Longitudinal Transitions Of Baryon Resonances in Constituent Quark Model

Abstract

Department of Physics, Beijing University, Beijing 100871, People's Republic of China

CCAST(World Laboratory), PO Box 8730, Beijing 100080, People's Republic of China
§
Institute of High Energy Physics, The Chinese Academy of Sciences, PO Box 918 Beijing, 100039, People's Republic of China

Full text

arXiv:hep-ph/9409431v1 28 Sep 1994
Longitudinal Transitions Of Baryon
Resonance s
In Constituent Quark Model
Zhenping Li
1
, Yubing Dong
2
and Weihsing Ma
2
1
Physics Department, Car negie-Mellon University
Pittsburgh, PA. 15213-3890
2
CCAST(World Laboratory), P. O. Box 8730, Beijing 100080
and
Institute of High Energy Physics, Academia Sinica
Beijing 100039, P.R. China
February 1, 2008
Abstract
A longitudinal transition operator that satisfies the gauge invariance
requirement is introduced in constituent quark model. The correspond-
ing longitudinal transitions between the nucleon and baryon resonances
are calculated. We show that the study of the longitudinal coupling
plays an important role in understanding the structure of baryons.
PACS numbers: 13.40.-f, 14.20.Gk, 12.40Qq, 13.40Hq
1

1. Introduction
The electromagnetic transition between the nucleon and excited baryons has
been shown to be a very important probe to the structure of nucleon and
baryon resonances. A significant progress has been made since the theoretical
investigations by Copley, Karl and Obryk[1], and Feynman, Kisslinger and
Ravndal[2], who presented first evidence of underlying SU(6) ⊗ O(3) sym-
metry of baryon spectrum. Recent studies have shown[3 ] that the relativistic
effects are required in order to give a consistent description of baryon spectrum
and its transitional properties, moreover, they are also essential to generate
the model independent results in the low energy limit, such as the low energy
theorem in the Compton scattering and corresponding Drell-Hearn-Gerasimov
sum rule[4]. The calculations including the relativistic effects in more realis-
tic potential quark models, such as the Isgur-Karl[5] model and its relativised
version[6], have shown that the successes of the nonrelativistic quark model
have been preserved, thus both spectroscopy and transitions of baryon reso-
nances can be described in the same framework.
However, these studies have mostly concentrated on the transverse transi-
tion amplitudes A
1/2
and A
3/2
, a nd there is an additional longitudinal transi-
tion amplitude S
1/2
in the electroproduction that has not been systematically
studied. Although attempts[7] have been made to investigate the longitudi-
nal transition, there is an important theoretical issue which was not treated
consistently in these investigations; the usual expression for the longitudinal
transition operator in a quark system
H
L
em
= ǫ
0
J
0
− ǫ
3
J
3
, (1)
where
J
0
=
X
j
s
2π
k
0
e
j
e
ik·r
j
(2)
and
J
3
=
X
j
s
2π
k
0
e
j
1
2m
q
h
p
3,j
e
ik·r
j
+ e
ik·r
j
p
3,j
i
(3)
and quark j at position r
j
has mass and charge m
j
and e
j
, does not satisfy
the gauge–invariance constraint
k
µ
J
µ
= 0, (4)
where k
µ
= {k
0
, 0, 0, k
3
} is the momentum of the photon. This was noticed
some time ago[10], and was emphasized by Bourdeau and Mukhopadhyay[8]
2

