Understanding the Flavor Symmetry Breaking and Nucleon
Flavor-Spin Structure within Chiral Quark Model
Zhan Shu, Xiao-Lin Chen, and Wei-Zhen Deng∗
Department of Physics, Peking University, Beijing 100871, China
Abstract
In χQM, a quark can emit Goldstone bosons. The flavor symmetry breaking in the Goldstone
boson emission process is used to intepret the nucleon flavor-spin structure. In this paper, we
study the inner structure of constituent quarks implied in χQM caused by the Goldstone boson
emission process in nucleon. From a simplified model Hamiltonian derived from χQM, the intrinsic
wave functions of constituent quarks are determined. Then the obtained transition probabilities
of the emission of Goldstone boson from a quark can give a reasonable interpretation to the flavor
symmetry breaking in nucleon flavor-spin structure.
PACS numbers: 12.39.-x, 12.39.Fe, 14.20.Dh
∗Electronic address: dwz@th.phy.pku.edu.cn
http://arxiv.org/abs/0704.0029v2
mailto:dwz@th.phy.pku.edu.cn
I. INTRODUCTION
The measurements of the polarized structure functions of the nucleon in deep inelastic
scattering(DIS) experiments[1, 2, 3, 4] show the complication in proton spin structure. Only
a portion of the proton spin is carried by valence quarks. Moreover, several experiments[5, 6,
7] clearly indicate the ū-d̄ asymmetry as well as the existence of the strange quark content s̄ in
the proton sea. Also the distribution of strange quark in the proton sea is polarized negative.
The DIS results deviate significantly from the näıve quark model (NQM) expectation.
NQM gives many fairly good descriptions of hadron properties. Why does NQM work?
It is a puzzle that the quarks inside a hadron could be treated as non-relativistic particles
in NQM. The chiral quark model (χQM) tries to bridge between QCD and NQM. It was
originated by Weinberg[8] and formulated by Manohar and Georgi[9]. Between the QCD
confinement scale (ΛQCD ≃200MeV) and a chiral symmetry breaking scale (ΛχSB ≃1GeV),
the strong interaction is described by an effective Lagrangian of quarks q, gluons g and
Numbu-Goldstone bosons Π. An important feature of the χQM is that, betweetn ΛQCD and
ΛχSB, the internal gluon effects in a hadron can be small compared to the internal Goldstone
bosons Π and quarks q, so the effective degrees of freedom in this region can be q and Π.
It is interesting that χQM can also be used to explain why NQM does not work in the
above DIS experiments. By the emission of Goldstone boson, χQM allows the fluctuation
of a quark q into a recoiling quark plus a Goldstone boson q → q′Π . The q′Π system then
further splits to generate quark sea through
• the helicity-flipping process
q↑ −→ Π+ q′↓ −→ (qq̄′) + q′↓ (1)
• and the helicity-non-flipping process
q↑ −→ Π+ q′↑ −→ (qq̄′) + q′↑ (2)
where the subscript indicates the helicity of quark. In both the process, q′Π is in a relative P-
wave state. In the helicity-flipping process (1), the orbital angular momentum along helicity
direction must be 〈lz〉 = +1. In the helicity-non-flipping process (2), 〈lz〉 = 0. The process
cause a modification of the spin content of the nucleon because a quark changes its helicity
in (1). Also it causes a modification of the flavor content because the generated quark sea
from Π is flavor dependent[10, 11].
χQM was first used to explain the nucleon sea flavor asymmetry and the smallness of
the quark spin fraction by Eichten, Hinchliffe and Quigg[10]. The flavor asymmetry of sea
quark distribution arises from the mass differences in different quark flavors and in different
Goldstone bosons. Only the lightest Goldstone Boson π was considered since its contribution
dominates. From a perturbation calculation, the probability for an up quark to emit a π+
was estimated to be a = 0.083. This would induce a flavor asymmetry in parton distributions
of nucleon and other hadrons.
However, the estimated transition probability is not enough to full account the flavor
asymmetry in DIS experiments. Contribution from other Π’s and even η′ was considered by
Cheng and Li[11]. Explicit SUf (3) breaking in the transition probabilities was later intro-
duced in refs. 12, 13 and further used by several authors[14, 15, 16, 17, 18, 19]. Nevertheless,
in all these calculations, the transition probabilities were put into model by hand. To fit the
experimental data, the probability of an up quark emitting π+ needs to be set to a >∼ 0.1,
which is about 20% larger than the perturbation calculation. Although the probability of π
emission can be enlarged by using a higher momentum cut off Λ > ΛχQM in the perturbation
calculation [20], however, the chiral quark model is no longer valid at arbitrary high energies
Λ ≫ ΛχQM.
We should not be surprised by this discrepancy since the χQM works in a region right
above the QCD confinement scale ΛQCD. There one may expect the confinement effect is
important and the perturbative calculation of QCD may contain large error. However, there
is another essential difference between the above model calculations and the perturbation
calcultion. In the perturbation calculation, the emitted Goldstone bosons are virtual par-
ticles. In the above model calculations which are closely related to NQM, however, the
Goldstone bosons are close to mass shell under the non-relativistic approximation.
Since χQM can be a bridge between NQM and QCD, it is interesting to explore χQM
from NQM side where we use the wave function method. This will give the above model cal-
culations a concrete foundation in NQM and help us further understand the flavor symmetry
breaking mechanism.
In this paper, we will use wave function method to investigate the flavor symmetry break-
ing in χQM. In a conventional quark model[21], a hadron consists of confined constituent
quarks and its wave function is constructed in the configuration space of the constituent
quarks. To incorporate the transition process of emitting Goldstone boson of χQM into the
quark model, the constituent quarks will have intrinsic wave functions within the configu-
ration q + q′Π.
In Sec. II, we first present the composite wave function of constituent quarks including
components of q′Π. The wave functions and the transition probabilities of q → q′Π are
determined from a simplified χQM Hamiltionian. In Sec. III and Sec. IV, the obtained
transition probabilities are used to calculate nucleon flavor-spin structure and baryon octet
magnetic moments respectively. The numerical results and a brief summary are presented
in Sec. V.
II. THE WAVE FUNCTION OF A CONSTITUENT QUARK
In χQM, the effective Lagrangian below the chiral symmetry breaking scale ΛχQM involves
quarks, gluons, and Goldstone bosons. The first few terms in this Lagrangian are[9]:
LχQM = ψ̄(iDµ + Vµ)γµψ + igAψ̄Aµγµγ5ψ
−mψ̄ψ +
f 2πtr∂
µΣ†∂µΣ+ ... (3)
where Dµ = ∂µ + igGµ is the gauge-covariant derivative of QCD, Gµ the gluon field and
g the strong coupling constant. The dimensionless axial-vector coupling gA = 0.7524 is
determined from the axial charge of the nucleon. m represents the constituent quark masses
due to chiral symmetry breaking. The pseudoscalar decay constant is fπ ≈ 93MeV. The Σ
field, vector currents Vµ and axial-vector currents Aµ are given in terms of the Goldstone
boson fields Φ
π0 + 1√
η π+ K+
π− − 1√
π0 + 1√
K− K̄0 − 2√
, (4)
Σ = exp(i
), (5)
(ξ†∂µξ ± ξ∂µξ†), (6)
ξ = exp(i
). (7)
An expansion of the currents in powers of Φ/fπ yields the effective interaction between Π
and q[10]
LI = −
ψ̄∂µΦγ
µγ5ψ. (8)
This allows the fluctuation of a quark into a recoil quark plus a Goldstone boson q → q′Π.
