Commun. Theor. Phys. (Beijing, China) 54 (2010) pp. 697–700
c Chinese Physical Society and IOP Publishing Ltd
Vol. 54, No. 4, October 15, 2010
Contributions of qqqq q̄ Components to Axial Charges of Proton and N (1440)∗
gf),
YUAN Si-Gang (
1,2
SS)),
AN Chun-Sheng (
3
Û)
and HE Jun (
1,†
1
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
2
Graduate University of Chinese Academy of Sciences, Beijing 100049, China
3
Institute of Hight Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
(Received January 8, 2010; revised manuscript received March 4, 2010)
Abstract The axial charges of the proton and N (1440) are studied in the framework of an extended constituent
quark model (CQM) including qqqq q̄ components. The cancellation between the contributions of qqq components and
qqqq q̄ components gives a natural explanation to the experimental value of the proton axial charge, which can not be
well reproduced in the traditional CQM even after the SU(6) ⊗ O(3) symmetry breaking is taken into account. The
experimental value of axial charge pins down the proportion of the qqqq q̄ component in the proton to about 20%, which is
consistent with the ones given by the strong decay widths and helicity amplitudes. Besides, an axial charge for N (1440)
about 1 is predicted with 30% qqqq q̄ component, which is obtained by the strong and electromagnetic decays.
PACS numbers: 12.39.Jh, 11.30.Rd, 14.20.Gk
Key words: axial charge, five-quark component, proton
1 Introduction
The axial charges of the proton and its resonances are
fundamental quantities in QCD. The proton axial charge
is related to the spontaneous chiral symmetry breaking in
the low energy QCD through the Goldberger–Treiman relation, gA =fπ gπN N /MN . Experimentally the proton axial
charge can be extracted from neutron beta decay, which
is 1.2695 ± 0.0029 (in unit of vector charge gV ).[1] The
fact that the proton axial charge is so accurately measured offers opportunity for understanding some key issues in nucleon structure. The traditional CQM following
the way of the Gell-Mann quark model assigns the proton
as uud, which are the three massive constituent quarks
including the effect of sea quarks and gluons. This naive
CQM gives an excellent description of the masses of octet
and decuplet.
However, in the SU(6) ⊗ O(3) symmetry scheme the
CQM predicts that the proton axial charge is 5/3 exactly, which is about 20% larger than the experimental
value. The axial charges of resonances can not be measured experimentally, but they can be calculated in the
lattice QCD. Recently the lattice results show that the axial charge of N (1535) is very small and the one of N (1650)
is about 0.55.[2] Compared with CQM values under the
SU(6) ⊗ O(3) symmetry, −1/9 and 5/9, the axial charge
of N (1535) is not well described by the naive CQM.
In the more complex models, such as, gluon exchange
model, extended Goldstone boson exchange model, and
MIT bag model, the proton axial charge gAp = 1.15,
1.11 or 1.09, respectively, which is improved on but still
not satisfying compared with experimental value even af∗ Support
ter taking relativistic effects into account.[3] In Ref. [4]
Di et al. pointed out that with about 15% sea quark
in the proton, an axial charge about 1.23 can be obtained in their valence-sea mixing model. Predicted values of lattice QCD for the proton axial charge fall in the
range of 1.1 ∼ 1.4.[5−10] If lattice simulation includes
staggered sea quark, lattice value for the proton axial
charge gAp = 1.226 ± 0.084.[6] In lattice QCD all sea quark
and gluon contributions are considered explicitly while in
CQM these contributions has been absorbed into the constituent quarks partly. Hence it is interesting to study
the importance of sea quark contribution in the proton
axial charges as indicted by valence-sea mixing model and
lattice QCD.
The past studies have suggested that CQM with only
three constituent quarks is not enough to reproduce the
experiment data. Many models are proposed to solve this
problem, such as the well-known meson cloud model.[11]
Recently Zou and Riska et al. suggested that many puzzles surrounding baryon resonances in CQM, such as, the
strange magnetic moment problem of the proton, may
be solved by extending three quark wave function to include multiquark configurations qqqq q̄.[12−18] This model
has been successfully used to explain the very small axial
charge of N (1535).[19]
In this work we will study the possibility to explain the
proton axial charge in CQM after including qqqq q̄ components.
