Nuclear Physics A577 (1994) 335c-340c North-Holland, Amsterdam

N U C L E A R PHYSICS A

S p i n - f l a v o r s t r u c t u r e o f t h e n u c l e o n

in t h e ch ira l q u a r k s o l i t o n m o d e l

M Wakamatsu

Depar tment of Physics, Facul ty of Science, Osaka University,

Toyonaka, Osaka 560, Japan

The spin-flavor s t ructure of the nucleon as mvestag,~ted within the chiral quark soh-

ton model, especially put t ing emphasis upon the 1/N~ lsorotatxonal correction to baryon

observables recently found in this model

1. I n t r o d u c t i o n

To unders tand what the chlral quark sohton model (CQSM) is, it is always instruc-

tive to compare it with more famlhar Skyrme model The one is an effective quark theory,

and the other is an effective meson theory Nonetheless, there are inseparable connections

between these models [1-3] This is eventual ly because in both models a nucleon xs in-

terpreted as a collective lsorotataonal s ta te of the hedgehog object Accordingly, physmal

pledlct lons of both rnodels are generally ra ther similar, which might tempt some people

to believe the so-called Cheshire Cat Principle, or approximate equivalence of a fermlon

theory and a boson theory

However, our recent analysis of the gA problem has revealed a crucial difference be-

tween these two effective theolaes [4] We showed that , within the CQSM, there exists a

crucially impor tan t 1/N~ lsorotat lonal correction to gA, while there is no corresponding

coriect lon in the Skyrme model It seems to resolve the long-standing puzzle why gA

is drast ical ly underes t imated in the Skyrme model This remarkable difference between

the two effective theoiles plays Impor tan t roles also an the physics of the nucleon spm

contents In this note, I t ry to show tha t the CQSM is one of the most promising low

energy models of QCD for investigating spin-flavor s tructure of baryons

2. T h e m o d e l : its bas i c s a n d p r e d i c t i o n s

The lagranglan of the CQM as extremely simple and given as [2,3]

13CQM : ~ ( z # -- M e"~sTzr(~)/f")%b ( I )

0375-9474/94/$07 00 © 1994 - Elsevier Science B V All nghts reserved SSDI 0375-9474(94)00393-9

336c M Wakamatsu / Spm-flavor ~tructure of the nucleon

It describes effective quark fields coupled to the Goldstone plon fields in a chlral lnvanant

way Here M denotes the effective quark mass, which plays the role of the quark-pion

coupling constant This is essentially only one parameter of the theory Note that the

absence of the plon kinetic term in this lagranglan imphes that it is not an independent

field of quarks, but eventually interpreted as a qq composite

There are two ways to t reat the above lagranglan The one is the functional bosomza-

tion approach If one integrates out the quark degrees of freedom with some s tandard

path integral techmque, one obtains an effective meson lagranglan, which greatly resem-

bles the Skyrmlon lagranglan Actually, we do not take this approach, because at the 4th

derivative order there appears a des tabdizmg term, which does not allow for the existence

of s table sohtons Nevertheless, the conceptually deep connection between the CQM and

the Skyrme model is clear from this consideration [1]

Our approach is based on the M F theory and the subsequent cranking quantizatlon,

which is familiar m nuclear physics [2,3] We star t with a stat ic plon field configuration

of hedgehog shape as,

,~(~) = ~ F ( r ) , (2)

which plays a role of the M F potent ia l for quarks The quark field in this M F potent ial

then obeys the following Dirac equation

H I ' ~ > = e m l m > , (3)

V H - + M / 3 ( c o s F ( r ) + zTs~- f ' s l n F ( r ) ) (4)

Z

Due to the presence of the term propor t ional to v i-, this M F potent ial destroys the

rotat ional symmet ry m the ordinary and isospm spaces A characterist ic feature of this

Dnac equation IS tha t one deep single-quark bound s tate appears from the positive energy

cont lnlmm We call it the ~alence quark orbital 4n object with baryon number one with

respect to the physmal vacuum is obtained by put t ing Arc (= 3) quarks Into this valence

orbi tal as well as all the negative energy (Dlrac-sea) orbltals Accordingly, the total energy

of this B = 1 object is given as a sum of the valence quark contr ibution and the vacuum

polarizat ion one The most probable plon field configuration is then determined from the

s ta t ionary requirement of this energy This requirement combined with the abo~e Dlrac

equation is now reduced to a self-consistent Hartree problem which can be solved by the

numerical method of Kahana and Rlpka

Unfortunately, the baryon number one mean-field solution obtained by solving the

Hartree problem cannot yet be identified with the physical nucleon, since it does not

have good spin and lsospm quantum numbers To obtain a baryon s tate with good spin

and lsospm, we use the familiar clanking method First , we consider a t ime-dependent

lsorotatlon of the hedgehog mean field Ug°(:r) = exp [z 7s ~- r F ( r ) ] as

U ~ ' ( x , t ) = A( t ) UgS(~e)At(t) (S)