In quark model, a hadron is built with constituent quarks. In accordance with χQM,
we should treat a constituent quark as a composite particle including such components q′Π.
Here we denote the wave function of a composite constituent quark as |q〉〉. At rest,
|q〉〉 = zq|q〉+
q′Π|q
′Π〉. (9)
In our paper, the state normalization relation is always taken as
〈p|p′〉 = δ3(p− p′). (10)
The above wave function is of essential importance in our work. The square of the mod-
ulus of the coefficient of each q′Π configuration is just the probability for the corresponding
Π emission process
Pq→q′Π = |xqq′Π|
2, (11)
|zq|2 = (1−
Pq→q′Π)
is the probability of no Π emission.
To determine the wave function (9), we first construct a simplified Hamiltonian in the
degrees of freedom q and Π,
H = H0 +HB +HI . (12)
H0 represents the kinetic energies of q and Π. It reads
ψ̄(iα · ∇+m)ψ +
Tr[Φ̇2 + (∇Φ)2] +
m2Π(Φ
, (13)
where mΠ is the physical mass of Π which is nonzero and nondegenerate.
HI = −
d3xLI , (14)
is the χQM interaction. HB is an accessary interaction which is needed to bind the q
together. In our simplified Hamiltonian, we will not disscuss the explicit formalism of HB.
Instead, we will put some physical restriction conditions on it later in this section, which is
sufficient to our calculation.
FromH0, we can expand free fields ψ and Π in terms of creation and annihilation operators
ψq(x) =
(2π)3/2
q(p, s)e−ip·x + bq†
ps(t)v
q(p, s)eip·x
, (15)
ΦΠ(x) =
(2π)3/2
e−ip·x + cΠ†
eip·x
p0=EΠ
, (16)
where
p2 +m2q
is the quark energy of flavor q,
p2 +m2Π
is the energy of Goldstone boson Π. aq†
ps and b
pr are the creation operators of quark q and
anti-quark q̄
pr, a
p′s} = {b
pr, b
p′s} = δ
(3)(p− p′)δrs. (17)
is the creation operator of Π
] = δ(3)(p− p′). (18)
Next, we will replace the field ψ and Φ in the Hamiltonian (12) with the free field of (15)
and (16). Then we can express the Hamiltonian in creation and annihilation operators, for
example
d3p Eq
ps + b
ps] +
d3p EΠ
. (19)
In all the model calculations [11, 12, 13, 14, 15, 16, 17, 18, 19], the emitted Π is assumed
bound to the quark source. To represent that q′Π are bound, we use the well known SHO
function as their spatial wave function
|qΠ〉 =
d3p|p|e−
2λ2 [Y1(θ, φ) c
]1/2 |0〉, (20)
|qΠ ↑〉 = 1√
d3p|p|e−
2λ2 Y11(θ, φ) c
p↓ |0〉
d3p|p|e−
2λ2 Y10(θ, φ) c
p↑ |0〉, (21)
where λ is the “characteristic radius” parameter in Gaussian function. 1/
N is the nor-
malization factor,
dp p4 e
πλ5. (22)
However, we need a binding interaction HB in the Hamiltonian. Yet we do not know how
to write out the explicit form of HB. However, HB should provide enough binding energy.
That is, for the q′Π system, we must have
〈qΠ|H0 +HB|qΠ〉 ≤ mq +mΠ. (23)
That is
EB = 〈qΠ|HB|qΠ〉 ≤ mq +mΠ − 〈qΠ|H0|qΠ〉 = mq − Eq +mΠ − EΠ. (24)
As a rough estimation, we will take the mininum value of EB
EB = −max
{Eq −mq + EΠ −mΠ} = −(Eu −mu + Eπ −mπ). (25)
Then the wave function of a composite constituent quark is determined from Schrödinger
equation
H|q〉〉 =Mq|q〉〉. (26)
After taking the above simplification, we need only solve a matrix eigen-value problem
 =Mq
 , (27)
where
aδ3(0) = 〈q|H|q〉,
Bq′Πδ
3(0) = 〈q|H|q′Π〉,
Cq′Π;q′′Π′δ
3(0) = 〈q′Π|H|q′′Π′〉,
q′Π = x
For example, let us consider the process u emitting Π. There are four possible |q′Π〉 states
generated by the fluctuations of a u quark: |uπ0〉, |uη〉, |dπ+〉 and |sK+〉. Thus
|u〉〉 = zu|u〉+ xuuπ0 |uπ0〉+ xuuη|uη〉+ xudπ+ |dπ+〉+ xusK+|sK+〉. (28)
Taking these wave functions as basis, we can calculate the matrix of the Hamiltonian in
(27).
a = mu. (29)
C is diagonalized. Its diagonal matrix elements are calculated from H0
Cuπ0;uπ0 =
dp p4 e
p2 +m2u +
) + EB, (30)
Cuη;uη =
dp p4 e
p2 +m2u +
p+m2η) + EB, (31)
Cdπ+;dπ+ =
dp p4 e
p2 +m2d +
) + EB, (32)
CsK+;sK+ =
dp p4 e
p2 +m2s +
) + EB. (33)
B is calculated from HI
Buπ0 = −
dp p4e
, (34)
Buη = −
dp p4e
, (35)
Bdπ+ = −
dp p4e
, (36)
BsK+ = −
dp p4e
. (37)
By diagonalizing this Hamiltonian matrix, we will obtain a new mass of the constituent u
quark Mu and its composite wave function. The constituent masses and wave functions of
d and s quarks can be obtained similarly. We have
|d〉〉 = zd|d〉+ xddπ0 |dπ0〉+ xddη|dη〉+ xduπ−|uπ−〉+ xdsK0|sK0〉, (38)
|s〉〉 = zs|s〉+ xssη|sη〉+ xsdK̄0 |dK̄
0〉+ xsuK−|uK−〉. (39)
From isospin symmetry, mu = md, we have
zd = zu; xddπ0 = −xuuπ0 ; xduπ− = xudπ+ ; ... (40)
However, since mu 6= ms, one should notice that
zs 6= zu; xsdK̄0 6= x
sK0; x
uK− 6= xusK+. (41)
After the diagonalization, the Goldstone bosons Π are separated from quarks q approx-
imately. With only degrees of freedom q one can rebuild the quark model and so Mu, Md,
Ms should be regarded as the constituent quark masses in quark model.
III. FLAVOR AND SPIN STRUCTURE OF PROTON
Having known the wave functions of constituent quark q and the transition amplitudes
of q emitting each Goldstone bosons Π, we are able to calculate the quark distribution in
a constituent quark following refs. 11, 12, 13. In χQM, Π will further split into a quark-
antiquark pair. By substituting the quark contents of Π into wave functions (28), (38) and
(39), we can rewrite the wave functions of constituent quark q as
|u〉〉 = zu|u〉+
xuuη√
|u(uū)〉+
xuuη√
|u(dd̄)〉
2xuuη√
|u(ss̄)〉+ xudπ+ |d(ud̄)〉+ xusK+|s(us̄)〉, (42)
|d〉〉 = zu|d〉+
xuuη√
|d(uū)〉+
xuuη√
|d(dd̄)〉
2xuuη√
|d(ss̄)〉+ xudπ+ |u(dū)〉+ xusK+|s(ds̄)〉, (43)
|s〉〉 = zs|s〉+
xssη√
|s(uū)〉+
xssη√
|s(dd̄)〉 −
2xssη√
|s(ss̄)〉
|d(sd̄)〉+ xsuK−|u(sū)〉. (44)
Then the antiquark and quark flavor contents of the proton (uud) are
ū = 2
xuuη√
xuuη√
+ |xudπ+ |2, u = ū+ 2, (45)
xuuη√
xuuη√
+ 2|xudπ+ |2, d = d̄+ 1, (46)
s̄ = 2|xuuη|2 + 3|xusK+|2, s = s̄. (47)
Some important quantities depending on the above quark distribution are: the Gottfried
sum rule IG =
(ū − d̄) whose deviation indicates the ū-d̄ asymmetry in proton sea;
ū/d̄ measured through the ratio of muon pair production cross sections; and the fractions
of quark flavors in proton fq =
Σ(q+q̄)
, f3 = fu − fd and f8 = fu + fd − 2fs.