The present paper is organized as follows. In Sec. 2,
the wave functions of the proton and N (1440) will be recalled. With these wave functions the axial charges will be
by National Natural Science Foundation of China under Grant No. 10905077
author, E-mail: junhe@impcas.ac.cn
† Corresponding
698
YUAN Si-Gang, AN Chun-Sheng, and HE Jun
calculated in Sec. 3. Conclusions will be given in Sec. 4.
2 Extended Wave Functions for Proton and
N (1440)
First we recall the extended wave functions for the
proton and its resonances as[18]
X
|p, N (1440), N (1710)i = A3 |qqqi +
Ai5 |qqqq q̄ii , (1)
i
Ai5
where A3 and
are the proportion factors for the 3quark and 5-quark components, respectively. The wave
functions for qqqq q̄ are taken as
X
X
Sz /2
JM
|qqqq q̄iiSz =
CJM,S
′ /2 C1m,SSz
z
a,b,c,d,e M,m,S,Sz
[14 ]
[31]
[F S]
× C[31]a [211]a C[31]ba[31]c C[F i ]dc[S i ]e
× [211]C (a)[31]X,m (b)[F i ]F (d)[S i ]Sz
× (e)χ̄y,tz ξ¯Sz′ ψ(κ~i ) ,
(2)
where the first summation runs over the indices in the
S4 Clebsch–Gordan coefficients for the color ([211]), orbital ([31]), flavor ([F ]), and spin ([S]) spaces of the qqqq
subsystem, and the second one runs over the spin indices of the standard SU(2) Clebsch–Gordan coefficient.
Here Wely tableaux is [31]X following the conventions in
Refs. [21–22]. The number of the qqqq q̄ configuration, i,
is as shown in Table 1. χ̄y,tz and ξ¯Sz′ represent the flavor
and spin wave functions of the antiquark respectively, and
ψ(κ~i ) represents the spatial wave function in the momentum space. The qqqq q̄ configurations with appropriate
quantum numbers of the P11 resonances, proton, N (1440)
and N (1710), are listed in Table 1.
Table 1 The qqqq q̄ configurations of proton, N (1440)
and N (1710). The second and third or fifth and sixth
rows are corresponding to the same flavor-spin configurations with the total angular momentum, L = 0 and 1,
of the four-quark subsystems, respectively.
i
Flavor-spin
hCFS i
Ci
1
[4]FS [22]F [22]S
−28
−2/9
2
3
[4]FS [31]F [31]S
[4]FS [31]F [31]S
−64/3
−64/3
−4/15
28/45
4
5
[31]FS [211]F [22]S
[31]FS [211]F [31]S
−16
−40/3
0
0
6
[31]FS [211]F [31]S
−40/3
4/9
7
[31]FS [22]F [31]S
−28/3
+17/18
Vol. 54
Now, these configurations can be ordered in the terms
of increasing matrix element hCFS i of chiral hyperfine interaction, which is assumed as[23]
X
~λF · ~λF ~σi · ~σj ,
CFS = −
(3)
i
j
i,j
where the λ and ~σ are the flavor and spin matrices respectively. The energy of the configuration with the spinflavor symmetry [4]FS [22]F [22]S is about 140–200 MeV
lower than others. Simultaneously, using this configuration the positive strange magnetic moment, strange electric radii and negative strange electric form factor are
well explained.[12,24] Hence in this work the convention
of Zou et al. will be followed that the configuration with
the lowest energy is the most important one of the all
five-quark components in the description of the physical
observables.[12]
3 Axial Charges of Proton and N (1440)
The axial charge operator is independent of spatial
P
wave function and given by b
gA = i σzi τzi .[20] With the
wave functions given in the previous section, the axial
charge can be given as a sum of the diagonal matrix elements,
X
gA = CP3 +
Ci P5i ,
(4)
i
where C or Ci is the axial charge for qqq or qqqq q̄ configuration, which is presented in Table 1. The proportion
P or Pi is the square of A or Ai in Eq. (1). Obviously
the axial charge is independent of adjustable parameters
appearing in the spatial wave function. In addition, another parameter, i.e., the relative phase factor δ between
the qqq and qqqq q̄ components, is not involved.
