#### Interpretation of the new \(\Omega _c^{0}\) states via their mass and width

Eur. Phys. J. C
0c states via their mass and width
S. S. Agaev 2
K. Azizi 1
H. Sundu 0
0 Department of Physics, Kocaeli University , 41380 Izmit , Turkey
1 Department of Physics, Dogˇus ̧ University , Acibadem-Kadiköy, 34722 Istanbul , Turkey
2 Institute for Physical Problems, Baku State University , 1148 Baku , Azerbaijan
The masses and pole residues of the ground and first radially excited c0 states with spin-parities J P = 1/2+, 3/2+, as well as P-wave c0 with J P = 1/2−, 3/2are calculated by means of the two-point QCD sum rules. The strong decays of c0 baryons are also studied and the widths of these decay channels are computed. The relevant computations are performed in the context of the full QCD sum rules on the light cone. The results obtained for the masses and widths are confronted with recent experimental data of the LHCb Collaboration, which allow us to interpret c(3000)0, c(3050)0, and c(3119)0 as the excited css baryons with the quantum numbers (1 P, 1/2−), (1 P, 3/2−), and (2S, 3/2+), respectively. The (2S, 1/2+) state can be assigned either to the c(3066)0 state or the c(3090)0 excited baryon.
1 Introduction
The observation by the LHCb Collaboration of new
narrow states c0 in the c+ K − invariant mass distribution is
one of the intriguing discoveries in physics of the heavy
baryons [
1
]. Preliminary analysis indicates that these five
neutral resonances are composed of css quarks and may be
orbitally/radially excited states of the c0 baryons with spins
1/2 and 3/2. Let us note that till the LHCb data experimental
information as regards baryons with css content was limited
to the masses of the c0 and c(2770)0 particles [
2
]
m = 2695.2 ± 1.7 MeV, m = 2765.9 ± 2.0 MeV, (1)
which were considered as the ground states with the spin–
parities J P = 1/2+ and 3/2+, respectively.
Theoretical investigations performed in the context of
different approaches, and predictions obtained for the
spectroscopic parameters provide incomparably more detailed
information on the features of the c0 baryons than
experimental data [
3–25
]. In fact, the masses of the ground-state
and radially/orbitally excited heavy baryons including the
c0 particles were calculated using the relativistic quark
models [
3,9,10
], the QCD sum rule method [
4,5,7,8,14–
16,21,22,24,25
], the heavy-quark effective theory (HQET)
[
6
], various quark models [
11–13,17,18,23
], and lattice
simulations [
19,20
]. The strong couplings and transitions of the
heavy flavored baryons, their magnetic moments and
radiative decays also attracted interest of physicists [
24–34
]. It
is worth noting that in some of these theoretical studies
different assumptions were made on the structure of the heavy
baryons. For example, in Refs. [
9,10
] a
heavy-quark–lightdiquark picture was employed in the relativistic quark model.
In other work, QCD sum rule calculations were carried out
in the context of the HQET [
7,8,21,22
].
The discovery of five new c0 particles by the LHCb
Collaboration changed the experimental situation and
stimulated theoretical activity to explain the observed states. These
states were seen as resonances in the c+ K − invariant mass
distribution. Their masses do not differ considerably from
each other and are within the range M = 3000−3150 MeV.
The transition c0 → c+ K − may be considered as main
decay modes of the c0 states, widths of which equal to a few
MeV.
The LHCb did not provide information on the spin–
parities of the new states, which is an important problem
of ongoing theoretical investigations. Thus, in [
35
] we have
calculated the masses of the ground states and first radial
excitations of c0 with J P = 1/2+ and 3/2+, and found
that the particles c(3066) and c(3119) can be
considered as the radially excited css baryons with the quantum
numbers (2S, 1/2+) and (2S, 3/2+), respectively. In
calculations we have employed the two-point QCD sum rule
method by invoking in the analysis general expressions for
the currents to interpolate the c0 baryons with spins 1/2 and
3/2. Our results correctly describe the masses of the ground
states c0 and c(2770)0, and they agree with two of the
recent experimental data of the LHCb Collaboration. It is
interesting that predictions obtained in some of previous
theoretical studies agree with new LHCb data and our results
(more detailed information can be found in Ref. [
35
], and in
references therein).
The problems connected with the c0 states have been
addressed in Refs. [
36–48
]. The new particles have been
assigned to the P-wave c baryons in Ref. [
36
], where
the authors evaluated the widths of their decay channels.
Calculations there have been performed in the framework
of HQET using the sum rule approach. In Refs. [
37,38
]
c(3000), c(3050), c(3066), c(3090) and c(3119)
have been interpreted as P-wave excited states of the c0
baryons with the spin–parities 1/2−, 1/2−, 3/2−, 3/2− and
5/2−, respectively. In Ref. [37] an alternative set of
assignments, namely 3/2−, 3/2−, 5/2−, 1/2+ and 3/2+ is made
for these states, as well. In this case 1/2− states are expected
around 2904 and 2978 MeV. In both of Refs. [
37,38
] the
authors utilized the heavy-quark–light-diquark model for c
baryons. On the basis of lattice simulations the same
conclusions have been drawn also in Ref. [39]. Attempts have
been made to classify new states as five-quark systems or
S-wave pentaquark molecules with J P = 1/2−, 3/2− and
5/2− [
40,41
]. The possible pentaquark interpretation of the
c0 baryons on the basis of the quark-soliton model has been
suggested also in Ref. [42].
The explorations carried out in the context of a constituent
quark model have allowed authors of Ref. [
43
] to conclude,
that c(3000) and c(3090) can be considered as states with
1/2−, c(3050) and c(3066) as the baryons with 3/2− and
5/2−, whereas the c(3119) might correspond to one of the
radial excitations (2S, 1/2+) or (2S, 3/2+). In Ref. [
44
] the
first three states from the LHCb range of excited c baryons
have been classified as P-wave states with 1/2−, 5/2− and
3/2−, whereas last two particles have been assigned to be
2S states with spin–parities 1/2+ and 3/2+, respectively.