We can further calculate the spin structure of proton. Here one should consider the effects
of configuration mixing generated by spin-spin forces[22]. We take the baryon wave functions
from the quark model calculation[23, 24, 25]. The proton wave function for example, is
expressed as
= 0.90|P 28SS〉 − 0.34|P 28S ′S〉 − 0.27|P 28SM〉 (48)
where the baryon SU(6)⊗O(3) wave functions are denoted as |B2S+1N Lσ〉, N is SU(3) mul-
tiplicity. S, L are the total spin and total orbital angular momentum while σ = S,M,A
denotes the permutation symmetry of SU(6). The spin polarization functions will be re-
markably affected by configuration mixing. Following refs. 15, 17, we define the number
operator by
N̂ = nu↑u↑ + nu↓u↓ + nd↑d↑ + nd↓d↓ + ns↑s↑ + ns↓s↓,
where nq↑, nq↓ are the number of q↑, q↓ quarks. The spin structure of the “mixed” proton is
given by
= (0.902 + 0.342)
+ 0.272
. (49)
The spin structure after considering Π-emission is obtained by replacing for every quark in
eq. (49) by
q↑,↓ −→ (1− ΣPi)q↑,↓ + Pflipping(q↑,↓) + Pnon−flipping(q↑,↓), (50)
where Pflipping(q↑,↓) and Pnon−flipping(q↑,↓)| are the probabilities of quark helicity flipping and
non-flipping for q↑,↓ respectively. For example, in the case of u↑ quark we have,
Pflipping(u↑) =
(|xuuπ0 |2 + |xuuη|2)u↓ + |xudπ+ |2d↓ + |xusK+|2s↓
Pnon−flipping(u↑) =
(|xuuπ0 |2 + |xuuη|2)u↑ + |xudπ+ |2d↑ + |xusK+|2s↑
Finally the spin polarization functions defined as ∆q = q↑ − q↓ are
∆u = (0.902 + 0.342)
114|xu
|2 + 48|xuuη|2 + 36|xusK+|2
+ 0.272
66|xu
|2 + 24|xuuη|2 + 18|xusK+|2
, (51)
∆d = (0.902 + 0.342)
|2 + 12|xuuη|2 + 9|xusK+|2
+ 0.272
42|xu
|2 + 12|xuuη|2 + 9|xusK+|2
, (52)
∆s = −
. (53)
There are several measured quantities which can be expressed in terms of the above spin
polarization functions. The quantities usually calculated are ∆3 = ∆u−∆d and ∆8 = ∆u+
∆d−2∆s, obtained from the neutron β-decay and the weak decays of hyperons respectively.
Another important quantity is the flavor singlet component of the total quark spin content
defined as 2∆Σ = ∆u + ∆d + ∆s . We also calculate some weak axial-vector form factors
which are also related to the spin polarization functions, (GA/GV )Λ→p =
(2∆u−∆d−∆s),
(GA/GV )Σ−→n = ∆d−∆s, and (GA/GV )Ξ−→Λ = 13(∆u+∆d− 2∆s).
IV. BARYON OCTET MAGNETIC MOMENTS
Considering the relative angular momentum between quark and Goldstone boson Π, the
magnetic moment operator of a qΠ system is
µ̂qΠ =
p2q +m
p2Π +m
p2q +m
p2Π +m
p2Π +m
p2q +m
p2q +m
p2Π +m
l̂ (54)
where eq and eΠ are the electric charges carried by q and Π respectively, ŝ the quark spin
operator and l̂ the relative angular momentum bewteen q and Π. The first term in Eq(54)
is the intrinsic magnetic moment of quark and the other two terms are the contribution of
the orbital angular momentum. Here we have to consider the relativistic effect since the
relative momentum of q or Π are comparable to their masses in the qΠ system
pq,Π ∼ Λ ∼ mq,Π.
With the SHO wave functions of (20), the magnetic moment of qΠ system (54) can be
readily calculated. Then we can recalculate the magnetic moments of constituent quarks
taking into account of the relativistic effect. For example, the magnetic moments of the u
quark is
µu = |zu|2〈u↑|µ̂|u↑〉+ Pu→uπ0〈uπ0|µ̂|uπ0〉+ Pu→uη〈uη|µ̂|uη〉
+ Pu→dπ+〈dπ+|µ̂|dπ+〉+ Pu→sK+〈sK+|µ̂|sK+〉, (55)
where
〈u↑|µ̂|u↑〉 =
, (56)
and the contribution from qΠ systems are
〈uπ0|µ̂|uπ0〉 = − eu
p2 +m2π
p2 +m2u +
p2 +m2π
p2 +m2u
λ2 , (57)
〈uη|µ̂|uη〉 = − eu
p2 +m2η
p2 +m2u +
p2 +m2η
p2 +m2u
λ2 , (58)
〈dπ+|µ̂|dπ+〉 = −
p2 +m2π
p2 +m2d +
p2 +m2π
p2 +m2d
p2 +m2d
p2 +m2d +
p2 +m2π
p2 +m2π
λ2 , (59)
〈sK+|µ̂|sK+〉 = −
p2 +m2K
p2 +m2s +
p2 +m2K
p2 +m2s
p2 +m2s
p2 +m2s +
p2 +m2K
p2 +m2K
λ2 . (60)
The magnetic moments of d and s quarks can be calculated similarly.
One can easily obtain the octet baryon magnetic moments by replacing the valence quarks
inside the baryons with the corresponding constituent quarks. Again we take proton as an
example,
µp = (0.90
2 + 0.342)
+ 0.272
. (61)
If we replace the µq by (55), µp can be further expressed as the baryon magnetic moment in
conventional quark model plus the contribution from the Goldstone boson emission process
[26]. The magnetic moments for other octet baryons can be calculated similarly.
V. NUMERICAL RESULTS AND CONCLUSIONS
In the numerical calculation, most of the parameters can be taken from the experimental
data or the chiral quark model. We collect these fixed input parameters of our calculation
in Table I. Here we have used the the physical masses of Goldstone bosons[27].
TABLE I: The fixed input parameters from chiral quark model and experimental data.
gA fπ(MeV) mπ(MeV) mK(MeV) mη(MeV)
0.7524 93 135 494 548
For the quark masses, since our work focuses on the inner context of the constituent quarks
in quark model, naturally we will refer to the quark masses from quark model, instead of
the chiral quark model values. Here we will use the quark mass values from the widely
accepted Isgur’s quark model[21] as shown in Table II. However, one should be cautious
that, in our model, it is the quark with the Goldstone boson mixing which corresponds to
the constituent quark in quark model. That is, mass values Mq after the diagonalization
process should be set to the quark masses in Isgur’s model. Our strategy is to adjust the
quark masses mq in the model Hamiltonian to fit the Mq values.
