentum 8. DVCS Analysis 9. Conclusions

5 8/27/ INTRODUCTION

6 8/27/17 Simonetta Liuti 6 Angular Momentum Budget 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% J g L q ΔΣ L g L q ΔG ΔΣ Ji Jaffe Manohar Gluon J Gluon L Quark L Gluon Spin Quark Spin

7 8/27/17 Simonetta Liuti 7 The Starting Point Jaffe and Manohar s field theoretical description of the quark and gluon orbital angular momentum through its relation to the QCD Energy Momentum Tensor, T µν M µνλ = x ν T µλ x λ T µν Angular Momentum density Energy density (mass) Momentum density T 00 T 01 T 02 T 03 T 10 T 11 T 12 T 13 T 20 T 21 T 22 T 23 T 30 T 31 T 32 T 33 Shear stress Pressure Hadronic observables have been studied precisely only for T 00,T 0i (4-momentum and OAM) EMT is studied much more extensively in nuclei (RHIC)

8 8/27/17 Simonetta Liuti 8 In QCD Jaffe Manohar: M +12 = ψ t σ 12 ψ + ψ t T µν = 1 4 iqψ ( γ µ D ν + γ ν D µ )ψ + Tr F µα F ν α 1 2 gµν F 2 ΔΣ x ( i ) [ ] 3 ψ + Tr(ε + ij F + j A j ) + 2iTrF + j ( x )A j L q ΔG L g * Ji: M +12 = ψ t σ 12 ψ + ψ t x i D ( ) 3 ψ + x E B ( ) 3 ΔΣ L q J g J q *Chen, Goldman et al., are consistent with this definition (see K.F.Liu et al.)

9 8/27/17 Simonetta Liuti 9 Quark sector : J q = L q q J q J u d Σ/2 =0.633(12) GPD model Lattice: GPD moments DVCS data d 2 data Lattice: F 14 A. Rajan et al, PRD (2016) arxiv: Ê HLzênL ê»lz Hh=0L ên Hh=0L» EXPERIMENT u-d quarks m p = 518 MeV z`=0.39 arxiv: M. Engelhardt Ê THEORY L u d -2.0 L- q h»v» Hlattice unitsl

10 8/27/17 10 IS IT POSSIBLE TO TEST OAM IN EXPERIMENTS?

11 8/27/17 Simonetta Liuti 11 Proposed Observables for L q Z 1 d 2 k T kt 2 F 14 (x, 0,kT 2, 0, 0) = hb T k T i 3 (x) M k T moment of a GTMD (lacks proof of observability ) (Lorce and Pasquini) L q (x) -L q -J q +S q 0 1 dx xg 2 Z 1 x moment of twist 3 GPD (Polyakov) 0 dx x(ẽ2t + H + E) = 1 2 Z 1 0 dx x(h + E)+ 1 2 Z 1 0 dx H Are the two connected and how?

12 8/27/17 12 Yes, they are connected through a generalized LIR 1 M Z d 2 k T k 2 T F 14 (x, 0,k 2 T, 0, 0) = Z 1 x dy h ee2t + H + Ei L q (x) A. Rajan, A. Courtoy, M. Engelhardt, S.L., PRD (2016), arxiv:

13 8/27/17 13 Measuring twist three GPDs gives us the same information on OAM as measuring k T integrals GTMDs, but..we have referred so far only to Ji s OAM

14 8/27/ DEFINITIONS

15 Partonic OAM: Wigner Distributions L U q = Z Z dx d 2 k T Z d 2 b (b k T ) z W U (x, k T, b) 8/27/17 15 Hatta Lorce, Pasquini, Xiong, Yuan Mukherjee k T b L P 8/27/17 Simonetta Liuti 15

16 8/27/17 16 Wigner Distribution U = 1 2 Z d 2 Z T (2 ) 2 ei T b dz d 2 z T e ikz hp, 0 q(0) + U(0,z)q(z) P, i z + =0 16 GTMD 8/27/17 p-p =Δ # Δ T Fourier conjugate: b = transverse position of the quark inside the proton # k T Fourier conjugate: z T = transverse distance traveled by the struck quark between the initial and final scattering

