The role of orbital angular momentum in the proton spin

1. The role of orbital angular momentum in the proton spin M. Wakamatsu (Osaka University) : KEK2010 based on collaboration with Y. Nakakoji, H. Tsujimoto Plan of Talk 1. The nucleon spin puzzle : current status 2. Burkardt and BC’s theoretical consideration 3. Thomas’ recent proposal toward the resolution of the puzzle 4. Our (nearly) model-independent analysis of the proton spin 5. Reminder : some distinctive predictions of the CQSM

2. 1. The nucleon spin puzzle : current status Two remarkable information from recent experimental progress : through new COMPASS & HERMES measurement by RHIC, COMPASS, HERMES, etc. After all the efforts, the following question still remains to be solved ! What carry the rest 2/3 of the nucleon spin ?

3. Two popular decomposition of the nucleon spin

4. gauge invariant, although cannot be decomposed further ! [Caution] • In these 2 decompositions, only part is common ! • There is still another decomposition. X.-S. Chen et al., Phys. Rev. Lett. 103, 062001 (2009). Phys. Rev. Lett. 100, 232002 (2008). [Advantages of the Ji decomposition] • can be made experimentally through DVCS and/or DVMP processes !

5. 2. Burkardt and BC’s theoretical consideration M. Burkardt and Hikmat BC, Phys. Rev. D79, 071501 (2009). Using two “toy models”, i.e. scalar diquark model & QED to order a they compared the fermion OAM (orbital angular momenta ) obtained from the Jaffe-Manohar decomposition and from the Ji decomposition, and found that • The two decompositions yield the same fermion OAM in scalar diquark model, but not in QED (gauge theory). • The x-distributions of the fermion OAM in the two decompositions are different even in scalar diquark model. • in QED and QCD to order a

6. 3. Thomas’ recent proposal toward the resolution of the puzzle Recently, Thomas carried out an analysis of the proton spin contents in the context of the refined cloudy bag (CB) model, and concluded that the modern spin discrepancy can well be explained in terms of the standard features of the nonperturbative structure of the nucleon, i.e. (1)relativistic motion of valence quarks (2)pion cloud required by chiral symmetry (3)exchange current contribution associated with the OGE hyperfine interactions supplemented with the QCD evolution or scale dependenceofand. (1) relativistic effect - lower p-wave components of relativistic wave functions -

7. bare nucleon probability reduction of quark spin fraction (2) pion cloud effects physical nucleon= “bare nucleon” + pion cloud valence quarks of quark core partial cancellation main factor of reduction !

8. (3) one-gluon exchange correction (Myhrer and Thomas)

9. predictions of the refined CB model • A. W. Thomas, Phys. Rev. Lett. 101 (2008) 102003. another characteristic feature • CB model prediction corresponds to low energy model scale. • The quark OAM is a strongly scale-dependent quantity !

10. Leading-order (LO) evolution equation for quark orbital angular momenta (OAM) • X. Ji, J. Tang, and P. Hoodbhoy, Phys. Rev. Lett. 76 (1996) 740. • flavor singlet channel • flavor non-singlet channel

11. crossover around 0.6 GeV scale • A. W. Thomas, Phys. Rev. Lett. 101 (2008) 102003.

12. Most remarkable is a crossover of and around 600 MeV scale ! This crossover is an inevitable consequence of the following two features ! (1) Refined CBM prediction at low energy scale : (2) Asymptotic boundary condition dictated by the QCD evolution equation : See next page Thomas then claims that, owing to this crossover, the predictions of the refined CB model after taking account of QCD evolution is qualitatively consistent with the recent lattice QCD data given at , which gives We shall show later that his statement is not necessarily justified !

13. Note on the asymptotic boundary condition of • M. W. and Y. Nakakoji, Phys. Rev. D77 (2008) 074011. Leading-order evolution eq. for with Since right-hand-side becomes 0 as , it immediatelly follows that neutron beta-decay coupling constant !

14. 4. Our (nearly) model-independent analysis of proton spin Strongly scale-dependent nature of the nucleon spin contents was repeatedly emphasized in a series of our investigations in the context of the CQSM. • M. W. and T. Kubota, Phys. Rev. D60 (1999) 034020. • M. W., Phys. Rev. D67 (2003) 034005. • M. W. and Y. Nakakoji, Phys. Rev. D74 (2006) 054006. • M. W. and Y. Nakakoji, Phys. Rev. D77 (2008) 074011. In the following, we try to carry out the analysis of the nucleon spin contents as model-independently as possible ! Most general nucleon spin sum rule in QCD The point is that this decomposition can be made purely experimentally through the GPD analyses (X. Ji).