The axial charges for the proton and N (1440) read
5
2
4
28
4
p,N (1440)
gA
= P3 − P51 − P52 + P53 + P55 . (5)
3
9
15
45
9
The axial changes in the SU(6) ⊗ O(3) symmetry scheme,
with mixings between qqq states with same quantum number due to symmetry breaking and with qqqq q̄ components
are calculated and presented in Table 2. The mixing coefficients for the three-quark wave functions of the proton
and N (1440) are taken from Ref. [25–26]. Here N (1710)
is not considered because there is neither experimental
value of the axial charge nor the theoretical prediction of
the qqqq q̄ component proportion.
Table 2 The axial charges of proton and N (1440). The first three columns are the results in
SU(6) ⊗ O(3) symmetry, with mixings and with qqqq q̄ components. The last two columns are the
lattice QCD and experimental values.
p
gA
N(1440)
gA
SU(6) ⊗ O(3)
with mixing
with qqqq q̄
LQCD
Exp.
5/3
1.60
1.25 ∼ 1.46
1.10 ∼ 1.40
1.2695 ± 0.0029
5/3
1.67
∼1
No. 4
Contributions of qqqq q̄ Components to Axial Charges of Proton and N (1440)
In Table 2, one can find that the theoretical value
of the proton axial charge in the traditional CQM in
SU(6)⊗ O(3) scheme is much larger than the experimental
value. After considering the mixings, the deviations of the
values of the axial charges for both proton and N (1440)
are very small. It is due to that the three-quark wave functions of two lowest P11 states corresponding to proton and
N (1440) are different only in the spatial part, which is not
relevant to the axial charge. Here we take the mixing angle of Isgur et al. for example. From above analysis, the
conclusion will be kept if other mixing angles are used.
Hence it is reasonable to neglect the effect from mixings
of three-quark wave functions in the following discussion.
In previous studies,[14−16,18] the proportion of fivequark component is restricted to a range from 10% to
20% by the investigations of the proton electromagnetic
and strong decays of ∆(1232), N (1440), and N (1535). If
this proportion is inserted into Eq. (5), the obtained numerical value for the proton axial charge is in the range
from 1.25 to 1.45, which brackets 1.26. In Fig. 1, the
dependence of the proton axial charge on the proportion
of qqqq q̄ component is presented. Here only the first two
terms in Eq. (5) are taken into consideration as discussed
in the previous section. One can find that the experimental value of the proton axial charge gives a very strong
constrain on the proportion of qqqq q̄ component. Using
the experimental value of gAp , one arrives at a narrow range
of the proportion from 19.6% to 20.7%, which is still in
the range from 10% to 20%. At the same time, if the
proportion of qqqq q̄ component in the proton is varied in
the range from 10% to 20% the deviations of helicity amplitudes for ∆(1232), N (1440), and N (1535) are only a
few percents.[14−16] Hence in the narrow range obtained
from the proton axial charges, these helicity amplitudes
will keep satisfying.
There is no experiment or lattice QCD value for the
axial charge of N (1440). However, the proportions of
qqqq q̄ components in N (1440) have been investigated. In
Ref. [14] the helicity amplitude and strong decay width
of N (1440) are reproduced by introducing of about 30%
qqqq q̄ component. Inserting P51 ∼ 30% into Eq. (5),
the axial charge of N (1440) is about 1, which is close
to the results of OGE and GBE model, 1.10 and 1.16
respectively.[3] As Ref. [14], the effect of contributions
from next-to-lowest-energy configurations with i = 2, 3
in Table 1 is considered here for completeness. Total 30%
qqqq q̄ component in N (1440) is adopted as before while
among qqqq q̄ component the proportions of configurations
with i = 1, 2, 3 are taken as 80%, 10%, and 10%, respectively. The axial charge of N (1440) is obtained as about
699
1.1. Comparing with axial charge obtained only with the
configuration i = 1, including of configuration i = 2, 3
does not lead to obvious deviation.
Fig. 1 The axial charge of proton as a function of the
proportion of qqqq q̄ component. The solid line represents
the result including qqqq q̄ component. The band is for
the experimental value with the uncertainty.