These states have been analyzed as the P-wave excitation of
the c0 baryons with spin–parities 1/2−, 1/2−, 3/2−, 3/2−
and 5/2− also in Ref. [
45
]. The studies have been performed
using the two-point sum rule method by introducing relevant
interpolating currents.
The newly discovered c0 states, their spin–parities has
been analyzed in Refs. [
46–48
], too. Thus, studies in Ref. [
46
]
showed that five resonances c0 can be grouped into the 1P
states with negative parity, i.e. the resonances c(3000) and
c(3090) have been considered there as (1 P, 1/2−) states,
c(3066) and c(3119) as resonances with (1 P, 3/2−),
and c(3050) as (1 P, 5/2−) state. The alternative
explanation has been suggested in Ref. [
47
], where the resonances
c(3066) and c(3090) have been interpreted as 1 P-wave
states with the spin–parity J P = 3/2− or J P = 5/2−.
Starting from decay features of the remaining three
resonances in Ref. [
47
] the authors have assigned them to be
1Dwave c0 states. Finally, in Ref. [
48
] the resonances c(3000)
and c(3066) have been classified as the (1 P, 1/2−) and
(1 P, 3/2−) states, respectively.
As is seen, a variety of suggestions made on the structures
of the c0 states, methods and schemes used to compute their
parameters, and the obtained predictions for the spin–parities
of these baryons is quite impressive. In the present work we
are going to extend our previous paper by including into the
analysis P-wave (1 P, 1/2−) and (1 P, 3/2−) states, as well.
We will evaluate the masses and pole residues of the ground
and four excited c0 states. We will also calculate the widths
of the c0 → + K − decays using the light-cone sum rule
(LCSR) method, which is one of the powerful
nonperturbative approaches to evaluating the parameters of exclusive
processes [
49
]. Calculations will be performed by taking into
account the K meson’s distribution amplitudes (DAs). The
states extracted from analysis mass and decay width of c0
will be confronted with existing LHCb data and the
predictions obtained in theoretical papers. This will allow us to
identify c(3000), c(3050), c(3066), and c(3119) by
fixing their quantum numbers.
This work is structured in the following way. In Sect. 2 we
calculate the masses and pole residues of the ground-state and
orbitally/radially excited c0 baryons with the quantum
numbers (1S, 1/2+) ⇒ c, (1 P, 1/2−) ⇒ c−, (2S, 1/2+)
⇒ c, and (1S, 3/2+) ⇒ c, (1 P, 3/2−) ⇒ c−,
(2S, 3/2+) ⇒ c . To this end, we employ the two-point
sum rule method. In Sect. 3 we analyze c− c+ K − and
c c+ K − vertices to evaluate the corresponding strong
couplings g − K and g K , and calculate widths of c− →
c+ K − and c → c+ K − decays. The similar
investigations are carried out in Sect. 4 for the vertices containing c0
baryons with J P = 3/2+ and J P = 3/2−. Here we find
widths of the processes c− → c+ K − and c → c+ K −.
In this section we also analyze the c → c+ K − decay,
which is kinematically allowed only for c baryon. Section
5 is reserved for a brief discussion of the obtained results. It
contains also our concluding remarks. Explicit expressions
of the correlation functions derived in the present work, as
well as the quark propagators used in the calculations are
presented in the appendix.
2 Masses and pole residues of the
0c states
In this section we evaluate the masses and pole residues of
the spin 1/2 and 3/2 ground-state and excited c (hereafter,
we omit the superscript 0 in c0) baryons by means of the
two-point sum rule method.
The sum rules necessary to find the masses and residues of
the c0 baryons can be derived using the two-point correlation
function
(μν)( p) = i
d4x ei px 0|T {η(μ)(x )η(ν)(0)}|0 ,
where η(x ) and ημ(x ) are the interpolating currents for c
states with spins J = 1/2 and J = 3/2, respectively. They
have the following forms:
η = − 21 abc{(saT C cb)γ5sc + β(saT C γ5cb)sc
− [(caT C sb)γ5sc + β(caT C γ5sb)sc]},
1
ημ = √3
+ (caT C γμsb)sc}.
abc{(saT C γμsb)cc + (saT C γμcb)sc
In the expressions above C is the charge conjugation operator.
The current η(x ) for the 1/2 baryons contains an arbitrary
auxiliary parameter β, where β = −1 corresponds to the
Ioffe current.
We start from the spin 1/2 baryons and calculate the
correlation function Phys( p) in terms of the physical parameters
of the states under consideration and determine OPE( p)
employing the quark propagators. Because, the current η(x )
couples not only to states c and c, but also to c−,
in the physical side of the sum rule we explicitly take
into account their contributions by adopting the
“groundstate+first orbitally+first radially excited states+continuum”
scheme: We follow an approach applied recently to a
calculation of the masses and residues of radially excited octet and
decuplet baryons in Refs. [
50,51
]. In this work the authors
got results which are compatible with existing experimental
data on the masses of the radially excited baryons, and they
demonstrated that besides ground-state baryons the QCD
sum rule method can be successfully applied to the
investigation of their excitations as well.
Thus, we find
Phys( p) =
0|η| c( p, s) c( p, s)|η|0
m2 − p2
+
+
0|η| c−( p, s) c−( p, s)|η||0
m2 − p2
0|η| c( p, s ) c( p, s )|η||0
m 2 − p2
+ · · · , (5)
where m, m, m and s, s, s are the masses and spins of
the c, c− and c baryons, respectively. The dots denote
contributions of higher resonances and continuum states. In
Eq. (5) the summations over the spins s, s, s are implied.
We proceed by introducing the matrix elements
0|η| (c )( p, s( )) = λ( )u( )( p, s( )),
0|η| c−( p, s) = λγ5u−( p, s).
