Sum rules for baryon decuplet magnetic moments

Sum rules for baryon decuplet magnetic moments

W. K. Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, California 91125, USA

and Department of Science Education and Research Institute for Basic Sciences, Ewha Womans University, Seoul 120-750, Korea

(Received 12 April 2007; published 29 November 2007)

In chiral models with SU(3) group structure, baryon decuplet, and octet magnetic moments are evaluated by constructing their sum rules to yield theoretical predictions. In these sum rules we exploit six experimentally known baryon magnetic moments. Sum rules for flavor components and strange form factors of the octet and decuplet magnetic moments and decuplet-to-octet transition magnetic moments are also investigated.

DOI:10.1103/PhysRevD.76.094029 PACS numbers: 12.39.
x, 11.30.Hv, 13.40.Em, 21.60.Fw

I. INTRODUCTION

The internal structure of the nucleon is still a subject of great interest to both experimentalists and theorists. In 1933, Frisch and Stern [1] performed the first measurement of the magnetic moment of the proton and obtained the earliest experimental evidence for the internal structure of the nucleon. However, it was not until 40 years later that the quark structure of the nucleon was directly observed in deep inelastic electron scattering experiments and we still lack a quantitative theoretical understanding of these prop-erties including the magnetic moments.

Since Coleman and Glashow [2] predicted the magnetic moments of the baryon octet about 40 years ago, there has been a lot of progress in both the theoretical paradigm and experimental verification for the baryon magnetic mo-ments. The measurements of the baryon decuplet magnetic moments were reported for

[3] and
_
[4] to yield a new avenue for understanding the hadron structure. The magnetic moments of baryon decuplet have been theoreti-cally investigated in several models such as the quenched lattice gauge theory [5], the quark models [6], the chiral bag model [7], the chiral perturbation theory [8], the QCD sum rules [9], the chiral quark model [10], and the chiral quark soliton model [11]. Moreover, by including the effect of decuplet intermediate states of spin-3=2 baryons explic-itly, the heavy baryon chiral expansion of baryon octet magnetic moments [12] and charge radii [13] were inves-tigated. The decuplet-to-octet transition magnetic mo-ments have been also analyzed in the 1=Nc expansion of

QCD [14–16] and in the chiral quark soliton model [17]. Quite recently, the SAMPLE Collaboration [18] re-ported the experimental data of the proton strange form factor through parity violating electron scattering [19]. To be more precise, they measured the neutral weak form

factors at a small momentum transfer Q2

S

0:1 GeV=c2 _{to yield the proton strange magnetic form} factor in units of Bohr nuclear magnetons (n.m.) Gs

M

0:37  0:20  0:26  0:07 n:m: [18]. The HAPPEX Collaboration later reported Gs_M 0:18  0:27 n:m: [20]. Moreover, McKeown [21] has shown that the strange form factor of proton should be positive by using the conjecture that the up-quark effects are generally dominant in the flavor dependence of the nucleon properties. The chiral bag model [22] predicted first the positive value for the proton strange form factor [23].

In this paper, we will exploit the chiral bag model to predict baryon decuplet and octet magnetic moments and their strange form factors. This model calculation can share those of other skyrmion extended models with SU(3) group structure if sum rules are properly used. More specifically, in the chiral models, we will investigate the magnetic moments of baryon decuplet and octet, together with the decuplet-to-octet transition magnetic moments, in terms of their sum rules. We will also study the sum rules for the flavor components and strange form factors of the baryon magnetic moments. In Sec. II, in the adjoint representation we construct the sum rules of the baryon decuplet and octet magnetic moments and the transition magnetic moments in the SU(3) chiral models, and in Sec. III the sum rules for their flavor components and strange form factors in a model independent way at least in the category of the chiral models with the SU(3) flavor group. In the Appendix, we list the u- and d-flavor components of the baryon magnetic moments and transition magnetic mo-ments and their sum rules.

II. MAGNETIC MOMENTS OF BARYON DECUPLET

We start with the chiral bag model with the broken U-spin symmetry whose Lagrangian is of the form *soonhong@ewha.ac.kr

