BLED Workshops	A	Proceedings of the Mini-Workshop
in physics	i'l	Problems in Multi-Ouark States
VOL. 10, No. 1	Bled, Slovenia, June 29 - July 6, 2009
p. 53
Multi-quark configurations in the baryons
D. O. Riska
Helsinki Institute of Physics, POB 64, 00014 University of Helsinki
Abstract. Several experiments have revealed the presence of antiquarks in the proton [1]. Extensive phenomenological studies of meson photoproduction on nucleons with unitary hadronic models with and without form factors have also revealed that the well known underproduction of the N A transition strengths by the conventional three quark model may be attributed to the missing "meson cloud" contributions [2]. The question thus arises of to what extent multi-quark configurations of the type qqqqq, qqqqq q q, ... explicitly contribute to the observable of baryons. Here the contribution of the 5-quark configurations qqqqq to the magnetic moments and the axial form factors of the nucleon and the lowest resonances are considered. The two conclusions that emerge are that (a) a combination of at least three different qqqqq configurations are required for a satisfactory description of the nucleon properties and (b) that the vanishing of the axial form factor of the N (1535) resonance is a natural consequence of the cancelation of the contributions of the qqq and qqqqq configurations [3].
1 The qqqqq configurations in the nucleon
The qqqq subsystem of a qqqqq configuration has to be completely antisymmetric. As there are only 3 colors, the most "antisymmetric" qqqq color configuration is the mixed symmetry configuration [211]C.
[211 ] c
[31]
XFS
(1)
The complete antisymmetry of the qqqq system therefore requires that the combined space-flavor-spin configuration has to have the (conjugate) mixed symmetry combination [31]XFS above. This can be achieved by either (1) choosing the spatial configuration to be completely symmetric [4]S, with the flavor-spin configuration [31]FS or (2) by choosing the latter to be completely symmetric [4]FS and the former to have the mixed symmetry [31 ]X:
(1):
FS
(2):
JFS
(2)
C
X
X
In the first case positive parity demands that the antiquark q be in the P—state, while in the latter case, the antiquark has to be in the ground (S—) state. A pion
Table 1. Magnetic moments for the qqqqq configurations in the nucleon
qqqq symmetry configuration	proton	neutron	
[31 ] x [4] f s [22]p[22]s	0	1/3	
[31 ] x [4] f s [31 ] f [31 ] s	2/9	-2/9	qqqq : J = 1
[31 ] x [4] f s [31 ] f [31 ] s	-1/3	0	qqqq : J = 0
[4] x [3 T ] f s [22]f[31]s	7/27	-23/27	q : J = 3/2
[4] x [3 T ] f s [22]f[31]s	-4/27	0	q: J = 1/2
[4] x [3 T ] f s [31]f[22]s	-2/9	0	
[4] x [3 T ] f s [31 ] f [31 ] s	-19/27	1/9	q : J = 3/2
[4] x [3 T ] f s [31 ] f [31 ] s	508/729	-95/729	q: J = 1/2
loop configuration would correspond to the antiquark in the P—state. The configuration with the q in the S—state is, however, that which is consistent with a positive strangeness magnetic moment [4,5]. Note that if the antiquark is in the P—state the required [31]XFS configuration can also be obtained with [31 ]FS and [22]fs flavor-spin configurations of higher energy [20].
No qqqqq component alone can achieve the remarkable —3/2 ratio between the proton and the neutron magnetic moments, which is characteristic of the basic qqq configuration in both its nonrelativistic and relativistic versions [6]. This may be inferred from Table 1, where the nucleon magnetic moments for the 7 possible qqqqq configurations in the nucleons are listed. This may also inferred from the comprehensive attempt in ref.[7] to combine only the first of these qqqqq configurations with the basic qqq configuration.
The desired -3/2 ratio can however be obtained with a linear combination of the qqq and the first 3 configurations in the table:
Tjj = V/P3<P[3][21][21] + y Ilb^-f 5b2 |	+!b2<P[4][22][22]
+ V/b7<p[4=[31][31] + v^pf^imi]}-	(3)
Here P3 and P5 are the probabilities for the qqq and (total) qqqqq components. The symmetry assignments [FS] [F] [S] in the wave functions represent flavor x spin, flavor and spin respectively. The q components in the qqqqq wavefunctions is to be understood. A combination of the form (3) with 68 % qqq and 32 % of these qqqqq can in fact be arranged to yield the empirical value for gA(n —> p), eg by taking bi = b2.