17 8/27/17 17 Which GTMD? The quark-quark correlator for a spin ½ hadron has been parametrized up to twist four in terms of GTMDs, TMDs and GPDs, in a complete way in:

18 Helicity Structure F 14 helicity non-flip UL correlation: unpolarized quark density in a longitudinally polarized proton

19 8/27/17 19 G 11 (L S)! 1 2 " W + 5 = 1 0 2M U(p0, 0 i ij T ) ki T apple i(k1 2 k 2 1 ) M 2 G 11 + G Z d 2 b (b k T ) z h q(0) + 5q(z)i # j T M 2 G 11 + i i+ 5 kt i P + G 12 + i i+ 5 i T P + G 13 + i + 5 G 14 U(p, ) apple 1 + i 2 G 13 + i (k 1 2 k 2 1 ) M 2M 2 G 11 + k 1 + i k 2 M G 12, 0 helicity non-flip UL correlation: longitudinally polarized quark density in an unpolarized proton

20 8/27/17 Simonetta Liuti 20 Integral relations Z 1 Z L q = dx 0 d 2 k T k 2 T M 2 F 14 = Z 1 0 dx F (1) 14 q S q = Z 1 0 dx Z d 2 k T k 2 T M 2 G 11 = Z 1 0 dx G (1) 11 Lorce, Pasquini, Xiong, Yuan Hatta, Yoshida Ji, Xiong, Yuan

21 8/27/ LIR

22 8/27/17 22 Generalized Lorentz Invariance Relations (LIR) LIR are relations between twist-3 PDFs and k T moments of TMDs (Metz, Pitoniak, Schlegel, Mulders, Goeke, ) LIR in the off-forward sector: relations between twist-3 GPDs (!PDFs) and k T moments of GTMDs (!TMDs) Based on the most general Lorentz invariant decomposition of the fully unintegrated quark-quark correlator LIRs are a consequence of there being a smaller number of independent unintegrated terms in the decomposition than the number of GTMDs

23 8/28/17 Simonetta Liuti 23 The completely unintegrated off-forward correlator 0 (k, ; U) = 1 2 Z d 4 z (2 ) 4 eikz hp 0, 0 z 2 U z 2, z 2 z 2 p, i! parametrized in terms of invariant functions A 1, A 2, The unintegrated (over k T ) off-forward correlator! parametrized in terms of GTMDs: F 11, F 12,, F 21, F 22

24 8/27/17 24

25 8/27/17 25 For any combination of A amplitudes one has: k T moment of GTMD Extension of Mulders Tangerman relation to off-forward configuration

26 8/27/17 26 Now look for GPDs combinations that give the rhs

27 8/28/17 27 OAM distribution emerges from LIR L q (x)!density L q!integrated F (1) 14 = Z 1 x dy (Ẽ2T + H + E) ) L q = Z 1 0 dxf (1) 14 = Z 1 0 dxxg 2 F (1) 14 and E ~ 2T x of OAM! give us the same information on the distribution in In addition : we confirm and corroborate the global/integrated OAM result deducible from Ji et al Different notation! G 2 E! 2T + H + E Polyakov et al. Meissner, Metz and Schlegel, JHEP(2009)

28 8/28/17 28 Generalized LIR for a staple link 0,0 z -,z T,z T,0 d dx Z d 2 k T k 2 T M 2 F 14 = Ẽ2T + H + E + A LIR violating term

29 8/27/ EQUATIONS OF MOTION

30 8/28/17 Simonetta Liuti 30 Equations of Motion (EoM) relation EoM in correlator for Γ= i σ +i γ 5 Z dz d 2 z T (2 ) 3 e ixp + z ik T z T hp 0, 0 ( z/2)( U i! 6D+i6D U) (z/2) p, i z+ =0 =0 symmetrized with respect to the argument + i 5 ] 2 W [ 0 + ik + ij W [ j ] 0 + i + 5 ] 2 W [ 0 i ij k j T W [ + ] 0 + Mi,S 0 =0 GTMDs in helicity combination for OAM: (++) - (- -) kt 2x 2 T T F 27 + F 28 + G 14 2 k2 T 2 T (k T T ) 2 M 2 2 F 14 + T i 2 T M i,s ++ M i,s =0

