Analysis of \(\Omega _c(3000)\) , \(\Omega _c(3050)\) , \(\Omega _c(3066)\) , \(\Omega _c(3090)\) and \(\Omega _c(3119)\) with QCD sum rules
Eur. Phys. J. C
ZhiGang Wang 0
respectively 0
where the two s quarks are in 0
relative Pwave 0
while 0
) can be assigned to the P wave baryon state with J P = 0
where the two s quarks are in relative Swave. 0
0 Department of Physics, North China Electric Power University , Baoding 071003 , People's Republic of China
In this article, we assign c(3000), c(3050), c(3066), c(3090) and c(3119) to the Pwave baryon states with J P = 21 −, 21 −, 23 −, 23 − and 25 −, respectively, and study them with the QCD sum rules by introducing an explicit relative Pwave between the two s quarks. The predictions support assigning c(3050), c(3066), c(3090) and c(3119) to the Pwave baryon states with J P = 21 −,

c(3050),
c(3066),
c(3090) and
c(3119)
1 Introduction
In the past years, several new charmed baryon states have
been observed, and the spectroscopy of the charmed baryon
states have reattracted much attention [1], the QCD sum
rules plays an important roles in assigning those new baryon
states. The masses of the heavy baryon states with J P = 21 ±,
23 ±, 5 ± have been studied with the full QCD sum rules [2–
2
18] or the QCD sum rules combined with the heavy quark
effective theory [19–28].
Recently, the LHCb collaboration studied the c+ K − mass
spectrum with a sample of pp collision data corresponding to
an integrated luminosity of 3.3 fb−1 collected by the LHCb
experiment, and one observed five new narrow excited c0
states, c(3000), c(3050), c(3066), c(3090), c(3119)
[29]. The measured masses and widths are
c(3000) : M = 3000.4 ± 0.2 ± 0.1 MeV,
= 4.5 ± 0.6 ± 0.3 MeV,
= 0.8 ± 0.2 ± 0.1 MeV,
c(3050) : M = 3050.2 ± 0.1 ± 0.1 MeV,
c(3066) : M = 3065.6 ± 0.1 ± 0.3 MeV,
= 3.5 ± 0.4 ± 0.2 MeV,
c(3090) : M = 3090.2 ± 0.3 ± 0.5 MeV,
= 8.7 ± 1.0 ± 0.8 MeV,
c(3119) : M = 3119.1 ± 0.3 ± 0.9 MeV,
= 1.1 ± 0.8 ± 0.4 MeV.
There have been several assignments for those new charmed
states. In Ref. [30], c(3066) and c(3119) are assigned
to the 2S c0 states with J P = 21 + and 23 +, respectively.
In Ref. [31], possible assignments of those c0 states to
the Pwave baryon states with J P = 21 −, 23 − and 25 −
are discussed. In Refs. [32–34], the c(3000), c(3050),
c(3066), c(3090) and c(3119) are assigned to the
Pwave baryon states with J P = 21 −, 21 −, 23 −, 23 − and 25 −,
respectively. In Refs. [35,36], those c0 states are assigned
to the pentaquark states or molecular pentaquark states with
J P = 21 −, 23 − or 25 −. In Ref. [37], c(3000), c(3050),
c(3066) and c(3090) are assigned to the Pwave baryon
states with J P = 21 −, 23 −, 3 − and 21 −, respectively. In Ref.
[38], c(3090) and c(31219) are assigned to the 2S c0
states with J P = 21 + and 23 +, respectively, while c(3000),
c(3066) and c(3050) are assigned to the Pwave baryon
states with J P = 21 −, 23 − and 25 −, respectively.
In this article, we tentatively assign c(3000), c(3050),
c(3066), c(3090) and c(3119) to the Pwave baryon
states with J P = 21 −, 21 −, 23 −, 23 − and 25 −, respectively, and
study their masses and pole residues with the QCD sum rules
in detail.
The ground state quarks have the spinparity 21 +, two
quarks can form a scalar diquark or an axialvector diquark
with the spinparity 0+ or 1+, the diquark then combines with
a third quark to form a positive parity baryon with spin 21 or
3 . We can construct the baryon currents η and ημ with
pos2
itive parity without introducing additional Pwave. As
multiplying i γ5 to the baryon currents changes their parity, the
currents i γ5η and i γ5ημ couple potentially to the negative
parity heavy baryon states. In Refs. [17,18], we construct
the currents without introducing relative Pwave to study
the negative parity heavy, doubly heavy and triply heavy
baryon states, and obtain satisfactory results. The predictions
M = 2.98 ± 0.16 GeV for the c0 states with J P = 21 −,
23 − are consistent with the masses of c(3000), c(3050),
c(3066), c(3090) from the LHCb collaboration [17].
