1. Proton Spin Decomposition : The Second Hot Debate in Proton Spin Physics Hai-Yang Cheng Academia Sinica Anomalous gluon & sea-quark interpretation of smallness of  Proton spin decomposition October 15, 2012 Journal Club, AS

2. 2 In DIS experiments, longitudinal proton spin sum rule is tested by measuring polarized parton distribution functions In non-relativistic QM it is expected that  =  u+  d=4/3-1/3=1 Relativistic QM   ~ 0.65, L q ~ 0.35 /2  q(x,Q 2 )=  is identical to flavor-singlet axial coupling g A 0 related to the axial vector current A  0 = u    5 u +d    5 d +s    5 s

3. 3 EMC (European Muon Collaboration ’87) measured g 1 p (x) = ½∑e i 2  q i (x) with 0.01 =10.7 GeV 2 and its first moment. Combining this with the couplings g A 3 =u-d, g A 8 =u+d-2s measured in low- energy neutron & hyperon  decays   = 0.14  0.18,  u = 0.77  0.06,  d = -0.49  0.06,  s = -0.15  0.06 Two surprises: strange sea polarization is sizable & negative very little of the proton spin is carried by quarks ⇒ Proton Spin Crisis (or proton helicity decomposition puzzle)

4. 4 Anomalous gluon interpretation Consider QCD corrections to order  s : Efremov, Teryaev; Altarelli, Ross; Leader, Anselmino; Carlitz, Collins, Muller (88’) Anomalous gluon contribution (  s /2  )  G arises from photon-gluon scattering. Since  G(Q 2 )  lnQ 2 and  s (Q 2 )  (lnQ 2 ) -1 ⇒  s (Q 2 )  G(Q 2 ) is conserved and doesn’t vanish in Q 2 →  limit from (a) from (b) Why is this pQCD correction so special ?

5. 5 QCD corrections imply that If  G is positive and large enough, one can have  s  0 and  =  u+  d  0.60 ⇒ proton spin problem is resolved provided that  G  (2  /  s )(0.08)  1.9 ⇒ L q + G also increases with lnQ 2 with fine tuning This anomalous gluon interpretation became very popular after 1988

6. 6 Operator Product Expansion moments of structure function=  1 0 x n-1 F(x)dx = ∑ C n (q) = short-distance  long-distance No twist-2, spin-1 gauge-invariant local gluonic operator for first moment OPE ⇒ Gluons do not contribute to  1 p ! One needs sea quark polarization to account for experiment (Jaffe, Manohar ’89) It is similar to the naïve parton model How to achieve  s  -0.08 ? Sea polarization (for massless quarks) cannot be induced perturbatively from hard gluons (helicity conservation ⇒  s=0 for massless quarks) J  5 has anomalous dimension at 2-loop (Kodaira ’79) ⇒  q is Q 2 dependent, against parton-model intuition

7. 7 A hot debate between anomalous gluon & sea quark interpretations before 1996 ! anomalous gluon sea quark Efremov, Teryaev Altarelli, Ross Carlitz, Collins, Muller Soffer, Perparata Strirling Roberts Ball, Forte Gluck, Reya, Vogelsang Lampe Mankiewicz Gehrmann …. Anselmino, Efremov, Leader [Phys. Rep. 261, 1 (1995)] Jaffe, Manohar Bodwin, Qiu Ellis, Karlinear Bass, Thomas … As a consequence of QCD, a measurement of  1 0 g 1 (x) does not measure . It measures only the superposition  -3  s /(2  )  G and this combination can be made small by a cancellation between quark and gluon contributions. Thus EMC results ceases to imply that  is small. - Anselmino,Efremov,Leader (’95)

8. 8 Two hot debates in the past years: 1988 ~ 1995: anomalous gluon or sea quark interpretation of smallness of  or g A 0 2008 ~ now: gauge-invariant decomposition of proton spin and gluon angular momentum J g = S g + L g

