#### Interpretation of Y(4390) as an isoscalar partner of Z(4430) from \({D^{*}}(2010){\bar{D}}_1(2420)\) interaction

Eur. Phys. J. C
Interpretation of Y (4390) as an isoscalar partner of Z(4430) from D∗(2010) D¯ 1(2420) interaction
0 School of Physics, Southeast University , Nanjing 210094 , China
1 Department of Physics and Institute of Theoretical Physics, Nanjing Normal University , Nanjing 210097 , China
Invoked by the recent observation of Y (4390) at BESIII, which is about 40 MeV below the D∗(2010)D¯ 1(2420) threshold, we investigate possible bound and resonance states from the D∗(2010)D¯ 1(2420) interaction with the one-bosonexchange model in a quasipotential Bethe-Salpeter equation approach. A bound state with quantum number 0−(1− −) is produced at 4384 MeV from the D∗(2010)D¯ 1(2420) interaction, which can be related to experimentally observed Y (4390). Another state with quantum number 1+(1+) is also produced at 4461 + i 39 MeV from this interaction. Different from the 0−(1− −) state, the 1+(1+) state is a resonance state above the D∗(2010)D¯ 1(2420) threshold. This resonance state can be related to the first observed charged charmonium-like state Z (4430), which has a mass about 4475 MeV measured above the threshold as observed at Belle and LHCb. Our result suggests that Y (4390) is an isoscalar partner of the Z (4430) as a hadronic-molecular state from the D∗(2010)D¯ 1(2420) interaction.
1 Introduction
a e-mail:
b e-mail:
about 130 MeV lower than Y (4390), has been interpreted as
a D D¯ 1 molecular state [4,5]. Considering the D∗ meson is
about 140 MeV heavier than the D meson, it is reasonable
to discuss an assignment of Y (4390) as a D∗ D¯ 1 molecular
state.
In the history of study of exotic states, the D∗ D¯ 1
molecular state has been applied to interpret the first
observed charged charmonium-like state near 4.43 GeV with
a mass of 4433 ± 4(stat) ± 2(syst) MeV and a width
of 45+−1183(stat)+−1330(syst) MeV reported by Belle Collabora
tion [6]. The mass measured by the Belle Collaboration,
about 4430 MeV, is close to the D∗ D¯ 1 threshold, so it had
ever been popular to explain Z (4430) as an S-wave D∗ D¯ 1
molecular state with spin parity J P = 0− [9–11].
However, a higher mass of 4485+−2222+−2181 MeV and a larger width
of 200−+4461−+3256 MeV were reported by a new measurement
at Belle Collaboration through a full amplitude analysis of
B0 → ψ K +π − decay and a spin parity of J P = 1+
was favored over other hypotheses [7]. A new LHCb
experiment in the B0 → ψ π − K + decay confirmed the
existence of the 1+ resonant structure Z (4430) with a mass of
4475 ± 7+−1255 MeV and a width of 172 ± 13+−3374 MeV [8].
The new Belle and LHCb results support that the spin
parity of the Z (4430) is 1+ instead of 0−, which was suggested
by previous hadronic-molecular-state studies. If insisting on
the interpretation of Z (4430) as a D∗ D¯ 1 molecular state,
one should go beyond S wave, at least to P wave, to
reproduce experimentally observed positive parity. Besides, the
new measured mass of Z (4430) is higher than the D∗ D¯ 1
threshold, which suggests that Z (4430) cannot be a bound
state. To explain the new observation of Z (4430), Barnes
et al. suggested that Z (4430) is either a D∗ D¯ 1 state
dominated by long-range π exchange, or a D D¯ ∗(1S, 2S) state
with short-range components [12]. It has also been suggested
that Z (4430) may be from the S-wave D D¯ 1∗(2600)
interaction, which has a threshold about 4470 MeV, to avoid the
difficulties mentioned above [13].
In Ref. [14], the D∗ D¯ 1 and D D¯ ∗(2600) interactions were
studied by solving the quasipotential Bethe–Salpeter
equation for vertex which is only valid for the bound state
problem. It is found that the D D¯ ∗(2600) interaction is too weak to
produce a bound state. An isovector bound state with
quantum number J P = 1+ can be produced from the D∗ D¯ 1
interaction, which corresponds to Z (4430). Such a picture
was confirmed by a lattice calculation where a state with
1+(1+−) is also produced from the D∗ D¯ 1 interaction [15].