Finally we are left only with one free parameter λ which describes the confinement of the
emitted Goldstone boson in our model. An overall fit to the experimental data of nucleon
flavor-spin structure and octet baryon magnetic moments shows that the best value should
be λ=152MeV. With this value of λ and a minimun binding energe EB = −218MeV, the
“bare” values of quark masses mq without Goldstone boson mixing are shown also in Table
Transition probabilities of the light and strange quarks to various q′Π systems are given
in Table III and IV respectively. The probability of a u quark emitting a π+ P (u →
TABLE II: The quark masses with vs. without Goldstone boson mixing.
λ EB(MeV) mu,d(MeV) ms(MeV) Mu,d(MeV) Ms(MeV)
152 −218 288 474 220 419
d + π+)=0.145 is significantly larger than the perturbation calculation a=0.083. One may
notice that the λ parameter value 152MeV in our wave function, which is below ΛQCD, is
rather small than another energy scale ΛχQM in chiral quark model. Surely this will weaken
the interaction between q and q′Π. However, one should also notice that the binding energy
EB = −218MeV will make the energy of a qΠ system much close to the single quark energy.
This will enhance the mixing of q′Π components in a constituent quark.
Also, we notice that the asymmetry between the probabilities of u(d) → s + K and
s → u(d) + K̄. Whether this asymmetry leads to any observable consequence in hadron
structure needs further investigation.
TABLE III: Transition probabilities of a u quark to various q′Π systems and the mass of constituent
u quark.
u → u+ π0 u → u+ η u → d+ π+ u → s+K+ no GB-emission Mu
0.072 0.003 0.145 0.010 0.770 220MeV
TABLE IV: Transition probabilities of a s quark to various q′Π systems and the mass of constituent
s quark.
s → s+ η s → u+K− s → d+ K̄0 no GB-emission Ms
0.012 0.071 0.071 0.846 419MeV
Next, we will compare our calculate results with the experimental data. Since our em-
phasis is on the substructure of a constituent quark in NQM, here we also quote the results
from NQM. In Table V, the calculated flavor and spin structures of the proton are shown.
It should be mentioned that the quark spin polarization functions can be further corrected
by the gluon anomaly[13, 15, 17, 28, 29, 30] through
∆q(Q2) = ∆q − αs(Q
∆g(Q2), (62)
and the flavor singlet component of the total helicity is modified accordingly as
∆Σ(Q2) = ∆Σ− 3αs(Q
∆g(Q2), (63)
where ∆q(Q2) and ∆Σ(Q2) are the experimentally measured quantities, ∆q and ∆Σ corre-
spond to the calculated quantities without gluon correction. Using the experimental data
Σ(Q2 = 5GeV2) = 0.19 ± 0.02[2], αs(Q2 = 5GeV2) = 0.285 ± 0.013[27], and our result
∆Σ=0.346, the gluon polarization ∆g(Q2) is estimated to be 2.293. Both the results with
and without gluon polarization corrections are presented in Table V. The inclusion of gluon
polarization leads to a better agreement with experimental data for the spin structure.
The calculated magnetic moments of octet baryons are given in Table VI. Although the
deviation is somewhat around 30% in the case of Ξ−, our overall fit to octet baryon magnetic
moments is in good agreement with experiments. Also it should be mentioned that even in
the case of Ξ− the fit can perhaps be improved if corrections due to pion loops are taken
into account[32, 33].
In the model calculations [11, 12, 13, 14, 15, 16, 17, 18, 19], the Goldstone boson sector in
χQM is usually extended to include the η′ meson with U(3) symmetry. According to Cheng
and Li[11], in the large Nc limit of QCD, there are nine Goldstone bosons including the usual
octet and the singlet η′. Thus an constituent quark can also transit to a quark-η′ system. We
have also made an U(3) calculation. With the inclusion of η′, we find that the probabilities
for η′-emission from light and strange quarks P (u → u + η′)=P (d → d + η′)=0.0021 and
P (s → s + η′)=0.0018 which are negligibly small as compared to those of octet Goldstone
boson emissions. We therefore conclude that the contribution of η′ is not important, due to
the obvious axial U(1) symmetry breaking in meson mass spectra mη′ > mK,η.
To summarize, the χQM builds a bridge between the QCD and low-energy quark model.
This allows us to understand the mechanism of flavor symmetry breaking and nucleon flavor-
spin structure in NQM through the consideration of the sea quark and Goldstone bosons
in the substructure of constituent quarks. Using the simple SHO wave function, we have
modeled the wave functions of the composite constituent quarks and thus estimated the
transition probabilities for Goldstone boson emissions. These transition probabilities indeed
reflect the flavor SU(3) symmetry breaking in χQM from the differences in quark masses
ms > mu,d and differences in Goldstone bosons masses mK,η > mπ and they are roughly
in agreement with the parametrizations of other model calculations [11, 12, 13, 14, 15, 16,
TABLE V: The calculated values for the quark flavor distribution functions and spin polarization
functions in proton, as compared with experimental data and NQM results.
Data NQM Our Model
With ∆g Without ∆g
∆u 0.85 ± 0.05[2] 1.33 0.864 0.968
∆d −0.41± 0.05[2] −0.33 −0.377 −0.274
∆s −0.07± 0.05[2] 0 −0.107 −0.003
∆3 = (GA/GV )n→p 1.270 ± 0.003[27] 1.67 1.242 1.242
(GA/GV )Λ→p 0.718 ± 0.015[27] 1 0.737 0.737
(GA/GV )Σ→n −0.340 ± 0.017[27] −0.33 −0.270 −0.270
(GA/GV )Ξ→Λ 0.25 ± 0.05[27] 0.33 0.234 0.234
∆8 0.58 ± 0.025[2] 1 0.701 0.701
∆Σ 0.19 ± 0.02[2] 0.5 0.190 0.346
ū − 0.264
d̄ − 0.392
s̄ − 0.036
ū− d̄ −0.118 ± 0.015[6] 0 −0.128
ū/d̄ 0.67 ± 0.06[6] 1 0.674
IG 0.254 ± 0.005[6] 0.33 0.248
fu − 0.577
fd − 0.407
fs 0.10 ± 0.06[31] 0 0.017
f3 − 0.170
f8 − 0.950
f3/f8 0.21 ± 0.05[14] 0.33 0.179
17, 18, 19]. The fit to both the flavor-spin structure of nucleon and octet baryon magnetic
moments are in good agreement with experiments.
TABLE VI: The caculated octet baryon magnetic moments in nuclear magneton, as compared with
experiments and the results of NQM.
Octet baryons Data[27] NQM[34] Our model
p 2.79± 0.00 2.72 2.73
n −1.91 ± 0.00 -1.81 −1.91
Σ− −1.16 ± 0.025 -1.01 −1.23
Σ+ 2.46± 0.01 2.61 2.67
Ξ0 −1.25± 0.0014 −1.41 −1.36
Ξ− −0.65 ± 0.002 −0.50 −0.44
Λ −0.61 ± 0.004 −0.59 −0.56
ΣΛ 1.61± 0.08 1.51 1.63
Acknowledgments
Zhan Shu would like to thank Fan-Yong Zou and Yan-Rui Liu for useful discussions. This
work was supported by the National Natural Science Foundation of China under Grants
10675008.
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	INTRODUCTION
	The Wave Function of a Constituent quark
	FLAVOR AND SPIN STRUCTURE OF PROTON
	BARYON OCTET MAGNETIC MOMENTS
	NUMERICAL RESULTS AND CONCLUSIONS
	Acknowledgments
	References

  In $\XQM$, a quark can emit Goldstone bosons. The flavor symmetry breaking in
the Goldstone boson emission process is used to intepret the nucleon
flavor-spin structure. In this paper, we study the inner structure of
constituent quarks implied in $\XQM$ caused by the Goldstone boson emission
process in nucleon. From a simplified model Hamiltonian derived from $\XQM$,
the intrinsic wave functions of constituent quarks are determined. Then the
obtained transition probabilities of the emission of Goldstone boson from a
quark can give a reasonable interpretation to the flavor symmetry breaking in
nucleon flavor-spin structure.