M Wakamatsu I Spm-flavor structure of the nucleon 337c

This leads to an a l ternat ive representat ion of the original lagrangian as follows

Here, ~ba and HA = H - f~ are respectively the quark field and the effective hamlltonlan

m the body-fixed lsorotat lng frame with f~ = z At( t ) (dA(t) / dr) an analog of the Conolis

force We then evaluate changes of the mtrinmc quark field reduced by this Corlohs

couphng and the assocmted effects to baryon observables Basle assumptmn here is that

the velocity of the collectwe lsorotatmn is much slower than tha t of the internal quark

motion Finally, this ro ta t ional motion is canonically quantized as

ao --~ - J o / i (7)

Note that the moment of Inert ia appearing in this quant lzahon rule is an order Nc quantity,

so that the Coriohs coupling itself is an order 1~No quant i ty

After all these steps, a baryon mat r ix element of a rb i t ra ry current operator can be eval-

uated by sandwltchmg an effective opera tor < O" > a with the wave functmns g2~Mr[A ]

describing the collective rota t ional motion as

< J 'M ' jM / r lO~ ' I JMJMT >

--/dA[

I te le the effechve opera tor < O" > a is obtained by fixmg the collective coordinate A and

by integrat ing out the internal quark motion, which amounts to evaluating the following

diagrams [1]

< 0 ~ >A ---- < + " " "/ / 0 n 0 u D

t - - T

0 ~

-t- ~ -I- <

0 ~ 0 ~ g~

+ <

338c M Wakamat~u / Spm-flavor structure of the nucleon

Not only the valence quarks but also the Dlrac-sea quarks contribute to thin effectwe

current operator It also contains the 0th and the 1st order contrlbutmns m the Corlohs

coupling Since f~ is a 1/N~ quantity, the latter IS an 1/N~ correction to the former

First let us show the model pxedlctlon for the nucleon axml-vector coupling constant

9A The effectwe quark mass used here is determined so as to roughly reproduce the

proton charge radms, which gives M ~_ 400 MeV As one can see, the contribution of

Table 1 Nucleon axial-vector couphng constant

gA(a0) g~(W) gA(a0 + a ~)

valence 0 74 0 41 1 15

sea 0 06 0 14 0 20

total 0 80 0 55 1 34

experiment 1 26

the 1st order term m f~ is sizable, and if it ~s added to the lowest one, the model roughly

reploduces the experimental value As already mentioned, there is no corresponding 1/N~ conect lon within the theoretical scheme of the Skyrme model Undoubtedly, this must

be the reason of the long-standing underest~matlon problem of 9A in the Skyrme model

The absence of the O(fP) te lm in the Skyrme model can be understood as fol-

lows Fns t I recall the general s t ructme of the effective meson action It consists of

the nonanomalous part and the anomalous one as

S~ H = F ('o + r (~) (8)

F~om the Lorentz structure of the action given as

r (~) ~ j(~ Tr [ UtOy U~O"U ] (9)

r(a) ~ e~"~e~ £o Tr[UtO.U UtcO~U UtO~U UtO~U UtO~g ] , (10)

it is clear that the nonanomalous part is an even function of the time derivative of the

pmn field Under the assumption of rigidly rotating hedgehog, this means that it ~s

an even function of f~ On the other hand, the anomalous part of the action, whmh is

essentially the Wess-Zummo action, is shown to be an odd function of f~ But the fact ~s

that it vanishes for the flavor SU(2) case Exactly the same can be said for the spatial

component of the vector or axial-vector current, which is enough to show that there is

no 1/N~ correction to 9A Itere I do not ha~e enough space to go rote the detail, but it

can be shown that this results from the fact that the usual bosomzatlon approach fails to

take m correct hme order of two relevant operators, f~ and R~b -- ½ tr [ A* ~-~ A rb ], whmh

are non-commuting aftel canonical quantizat lon [5]