in their study of the transition γ
v
N → ∆ in the Isgur–Karl[5] and the Vent–
Baym–Jackson models[9]. One solution[10] to this problem is to add an ad hoc
current
J
′
3
= −
k
3
J
3
− k
0
J
0
|k
3
|
2
k
3
(5)
to Eq. 3, which was used in the calculation of the longitudinal transitions
between the nucleon and baryon r esonances[7 ].
The focus of this paper is t o derive a longitudinal transition operator that
satisfies the gauge invariant condition, and use this transition operator to study
the longitudinal transitions between the nucleon and baryon resonances. In
Ref. [3], we shown that the current conservation in the nonrelativistic limit
is equivalent to the energy conservation with the nonrelativistic kinematics.
Thus, in addition to the truncated model space problem discussed in Ref. [8],
the current conservation will break down due to the nonrelativistic treatment
of the recoil effects. In next section, we will show that the relativistic elec-
tromagnetic current does satisfy the gauge invariant condition, assuming that
the wavefunction is the eigenstate of the relativistic Hamiltonian. Thus, the
problem could be avoided by imposing the current conservation in the rela-
tivistic limit and then extracting t he appropriate nonrelativistic expression,
the resulting longitudinal transition operator will be gauge invariant.
In section 3, we calculate the longitudinal transition amplitudes S
1/2
using
the new transition operator derived in section 2. We find significant differences
between our results with those in R ef. [7], in which the current in Eq. 5
was added to Eq. 1. We present our results in terms o f the ratio between
the longitudinal and the transverse cross sections, which would be easier to
compare with the experiment al data. Finally, the conclusions will be given in
section 4.
2. A gauge invariant longitudinal transition operator
The H
L
em
in Eq. 1 follows from a nonrelativistic approximation to the longitu-
dinal quark–photon vertex,
H
rel
em
= ǫ
0
J
rel
0
− ǫ
3
J
rel
3
, (6)
where
J
rel
0
=
s
2π
k
0
3
X
j=1
e
j
e
ik·r
j
(7)
and
J
rel
3
=
s
2π
k
0
3
X
j=1
e
j
α
3,j
e
ik·r
j
, (8)
3

and α
3,j
are Dirac matrices. We can rewrite t he current J
3
as
J
rel
3
=
s
4π
2k
0


3
X
j=1
α
j
· p
j
,
3
X
j=1
e
j
e
ik·r
j
1
k
3


=
s
4π
2k
0


H
b
,
3
X
j=1
e
j
e
ik·r
j
1
k
3


, (9)
where the Hamiltonian[11] in Eq. 2-11 for a three-body system is
H =
3
X
i=1
{α
i
· p
i
+ β
i
m
i
} +
X
i<j

V
v
(r)(1 −
1
2
α
i
· α
j
)
+
1
2
α
i
· r α
j
· r
V
′
v
(r)
|r|
+ β
i
β
j
V
s
(r}, (10)
where r = r
i
− r
j
, V
′
v
=
dV
v
(r)
dr
, and V
v
(r) and V
s
(r) denote vector and scalar
binding potentia ls for the quark system. Typically, V
s
(r) could be a long range
scalar linear potential and V
v
(r) a single-gluon exchang e potential. Thus
hΨ
rel
f
|J
rel
3
|Ψ
rel
i
i = hΨ
rel
f
|


H
b
,
3
X
j=1
e
j
e
ik·r
j
1
k
3


|Ψ
rel
i
i
= (E
rel
f
− E
rel
i
)hΨ
rel
f
|
3
X
j=1
e
j
e
ik·r
j
|Ψ
rel
i
i
1
k
3
=
(E
rel
f
− E
rel
i
)
k
3
hΨ
rel
f
|J
rel
0
|Ψ
rel
i
i, (11)
where the initial- and final-state wavefunctions |Ψ
rel
i
i and |Ψ
rel
f
i must be eigen-
functions of the Hamiltonian H
b
. In a radiative transition the energy difference
between initial and final states equals the photon energy, that is
E
rel
f
− E
rel
i
= k
0
. (12)
Note that Eq. 12 is exact in relativistic limit, so we have the gauge invariance
constraint
k
µ
J
rel
µ
= k
0
J
0
− k
3
J
3
= 0. (13)
Since the currents J
rel
0
and J
rel
3
have different transformation to the nonrela-
tivistic limit, in par ticular the nonrelativistic kinematics is used in Eq. 1 2, the
current are no longer conserved in t he nonrelativistic limit. Moreover, Eq. 11
shows that the binding potential plays an important role in the current J
rel
3
,
4