Understanding the Flavor Symmetry Breaking and Nucleon
Flavor-Spin Structure within Chiral Quark Model
Zhan Shu, Xiao-Lin Chen, and Wei-Zhen Deng∗
Department of Physics, Peking University, Beijing 100871, China
Abstract
In χQM, a quark can emit Goldstone bosons. The flavor symmetry breaking in the Goldstone
boson emission process is used to intepret the nucleon flavor-spin structure. In this paper, we
study the inner structure of constituent quarks implied in χQM caused by the Goldstone boson
emission process in nucleon. From a simplified model Hamiltonian derived from χQM, the intrinsic
wave functions of constituent quarks are determined. Then the obtained transition probabilities
of the emission of Goldstone boson from a quark can give a reasonable interpretation to the flavor
symmetry breaking in nucleon flavor-spin structure.
PACS numbers: 12.39.-x, 12.39.Fe, 14.20.Dh
∗Electronic address: dwz@th.phy.pku.edu.cn
http://arxiv.org/abs/0704.0029v2
mailto:dwz@th.phy.pku.edu.cn
I. INTRODUCTION
The measurements of the polarized structure functions of the nucleon in deep inelastic
scattering(DIS) experiments[1, 2, 3, 4] show the complication in proton spin structure. Only
a portion of the proton spin is carried by valence quarks. Moreover, several experiments[5, 6,
7] clearly indicate the ū-d̄ asymmetry as well as the existence of the strange quark content s̄ in
the proton sea. Also the distribution of strange quark in the proton sea is polarized negative.
The DIS results deviate significantly from the näıve quark model (NQM) expectation.
NQM gives many fairly good descriptions of hadron properties. Why does NQM work?
It is a puzzle that the quarks inside a hadron could be treated as non-relativistic particles
in NQM. The chiral quark model (χQM) tries to bridge between QCD and NQM. It was
originated by Weinberg[8] and formulated by Manohar and Georgi[9]. Between the QCD
confinement scale (ΛQCD ≃200MeV) and a chiral symmetry breaking scale (ΛχSB ≃1GeV),
the strong interaction is described by an effective Lagrangian of quarks q, gluons g and
Numbu-Goldstone bosons Π. An important feature of the χQM is that, betweetn ΛQCD and
ΛχSB, the internal gluon effects in a hadron can be small compared to the internal Goldstone
bosons Π and quarks q, so the effective degrees of freedom in this region can be q and Π.
It is interesting that χQM can also be used to explain why NQM does not work in the
above DIS experiments. By the emission of Goldstone boson, χQM allows the fluctuation
of a quark q into a recoiling quark plus a Goldstone boson q → q′Π . The q′Π system then
further splits to generate quark sea through
• the helicity-flipping process
q↑ −→ Π+ q′↓ −→ (qq̄′) + q′↓ (1)
• and the helicity-non-flipping process
q↑ −→ Π+ q′↑ −→ (qq̄′) + q′↑ (2)
where the subscript indicates the helicity of quark. In both the process, q′Π is in a relative P-
wave state. In the helicity-flipping process (1), the orbital angular momentum along helicity
direction must be 〈lz〉 = +1. In the helicity-non-flipping process (2), 〈lz〉 = 0. The process
cause a modification of the spin content of the nucleon because a quark changes its helicity
in (1). Also it causes a modification of the flavor content because the generated quark sea
from Π is flavor dependent[10, 11].
χQM was first used to explain the nucleon sea flavor asymmetry and the smallness of
the quark spin fraction by Eichten, Hinchliffe and Quigg[10]. The flavor asymmetry of sea
quark distribution arises from the mass differences in different quark flavors and in different
Goldstone bosons. Only the lightest Goldstone Boson π was considered since its contribution
dominates. From a perturbation calculation, the probability for an up quark to emit a π+
was estimated to be a = 0.083. This would induce a flavor asymmetry in parton distributions
of nucleon and other hadrons.
However, the estimated transition probability is not enough to full account the flavor
asymmetry in DIS experiments. Contribution from other Π’s and even η′ was considered by
Cheng and Li[11]. Explicit SUf (3) breaking in the transition probabilities was later intro-
duced in refs. 12, 13 and further used by several authors[14, 15, 16, 17, 18, 19]. Nevertheless,
in all these calculations, the transition probabilities were put into model by hand. To fit the
experimental data, the probability of an up quark emitting π+ needs to be set to a >∼ 0.1,
which is about 20% larger than the perturbation calculation. Although the probability of π
emission can be enlarged by using a higher momentum cut off Λ > ΛχQM in the perturbation
calculation [20], however, the chiral quark model is no longer valid at arbitrary high energies
Λ ≫ ΛχQM.
We should not be surprised by this discrepancy since the χQM works in a region right
above the QCD confinement scale ΛQCD. There one may expect the confinement effect is
important and the perturbative calculation of QCD may contain large error. However, there
is another essential difference between the above model calculations and the perturbation
calcultion. In the perturbation calculation, the emitted Goldstone bosons are virtual par-
ticles. In the above model calculations which are closely related to NQM, however, the
Goldstone bosons are close to mass shell under the non-relativistic approximation.
Since χQM can be a bridge between NQM and QCD, it is interesting to explore χQM
from NQM side where we use the wave function method. This will give the above model cal-
culations a concrete foundation in NQM and help us further understand the flavor symmetry
breaking mechanism.
In this paper, we will use wave function method to investigate the flavor symmetry break-
ing in χQM. In a conventional quark model[21], a hadron consists of confined constituent
quarks and its wave function is constructed in the configuration space of the constituent
quarks. To incorporate the transition process of emitting Goldstone boson of χQM into the
quark model, the constituent quarks will have intrinsic wave functions within the configu-
ration q + q′Π.
In Sec. II, we first present the composite wave function of constituent quarks including
components of q′Π. The wave functions and the transition probabilities of q → q′Π are
determined from a simplified χQM Hamiltionian. In Sec. III and Sec. IV, the obtained
transition probabilities are used to calculate nucleon flavor-spin structure and baryon octet
magnetic moments respectively. The numerical results and a brief summary are presented
in Sec. V.
II. THE WAVE FUNCTION OF A CONSTITUENT QUARK
In χQM, the effective Lagrangian below the chiral symmetry breaking scale ΛχQM involves
quarks, gluons, and Goldstone bosons. The first few terms in this Lagrangian are[9]:
LχQM = ψ̄(iDµ + Vµ)γµψ + igAψ̄Aµγµγ5ψ
−mψ̄ψ +
f 2πtr∂
µΣ†∂µΣ+ ... (3)
where Dµ = ∂µ + igGµ is the gauge-covariant derivative of QCD, Gµ the gluon field and
g the strong coupling constant. The dimensionless axial-vector coupling gA = 0.7524 is
determined from the axial charge of the nucleon. m represents the constituent quark masses
due to chiral symmetry breaking. The pseudoscalar decay constant is fπ ≈ 93MeV. The Σ
field, vector currents Vµ and axial-vector currents Aµ are given in terms of the Goldstone
boson fields Φ
π0 + 1√
η π+ K+
π− − 1√
π0 + 1√
K− K̄0 − 2√
, (4)
Σ = exp(i
), (5)
(ξ†∂µξ ± ξ∂µξ†), (6)
ξ = exp(i
). (7)
An expansion of the currents in powers of Φ/fπ yields the effective interaction between Π
and q[10]
LI = −
ψ̄∂µΦγ
µγ5ψ. (8)
This allows the fluctuation of a quark into a recoil quark plus a Goldstone boson q → q′Π.