L  LCS LCSB LFSB; LCS  i
@
B
1 2 U 5
B

1 4f 2 trl
l
  1 32e2l
; l 2_{ L} WZW   B; LCSB
 M B 1 4f 2 m2trU  Uy
2 B; LFSB 1 6f 2 2m2K
m2 tr1
 3 p 8U  Uy
2 B
1 12f 2 2
1 tr1
 3 p 8Ul
l
 l
l
Uy B; (2.1)

where the quark field has SU(3) flavor degrees of free-dom and the chiral field U  eiaa=f 2 SU3 is

de-scribed by the pseudoscalar meson fields a

(a  1; . . . ; 8) and Gell-Mann matrices a with ab 

2

3ab ifabc dabcc, and B 1
B is the bag

theta function (one inside the bag and zero outside the bag). In the limit of vanishing bag radius, the chiral bag model is reduced to the Skyrmion model. Here l
 Uy@
U and LWZW stands for the topological Wess-Zumino-Witten (WZW) term. The chiral symmetry (CS) is broken by the quark masses M  diagm_u; m_d; m_s and pion mass m_ in LCSB. Furthermore the SU(3) flavor symmetry breaking (FSB) with m_K=m_
1 and   f_K=f_
1 is included in LFSB. Even though the mass terms in LCSB and LFSB break both the SUL3

SUR3 and diagonal SU(3) symmetry so that chiral

sym-metry cannot be conserved, these terms without derivatives yield no explicit contribution to the electromagnetic (EM) currents J
_{and at least in the adjoint representation of the}

SU(3) group the EM currents are conserved and of the same form as the chiral limit result J_CS
to preserve the U-spin symmetry. However the derivative-dependent term inLFSB gives rise to the U-spin symmetry breaking con-served EM currents J_FSB
so that J
_{ J}

CS J

FSB. Assuming that the hedgehog classical solution in the meson phase U0  eii^rir (i  1; 2; 3) is embedded in

the SU(2) isospin subgroup of SU(3) and the Fock space in the quark phase is described by the Nc valence quarks

and the vacuum structure composed of quarks filling the negative energy sea, the chiral model generates the zero mode with the collective variable At 2 SU3 by per-forming the slow rotation U ! AU₀Ay and ! A on SU(3) group manifold. Given the spinning chiral model ansatz, the EM currents yield the magnetic moment opera-tors
^i_{ ^}
i3_ 1_

3

p
_^i8_{, where
^}ia_{ ^}
ia CS  ^
ia FSB with ^
ia_CS 
N D8 ai
N 0_d ipqD8apT^Rq Nc 2p MD3 8 a8J^i; ^
ia_FSB
P D8 ai1
D888  Q  3 p 2 dipqD 8 apD88q  RD8 a8D88i; (2.2)

whereM, N , N0,P , Q, and R are the inertia parame-ters calculable in the chiral models [23,24]. Using the theorem that the tensor product of the Wigner D functions can be decomposed into the sum of the single D functions, the isovector and isoscalar parts of the operator ^
ia_FSB are then rewritten as ^
i3_FSB P

4 5D 8 3i 1 4D 10 3i  1 4D  10 3i  3 10D 27 3i   Q
₃ 10D 8 3i
3 10D 27 3i   R
₁ 5D 8 3i 7 10D 27 3i  ; ^
i8_FSB P

6 5D 8 8i 9 20D 27 8i   Q

3 10D 8 8i
9 20D 27 8i   R

1 5D 8 8i 9 20D 27 8i  : (2.3)

Here one notes that the 1, 10, and 10 irreducible representations (IRs) do not occur in the decuplet baryons while 10 and 10 IRs appear together in the isovector channel of the baryon octet to conserve the hermiticity of the operator.

Using the above operator ^
i_{together with the decuplet baryon wave function } B

 dim p

D

ab with the quantum

numbers a  Y; I; I₃ (Y, hypercharge; I, isospin) and b  YR; J;
J3 (YR, right hypercharge; J, spin) and  the

dimension of the representation, the baryon decuplet magnetic moments for 10 (J₃ 3=2) and transition magnetic moments for 10 J₃ 1=2 ! 8 J3  1=2   have the following hyperfine structure:

  1 8M  1 2
N
1 2p N3 0_3 7P
3 56Q
1 14R;

 1 16M  1 4
N
1 2p N3 0_ 5 21P  1 84Q
1 84R;

0  1 21P  13 168Q  1 21R;



1 16M
1 4
N
1 2p N3 0
1 7P  1 7Q  3 28R;
_  1 16M  1 4
N
1 2p N3 0_19 84P
17 168Q
1 42R;
0  1 84P
1 84Q  1 84R;
_

1 16M
1 4
N
1 2p N3 0
17 84P  13 168Q  1 21R;
_0 
1 42P
17 168Q
1 42R;
_

1 16M
1 4
N
1 2p N3 0
11 42P  1 84Q
1 84R;
_

1 16M
1 4
N
1 2p N3 0
9 28P
3 56Q
1 14R; 1  5 p
_p
 1  5 p
_n
0 
2 15
N 2
 2 p 8 N 0
4 45P  7 180Q  79 2700R; 1  5 p
_   2 15
N 2
 2 p 8 N 0_13 90P  1 180Q
1 100R; 1  5 p
_0_0  1 15
N 2
 2 p 8 N 0_ 7 90P  1 45Q  1 90R; 1  5 p