Here λ, λ and λ are the pole residues of the c, c− and c
states, respectively. Using Eqs. (5) and (6) and carrying out
the summation over the spins of the 1/2 baryons
(2)
(3)
(4)
(6)
u( p, s)u( p, s) = /p + m,
s
we obtain
Phys( p) =
λ2( /p + m)
m2 − p2 +
λ2( /p − m)
m2 − p2
+
λ 2( /p + m )
m 2 − p2
+ · · ·
(7)
(8)
The Borel transformation of this expression is
B
Phys( p) = λ2e− M2 ( /p + m)
m2
m2 m 2
+ λ2e− M2 ( /p − m) + λ 2e− M2 ( /p + m ). (9)
As is seen, it contains the structures ∼ /p and ∼ I . In order
to derive the sum rules we use both of them and find from
the terms ∼ /p
m2 m2 m 2
λ2e− M2 + λ2e− M2 + λ 2e− M2 = B 1OPE( p),
(10)
and from the terms ∼ I
m2 m2 m 2
λ2me− M2 − λ2me− M2 + λ 2m e− M2 = B 2OPE( p), (11)
where B 1OPE( p) and B 2OPE( p) are the Borel
transformations of the same structures in OPE( p) computed
employing the quark propagators, as has been explained above. It
is assumed that continuum contributions are subtracted from
the right-hand sides of Eqs. (10) and (11) utilizing the quark–
hadron duality assumption.
The derived sum rules contain six unknown parameters of
the ground-state and excited baryons. Therefore, from Eqs.
(10) and (11) we determine the parameters (m, λ) of the
ground-state c baryon by keeping there only the first terms,
and choosing accordingly the continuum threshold parameter
s0 in 1OPE(M 2, s0) and 2OPE(M 2, s0): this is the sum rule
computation within the “ground state+continuum” scheme.
At the next step, we retain in the sum rule terms
corresponding to c and c− baryons, but we treat (m, λ) as input
parameters to extract (m, λ): the continuum threshold now
is chosen as s0 > s0. Finally, the set of (m, λ) and (m, λ) is
utilized in the full version of the sum rules to find parameters
(m , λ ) of the c baryon, with s0 > s0 being the relevant
continuum threshold.
A similar analysis with additional technical details is valid
also for the spin 3/2 baryons. Indeed, in this case we use the
matrix elements
0|ημ| c∗( )( p, s( )) = λ( )u(μ)( p, s( )),
0|ημ| c∗−( p, s) = λγ5uμ−( p, s)
(12)
where uμ( p, s) are the Rarita–Schwinger spinors, and we
carry out the summation over s by means of the formula
The interpolating current ημ couples to spin-1/2 baryons,
therefore the sum rules contain contributions arising from
these terms. Their undesired effects can be eliminated by
applying a special ordering of the Dirac matrices (see for
example Ref. [
50
]). It is not difficult to demonstrate that the
structures ∼ /pgμν and ∼gμν are free of contaminations and
formed only due to contributions of spin-3/2 baryons. In
order to derive the sum rules for the masses and pole residues
of the ground-state and excited c0 baryons with spin–parities
3/2− and 3/2+, we employ only these structures and the
corresponding invariant amplitudes.
The correlation functions ( p) and μν ( p) should be
found using the quark propagators: this is necessary to get
the QCD side of the sum rules. We calculate them employing
the general expression given by Eq. (2) and currents defined
in Eqs. (3) and (4). The results for OPE( p) and μOνPE( p) in
terms of the s and c-quarks’ propagators are written down in
the appendix. Here we also present analytic expressions of
the propagators, themselves. Manipulations to calculate
correlators using propagators in the coordinate representation,
to extract relevant two-point spectral densities and perform
the continuum subtraction, are well known and were
extendedly described in the existing literature. Therefore, we do
not concentrate further on the details of these rather lengthy
computations.
The sum rules contain the vacuum expectations values
of the different operators and masses of the s and c-quarks,
which are input parameters in the numerical calculations. The
vacuum condensates are well known: for the quark and mixed
condensates we use ss = −0.8 × (0.24 ± 0.01)3 GeV3,
sgs σ Gs = m20 ss , where m02 = (0.8 ± 0.1) GeV2,
whereas for the gluon condensate we utilize αs G2/π =
(0.012 ± 0.004) GeV4. The masses of the strange and
charmed quarks are chosen equal to ms = 96+−84 MeV and
mc = (1.27 ± 0.03) GeV, respectively. These parameters
and their different products determine an accuracy of
performed numerical computations: In the present work we take
into account terms up to ten dimensions.
The sum rules depend also on the auxiliary parameters M 2
and s0, which are not arbitrary, but can be changed within
special regions. Inside of these working regions the
convergence of the operator product expansion, dominance of
the pole contribution over remaining terms should be
satis(n, J P )
M2 (GeV2)
s0 (GeV2)
m c (MeV)
λ c · 102 (GeV3)
(n, J P )
M2 (GeV2)
s0 (GeV2)
m c (MeV)
λ c · 102 (GeV3)
fied. The prevalence of the perturbative contribution in the
sum rules, and the relative stability of the extracted results
are also among the restrictions of the calculations. At the
same time, the Borel and continuum threshold parameters
are the main sources of ambiguities, which affect the final
predictions considerably. These uncertainties may amount
to 30% of the results and are unavoidable features of the sum
rules’ predictions. For spin-1/2 particles there is an
additional dependence on β, stemming from the expression of
the interpolating current η(x ). The choice of an interval for
β should also obey the clear requirement: we fix the
working region for β by demanding a weak dependence of our
results on its choice. The results for the spin-1/2 particles
are obtained by varying β = tan θ within the limits
−0.75 ≤ cos θ ≤ −0.45, 0.45 ≤ cos θ ≤ 0.75,
(14)
where we have achieved the best stability of our predictions.
Let us note that for the famous Ioffe current cos θ = −0.71.
Results obtained in this work for the masses and residues
of the spin-1/2 and 3/2 c baryons are presented in Tables
1 and 2, respectively. Here we also provide the working
windows for the parameters M 2 and s0 used in extracting m
and λ. The masses and pole residues of the radially excited
baryons (2S, 1/2+) and (2S, 3/2+) slightly differ from
predictions obtained for these states in our previous work
[
35
]. These unessential differences can be explained by
features of schemes adopted in Ref. [
35
] and in the present
work. In fact, in Ref. [
35
] the parameters of the radially
excited states were extracted within the
“ground-state+2Sstate+continuum” approximation, whereas now we apply the
“ground-state+1P+2S-states+continuum” scheme: an
additional baryon included into analysis, naturally affects final
predictions.
