The Quark-Antiquark Asymmetry of
the Nucleon Strange Sea
M. Wakamatsu
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, JAPAN
[email protected]
Abstract. Theoretical predictions are given for the light-flavor sea-quark distributions in the nucleon including the strange quark ones on the basis of the flavor SU(3) version of the chiral quark
soliton model. Careful account is taken of the SU(3) symmetry breaking effects due to the mass
difference between the strange and nonstrange quarks, which is the only one parameter necessary
for the flavor SU(3) generalization of the model. A particular emphasis of study is put on the lightflavor sea-quark asymmetry as well as the particle-antiparticle asymmetry of the strange quark
distributions in the nucleon.
INTRODUCTION
The key observation that motivates our investigation here is that, in their semiphenomenological fit, Glück, Reya and Vogt prepared the initial PDF at pretty low
energy scale around 600 MeV, in contrast to the common consensus of perturbative
QCD [1], and concluded that sea-quark (or antiquark) components are absolutely
necessary even at this low energy scale. Moreover, even the isospin asymmetry of the
sea-quark distributions are established by the celebrated NMC measurement. The origin
of this sea-quark asymmetry is definitely nonperturbative, and cannot be radiatively
generated through the perturbative QCD evolution processes. We certainly need some
low energy (nonperturbative) mechanism which generates sea-quark excitations.
In our opinion, the Chiral Quark Soliton Model (CQSM) is the simplest and most
powerful effective model of QCD, which fulfills the above requirement. Although it is
still a toy model in the sense that the gluon degrees of freedom are only implicitly treated,
it has several nice features that are not possessed by many other effective models of QCD
like the MIT bag model [2, 3]. Among others, most important in the above-explained
context would be its field theoretical nature, i.e. the proper account of the polarization of
Dirac sea quarks, which enables us to make reasonable estimation not only of quark
distributions but also of antiquark distributions [4, 5]. We have already shown that,
without introducing any adjustable parameter, it reproduces almost all qualitatively
noticeable features of the recent DIS observables including the NMC measurement as
well as the famous EMC finding [6]. What was lacking for the flavor SU(2) CQSM is the
neglect of hidden strange quark degrees of freedom in the nucleon. Here, we attack this
problem by using the flavor SU(3) generalization of the CQSM, which is constructed on
the basis of the SU(2) model with some additional dynamical assumptions [7, 8].
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FLAVOR SU(3) CHIRAL QUARK SOLITON MODEL
The model lagrangian is a straightforward generalization of the SU(2) model with flavor
octet collective meson fields, except for one important new feature, i.e., the existence
of the sizably large SU(3) symmetry breaking term due to the mass difference between
the strange and nonstrange quarks. This mass difference ∆ms is the only one additional
parameter of our effective model.
The basic dynamical assumption of the flavor SU(3) CQSM is as follows. First,
the lowest energy classical solution is obtained by the embedding of the SU(2) selfconsistent mean-field into the SU(3) matrix, analogous to the flavor SU(3) Skyrme
model. The next is the collective quantization of the symmetry restoring rotational
motion of the soliton in SU(3) collective coordinate space. Finally, we assume that the
SU(3) symmetry breaking effects can be treated perturbatively. Actually, we have taken
account of several possible SU(3) breaking corrections consistently, which are all first
order in ∆ms . The detail may be found in our recent article [8].
COMPARISON WITH HIGH ENERGY DATA
The model predictions are compared with the available high-energy data in the following
way. First, after some trial, only one parameter of the SU(3) CQSM, i.e. ∆ms , is fixed to
be 100 MeV. Then, we use the predictions of the model as initial distributions given at
the low energy model scale, simply assuming the smallness of the gluon distributions at
this energy scale. The scale dependence of the PDF is then taken into account by solving
the standard DGLAP equation at the NLO. The intial energy scale of this Q 2 -evolution
is fixed to be Q2ini 030 GeV2 throughout the study.
Skipping the details, we show in Fig.1 the final predictions of the SU(3) CQSM for
the unpolarized s- and s̄-quark distributions at the model energy scale. The left figure
shows the result obtained in the chiral limit, i.e. by neglecting the SU(3) symmetry
breaking effects, while the right figure is obtained after introducing ∆m s correction. One
sees that the s s̄ asymmetry of the unpolarized distribution functions certainly exists.
The difference function sx s̄x has some oscillatory behavior with several zeros as
a function of x. This is of course due to the two general constraints of the PDF, i.e.
the positivity constraint for the unpolarized distributions and the strangeness quantum
number conservations. Comparing the two figures, one observes that sx s̄x is
extremely sensitive to the SU(3) breaking effects.
Fig.2 shows the theoretical predictions for the longitudinally polarized strange quark
distributions. In the chiral limit case, the s and s̄ are both negatively polarized. After introducing ∆ms correction, ∆sx remains large and negative, while ∆s̄x becomes very
small. As a consequence, the s-s̄ asymmetry of the longitudinally polarized distributions
is much more profound than that of the unpolarized distributions. This is reasonable because, for the spin-dependent distributions, there is no conservation laws which prevents
the generation of asymmetry.