4 Conclusions
The traditional CQM can not give a satisfying description of the proton axial charge extracted from experiment even though the breakdown of the SU(6) ⊗ O(3)
symmetry is taken into account. In this work the proton
axial charge is investigated in an extended CQM including the five-quark component and found it is well reproduced through the cancellation between the contributions
from three-quark and five-quark components. The proportion of the five-quark component in the proton, about
20%, is extracted, which is consistent with the previous
results.[14−18] The consistency of the values of five-quark
component proportions in the studies of different physical
quantities suggests the rationality of including five-quark
components. Besides, the axial charge for N (1440) is predicted after the contribution from about 30% five-quark
component is considered. The results in the valence-sea
mixing model[4] and this work indicate that the sea quark
components in CQM have an essential contribution to the
proton axial charge. The future wealth lattice QCD results of axial charge are expected to deepen our understanding about the sea quark components in Roper state
and other nucleon resonances.
Acknowledgments
S.G. Yuan acknowledges the hospitality of the Theoretical Physics Center for Science Facilities of Chinese
Academy of Sciences during the course of this work.
700
YUAN Si-Gang, AN Chun-Sheng, and HE Jun
References
[1] C. Amsler, et al., Phys. Lett. B 667 (2008) 1.
[2] T.T. Takahashi and T. Kunihiro, Phys. Rev. D 78 (2008)
011503.
[3] K.S. Choi, W. Plessas, and R.F. Wagenbrunn, arXiv:
0908.3959.
[4] D. Qing, X.S. Chen, and F. Wang, Phys. Rev. D 58 (1998)
114032.
[5] D. Dolgov, et al., Phys. Rev. D 66 (2002) 034506.
[6] R.G. Edwards, et al., Phys. Rev. Lett. 96 (2006) 052001.
[7] A.A. Khan, et al., Phys. Rev. D 74 (2006) 094508.
[8] T. Yamazaki, et al., Phys. Rev. Lett. 100 (2008) 171602.
[9] H.W. Lin, T. Blum, S. Ohta, S. Sasaki, and T. Yamazaki,
Phys. Rev. D 78 (2008) 014505.
[10] C. Alexandrou, et al., PoS (LATTICE (2008)) 139.
[11] S. Theberge, A.W. Thomas, and G.A. Miller, Phys. Rev.
D 22 (1980) 2838; [Erratum-ibid. D 23 (1981) 2106].
[12] B.S. Zou and D.O. Riska, Phys. Rev. Lett. 95 (2005)
072001.
Vol. 54
[13] C.S. An, D.O. Riska, and B.S. Zou, Phys. Rev. C 73
(2006) 035207.
[14] Q.B. Li and D.O. Riska, Phys. Rev. C 74 (2006) 015202.
[15] Q.B. Li and D.O. Riska, Nucl. Phys. A 766 (2006) 172.
[16] Q.B. Li and D.O. Riska, Nucl. Phys. A 791 (2007) 406.
[17] C.S. An, Q.B. Li, D.O. Riska, and B.S. Zou, Phys. Rev.
C 74 (2006) 055205; [Erratum ibid. C 75 (2007) 069901].
[18] C.S. An and B.S. Zou, Eur. Phys. J. A 39 (2009) 195.
[19] C.S. An and D.O. Riska, Eur. Phys. J. A 37 (2008) 263.
[20] L. Ya. Glozman and A.V. Nefediev, Nucl. Phys. A 807
(2009) 38.
[21] J.Q. Chen, M.J. Gao, and G.Q. Ma, Rev. Mod. Phys. 57
(1985) 211.
[22] J.Q. Chen, J.L. Ping, and F. Wang, Group Representation
Theory for Physicists, World Scientific, Singapore (2002).
[23] C. Helminen and D.O. Riska, Nucl. Phys. A 699 (2002)
624.
[24] D.O. Riska and B.S. Zou, Phys. Lett. B 636 (2006) 265.
[25] N. Isgur and G. Karl, Phys. Lett. B 72 (1977) 109.
[26] J. He and B. Saghai, Phys. Rev. C 78 (2008) 035204.