31 8/28/17 31 Integrate over k T x e E 2T (x) = e H(x)+F (1) 14 (x) M F 14 Insert LIR d dx Z d 2 k T k 2 T M 2 F 14 = Ẽ2T + H + E + A x 1 Ẽ 2T = dy y A F 14 Z 1 x dy y (H + E) + " H x Z 1 x # dy y H 2 + apple 1 x M F 14 Z 1 x dy y 2 M F 14

32 8/28/17 32 Angular Momentum Sum Rule 2T = Z 1 x dy y (H + E)+ " H x Z 1 x # dy y H 2 + apple 1 x M F 14 Z 1 x dy y 2 M F 14 L = = J - S + 0

33 Integrating over x we re-obtain the OPE based relation Z 1 0 dx xg 2 = 1 2 Z 1 0 Polyakov et al.(2000), Hatta(2012) Z 1 dx x(h + E) dx H J q = L q q A generalized Wandzura Wilczek relation obtained using OPE for twist 2 and twist 3 operators for the off-forward matrix elements

34 8/27/17 34 Other integrated relations Z dxx E2T 0 +2 H 2T 0 = 1 2 Z dxx H 1 2 Z dxh + m 2M Z dx(e T +2 H T ) (L z S z ) q = Z dxx E2T 0 +2 H 2T 0 + H, apple T = Z dx (E T +2 H T ), e q = Z dx H 1 2 Z dxx H =(L z S z ) q e q m q 2M appleq T # Integral relation without connecting to spin-orbit Polyakov et al. (2000) # Integral relation with educated guess for spin-orbit Lorce (2015) Chiral symmetry breaking test!

35 8/27/17 35 Transverse proton spin (unpolarized quark) x F 23 + k2 T 2 T (k T T ) 2 M 2 2 F 24 + T + 1 2M 2 2 T G 13 + k T T G 12 + k2 T i T 2M 2 T 2 T (k T T ) 2 M 2 ( 1 i 2)M i,s + +( 1 + i 2)M i,s + 2 T F 12 =0. f?(1) 1T = F o(1) 12 = M F12 T =0 Sivers function Qiu-Sterman term

36 8/28/17 36 H 0 2T 2 T 4M 2 E0 2T = P 2 M 2 2 T + 4M 2 Z 1 x apple 1 (H + E) x dy y H + m M Z 1 x apple 1 x H T dy (H + E) + y2 2 T 4M 2 E T apple MG12 x Z 1 x Z 1 x dy y 2 H T dy y 2 M G 12 2 T 4M 2 E T Z 1 x dy y A G 12 2 = g 1 Z 1 x dy y g 1 + m M 1 x h 1 Z 1 x Z dy 1 y 2 h 1 + eg T x Z dy 1 y eg T + x dy y bg T Original Wandzura Wilczek relation in forward limit Chiral symmetry breaking test!

37 8/27/ A PROBE OF QCD AT THE AMPLITUDE LEVEL

38 8/27/17 38 PT transformation Forward case: Sivers function (J. Collins, 2002) PT: hp, S (0) + (z) P,Si = hp, S (0) + (z) P, Si M + f 1T perp = M + -M - = 0 M - z -,z T,z T -,z T z -,z T PT: 0,0,0 -,0 0,0 P, S (0) + U(v, z) (z) P,Si = hp, S (0) + U( v, z) (z) P, Si M +v -M - -v = 0 f 1T perp,sidis = M +v -M -v = -f 1T perp,dy =-M + -v +M - -v

39 8/27/17 39 Off forward case: F 14 PT: P, S (0) + U(v, z) (z) P,Si = hp, S (0) + U( v, z) (z) P, Si L + v,δ L - -v,-δ L + v,δ -L - -v,-δ = 0 (k T xδ T ) F 14 SIDIS = L + v, Δ -L - v, Δ = (k T xδ T ) F 14 DY =L + -v, Δ -L - -v, Δ

40 40 -,z T z -,z T,0 z -,z T,z T -, ,0 0,0,0 HLzênL ê»lz Hh=0L ên Hh=0L» Ê u-d quarks m p = 518 MeV z`=0.39 Ê h»v» Hlattice unitsl

41 8/27/17 41 q k+q q =q+δ k l k-l-δ SIDIS-like P P-k P-k+l P =P-Δ -k+q q k l k-l-δ DrellYan-like P P-k P-k+l P =P-Δ

