Z. Phys. A - Atomic Nuclei 337, 447~450 (1990) Zeitsehrift for PhysikA

Atomic Nuclei �9 Springer-Verlag 1990

Spin content of the nucleon in a non-topological chiral soliton model* A. Hosaka 1,**, T. Sch~ifer i and U. Kalmbach 2

1 Institut fiir Theoretische Physik, Universit~it Regensburg Universitfitsstrasse 31, W-8400 Regensburg, Federal Republic of Germany 2 Institut fiir Theoretische Physik, Universitfit GieBen, W-6300 Giegen, Federal Republic of Germany

Received April 10, 1990; revised version June 18, 1990

The spin content of the proton is investigated by study- ing the flavor singlet axial structure of the nucleon in a non-topological chiral soliton model. In order to con- struct a nucleon state we used the generator coordinate projection method as well as a coherent state for the meson wave function. Using a standard set of parameters we found the value gO ~ 0.44 for the flavor singlet axial vector coupling constant. This result is not far from that of a typical valence quark model.

PACS: 12.35.Ht

Spin observables provide us with a good chance to test different models of the nucleon. Particular interest has been paid to the flavor singlet axial vector coupling con- stant gO of the nucleon. It is defined by

i 'd 3 x (N [Au[ N) = i 'd 3 x (Ul ~ff�89 ~ IN) - ~ (NI o-~ IN) gO, (1)

where the flavor singlet axial vector current A u is given in terms of the quark field ip, and o% is the standard spin operator for the nucleon. The matrix elements are taken in the nucleon state [N). Ignoring effects caused by anomalous gluon contributions the parameter go can be interpreted as the fraction of the total nucleon spin carried by the quarks.

Recent measurements of polarized muon-proton scattering by the European Muon Collaboration (EMC) imply that I-1]

gO = 0.00_+ 0.24, (p2 = 11 GeV), (2)

where # is the scale at which the operator in (1) is renor- malized. Apparently the result (2) contradicts naive va- lence quark model predictions, where gO comes out to

* Work supported in part by DFG grant We 665/9-3 ** Present address: Department of Physics, University of Pennsyl- vania, Philadelphia, PA 19104, USA

be essentially one: the nucleon structure appears to be more complex than expected in a naive quark model. On the other hand, the Skyrme model predicts gO = 0 [2], which seems to be consistent with (2). The impor- tance of the anomalous Ua(1) breaking due to the cou- pling of gluons to the axial vector current A u has also been pointed out [3].

The purpose of this paper is to study g~ A in a non- topological chiral soliton model [4], which has been suc- cessfully applied to the description of nucleon structure. An advantage of this model is that quantum effects in the soliton sector can, at least to some extent, be taken into account in the framework of the variational method, in which the generator coordinate projection method [-5, 7] is applied to construct a physical nucleon state instead of the semiclassical cranking method usually ap- plied to the Skyrme model [8J. We calculate gO for sever- al different model parameters and discuss the properties of gO. We compare our results with those of the chiral bag model [-9] and point out the importance of vacuum polarization effects on the quark field.

The model is based on the lagrangian of the linear sigma model with quark degrees of freedom

Ae = ~[iO,?u + g(a + i'c'~r75)] 22

- v ) -f~rn~cr, (3) + � 8 9 1 8 9 2 ~

where f~, g and 2, are the pion decay constant, quark meson coupling constant and four point meson coupling constant, respectively. Using the value v2=fZ--m~/22 the classical minimum of the quartic potential is fixed at (a, n)= ( - f~, 0). The last term in (3) is introduced to explicitly break chiral symmetry and gives the pion its observed mass, m~ = 139 MeV. We restrict our discussion to the SU(2) sector. The effects of strange quarks and SU(3) symmetry breaking are expected to be small [10].

Static properties of the nucleon derived from (3) have been studied extensively by a number of authors [4, 5]. Here we are interested in the flavor singlet axial proper-

4 4 8

ties (1). The flavor singlet axial vector current A, the system (3) is

for

A~ = �89 ~. (4)

We note that A u is not conserved, O~,A"~ O. Axial vector current conservation can be recovered by introducing the flavor singlet pseudoscalar meson t/' which couples to the quark field ~ in a chirally invariant way. This is done in a way analogous to the introduction of the pion field in the isovector sector. The total axial vector current then consists of a quark term (4) and an t/' piece. Since the ~' has a non-zero mass m,, 4=0, the total axial vector current is not conserved�9 This non-conservation is now related to the UA(1) anomaly through the QCD anomaly equation [11]

(5) O~sN f OuAU= 2n trFu~/~"~

and the current algebra relation

O,AU=f,,mZ, tl, (6)

where cr Nr and f,, are the quark-gluon coupling con- stant, number of flavors and the decay constant for the t/'. Furthermore, Fu~ is the gluon field tensor and flu, its dual tensor.