In Ref. [39], we construct the interpolating currents by
introducing the relative Pwave explicitly, study the negative
parity charmed baryon states c(2625) and c(2815) with
the full QCD sum rules, and reproduce the experimental
values of the masses. In this article, we extend our previous
work to study c(3000), c(3050), c(3066), c(3090)
and c(3119) with QCD sum rules by introducing the
relative Pwave explicitly.
The article is arranged as follows: we derive the QCD
sum rules for the masses and pole residues of the c0 states
in Sect. 2; in Sect. 3, we present the numerical results and
discussions; and Sect. 4 is reserved for our conclusions.
2 QCD sum rules for the
In the following, we write down the twopoint correlation
functions ( p), μν ( p), μναβ ( p) in the QCD sum rules,
( p) = i
d4x ei p·x 0T J (x ) J¯(0) 0 ,
+ siT (x )C γμ∂ν s j (x ) ck (x ),
gμν = gμν − 41 γμγν , i , j , k are color indices, C is the charge
conjugation matrix. We construct the currents with the light
diquarks Sμiν = εi jk [∂μsiT C γν s j + siT C γν ∂μs j ]. The Sμiν
have two Lorentz indices μ and ν, but they are neither
symmetric nor antisymmetric when interchanging the indices
μ and ν. The scalar components Sμiν gμν and Sμiν σ μν
couple potentially to the spin0 diquarks. The Dirac matrices
gαμγ ν − gαν γ μ and gαμγ ν + gαν γ μ − 21 gμν γ α are
antisymmetric and symmetric, respectively, when interchanging
the indices μ and ν, the vector components Sμiν (gαμγ ν −
gαν γ μ) and Sμiν (gαμγ ν + gαν γ μ − 21 gμν γ α) couple
potentially to the spin1 diquarks. The symmetric components
Sμiν + Sνiμ couple potentially to the spin0 and 2 diquarks.
So we choose the currents J (x ), Jμ(x ) and Jμν (x ) to study
the spin 21 , 23 and 25 baryon states, respectively.
The currents J (0), Jμ(0) and Jμν (0) couple potentially to
the 21 −, 21 +, 3 − and 21 −, 23 +, 5 − charmed baryon states B−,
2 2 21
B +1, B −3 and B −1, B +3, B −5, respectively,
2 2 2 2 2
0 J (0)B −1( p) = λ −1U −( p, s),
2 2
0 Jμ(0)B +1( p) = f 1+ pμU +( p, s),
2 2
0 Jμ(0)B −3( p) = λ −3Uμ−( p, s),
2 2
0 Jμν (0)B −1( p) = g −1 pμ pνU −( p, s),
2 2
0 Jμν (0)B +3( p) = f 3+ pμUν+( p, s) + pνUμ+( p, s) ,
2 2
0 Jμν (0)B −5( p) = λ −5Uμ−ν ( p, s).
2 2
tially to the 21 +, 21 −, 23 + and 21 +, 23 −, 25 + charmed baryon
states B +1, B−, B +3 and B +1, B−, B +5, respectively [42–44],
2 21 2 2 23 2
0 J (0)B +1( p) = λ +1i γ5U +( p, s), (7)
2 2
0 Jμ(0)B −1( p) = f 1− pμi γ5U −( p, s),
2 2
0 Jμ(0)B +3( p) = λ +3i γ5Uμ+( p, s), (8)
2 2
0 Jμν (0)B +1( p) = g +1 pμ pν i γ5U +( p, s),
2 2
0 Jμν (0)B −3( p) = f 3−i γ5 pμUν−( p, s) + pνUμ−( p, s) ,
2 2
0 Jμν (0)B +5( p) = λ +5i γ5Uμ+ν ( p, s). (9)
2 2
The spinors U ±( p, s) satisfy the Dirac equations ( p − M±)
U ±( p) = 0, while the spinors Uμ±( p, s) and Uμ±ν ( p, s)
satisfy the Rarita–Schwinger equations ( p − M±)Uμ±( p) = 0
and ( p − M±)Uμ±ν ( p) = 0, and the relations γ μUμ±( p, s) =
0, pμUμ±( p, s) = 0, γ μUμ±ν ( p, s) = 0, pμUμ±ν ( p, s) = 0,
Uμ±ν ( p, s) = Uν±μ( p, s). The λ ±21/ 23 / 25 , f 21±/ 23 and g ±21 are the
pole residues or currentbaryon coupling constants.