thus the truncated model space will further destroy the current conservation,
which has b een discussed in detail in Ref. [8].
This problem could be avoid if we take a different approach to transform
H
rel
em
in Eq. 6 into the nonrelativistic limit; since the current conservation is
exact in the relativistic limit, we substitute Eq. 13 into Eq. 6;
H
rel
em
=
"
ǫ
0
− ǫ
3
k
0
k
3
#
J
rel
0
, (14)
and we chose the longitudinal polarization vector ǫ
L
µ
ǫ
L
µ
= {ǫ
0
, 0, 0, ǫ
z
} =
(
k
3
√
Q
2
, 0, 0,
k
0
√
Q
2
)
, (15)
so the gauge invariant condition,
k
µ
· ǫ
µ
= 0, (16)
is satisfied (Q is the virtual photon mass). Note also that
ǫ
0
−
ǫ
z
k
0
k
z
=
√
Q
2
k
z
, (17)
which leads to
hΨ
rel
f
|H
rel
e,m
|Ψ
rel
i
i =
X
j
√
Q
2
k
3
hΨ
rel
f
|J
rel
0
|Ψ
rel
i
i. (18)
Of course, the longitudinal electromagnetic interaction should be proportional
to
√
Q
2
, and vanishes in the real photon limit Q
2
= 0. This is a direct
consequence of the gauge invariance.
The nonrelativistic expansion of Eq. 18 has been given in Ref. [12 ];
J
0
=
s
2π
k
0

X
j
e
j
+
ie
j
4m
2
j
k · (σ
j
× p
j
)
!
e
ik·r
j
−
X
j<l
i
4M
T
σ
j
m
j
−
σ
l
m
l
!
·

e
j
k × p
l
e
ik·r
j
− e
l
k × p
j
e
ik·r
l


, (19)
where the first term is the charge operator, which is conventionally used in the
calculation of longitudinal helicity amplitudes; the second a nd third terms a r e
spin–orbit and nonadditive terms which have counterparts in the transverse
electromagnetic transition[3]. The spin–orbit and nonadditive terms represent
5

O(v
2
/c
2
) relativistic corrections to t he first term, which have long been known
to be necessary even for systems of free particles, if low-energy theorem and
Drell-Hearn-Gerasimov sum rule are to be satisfied[13] for the real photon case.
The longitudinal helicity amplitude S
1
2
is defined by
S
1
2
= hΨ
f
|J
0
|Ψ
i
i (20)
where J
0
is given by Eq. 19. The group structure of J
0
is
J
0
= AI + B(S
+
L
−
− S
−
L
+
), (21)
where I is the identity operator, A and B are the coefficients determined by Eq.
19. The second term corresponds t o the spin-orbit and nonadditive terms, and
require that the spin and orbital angular momentum z-component change by
±1 unit in a transition. Thus, if there is no orbital angular momentum in the
initial and final state wavefunctions, the contribution from the second term
vanishes. In particular, the selection rule[14] that t he longitudinal helicity
amplitudes vanish for the transition between the nucleon and hybrid states
survives these relativistic corrections if the quark spatial wavefunction of a
hybrid state is essentially the same as that of the nucleon and does not have
orbital angular momentum.
It should be noted that the expression for H
em
may not be unique in the
nonrelativistic limit; for example, H
em
can also be written as
H
em
=
√
Q
2
k
0
J
3
(22)
due to the current conservation in the relativistic limit. The nonrelativis-
tic expression of J
3
, however, is much more complicated due to the explicit
presence of the binding potential shown in Eq. 11, and the problem of the
truncated model space becomes important. Moreover, the recoil effects explic-
itly depend on the choice of the frame, which is also a source of the theoretical
uncertainty. This is why Eq. 20 is more convenient and simpler to use, as the
explicit dependence o f the recoil effects on the choice of frame is eliminated.
3. The Longitudinal Coupling of Baryon Resonances
In Table 1, we show the ana lytical expressions of the longitudinal transition
between the nucleon and the baryon resonances in the SU(6)⊗O(3) symmetry
limit. The terms proportional to
α
2
m
2
q
represent the relativistic contributions
that come fro m the spin-orbit and nonaddtive term in Eq. 19. The relativistic
6