9. 9 Factorization scheme dependence It was realized by Bodwin, Qiu (’90) and by Manohar (’90) that hard gluonic contribution to  1 p is a matter of convention used for defining  q Consider polarized photon-gluon cross section 1.Its hard part contributes to  C G and soft part to  q s. This decomposition depends on the choice of factorization scheme 2.It has an axial QCD anomaly that breaks down chiral symmetry fact. scheme dependent Int. J. Mod. Phys. A11, 5109 (1996)

10. 10 Two extreme schemes of interest (HYC, ’95) gauge-invariant (GI) scheme (or MS scheme) -- Axial anomaly is at soft part, i.e.  q G, which is non-vanishing due to chiral symmetry breaking and  1 0  C G (x)=0 (but  G  0 !) -- Sea polarization is partially induced by gluons via axial anomaly chiral-invariant (CI) scheme (or “jet”, “parton-model”, “k T cut-off’, “Adler-Bardeen” scheme) Axial anomaly is at hard part, i.e.  C G, while hard gluons do not contribute to  q s due to chiral symmetry  Hard gluonic contribution to  g 1 p is matter of factorization convention used for defining  q.  It is necessary to specify the factorization scheme for data analysis. It is usually done in MS scheme. HYC (’95); Muller, Teryaev (’97) parton model OPE

11. 11 In retrospect, the dispute among the anomalous gluon and sea-quark explanations…before 1996 is considerably unfortunate and annoying since the fact that  g 1 p (x) is independent of the definition of the quark spin density and hence the choice of the factorization scheme due to the axial- anomaly ambiguity is presumably well known to all the practitioners in the field, especially to those QCD experts working in the area. hep-ph/0002157 My conclusion: Dust is settled down after 1995 !

12. 12 How to probe gluon polarization ? DIS via scaling violation in g 1 (x,Q 2 ) photon or jet or heavy quark production in polarized pp collider, lepton- proton collider or lepton-proton fixed target RHIC (at BNL): via direct high-p T prompt   production, jet production HERMES (at DESY): via open charm production COMPASS (at CERN): via open charm production Direct measurement of  G: Photon-Gluon-Fusion process

13. 13 Adolph et al. arXiv:1202.4064  G/G is very small and cannot explain the smallness of  via anomalous gluon effect, but  G  0.2 - 0.3 makes a significant contribution to proton spin

14. 14 SU(3) symmetry implies g A 8 = 3F-D = 0.585 while g A 3 = F+D = 1.2701 Using g A 8 =0.585, COMPASS (’07) & HERMES (’07) obtained g A 0 (3 GeV 2 ) = 0.35  0.03  0.05 (COMPASS) g A 0 (5 GeV 2 ) = 0.330  0.0011  0.025  0.028 (HERMES)   u = 0.85,  d = -0.42,  s = -0.08 at Q 2  4 GeV 2 Semi-inclusive DIS data of COMPASS & HERMES show no evidence of large negative  s:  s = -0.02  0.03 by COMPASS Sea polarization should be small due to smallness of  G

15. 15 When SU(3) symmtry is broken, g A 8 may be reduced. For example, g A 8 = 0.46  0.05 in cloudy bag model g A 8   g A 0   -  s = 1/3 (g A 8 - g A 0 )  For g A  = 0.46,  s = - 0.03  sensitive to SU(3) breaking Three lattice calculations in 2012 : 1.QCDSF  s = - 0.020  0.010  0.004 at Q = 2.7 GeV 2.Engelhardt  s = - 0.031  0.017 at Q = 2 GeV 3.Babich et al  s = G A s (0 ) = - 0.019  0.017 not renormalized yet Bass, Thomas (’10) The smallness of  G implies a small  s. Hence, SU(3) symmetry should be broken in g A 8

16. 16 Second hot debate on gauge-invariant decomposition of the proton spin

17. 17 Conservation of energy-momentum is governed by T , while conservation of angular momentum is described by rank-3 angular momentum tensor M  = T  x  -T  x  with symmetric T  which can be achieved by Belinfante symmetrized expression