42 42 Two additional processes: DVCS and TCS twist three contributions spacelike q k+q DVCS -k+q q timelike TCS Extracting twist 3 GPDs from these processes will allow us to zoom into aspects of the sign change

43 43 8/28/17 Genuine/intrinsic twist-three terms

44 8/27/17 Simonetta Liuti 44 i 0 = 1 4 Z dz d 2 z T (2 ) 3 hp 0, 0 ( z/2) e ixp + z ik T z T apple (! /@ ig/a)u z/2 + U( /@ + ig/a) z/2 (z/2) p, i z+ =0 xẽ2t = H xẽ2t = H Z Z d 2 kt 2 k T M 2 F staple 14 + M staple F 14 d 2 kt 2 k T M 2 F straight 14 + M straight F 14

45 8/28/17 45 By subtracting the two expressions F (1) 14 staple F (1) 14 staple = M F 14 staple M F14 straight integrating Z dx F (1) 14 di T i i ij gv 1 2P + Z 1 0 ds hp 0, + (0) + U(0,sv)F +j (sv)u(sv, 0) (0) p, +i T =0 Difference between Jaffe-Manohar and Ji (Hatta, Burkardt, 2013) A = d dx Mstaple M straight LIR violating term is the difference between JM and Ji

46 8/27/17 46 Generalized Qiu Sterman term Z d 2 k T kt 2 M 2 F 14 JM Z d 2 k T k 2 T M 2 F Ji 14 = T F (x, x, )

47 8/28/17 47 x Use the connection with twist-three GPD! 1 Ẽ 2T = dy y A F 14 Z 1 x dy y (H + E) + " H x Z 1 x # dy y H 2 + apple 1 x M F 14 Z 1 x dy y 2 M F 14 Ẽ 2T = ẼWW 2T + Ẽ(3) 2T + ẼLIR 2T Ẽ WW 2T = Z 1 x dy (H + E) y " H x Z 1 x # dy y H 2 apple 1 x M F 14 Z 1 x dy y 2 M F 14 Z 1 x dy y A F 14

48 8/28/17 48 Can we measure this and how?

49 8/28/17 49 Direct measurements cannot distinguish between LIRgenerated and EoM-generated genuine twist-three/qgq terms so in principle we cannot measure the difference between Jaffe-Manohar and Ji through DVCS but. Caveat Assuming that twist-three GPDs are process independent, or that they behave like twist-two collinear objects

50 8/28/17 50 x-moments M 0 M 1 M 2 Z Z Z Z dxẽ2t = dx(h + E) ) dx Ẽ2T + H + E =0 Z dxxẽ2t = 1 Z Z 1 dxx(h + E) dx 2 2 H dxx 2 Ẽ 2T = 1 Z Z dxx 2 2 (H + E) dxx 3 3 H 2 Z dxx M F14 3 OAM Sum Rule

51 8/28/17 51 Interpretation M 1 Z dx Z dx d 2 k T M i,s 0 = i ij gv 1 2P + d 2 k T M i,a 0 = gv 1 2P + Z 1 0 Z 1 0 ds hp 0, 0 (0) + U(0,sv)F +j (sv)u(sv, 0) (0) p, i ds hp 0, 0 (0) + Force acting on quark 5 U(0,sv)F +i (sv)u(sv, 0) (0) p, i Non zero only for staple link M 2 Z Z Z dx x Z dx x d 2 k T M i,s 0 = d 2 k T M i,a 0 = ig 4(P + ) 2 hp0, 0 (0) + 5 F +i (0) (0) p, i g 4(P + ) 2 ij hp 0, 0 (0) + F +j (0) (0) p, i rest frame interaction d 2

52 8/28/17 52 Situation is analogous to studies of g 2 Accardi, Bacchetta, Melnitchouk, Schlegel JHEP (2009) The effect of the two twist-three terms combined might be small but each individual contribution can be large D. Flay et al, PRC 2016

53 8/27/17 53 Large effect from lattice (M. Engelhardt, arxiv: ) 0.0 HLzênL ê»lz Hh=0L ên Hh=0L» Ê u-d quarks m p = 518 MeV z`=0.39 F 14 Ê PRD, arxiv: h»v» Hlattice unitsl f 1T perp

54 8/27/17 54

55 8/27/17 55

56 8/27/17 56 $ Cancellation of L u and L d is found consistently with previous calculations $ It persists for Jaffe Manohar OAM!

