Analysis of Ω∗
(css) and Ω∗
(bss) with QCD sum rules
Zhi-Gang Wang 1
Department of Physics, North China Electric Power University, Baoding 071003,
P. R. China
Abstract
In this article, we calculate the masses and residues of the heavy baryons
Ω∗c(css) and Ω
(bss) with spin-parity 3
with the QCD sum rules. The nu-
merical values are compatible with experimental data and other theoretical
estimations.
PACS number: 14.20.Lq, 14.20.Mr
Key words: Ω∗c(css), Ω
b(bss), QCD sum rules
1 Introduction
Several new excited charmed baryon states have been observed by BaBar, Belle and
CLEO Collaborations, such as Λc(2765)
+, Λ+c (2880), Λ
c (2940), Σ
c (2800), Ξ
c (2980),
Ξ+c (3077), Ξ
c(2980) , Ξ
c(3077) [1, 2]. The charmed baryons provide a rich source
of states, including possible candidates for the orbital excitations. They serve as an
excellent ground for testing predictions of the constituent quark models and heavy
quark symmetry [3]. The charmed and bottomed baryons, which contain a heavy
quark and two light quarks, provides an ideal tool for studying dynamics of the light
quarks in the presence of a heavy quark. The u, d and s quarks form an SU(3) flavor
triplet, 3× 3 = 3̄ + 6, two light quarks can form diquarks with a symmetric sextet
and an antisymmetric antitriplet. For the S-wave baryons, the sextet contains both
spin-1
and spin-3
states, while the antitriplet contains only spin-1
states. By now,
the 1
antitriplet states (Λ+c , Ξ
c), and the
and 3
sextet states (Ωc,Σc,Ξ
and (Ω∗c ,Σ
c ) have been established.
The baryon Ω∗c , a css candidate for the
partner of the strange baryon Ω(sss),
was observed by BaBar collaboration in the radiative decay Ω∗c → Ωcγ [4]. The
baryon Ωc(css) was reconstructed in decays to the final states Ω
−π+, Ω−π+π0,
Ω−π+π−π+ and Ξ−K−π+π+. It lies about 70.8±1.0±1.1MeV above the Ωc, and it is
the last singly-charmed baryon with zero orbital momentum observed experimentally
In this article, we calculate the mass and residue of the Ω∗c (and Ω
b as byproduct,
the Ω∗b has not been observed experimentally yet) with the QCD sum rules [6, 7]. In
the QCD sum rules, operator product expansion is used to expand the time-ordered
currents into a series of quark and gluon condensates which parameterize the long
distance properties of the QCD vacuum. Based on current-hadron duality, we can
1E-mail,wangzgyiti@yahoo.com.cn;wangzg@yahoo.cn.
http://arxiv.org/abs/0704.1106v3
obtain copious information about the hadronic parameters at the phenomenological
side.
The article is arranged as follows: we derive the QCD sum rules for the masses
and residues of the Ω∗c and Ω
b in section 2; in section 3, numerical results and
discussions; section 4 is reserved for conclusion.
2 QCD sum rules for the Ω∗c and Ω
In the following, we write down the two-point correlation functions Πaµν(p
2) in the
QCD sum rules approach,
Πaµν(p
2) = i
d4xeip·x〈0|T
Jaµ(x)J̄
ν (0)
|0〉 , (1)
Jaµ(x) = ǫijks
i (x)Cγµsj(x)Q
k(x) , (2)
λaNµ(p, s) = 〈0|Jaµ(0)|Ω∗a(p, s)〉 , (3)
where the upper index a represents the c and b quarks respectively; the Nµ(p, s)
and λa stand for the Rarita-Schwinger spin vector and residue of the baryon Ω
respectively. i, j and k are color indexes, C is charge conjunction matrix, and µ and
ν are Lorentz indexes.
The correlation functions Πaµν(p) can be decomposed as follows:
Πaµν(p) = −gµν
6pΠa1(p2) + Πa2(p2)
+ · · · , (4)
due to Lorentz covariance. The first structure gµν 6p has an odd number of γ-matrices
and conserves chirality, the second structure gµν has an even number of γ-matrices
and violate chirality. In the original QCD sum rules analysis of the nucleon masses
and magnetic moments [8], the interval of dimensions (of the condensates) for the
odd structure is larger than the interval of dimensions for the even structure, one
may expect a better accuracy of results obtained from the sum rules with the odd
structure.
In this article, we choose the two tensor structures to study the masses and
residues of the heavy baryons Ω∗c and Ω
b , as the masses of the heavy quarks break
the chiral symmetry explicitly.