Fig. 1 The mass of the
ground-state c baryon as a
function of the Borel parameter
M2 at fixed s0 (left panel), and
as a function of the continuum
threshold s0 at fixed M2 (right
panel)
Fig. 2 The dependence of the
c baryon’s residue λ c on the
Borel parameter M2 at chosen
values of s0 (left panel), and on
the s0 at fixed M2 (right panel)
Fig. 3 The same as in Fig. 1,
but for the orbitally excited c−
baryon
In order to explore the sensitivity of the obtained results on
the Borel parameter M 2 and continuum threshold s0, in Figs.
1, 3 and 4 we depict the c, c− and c baryon masses as
functions of these parameters. It is seen that the dependence
of the masses on the parameters M 2 and s0 is mild. In Fig. 2
we show, as an example, the dependence of the ground-state
c baryon’s residue on the auxiliary parameters of the sum
rule computations. The observed behavior of λ on M 2 and
s0 is typical for such kind of quantities: the systematic errors
are within limits accepted in the sum rule method. The sum
rule predictions for the masses and residues of the spin-1/2
baryons c, c− and c demonstrate a similar dependence on
the Borel parameter M 2 and continuum threshold s0;
therefore we refrain from providing corresponding graphics here.
It is instructive to explore the “convergence” of the
iterative process used in the present work to evaluate parameters
of the c baryons. It is well known that the ground state
contributes dominantly to the spectral density. The excited
states included into the sum rules are sub-leading terms. To
quantify this statement we calculate the pole contribution
(PC) to the sum rules in the successive stages of the
iterative process to reveal effects due to the ground-state and
excited baryons. To this end, we fix the Borel parameter
M 2 = 4.5 GeV2 (for spin-1/2 baryons also cos θ = −0.5)
and compute the PC at each stage using for the continuum
threshold s0 its upper limit from the relevant intervals (see
Tables 1, 2). We start from the spin-1/2 baryons and from
the “ground-state+continuum” phase, and find that PC
arising from c equals 44% of the result. Computations in the
“ground-state+1P state+continuum” step allows us to fix the
total PC from c and c− baryons at the level of 58% of the
whole prediction, or 14% effect appearing due to c−. Finally,
in the “ground-state+1P+2S states+continuum” stage the
PC arising from the c, c− and c baryons amounts to
68% of the sum rules, which indicates 10% contribution of
the c baryon. The same analysis carried out for the
spin3/2 baryons leads to the following results: the ground-state
baryon c forms 41% of the sum rule, whereas the excited
Fig. 4 The same as in Fig. 1,
but for the radially excited c
baryon
states c− and c constitute 15 and 9% of the whole
prediction, respectively. It is worth noting that the dependence
of the estimations presented on M 2 and cos θ is negligible.
It is seen that the procedure adopted in the present work
is consistent with general principles of the sum rule
calculations. Because contributions of the higher excited states
decrease, it is legitimate to restrict analysis by considering
only two of them. But there are other reasons to truncate
the iterative process at this phase. Indeed, the next
spin1/2 excited baryons in this range should be (2 P, 1/2−) and
(3S, 1/2+) states. By taking into account the mass splitting
between c and the first orbitally and radially excited c−
and c baryons, it is not difficult to anticipate that the masses
of the (2 P, 1/2−) and (3S, 1/2+) states will be higher than
recent LHCb data. The same arguments are valid for the
spin3/2 baryons. The parameters of the higher excited states of c
and c baryons may provide valuable information on their
properties, which are interesting for hadron spectroscopy;
nevertheless, this task is beyond the scope of the present
investigation.
Based on the results for the masses of c0 baryons, taking
into account the central values in the sum rules’ predictions,
and comparing them with the LHCb data we assign, at this
stage of our investigations, the orbitally and radially excited
c0 baryons to the newly discovered states, as is shown in
Table 3. Thus, we have correlated the excited c0 baryons to
states which were recently observed by the LHCb
Collaboration. Nevertheless, we consider this assignment as a
preliminary one, because the systematic errors in the sum rule
calculations are significant, and robust conclusions can be drawn
onc0ly→afterc+anKa−ly.sis of the width of decays c0 → c+ K − and
3
−c and
c transitions to
+c K −
The results for the masses of the excited c0 baryons show
that all of them are above the c+ K − threshold. Hence, these
four states can decay through the c0 → c+ K − channels.
In this section we study the vertices c− c+ K − and
c c+ K −, and calculate the corresponding strong couplings
g − K and g K (the index c is omitted from the baryons
for simplicity), which are necessary to calculate the widths
of the decays c− → c+ K − and c → c+ K −. To this end
we introduce the correlation function
( p, q) = i
d4x ei px K (q)|T {η c (x )η(0)}|0 ,
(15)
where η c (x ) is the interpolating current for the c+ baryon.
The c+ belongs to the anti-triplet configuration of the
heavy baryons with a single heavy quark. Its current is
antisymmetric with respect to exchange of the two light quarks,
and it has the form
1
η c = √6
+ (uaT C cb)γ5sc + β(uaT C γ5cb)sc
+ (caT C sb)γ5uc + β(caT C γ5sb)uc}.
abc{2(uaT C sb)γ5cc + 2β(uaT C γ5sb)cc
(16)
We first represent the correlation function ( p, q) using the
parameters of the involved baryons and in this manner
determine the phenomenological side of the sum rules. We get
Phys( p, q) =
0|η c | c( p, s)
p2 − m2 c
×
c−( p , s )|η|0
p 2 − m2
+
0|η c | c( p, s)
p2 − m2 c
× K (q) c( p, s)| c( p , s )
c( p , s )|η|0
p 2 − m 2
K (q) c( p, s)| c−( p , s )
+ · · · ,
(17)
where p = p + q, p and q are the momenta of the c, c
baryons and K meson, respectively. In the last expression
m c is the mass of the c+ baryon. The dots in Eq. (17)
stand for contributions of the higher resonances and
continuum states. Note that in principle the ground-state c0 baryon
can also be included into the correlation function. However,
its mass remains considerably below the threshold c+ K −
and its decay to the final state c+ K − is not kinematically
allowed.