_
 1 90P  7 180Q  29 900R; 1  5 p
_0_0  2 15
N 2
 2 p 8 N 0_ 7 45P
1 180Q  23 2700R; 1  5 p
_

 1 90P  7 180Q  67 2700R; 1  15 p
_0 
1 15
N 2
 2 p 8 N 0
1 18P  1 45Q  11 1350R: (2.4)

Here the coefficients are solely given by the SU(3) group structure of the chiral models and the physical information such as decay constants and masses are included in the above inertia parameters, such asM, N , and so on. Note that the SU(3) group structure in the coefficients is generic property shared by the chiral models which exploit the hedgehog ansatz solution corresponding to the little group SU2 Z2 [25]. In the chiral perturbation theory [8] and in the 1=Nc expansion of QCD [14–16] to which the

hedgehog ansatz does not apply, one can thus see the coefficients different from those in (2.4) even though the

SU(3) flavor group is used in the theory. Now, it seems appropriate to comment on the 1=N_c expansion [25–28]. In the above relations (2.4), the inertia parametersN , N0, P , Q, and R are of order Nc while M is of order Nc
1.

However, since the inertia parameter M is multiplied by an explicit factor Ncin (2.2), the terms withM are of order

N0

c. (For details of further 1=Nc, see [25,28].)

In the SU(3) flavor symmetric limit with the chiral symmetry breaking masses mu md ms, mK m

and decay constants f_K f_, the magnetic moments of the decuplet baryons are simply given by [29]

B  QEM
₁ 16M  1 4
N
1 2p N3 0_{; (2.5)} where Q_EM is the EM charge. Here one notes that in the chiral model in the adjoint representation the prediction of the baryon magnetic moments with the chiral symmetry is the same as that with the SU(3) flavor symmetry since the mass-dependent term inLCSB andLFSB do not yield any contribution to J_FSB
so that there are no terms withP , Q, andR in (2.4). Because of the degenerate d- and s-flavor charges in the SU(3) EM charge operator ^QEM, the chiral model possesses the U-spin symmetry relations in the baryon decuplet magnetic moments, similar to those in the octet baryons [30]


_

_

_
;

0
_0 
_0;

 
_; (2.6)

which are a subset of the stronger symmetry relations (2.5).

Next, since the SU(3) FSB quark masses do not affect the magnetic moments of the baryon decuplet in the adjoint representation of the chiral model, in the more general SU(3) flavor symmetry broken case with m_u m_d
m_s, m_
m_K and f_
f_K, the decuplet baryon magnetic moments with all the inertia parameters satisfy the follow-ing symmetric sum rules:

_0  1 2
 1 2

;




 

0

; X B2decuplet
B 0: (2.7)

Now, in order to predict baryon decuplet magnetic mo-ments we proceed to derive sum rules for the baryon magnetic moments in terms of the experimentally known baryon magnetic moments. To do this, we first consider the baryon octet magnetic moments of the form

p 1 10M  4 15
N 1 2N 0_ 8 45P
2 45Q
1 45R;
_n 1 20M
1 5
N 1 2N 0
1 9P  7 90Q  1 45R;
_ 1 10M  4 15
N 1 2N 0_13 45P
1 45Q
2 45R;
_0 
1 40M  1 10
N 1 2N 0_11 90P  1 36Q  1 45R;


3 20M
1 15
N 1 2N 0
2 45P  7 90Q  4 45R;
0  1 20M
1 5
N 1 2N 0
11 45P
1 45Q  1 45R;
_

3 20M
1 15
N 1 2N 0
4 45P
2 45Q
4 45R;
_  1 40M
1 10
N 1 2N 0
1 10P
1 20Q: (2.8)

Since we have effectively five inertia parameters M, N 1 2N

0_,

P , Q, and R, we can derive sum rules for three magnetic moments
_0,
_, and
_
in terms of five experimentally known octet magnetic moments,
_p,
_n,
_,
_
, and
_0 as follows:
_0  1 2
 1 2

;


1 3
n
8 3

5 3


7 3
0;
_

_p
2 3
n 7 6
 1 6

 4 3
0: (2.9)

Using the above sum rules we can predict the magnetic moments as in Table I. One notes that the value of
 is comparable to the experimental data
exp_ 
0:61, while the value of

is not so comparable to
exp



0:65. Similarly we can derive sum rules for the decuplet magnetic moments in terms of six experimentally known magnetic moments,
_p,
_n,
_,
_
,

0, and

 to arrive at

  5 28
p 55 168
n 5 42
 25 84


5 168
0 1 2

;

0  5 14
p 55 84
n 5 21
 25 42


5 84
0;


 15 28
p 55 56
n 5 14
 25 28


5 56
0
1 2

;
 