According to basic assumption of current-hadron duality in the QCD sum rules
approach [6], we insert a complete series of intermediate states satisfying unitarity
principle with the same quantum numbers as the current operator Jaµ(x) into the
correlation functions in Eq.(1) to obtain the hadronic representation. After isolating
the pole terms of the lowest states Ω∗a, we obtain the following result:
Πaµν(p
2) = −gµνλ2a
MΩ∗a+ 6p
+ · · · , (5)
where we have used the relation to sum over the Rarita-Schwinger spin vector,
Nµ(p, s)N̄ν(p, s) = −(6p +MΩ∗a)
gµν −
− 2pµpν
pµγν − pνγµ
3MΩ∗a
. (6)
In the following, we briefly outline operator product expansion for the correlation
functions Πaµν(p) in perturbative QCD theory. The calculations are performed at
large space-like momentum region p2 ≪ 0, which corresponds to small distance
x ≈ 0 required by validity of operator product expansion. We write down the ”full”
propagators Sij(x) and S
(x) of a massive quark in the presence of the vacuum
condensates firstly [6]2,
Sij(x) =
iδij 6x
2π2x4
− δijms
4π2x2
− δij
〈s̄s〉+ iδij
ms〈s̄s〉 6x−
〈s̄gsσGs〉
iδijx
ms〈s̄gsσGs〉 6x−
32π2x2
Gijµν(6xσµν + σµν 6x) + · · · ,
Q (x) =
(2π)4
d4ke−ik·x
6k −mQ
σαβ(6k +mQ) + (6k +mQ)σαβ
(k2 −m2Q)2
〈αsGG
〉δijmQ
k2 +mQ 6k
(k2 −m2Q)4
+ · · ·
, (7)
where 〈s̄gsσGs〉 = 〈s̄gsσαβGαβs〉 and 〈αsGGπ 〉 = 〈
αsGαβG
〉, then contract the quark
fields in the correlation functions Πaµν(p) with Wick theorem, and obtain the result:
Πaµν(p) = 2iǫijkǫi′j′k′
d4x eip·xTr
γµSii′(x)γνCS
jj′(x)C
Q (x) . (8)
Substitute the full s, c and b quark propagators into above correlation functions
and complete the integral in coordinate space, then integrate over the variable k,
we can obtain the correlation functions Πai (p
2) at the level of quark-gluon degree of
freedom:
2One can consult the last article of Ref.[6] for technical details in deriving the full propagator.
Πa1(p
2) = −
dxx(1 − x)2(x+ 2)
m̃2a − p2
m̃2a − p2
−ms〈s̄s〉
dxx(x− 2) log
m̃2a − p2
192π2
〈αsGG
dxx(x − 2) log
m̃2a − p2
ms〈s̄gsσGs〉
m̃2a − p2
576π2
(1− x)2(x+ 2)
x2(m̃2a − p2)
〈s̄s〉2
m2a − p2
ms〈s̄gsσGs〉
m2a − p2
+ · · · , (9)
Πa2(p
2) = −
dx(1− x)2(x+ 2)
m̃2a − p2
m̃2a − p2
mams〈s̄s〉
dx(x− 2) log
m̃2a − p2
576π2
〈αsGG
− 3x2 + 2x+ 9) log
m̃2a − p2
576π2
〈αsGG
x4 − x3 − 3x2 + 5x− 2
x(1− x)
m̃2a − p2
mams〈s̄gsσGs〉
m̃2a − p2
ma〈s̄s〉2
m2a − p2
mams〈s̄gsσGs〉
m2a − p2
+ · · · , (10)
where m̃2a =
We carry out operator product expansion to the vacuum condensates adding up
to dimension-6. In calculation, we take assumption of vacuum saturation for high
dimension vacuum condensates, they are always factorized to lower condensates
with vacuum saturation in the QCD sum rules, factorization works well in large
Nc limit. In this article, we take into account the contributions from the quark
condensate 〈s̄s〉, mixed condensate 〈s̄gsσGs〉, gluon condensate 〈αsGGπ 〉, and neglect
the contributions from other high dimension condensates, which are suppressed by
large denominators and would not play significant roles.