In order to incorporate these aspects, the t/' field must be properly introduced in the lagrangian (3). However, as shown in the chiral bag calculation [9], the q' contri- bution to gO is small as compared to the quark piece if the large physical t/' mass m,, = 958 MeV is used. Fur- thermore, the ~/'-quark coupling is not strong enough to modify the quark wave function significantly. We ex- pect a similar situation in the chiral soliton model and assume that the major contribution to gO comes from the explicit quark piece (4) whereas the q'-quark coupling is negligible�9

The system (3) allows a classical soliton solution within the hedgehog ansatz:

(7)

[ u(r), \ 1 n=~ia.fv(r))Z, Z = ~ ( l u $ > - I d T > ) ,

r = G(r),

(a i = Pih(r),

with spherically symmetric functions u(r), v(r), a(r) and h(r). The valence quark wave function is normalized as

4n ~ r 2 dr(u 2 + v 2) = 1, and the spin-isospin spinor ){ has 0

good quantum numbers KP=0 +, where K = L + S + I with L, S and I being an orbital angular momentum, spin and isospin, respectively, and P is the parity�9 u(d) and 1" ($) denote the isospin up (down) and spin up (down) states, respectively�9

A possible quantum interpretation of the classical meson configuration in (7) is that they are expectation

values of the corresponding field operators in the hedge- hog state Ih):

(hi q~o (x) Ih) = a(r), (hi c~i(x) Ih ) = fih(r). (8)

A state which has the property (8) can be constructed using a coherent state�9 Introducing the canonical mo- mentum operators n, conjugate to ~b~, the coherent state for the meson sector Ihmes) is written as [5]

[hm~)=exp{i~dax(no(X)a(r)+ndx)~ih(r))}lO). (9)

It is convenient to expand the operators n o in momen- tum space:

"" d 3 k / - ~ k t - e a k ~+e +k,~J, 7 ~ a ( X ) = I J ~ V ~ _ : i k ' x , - i k . x a "~ (10)

where C0k= ~ . The vacuum state 10) in (3) is then defined as the state which is annihilated by all the opera- tors ak,~: ak,~10)=0.

The total hedgehog state Ih) is a direct product of the meson and quark wave functions, Ihm~+) and Ihq),

Ih)=]hq)lhm~+), Ihq) =(b*o +)~c[0), (11)

where the operator b*o+ creates the quark state On in (7) and Nc denotes the number of colors�9 The c-number functions u(r), v(r), a(r) and h(r) are regarded as varia- tional parameters to be determined by energy minimiza- tion: 3<hlHlh)=O.

Because of its K-symmetry, the hedgehog state is written as a superposition of states with various spins J and isospins I. A proper nucleon state with definite spin and isospin is obtained by applying the projection method. For example, a proton spin-down state can be written as

[P $) ~ ~d [f2] DI12",1/2 (~)R (O)Ih), (12)

where R(f2) is the SU(2) isospin rotational operator with the angle f2 and D~,t(f2) is a representation of R(f2). Once we have the proton wave function (12), it is straightforward to calculate the matrix element (4). Using some elementary angular momentum algebra, we find

(P$1A31p$) (P$1P$)

_ Nc [.d[Q-lcfc+lc~zC-tse(hmeslR(~)lhmes) 2 yd[_fflcTC+lc'dc+lfhmodR(O)lhm,+)

l=COS( ) c =cos( ) �9 / c ~ + ? \

I,

(13)

where the angle f2 is represented by the Euler angles ~, fl and 7. Furthermore, I denotes the radial integral including the quark wave functions

oo

I=4rc ~ r2dr(u2-�89 2) (14) 0

whereas the meson overlap function is given by

(hmeslR (f2) I hines) = exp { - a (1 - c~ z c2)}

with

(15)

O9

a= _~_f2 ~ k2dkcok~2(k), 0

oo

~(k)= ~ r2drj~(kr)h(r). 0

(16)

/ \ I T .

Dividing (11) by the factor ~p ~ ~ p ~)= -1 /2 , one ob- /

tains gO. We have calculated gO for several different mod- el parameters. In Fig. 1 we show gO as a function of the quark-meson coupling constant g with the hedgehog mass MH fixed at 1088 MeV. To keep Mn=cons t , the pion decay constant f~ is varied as shown on top of the figure. The four point meson coupling constant 2 is fixed at 2 = 10. We found that gO depends rather weak- ly on g. At the physical value of f~--93 MeV where g -~ 5.6, gO comes out to be 0.44 which is somewhat larger than the upper limit of the EMC result (2).