18. 18 Ji spin sum rule (PRL ’97) Gauge-invariant decomposition, but J g cannot be further decomposed into spin and OAM parts. However, gluon spin  G has been measured in many experiments. In QED, S  & L  are measurable. Using the identity Jaffe-Manohar decomposition (’90)

19. 19 Each term is not separately gauge invariant except for S q. Xiang-Song Chen, Xiao-Fu Lu, Wei-Min Sun, Fan Wang, T. Goldman (PRL, 2008, 2009) proposed to solve the gauge-invariance problem by decomposing gauge field into A  phys carries the physical d.o.f. while A  pure carries the pure gauge d.o.f. To achieve this goal, they demand A  phys transforms covariantly as F  A  pure transforms as A  and gives zero F , i.e. F  pure =0

20. 20 A pure is used to construct covariant derivative & it doesn’t contribute to F 

21. 21 In QED, one can impose the conditions to ensure A phys has no longitudinal component to ensure A pure has only longitudinal component Under gauge transformation: A  phys  A  phys, A  pure  A  pure +    In QCD, replace the two conditions in QED in terms of covariant derivatives Under gauge transformation: A  phys  U(x)A  phys U(x ) +, A  pure  U(A  pure +i/g   )U(x ) +  A phys =A , A pure =A ‖ Gauge fixing  A = 0   A pure = 0. Hence we can set A pure =0

22. 22 The separation of A  into physical and pure gauge parts is possible at the cost of introducing nonlocality. A phys & A pure are nonlocally related to the total A. For example, in QED Chen et al. decomposition (’08) Each term is gauge invariant, S g and L g have operator definition ! satisfies the commutation relation Momentum p-gA pure is neither the canonical momentum p nor the dynamical momentum p-gA; it is gauge-invariant and satisfies canonical commutation relation

23. 23 Criticism (mainly due to Ji) clashes with locality, A phys is non-local lack of Lorentz covariance: The decomposition A=A phys +A pure is not Lorentz covariant. In other Lorentz frame, A phys and A pure may mix together A phys is not unique even after a gauge fixing How to quantize the theory with both A phys & A pure as quantum mechanical degrees of freedom ? It seems quantization makes sense only if A pure =0. gauge invariance not in the textbook sense limited physical significance; cannot be measured experimentally. For example, S g is not the gluon spin  G measured in high energy DIS experiments or in pp collisions Gluons carry only 1/5 of the proton momentum in Q 2  limit

24. 24 Lorentz covariance Q: Is the separation A=A phys +A pure Lorentz covariant ? In the other Lorentz frame, will A phys and A pure remain physical & pure gauge ? A: A  cannot transform as a 4-vector Chen et al.; Leader, Lorce,… One can choose   in such a way that A phys remains physical in any Lorentz frame. However, the Lorentz transformation law is complicated. For example,  A phys =0 in one frame, then    restores  ’  A’ phys =0 in a transformed frame

25. 25 Uniqueness of A phys ? Ji: A phys is not unique even after a gauge fixing In QED,  A phys =0 doesn’t fix A phys. Consider A phys  A phys +  with  2  =0. Hence, there are infinite numbers of A phys Chen et al.: Since we demand A pure transforms as A, A ph ys is invariant under guage transformation. With A phys  0 at spatial infinity, it can be solely expressed in terms of E & B fields in QED Hence, A phys is as measurable (physical) as E and B are.

26. 26 Different decompositions Ji (’97) Jaffe-Manohar (’90): Barshinsky-Jaffe LC gauge (’98): Chen et al. (’08,’09): Wakamatsu (’10,’11): Also decompositions by Cho, Ge, Zhang; Leader; Guo, Schmidt; Lorce,…

27. 27 Wakamatsu decomposition To decompose A  =A  phys + A  pure, Wakamatsu imposed two conditions alone: (i) F  pure =0, (ii) A  phys transforms as F  and A pure as A . Gauge fixing will be done in a later stage canonical OAM potential OAM His decomposition differs from that of Chen et al. in quark & gluon OAMs.