25 14
p
235 168
n 215 84
 65 84

 365 168
0 1 2

;
_0 
15 28
p
5 14
n 25 28
 5 14

 5 7
0;

 5 7
p 115 168
n
65 84

5 84


125 168
0
1 2

;
_0 
10 7
p
115 84
n 65 42
 5 42

 125 84
0;

 25 28
p 65 168
n
40 21

85 84


235 168
0
1 2

;

 15 14
p 5 56
n
85 28

55 28


115 56
0
1 2

; 1  5 p
p
  1  5 p
n
0 

1 189 17p2 140
17p3 210
17p6 420 
p
₁₇₃ 567  2 p 168
 3 p 252
 6 p 504 
n 
₃₃₈₃ 11 340
 2 p 420  3 p 630  6 p 1260 
_
₂₁₈₉ 11 340
13p2 420  13p3 630  13p6 1260 
_

₂₁₃₁ 11 340
73p2 840  73p3 1260 73p6 2520 
_0

1 5
 2 p 10  3 p 15  6 p 30 

; 1  5 p
_ 

83 630
17p2 140  17p3 210  17p6 420 
p

1 378
 2 p 168  3 p 252  6 p 504 
n 
₁₃₁ 756  2 p 420
 3 p 630
 6 p 1260 

₁₀₇ 756 13p2 420
13p3 630
13p6 1260 



589 3780 73p2 840
73p3 1260
73p6 2520 
_0
₁ 5  2 p 10
 3 p 15
 6 p 30 

; 1  5 p
_0_0 
₁₁ 140
17p2 280  17p3 420  17p6 840 
p
₉ 56
 2 p 336  3 p 504  6 p 1008 
n
₁₉ 140  2 p 840
 3 p 1260
 6 p 2520 
 
₃₇ 140 13p2 840
13p3 1260
13p6 2520 



13 280 73p2 1680
73p3 2520
73p6 5040 
_0 
₁ 10  2 p 20
 3 p 30
 6 p 60 

; 1  5 p


 13 45
p 35 108
n 53 540
 209 540

 17 270
0; 1  5 p
0_0 

667 945
17p2 140  17p3 210  17p6 420 
_p

451 1134
 2 p 168  3 p 252  6 p 504 
_n 
₂₇₅₉ 2268  2 p 420
 3 p 630
 6 p 1260 
_
₁₃₈₅ 2268 13p2 420
13p3 630
13p6 1260 
_

₈₀₃₃ 11 340 73p2 840
73p3 1260
73p6 2520 
_0
₁ 5  2 p 10
 3 p 15
 6 p 30 

;

1  5 p


 49 135
p 125 324
n
181 1620
 407 1620


109 810
0; 1  15 p
_0 
₂₁₅ 756 17p2 280
17p3 420
17p6 840 
_p
₁₃₉₉ 4536  2 p 336
 3 p 504
 6 p 1008 
_n 

3751 11 340
 2 p 840  3 p 1260  6 p 2520 
_

1093 11 340
13p2 840  13p3 1260 13p6 2520 
_


5779 22 680
73p2 1680 73p3 2520  73p6 5040 
_0

1 10
 2 p 20  3 p 30  6 p 60 

: (2.10)

Here we have used the additional magnetic moment

 since we have six inertia parametersM, N , N0_{,P , Q,} andR. We list the predictions for the magnetic moments of the decuplet and octet baryons in TableI, and those for the decuplet-to-octet transition magnetic moments in Table II, by using their sum rules (2.10). Here note that

our prediction of

is comparable to its experimental value
exp_

1:94. The predicted values of the decuplet-to-octet transition magnetic moments in Table II are also comparable to the previous ones obtained by using the chiral quark soliton model [17].

TABLE II. The transition magnetic moments and their flavor components for 10J3 1=2 !

8J3 1=2  . B8B10
B8B10
s B8B10
u B8B10
d B8B10 p

2:76
0:54
0:76
1:46 n
0
_{2:76
0:54
0:76
1:46}  2.24 0.13 2.58
0:47 0_0 _{1.01 0.13 1.76
0:88} 


0:22 0.13 0.94
1:29 0_0 _{2.46 0.19 2.72
0:45} 


0:27 0.19 0.90
1:36 0
_{2:46
0:54
0:56
1:36}

TABLE I. The magnetic moments, their flavor components, and strange form factors of the decuplet and octet baryons. The quantities used as input parameters are indicated by .