Once analytical results are obtained, then we can take current-hadron duality
below the threshold s0a and perform Borel transformation with respect to the variable
P 2 = −p2, finally we obtain the following sum rules:
λ2a exp
∫ s0a
dxx(1− x)2(x+ 2)
m̃2a − s
ms〈s̄s〉
∫ s0a
dxx(x− 2) exp
192π2
∫ s0a
dxx(x− 2) exp
ms〈s̄gsσGs〉
dxx exp
576π2
(1− x)2(2 + x)
〈s̄s〉2
ms〈s̄gsσGs〉
, (11)
MΩ∗aλ
a exp
∫ s0a
dx(1− x)2(x+ 2)
m̃2a − s
mams〈s̄s〉
∫ s0a
dx(x− 2) exp
576π2
∫ s0a
− 3x2 + 2x+ 9) exp
576π2
〈αsGG
x4 − x3 − 3x2 + 5x− 2
x(1− x)
m̃2a exp
mams〈s̄gsσGs〉
dx exp
ma〈s̄s〉2
mams〈s̄gsσGs〉
, (12)
where th = (ma + 2ms)
2 and ∆a =
Differentiate the above sum rules with respect to the variable 1
, then eliminate
the quantity λΩ∗a , we obtain two QCD sum rules for the masses MΩ∗a :
∫ s0a
dxx(1 − x)2(x+ 2)
m̃2a − s
s exp
ms〈s̄s〉
∫ s0a
dxx(x− 2)s exp
192π2
∫ s0a
dxx(x− 2)s exp
a〈s̄gsσGs〉
dx exp
576π2
(1− x)2(2 + x)
m2a〈s̄s〉2
a〈s̄gsσGs〉
∫ s0a
dxx(1 − x)2(x+ 2)
m̃2a − s
ms〈s̄s〉
∫ s0a
dxx(x− 2) exp
192π2
∫ s0a
dxx(x− 2) exp
ms〈s̄gsσGs〉
dxx exp
576π2
〈αsGG
(1− x)2(x+ 2)
〈s̄s〉2
ms〈s̄gsσGs〉
, (13)
M2Ω∗a =
∫ s0a
dx(1− x)2(x+ 2)
m̃2a − s
s exp
mams〈s̄s〉
∫ s0a
dx(x− 2)s exp
576π2
〈αsGG
∫ s0a
− 3x2 + 2x+ 9)s exp
576π2
x4 − x3 − 3x2 + 5x− 2
x(1 − x)
m̃4a exp
m3ams〈s̄gsσGs〉
m3a〈s̄s〉2
m3ams〈s̄gsσGs〉
∫ s0a
dx(1− x)2(x+ 2)
m̃2a − s
mams〈s̄s〉
∫ s0a
dx(x− 2) exp
576π2
∫ s0a
− 3x2 + 2x+ 9) exp
576π2
x4 − x3 − 3x2 + 5x− 2
x(1 − x)
m̃2a exp
mams〈s̄gsσGs〉
dx exp
ma〈s̄s〉2
mams〈s̄gsσGs〉
. (14)
3 Numerical results and discussions
The input parameters are taken to be the standard values 〈q̄q〉 = −(0.24±0.01GeV)3,
〈s̄s〉 = (0.8 ± 0.2)〈q̄q〉, 〈s̄gsσGs〉 = m20〈s̄s〉, m20 = (0.8 ± 0.2)GeV2, 〈αsGGπ 〉 =
(0.33GeV)4, ms = (0.14±0.01)GeV, mc = (1.4±0.1)GeV and mb = (4.8±0.1)GeV
[6, 7, 9]. The contribution from the gluon condensate 〈αsGG
〉 is less than 4%, and
the uncertainty is neglected here.
For the octet baryons with I(JP ) = 1
), the mass of the proton (the ground
state) is Mp = 938MeV, and the mass of the first radial excited state N(1440)
(the Roper resonance) is M1440 = (1420 − 1470)MeV ≈ 1440MeV [10]. For the
decuplet baryons with I(JP ) = 3
) , the mass of the ∆(1232) (the ground state)
is M1232 = (1231 − 1233)MeV ≈ 1232MeV, and the mass of the first radial excited
Eq.(11) Eq.(12)
perturbative term +80% +83%
〈s̄s〉 +12% +10%
〈s̄gsσGs〉 −4% −2%
〈s̄s〉2 +12% +7%
〈αsGG
〉 +1% +2%
Table 1: The contributions from different terms in the sum rules for the Ω∗c with
the central values of the input parameters.
Eq.(11) Eq.(12)
perturbative term +78% +80%
〈s̄s〉 +10% +10%
〈s̄gsσGs〉 −4% −3%
〈s̄s〉2 +15% +12%
〈αsGG
〉 +1% +1%
Table 2: The contributions from different terms in the sum rules for the Ω∗b with
the central values of the input parameters.
state ∆(1600) is M1600 = (1550 − 1700)MeV ≈ 1600MeV [10]. The separation
between the ground states and first radial excited states is about 0.5GeV. So in the
QCD sum rules for the baryons with the light quarks, the threshold parameters s0
are always chosen to be
s0 = Mgr + 0.5GeV [8, 11], here gr stands for the ground
states. The threshold parameters for the heavy baryons Ω∗c and Ω
b can be chosen
to be s0
= (2.8 + 0.5)2GeV2 and s0
= (6.1 + 0.5)2GeV2, respectively. The mass
of the bottomed baryon Ω∗b with spin-parity
is about MΩ∗
= (6.04 − 6.09)GeV,
which is predicted by the quark models and lattice QCD [12, 13].
In this article, the threshold parameters and Borel parameters are taken as
= 11.0GeV2 and M2 = (2.5 − 3.5)GeV2 for the charmed baryon Ω∗c , and
= 45.0GeV2 and M2 = (5.0 − 6.0)GeV2 for the bottomed baryon Ω∗b . The
contributions from different terms for the central values of the input parameters
are presented in Table.1 and Table.2, respectively. From the two tables, we can
expect convergence of the operator product expansion. In the two sum rules in
Eqs.(11-12), the contributions from the terms proportional to the quark conden-
sate 〈s̄s〉 and mixed condensate 〈s̄gsσGs〉 are suppressed due to the small mass ms
comparing with the terms proportional to the 〈s̄s〉2. Furthermore, from the ’full’
propagator of the s quark, we can see that the mixed condensate 〈s̄gsσGs〉 is com-
panied with additional large denominators, its contribution is even smaller. In the
right-hand side of Eqs.(11-12), the terms proportional to the 〈s̄s〉2 are suppressed
2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5
M2(GeV2)
 Central value;
 Upper bound;
 lower bound.