effects only scale the longitudinal coupling amplitudes, and do not affect the
general behaviour of Q
2
dependence of S
1/2
(Q
2
). Therefore, the ratio between
the longitudinal couplings of the resonances S
11
(1530) and D
13
(1520) would
be independent of Q
2
since their masses are approximately equal. This ratio
is determined by the Clebsch-Gordon coefficients in the nonrelativistic limit;
S
1/2
(S
11
(1530))
S
1/2
(D
13
(1520))
= −
1
√
2
, (23)
and the relativistic effects would change this ratio by a factor of
1+
α
2
6m
2
q
1−
α
2
12m
2
q
. The
standard quark model parameters m
q
= 0.33 GeV and α
2
= 0.17 GeV
2
give
1 +
α
2
6m
2
q
1 −
α
2
12m
2
q
= 1.45, (24)
thus, this give an approximately −1 ratio with the relativistic corrections. The
relativistic effects also lead to a nonzero longitudinal transitions between the
resonance D
15
(1670), which vanishes for the nonrelativistic transition operator.
This gives us an important experimental test for the spin-orbit and nonadditive
term in the lo ngitudinal transition operator.
The calculation of the Q
2
dependence of S
1/2
(Q
2
) f ollows the procedure
of Foster and Hughes[15]; a Lorentz boost factor in the spatial integrals are
introduced so that
R(k) →
1
γ
2
R
k
γ
!
, (25)
where the Lorentz boost factor γ can be written as
γ
2
= 1 +
k
2
(M
r
+ M
p
)
2
(26)
in the equal velocity frame and
k
2
(EV F ) =
(M
2
r
− M
2
p
)
2
2M
r
M
p
+
Q
2
(M
2
r
+ M
2
p
)
4M
r
M
p
(27)
for the initial nucleon mass M
p
and final resonance mass M
r
. The correspond-
ing Q
2
dependence of S
1/2
(Q
2
) for S
11
(1530) is given by
S
1/2
(Q
2
) =
2
3
s
π
k
0
µm
q
k
γ
3
α
1 +
α
2
6m
2
q
!
1
1 + Q
2
/0.8
e
−
k
2
6α
2
γ
2
, (28)
7

where an ad hoc form factor
1
1+Q
2
/0.8
is being a dded, since it gives a better quan-
titative description of the Q
2
dependence of transverse helicity a mplitudes[15]
for α
2
= 0.17 GeV
2
, and it becomes unnecessary with α
2
= 0.09 GeV
2
[3].
In Fig.1, we show the Q
2
dependence of the longitudinal amplitude S
1/2
(Q
2
)
for the resonance S
11
(1530) in the SU(6) ⊗ O(3) symmetry limit. The rela-
tivistic effects increase S
1/2
(Q
2
) by about 25 percent. The resulting S
1/2
(Q
2
)
is significantly larger than that in Ref. [7], and in better agreement with the
analysis by Gerhardt[16], who extracted the long itudinal transition amplitudes
from the electroproduction data. This shows the importance of choosing the
correct transition operator for the longitudinal coupling.
A more important quantity is the ratio between the longitudinal coupling
and transverse coupling amplitudes,
R =
S
2
1/2
(Q
2
)
A
2
1/2
(Q
2
) + A
2
3/2
(Q
2
)
, (29)
in which the common factors, such as the ad hoc form factor in Eq. 28,
may cancel out, thus provides us a direct probe of the underlying structure
of the resonance. The analytic expressions f or tr ansverse helicity amplitudes
A
1/2
and A
3/2
are given in Ref. [3]. The Q
2
dependence of this ratio f or
the resonance S
11
(1530) is shown in Fig. 2, it shows a strong presence of
the longitudinal transitions for this resonance. Moreover, the result for the
transverse helicity amplitude A
1/2
(Q
2
)[3] in t he symmetry limit is twice larger
than the experimental data and it decreases too fast as Q
2
increases, this
indicates a strong configuration mixing for the resonance S
11
(1530). The data
of the Q
2
dependence of this ratio provide another crucial test to the various
potential quark models with different binding potentials. The extension of t his
investigation to include the configuration mixing is in progress. It is interesting
to note that the configuration mixings in the Isgur-Karl model[3] do not change
the result of naive quark model significantly due to the strong presence of the
70 multiplet state in the nucleon wavefunction.
It is straightforward to obtain the Q
2
dependence of the longitudinal tran-
sition for the resonance D
13
(1520), which Eq. 24 shows that its longitudinal
transition approximately equals to that of the resonance S
11
(1520) with a op-
posite sign. Thus, strong contributions from the longitudinal tra nsitions are
expected for the P-wave resonances, and furthermore, the r elativistic effects
contributes significantly to S
1/2
(Q
2
) because of the nonzero orbital angular
momentum in the wavefunction of P-wave resonances.
In Fig. 3, we show the Q
2
dependence of S
1/2
(Q
2
) for the resonance
F
16
(1688), the relativistic effects reduces the longitudinal transitions signif-
icantly, which are in better agreement with the result of Gerhardt[16]. The Q
2
8