B
B
sB
u B
d B F s 2B
 4:52 0.07 4.10 0.35
0:21
 _{2.12 0.07 2.50
0:45
0:21}
0
_{0:29 0.07 0.89
1:25
0:21}


2:69 0.07
0:71
2:05
0:21  2.63
0:49 2.84 0.28 0.47 0 0.08
0:49 1.14
0:57 0.47 

2:48
0:49
0:57
1:42 0.47 0 0.44
1:15 1.38 0.21 1.45 

2:27
1:15
0:43
0:69 1.45 

2:06
1:92
0:29 0:15 2.76 p 2:79
0:25 2.94 0.10 0.75 n
1:91
_{0:25
0:19
1:47 0.75}  _{2:46
0:11 2.72
0:15
0:67} 0 _{0.65
0:11 1.52
0:76
0:67} 

1:16
_{0:11 0.31
1:36
0:67} 0
_{1:25
1:32 0.25
0:18 1.95} 

1:07
1:32 0.37
0:12 1.95 
0:51
0:88 0.75
0:38 1.64

III. STRANGE FLAVOR MAGNETIC MOMENTS OF BARYON DECUPLET

In the SU(3) flavor symmetry broken case we decompose the EM currents into three pieces J
_{ J}
u_{ J}
d_{ J}
s

where the q-flavor currents J
q J
q_CS  J
q_FSB are given by substituting the charge operator ^Qwith the q-flavor charge operator ^Qq J_CS
q  
_Q^ q B

i 2f 2 tr ^Qql
  i 8e2 tr ^Qq; l _{l
; l}_{  U $ U}y_ B  Nc 482
_{tr ^Q} qlll
U $ Uy B; J_FSB
q
i 12f 2 2
1 tr1
 3 p ₈U ^Q_ql
_{ l
Q}^ qUy  U $ Uy B (3.1)

to obtain the baryon magnetic moments and transition magnetic moments in the s-flavor channel

s

7 48M  1 12
N
1 2p N3 0_ 2 21P  5 168Q  1 84R;
s_ 
1 6M  1 126P
1 126Q  1 126R;
s_ 
3 16M
1 12
N
1 2p N3 0
2 21P
5 168Q
1 84R;
s_ 
5 24M
1 6
N
1 2p N3 0
3 14P
1 28Q
1 21R;
s_N 
7 60M  1 45
N 1 2N 0_ 1 45P  1 90Q;
s_ 
11 60M  1 15
N 1 2N 0_ 11 135P  1 54Q  2 135R;
s_ 
1 5M
4 45
N 1 2N 0
1 9P
1 45Q
1 45R;
s_ 
3 20M
1 15
N 1 2N 0
1 15P
1 30Q; 1  5 p
s_N

1 6p M;5 1  5 p
s_ 
1 6p M 5 2 45
N 2
 2 p 8 N 0_ 7 135P  2 135Q  1 135R; 1  5 p
s_ 
1 6p M 5 2 45
N 2
 2 p 8 N 0_ 1 18P  1 90Q  1 90R; 1  15 p
s 0 
1 6p15M: (3.2)

Similarly, in the u- and d-flavor channels of the adjoint representation we obtain the baryon magnetic moments and transition magnetic moments (A1) in the Appendix. Here one notes that in general all the baryon decuplet and octet magnetic moments fulfill the model independent relations in the u- and d-flavor components and the I-spin symmetry of the isomultiplets with the same strangeness in the s-flavor channel

d_B Qd Q_u
u  B ;
s B 
s  B ; (3.3)

with Bbeing the isospin conjugate baryon in the isomultiplets of the baryon.

As in the previous sum rules of the magnetic moments (2.9) and (2.10), we find the sum rules of the strange components of the magnetic moments and transition magnetic moments in terms of the six experimentally known magnetic moments:

s

13 84
p
115 504
n
83 126
 47 252


463 504
0 1 6

;
s_ 
29 42
p
50 63
n
23 126
 8 63


26 63
0;
s_ 
43 84
p
445 504
n
113 126

103 252


433 504
0
1 6

;
s_  8 21
p
125 252
n
353 126

179 126


569 252
0
1 3

;
s_N 
2 9
n
7 9

1 9


8 9
0;
s_ 
1 3
p
5 9
n
4 9
 2 9


8 9
0;
s_ 
1 3
p
2 3
n
5 3

2 3


4 3
0;
s_ 

p

n; 1  5 p
s_N

 5 p 15
p
 5 p 9
n
7p5 45

 5 p 45


8p5 45
0; 1  5 p
s_ 
₁₁ 210
17p2 420  17p3 630
 5 p 15 17p6 1260 
_p
₃ 28
 2 p 504  3 p 756
 5 p 9   6 p 1512 
_n 
₁₉ 210  2 p 1260
 3 p 1890
7p5 45
 6 p 3780 
_
₃₇ 210 13p2 1260
13p3 1890
 5 p 45
13p6 3780 
_