2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5
M2(GeV2)
 Central value;
 Upper bound;
 Lower bound.
Figure 1: MΩ∗c with Borel parameter M
2, A from Eq.(13) and B from Eq.(14).
by the exponents exp[−m2a/M2], which is balanced by the factor exp[−M2Ω∗a/M
in the left-hand side. Although the masses of the c quark and Ω∗c baryon are
much smaller than the corresponding ones of the b quark and Ω∗b baryon, the Borel
parameters M2 are different, for the central values of the Borel parameters M2,
exp[M2
/M2−m2b/M2] > exp[M2Ω∗c/M
2−m2c/M2]. It is not unexpected, the contri-
butions from the 〈s̄s〉2 are larger in the sum rules for the Ω∗b baryon than the ones
for the Ω∗c baryon.
If we approximate the phenomenological spectral density with the perturbative
term, the contribution from the pole term is as large as (28− 54)% for the charmed
baryon Ω∗c and (33 − 50)% for the bottomed baryon Ω∗b . We can choose smaller
Borel parameter M2 or larger threshold parameters s0a to enhance the contributions
from the ground states. However, if we take larger threshold parameter s0a, the
contribution from the first radial excited state maybe included in; on the other hand,
for smaller Borel parameterM2, the sum rules are not stable enough, the uncertainty
with variation of the Borel parameter is large. In the case of the multiquark states,
the standard criterion of the lowest pole dominance cannot be satisfied, we have to
resort to new criterion to overcome the problem, for detailed discussions about this
subject, one can consult Ref.[14].
Taking into account all uncertainties of the input parameters, finally we obtain
the values of the masses and residues of the heavy baryons Ω∗c and Ω
b , which are
shown in Figs.1-4 respectively,
MΩ∗c = (2.72± 0.12)GeV ,
= (6.04± 0.13)GeV ,
λΩ∗c = (0.047± 0.008)GeV ,
= (0.057± 0.011)GeV , (15)
2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5
M2(GeV2)
 Central value;
 Upper bound;
 Lower bound.
2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5
M2(GeV2)
 Central value;
 Upper bound;
 Lower bound.
Figure 2: λΩ∗c with Borel parameter M
2, A from Eq.(11) and Eq.(13), and B from
Eq.(12) and Eq.(14).
5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0
M2(GeV2)
 Central value;
 Upper bound;
 Lower bound.
5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0
M2(GeV2)
 Central value;
 Upper bound;
 Lower bound.
Figure 3: MΩ∗
with Borel parameter M2, A from Eq.(13) and B from Eq.(14).
5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0
M2(GeV2)
 Central value;
 Upper bound;
 Lower bound.
5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0
M2(GeV2)
 Central value;
 Upper bound;
 Lower bound.
Figure 4: λΩ∗
with Borel parameter M2, A from Eq.(11) and Eq.(13), and B from
Eq.(12) and Eq.(14).
from A from Eq.(11) and Eq.(13), and
MΩ∗c = (2.80± 0.08)GeV ,
= (6.08± 0.12)GeV ,
λΩ∗c = (0.046± 0.007)GeV ,
= (0.060± 0.011)GeV , (16)
from Eq.(12) and Eq.(14). The average values are about
MΩ∗c = (2.76± 0.10)GeV ,
= (6.06± 0.13)GeV ,
λΩ∗c = (0.047± 0.008)GeV ,
= (0.058± 0.011)GeV . (17)
The value of the mass MΩ∗c is compatible with the experimental data MΩ∗c =
(2.768±0.003)GeV [10], the interpolating current Jcµ(x) can couple with the charmed
baryon Ω∗c and give reasonable mass. The value of the mass MΩ∗b for the bottomed
baryon Ω∗b with
is compatible with other theoretical calculations, MΩ∗
= (6.04−
6.09)GeV, such as the quark models and lattice QCD [12, 13]. Once reasonable
values of the residues λΩ∗c and λΩ∗b are obtained, we can take them as basic input
parameters and study the hadronic processes [15], for example, the radiative decay
Ω∗c → Ωcγ, with the light-cone QCD sum rules or the QCD sum rules in external
field.
4 Conclusion
In this article, we calculate the masses and residues of the heavy baryons Ω∗c(css)
and Ω∗b(bss) with the QCD sum rules. The numerical values are compatible with
the experimental data and other theoretical estimations. Once reasonable values of
the residues λΩ∗c and λΩ∗b are obtained, we can take them as basic parameters and
study the hadronic processes, for example, the radiative decay Ω∗c → Ωcγ, with the
light-cone QCD sum rules or the QCD sum rules in external field.
Acknowledgments
This work is supported by National Natural Science Foundation, Grant Number
10405009, 10775051, and Program for New Century Excellent Talents in University,
Grant Number NCET-07-0282, and Key Program Foundation of NCEPU.