15. Once is known, is automatically known from Ji’s angular momentum sum rule with For flavor decomposition, we also need nonsinglet combination with

16. and similarly for Ji showed that and obey exactly the same evolution equation ! At the leading order (LO) with and similarly for

17. A key observation now is that the quark and gluon momentum fractions are basically known quantities at least above , where the framework of pQCD can safely be justified ! Neglecting small contribution of strange quarks, which is not essential for the present qualitative discussion, we are then left with two unknowns : An interesting idea is then to use the QCD evolution equation to estimate the nucleon spin contents at lower energy scales of nonperturbative QCD. Fortunately, the available predictions of lattice QCD for these quantities corresponds to the renormalization scale , which is high enough for the framework of pQCD to work. inverse or downward evolution !

18. A natural question is how far down to the low energy scale we can trust the framework of perturbative renormalization group equations. As advocated by Mulders and Pollock, these scales may be regarded as a matching scale with low energy effective quark models. Here, we take a little more conservative viewpoint that matching scale would be somewhere between and . Leaving this fundamental question aside, one may continue the downward evolution up to the scale , where By starting with the MRST2004 values, at , we find : Nevertheless, one can say at the least that downward evolution below this unitarity-violating limit is absolutely meaningless ! : unitarity-violating limit

19. (A) Isovector part • newest lattice QCD results given at Now, we concentrate on getting reliable information on two unkowns : close to each other ! • CQSM 2008 (M.W. and Y. Nakakoji) To avoid initial energy dependence of CQSM estimate, here we simply use the central value of LHPC 2008 :

20. (B) Isoscalar part From the analysis of forward limit of unpolarized GPD within the CQSM, the 2nd moment of which gives , a reasonable theoretical bound for is obtained ( M.W. and Y. Nakakoji, 2008 ) Lattice QCD predictions are sensitive to the used method of cPT and dispersed ! This works to exclude some range of lattice QCD predictions ! In the following analysis, we therefore regard as an uncertain constant within the above bound.

21. Once is known, the quark OAM is easily obtained by subtracting the known longitudinal quark polarization : For , we simply use here the central value of HERMES analysis : complete initial conditions at

22. Data at are from LHPC2008. • Ph. Hagler et al. (LHPC Collaboration), Phys. Rev. D77 (2008) 094502.

23. The most significant difference appears in the quark OAM ! Thomas’ analysis shows a crossover of and around 0.6 GeV scale. • refined CB model predicts at low energy model scale. • QCD evolution dictates that . In contrast, no crossover of and is observed in our analysis. remains to be larger than even down to the unitarity-violating limit. One sees that the difference between results of the two analyses is quite large. This also means that the agreement between Thomas’ results and the lattice QCD predictions is not so good as he claimed. It comes from the fact that

24. One might suspect that the uncertainties of the initial conditions given at might alter this remarkable conclusion. It is clear by now, however, that the problem exists rather in the isovector channel. Main uncertainty comes from the isovector AGM ! since

25. remains negative even down to the lower energy scale close to the unitarity-violating bound, in contrast to the prediction of the CB model !

26. Also interesting would be a direct comparison with the empirical information on and extracted from the recent GPD analyses. See figure in the next page. One sees that, by construction, the result of our analysis is fairly close to that of the lattice QCD simulations. On the other hand, the result of Thomas’ analysis significantly deviates from the other two, and outside the error-band of JLab data. General trend of Thomas’ predictions

27. Comparison with GPD extraction of

28. Anyhow, our semi-phenomenological analysis, which is consistent with empirical information as well as the lattice QCD data at high energy scale indicates that remains large and negative even at low energy scale of nonperturbative QCD ! If this is really confirmed, it is a serious challenge to any low energy models of nucleon, because they must now explain new ! simultaneously ! One may really call it new or another proton spin puzzle in the sense that it is totally incompatible with the picture of the standard quark model, including the refined CB model of Thomas and Myhrer.

29. Very interestingly, its prediction for given in Is there any low energy model which can explain this peculiar feature ? Interestingly or strangely , the CQSM can ! It has been long claimed that it can explain very small quark spin fraction : because of the very nature of the model ( i.e. the nucleon as a rotating hedgehog ) • M. W. and H. Tsujimoto, Phys. Rev. D71 (2005) 074001. perfectly matches the scenario emerged from the present semi-empirical analysis ! But why ?