57 8/27/17 57 F 14 (1) with B. Kriesten and A. Rajan, using diquark model Ji Ratio L Ji /L JM =0.72 JM Lattice Ratio L Ji /L JM =0.62±0.16(stat) (extrapolated at ζ= )

58 8/27/17 Simonetta Liuti 58 Low x! L q (x) F 14! G Abha Rajan et al., arxiv: Q 2 = 0.1 GeV 2 genuine u quark tw WW Sum Rule Integrand LC model xg WW total x -1 xg 2 xg2 tw 3 xg2 tw x

59 8/29/ NUCLEI

60 8/27/17 60 Finally, nuclei

61 8/29/17 61 Original studies of spin 0 GPDs: SL & Taneja, PRD70 (2004), PRC72 (2005), PRC72R (2005) W + 0 = A 0 +, + + A 0, W i +i 5 0 = A 0 +, + A 0, +. Twist-three: W i 0 = A(3) 0 +, i A(3) 0 +, = A(3) 0 +, A (3) 0 +, + A(3) 0, + + A(3) 0 + A(3) 0 +, A (3) 0 +, H A H A T, + H (3) A H (3) A The spin-orbit term can shed light on the origin of the polarized EMC effect (work in progress with I. Cloet) I. Cloet et al PLB (2006)

62 8/28/17 62 Physics of the D-term d represents the spatial distribution of the shears forces (Polyakov Shuvaev)

63 8/29/17 63 From S.L. and S.K. Taneja, PRC72(2005) Is this factorization broken? First signature of non-nucleonic effects

64 8/29/17 64 In liquid drop model d A 1 (0) / A 7/3 d A 1 (0) 1 hei A 1 M hp 2 µi A M 2 / A ln A Nuclear model taking into account virtuality

65 8/27/17 65 # Spin and 3D structure of Deuteron 1 2 Z 1 dxx [H q (x, 0, 0) + E q (x, 0, 0)] = J q 1 Nucleon (Ji, 1997) 1 2 Z 1 dxx H q 2 (x, 0, 0) = J q 1 65 Deuteron (S.K. Taneja et al, 2012)

66 8/29/ GLUON SPIN AND HELICITY IN THE PROTON

67 Jaffe Manohar s Sum Rule:! four independently measured quantities 1 2 ( G + L JM g )=L JM q q 67 L g =0 L g =-0.4 Using the measured value of ΔΣ u+d Using the measured value of ΔG M. Engelhardt, preliminary Lattice QCD evaluation of GTMD F 14 + gauge link 8/27/17 Simonetta Liuti 67

68 8/29/ DVCS ANALYSIS

69 How do we detect all this? Dustin Keller & U.Va. Polarized Target Group

70 Deeply Virtual Exclusive Photoproduction d 5 dx Bj dq 2 d t d d S = (s M 2 ) 2p 1+ 2 T 2, T (k, p, k 0,q 0,p 0 )=T DV CS (k, p, k 0,q 0,p 0 )+T BH (k, p, k 0,q 0,p 0 ),

71 8/27/17 71 From BKM formalism to exact Rosenbluth-like separation Example 1 BH unpolarized cross section BH = apple A(y, t,,q 2, ) F 1 + F 2 2 M 2 + B(y, t,,q 2, ) G 2 M (t)

72 Example 2 DVCS unpolarized cross section Twist 2 Twist 4 Twist 3 Photon helicity flip: transverse gluons

73 Helicity amplitudes Virtual Photon helicities Λ (1) (2 ) F γ * Λ γ * ΛΛ' ( Λ (1) = f γ * Λ γ ' ΛΛ' )* (2 Λ ) f γ * Λ γ ' ΛΛ' Λ γ ' Initial and final proton helicities

74 Phase dependence f e i Λ γ * Λ γ ' (Λ Λ') φ The phase is determined by the virtual photon helicity which can be different for the amplitude and its conjugate

