#### Possible open-charmed pentaquark molecule \(\varOmega _c(3188)\) – the \(D \varXi \) bound state – in the Bethe–Salpeter formalism

Eur. Phys. J. C
Possible open-charmed pentaquark molecule ?c(3188) - the D? bound state - in the Bethe-Salpeter formalism
Chao Wang 0
Liang-Liang Liu 1
Xian-Wei Kang 2 3
Xin-Heng Guo 2
Rui-Wu Wang 0
0 Center for Ecological and Environmental Sciences, Key Laboratory for Space Bioscience and Biotechnology, Northwestern Polytechnical University , Xi'an 710072 , China
1 College of Physics and Information Engineering, Shanxi Normal University , Linfen 041004 , China
2 College of Nuclear Science and Technology, Beijing Normal University , Beijing 100875 , China
3 Institute of Physics , Academia Sinica, Taipei 115 , Taiwan
We study the S-wave D? bound state in the Bethe-Salpeter formalism in the ladder and instantaneous approximations. With the kernel generated by the hadronic effective Lagrangian, two open-charmed bound states, which quantum numbers are I = 0, J P = ( 21 )? and I = 1, J P = ( 21 )?, respectively, are predicted as new candidates of hadronic pentaquark molecules in our formalism. If existing, they could contribute to the broad 3188 MeV structure near the five new narrow ?c states observed recently by the LHCb Collaboration.
1 Introduction
In the last decade, many important experimental progresses
were made in the study of charm hadrons. Several charm
baryons and their excited states have been reported and this
stimulates great interest in understanding the structures of
charm baryons. In these experimental observations, there
are some non-conventional states, which are more
complicated than the hadronic states in the classical quark model.
These states can be interpreted as five-quark or meson-baryon
bound states. In 2015, the LHCb Collaboration discovered
two hidden-charmed pentaquark-like structures Pc(4380)+
and Pc(4450)+ [
1?3
], which are considered as ?c D? ?, ?c? D? ,
or ?c? D? ? pentaquark molecule [
4?7
]. Besides the
hiddencharmed pantaquarks, only a few open-charmed baryons
are treated as candidates for pentaquark molecules which
have not yet been confirmed by experiments. For example,
?c(2800) and ?c(2940)+ have been suggested to be S-wave
D N and D? N molecular states, respectively [
11?14
].
Very recently, the LHCb Collaboration declared that they
observed five new narrow ?c states, which are ?c(3000)0,
?c(3050)0, ?c(3066)0, ?c(3090)0, and ?c(3119)0 [
15
].
Actually, there are six structures existing in the spectrum
of invariant mass of ?c+ K ?. The data also indicate the
presence of a broad structure around 3188 MeV that is fitted as
single resonance [shortly denoted by ?c(3188) in our work].
In the experiment, this resonance is described as the sum of
four incoherent Breit-Wigner functions to be left into the
systematic uncertainties of the others five resonances [
15
].
However, one notes that this structure is located at the D?
threshold (3179?3191 MeV) and therefore can also be
interpreted as the D? bound state naturally. At the same time,
since the width of corresponding peak is quite broad [
15
],
if this structure is a real signal of resonance, there is
reason to believe that it would be a new pentaquark molecule
with a single charm quark. At present, there is no enough
information concerning this structure, such as spin-parity, it
is interesting and significative to make efforts on the
theoretical side to confirm its existence and reveal its properties.
In this work, we will focus on this structure and study the
possible S-wave D? bound state.
The Bethe?Salpeter (BS) equation is a formally exact
equation to describe the relativistic bound state. This
technique was developed by Feynman, Bethe, and Salpeter et al.
[
23?25
]. It has been applied to theoretical studies concerning
heavy baryons and molecular bound states [
26?31
]. In
previous studies, the possible bound states of K K? , D K , B K? , and
K ? p have been investigated in the BS formalism in the
ladder and instantaneous approximations [
27?30
]. We will try
to study S-wave D? molecular bound state with the kernel
introduced by the vector meson exchange interactions in this
framework. We will investigate whether this state exists or
not and study its decay. We will also discuss the possibility
of the ?c(3188) structure to be the D? bound state.
