The role of orbital angular momentum in the proton spin

based on collaboration with Y. Nakakoji, H. Tsujimoto

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

Plan of Talk

M. Wakamatsu (Osaka University) : KEK2010

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 ?

Two popular decomposition of the nucleon spin

[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]

• gauge invariant, although cannot be decomposed further !

• can be made experimentally through DVCS and/or DVMP processes !

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

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,

• The x-distributions of the fermion OAM in the two decompositions are

• in QED and QCD to order

but not in QED (gauge theory).

different even in scalar diquark model.

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

(1) relativistic effect

- lower p-wave components of relativistic wave functions -

supplemented with the QCD evolution or scale dependence of and .

(2) pion cloud effects

physical nucleon 　 = 　 　　“ bare nucleon” 　 + 　pion cloud

valence quarks of quark core

reduction of quark spin fraction

bare nucleon probability

main factor of reduction !

partial cancellation

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

predictions of the refined CB model

• A. W. Thomas, Phys. Rev. Lett. 101 (2008) 102003.

• CB model prediction corresponds to low energy model scale.

• The quark OAM is a strongly scale-dependent quantity !

another characteristic feature

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

• A. W. Thomas, Phys. Rev. Lett. 101 (2008) 102003.

crossover around 0.6 GeV scale

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 :

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 !

See next page

neutron beta-decay coupling constant !

with

• Note on the asymptotic boundary condition of

• M. W. and Y. Nakakoji, Phys. Rev. D77 (2008) 074011.

Since right-hand-side becomes 0 as , it immediatelly follows that

Leading-order evolution eq. for

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).

Ji’s angular momentum sum rule

with

For flavor decomposition, we also need nonsinglet combination

with

Once is known, is automatically known from

Ji showed that and obey exactly the same evolution equation !

At the leading order (LO)

with and similarly for

and similarly for

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 :

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.

An interesting idea is then to use the QCD evolution equation to estimate the nucleon spin contents at lower energy scales of nonperturbative QCD.

inverse or downward evolution !

A natural question is how far down to the low energy scale we can trust the framework of perturbative renormalization group equations.

Leaving this fundamental question aside, one may continue the downward evolution up to the scale , where

As advocated by Mulders and Pollock, these scales may be regarded as a matching scale with low energy effective quark models.

By starting with the MRST2004 values, at , we find :

Here, we take a little more conservative viewpoint that matching scale would be somewhere between and .

Nevertheless, one can say at the least that downward evolution below this unitarity-violating limit is absolutely meaningless !

: unitarity-violating limit

Now, we concentrate on getting reliable information on two unkowns :

(A) Isovector part

• newest lattice QCD results given at

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 :

(B) Isoscalar part

Lattice QCD predictions are sensitive to the used method of PT 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.

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 )

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

• Ph. Hagler et al. (LHPC Collaboration), Phys. Rev. D77 (2008) 094502.

Data at are from LHPC2008.

The most significant difference appears in the quark OAM !

It comes from the fact that

• 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.

Thomas’ analysis shows a crossover of and around 0.6 GeV scale.

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.

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

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

Also interesting would be a direct comparison with the empirical information on

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.

See figure in the next page.

General trend of Thomas’ predictions

and extracted from the recent GPD analyses.

Comparison with GPD extraction of

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

simultaneously !

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.

One may really call it

new !

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 !

Very interestingly, its prediction for given in

But why ?

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.

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).

where

with

Isoscalar case : 2nd moment of

with

Isovector case : 2nd moment of

We found that

extra piece

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

Summary at this point

another or 2nd nucleon spin puzzle ?

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.

because it absolutely contradicts the picture of standard quark model !

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

standard scenario

parameter-free predictions for PDFs

• field theoretical nature of the model (nonperturbative inclusion of polarized Dirac-sea quarks) enables reasonable estimation of antiquark distributions.

Default

Lack of explicit gluon degrees of freedom

• only 1 parameter of the model (dynamical quark mass M) was already fixed from low energy phenomenology

5. Reminder : some distinctive predictions of the CQSM

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 !

• use predictions of CQSM as initial-scale distributions of NLO DGLAP eq.

• initial energy scale is fixed to be (according to the GRV PDF fitting program)

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.

How to choose the starting energy of evolution

Our general strategy

Parameter free predictions ofCQSM for 3 twist-2 PDFs

• unpolarized PDFs

• longitudinally polarized PDFs

• transversities

totally different behavior ofthe Dirac-sea contributions

in different PDFs !

Isoscalar unpolarized PDF : D. I. Diakonov et al. (1997)

positivity

sea-like soft component

Isovector unpolarized PDF

ratio in comparison with neutrino scattering data

old fits

CQSM parameter free prediction

new fit

Isoscalar longitudinally polarized PDF

New COMPASS data

deuteron

sign change inlow region !

Isovector longitudinally polarized PDF

CQSM predicts

This means that antiquarks gives sizable positive contribution to Bjorken S.R.

1st moment or Bjorken sum rule in CQSM

A recent global fit including polarized pp data at RHIC

• D. Florian, R. Sassot, M. Strattmann, W. Vogelsang, Phys. Rev. D80, 034030 (2009).

DSSV2009

consistent with CQSM ?

but

in the valence region

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 !

[Backup slides]

[ 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

no net Dirac sea contribution

small valence contribution to

cancel !

CQSM prediction for

Dirac sea contribution valence contribution

schematic representation

On the theoretical bound on

1st and 2nd moment sum rules

From these, we have

Since

New compass data (2005)

CQSM predictions for longitudinally polarized structure functions for p, n, D

old SMC data

old SMC data

New COMPASS and HERMES fits for in comparison with CQSM prediction

COMPASS preliminary, A. Korzenev, arXiv : 0909.3729 [hep-ex]

polarized strange quark distributions : M.W., Phys. Rev. D67, 034005 (2003)