In quark model, a hadron is built with constituent quarks. In accordance with χQM,
we should treat a constituent quark as a composite particle including such components q′Π.
Here we denote the wave function of a composite constituent quark as |q〉〉. At rest,
|q〉〉 = zq|q〉+
q′Π|q
′Π〉. (9)
In our paper, the state normalization relation is always taken as
〈p|p′〉 = δ3(p− p′). (10)
The above wave function is of essential importance in our work. The square of the mod-
ulus of the coefficient of each q′Π configuration is just the probability for the corresponding
Π emission process
Pq→q′Π = |xqq′Π|
2, (11)
|zq|2 = (1−
Pq→q′Π)
is the probability of no Π emission.
To determine the wave function (9), we first construct a simplified Hamiltonian in the
degrees of freedom q and Π,
H = H0 +HB +HI . (12)
H0 represents the kinetic energies of q and Π. It reads
ψ̄(iα · ∇+m)ψ +
Tr[Φ̇2 + (∇Φ)2] +
m2Π(Φ
, (13)
where mΠ is the physical mass of Π which is nonzero and nondegenerate.
HI = −
d3xLI , (14)
is the χQM interaction. HB is an accessary interaction which is needed to bind the q
together. In our simplified Hamiltonian, we will not disscuss the explicit formalism of HB.
Instead, we will put some physical restriction conditions on it later in this section, which is
sufficient to our calculation.
FromH0, we can expand free fields ψ and Π in terms of creation and annihilation operators
ψq(x) =
(2π)3/2
q(p, s)e−ip·x + bq†
ps(t)v
q(p, s)eip·x
, (15)
ΦΠ(x) =
(2π)3/2
e−ip·x + cΠ†
eip·x
p0=EΠ
, (16)
where
p2 +m2q
is the quark energy of flavor q,
p2 +m2Π
is the energy of Goldstone boson Π. aq†
ps and b
pr are the creation operators of quark q and
anti-quark q̄
pr, a
p′s} = {b
pr, b
p′s} = δ
(3)(p− p′)δrs. (17)
is the creation operator of Π
] = δ(3)(p− p′). (18)
Next, we will replace the field ψ and Φ in the Hamiltonian (12) with the free field of (15)
and (16). Then we can express the Hamiltonian in creation and annihilation operators, for
example
d3p Eq
ps + b
ps] +
d3p EΠ
. (19)
In all the model calculations [11, 12, 13, 14, 15, 16, 17, 18, 19], the emitted Π is assumed
bound to the quark source. To represent that q′Π are bound, we use the well known SHO
function as their spatial wave function
|qΠ〉 =
d3p|p|e−
2λ2 [Y1(θ, φ) c
]1/2 |0〉, (20)
|qΠ ↑〉 = 1√
d3p|p|e−
2λ2 Y11(θ, φ) c
p↓ |0〉
d3p|p|e−
2λ2 Y10(θ, φ) c
p↑ |0〉, (21)
where λ is the “characteristic radius” parameter in Gaussian function. 1/
N is the nor-
malization factor,
dp p4 e
πλ5. (22)
However, we need a binding interaction HB in the Hamiltonian. Yet we do not know how
to write out the explicit form of HB. However, HB should provide enough binding energy.
That is, for the q′Π system, we must have
〈qΠ|H0 +HB|qΠ〉 ≤ mq +mΠ. (23)
That is
EB = 〈qΠ|HB|qΠ〉 ≤ mq +mΠ − 〈qΠ|H0|qΠ〉 = mq − Eq +mΠ − EΠ. (24)
As a rough estimation, we will take the mininum value of EB
EB = −max
{Eq −mq + EΠ −mΠ} = −(Eu −mu + Eπ −mπ). (25)
Then the wave function of a composite constituent quark is determined from Schrödinger
equation
H|q〉〉 =Mq|q〉〉. (26)
After taking the above simplification, we need only solve a matrix eigen-value problem
 =Mq
 , (27)
where
aδ3(0) = 〈q|H|q〉,
Bq′Πδ
3(0) = 〈q|H|q′Π〉,
Cq′Π;q′′Π′δ
3(0) = 〈q′Π|H|q′′Π′〉,
q′Π = x
For example, let us consider the process u emitting Π. There are four possible |q′Π〉 states
generated by the fluctuations of a u quark: |uπ0〉, |uη〉, |dπ+〉 and |sK+〉. Thus
|u〉〉 = zu|u〉+ xuuπ0 |uπ0〉+ xuuη|uη〉+ xudπ+ |dπ+〉+ xusK+|sK+〉. (28)
Taking these wave functions as basis, we can calculate the matrix of the Hamiltonian in
(27).
a = mu. (29)
C is diagonalized. Its diagonal matrix elements are calculated from H0
Cuπ0;uπ0 =
dp p4 e
p2 +m2u +
) + EB, (30)
Cuη;uη =
dp p4 e
p2 +m2u +
p+m2η) + EB, (31)
Cdπ+;dπ+ =
dp p4 e
p2 +m2d +
) + EB, (32)
CsK+;sK+ =
dp p4 e
p2 +m2s +
) + EB. (33)
B is calculated from HI
Buπ0 = −
dp p4e
, (34)
Buη = −
dp p4e
, (35)
Bdπ+ = −
dp p4e
, (36)
BsK+ = −
dp p4e
. (37)
By diagonalizing this Hamiltonian matrix, we will obtain a new mass of the constituent u
quark Mu and its composite wave function. The constituent masses and wave functions of
d and s quarks can be obtained similarly. We have
|d〉〉 = zd|d〉+ xddπ0 |dπ0〉+ xddη|dη〉+ xduπ−|uπ−〉+ xdsK0|sK0〉, (38)
|s〉〉 = zs|s〉+ xssη|sη〉+ xsdK̄0 |dK̄
0〉+ xsuK−|uK−〉. (39)
From isospin symmetry, mu = md, we have
zd = zu; xddπ0 = −xuuπ0 ; xduπ− = xudπ+ ; ... (40)
However, since mu 6= ms, one should notice that
zs 6= zu; xsdK̄0 6= x
sK0; x
uK− 6= xusK+. (41)
After the diagonalization, the Goldstone bosons Π are separated from quarks q approx-
imately. With only degrees of freedom q one can rebuild the quark model and so Mu, Md,
Ms should be regarded as the constituent quark masses in quark model.
III. FLAVOR AND SPIN STRUCTURE OF PROTON
Having known the wave functions of constituent quark q and the transition amplitudes
of q emitting each Goldstone bosons Π, we are able to calculate the quark distribution in
a constituent quark following refs. 11, 12, 13. In χQM, Π will further split into a quark-
antiquark pair. By substituting the quark contents of Π into wave functions (28), (38) and
(39), we can rewrite the wave functions of constituent quark q as
|u〉〉 = zu|u〉+
xuuη√
|u(uū)〉+
xuuη√
|u(dd̄)〉
2xuuη√
|u(ss̄)〉+ xudπ+ |d(ud̄)〉+ xusK+|s(us̄)〉, (42)
|d〉〉 = zu|d〉+
xuuη√
|d(uū)〉+
xuuη√
|d(dd̄)〉
2xuuη√
|d(ss̄)〉+ xudπ+ |u(dū)〉+ xusK+|s(ds̄)〉, (43)
|s〉〉 = zs|s〉+
xssη√
|s(uū)〉+
xssη√
|s(dd̄)〉 −
2xssη√
|s(ss̄)〉
|d(sd̄)〉+ xsuK−|u(sū)〉. (44)
Then the antiquark and quark flavor contents of the proton (uud) are
ū = 2
xuuη√
xuuη√
+ |xudπ+ |2, u = ū+ 2, (45)
xuuη√
xuuη√
+ 2|xudπ+ |2, d = d̄+ 1, (46)
s̄ = 2|xuuη|2 + 3|xusK+|2, s = s̄. (47)
Some important quantities depending on the above quark distribution are: the Gottfried
sum rule IG =
(ū − d̄) whose deviation indicates the ū-d̄ asymmetry in proton sea;
ū/d̄ measured through the ratio of muon pair production cross sections; and the fractions
of quark flavors in proton fq =
Σ(q+q̄)
, f3 = fu − fd and f8 = fu + fd − 2fs.