The remainder of this paper is organized as follows. In
Sect. 2, we give the generalized formalism of the BS
equation for the fermion-scalar system. In Sect. 3, we derive the
BS equation for the D? system in detail and present the
normalization condition of the corresponding BS amplitude.
In Sect. 3, the decay of the D? bound state to ?c+ K ? is
discussed. The numerical results are presented in Sect. 4. In
the last section, we give a summary and some discussions.
2 The Bethe?Salpeter equation for the fermion-scalar system
In this section, we will present the BS equation for the
fermion-scalar system. We assume that the bound state exists
in a fermion-scalar system and its mass is M . The BS wave
function can be defined as [
25?30
]
? (x1, x2, P) = 0|T ? (x1)? (x2)| P ,
with ? (x1) and ? (x2) being field operators of the fermion
and scalar particles, respectively, and P being the momentum
of the system. In momentum space, the BS wave function,
?P ( p), is related to ? (x1, x2, P) through the following
equation [25]:
? (x1, x2, P) = eiPX
d4 p
(2? )4 ?P ( p)eipx,
where p and x (= x1 ? x2) are the relative momentum and
the relative coordinate of two constituents, respectively, and
X is the center of mass coordinate which is defined as X =
?1x1 + ?2x2, where ?1 = m1m+1m2 , ?2 = m1m+2m2 , with m1 and
m2 being the masses of the fermion and the scalar constituent,
respectively. The momentum of the fermion is p1 = ?1 P + p
and that of the scalar particle is p2 = ?2 P ? p. The derivation
of the BS formalism for the two fermion system can be found
in the textbook [
25
]. In the same way, one can prove that the
form of the BS equation is still valid for the fermion and scalar
object system. The BS wave function in our case satisfies the
follow homogeneous integral equation [
25?30
]:
?P ( p) = sF (?1 P + p)
K ( P, p, q)?P (q)
where sF and sS are propagators of the fermion and the scalar
particle, respectively, and K ( P, p, q) is the interaction kernel
which can be described by the sum of all the irreducible
graphs which cannot be split into two pieces by cutting two
particle lines as defined in Ref. [
25
]. For convenience, we
also define the relative longitudinal momentum pl (= v ? p)
and the relative transverse momentum pt [= p ? (v ? p)v]
with v(= P/M ) being the four velocity of the bound state.
3 The Bethe?Salpeter formalism for the D? bound
state
3.1 Isospin structure of the possible D? bound state
The isospin field doublets, D = (?D+, D0)T and ? =
(? 0, ? ?)T , have the following expansions in momentum
space:
D1(x) =
D2(x) =
?1(x) =
?2(x) =
d3 p
d3 p
d3 p
d3 p
(2? )3 2Ep
(2? )3 2Ep
(2? )3 2Ep
(2? )3 2Ep
aD+ e?ipx + a?D? eipx ,
aD0 e?ipx + a?D? 0 eipx ,
? eipx v? ( p) ,
a?0 e?ipx u? ( p) + a?? 0
a?? e?ipx u? ( p) + a??+ eipx v? ( p) ,
(4)
(5)
(6)
(7)
(8)
(9)
with Ep = |p|2 + m2.
The isoscalar bound state can be written as
1
| P (0, 0) = ? |D0? 0 + D+? ? ,
2
while the isovector one is
1
| P (1, 0) = ? |D0? 0 ? D+? ? ,
2
| P (1, 1) = ?|D+? 0 ,
| P (1, ?1) = |D0? ? ,
where the subscripts (I, I3) denote the isospin and the third
component of the isospin. Since the interaction between the
two constituents is dominated by strong interaction, the BS
wave function depends only on the isospin I . Therefore, the
BS wave function of the D? system can be defined as
0|T{Di (x1)? j (x2)}| P I = C(iIj, I3)? PI (x1, x2),
where the isospin coefficients C(iIj, I3) are
1
C(102, 0) = C(201, 0) = ? ,
2
for the isoscalar state and
1
C(112, 0) = ? ?2 ,
C(112, 1) = ?1,
for the isovector state.
else = 0,
1
C(211, 0) = ? ,
2
C(211, ?1) = 1,
else = 0,
d4q
(2? )4
?