28. 28 Debate between canonical & dynamical variables Quark OAM extracted from GPD analysis is dynamical OAM; useful in classical picture canonical variables dynamical variables quark momentum: quark OAM: Jaffe-Manohar; Chen et al. Bashinsky-Jaffe; Cho et al. Leader,… Ji; Wakamatsu In QM, they correspond to generators of translation & rotation; suitable for quantization

29. 29 In the gauge A pure =0, A=A phys, Bashinsky-Jaffe and Chen et al. decompositions are reduced to Jaffe-Manohar one as D pure  , E  A phys  E  A. Wakamatsu claimed that these 3 decompositions are all gauge equivalent, provided that gauge fixing procedure is done consistently with the general conditions for A phys and A pure Wakamatsu argued that there exist only two physically inequivalent decompositions (I) & (II) (Wakamatsu)

30. 30 Gauge invariant extension (GIE) Ji et al. claimed that gauge-invariant decomposition of proton spin is just a GIE of gauge-variant quantities which generalizes the fixed gauge result extrapolated to any other gauge. Consider gaue-variant Jaffe-Manohar decomposition GIE at Coulomb gauge: Chen et al at LC gauge : Bashinsky-Jaffe Coulomb GIE LC GIE The decompositions of Chen et al. and Bashinsky-Jaffe correspond to different GIE and hence they are not necessarily gauge equivalent

31. 31 In Jaffe-Manohar decomposition, gluon spin S g = d 3 x (E  A) 3 is gauge dependent. Its values in light-cone, covariant and Coulomb gauge fixings are different. According to Ji, Chen et al. decomposition is a GIE of JM in Coulomb gauge, while Bashinsky-Jaffe is a GIE of JM in LC gauge. Hence, S g is different in these two schemes. Indeed, Chen et al. found S g (Chen) = 5/9 S g (BJ) Wakamatsu argued that since S g =d 3 x(EA phys ) 3 in the gauge- invariant decomposition is gauge invariant, it should be same in Chen et al and Bashinsky-Jaffe, provided that gauge fixing procedure is done consistently with the general conditions for A phys and A pure Hoodbhoy, Ji, Lu (’99)

32. 32 Gluon momentum fraction Conventional approach (Ji) Chen et al. (PRL, 2009) Chen et al. thus claimed the standard textbook statement that gluons carry half of the nucleon momentum is wrong !

33. 33 T ++ = T ++ q + T ++ g   + i  +  + 2Tr(  + A  ) 2 Momentum sum rule follows from /2(P + ) 2 =1 In A + =0 gauge, D + =  + -igA +   + F +  =  + A  -   A + + g[A +,A  ]   + A  Hence, 1 =  dx x [q(x)+g(x)] = q + g Wakamatsu (’02) has shown explicitly that the anomalous dimension matrix in Chen et al. decomposition is the same as in the conventional approach. This makes him wondering if the result S g (Chen ) = 5/9 S g (BJ ) obtained by Chen et al. is also wrong. T ++ q (Chen)-T ++ q (Ji) = g  +  A + phys  = 0 in LC gauge

34. 34 Observables must be gauge-invariant. Doesn’t it mean that GIE of gauge-variant quantity becomes measurable experimentally if one is lucky enough ? Ji: problems with GIE: GIE operators are in general nonlocal and hence doesn’t have clear physical meaning in general gauges, although they do in the fixed gauge; cannot be calculated in lattice QCD Do not transform simply under Lorentz transformation infinite number of non-local operators Wakamatsu: GIE is not a correct way of handling gauge symmetry. Color SU(3) symmetry is an intrinsic property of QCD Lagrangian, no need of GIE.