We can further calculate the spin structure of proton. Here one should consider the effects
of configuration mixing generated by spin-spin forces[22]. We take the baryon wave functions
from the quark model calculation[23, 24, 25]. The proton wave function for example, is
expressed as
= 0.90|P 28SS〉 − 0.34|P 28S ′S〉 − 0.27|P 28SM〉 (48)
where the baryon SU(6)⊗O(3) wave functions are denoted as |B2S+1N Lσ〉, N is SU(3) mul-
tiplicity. S, L are the total spin and total orbital angular momentum while σ = S,M,A
denotes the permutation symmetry of SU(6). The spin polarization functions will be re-
markably affected by configuration mixing. Following refs. 15, 17, we define the number
operator by
N̂ = nu↑u↑ + nu↓u↓ + nd↑d↑ + nd↓d↓ + ns↑s↑ + ns↓s↓,
where nq↑, nq↓ are the number of q↑, q↓ quarks. The spin structure of the “mixed” proton is
given by
= (0.902 + 0.342)
+ 0.272
. (49)
The spin structure after considering Π-emission is obtained by replacing for every quark in
eq. (49) by
q↑,↓ −→ (1− ΣPi)q↑,↓ + Pflipping(q↑,↓) + Pnon−flipping(q↑,↓), (50)
where Pflipping(q↑,↓) and Pnon−flipping(q↑,↓)| are the probabilities of quark helicity flipping and
non-flipping for q↑,↓ respectively. For example, in the case of u↑ quark we have,
Pflipping(u↑) =
(|xuuπ0 |2 + |xuuη|2)u↓ + |xudπ+ |2d↓ + |xusK+|2s↓
Pnon−flipping(u↑) =
(|xuuπ0 |2 + |xuuη|2)u↑ + |xudπ+ |2d↑ + |xusK+|2s↑
Finally the spin polarization functions defined as ∆q = q↑ − q↓ are
∆u = (0.902 + 0.342)
114|xu
|2 + 48|xuuη|2 + 36|xusK+|2
+ 0.272
66|xu
|2 + 24|xuuη|2 + 18|xusK+|2
, (51)
∆d = (0.902 + 0.342)
|2 + 12|xuuη|2 + 9|xusK+|2
+ 0.272
42|xu
|2 + 12|xuuη|2 + 9|xusK+|2
, (52)
∆s = −
. (53)
There are several measured quantities which can be expressed in terms of the above spin
polarization functions. The quantities usually calculated are ∆3 = ∆u−∆d and ∆8 = ∆u+
∆d−2∆s, obtained from the neutron β-decay and the weak decays of hyperons respectively.
Another important quantity is the flavor singlet component of the total quark spin content
defined as 2∆Σ = ∆u + ∆d + ∆s . We also calculate some weak axial-vector form factors
which are also related to the spin polarization functions, (GA/GV )Λ→p =
(2∆u−∆d−∆s),
(GA/GV )Σ−→n = ∆d−∆s, and (GA/GV )Ξ−→Λ = 13(∆u+∆d− 2∆s).
IV. BARYON OCTET MAGNETIC MOMENTS
Considering the relative angular momentum between quark and Goldstone boson Π, the
magnetic moment operator of a qΠ system is
µ̂qΠ =
p2q +m
p2Π +m
p2q +m
p2Π +m
p2Π +m
p2q +m
p2q +m
p2Π +m
l̂ (54)
where eq and eΠ are the electric charges carried by q and Π respectively, ŝ the quark spin
operator and l̂ the relative angular momentum bewteen q and Π. The first term in Eq(54)
is the intrinsic magnetic moment of quark and the other two terms are the contribution of
the orbital angular momentum. Here we have to consider the relativistic effect since the
relative momentum of q or Π are comparable to their masses in the qΠ system
pq,Π ∼ Λ ∼ mq,Π.
With the SHO wave functions of (20), the magnetic moment of qΠ system (54) can be
readily calculated. Then we can recalculate the magnetic moments of constituent quarks
taking into account of the relativistic effect. For example, the magnetic moments of the u
quark is
µu = |zu|2〈u↑|µ̂|u↑〉+ Pu→uπ0〈uπ0|µ̂|uπ0〉+ Pu→uη〈uη|µ̂|uη〉
+ Pu→dπ+〈dπ+|µ̂|dπ+〉+ Pu→sK+〈sK+|µ̂|sK+〉, (55)
where
〈u↑|µ̂|u↑〉 =
, (56)
and the contribution from qΠ systems are
〈uπ0|µ̂|uπ0〉 = − eu
p2 +m2π
p2 +m2u +
p2 +m2π
p2 +m2u
λ2 , (57)
〈uη|µ̂|uη〉 = − eu
p2 +m2η
p2 +m2u +
p2 +m2η
p2 +m2u
λ2 , (58)
〈dπ+|µ̂|dπ+〉 = −
p2 +m2π
p2 +m2d +
p2 +m2π
p2 +m2d
p2 +m2d
p2 +m2d +
p2 +m2π
p2 +m2π
λ2 , (59)
〈sK+|µ̂|sK+〉 = −
p2 +m2K
p2 +m2s +
p2 +m2K
p2 +m2s
p2 +m2s
p2 +m2s +
p2 +m2K
p2 +m2K
λ2 . (60)
The magnetic moments of d and s quarks can be calculated similarly.
One can easily obtain the octet baryon magnetic moments by replacing the valence quarks
inside the baryons with the corresponding constituent quarks. Again we take proton as an
example,
µp = (0.90
2 + 0.342)
+ 0.272
. (61)
If we replace the µq by (55), µp can be further expressed as the baryon magnetic moment in
conventional quark model plus the contribution from the Goldstone boson emission process
[26]. The magnetic moments for other octet baryons can be calculated similarly.
V. NUMERICAL RESULTS AND CONCLUSIONS
In the numerical calculation, most of the parameters can be taken from the experimental
data or the chiral quark model. We collect these fixed input parameters of our calculation
in Table I. Here we have used the the physical masses of Goldstone bosons[27].
TABLE I: The fixed input parameters from chiral quark model and experimental data.
gA fπ(MeV) mπ(MeV) mK(MeV) mη(MeV)
0.7524 93 135 494 548
For the quark masses, since our work focuses on the inner context of the constituent quarks
in quark model, naturally we will refer to the quark masses from quark model, instead of
the chiral quark model values. Here we will use the quark mass values from the widely
accepted Isgur’s quark model[21] as shown in Table II. However, one should be cautious
that, in our model, it is the quark with the Goldstone boson mixing which corresponds to
the constituent quark in quark model. That is, mass values Mq after the diagonalization
process should be set to the quark masses in Isgur’s model. Our strategy is to adjust the
quark masses mq in the model Hamiltonian to fit the Mq values.