75 BASIC MODULE (based on helicity amplitudes) 1 1 Q 2 1 X 0, 0 T h 0 DV CS, 0 T h 0 DV CS, 0 = (F F 11 + F F 1 1 )+ (F F 00 ) +2 p (1 + )Re( F + 01 F 01 + F F )+2 Re (F + + F 1 1 hp ) +(2h) 1 2 (F F 11 F F 1 1 ) 2 p io polarized lepton (1 )Re(F F 01 + F F 0 1 )

76

77 Twist 3 Bad component (exchanged gluon flips the quark chirality) q, Λ γ q, Λ γ g Λ γλ γ λλ k, λ k, λ q, Λ γ =0 g 0Λ γ q, Λ λλ γ k, λ l, Λ g k,λ P, Λ P, Λ A Λ λ,λλ A Λ λ,λλ P, Λ P, Λ

78 Connecting the DVCS formalism with the TMD/GPD/GTMD comprehensive parametrizations Bacchetta et al JHEP02 (2007), Meissner Metz and Schlegel, JHEP08 (2009) Example A +,++ = 1 2 Ẽ2T E 2T + Ẽ0 2T + E 0 2T A +,++ = 1 2 Ẽ2T + E 2T + Ẽ0 2T + E 0 2T % % % Orbital angular momentum Spin Orbit interaction

79 DVCS: bilinears of tw 2 and tw 3 CFFs F = P h H (Ẽ2T i Ē 2T +...),... Extraction from experiment using Wandzura Wilczek approximation A.Courtoy, G.Goldstein, O.Gonzalez Hernandez, S.L. and A.Rajan, PLB 731(2014)

80 8/27/17 80 Measuring directly GTMDs

81 8/27/17 81 Double DVCS (Bhattacharya, Metz, Zhou, PLB 2017)

82 8/27/17 GTMDs from Double DVCS hadron production (off-forward SIDIS) Simonetta Liuti 82 ep! e 0 + µ + µ p 0 P h t P h -Δ q t,t,p h2 <Q 2, Q 2 k p Δ Φ k-δ p-δ µ + µ - q =q+δ-δ P P t

83 8/27/17 83 Helicity amplitude formalism for DDVCS hadron production # To measure F 14 one has to be in a frame where the reaction cannot be viewed as a two-body quark-proton scattering # In the CoM the amplitudes are imaginary!ul term goes to 0 unless one defines two hadronic planes y photon-proton plane φ Δ p Δ e iϕ Δ k T eiϕ k Δ 1 iδ 2 ( )( k 1 ik 2 ) ik T Δ T z θ e k q h 1 φ k k h 2 two hadrons plane x q

84 8/27/17 Simonetta Liuti 84 Conclusions and Outlook The connection we established through the new relations between (G)TMDs and Twist 3 GPDs, not only allows us to evaluate the angular momentum sum rule, it also opens many interesting avenues: It allows us to study in detail the role of quark-gluon correlations, in a framework where the role of k T and off-shellness, k 2, is manifest. OAM was obtained so far by subtraction (also in lattice). We can now both calculate OAM on the lattice (GTMD) and validate this through measurements (twist 3 GPD) It provides an ideal setting to test renormalization issues, evolution etc QCD studies at the amplitude level shed light on chiral symmetry breaking TWIST THREE GPDs ARE CRUCIAL TO STUDY QCD AT THE AMPLITUDE LEVEL

85 8/29/17 85 Hopefully experimental studies of the hard exclusive processes will fill the gap in our understanding of the strong forces creating our world as we see it. Maxim Polyakov (hep-ph/ )

86 8/27/17 86 Back up

87 8/27/17 Simonetta Liuti 87 Helicity and Transverse Spin Structures of H+E ( A ++,++ + A +,+ + A +, + + A, ) + ( A ++, + + A +, A,+ A +,++ ) ( A X + ++,++ AX + +,+ AX + +, + AX, ) + ( A X + ++,++ AX +,+ AX +, + AX, ) H iδ 2 E = Helicity flips Transv. spin conserved non flip E measures J, not L, but a change of one unit of L (because of the helicity flip) S z = 1/2! 1/2 ) L z =1 at fixed J k, λ k = k Δ, λ P, Λ P = P-Δ, Λ Brodsky and Drell 80s, Belitsky, Ji and Yuan, 90 s