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http://arxiv.org/abs/0704.0075
	Introduction
	QCD sum rules for the c* and b*
	Numerical results and discussions
	Conclusion

  In this article, we calculate the masses and residues of the heavy baryons
$\Omega_c^*(css)$ and $\Omega_b^*(bss)$ with spin-parity ${3/2}^+$ with the QCD
sum rules. The numerical values are compatible with experimental data and other
theoretical estimations.

Introduction
Several new excited charmed baryon states have been observed by BaBar, Belle and
CLEO Collaborations, such as Λc(2765)
+, Λ+c (2880), Λ
c (2940), Σ
c (2800), Ξ
c (2980),
Ξ+c (3077), Ξ
c(2980) , Ξ
c(3077) [1, 2]. The charmed baryons provide a rich source
of states, including possible candidates for the orbital excitations. They serve as an
excellent ground for testing predictions of the constituent quark models and heavy
quark symmetry [3]. The charmed and bottomed baryons, which contain a heavy
quark and two light quarks, provides an ideal tool for studying dynamics of the light
quarks in the presence of a heavy quark. The u, d and s quarks form an SU(3) flavor
triplet, 3× 3 = 3̄ + 6, two light quarks can form diquarks with a symmetric sextet
and an antisymmetric antitriplet. For the S-wave baryons, the sextet contains both
spin-1
and spin-3
states, while the antitriplet contains only spin-1
states. By now,
the 1
antitriplet states (Λ+c , Ξ
c), and the
and 3
sextet states (Ωc,Σc,Ξ
and (Ω∗c ,Σ
c ) have been established.
The baryon Ω∗c , a css candidate for the
partner of the strange baryon Ω(sss),
was observed by BaBar collaboration in the radiative decay Ω∗c → Ωcγ [4]. The
baryon Ωc(css) was reconstructed in decays to the final states Ω
−π+, Ω−π+π0,
Ω−π+π−π+ and Ξ−K−π+π+. It lies about 70.8±1.0±1.1MeV above the Ωc, and it is
the last singly-charmed baryon with zero orbital momentum observed experimentally
In this article, we calculate the mass and residue of the Ω∗c (and Ω
b as byproduct,
the Ω∗b has not been observed experimentally yet) with the QCD sum rules [6, 7]. In
the QCD sum rules, operator product expansion is used to expand the time-ordered
currents into a series of quark and gluon condensates which parameterize the long
distance properties of the QCD vacuum. Based on current-hadron duality, we can
1E-mail,wangzgyiti@yahoo.com.cn;wangzg@yahoo.cn.
http://arxiv.org/abs/0704.1106v3
obtain copious information about the hadronic parameters at the phenomenological
side.
The article is arranged as follows: we derive the QCD sum rules for the masses
and residues of the Ω∗c and Ω
b in section 2; in section 3, numerical results and
discussions; section 4 is reserved for conclusion.
2 QCD sum rules for the Ω∗c and Ω
In the following, we write down the two-point correlation functions Πaµν(p
2) in the
QCD sum rules approach,
Πaµν(p
2) = i
d4xeip·x〈0|T
Jaµ(x)J̄
ν (0)
|0〉 , (1)
Jaµ(x) = ǫijks
i (x)Cγµsj(x)Q
k(x) , (2)
λaNµ(p, s) = 〈0|Jaµ(0)|Ω∗a(p, s)〉 , (3)
where the upper index a represents the c and b quarks respectively; the Nµ(p, s)
and λa stand for the Rarita-Schwinger spin vector and residue of the baryon Ω
respectively. i, j and k are color indexes, C is charge conjunction matrix, and µ and
ν are Lorentz indexes.
The correlation functions Πaµν(p) can be decomposed as follows:
Πaµν(p) = −gµν
6pΠa1(p2) + Πa2(p2)
+ · · · , (4)
due to Lorentz covariance. The first structure gµν 6p has an odd number of γ-matrices
and conserves chirality, the second structure gµν has an even number of γ-matrices
and violate chirality. In the original QCD sum rules analysis of the nucleon masses
and magnetic moments [8], the interval of dimensions (of the condensates) for the
odd structure is larger than the interval of dimensions for the even structure, one
may expect a better accuracy of results obtained from the sum rules with the odd
structure.
In this article, we choose the two tensor structures to study the masses and
residues of the heavy baryons Ω∗c and Ω
b , as the masses of the heavy quarks break
the chiral symmetry explicitly.