Finally we are left only with one free parameter λ which describes the confinement of the
emitted Goldstone boson in our model. An overall fit to the experimental data of nucleon
flavor-spin structure and octet baryon magnetic moments shows that the best value should
be λ=152MeV. With this value of λ and a minimun binding energe EB = −218MeV, the
“bare” values of quark masses mq without Goldstone boson mixing are shown also in Table
Transition probabilities of the light and strange quarks to various q′Π systems are given
in Table III and IV respectively. The probability of a u quark emitting a π+ P (u →
TABLE II: The quark masses with vs. without Goldstone boson mixing.
λ EB(MeV) mu,d(MeV) ms(MeV) Mu,d(MeV) Ms(MeV)
152 −218 288 474 220 419
d + π+)=0.145 is significantly larger than the perturbation calculation a=0.083. One may
notice that the λ parameter value 152MeV in our wave function, which is below ΛQCD, is
rather small than another energy scale ΛχQM in chiral quark model. Surely this will weaken
the interaction between q and q′Π. However, one should also notice that the binding energy
EB = −218MeV will make the energy of a qΠ system much close to the single quark energy.
This will enhance the mixing of q′Π components in a constituent quark.
Also, we notice that the asymmetry between the probabilities of u(d) → s + K and
s → u(d) + K̄. Whether this asymmetry leads to any observable consequence in hadron
structure needs further investigation.
TABLE III: Transition probabilities of a u quark to various q′Π systems and the mass of constituent
u quark.
u → u+ π0 u → u+ η u → d+ π+ u → s+K+ no GB-emission Mu
0.072 0.003 0.145 0.010 0.770 220MeV
TABLE IV: Transition probabilities of a s quark to various q′Π systems and the mass of constituent
s quark.
s → s+ η s → u+K− s → d+ K̄0 no GB-emission Ms
0.012 0.071 0.071 0.846 419MeV
Next, we will compare our calculate results with the experimental data. Since our em-
phasis is on the substructure of a constituent quark in NQM, here we also quote the results
from NQM. In Table V, the calculated flavor and spin structures of the proton are shown.
It should be mentioned that the quark spin polarization functions can be further corrected
by the gluon anomaly[13, 15, 17, 28, 29, 30] through
∆q(Q2) = ∆q − αs(Q
∆g(Q2), (62)
and the flavor singlet component of the total helicity is modified accordingly as
∆Σ(Q2) = ∆Σ− 3αs(Q
∆g(Q2), (63)
where ∆q(Q2) and ∆Σ(Q2) are the experimentally measured quantities, ∆q and ∆Σ corre-
spond to the calculated quantities without gluon correction. Using the experimental data
Σ(Q2 = 5GeV2) = 0.19 ± 0.02[2], αs(Q2 = 5GeV2) = 0.285 ± 0.013[27], and our result
∆Σ=0.346, the gluon polarization ∆g(Q2) is estimated to be 2.293. Both the results with
and without gluon polarization corrections are presented in Table V. The inclusion of gluon
polarization leads to a better agreement with experimental data for the spin structure.
The calculated magnetic moments of octet baryons are given in Table VI. Although the
deviation is somewhat around 30% in the case of Ξ−, our overall fit to octet baryon magnetic
moments is in good agreement with experiments. Also it should be mentioned that even in
the case of Ξ− the fit can perhaps be improved if corrections due to pion loops are taken
into account[32, 33].
In the model calculations [11, 12, 13, 14, 15, 16, 17, 18, 19], the Goldstone boson sector in
χQM is usually extended to include the η′ meson with U(3) symmetry. According to Cheng
and Li[11], in the large Nc limit of QCD, there are nine Goldstone bosons including the usual
octet and the singlet η′. Thus an constituent quark can also transit to a quark-η′ system. We
have also made an U(3) calculation. With the inclusion of η′, we find that the probabilities
for η′-emission from light and strange quarks P (u → u + η′)=P (d → d + η′)=0.0021 and
P (s → s + η′)=0.0018 which are negligibly small as compared to those of octet Goldstone
boson emissions. We therefore conclude that the contribution of η′ is not important, due to
the obvious axial U(1) symmetry breaking in meson mass spectra mη′ > mK,η.
To summarize, the χQM builds a bridge between the QCD and low-energy quark model.
This allows us to understand the mechanism of flavor symmetry breaking and nucleon flavor-
spin structure in NQM through the consideration of the sea quark and Goldstone bosons
in the substructure of constituent quarks. Using the simple SHO wave function, we have
modeled the wave functions of the composite constituent quarks and thus estimated the
transition probabilities for Goldstone boson emissions. These transition probabilities indeed
reflect the flavor SU(3) symmetry breaking in χQM from the differences in quark masses
ms > mu,d and differences in Goldstone bosons masses mK,η > mπ and they are roughly
in agreement with the parametrizations of other model calculations [11, 12, 13, 14, 15, 16,
TABLE V: The calculated values for the quark flavor distribution functions and spin polarization
functions in proton, as compared with experimental data and NQM results.
Data NQM Our Model
With ∆g Without ∆g
∆u 0.85 ± 0.05[2] 1.33 0.864 0.968
∆d −0.41± 0.05[2] −0.33 −0.377 −0.274
∆s −0.07± 0.05[2] 0 −0.107 −0.003
∆3 = (GA/GV )n→p 1.270 ± 0.003[27] 1.67 1.242 1.242
(GA/GV )Λ→p 0.718 ± 0.015[27] 1 0.737 0.737
(GA/GV )Σ→n −0.340 ± 0.017[27] −0.33 −0.270 −0.270
(GA/GV )Ξ→Λ 0.25 ± 0.05[27] 0.33 0.234 0.234
∆8 0.58 ± 0.025[2] 1 0.701 0.701
∆Σ 0.19 ± 0.02[2] 0.5 0.190 0.346
ū − 0.264
d̄ − 0.392
s̄ − 0.036
ū− d̄ −0.118 ± 0.015[6] 0 −0.128
ū/d̄ 0.67 ± 0.06[6] 1 0.674
IG 0.254 ± 0.005[6] 0.33 0.248
fu − 0.577
fd − 0.407
fs 0.10 ± 0.06[31] 0 0.017
f3 − 0.170
f8 − 0.950
f3/f8 0.21 ± 0.05[14] 0.33 0.179
17, 18, 19]. The fit to both the flavor-spin structure of nucleon and octet baryon magnetic
moments are in good agreement with experiments.
TABLE VI: The caculated octet baryon magnetic moments in nuclear magneton, as compared with
experiments and the results of NQM.
Octet baryons Data[27] NQM[34] Our model
p 2.79± 0.00 2.72 2.73
n −1.91 ± 0.00 -1.81 −1.91
Σ− −1.16 ± 0.025 -1.01 −1.23
Σ+ 2.46± 0.01 2.61 2.67
Ξ0 −1.25± 0.0014 −1.41 −1.36
Ξ− −0.65 ± 0.002 −0.50 −0.44
Λ −0.61 ± 0.004 −0.59 −0.56
ΣΛ 1.61± 0.08 1.51 1.63
Acknowledgments
Zhan Shu would like to thank Fan-Yong Zou and Yan-Rui Liu for useful discussions. This
work was supported by the National Natural Science Foundation of China under Grants
10675008.
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	INTRODUCTION
	The Wave Function of a Constituent quark
	FLAVOR AND SPIN STRUCTURE OF PROTON
	BARYON OCTET MAGNETIC MOMENTS
	NUMERICAL RESULTS AND CONCLUSIONS
	Acknowledgments
	References