35. Gluon helicity Experimentally, the polarized gluon distribution is given by In light-cone gauge A + =0,  g(x) has a simple interpretation: it measures the distribution of gluon polarization Manohar; Collins, Soper first moment: Bashinsky & Jaffe claimed that  G is gauge invariant, but it is not obvious why it is so. Wakamatsu showed that A above can be replaced by A phys without making any approximation. Gluon spin (  E  A) & OAM are not gauge invariant, but helicity is. Gluon polarization in IMF  G=  dx  g(x) is a measurable quantity. where is a gauge link to ensure local gauge invariance

36. 36 In gauge-invariant decomposition of the proton spin This leads to The above expression is not in contradiction to the usual statement: Gluon helicity cannot be expressed in terms of gauge-invariant twist-2 local operator as A phys is not a local operator However, Ji et al. claimed that  G is meaningful only in LC gauge and in infinite momentum frame

37. 37 Criticism from Leader: None of these (Chen et al. Ji, Wakamatsu,…) is acceptable as not enough attention is paid to the difference between classical & quantum field theory. Gauge invariance of operators is not an important criterion. Physical m.e. of measurable operators must be gauge invariant Previous treatment is classical and use has been made of classical EOM. It is OK to use non-local field operator, but not OK if they are dynamical variables. In Coulomb gauge A 0 is not an independent dynamical variable.

38. 38 Quark orbital angular momentum At Q 2 → , Ji, Tang & Hoodbhoy found (’96) Analogous to the nucleon’s momentum partition: half of the proton’s momentum is carried by gluons for n f =6 Experimentally, how to measure J q ?

39. 39 J q is related to the GPDs by the Ji sum rule Ji (’97) Study of hard exclusive processes leads to a new class of PDFs: four independent GPDs (at twist-2): DVCS in large s and small t region can probe GPDs

40. 40 Quark OAM extracted from GPD analysis is dynamical OAM not canonical OAM

41. 41 Recent development: relate OAM to Wigner or phase-space operator as OAM is a correlation between position & momentum Lorce, Pasquini [1106.0139] Hatta [1111.3547] Lorce, Pasquini, Xiong, Yuan [1111.4827] Ji, Xiong, Yuan [1202.2843] Burkardt [1205.2916] Lorce [1205.6483] Lorce, Pasquini [1206.3143] Hatta, Yoshida [1207.5332] Ji, Xiong, Yuan [1207.5221] Lorce, Pasquini [1208.3065] Lorce [1210.2581] Ji (’03)

42. 42 Just like the debate between anomalous gluon & sea quark interpretation of the proton spin, it appears all the different decompositions are correct. Controversies mainly concern the physical interpretation. As to which one is more convenient and more physical is most likely a matter of taste.

43. 43 Conclusions  & L q are factorization scheme dependent, but not J q DIS data ⇒  GI  0.34,  s GI  -0.03 RHIC, COMPASS & SIDIS data imply small  G &  q s  dxg 1 p (x) is independent of the definition of quark spin density and the choice of the factorization scheme due to axial-anomaly ambiguity Several different gauge-invariant decompositions of proton spin have been proposed. Controversies mainly concern the physical interpretation. As to which decomposition is more convenient and more physical is a matter of taste. What do we learn in past 25 years about the proton helicity decomposition ?

44. 44 Extra slides

45. 45 Lattice QCD Can lattice QCD shed some light on the protn spin content ? Sea polarization from disconnected insertion ⇒  u s =  d s =  s = -0.12±0.01

46. 46 J u = ½  u+L u J d = ½  d+L d HERMES: hep-ex/0606061 JLab: nucl-ex/0709.0450 p-DVCS sensitive to J u n-DVCS sensitive to J d

47. 47 Lattice calculations of GPDs arXiv:0705.4295 (LHPC,MILC): Hagler, Schroers,… arXiv:0710.1534 (QCDSF,UKQCD): Brommel, Gockeler, Schroers,… L u+d ~0 & J d ~0 ) cancellation between L u & L d ; ½¢d & L d From J u =0.230, J d = -0.004, L u+d =0.025, ) L u =- 0.190, L d = 0.215 How about L s ? LHPC QCDSF ½  u+d L u+d JuJu JdJd LuLu LdLd

48. 48 Alexandrou et al. (ETM, European twisted mass) 1104.1600 Syritsyn et al. 1111.0718