D
Considering the isospin structure, the BS equation for the
D? system can be written as
C(iIj )? PI ( p) = s? (?1 P + p)
K i j, lk ( P, p, q)
?C(lkI)? PI (q)sD(?2 P ? p),
where i ( j ) and l(k) refer to the components of the D(? )
field doublets, we neglect the effect of isospin violation and
take m? = 21 (m?0 + m?? ) and m D = 21 (m D+ + m D0 ).
Explicitly, we give the isoscalar case as an example:
? PI=0( p) = s? (?1 P + p)
d4q
? (2? )4 K 12, 12(P, p, q) + K 12, 21(P, p, q)
(10)
(11)
?? PI=0(q)sD(?2 P ? p),
in which we can see that K ( P, p, q) = K 12, 12( P, p, q) +
K 12, 21( P, p, q).
3.2 The Bethe?Salpeter equation for the D? bound state
In general, considering v/u(v, s) = u(v, s) , ?P ( p) can be
written as [
32?34
]
?P ( p) = (g1 + g2?5 + g3?5 p/t + g4 p/t )u(v, s),
(12)
where u(v, s) is the spinor of the bound state with helicity s
and gi (i = 1, 2 . . . 4) are Lorentz-scalar functions.
According to our previous works, the momentum transfer between
the two constituents in a hadronic bound state is in the order
of 0.1 GeV [
27?30
] and this quantity is quite smaller than the
mass of D. Thus, we can apply the heavy quark symmetry
and have v/?P ( p) = ?P ( p). With the constraints imposed by
parity and Lorentz transformations, it is easy to prove that
?P ( p) can be simplified as [
32,33
]
?P ( p) = f ( p)u(v, s),
in which f ( p) is a Lorentz-scalar function of p.
In this work, we describe the D? interaction by
oneparticle exchange diagrams as shown in Fig. 1. In the chiral
limit and the heavy quark limit, the effective Lagrangian of
the interacting vertices involved are [
29,35
]
(13)
? ?
LD DV = ?igD DV Di ? ? D j Vi j?,
L?? V = ?g?? V ??i ??Vi j?? j ,
where gD DV and g?? V are the coupling constants, and V
refers to the fields of the vector mesons and have the
following form:
V = ?
? ??02 + ??2
??
?+
? ?2 + ??2 ? ,
?0
?
where the ? meson exchange process is neglected because
of the OZI suppression. In order to include the finite-size
effects of these hadrons, we also introduce a form factor at
each interacting vertex of hadrons. Following Refs. [
27?29
],
we take the monopole form:
F (k2) =
?2 ? m2
?2 ? k2
V ,
where mV is the mass of the exchanged meson, k is the
momentum transfer carried by the exchanged meson, and
? is a cutoff parameter. In the instantaneous approximation,
we have k = kt = pt ? qt .
The BS equation can be treated in the so-called ladder
approximation [
25
]. In this approximation, K ( P, p, q) is
replaced by its lowest order form. Using these interaction
vertices and the form factor, one can get the following kernel:
K ( P, p, q) =
?igD DV ? g?? V ? cV F (kt2) 2
(qt ? pt )2 ? m2
V
where we have used the covariant instantaneous
approximation [
26?30
], pl = ql , and c?(?) = 21 (? 21 ) and 23 ( 21 )
corresponding to isoscalar and isovector mesons, respectively.
In the chiral limit and the heavy quark limit, the propagator
of the D meson can be expressed at the leading order of the
1/m D expansion as follows [36]:
i
sD(?2 P ? p) = 2m2( pl + M ? m2 + i ) .
The propagator of ? has the form [
30,33
]
s? (?1 P + p) = (?1 M + ip[l(??1?M1 ++ ipl))(v/?+1Mp/t++pml+1] ?1 ? i ) .
(18)
(14)
(15)
(16)
(17)
f ( pl , pt )
=
Then, we substitute Eqs. (13), (16)?(18) into Eq. (10) and
obtain the BS equation for the D? bound state:
igDDV ? gV ?? ? cV [(?1 M + pl )v/ + p/t + m1]
2m2( pl + M ? m2 + i )
1
? (?1 M + pl ? ?1 + i )(?1 M + pl + ?1 ? i )
d4q 1
? (2? )4 F(kt2) 2 (qt ? pt )2 ? m2 2(?2 M + pl )v/
V
1
+ p/t + q/t ? m2 (qt2 ? pt2)(q/t ? p/t ) f (ql , qt ).