If Z (4430) is from the D∗ D¯ 1 instead of D D¯ ∗(2600)
interaction, the new observed Z (4430) mass at Belle and LHCb
above the D∗ D¯ 1 threshold suggests that Z (4430) should be
a resonance state above the threshold instead of a bound state
below the threshold.
In Refs. [16,17], we develop a quasipotential Bethe–
Salpeter equation for amplitude to study the resonance state
above the threshold. With such formalism, it is found that
a state corresponding to P wave should be taken as serious
as these corresponding to S wave [18]. Such an idea was
applied to interpret the puzzling parities of two LHCb
hiddencharmed pentaquarks and Y (4274) [18]. It was found that a
P-wave state is usually higher than an S-wave state because
of weaker interaction but is still hopefully to be observed. If
we turn to the case of Y (4390) and Z (4430), it is very
natural to assign this two states as an S-wave D∗ D¯ 1 bound state
and a P-wave D∗ D¯ 1 resonance state, respectively. Hence, in
this work we will investigate the D∗ D¯ 1 interaction with the
Bethe–Salpeter equation for the amplitude to study the
possibility of interpreting the Y (4390) and Z (4430) as
hadronicmolecular states form the D∗ D¯ 1 interaction.
In the next section, the formalism adopted in the current
work is presented. The interaction potential is constructed
with an effective Lagrangian and the quasipotential Bethe–
Salpeter equation will be introduced briefly. The numerical
results are given in Sect. 3. A brief summary is given in the
last section.
2 Formalism
In the energy region of Y (4390) and Z (4430), besides the
D∗ D¯ 1 threshold, there are other three thresholds of
channels, D∗ D¯ 1(2430), D D¯ ∗(2600), and D∗ D¯ ∗(2550). The
large width of D1(2430), = 384−+111300 MeV [19], which
means a very short lifetime, makes it difficult to bind the
D∗ meson and itself together to form a state with a width
of about 170 MeV. The D∗ D¯ ∗(2550) interaction has also
been related to the Z (4430) in the literature. However, its
threshold is about 100 MeV higher than the Z (4430) mass.
The calculation in Ref. [14] suggested that the D D¯ 1∗(2600)
interaction and its coupling to the D∗ D¯ 1 interaction are very
weak. Hence, in this work, we only consider the D∗ D¯ 1
interaction.
D∗(p1)
D∗(p1)
For a loosely bound system, long-range interaction by the
π exchange should be more important than short-range
interaction by exchanges of heavier mesons. The Z (4430) locates
higher than the D∗ D¯ 1 threshold, so in the current work it will
be seen as a resonance state where the interaction is even
weaker than a loosely bound state. Hence, the dominance of
the π exchange is well satisfied in the case of Zc(4430).
However, in the case of Y (4390), the exchanges by heavier mesons
may be involved because a binding energy about 40 MeV is
found. For the heavier pseudoscalar mesons which can easily
be introduced as π meson in the frame of this work, their
contribution is obviously much smaller than the π meson because
their mass is much heavier than the π meson and the coupling
constants are the same as those of the π meson. An explicit
calculation suggested the medium-range σ exchange is very
small partly due to its larger mass [14]. For the vector-meson
exchange, it is difficult to determine the coupling constants
involved with the existent information in the literature. And
it is beyond the scope of this work to calculate such coupling
constants. Furthermore, all vector mesons have much larger
mass than π meson, which also leads to a suppression effect
on their contributions as in the case of the σ meson exchange.
Hence, in the current work, we do not include heavier-meson
exchanges to avoid more not-well-determined coupling
constants being introduced in the calculation with an assumption
that the contributions form the heavier-meson exchanges are
suppressed by the heavier mass as the σ meson exchange.
The direct diagram of the π exchange was also found
negligible compared with cross diagram by the π exchange in an
explicit calculation [14]. Hence, in this work, we will only
consider the cross diagram of the D∗ D¯ 1 interaction by π
exchange as shown in Fig. 1.