Now we make some preliminary comparison with the existing high energy data. In
Fig.3, we compare the theoretical strange-quark distributions evolved to Q 2 4 GeV2
386
ms = 0 MeV
ms = 100 MeV
1.8
1.0
1.6
s (x)
s- (x)
1.4
s (x)
s- (x)
0.8
1.2
0.6
1.0
0.8
0.4
0.6
0.4
0.2
0.2
0.0
0.0
0.5
1.0
x
0.0
0.0
0.5
1.0
x
FIGURE 1. Theoretical unpolarized s- and s̄- distributions at the model energy scale.
ms = 0 MeV
ms = 100 MeV
0.0
0.0
-0.02
-0.05
-0.04
-0.06
-0.1
-0.08
-0.15
-0.1
s(x)
s(x)
-0.2
-0.25
0.0
-0.12
0.5
x
s(x)
s(x)
-0.14
1.0
-0.16
0.0
0.5
x
1.0
FIGURE 2. Theoretical longitudinally polarized s- and s̄- distributions at the model energy scale.
with the corresponding CCFR (NLO) fit of the neutrino-induced charm production,
which was carried out under the constraint s̄x sx [9]. One can say that, after inclusion of the SU(3) symmetry breaking corrections, the theory reproduces the qualitative
feature of CCFR fit.
Recently, Barone et al. carried out quite elaborate global analysis of the DIS data,
especially by using all the available neutrino data [10]. This enables them to obtain
some interesting information even for the asymmetry of the s and s̄ distributions. The
thick and thin shaded areas in Fig. 4 are the allowed regions, respectively obtained by
Barone et al. and by CCFR analysis. One clearly sees that the theory reproduces the
qualitative tendency of the data only after including the SU(3) breaking effects.
Although we have no space to show them, we also find that the predictions of the
SU 3 CQSM for the longitudinally polarized distributions including the strange quarks
are qualitatively consistent with the recent elaborate analyses carried out by Leader,
Sidorov and Stamenov [11].
387
without
ms correction
with
0.25
x s (x)
x s- (x)
0.2
2
2
2
CCFR at Q = 4 GeV
0.15
0.15
0.1
0.1
0.05
0.05
-2
10
2
5
10
-1
x s (x)
x s- (x)
0.2
CCFR at Q = 4 GeV
0.0
ms correction
0.25
2
5
0.0
1
10
-2
2
x
5
10
-1
2
5
2
1
x
FIGURE 3. The longitudinally polarized s- and s̄- distributions at the model energy scale.
2
2
s(x) / -s(x) at Q = 20 GeV
5.0
without ms correction
with ms correction
4.5
4.0
Barone et al.
3.5
3.0
2.5
2.0
CCFR
1.5
1.0
0.5
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
x
FIGURE 4. The longitudinally polarized s- and s̄- distributions at the model energy scale.
So far, we have shown that the SU(3) CQSM can give reasonable and unique predictions for the hidden strange quark distributions in the nucleon. A natural question is
whether it does not destroy the success of the SU(2) CQSM already obtained for u dquark dominated observables. To verify it, we have compared the predictions of the both
versions of the CQSM with the corresponding EMC and SMC data for the longitudinally polarized structure functions for the proton, the neutron and the deuteron to find
that both models reproduces equally well the general tendency of the experimental data.
Next, turning to the problem of isospin asymmetry of sea quark distribution, we recall
that the SU(2) CQSM predicts that ūx d¯x 0, while ∆ūx ∆d¯x 0. Now
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the question is what the predictions of the SU(3) CQSM is like. We have compared
the predictions of both versions of the CQSM. We find that, for the unpolarized sea
quark distributions, both models give nearly the same answer, which we already know
is consistent with the NMC observation. In contrast to the unpolarized distributions,
the magnitude of ∆ūx ∆d¯x turns out to become sizably smaller when going from
the SU(3) model to the SU(2) one. Still, the positive polarization of ∆ū and negative
polarization of ∆d¯ in the proton is a definite prediction of the both version of the CQSM,
which should be contrasted with the prediction of other models like the naive meson
cloud convolution model.
CONCLUSION
To summarize, an incomparable feature of the CQSM as compared with many other
effective models like the MIT bag model is that it can give reasonable predictions also
for the antiquark distribution functions. We emphasize that this feature is essential for
giving any reliable predictions for strange distributions in the nucleon, which totally
have non-valence character.
With a single parameter, the SU(3) CQSM predicts that the sx s̄x difference
function has some oscillatory x-dependence, due to the positivity constraint for the
spin-averaged distributions and the strangeness quantum number conservation. We have
also shown that, after introducing the SU(3) breaking effects, the x dependence of
sx s̄x and sxs̄x are qualitatively consistent with the global analysis of Barone
et al. The s s̄ asymmetry of longitudinally polarized sea is more profound than that
of unpolarized sea. The model predicts that the polarization of s-quark is large and
negative, while the polarization of s̄-quark is very small. The model also predicts quite
large isospin asymmetry of the sea-quark distributions not only for the unpolarized
distributions but also for the longitudinally polarized ones.
An important lesson learned from our investigation is that the nonperturbative QCD
dynamics due to the spontaneous chiral-symmetry breaking manifest most clearly in
the spin and isospin dependence of antiquark distributions in the nucleon. What is
absolutely required for future experiments is therefore the flavor and valence seaquark decomposition of PDF.
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