V
(19)
In the bound state rest frame, one has pt = (0, ?pt ),
pt2 = ? pt2 and pt ? qt = ? pt ? qt . Performing the integration
over pl on both sides with the residue theorem, we have
f?(pt ) = ?
?
?
? gD DV ? g?? V ? cV
V =?, ? 2m2?1(M ? ?1 M ? ?1 ? m2)
d3qt
(2? )3
F (kt2)
2
1
(qt ? pt )2 + m2
V
? 2?1 (?2 ? ?1)M ? ?1 ? pt2 ? pt ? qt
+ m12 (qt2 ? pt2)(pt ? qt ? pt2) f?(qt ),
V
(20)
where we have defined the function f?(pt ) = d2?pl f ( pl , pt ).
One may note that this equation involves the integration of
qt and it looks like a divergent integration since qt varies
from 0 to +?. However, the Lorentz-scalar function, f?(pt ),
decreases to zero rapidly at the large momentum transfer and
thus there is no divergence in practice [
27?30
]. Since we
study the ground state of the D? bound state, the BS wave
function is in fact rotationally invariant and depends only on
the norm of the three momentum, |pt |. Then, the BS equation
becomes a one-dimensional integral equation.
3.3 Normalization condition
In Eq. (20), we leave the normalization undetermined.
Following Ref. [25], the normalization condition for the BS
equation can be written as
i
? (2? )4
d4 pd4q?? P ( p) ?
? P0
??P (q) = 1, (P0 = E P ),
[I (P, p, q) + K (P, p, q)]
(21)
where I ( P, p, q) is the inverse of the four-point propagator
I (P, p, q) = ?(4)( p ? q)[s? (?1 P + p)]?1[sD(?2 P ? p)]?1.
?0
D0
Ds?
?+
c
K?
In the D? bound state rest frame, the normalization
condition can be written in the following form:
i
? (2? )4
+K ( P, p, q)]?P (q)
8i?1 M m2
= ? (2? )4
+ i ) f 2( pl , pt ) = 1.
?
d4 pd4q?? P ( p) ? P0 [I ( P, p, q)
d4 p( pl + M ? m2
According to Eqs. (19) and (20), we have
? f?2(pt ) = 1.
i 2?1(?2 M ? ?1 ? m2)
f ( fl , pt ) = ? ? ( pl + M ? m2 + i )(?1 M + pl ? ?1 + i )
1
? (?1 M + pl + ?1 ? i )
f?(pt ).
Then, One can recast the normalization condition for the BS
wave function into the form
?
?1 M m2
? 5
3 4?12(?2 M ? ?1 ? m2)2
d pt (?2 M + m2 ? ?1)2(?2 M + m2 + ?1)2
We note that the unit of f?(pt ) is GeV?5/2.
(23)
(24)
(25)
4 The decay width of the D? bound state
In this section, we will proceed to calculate the decay of the
D? bound state through BS technique. In the experiments,
the LHCb collaboration observed the new ?c0 resonances in
the ?c+ K ? spectroscopy [
15
]. In order to compare with the
experimental results, we will study the process of the D?
bound state decay into above final state. The interaction is
described via the vector meson exchange (the Ds? meson)
as shown in Fig. 2. The D0 K ? Ds? coupling vertex is given
below in the heavy quark limit and the chiral limit [37?40]
(22)
LD0 K ? Ds? = gD0 K ? Ds? (D0? D?? s
s + D??? D0)?? K ?.
(26)
(27)
The effective Lagrangian for the ? 0?c+ Ds? vertex is
L?0?c+ Ds? = g?0?c+ Ds? ??c+?? Ds??? 0 + h.c.,
where SU(3) flavor symmetry is extended to the more general
SU(4) symmetry including the charm quark [41].