We introduce the matrix element of the
c+ baryon
0|η | c( p, s) = λ c u( p, s),
)
V
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M
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Then using the matrix elements of the c− and c baryons,
and performing the summation over s and s , we recast the
function Phys( p, q) into the form
Phys( p, q) = − ( p2 −gm−2 c )K(λp 2c λ− m2) (/p + m c )
g K λ c λ
× (/p + q/ + m)γ5 + ( p2 − m2 c )( p 2 − m 2)
× (/p + m c )γ5(/p + q/ + m ) + . . . .
(19)
The double Borel transformation on the variables p2 and p 2
applied to Phys( p, q) yields
B Phys( p, q) = g − K λ c λe−m2/M12 e−m2 c /M22
× {q/ /pγ5 − m c q/γ5 − (m + m c ) /pγ5
+ [m2K − m(m + m c )]γ5} + g K λ c λ
× e−m 2/M12 e−m2 c /M22 {q/ /pγ5 − m c q/γ5
+ (m − m c ) /pγ5
+ [m2K − m (m − m c )]γ5},
where m2K = q2 is the mass of the K meson, and M12 and
M22 are the Borel parameters.
As is seen, there are different structures in Eq. (20), which
can be used to derive the sum rules for the strong couplings.
We work with the structures q/ /pγ5 and /pγ5. Separating the
relevant terms in the Borel transformation of the correlation
function OPE( p, q) computed employing the quark–gluon
degrees of freedom we get
em2/M12 em2 c /M22
g − K = λ c λ(m + m) [(m − m c )B 1OPE
− B 2
(20)
OPE
and
g
em 2/M12 em2 c /M22
K = λ c λ (m + m) [(m + m c )B 1
OPE
+ B 2OPE],
(22)
where 1OPE( p2, p 2) and 2OPE( p2, p 2) are the
invariant amplitudes corresponding to structures q/ /pγ5 and /pγ5,
respectively.
The general expressions obtained above contain two Borel
parameters M12 and M12. But in our analysis we choose
M12(2) = 2M 2,
which is traditionally justified by a fact that the masses of the
involved heavy baryons c0 and c+ are close to each other.
Using the couplings g − K and g K we can easily
calculate the widths of the c− → c+ K − and c → c+ K −
decays. After some computations we obtain
( c →
c+ K −) = 8gπ2 mK2 [(m − m c )2 − m2K ]
× f (m , m c , m K ).
(23)
(24)
(25)
In the expressions above the function f (x , y, z) is given by
1
f (x, y, z) = 2x
x4 + y4 + z4 − 2x2 y2 − 2x2z2 − 2y2z2.
The QCD side of the correlation function OPE( p, q) can
be found by contracting quark fields, and inserting into the
obtained expression the relevant propagators. The remaining
non-local quark fields saαubβ have to be expanded using
α β = 41 βiα(sa i ub),
a b
s u
where i = 1, γ5, γμ, i γ5γμ, σμν /√2 is the full set
of Dirac matrices. Sandwiched between the K-meson and
vacuum states these terms, as well as the ones generated by
insertion of the gluon field strength tensor Gλρ (uv) from
quark propagators, give rise to the K-meson’s distribution
amplitudes of various quark–gluon contents and twists. Both
in analytical and numerical calculations we take into account
the K-meson DAs up to twist-4 and employ their explicit
expressions from Ref. [
52
].
Apart from the parameters in the distribution amplitudes,
the sum rules for the couplings depend also on numerical
values of the c+ baryon’s mass and pole residue. In numerical
calculations we utilize
m c = 2467.8+−00..46 MeV, λ c = 0.054 ± 0.020 GeV3, (26)
from Refs. [
2,53
], respectively. The Borel and threshold
parameters for the decay of a baryon are chosen exactly
as in computations of its mass. The auxiliary parameters
β in the interpolating currents of c0 and c+ baryons are
taken equal to each other and varied within the limits cos θ ∈
[−0.75, −0.3] and [0.3, 0.75], which are a little bit extended
compared to the mass rules (see Eq. (14)).
Numerical calculations lead to the following values for
the strong couplings:
g − K = 0.48 ± 0.09, g
The predictions for the widths of c− → c+ K − and c →
c+ K − decays are collected in Table 4 and compared with
the LHCb data and results of other theoretical work.
The decays of the spin-3/2 baryons c and c− to c+ K −
can be analyzed as has been done for the spin-1/2 baryons.
Additionally, we take into account that the radially excited
c baryon can decay through the channel c → c+ K −, as
well. c+ is a spin-1/2 ground-state baryon, and it belongs to
the sextet part of the heavy baryons. Its interpolating current
should be symmetric under exchange of the two light quarks.
In this section we consider these three decay processes.
Again we start from the same correlation function, but
with the current η(x ) replaced by ημ(x ):
μ( p, q) = i
d4x ei px K (q)|T {η c (x )ημ(0)}|0 .
(28)
We define the strong couplings g − K and g
the matrix elements
K through
K (q) c( p, s)| c∗−( p , s ) = g − K u( p, s)γ5uα( p , s )qα,
K (q) c( p, s)| c∗ ( p , s ) = g K u( p, s)uα( p , s )qα,
and for
Phys( p, q) we obtain the following expression:
μ
g − K λ c λ
Pμhys( p, q) = ( p2 − m2 c )( p 2 − m2)
qα( /p + m c )γ5
g K λ c λ
× ( /p + q/ + m)Fαμ(m)γ5 − ( p2 − m2 c )( p 2 − m 2)
× qα( /p + m c )( /p + q/ + m )Fαμ(m ) + · · · ,
where we have used the shorthand notation
1 2
Fαμ(m) = gαμ − 3 γαγμ − 3m2 ( pα + qα)( pμ + qμ)
1
+ 3m [( pα + qα)γμ − ( pμ + qμ)γα].