M Wakamatsu / Sp,n-flavor structure of the nucleon 339c

A big difference between the CQSM and the Skyrme model can also be seen in the

spin contents of the proton [1] The CQSM predicts that about a half of the proton

spin comes from the quark spin, while remaining half comes from the orbital angular

momentum of quarks This should be compared with the corresponding prediction

Table 2 Spin contents of the proton

contents < E3 > < 2 L3 > < 2,]3 >

valence 0 48 0 28 0 76

sea 0 O0 0 24 0 24

total 0 48 0 52 1 00

of the naive quark model, < E3 > = 1, and that of the Skyrme model, < Es > = 0

One sees that the prediction of the CQSM hes something between these two extremes

At first sight, the predlctaon of the Skyrme model is most consistent with the EMC

result The truth is far flora clear, however, because what EMC group determmed is the

proton matrix element of the flavor smglet axial-vector current, which cannot be ldentafied

with the quark spm expectation value under the presence of the UA(t) anomaly of QCD

Despite this somewhat unsettled situation, a lesson learned from our analysis is clear The predmtlon of the Skyrme model for < E3 > should not be taken too seriously, because it

seems to stem from Its problematical aspect According to the CQSM, the contribution of

the three valence qualks to the total nucleon spm defimtely exists, although it is pamally

cancelled by the polarization of the sea quarks

There are several other charactenstm predictions of the CQSM, which I beheve has much to do with the essence of the model They are the umque spatial structure of the quark condensates around the nucleon center, the sea quark contribution to the neutron

charge distribution, the nonelectromagnetlc neutron-proton mass difference, and the Got-

fined sum So etc [6] Here, because of limitation of space, I confine to the last quantity

Roughly speaking, the deviation of SG from 1/3 is proportional to the number difference

between the dd pairs and the u~ ones inside the proton The CQSM ~s just able to cal-

culate such a quantity Given below are the theoretical prediction for SG m comparison

with the experimental value by the NMC group

s a ~- 0 2 5 7 . = . 5~ xp = 0 2 4 0 ± 0016 (11)

One sees that the prediction of the CQSM is qualitatively consistent wath the NMC measurement This means that the lsospm asymmetry of the q~ sea inside the nucleon

is a automatical dynamical consequence of the CQSM I emphasize that this lsospm

asymmetry of the q~/ sea inside the nucleon is a natural consequence of two dynamical factors [6]

340c M Wakamatsu / Sptn-flavor structure of the nucleon

• the flavor asymmet ry of the valence quark numbers in the nucleon

• the spontaneous chlral symmet ry breaking of the QCD vacuum which inevitably

generates the plonic qq excitat ions

3. Conclusion

To summarize, the CQSM is a umque model of the nucleon based on an extremely

simple effective lagrangIan of QCD given by eq (1) Who can ,magzne szrnpler effechve

lagrangzan than th~s 9 A prominent feature of the model is that a nucleon bound-sta te

problem can be solved nonper turba t ive ly with full inclusion of the valence and Dirac-sea

quarks After solving it, we are led to a very natura l nucleon picture, i e the core of N~

valence quarks surrounded by the cloud of plons or the plonlc qc~ e×cltatlon We find that

the mduced q~ sea IS defimtely lsospln asymmetr ic , m consistent with the recent NMC

measurement of the Got t fned sum The model also seems to predmt physically plausible

spin s t ructure of the proton, in shalp contrast to the deeply-related Skyrme model In

fact, our solution of the long-standing gA problem throws serious doubts on the widely-

beheved Cheshire Cat Picture It appears to break down at 1~No order This means that

Wl t t en ' s conjecture tha t the nucleon is a topologmal soliton of the nonhnear sigma model

might also be justified only m the fictitious world with Nc = e~

Acknowledgement

The talk is par t la l ly based on the collaborat ion with K Goeke, A Blotz, Ch Christo~

at RuhI-Umvers i ta t Bochum, V Yu Petrov, P V Pobyhtsa at St Petersburg Nuclear

Physms Inst i tute, and T W a t a b e a t T o k y o M e l l o p o h t a n Unlvmslty

References

1 M Wakamatsu , Prog Theor Phys Suppl 109 (1992)115

2 D I Dlakonov, V Yu Petzov and P V Pobyli tsa, Nucl Phys B306 (1988) 809

3 M Wakamatsu and H Yoshlkl, Nucl Phys A524 (1991) 561

4 M Wakamatsu and T Watabe , Phys Lett B312 (1993) 184

5 A Blotz, Ch Chrlstov, K Goeke, V Yu Petrov, P V Pobylitsa, M Wakamatsu and

T Watabe , "l /Nc rotat ional correction to the gA and lsovector magnetic moment

of the nucleon", Phys Lett B (1994) m print

6 M Wakamatsu Phys Rev D46 (1992) 3762