Considering the charm quark, ? 0 and ?c+ are collected into SU(3)
multiplet fields 8F and 6F , respectively, and the Ds? meson
belongs to SU(3) multiplet fields 3?F . The coupling strength
among baryon 8F -plet, 6F -plet and meson 3?F -plet is
g83?6 = ?22 g, (28)
where g is a universal coupling constant in the SU(3)
symmetry without the charm quark. One may doubt the validity
of this relation with the limit of a light charm quark mass.
In Ref. [41]. the authors demonstrated that
KawarabayashiSuzuki-Riazuddin-Fayyazuddin (KSRF) relation can be
derived by insisting on SU(4) symmetric coupling constants.
They found empirically that the relation (28) can provide
magnitudes for the coupling reliable within a factor two, i.e.
g83?6 = ?2g.
We define pa [= (Ea , ?pa )] and pb[= (Eb, ?pb)] to be
the momenta of ?c+ and K ?, respectively. p (= ?2 pa ?
?1 pb) is defined as the relative momentum between ?c+ and
K ? where ?1 = mam+amb , ?2 = mam+bmb , with ma and mb
being the masses of ?c+ and K ?, respectively. According to
the kinematics of the two-body decay, in the rest frame of the
bound state one has
Eb =
M 2 ? ma2 + mb2 ,
2M
Ea =
M 2 ? m2b + ma2
2M
,
M 2 ? (ma + mb)2
M 2 ? (ma ? mb)2
|pa | = |pb| =
The differential decay width reads
1 2 |pa | d?,
d? = 32? 2 |M| M 2
2M
.
(29)
(30)
where ? is the solid angle of ?c+, and the amplitude, M,
can be obtained based on the BS techniques [
25,30
]. We will
consider the instantaneous approximation, pl = pl , again.
According to the BS technique, the decay amplitude of
the isoscalar D? bound state to ?c+ K ? can be written as
[
30
]
MI =0 = gD0 K ? Ds? ? g?0?c+ Ds? ? C(201, 0)
?
?
d3 pt F 2( pt )u??c ( pa )? ? f I =0( pt )u?c ( P)
(2? )3
g?? +
( pt ? pt )?( pt ? pt )?
m2Ds?
( p2 + pb)? ,
(31)
where pt [= p ? (v ? p )v] is the relative transverse
momentum between ?c+ and K ?, P(= mv) is the momentum of
the D? bound state, pt is the relative transverse
momentum between D and ? in the bound state, u?c and u?c+ and
are the Dirac spinors for the bound state ?c(3188) and ?c+,
respectively, and m Ds? is the mass of D?.
s
For the isovector channel, considering the ?c+ K ? final
state, we calculate the I3 = 0 state decay and the
corresponding amplitude is
MI =1 = gD0 K ? Ds? ? g?0?c+ Ds? ? C(211, 0)
?
?
d3 pt F 2( pt )u??c ( pa )? ? f I =1( pt )u?c ( P)
(2? )3
g?? +
( pt ? pt )?( pt ? pt )?
m2Ds?
( p2 + pb)? .
(32)
5 Numerical result
In this part, we will solve the BS equation numerically and
study whether the S-wave D? bound state exists or not.
Since the BS wave function for the ground state is in fact
rotationally invariant, f? depends only on | pt |. Generally, | pt |
varies from 0 to +? and f? would decrease to zero when
| pt | ? +?. We replace | pt | by the variable, t :
MeV,
(33)
| pt | =
1 + t
+ 50 ? ln 1 + 1 ? t
where is a small parameter and is introduced to avoid
divergence in numerical calculations and t varies from ? 1 to 1.
We then discretize Eq. (20) into n pieces (n is large enough)
through the Gauss quadrature rule. The BS wave function can
be written as n-dimension vectors, f (n). The coupled integral
equation becomes a matrix equation f (n) = A(n?n) ? f (n) [ A
corresponds to the coefficients in Eq. (20)].
Suppose now that the bound state may exist and its mass is
M . One gets the numerical results by solving the eigenvalue
equation obtained from the matrix equation. When the
solution of the matrix equation exists, i.e. the eigenvalue is 1.0,
we say that D? forms a S-wave ground bound state with the
mass being M .