For the Borel transformation of
Phys( p, q) we get
μ
(29)
(30)
(31)
c(3000)0 (MeV)
To extract the sum rules we choose the structures q/ /pγμ and
q/qμ. The same structures should be isolated in B μQCD( p, q)
and matched with the ones from B Pμhys( p, q). The final
formulas for the strong couplings are rather lengthy; therefore
we refrain from providing their explicit expressions.
Knowledge of the strong couplings allows us to find the
widths of the corresponding decay channels. Thus, the width
of the c∗− → c+ K − decay can be obtained:
whereas for ( c∗ →
c+ K −) we get
( c∗− →
( c∗ →
c+ K −) = g224π−mK2 [(m − m c )2 − m2K ]
× f 3(m, m c , m K ),
c+ K −) = 2g42π mK2 [(m + m c )2 − m2K ]
× f 3(m , m c , m K ).
In order to find g K corresponding to the vertex
c c+ K −, we again use the correlation function μ( p, q),
but with the current η c ,
1
η c = − √2
−[(caT C sb)γ5uc + β(caT C γ5sb)uc]}.
abc{(uaT C cb)γ5sc + β(uaT C γ5cb)sc
We skip the details and provide below only the final
expression for the double Borel transformation of the term ∼ q/ /pγμ
in Pμhys( p, q), which is utilized to derive the required sum
rule
Phys( p, q) = −
B μ
g
K λ c λ e−m 2/M12 e−m2 c /M22
6m
× [(m + m c )2 − m2K ]q/ /pγμ.
In Eq. (36) m c and λ c are the
residue, respectively.
c+ baryon’s mass and pole
(33)
(34)
(35)
(36)
The coupling g K and widths of the decay
c+ K − are given by the expressions
and
( c →
c+ K −) = 2g42π mK2 [(m + m c )2 − m2K ]
× f 3(m , m c , m K ).
In numerical computations for the mass and residue of the
c+ baryon we use
m c = 2575.6 ± 3.1 MeV, λ c = 0.055 ± 0.016 GeV3,
(38)
which are borrowed from Refs. [
2,16
], respectively.
Numerical computations for the strong couplings yield (in
GeV−1)
( c →
( c →
c+ K −) = 0.6 ± 0.2 MeV,
c+ K −) = 1.3 ± 0.4 MeV,
c+ K −) = 0.6 ± 0.2 MeV.
K = 0.75 ± 0.20,
The predictions obtained for the widths of the c− and c
baryons are shown in Table 4: for c we present there a sum
of its two possible decay channels.
5 Discussion and concluding remarks
In the present work we have investigated the newly
discovered c0 baryons by means of QCD sum rule method. We
have calculated masses and pole residues of the ground-state
and first orbitally and radially excited c0 baryons with the
spin-1/2 and -3/2. To this end, we have employed two-point
QCD sum rule method and started from the ground-state
baryons. We have derived required sum rules for m c and
(37)
(39)
(40)
λ c using two different structures in the relevant correlation
functions. The masses and residues of the ground states have
been treated as input information in the sum rules obtained
to evaluate parameters of the first orbitally excited baryons.
The same manipulations have been made in the case of the
radially excited states.
The predictions for the masses and residues obtained in
the present work almost coincide with results of our
previous paper [
35
] excluding numerically small modifications
in parameters of the radially excited baryons. This may be
expected, because in the present work we have employed
a more sophisticated iterative scheme. Nevertheless, the
assignments for c0 made in Ref. [
35
] remain valid here as
well (see Table 3).
The widths of the c0 → c+ K − decays, calculated in
the context of the QCD full LCSR method, have allowed
us to confirm an essential part of our previous conclusions.
Thus, the mass and width of the (1 P, 1/2−) and (2S, 3/2+)
states are in a nice agreement with the same parameters of the
c(3000) and c(3119) baryons, respectively. The mass of
the orbitally excited state (1 P, 3/2−) is close to c(3050).
But it may be considered also as the c(3066) baryon. A
decisive argument in favor of c(3050) is the width of the state
(1 P, 3/2−), which is in excellent agreement with LHCb
measurements. As a result, we do not hesitate to confirm
our previous assignment of c(3050) to be the baryon with
J P = 3/2−. The situation with the orbitally excited state
(2S, 1/2+) is not quite clear. In fact, its mass and width
allow one to interpret it either as c(3066) or c(3090). We
have kept in Tables 3 and 4 our previous classification of the
(2S, 1/2+) state as the c(3066) baryon, but its
interpretation as c(3090) is also legitimate.
The masses of the excited c0 baryons were predicted in
the theoretical literature long before the recent LHCb data.
Most of them were made in the framework of different quark
models (see, for example, Refs. [
10, 12, 17, 23
]). Within the
two-point QCD sum rule method problems of the c0 baryons
were addressed in Refs. [
14–16, 25
], where the masses of the
ground-state and excited c0 were found. Obtained in Refs.
[
25
] mass of c0 baryon with J P = 3/2−
m c = 3080 ± 120 GeV
(41)
within errors agrees both with LHCb data and our present
result for (1 P, 3/2−) state.
After discovery of the LHCb Collaboration, parameters
of new states in the context of QCD sum rule approach have
also been investigated in Refs. [
43, 48
]. In Ref. [43] all of
five states have been considered as negative-parity baryons,
whereas in Ref. [
48
] only two of them have been classified
as negative-parity states. But lack of information as regards
the widths of c0 makes the comparison of their results with
available LHCb data incomplete.
The situation around excited c0 states remains
controversial and unclear. Additional efforts of experimental
collaborations are necessary to explore the c0 states, mainly to fix
their spin–parities.
Acknowledgements K. A. thanks Dogˇus¸ University for the partial
financial support through the Grant BAP 2015-16-D1-B04. The work
of H. S. was supported partly by BAP Grant 2017/018 of Kocaeli
University.