dependence of the longitudinal and transverse transitions for this resonance
is shown in Fig . 4; the longitudinal transitions are much smaller in this case,
in particular, it is less t han 0.1 with the relativistic corrections. The naive
quark model[1, 17] does give a good description of the transverse helicity am-
plitudes even quantitatively. More precise data for the longitudinal transition
would provide us more insights into the structure of this resonance; deviations
from the prediction in Figs. 3 and 4 would be evidences for the configuration
mixings.
The longitudinal transition between the resonance P
33
(1232) and the nu-
cleon vanishes in t he symmetry limit. However, if there is a small component
of the orbital angular momentum in both wavefunctions of the nucleon and
the resonance P
33
(1232), the spin-orbit and the nonadditive term in the lon-
gitudinal transition op erator would lead to a nonzero longitudinal transition
between these two states. In Fig. 5, we show the ratio
R = −
S
1/2
(Q
2
)
√
2M1
=
√
2S
1/2
(Q
2
)
√
3A
3/2
(Q
2
) + A
1/2
(Q
2
)
(30)
(notice S
1/2
(Q
2
) in Eq. 20 differs by a factor of −
1
√
2
from Ref. [8]) for
the Isgur-Karl model, whose wavefunctions for the nucleon and the resonance
P
33
(1232) are obtained f r om Ref. [18], and the transverse helicity amplitudes
A
3/2
(Q
2
) and A
1/2
(Q
2
) are given in Ref. [3]. We find that the relativistic
corrections approximately double this ratio. While the experimental data[19]
are inconclusive for a finite ratio E1/M1, they do suggest a finite and negative
ratio S
1/2
/M1. This suggests that the longitudinal transitions between the
nucleon and the resonance P
33
(1232) might b e a better probe to the orbital
angular moment um in the nucleon wavefunction than the E1 transition, which
certainly deserves more att ention.
4. Conclusions
We have derived a gauge independent long itudinal transitions operator, in
which the relativistic effects are included. The calculations of longitudinal
transitions between the nucleon and baryon resonances are made in the sym-
metry limit. We show that the relativistic effects in the transition operator
give important contributions to the longitudinal helicity amplitudes, especially
in the transition b etween the nucleon and the resonance P
33
(1232).
We show that the longitudinal transitions of baryon resonances play an
important role in understanding the structure of baryons. The longitudinal
transition amplitude S
1/2
(Q
2
) decreases as the Q
2
increases. Thus, it is im-
portant in the small Q
2
regions, in particular, f or the transitions between the
9