At the phenomenological side, we insert a complete set
of intermediate charmed baryon states with the same
quantum numbers as the current operators J (x ), i γ5 J (x ), Jμ(x ),
i γ5 Jμ(x ), Jμν (x ) and i γ5 Jμν (x ) into the correlation
functions ( p), μν ( p) and μναβ ( p) to obtain the hadronic
representation [40,41]. After isolating the pole terms of the
lowest states of the charmed baryon states, we obtain the
following results:
( p) = λ −212 Mp −2+−Mp−2 + λ +212 Mp +2−−Mp+2 + . . . ,
2 M −2 − p2
2 M +2 − p2
2 M +2 − p+2 pμ pν
+ f 1−2 p − M
2 M −2 − p−2 pμ pν + . . . ,
2 M+2 − p+2 pμ pα −gνβ +
2 M−2 − p−2 pμ pα −gνβ +
2 M−2 − p−2 pμ pν pα pβ
+ g +12 p − M
2 M+2 − p+2 pμ pν pα pβ + . . . ,
U U = ( p + M±) ,
where gμν = gμν − pμp2pν . In calculations, we have used the
following summations [45]:
and p2 = M 2 on the mass shell.
±
andWeμcνaαnβ (repw)riintteotthhee cfoorllroewlaitniognfofurmncaticocnosrdin(gp)to, Loμrνe(nptz)
covariance:
( p) =
μναβ ( p) = 25 ( p2) gμα gνβ +2 gμβ gνα + . . . . (18)
In this article, we choose the tensor structures gμν and
gμα gνβ + gμβ gνα for analysis, and separate the contributions
of the 3 ± and 25 ± charmed baryon states unambiguously. For
2
a detailed discussion of this subject, one can consult Ref. [44].
We obtain the hadronic spectral densities at
phenomenological side through the dispersion relation,
Im j (s)
= p[λ −j2δ(s − M −2) + λ +j2δ(s − M 2 )
+ ]
+[M−λ −j2δ(s − M −2) − M+λ +j2δ(s − M 2 ) ,
+ ]
= p ρ 1j,H (s) + ρ 0j,H (s), (19)
where j = 21 , 23 , 25 , the subscript H denotes the hadron side,
then we introduce the weight function exp(− Ts2 ) to obtain
the QCD sum rules at the phenomenological side,
s0
s
√sρ 1j,H (s) + ρ 0j,H (s) exp − T 2
where the s0 are the continuum thresholds and the T 2 are the
Borel parameters [44].
At the QCD side, we calculate the light quark parts of the
correlation functions ( p), μν ( p), μναβ ( p) with the full
light quark propagators in the coordinate space and take the
momentum space expression for the full cquark
propagator. It is straightforward but tedious to compute the integrals
both in the coordinate and momentum spaces to obtain the
correlation functions j ( p2), therefore we obtain the QCD
spectral densities through the dispersion relation,
= p ρ 1j,QC D(s) + ρ j,QC D(s),
0
We derive Eq. (22) with respect to T12 , then eliminate the
pole residues λ −j and obtain the QCD sum rules for the masses
of the charmed baryon states,
M−2 =
− d(1/dT 2) ms0c2 ds √sρ 1j,QC D(s) + ρ 0j,QC D(s) exp − Ts2
ms0c2 ds √sρ 1j,QC D(s) + ρ 0j,QC D(s) exp − Ts2
3 Numerical results and discussions
In the article, we take the M S masses mc(mc) = (1.275 ±
0.025) GeV and ms (μ = 2 GeV) = (0.095 ± 0.005) GeV
from the particle data group [1], and take into account the
energyscale dependence of the M S masses from the
renormalization group equation,
where t = log μ22 , b0 = 331−22πn f , b1 = 1532−4π192n f ,
b2 = 2857− 51093238nπf3+ 32275 n2f , = 213, 296 and 339 MeV for
the flavors n f = 5, 4 and 3, respectively [1].
In Refs. [47–51], we study the acceptable energy scales
of the QCD spectral densities for the hiddencharm
(bottom) tetraquark states and molecular states in the QCD sum
rules for the first time, and we suggest the empirical formula
μ = M X2/Y/Z − (2MQ )2 to determine the optimal energy
scales, where X , Y , Z denote the fourquark states, and MQ
is the effective heavy quark mass. The empirical energyscale
formula also works well in studying the hiddencharm
pentaquark states [44]. In Ref. [39], we use the diquark–quark
model to construct the interpolating currents, and take the
analogous formula μ = M 2 c/ c − Mc2 to determine the
energy scales of the QCD spectral densities of the QCD sum
rules for the charmed baryon states c(2625) and c(2815),
and obtain satisfactory results. In this article, we use the
formula μ = M 2 c − Mc2 to determine the energy scales of
the QCD spectral densities. If we take the updated value
Mc = 1.82 GeV [52], then μ ≈ 2.5 GeV. In calculations,
we set the energy scales of the QCD spectral densities to be
μ = 2.5 GeV.
Now we search for the Borel parameters T 2 and
continuum threshold parameters s0 to satisfy the following three
criteria:
1. pole dominance at the phenomenological side;
2. convergence of the operator product expansion;
3. appearance of the Borel platforms.