The identification of specific multi-quark contributions in the nucleon form factors is difficult because of their smooth behavior, which may be reproduced by a large variety of models. The prospective node in the region above Q2 ~ 6
GeV2 in GE [8] does for example arise naturally already in the case of the qqq configuration if calculated with front form kinematics [9], although it also arises if a qqqqq component is included, the magnitude and form of which are set by the empirical values for G£ [10]. The electric form factor of the neutron G£, which vanishes in the nonrelativistic qqq model, can in fact be brought into agreement with the empirical values by including a mixed symmetry S—state in the nucleon wave function with a probability of 1 — 2 % [9].
2 The qqqqq configurations in the nucleon resonances
While it is possible to achieve a qualitative description of the lowest baryon resonances with the basic qqq model with spin and flavor dependent interactions [11], that model does not describe the systematics of the resonance decay widths. In the case of the A(1232) and the N(1440) resonances it has been shown that the inclusion of a qqqqq component in the wave function makes it possible to overcome the underpredictions of the electromagnetic and strong decay widths [12-14]. Such calculations are however only qualitative in that the cross term matrix elements between the qqq and qqqqq components are very sensitive to the wave function models.
The cross terms between the qqq and the qqqqq configurations are large when the operator, which connects the annihilating qq pair and the meson or the Y ray involves the "large" components of the Dirac spinors. When the operator involves the small components, which is the case of the axial charge operator, the cross terms are suppressed.
In this context the recent lattice result that the axial charge of the N (1535) is very small - if not 0 - is particularly interesting [15]. If the corresponding result for the (near) parity partner N (1440) would also be close to 0, that might actually indicate the onset of restored chiral symmetry [16]. As the configuration mixing between the N (1535) and the following 1 /2- resonance N (1650) is expected to be small [17,18], these resonances may be considered separately.
The general expression for the axial charge of the N (1535) is
9a -X AnPn , n = 3,5,..	(4)
n
where n is the number of constituents ((n+3)/2 is the number of quarks and (n— 3)/2 the number of antiquarks). Since the qqq model value for gA is —1 /9 [16], it follows that if indeed the axial charge of the N (1535) vanishes, the multiquark configurations with n > 3 have to cancel that value.
Consideration of the qqqqq components indicates that this would be a very natural result [3]. In Table 2 all the possible qqqqq configurations in the N(1535) and the corresponding coefficients An in the axial charge expression 4 are listed. These are listed in order of increasing energy under the assumption that the interaction between the quarks depend on spin and flavor or color.
Inclusion of these qqqqq components in addition to the qqq component leads to the axial charge expression,
9a(n(1535)) = -1p3 + ^p<2> - lp<3> - ±p<4> + ^p<5> ,	(5)
Table 2. The qqqqq configurations in the N(1535) and the corresponding axial charge coefficient An (4) [19].
configuration	q q q q flavor-spin	q q q q color-spin	An
1	[31 ]FS [21 1]F [22]s	[31]cs [211 ]c [22]s	0
2	[31 ]FS [21 1]F [31 ]s	[31]cs [21 l]c [31 ]s	+5/6
3	[31 ] F s [22] F [31 ] s	[22] c s [211 ] c [31 ] s	-1/9
4	[31 ] F s [31 ] F [22] s	[211 ] es [211 ] c [22] s	-4/15
5	[31 ] F s [31 ] F [31 ] s	[211 ]es [211 ]c [31 ]s	+17/18
where the coefficients P indicate the corresponding probabilities. Because two of the qqqqq components have large positive coefficients, while the qqq contribution has a small negative coefficient it is possible to cancel the latter contribution altogether with only modest probabilities of the qqqqq components [19].
Combination of this result with the lattice calculation result for the axial charge of the N ( 1650) resonance [15], which is close to the qqq quark model value 5/9 [16], suggests the conclusion that the smallness of the axial charge of the N ( 1535) is a natural consequence of its quark configuration and (possibly also) the cancelation between the contributions of the qqq and the qqqqq components [19] rather than an indication of restored chiral symmetry.
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