Open Access This article is distributed under the terms of the Creative
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ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
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Appendix: The correlation functions and quark
propagators
The correlation function for the spin 1/2 baryons
( p) = i
d4 x ei px 0|T {η(x )η(0)}|0 ,
in terms of the quark propagators takes the following form:
OPE( p) =
d4xeipx 4 {−γ5 Ssca (x)Scab (x)Ssbc (x)γ5
+ γ5 Ssca (x)Scbb (x)Ssac (x)γ5 + γ5 Sscb (x)Scaa (x)Ssbc (x)γ5
− γ5 Sscb (x)Scba (x)Ssac (x)γ5 + γ5 Sscc (x)γ5
× [−Tr[Scaa (x)Ssbb (x)] + Tr[Scab (x)Ssba (x)] − Tr[Ssaa (x)Scbb (x)]
+ Tr[Ssab (x)Scba (x)]] + β[−γ5 Ssca (x)γ5 Scab (x)Ssbc (x)
+ γ5 Ssca (x)γ5 Scbb (x)Ssac (x) + γ5 Sscb (x)γ5 Scaa (x)Ssbc (x)
−γ5 Sscb (x)γ5 Scba (x)Ssac (x) − Ssca (x)Scab (x)γ5 Ssbc (x)γ5
+ Ssca (x)Scbb (x)γ5 Ssac (x)γ5 + Sscb (x)Scaa (x)γ5 Ssbc (x)γ5
−Sscb (x)Scba (x)γ5 Ssac (x)γ5 + γ5 Sscc (x)[−Tr[Scaa (x)γ5 Ssbb (x)]
+Tr[Scab (x)γ5 Ssba (x)] − Tr[Ssaa (x)γ5 Scbb (x)]
+Tr[Ssab (x)γ5 Scba (x)]] + Sscc (x)γ5[−Tr[Scaa (x)Ssbb (x)γ5]
+ Tr[Scab (x)Ssba (x)γ5] − Tr[Ssaa (x)Scbb (x)γ5]
+Tr[Ssab (x)Scba (x)γ5]]] + β2[−Ssca (x)γ5 Scab (x)γ5 Ssbc (x)
+ Ssca (x)γ5 Scbb (x)γ5 Ssac (x) + Sscb (x)γ5 Scaa (x)γ5 Ssbc (x)
−Sscb (x)γ5 Scba (x)γ5 Ssac (x) + Sscc (x) × [Tr[Scba (x)γ5 Ssab (x)γ5]
−Tr[Scbb (x)γ5 Ssaa (x)γ5] + Tr[Ssba (x)γ5 Scab (x)γ5]
− Tr[Scbb (x)γ5 Scaa (x)γ5]]}.
(A.1)
For the correlation function of spin 3/2 baryons we get
μOνPE( p) =
d4xei px 3 {Scca (x)γν Ssab (x)γμ Ssbc (x)
− Scca (x)γν Ssbb (x)γμ Ssac (x) − Sccb (x)γν Ssaa (x)γμ Ssbc (x)
+ Sccb (x)γν Ssba (x)γμ Ssac (x) + Ssca (x)γν Scab (x)γμ Ssbc (x)
− Ssca (x)γν Scbb (x)γμ Ssac (x) + Ssca (x)γν Ssab (x)γμ Scbc (x)
− Ssca (x)γν Ssbb (x)γμ Scac (x) − Sscb (x)γν Scaa (x)γμ Ssbc (x)
+ Sscb (x)γν Scba (x)γμ Ssac (x) − Sscb (x)γν Ssaa (x)γμ Scbc (x)
+ Sscb (x)γν Ssba (x)γμ Scac (x) − Sscc (x)[Tr[Scba (x)γν Ssab (x)γμ]
− Tr[Scbb (x)γν Ssaa (x)γμ] + Tr[Ssba (x)γν Scab (x)γμ]
− Tr[Ssbb (x)γν Scaa (x)γμ]] − Sccc (x)[Tr[Ssba (x)γν Ssab (x)γμ]
− Tr[Ssbb (x)γν Ssaa (x)γμ]]}.
(A.2)
In Eqs. (A.1) and (A.2)
C SsT(c)(x )C .
The quark propagators are important ingredients of sum
rules calculations. Below we provide explicit expressions of
the light- and heavy-quark propagators in the x -representation.
For the light q = u, d, s quarks we have
=
abc a b c and Ss(c)(x ) =
i x/ mq qq
Sqab(x) = 2π 2x4 δab − 4π 2x2 δab − 12
x2
− 192 m02 qq 1 − i 6
0
1
du
x/ iuxμ
16π 2x2 Gaμbν (ux)σμν − 4π 2x2 Gaμbν (ux)γν
− 3i2mπq2 Gaμbν (ux)σμν ln
,
(A.3)
where γE 0.577 is the Euler constant and is the QCD
μν
scale parameter. We have also used the notation Gab ≡
GμAν taAb, A = 1, 2, . . . 8, and t A = λA/2, with λA being
the Gell–Mann matrices.
The heavy Q = c, b quark propagators we get
√
m2Q K1 m Q −x2
SQab(x) = 4π 2 √
−x2
√
m2Q x/ K2 m Q −x2
−x2
2
⎤
gs m Q 1
− 16π 2 0
√
K1 m Q −x2
× √
−x2
(A.4)
The first two terms above is the free heavy-quark propagator
in the coordinate representation, and Kn(z) are the modified
Bessel functions of the second kind.