According to basic assumption of current-hadron duality in the QCD sum rules
approach [6], we insert a complete series of intermediate states satisfying unitarity
principle with the same quantum numbers as the current operator Jaµ(x) into the
correlation functions in Eq.(1) to obtain the hadronic representation. After isolating
the pole terms of the lowest states Ω∗a, we obtain the following result:
Πaµν(p
2) = −gµνλ2a
MΩ∗a+ 6p
+ · · · , (5)
where we have used the relation to sum over the Rarita-Schwinger spin vector,
Nµ(p, s)N̄ν(p, s) = −(6p +MΩ∗a)
gµν −
− 2pµpν
pµγν − pνγµ
3MΩ∗a
. (6)
In the following, we briefly outline operator product expansion for the correlation
functions Πaµν(p) in perturbative QCD theory. The calculations are performed at
large space-like momentum region p2 ≪ 0, which corresponds to small distance
x ≈ 0 required by validity of operator product expansion. We write down the ”full”
propagators Sij(x) and S
(x) of a massive quark in the presence of the vacuum
condensates firstly [6]2,
Sij(x) =
iδij 6x
2π2x4
− δijms
4π2x2
− δij
〈s̄s〉+ iδij
ms〈s̄s〉 6x−
〈s̄gsσGs〉
iδijx
ms〈s̄gsσGs〉 6x−
32π2x2
Gijµν(6xσµν + σµν 6x) + · · · ,
Q (x) =
(2π)4
d4ke−ik·x
6k −mQ
σαβ(6k +mQ) + (6k +mQ)σαβ
(k2 −m2Q)2
〈αsGG
〉δijmQ
k2 +mQ 6k
(k2 −m2Q)4
+ · · ·
, (7)
where 〈s̄gsσGs〉 = 〈s̄gsσαβGαβs〉 and 〈αsGGπ 〉 = 〈
αsGαβG
〉, then contract the quark
fields in the correlation functions Πaµν(p) with Wick theorem, and obtain the result:
Πaµν(p) = 2iǫijkǫi′j′k′
d4x eip·xTr
γµSii′(x)γνCS
jj′(x)C
Q (x) . (8)
Substitute the full s, c and b quark propagators into above correlation functions
and complete the integral in coordinate space, then integrate over the variable k,
we can obtain the correlation functions Πai (p
2) at the level of quark-gluon degree of
freedom:
2One can consult the last article of Ref.[6] for technical details in deriving the full propagator.
Πa1(p
2) = −
dxx(1 − x)2(x+ 2)
m̃2a − p2
m̃2a − p2
−ms〈s̄s〉
dxx(x− 2) log
m̃2a − p2
192π2
〈αsGG
dxx(x − 2) log
m̃2a − p2
ms〈s̄gsσGs〉
m̃2a − p2
576π2
(1− x)2(x+ 2)
x2(m̃2a − p2)
〈s̄s〉2
m2a − p2
ms〈s̄gsσGs〉
m2a − p2
+ · · · , (9)
Πa2(p
2) = −
dx(1− x)2(x+ 2)
m̃2a − p2
m̃2a − p2
mams〈s̄s〉
dx(x− 2) log
m̃2a − p2
576π2
〈αsGG
− 3x2 + 2x+ 9) log
m̃2a − p2
576π2
〈αsGG
x4 − x3 − 3x2 + 5x− 2
x(1− x)
m̃2a − p2
mams〈s̄gsσGs〉
m̃2a − p2
ma〈s̄s〉2
m2a − p2
mams〈s̄gsσGs〉
m2a − p2
+ · · · , (10)
where m̃2a =
We carry out operator product expansion to the vacuum condensates adding up
to dimension-6. In calculation, we take assumption of vacuum saturation for high
dimension vacuum condensates, they are always factorized to lower condensates
with vacuum saturation in the QCD sum rules, factorization works well in large
Nc limit. In this article, we take into account the contributions from the quark
condensate 〈s̄s〉, mixed condensate 〈s̄gsσGs〉, gluon condensate 〈αsGGπ 〉, and neglect
the contributions from other high dimension condensates, which are suppressed by
large denominators and would not play significant roles.
Once analytical results are obtained, then we can take current-hadron duality
below the threshold s0a and perform Borel transformation with respect to the variable
P 2 = −p2, finally we obtain the following sum rules:
λ2a exp
∫ s0a
dxx(1− x)2(x+ 2)
m̃2a − s
ms〈s̄s〉
∫ s0a
dxx(x− 2) exp
192π2
∫ s0a
dxx(x− 2) exp
ms〈s̄gsσGs〉
dxx exp
576π2
(1− x)2(2 + x)
〈s̄s〉2
ms〈s̄gsσGs〉
, (11)
MΩ∗aλ
a exp
∫ s0a
dx(1− x)2(x+ 2)
m̃2a − s
mams〈s̄s〉
∫ s0a
dx(x− 2) exp
576π2
∫ s0a
− 3x2 + 2x+ 9) exp
576π2
〈αsGG
x4 − x3 − 3x2 + 5x− 2
x(1− x)
m̃2a exp
mams〈s̄gsσGs〉
dx exp
ma〈s̄s〉2
mams〈s̄gsσGs〉
, (12)
where th = (ma + 2ms)
2 and ∆a =
Differentiate the above sum rules with respect to the variable 1
, then eliminate