At this point, we would like to comment on possible corrections which would arise from a more refined treat- ment of the projection. We have used exact Peiels Yoccoz

f~ [NeVl 75 85 106 153 280

0.8 , , , , ,

0.6

go

0.2

0 I I I I I t~ 5 6 7 8 9

g Fig. 1. Flavor singlet axial vector coupling constant gO as a function of the quark-meson coupling constant g. The hedgehog mass is fixed at Mn= 1088 MeV. The pion decay constant is shown on the upper scale of the figure

1.0

449

go 0.8

0.6

0.t,

0.2

--7

/ /

/ /

/ j

I I I i i 0 0.2 O.L. 0.6 0.8 1.0 1.2

<r2> I/2 [fro]

Fig. 2. Flavor singlet axial vector coupling constant gO as a function of the quark piece of the baryon charge distribution (rZ) x/2. The chiral bag values are given by the dashed line

projection but have varied the hamiltonian before rather than after the projection. Furthermore, as discussed in great detail by Fiolhais et al. I-6] the Goldberger-Trei- man relation is not satisfied if the standard hedgehog ansatz is used. This problem is however essentially due to ambiguities arising in the process of extracting the pion nucleon coupling constant from the shape of the pion profile. It can be solved if the extended hedgehog ansatz of Ref. 1-6] is used in conjunction with variation after projection. It is nevertheless important to point out that the isovector axial coupling constant gA as well as the quark and meson contributions to it are rather insen- sitive to the method used. Roughly speaking, the differ- ence between gA and gO is just the isospin structure of the nucleon matrix element which leads to a different weight function in the integral over the Euler angles whereas the way the quark and meson profiles enter in (14) and (15) is identical in the isovector case. There- fore, we feel confident that an extension of the simple hedgehog ansatz would not very much affect the results presented here.

Now we study the properties of gO in more detail. To do this, it is convenient to plot gO as a function of the baryon charge radius carried by the quarks

co ) 1 / 2 . ( / , 2 ) 1 / 2 = i daxr2(l~fO) ( 1 7 )

\ o

We use this parameter rather than the total baryon charge radius since in the chiral bag model the isosinglet axial vector coupling is essentially due the quark part of the axial vector current [-9]. Comparing the curves calculated in the two models as given in Fig. 2 it is inter-

1 In the non topological chiral solition model the baryon charge is carried entirely by the quarks, whereas in the chiral bag it is shared by the quarks inside the bag and the mesons fields outside

450

esting to see an apparent correlation between them for (r2) 1/2 larger than 0.6fm. For small (r2) 1/2, however, they tend to deviate from each other. We argue that this is because the vacuum polarization effect [12] on the quark wave function is totally ignored in a standard treatment of the chiral soliton model. In fact, the point (r 2) 1/2 ,,, 0.3 fm corresponds to a bag radius R ~- 0.5 fm in the chiral bag model, where about half of the baryon number in the quark bag is carried by the vacuum. Simi- larly it is expected that the vacuum contribution to gO can no longer be neglected and gO would be reduced [133.

At the physically realistic point where f~ = 93 MeV, the radius of the quark distribution in the chiral soliton model is (r2)l/2-~0.7fm and the corresponding chiral bag size is R~-0.8fm. In this case physical quantities in the quark bag are dominated by the valence quarks. Therefore, for any realistic parameter set, the non-topo- logical chiral soliton model behaves like a valence quark model rather than like the Skyrme model.

Finally, we would like to briefly discuss how the va- lence quark (MIT bag) model and the large Nc limits are reached in the non-topological chiral soliton model. The MIT bag model limit is obtained by simply setting a = 0 in (13). Physically this implies the disappearance of the pion cloud around the nucleon, since a is related to the expectation value of the pion number by [-5]

a= -4(hmeslN~lhmes). (18)

In this case, the integral (13) can be performed analyti- cally. We find g ~ independent of Nc, as expected. In order to take the large Nc limit properly, we have to take into account the large Nc behavior of the pion decay constant: f~o@/Nc. If f~ is scaled at the same time as the number of valence quarks is increased, we find from (13) that gO approaches some finite albeit small value in this limit. Hence gO is a quantity of order one in the large Nc limit. This behavior is essentially the same as the one in a naive valence quark model.

Let us note that, contrary to the claims made in refer- ence [2], this is also the scaling behavior expected for the Skyrme model. This point has been clarified in [14].

If the consequences of the anomaly are correctly incorpo- rated the Nc counting rules only imply that g~ is of order (9(1/Nc) in the large Nc limit. This behavior does not distinguish the skyrmion from a constituent model.

In summary we have studied the flavor singlet axial vector coupling constant gO in a nontopological chiral soliton model. The model predicts gO___ 0.44 for its stan- dard parameter set, a result reminiscent of a valence quark model.

We thank W. Weise for useful discussions and comments. A.H. is grateful to R. Amado and M. Oka for discussions and a careful reading of the manuscript.

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