In calculations, we observe that no stable QCD sum rules
can be obtained for the current J 2(x ). The resulting Borel
parameters T 2, continuum threshold parameters s0, pole
contributions and perturbative contributions (per) are shown
explicitly in Table 1, where the perturbative contributions
are defined by
per =
Table 1 The Borel parameters
T 2, continuum threshold
parameters s0, pole
contributions (pole) and
perturbative contributions
(perturbative)
T 2 (GeV2)
√s0 (GeV)
ρper(s) and ρtot(s) denote the perturbative and total QCD
spectral densities, respectively. From the table, we can see
that the criteria 1 and 2 can be satisfied.
We take into account all uncertainties of the relevant
parameters, and we obtain the values of the masses and
pole residues of the c0 baryon states, which are shown
in Figs.1, 2 and Table 2. In Figs.1, 2, we plot the masses
and pole residues with variations of the Borel parameters at
much larger intervals than the Borel windows shown in Table
1. In the Borel windows, the uncertainties originating with
the Borel parameters in the Borel windows are very small,
δ M c /M c = (1.2 − 1.6)%, the criterion 3 is also satisfied.
The three criteria are all satisfied, we expect to make reliable
predictions. In Figs. 1 and 2 and Table 2, we also present
the possible assignments of the c0 states according to the
masses.
In Ref. [17], we choose the currents without introducing
the relative Pwave to study the negative parity heavy and
doubly heavy baryon states, and we obtain the predictions
M = 2.98 ± 0.16 GeV for the c0 states with J P = 21 −,
23 −, where the diquark constituent εi jk s Tj C γμsk is taken to
construct the currents. Multiplying i γ5 to the baryon currents
changes their parity, we can choose currents without
introducing relative Pwave to study the Pwave baryon states. The
current εi jk s Tj C γμsk γ μci couples potentially to the c0 state
with J P = 21 − [17], the mass of the c(3000) is in
excellent agreement with the prediction M = 2.98 ± 0.16 GeV
[17] or the prediction M = 2.990 ± 0.129 GeV based on a
more general interpolating current with additional parameter
[53], the c(3000) can be assigned to the Pwave charmed
baryon state with J P = 21 −, where two s quarks are in
relative Swave. In Table 3, we present some predictions for the
masses of the Pwave c0 baryon states from the full QCD
sum rules [17, 53, 54] and potential quark models [55–58].
Table 2 The masses M, pole residues λ and possible assignments of the
charmed baryon states, where jl denotes the total angular momentum
of the light degree of freedom
3.05 ± 0.11
3.06 ± 0.11
3.06 ± 0.10
3.11 ± 0.10
Table 3 The masses of the
Pwave c baryon states, where
the unit is GeV, jl denotes the
total angular momentum of the
light degree of freedom. We
neglect the mixing effects of the
2 0 − 21 1− and 23 1− − 23 2− in the
1 −
potential quark models for
simplicity
2.34 ± 0.50
1.03 ± 0.23
2.47 ± 0.47
1.07 ± 0.17
c(3066/3090)
c(3066/3090)
3.05 ± 0.11
3.06 ± 0.11
3.06 ± 0.10
3.11 ± 0.10
2.98 ± 0.16
2.98 ± 0.16
2.990 ± 0.129
3.056 ± 0.103
3.08 ± 0.12
We cannot identify a baryon state unambiguously with the
mass alone; it is necessary to study the decay widths of those
Pwave baryon states with the QCD sum rules. In Ref. [53],
Agaev, Azizi and Sundu study the masses and widths of the
1P 21 −, 23 − and 2S 21 +, 23 + c0 baryon states with the full
QCD sum rules, and they assign c(3000), c(3050) and
c(3119) to the c0 baryon states with the quantum numbers
(1P, 21 −), (1P, 23 −) and (2S, 23 +), respectively, and assign
the c(3066) or c(3090) to the c0 baryon state with the
quantum numbers (2S, 21 +).
4 Conclusion
In this article, we assign c(3000), c(3050), c(3066),
c(3090) and c(3119) to the Pwave charmed baryon states
with J P = 21 −, 21 −, 23 −, 23 − and 25 −, respectively, and we
study their masses and pole residues with the QCD sum
rules in detail by introducing an explicit relative Pwave
between the two constituents of the light diquarks. The
predictions support assigning c(3050), c(3066), c(3090)
and c(3119) to the Pwave baryon states with J P = 21 −,
23 −, 3 − and 25 −, respectively, where the two constituents of
2
the light diquark are in relative Pwave; while the c(3000)
can be assigned to the Pwave charmed baryon state with
J P = 21 −, where the two constituents of the light diquark
are in relative Swave.
Acknowledgements This work is supported by National Natural
Science Foundation, Grant Number 11375063.
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