30. The problem may have deep connection with how to define quark OAM ! Remember the fact that there are two popular decomposition of the nucleon spin, i.e. the Jaffe-Manohar decomposition and the Ji decomposition. It has been long recognized that the quark OAM in the Ji decomposition is manifestly gauge invariant, so that it contains interaction term with the gluon. Since the CQSM is an effective quark theory that contains no gauge field, one might naively expect that there is no such ambiguity problem in the definition of the quark OAM. However, it turns out that this is not necessarily the case. The point is that it is a highly nontrivial interaction theory of quark fields. To explain it, we recall the past analyses of GPD sum rules within the CQSM.

31. Analyses of GPD sum rules in the CQSM : • Isoscalar channel : J. Ossmann et al., Phys. Rev. D71,034001 (2005). • Isovector channel : M. W. and H. Tsujimoto, Phys. Rev. D71,074001 (2005). Isoscalar case : 2nd moment of where with

32. Isovector case : 2nd moment of We found that extra piece with

33. This extra term is highly model-dependent and its physical interpretation is not obvious at all. It is nevertheless clear that there is no compelling reason to believe that the quark OAM defined through Ji's sum rule must coincide with the canonical OAM , i.e. the proton matrix element of the free-field OAM operator. Since the CQSM is a nontrivial interacting theory of effective quarks, which mimics the important chiral-dynamics of QCD, it seems natural to interpret this peculiar term as a counterpart of the interaction dependent part of the quark OAM in the Ji decomposition of the nucleon spin. Numerically

34. We have estimated the orbital angular momentum of up and down quarks in the proton as functions of the energy scale, by carrying out a downward evolution of available information at high energy, to find that remains to be large and negative even at low energy scale of nonperturbative QCD ! We emphasized that, if it is really confirmed, it may be called Although the quark OAM are not directly measurable, they can well be extracted since and are measurable quantities from GPD analysis and since the intrinsic quark polarizations are basically known by now. Summary at this point another or 2nd nucleon spin puzzle ? because it absolutely contradicts the picture of standard quark model !

35. The key is a precise measurement of at a few GeV scale. standard scenario

36. 5. Reminder : some distinctive predictions of the CQSM • field theoretical nature of the model (nonperturbative inclusion of polarized • Dirac-sea quarks) enables reasonable estimation of antiquark distributions. • only 1 parameter of the model (dynamical quark mass M) was • already fixed from low energy phenomenology parameter-free predictions for PDFs It may be better to say that the matching energy or the initial energy scale of evolution is a parameter of this effective model, although we do not treat it as a fitting parameter ! Default Lack of explicit gluon degrees of freedom

37. How to choose the starting energy of evolution Follow the spirit of the PDF fit by Glueck et. al. (GRV) • They start the QCD evolution at the extraordinary low energy scales like • They found that, even at such low energy scales, one needs nonperturbatively • generated sea-quarks, which may be identified with effects of meson clouds. Our general strategy • use predictions of CQSM as initial-scale distributions of NLO DGLAP eq. • initial energy scale is fixed to be (according to the GRVPDF fitting program)

38. Parameter free predictions of CQSM for 3 twist-2 PDFs • unpolarized PDFs • longitudinally polarized PDFs • transversities totally different behavior of the Dirac-sea contributions in different PDFs !

39. Isoscalar unpolarized PDF : D. I. Diakonov et al. (1997) sea-like soft component positivity

40. Isovector unpolarized PDF

41. ratio in comparison with neutrino scattering data old fits CQSM parameter free prediction new fit

42. sign change in low region ! Isoscalar longitudinally polarized PDF deuteron New COMPASS data

43. CQSM predicts Isovector longitudinally polarized PDF This means that antiquarks gives sizable positive contribution to Bjorken S.R. 1st moment or Bjorken sum rule in CQSM

44. A recent global fit including polarized pp data at RHIC • D. Florian, R. Sassot, M. Strattmann, W. Vogelsang, Phys. Rev. D80, 034030 (2009).

45. DSSV2009 consistent with CQSM ? but in the valence region

46. To obtain more definite information on the polarized sea-quark distribution in the nucleon, we need more and more efforts ! • semi-inclusive reactions • Drell-Yan processes • neutrino scatterings • etc. Thank you for your attention !

47. [Backup slides]

48. [ Relativistic reduction factor due to lower component of w.f. ] ground state w.f. of MIT bag model isoscalar and isovector axial charge where To reproduce Thomas’ estimate

49. CQSM prediction for Dirac sea contribution valence contribution no net Dirac sea contribution cancel ! to smallvalence contribution

50. On the theoretical bound on Since schematic representation 1st and 2nd moment sum rules From these, we have