In principle, the cutoff parameter ? in our model is not a
free parameter. It contains the information of the non-point
interaction among the hadrons. However, ? cannot be exactly
determined and depends on the specific process in practice. It
should be the typical hadronic scale, which is about 1 GeV. In
Ref. [
35
] one takes ?? N N = 1.3 GeV and ?? N N = 1.4 GeV.
For the mesons with heavy quarks, the value of ? can be as
large as 3 GeV [43]. ? is varied in the ranges 0.8?4.5 GeV
[
28
] and 1?4 GeV [
29
] when studing the D K bound state. In
the present paper, we vary ? from 1 to 5 GeV.
In our calculation, we take the mass of the mesons and
baryons from the Review of PDG [42]. There are several
coupling constants in our calculations. Following Ref. [36], we
have gD DV = ?gV /?2 where gV is determined by the KSRF
relation, gV = m? / f? ( 5.8) and ? = ?2mV /(gV fV )(
0.9). The coupling constants for the ? ? V vertex in Fig. 1
and the D0 K ? Ds? vertex in Fig. 2 are obtained in the
framework of light-cone QCD sum rules, g?? V = 1.5 [44],
and gD0 K ? Ds? = 1.84 [45]. As for the coupling constant
g?0?c+ Ds? , we estimate its value by extending to the more
general SU(4) symmetry including the charm quark as
mentioned in Sect. 4. We take g?0?c+ Ds? = ?26 g83?6( 7.6) with
g 6.6 [41].
Then, we vary the mass of the possible bound state, M ,
from 3100 to 3200 MeV and look for the proper values of M
and ? for which the non-trivial solutions of the eigenvalue
equation exist. Let us first consider the isoscalar case. We
find that there are three values of ? corresponding to each
M . In Table 1, we list the minimum values of ? and the
corresponding situations when M = 3100, 3125, 3150, 3175,
3200 MeV. When the values of ? is less than the minimum
values, there is no numerical solution of the BS wave
function, i.e., D? cannot form a bound state. In order to study
the ?c(3188) structure, we keep the mass of the bound state
fixed (M = 3188 MeV) and find that there are three values
of ?, 1.439, 3.734, and 4.028 GeV, for which we can get the
solutions. The numerical results of the BS wave functions are
displayed in Fig. 3. Then, we apply the numerical solutions
of the BS wave function corresponding to M = 3188 MeV
to calculate the width of D? decay into ?c+ K ?. The decay
widths are given below
? I =0 = 900 keV,
? I =0 = 10 MeV,
? I =0 = 15 MeV,
when ? = 1.439 GeV,
when ? = 3.734 GeV,
when ? = 4.028 GeV.
For the isovector bound state, we list the minimum values
of ? and corresponding M in Table 2. When keeping the
mass of the bound state fixed (M = 3188 MeV), we can
get two values of ?, 2.933 and 4.254 GeV. The numerical
results of the BS wave functions are displayed in Fig. 4. The
corresponding decay widths are
? I =1 = 3 MeV,
? I =1 = 32 MeV,
when ? = 2.933 GeV,
when ? = 4.254 GeV.
25
20
0.10
|pt|(GeV)
0.05
According to the LHCb collaboration experiments, the
peak and width of ?c(3188) is 3188 ? 5 ? 13 and 60 ? 15 ?
11 MeV, respectively [
15
]. We can see the two D? bound
states in our calculations are located in the range of ?c(3188)
and its decay width is smaller than that of ?c(3188). That is
to say, the D? bound states are likely to exist and could
contribute to the observed ?c(3188). However, ?c(3188) might
also be produced by a superposition of others resonances.
We still need more experimental data to support our results.
If ?c(3188) corresponds to the S-wave D? bound states, its
quantum numbers should be I ( J P ) = 0( 21 ?) or 1( 21 ?) and
one can verify this resonance by its quantum numbers. This
is important to ascertain the structure of ?c(3188).
6 Summary and discussion
In this paper, we studied the possible S-wave molecular
bound states of the D? system in the BS formalism.