1. R. Aaij et al. [LHCb Collaboration], arXiv:1703 .04639 [hep-ex]
2. C. Patrignani et al., Particle Data Group. Chin. Phys. C 40 , 100001 ( 2016 )
3. S. Capstick , N. Isgur , Phys. Rev. D 34 , 2809 ( 1986 )
4. E. Bagan , M. Chabab , H.G. Dosch , S. Narison, Phys. Lett. B 278 , 367 ( 1992 )
5. E. Bagan , M. Chabab , H.G. Dosch , S. Narison, Phys. Lett. B 287 , 176 ( 1992 )
6. G. Chiladze , A.F. Falk , Phys. Rev. D 56 , R6738 ( 1997 )
7. C.S. Huang , A. l. Zhang, S.L. Zhu , Phys. Lett. B 492 , 288 ( 2000 )
8. D.W. Wang , M.Q. Huang , C.Z. Li , Phys. Rev. D 65 , 094036 ( 2002 )
9. D. Ebert , R.N. Faustov , V.O. Galkin , Phys. Lett. B 659 , 612 ( 2008 )
10. D. Ebert , R.N. Faustov , V.O. Galkin , Phys. Rev. D 84 , 014025 ( 2011 )
11. H. Garcilazo , J. Vijande , A. Valcarce , J. Phys . G 34 , 961 ( 2007 )
12. A. Valcarce , H. Garcilazo , J. Vijande , Eur. Phys. J. A 37 , 217 ( 2008 )
13. W. Roberts, M. Pervin , Int. J. Mod. Phys. A 23 , 2817 ( 2008 )
14. Z.G. Wang , Eur. Phys. J. C 54 , 231 ( 2008 )
15. Z.G. Wang , Eur. Phys. J. C 61 , 321 ( 2009 )
16. Z.G. Wang , Phys. Lett. B 685 , 59 ( 2010 )
17. J. Vijande , A. Valcarce , T.F. Carames , H. Garcilazo , Int. J. Mod. Phys. E 22 , 1330011 ( 2013 )
18. T. Yoshida , E. Hiyama , A. Hosaka , M. Oka , K. Sadato , Phys. Rev. D 92 , 114029 ( 2015 )
19. R.G. Edwards et al., Hadron Spectrum Collaboration. Phys. Rev. D 87 , 054506 ( 2013 )
20. M. Padmanath , R.G. Edwards , N. Mathur , M. Peardon , arXiv: 1311 .4806 [hep-lat]
21. H.X. Chen , W. Chen , Q. Mao , A. Hosaka , X. Liu , S.L. Zhu , Phys. Rev. D 91 , 054034 ( 2015 )
22. H.X. Chen , Q. Mao , A. Hosaka , X. Liu , S.L. Zhu , Phys. Rev. D 94 , 114016 ( 2016 )
23. Z. Shah , K. Thakkar , A.K. Rai , P.C. Vinodkumar , Chin. Phys. C 40 , 123102 ( 2016 )
24. T.M. Aliev , K. Azizi , T. Barakat , M. Savci , Phys. Rev. D 92 , 036004 ( 2015 )
25. K. Azizi , H. Sundu , Eur. Phys. J. Plus 132 , 22 ( 2017 )
26. S.L. Zhu , W.Y.P. Hwang , Z.S. Yang , Phys. Rev. D 56 , 7273 ( 1997 )
27. T.M. Aliev , K. Azizi , A. Ozpineci , Nucl. Phys. B 808 , 137 ( 2009 )
28. T.M. Aliev , K. Azizi , A. Ozpineci , Phys. Rev. D 79 , 056005 ( 2009 )
29. T.M. Aliev , K. Azizi , M. Savci , Nucl. Phys. A 852 , 141 ( 2011 )
30. T.M. Aliev , K. Azizi , M. Savci , Eur. Phys. J. C 71 , 1675 ( 2011 )
31. T.M. Aliev , K. Azizi , M. Savci , V.S. Zamiralov , Phys. Rev. D 83 , 096007 ( 2011 )
32. T.M. Aliev , K. Azizi , M. Savci , Nucl. Phys. A 870-871 , 58 ( 2011 )
33. T.M. Aliev , K. Azizi , M. Savci , Eur. Phys. J. A 47 , 125 ( 2011 )
34. T.M. Aliev , K. Azizi , M. Savci , Phys. Lett. B 690 , 164 ( 2010 )
35. S.S. Agaev , K. Azizi , H. Sundu, arXiv: 1703 .07091v2 [hep-ph]
36. H.X. Chen , Q. Mao , W. Chen , A. Hosaka , X. Liu , S.L. Zhu , arXiv: 1703 .07703 [hep-ph]
37. M. Karliner , J.L. Rosner , arXiv: 1703 .07774 [hep-ph]
38. W. Wang , R.L. Zhu , arXiv: 1704 .00179 [hep-ph]
39. M. Padmanath , N. Mathur , arXiv: 1704 .00259 [hep-ph]
40. G. Yang, J. Ping , arXiv: 1703 .08845 [hep-ph]
41. H. Huang , J. Ping , F. Wang , arXiv: 1704 .01421 [hep-ph]
42. H.C. Kim , M.V. Polyakov , M. Praszałowicz , arXiv: 1704 .04082 [hep-ph]
43. K.L. Wang , L.Y. Xiao , X.H. Zhong , Q. Zhao , arXiv: 1703 .09130 [hep-ph]
44. H.Y. Cheng, C.W. Chiang, arXiv: 1704 .00396 [hep-ph]
45. Z.G. Wang , arXiv: 1704 . 01854 [hep-ph]
46. B. Chen , X. Liu, arXiv: 1704 .02583 [hep-ph]
47. Z. Zhao , D.D. Ye , A . Zhang, arXiv: 1704 .02688 [hep-ph]
48. T.M. Aliev , S. Bilmis , M. Savci , arXiv: 1704 .03439 [hep-ph]
49. I.I. Balitsky , V.M. Braun , A.V. Kolesnichenko , Nucl. Phys. B 312 , 509 ( 1989 )
50. T.M. Aliev , K. Azizi , H. Sundu, arXiv: 1612 .03661 [hep-ph]
51. T.M. Aliev , S. Bilmis , Adv. High Energy Phys . 2017 , 1350140 ( 2017 )
52. P. Ball , V.M. Braun , A. Lenz , JHEP 0605 , 004 ( 2006 )
53. K. Azizi , N. Er , H. Sundu , Nucl. Phys. A 960 , 147 ( 2017 )