the quantity λΩ∗a , we obtain two QCD sum rules for the masses MΩ∗a :
∫ s0a
dxx(1 − x)2(x+ 2)
m̃2a − s
s exp
ms〈s̄s〉
∫ s0a
dxx(x− 2)s exp
192π2
∫ s0a
dxx(x− 2)s exp
a〈s̄gsσGs〉
dx exp
576π2
(1− x)2(2 + x)
m2a〈s̄s〉2
a〈s̄gsσGs〉
∫ s0a
dxx(1 − x)2(x+ 2)
m̃2a − s
ms〈s̄s〉
∫ s0a
dxx(x− 2) exp
192π2
∫ s0a
dxx(x− 2) exp
ms〈s̄gsσGs〉
dxx exp
576π2
〈αsGG
(1− x)2(x+ 2)
〈s̄s〉2
ms〈s̄gsσGs〉
, (13)
M2Ω∗a =
∫ s0a
dx(1− x)2(x+ 2)
m̃2a − s
s exp
mams〈s̄s〉
∫ s0a
dx(x− 2)s exp
576π2
〈αsGG
∫ s0a
− 3x2 + 2x+ 9)s exp
576π2
x4 − x3 − 3x2 + 5x− 2
x(1 − x)
m̃4a exp
m3ams〈s̄gsσGs〉
m3a〈s̄s〉2
m3ams〈s̄gsσGs〉
∫ s0a
dx(1− x)2(x+ 2)
m̃2a − s
mams〈s̄s〉
∫ s0a
dx(x− 2) exp
576π2
∫ s0a
− 3x2 + 2x+ 9) exp
576π2
x4 − x3 − 3x2 + 5x− 2
x(1 − x)
m̃2a exp
mams〈s̄gsσGs〉
dx exp
ma〈s̄s〉2
mams〈s̄gsσGs〉
. (14)
3 Numerical results and discussions
The input parameters are taken to be the standard values 〈q̄q〉 = −(0.24±0.01GeV)3,
〈s̄s〉 = (0.8 ± 0.2)〈q̄q〉, 〈s̄gsσGs〉 = m20〈s̄s〉, m20 = (0.8 ± 0.2)GeV2, 〈αsGGπ 〉 =
(0.33GeV)4, ms = (0.14±0.01)GeV, mc = (1.4±0.1)GeV and mb = (4.8±0.1)GeV
[6, 7, 9]. The contribution from the gluon condensate 〈αsGG
〉 is less than 4%, and
the uncertainty is neglected here.
For the octet baryons with I(JP ) = 1
), the mass of the proton (the ground
state) is Mp = 938MeV, and the mass of the first radial excited state N(1440)
(the Roper resonance) is M1440 = (1420 − 1470)MeV ≈ 1440MeV [10]. For the
decuplet baryons with I(JP ) = 3
) , the mass of the ∆(1232) (the ground state)
is M1232 = (1231 − 1233)MeV ≈ 1232MeV, and the mass of the first radial excited
Eq.(11) Eq.(12)
perturbative term +80% +83%
〈s̄s〉 +12% +10%
〈s̄gsσGs〉 −4% −2%
〈s̄s〉2 +12% +7%
〈αsGG
〉 +1% +2%
Table 1: The contributions from different terms in the sum rules for the Ω∗c with
the central values of the input parameters.
Eq.(11) Eq.(12)
perturbative term +78% +80%
〈s̄s〉 +10% +10%
〈s̄gsσGs〉 −4% −3%
〈s̄s〉2 +15% +12%
〈αsGG
〉 +1% +1%
Table 2: The contributions from different terms in the sum rules for the Ω∗b with
the central values of the input parameters.
state ∆(1600) is M1600 = (1550 − 1700)MeV ≈ 1600MeV [10]. The separation
between the ground states and first radial excited states is about 0.5GeV. So in the
QCD sum rules for the baryons with the light quarks, the threshold parameters s0
are always chosen to be
s0 = Mgr + 0.5GeV [8, 11], here gr stands for the ground
states. The threshold parameters for the heavy baryons Ω∗c and Ω
b can be chosen
to be s0
= (2.8 + 0.5)2GeV2 and s0
= (6.1 + 0.5)2GeV2, respectively. The mass
of the bottomed baryon Ω∗b with spin-parity
is about MΩ∗
= (6.04 − 6.09)GeV,
which is predicted by the quark models and lattice QCD [12, 13].
In this article, the threshold parameters and Borel parameters are taken as
= 11.0GeV2 and M2 = (2.5 − 3.5)GeV2 for the charmed baryon Ω∗c , and
= 45.0GeV2 and M2 = (5.0 − 6.0)GeV2 for the bottomed baryon Ω∗b . The
contributions from different terms for the central values of the input parameters
are presented in Table.1 and Table.2, respectively. From the two tables, we can
expect convergence of the operator product expansion. In the two sum rules in
Eqs.(11-12), the contributions from the terms proportional to the quark conden-
sate 〈s̄s〉 and mixed condensate 〈s̄gsσGs〉 are suppressed due to the small mass ms
comparing with the terms proportional to the 〈s̄s〉2. Furthermore, from the ’full’
propagator of the s quark, we can see that the mixed condensate 〈s̄gsσGs〉 is com-
panied with additional large denominators, its contribution is even smaller. In the
right-hand side of Eqs.(11-12), the terms proportional to the 〈s̄s〉2 are suppressed
2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5
M2(GeV2)
 Central value;
 Upper bound;
 lower bound.