Considering the interaction kernel based on vector meson exchange
diagrams, we established the BS equation for the D?
system in the ladder and instantaneous approximations. Then,
we discretized the integral equation and solved the
eigenvalue equation numerically. We confirmed the existence of
the S-wave isoscalar and isovector D? bound states in this
formalism and obtained their BS wave functions. We also
calculated the decay widths of the bound states by using the
BS wave functions. According to our results, the D? bound
states are compatible with the new observed ?c(3188)
structure. Or, at least, we can say safely that these two bound states
could contribute to the ?c(3188) peak. Further experimental
data are expected to verify ?c(3188) to be the D? bound
state with its quantum numbers determined.
We studied the D? bound states since the sum of D and
? is just near the 3188 MeV threshold. One may note that
other channels, including ? 0?c, ??c, K ??c+, K 0?c0, and
Ds+??et al., would couple to the D? system in the
mesonbaryon scattering with the strangeness S = 2 and the charm
number C = 1. The Ds+?? state is forbidden to couple to
the S-wave D? system by its spin quantum number. For
other channels including light meson when they couple to
D? , a charm meson should be exchanged. Then the coupled
channel effects will be suppressed in comparison with the
?(?) exchange D? interaction. Therefore, we omitted the
coupled channel effects in our calculation.
In this work, we predicted the existence of the D? bound
states and discussed the possibility of ?c(3188) to be the D?
bound states. Unfortunately, there are several parameters not
being well determinated in the calculations. These
uncertainties reduced the predictability of our model. In order to
describe non-pointlike effects, we introduced a cutoff
parameter ?. This cutoff parameter is dominated by
nonperturbative QCD and cannot be determined at present, which leads to
large uncertainties. For a fixed M , we vary ? from 1 to 5 GeV
in practice. We find that the solutions exist with three (two)
values of ? for a fixed M in the case of isoscalar (isovector)
bound state. We cannot determine ? exactly at present and
thus give a range of ?, which causes the uncertainties of the
results. However, there are only three (two) different values
of ? for a fixed M . Hence our results do not rely on ?
heavily. With more experimental data available in the future, we
will be able to reduce the uncertainty of ? further.
Acknowledgements Chao Wang is grateful to Zhen-Yang Wang for
helpful discussions. This work was supported by the
Fundamental Research Funds for the Central Universities of China (Project
No. 31020170QD052), NSFC-Yunnan United Fund (Project No.
U1302267), the National Natural Science Foundation of China (Project
Nos. 11275025, 11575023, and 11775024), and the National Science
Fund for Distinguished Young Scholars (Project No. 31325005).
Open Access This article is distributed under the terms of the Creative
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Funded by SCOAP3.
36. G.-Q. Feng, X.-H. Guo, Phys. Rev. D 86, 036004 (2012)
37. G. Burdman, J.F. Donoghue, Phys. Lett. B 280, 287 (1992)
38. T.-M. Yan, H.-Y. Cheng, C.-Y. Cheung, G.-L. Lin, Y.C. Lin,