nucleon and the P-wave resonance, which is accesible to the experiments at
CEBAF.
An extension of t his investigation is to study the spin structure function
g
1,2
(x, Q
2
) in the resonance region, where the studies[20] have shown that the
Q
2
dependence of the spin-structure function is very significant, and the lon-
gitudinal tra nsitions amplitudes provide important contributions to the spin-
structure functions in the low Q
2
region.
This work was supported in part by the United Sta t es National Science
Foundation grant PHY-9023586.
References
[1] L.A. Coplay, G. Karl and E. Obryk, Nucl. Phys. B13, 303(1969).
[2] R. Feynamn, M. Kisslinger and F. Ravndal, Phys. Rev. D3, 2706(1971).
[3] F.E.Close and Zhenping Li, Phys. Rev. D42, 2194 (1990), ibid D42, 2207
(1990).
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V. Vento and J. Navarro, Phys. Lett. 141B, 28 (1984).
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[12] F. E. Close and Zhenping Li, Phys. Lett. B289, 143(1992).
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2127 (1970).
10

[14] Zhenping Li, V. Burkert and Zhujun Li, Phys. Rev. D46, 7 0(1992).
[15] F. Foster and G. Hughes, Z. Phys. C41, 123(1982).
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[19] N. C. Mukhopadhyay, Excited Baryon 1988, in proceedings of the Topi-
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11

Figure Caption
1. The Q
2
dependence of S
1/2
(Q
2
) for the resonance S
11
(1535), the
solid line represents the nonrelativistic result and the dashed line
includes the relativisitc corrections.
2. The ratio between the logitudinal and transverse cross sections for
the resonance S
11
(1535). The solid line represents the nonrelativis-
tic result, and the dashed line includes the relativistic corrections.
3. The same as Fig. 1 for the resonance F
15
(1688).
4. The same as Fig. 2 for the resonance F
15
(1688).
5. The ratio between the longitudinal and M1 transitions for the res-
onance P
33
(1232) in the Isgur-Karl model, see text.
12

Table 1: Transition matrix elements between the nucleon and baryon reso-
nances in the SU(6) ⊗ O(3) symmetry limit. The full matrix elements are
obtained by multiplying the entries in this table by a factor
q
2π
k
0
2µm
q
e
−
k
2
6α
2
,
and S
n
1
2
= S
p
1
2
for ∆ states.
Multiplet States Proton Neutron
[70, 1
−
]
1
N(
2
P
M
)
1
2
−1
1
3
√
2
|k|
α

1 +
α
2
6m
2
q

−
1
3
√
2
|k|
α

1 +
α
2
6m
2
q

N(
2
P
M
)
3
2
−1
−
1
3
|k|
α

1 −
α
2
12m
2
q

1
3
|k|
α

1 −
α
2
12m
2
q

N(
4
P
M
)
1
2
−1
1
36
√
2
α|k|
m
2
q
−
1
108
√
2
α|k|
m
2
q
N(
4
P
M
)
3
2
−1
1
9
√
10
α|k|
m
2
q
−
5
27
√
10
α|k|
m
2
q
N(
4
P
M
)
5
2
−1
1
12
√
10
α|k|
m
2
q
−
5
36
√
10
α|k|
m
2
q
∆(
2
P
M
)
1
2
−1
−
1
3
√
2
|k|
α

1 −
α
2
6m
2
q

∆(
2
P
M
)
3
2
−1
1
3
|k|
α

1 +
α
2
12m
2
q

[56, 0
+
]
2
N(
2
S
S
′
)
1
2
+
−
1
3
√
6
k
2
α
2
0
∆(
4
S
S
′
)
3
2
+
0
[56, 2
+
]
2
N(
2
D
S
)
3
2
+
−
1
3
√
15
k
2
α
2

1 +
α
2
2m
2
q

−
k
2
12
√
15m
2
q
N(
2
D
S
)
5
2
+
−
1
3
√
10
k
2
α
2

1 −
α
2
3m
2
q

k
2
9
√
10m
2
q
∆(
4
D
S
)
1
2
+
−
5k
2
72
√
15m
2
q
∆(
4
D
S
)
3
2
+
0
∆(
4
D
S
)
5
2
+
5
√
5k
2
216
√
7m
2
q
∆(
4
D
S
)
7
2
+
5k
2
36
√
105m
2
q
[70, 0
+
]
2
N(
2
S
M
′
)
1
2
+
1
18
k
2
α
2
−
1
18
k
2
α
2
13