2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5
M2(GeV2)
 Central value;
 Upper bound;
 Lower bound.
Figure 1: MΩ∗c with Borel parameter M
2, A from Eq.(13) and B from Eq.(14).
by the exponents exp[−m2a/M2], which is balanced by the factor exp[−M2Ω∗a/M
in the left-hand side. Although the masses of the c quark and Ω∗c baryon are
much smaller than the corresponding ones of the b quark and Ω∗b baryon, the Borel
parameters M2 are different, for the central values of the Borel parameters M2,
exp[M2
/M2−m2b/M2] > exp[M2Ω∗c/M
2−m2c/M2]. It is not unexpected, the contri-
butions from the 〈s̄s〉2 are larger in the sum rules for the Ω∗b baryon than the ones
for the Ω∗c baryon.
If we approximate the phenomenological spectral density with the perturbative
term, the contribution from the pole term is as large as (28− 54)% for the charmed
baryon Ω∗c and (33 − 50)% for the bottomed baryon Ω∗b . We can choose smaller
Borel parameter M2 or larger threshold parameters s0a to enhance the contributions
from the ground states. However, if we take larger threshold parameter s0a, the
contribution from the first radial excited state maybe included in; on the other hand,
for smaller Borel parameterM2, the sum rules are not stable enough, the uncertainty
with variation of the Borel parameter is large. In the case of the multiquark states,
the standard criterion of the lowest pole dominance cannot be satisfied, we have to
resort to new criterion to overcome the problem, for detailed discussions about this
subject, one can consult Ref.[14].
Taking into account all uncertainties of the input parameters, finally we obtain
the values of the masses and residues of the heavy baryons Ω∗c and Ω
b , which are
shown in Figs.1-4 respectively,
MΩ∗c = (2.72± 0.12)GeV ,
= (6.04± 0.13)GeV ,
λΩ∗c = (0.047± 0.008)GeV ,
= (0.057± 0.011)GeV , (15)
2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5
M2(GeV2)
 Central value;
 Upper bound;
 Lower bound.
2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5
M2(GeV2)
 Central value;
 Upper bound;
 Lower bound.
Figure 2: λΩ∗c with Borel parameter M
2, A from Eq.(11) and Eq.(13), and B from
Eq.(12) and Eq.(14).
5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0
M2(GeV2)
 Central value;
 Upper bound;
 Lower bound.
5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0
M2(GeV2)
 Central value;
 Upper bound;
 Lower bound.
Figure 3: MΩ∗
with Borel parameter M2, A from Eq.(13) and B from Eq.(14).
5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0
M2(GeV2)
 Central value;
 Upper bound;
 Lower bound.
5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0
M2(GeV2)
 Central value;
 Upper bound;
 Lower bound.
Figure 4: λΩ∗
with Borel parameter M2, A from Eq.(11) and Eq.(13), and B from
Eq.(12) and Eq.(14).
from A from Eq.(11) and Eq.(13), and
MΩ∗c = (2.80± 0.08)GeV ,
= (6.08± 0.12)GeV ,
λΩ∗c = (0.046± 0.007)GeV ,
= (0.060± 0.011)GeV , (16)
from Eq.(12) and Eq.(14). The average values are about
MΩ∗c = (2.76± 0.10)GeV ,
= (6.06± 0.13)GeV ,
λΩ∗c = (0.047± 0.008)GeV ,
= (0.058± 0.011)GeV . (17)
The value of the mass MΩ∗c is compatible with the experimental data MΩ∗c =
(2.768±0.003)GeV [10], the interpolating current Jcµ(x) can couple with the charmed
baryon Ω∗c and give reasonable mass. The value of the mass MΩ∗b for the bottomed
baryon Ω∗b with
is compatible with other theoretical calculations, MΩ∗
= (6.04−
6.09)GeV, such as the quark models and lattice QCD [12, 13]. Once reasonable
values of the residues λΩ∗c and λΩ∗b are obtained, we can take them as basic input
parameters and study the hadronic processes [15], for example, the radiative decay
Ω∗c → Ωcγ, with the light-cone QCD sum rules or the QCD sum rules in external
field.
4 Conclusion
In this article, we calculate the masses and residues of the heavy baryons Ω∗c(css)
and Ω∗b(bss) with the QCD sum rules. The numerical values are compatible with
the experimental data and other theoretical estimations. Once reasonable values of
the residues λΩ∗c and λΩ∗b are obtained, we can take them as basic parameters and
study the hadronic processes, for example, the radiative decay Ω∗c → Ωcγ, with the
light-cone QCD sum rules or the QCD sum rules in external field.
Acknowledgments
This work is supported by National Natural Science Foundation, Grant Number
10405009, 10775051, and Program for New Century Excellent Talents in University,
Grant Number NCET-07-0282, and Key Program Foundation of NCEPU.
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http://arxiv.org/abs/0704.0075
	Introduction
	QCD sum rules for the c* and b*
	Numerical results and discussions
	Conclusion