13 420 73p2 2520
73p3 3780
8p5 45
73p6 7560 
_0
₁ 15  2 p 30
 3 p 45
 6 p 90 

; 1  5 p
s_ 

4 35
17p2 420  17p3 630
 5 p 15 17p6 1260 
p

1 252
 2 p 504  3 p 756
 5 p 9   6 p 1512 
n 
₁₁₆ 315  2 p 1260
 3 p 1890
7p5 45
 6 p 3780 

₁₈₁ 630 13p2 1260
13p3 1890
 5 p 45
13p6 3780 


₂₄₁ 1260 73p2 2520
73p3 3780
8p5 45
73p6 7560 
_0
₁ 15  2 p 30
 3 p 45
 6 p 90 

; 1  15 p
s_0 
 15 p 45
p
 15 p 27
n
7p15 135

 15 p 135


8p15 135
0: (3.4)

Similarly, the sum rules of the u- and d-flavor components of the magnetic moments and transition magnetic moments are given by (A2) in the Appendix. Here one notes that the flavor components of the transition magnetic moments
p
 and

n
0 satisfy the identities
q_p


q

n
0; q  u; d; s; (3.5) to yield
_p
 

n
0 as in (2.10). The identities (3.5) are consistent with the previous ones in Refs. [14,16]. We list the predictions for the u-, d-, and s-flavor components of the decuplet and octet magnetic moments in TableI, and those for the decuplet-to-octet transition magnetic mo-ments in TableII, by using the sum rules (3.4) and (A2).

Next, the form factors of the decuplet baryons, with internal structure, are defined by the matrix elements of the EM currents hp  qjJ
_{jpi  up  q

F} 1Bq2  i 2mB 
_q_F 2Bq2  up; (3.6)

where up is the spinor of the baryons and q is the momentum transfer. Using the s-flavor charge operator in the EM currents as before, in the limit of zero momentum transfer, one can obtain the strange form factors of baryon decuplet and octet

F_1Bs0  S; Fs_2B0 
3
s_B
S (3.7) in terms of the strangeness quantum number of the baryon S 1
Y (Y, hypercharge) and the strange components of the baryon decuplet and octet magnetic moments
s_B . The predictions for the strange form factors of the decuplet

and octet baryons are listed in TableIby using the relation (3.7).

IV. CONCLUSIONS

In summary, we have derived sum rules for the baryon decuplet and octet magnetic moments and the decuplet-to-octet transition magnetic moments, in the chiral models with the SU(3) flavor group. These sum rules are explicitly constructed in terms of the six experimentally known baryon magnetic moments
p,
n,
,
_
,

0, and

 to yield the theoretical predictions for the remnant baryon magnetic moments. Especially in the case of using the experimental data for the six baryon magnetic moments as input data of the sum rules, the predicted value of

is comparable to its experimental datum.

Next, we have extended the above algorithm to flavor components and strange form factors of the baryon dec-uplet and octet magnetic moments to find their sum rules in terms of the six baryon magnetic moments. The sum rules for the decuplet-to-octet transition magnetic moments and their flavor components have also been obtained. It is also shown that all the baryon decuplet and octet magnetic moments fulfill the model independent relations in the u-and d-flavor components u-and the I-spin symmetry of the isomultiplets with the same strangeness in the s-flavor channel. Moreover, the s-flavor components of the decuplet-to-octet transition magnetic moments respect the I-spin symmetry of the isomultiplets with the same strangeness. However, some transition magnetic moments such as
u_p
,
u_n
0, and
u_0 do not satisfy the model

independent relations in the u- and d-flavor components. It is also interesting to see that
q

p
 and

q

n
0 are equal to each other in the q-flavor (q  u; d; s) channels, which are consistent with the previous results [14,16].

It would be desirable if the SU(3) representation mixing effects on the baryon magnetic moments [24] could be investigated by exploiting the multiquark structure associ-ated with the highly nontrivial nonlinear symmetry break-ing terms.

ACKNOWLEDGMENTS

The author would like to deeply thank R. D. McKeown for the warm hospitality at Kellogg Radiation Laboratory, Caltech where a part of this work has been done. He is also grateful to G. E. Brown, H. C. Kim, R. D. McKeown, D. P. Min, B. Y. Park, M. Ramsey-Musolf, M. Rho, and P. Wang for helpful discussions and encouragements. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund), Grant No. KRF-2006-331-C00071, and by the Korea Research Council of Fundamental Science and Technology (KRCF), Grant No. C-RESEARCH-2006-11-NIMS.