H.L. Yu, Phys. Rev. D 46, 1148 (1992). Erratum: [Phys. Rev. D 55,
5851 (1997)]
39. M.B. Wise, Phys. Rev. D 45, 2188 (1992)
40. X.-W. Kang, B. Kubis, C. Hanhart, Ulf-G Mei?ner, Phys. Rev. D
89, 053015 (2014)
41. J. Hofmann, M.F.M. Lutz, Nucl. Phys. A 763, 90 (2005)
42. C. Patrignani et al., [Particle Data Group]. Chin. Phys. C 40, 100001
(2016)
43. Y.-J. Zhang, H.-C. Chiang, P.-N. Shen, B.-S. Zou, Phys. Rev. D 74,
014013 (2006)
44. Z.-G. Wang, Phys. Rev. D 75, 054020 (2007)
45. Z.G. Wang, S.L. Wan, Phys. Rev. D 74, 014017 (2006)
1. R. Aaij et al., [LHCb Collaboration], Phys. Rev. Lett . 115 , 072001 ( 2015 )
2. R. Aaij et al., [LHCb Collaboration], Phys. Rev. Lett . 117 , 082002 ( 2016 )
3. R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett . 117 , 082003 ( 2016 ). Errata [Phys. Rev. Lett . 117 , 109902 ( 2016 ); Phys. Rev. Lett . 118 , 119901 ( 2017 )]
4. R. Chen , X. Liu , X.-Q. Li , S.-L. Zhu , Phys. Rev. Lett . 115 , 132002 ( 2015 )
5. H. -X. Chen , W. Chen , X. Liu , T.G. Steele, S.-L. Zhu, Phys. Rev. Lett . 115 , 172001 ( 2015 )
6. L. Roca , J. Nieves , E. Oset, Phys. Rev. D 92 , 094003 ( 2015 )
7. J. He , Phys. Lett. B 753 , 547 ( 2016 )
8. H. -X. Chen , W. Chen , X. Liu , S.-L. Zhu, Phys. Rept . 639 , 1 ( 2016 )
9. F.-K. Guo et al., Rev. Mod. Phys . 90 , 015004 ( 2018 ). arXiv: 1705 .00141 [hep-ph]
10. H. -X. Chen , W. Chen , X. Liu , Y.-R. Liu , S.-L. Zhu , Rept. Prog. Phys . 80 , 076201 ( 2017 )
11. Y. Dong , A. Faessler , T. Gutsche , S. Kumano , V.E. Lyubovitskij , Phys. Rev. D 82 , 034035 ( 2010 )
12. J. He , X. Liu, Phys. Rev. D 82 , 114029 ( 2010 )
13. J.-R. Zhang , Phys. Rev. D 89 , 096006 ( 2014 )
14. R. Aaij et al., [LHCb Collaboration], JHEP 1705 , 030 ( 2017 )
15. R. Aaij et al., [LHCb Collaboration], Phys. Rev. Lett . 118 , 182001 ( 2017 )
16. H.-Y. Cheng, C.-W. Chiang, Phys. Rev. D 95 , 094018 ( 2017 )
17. W. Wang , R.-L. Zhu, Phys. Rev. D 96 , 014024 ( 2017 )
18. H. -X. Chen , Q. Mao , W. Chen , X. Liu , S.-L. Zhu, Phys. Rev. D 96 , 031501 ( 2017 )
19. Q. Mao , H.-X. Chen , A. Hosaka , X. Liu , S.-L. Zhu, Phys. Rev. D 96 , 074021 ( 2017 )
20. V.R. Debastiani , J.M. Dias , W.H. Liang , E. Oset, arXiv: 1710 .04231 [hep-ph]
21. G. Monta?a, A. Feijoo , ?. Ramos, Eur. Phys. J. A 54 , 64 ( 2018 ). arXiv: 1709 .08737 [hep-ph]
22. M. Karliner , J.L. Rosner , Phys. Rev. D 95 , 114012 ( 2017 )
23. R.P. Feynman , Phys. Rev . 76 , 749 ( 1949 )
24. E.E. Salpeter , H.A. Bethe , Phys. Rev . 84 , 1232 ( 1951 )
25. D. Lurie , Particles and Fields (Interscience Publishers, New York, 1968 ), Chap. 9
26. X. -H. Guo , T. Muta, Phys. Rev. D 54 , 4629 ( 1996 )
27. X. -H. Guo , X. -H. Wu , Phys. Rev. D 76 , 056004 ( 2007 )
28. Z.-X. Xie , G.-Q. Feng , X.-H. Guo , Phys. Rev. D 81 , 036014 ( 2010 )
29. G.-Q. Feng , Z.-X. Xie , X.-H. Guo , Phys. Rev. D 83 , 016003 ( 2011 )
30. C. Wang , L.-L. Liu , X.-H. Guo , Phys. Rev. D 96 , 056002 ( 2017 )
31. H. -W. Ke , L. Gao , X.-Q. Li , Eur. Phys. J. C 77 , 285 ( 2017 )
32. L. Zhang , X.-H. Guo , Phys. Rev. D 87 , 076013 ( 2013 )
33. Y. Liu , X.-H. Guo , C. Wang , Phys. Rev. D 91 , 016006 ( 2015 )
34. L. -L. Liu , C. Wang , Y. Liu , X.-H. Guo , Phys. Rev. D 95 , 054001 ( 2017 )
35. W. Liu, C.M. Ko , Z.W. Lin , Phys. Rev. C 65 , 015203 ( 2001 )