APPENDIX

In the u- and d-flavor channels of the adjoint represen-tation, the baryon magnetic moments and transition mag-netic moments are given by

u
 
2
d

 5 12M  1 3
N
1 2p N3 0_2 7P
1 28Q
1 21R;
u
 
2
d
0  3 8M  1 6
N
1 2p N3 0_10 63P  1 126Q
1 126R;
u
0 
2
d
  1 3M  2 63P  13 252Q  2 63R;
u


2
d
  7 24M
1 6
N
1 2p N3 0
2 21P  2 21Q  1 14R;
u_ 
2
d_
 3 8M  1 6
N
1 2p N3 0_ 19 126P
17 252Q
1 63R;
u_0 
2
d 0  1 3M  1 126P
1 126Q  1 126R;
u_

2
d   7 24M
1 6
N
1 2p N3 0
17 126P  13 252Q  2 63R;
u_0 
2
d 
 1 3M
1 63P
17 252Q
1 63R;
u_

2
d 0  7 24M
1 6
N
1 2p N3 0
11 63P  1 126Q
1 126R;
u_

2
d 
 7 24M
1 6
N
1 2p N3 0
3 14P
1 28Q
1 21R;

up 
2
dn  2 5M  8 45
N 1 2N 0_ 16 135P
4 135Q
2 135R;
un 
2
dp  11 30M
2 15
N 1 2N 0
2 27P  7 135Q  2 135R;
u_ 
2
d 
 2 5M  8 45
N 1 2N 0_ 26 135P
2 135Q
4 135R;
u_0 
2
d 0  19 60M  1 15
N 1 2N 0_ 11 135P  1 54Q  2 135R;
u_

2
d   7 30M
2 45
N 1 2N 0
4 135P  7 135Q  8 135R;
u 0 
2
d 
 11 30M
2 15
N 1 2N 0
22 135P
2 135Q  2 135R;
u_

2
d 0  7 30M
2 45
N 1 2N 0
8 135P
4 135Q
8 135R;
u_ 
2
d_  7 20M
1 15
N 1 2N 0
1 15P
1 30Q; 1  5 p
u_p
  1  5 p
u_n
0  1 3p M
5 4 45
N 2
 2 p 8 N 0
8 135P  7 270Q  79 4050R; 1  5 p
u_  
2  5 p
d_

 1 3p M 5 4 45
N 2
 2 p 8 N 0_ 13 135P  1 270Q
1 150R; 1  5 p
u_0_0 
2  5 p
d_0_0  1 3p M 5 2 45
N 2
 2 p 8 N 0_ 7 135P  2 135Q  1 135R; 1  5 p
u_


2  5 p
d_   1 3p M 5 1 135P  7 270Q  29 1350R; 1  5 p
u_0_0 
2  5 p
d_

 1 3p M 5 4 45
N 2
 2 p 8 N 0_ 14 135P
1 270Q  23 4050R; 1  5 p
u_


2  5 p
d_0_0  1 3p M 5 1 135P  7 270Q  67 4050R; 1  15 p
u_0  1 3p15M
2 45
N 2
 2 p 8 N 0
1 27P  2 135Q  11 2025R; 1  5 p
d_p
  1  5 p
d_n
0 
1 6p M
5 2 45
N 2
 2 p 8 N 0
4 135P  7 540Q  79 8100R; 1  15 p
d 0 
1 6p15M
1 45
N 2
 2 p 8 N 0
1 54P  1 135Q  11 4050R: (A1)

Next, we list the sum rules of the u- and d-flavor components of the magnetic moments and transition magnetic moments

u
  2 3
p 10 9
n 14 9
 2 9

 16 9
0 2 3

;
u
 11 14
p 335 252
n 103 63
 53 126

 443 252
0 1 3

;
u
0  19 21
p 65 42
n 12 7
 13 21

 73 42
0;
u

 43 42
p 445 252
n 113 63
 103 126

 433 252
0
1 3

;
u_ 
11 21
p 5 28
n 137 42
 31 42

 271 84
0 1 3

;
u_0 13 42
p 55 63
n 271 126
 29 63

 142 63
0;
u_
 8 7
p 395 252
n 131 126
 23 126

 323 252
0
1 3

;
u_0
2 7
p 25 126
n 163 63
 19 63

 349 126
0;
u_
 53 42
p 115 84
n 2 7

19 42

 71 84
0
1 3

;
u_
 29 21
p 295 252
n
59 126

137 126

 103 252
0
1 3

;
up  4 3
p 10 9
n 14 9
 2 9

 16 9
0;
un  2 3
p 16 9
n 14 9
 2 9

 16 9
0;
u_ 2 3
p 10 9
n 20 9
 2 9

 16 9
0;
u 0  2 3
p 10 9
n 17 9
 5 9

 16 9
0;
u_
 2 3
p 10 9
n 14 9
 8 9

 16 9
0;
u 0  2 3
p 10 9
n 14 9
 2 9

 22 9
0;
u_
 2 3
p 8 9
n
2 9

8 9

 2 9
0;
u_ 2 3
n 7 3
 1 3

 8 3
0; 1  5 p
u_p
 1  5 p
u_n
0 

2 567 17p2 210
17p3 315  2p5 15
17p6 630 
p
₃₄₆ 1701  2 p 252
 3 p 378 2p5 9
 6 p 756 
n 
₃₃₈₃ 17 010
 2 p 630  3 p 945 14p5 45   6 p 1890 
_
₂₁₈₉ 17 010
13p2 630  13p3 945  2p5 45  13p6 1890 
_

₂₁₃₁ 17 010
73p2 1260 73p3 1890  16p5 45  73p6 3780 
_0

2 15
 2 p 15 2p3 45   6 p 45 

;

1  5 p
u__ 

83 945
17p2 210  17p3 315  2p5 15  17p6 630 
p

1 567
 2 p 252  3 p 378 2p5 9   6 p 756 
n 
₁₃₁ 1134  2 p 630
 3 p 945 14p5 45
 6 p 1890 

₁₀₇ 1134 13p2 630
13p3 945  2p5 45
13p6 1890 



589 5670 73p2 1260
73p3 1890 16p5 45
73p6 3780 
_0
₂ 15  2 p 15
2p3 45
 6 p 45 

; 1  5 p
u_0_0
₁₁ 210
17p2 420  17p3 630  2p5 15  17p6 1260 
_p
₃ 28
 2 p 504  3 p 756 2p5 9   6 p 1512 
_n 
₁₉ 210  2 p 1260
 3 p 1890 14p5 45
 6 p 3780 
_
₃₇ 210 13p2 1260
13p3 1890  2p5 45
13p6 3780 
_


13 420 73p2 2520
73p3 3780 16p5 45
73p6 7560 
_0
₁ 15  2 p 30
 3 p 45
 6 p 90 

; 1  5 p
u_


₂₆ 135 2p5 15 
p
₃₅ 162 2p5 9 
n
₅₃ 810 14p5 45 

₂₀₉ 810 2p5 45 


₁₇ 405 16p5 45 
_0; 1  5 p
u_0_0

1334 2835
17p2 210  17p3 315  2p5 15  17p6 630 
p

451 1701
 2 p 252  3 p 378 2p5 9   6 p 756 
n 
₂₇₅₉ 3402  2 p 630
 3 p 945 14p5 45
 6 p 1890 

₁₃₈₅ 3402 13p2 630
13p3 945  2p5 45
13p6 1890 


₈₀₃₃ 17 010 73p2 1260
73p3 1890  16p5 45
73p6 3780 
0
₂ 15  2 p 15
2p3 45
 6 p 45 

; 1  5 p
u_


₉₈ 405 2p5 15 
_p
₁₂₅ 486 2p5 9 
_n

181 2430 14p5 45 
_
₄₀₇ 2430 2p5 45 
_


109 1215 16p5 45 
_0; 1  15 p
u_0
₂₁₅ 1134 17p2 420
17p3 630
17p6 1260 2p15 45 
p
₁₃₉₉ 6804  2 p 504
 3 p 756
 6 p 1512 2p15 27 
n 

3751 17 010
 2 p 1260  3 p 1890  6 p 3780 14p15 135 


1093 17 010
13p2 1260  13p3 1890  13p6 3780 2p15 135 



5779 34 020
73p2 2520  73p3 3780  73p6 7560 16p15 135 
0

1 15
 2 p 30  3 p 45  6 p 90 

; 1  5 p
d_p
 1  5 p
d_n
0 

1 567 17p2 420
17p3 630
 5 p 15
17p6 1260 
_p
₁₇₃ 1701  2 p 504
 3 p 756
 5 p 9
 6 p 1512 
_n 
₃₃₈₃ 34 020
 2 p 1260  3 p 1890
7p5 45   6 p 3780 
_
₂₁₈₉ 34 020
13p2 1260  13p3 1890
 5 p 45 13p6 3780 


₂₁₃₁ 34 020
73p2 2520 73p3 3780
8p5 45  73p6 7560 
_0

1 15
 2 p 30  3 p 45  6 p 90 

;

1  15 p
d_0 
₂₁₅ 2268 17p2 840
17p3 1260
17p6 2520
 15 p 45 
p
₁₃₉₉ 13 608  2 p 1008
 3 p 1512
 6 p 3024
 15 p 27 
n 

3751 34 020
 2 p 2520  3 p 3780  6 p 7560
7p15 135 


1093 34 020
13p2 2520  13p3 3780 13p6 7560
 15 p 135 



5779 68 040
73p2 5040 73p3 7560 73p6 15120
8p15 135 
_0

1 30
 2 p 60  3 p 90  6 p 180 

: (A2)

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