The explicit flavor structures for isovectors (T ) or
isoscalars (S) |D∗ D¯ 1 are [10]
+ c(|D1+ D∗−
− |D10 D¯ ∗0 ) ,
+ c(|D1+ D∗−
+ |D10 D¯ ∗0 ) ,
where c = ± corresponds to C -parity, C = ∓. For the
isovector state, c is related to the G-parity.
The involved effective Lagrangians describing the
interaction between a light pseudoscalar meson P and heavy flavor
mesons can be constructed with the help of the chiral and
heavy quark symmetries [20,21],
LD1 D∗P = i
where p1(,)2 and λ(1,)2 are the initial (final) momentum and
the helicity for constituent 1 or 2. And the flavor factor
f I = −c/2 and 3c/2 for I = 1 and 0, respectively.
With available experimental information, Casalbuoni et al.
extracted h = 0.55 GeV−1 from the old data of decay width
tot(D1(2420)) ≈ 6 MeV [21]. Compared with the new
suggested value of the decay width in PDG, 25 ± 6 MeV [19],
a value of 1.1 GeV−1 can be obtained for the coupling
constant h . In this work, we will adopt this new value of h in
the calculation. The adoption of such a value of h does not
affect the analysis above as regards the relative magnitude
of the contributions from different interaction channels and
different exchanges.
The scattering amplitude of the D∗ D¯ 1 interaction can be
obtained by solving Bethe–Salpeter equation with the above
potential. The Bethe–Salpeter equation is usually reduced
to three-dimensional equation with a quasipotential
approximation. To avoid the unphysical singularity from the OBE
With the above Lagrangians, we can obtain the potential
for the cross diagram by the π exchange,
− 4m D1 m D∗
− 4m D1 m D∗
−D1αb Dα∗†a ∂ρ ∂ρ Pba + 3D1αb Da∗†β ∂α∂β Pba
−Dα∗†a D1αb∂ρ ∂ρ Pab + 3Da∗†β D1αb∂α∂β Pab
G0 =
= 2E2( p )[(W − E2( p ))2 − E12( p )]
interaction below the threshold, the off-shellness of two
constituent hadrons should be kept. Here we adopt the most
economic treatment, that is, the covariant spectator theory [22–
24], which was explained explicitly in the appendices of
Ref. [17] and applied to a study of X (3250), Z (3900) and
the LHCb pentaquarks and its strange partners [25–28]. In
such a treatment, we put the heavier constituent, D1 meson
here, on shell [29,30]. Then the partial-wave Bethe–Salpeter
equation with fixed spin parity J P reads [17]
· i VλJ1Pλ2,λ1λ2 (p , p )G0(p )i Mλ1λ2,λ1λ2 (p , p).
J P
Written down in the center-of-mass frame where P =
(W, 0), the reduced propagator is
gρσ − 3qρ qσ , (3) i VλJ1Pλ2λ1λ2 (p , p) = 2π
where the momentum of D∗ meson k1 = (k10, − p ) =
(W−E2(p ),−p ) and the momentum of the D1 meson k2 =
(k20, p ) = (E2(p ), p ) with E1,2(p ) = M12,2 + p 2.
Here and hereafter we will adopt a definition p = | p|. The
potential kernel Vλ1λ2λ1λ2 obtained in previous section, the
partial-wave potential with fixed spin parity J P can be
calculated as
+ ηdλJ2−λ1λ1−λ2 (θ )i Vλ1λ2−λ1−λ2 ( p , p)],
where η = P P1 P2(−1)J −J1−J2 with P(1,2) and J(1,2) being
the parity and spin of constituent 1 or 2. Here without loss
of generality the initial and final relative momenta can be
chosen as p = (0, 0, p) and p = (p sin θ , 0, p cos θ ), and
the dλJλ (θ ) is the Wigner d-matrix.
To guarantee the convergence of the integral in Eq. (2),
a regularization should be introduced. In this work we will
introduce an exponential regularization by a replacement of
the propagator as
where k1 and m1 are the momentum and mass of the lighter
one of two constituent mesons. We would like to recall that
the exponential factor e−(k22−m22)2/ 4 for particle 2 vanishes,
which is only because the particle 2 is put on shell in the
quasipotential approximation adopted in the current work.
With such treatment, the contributions at large momentum
p will be suppressed heavily at the energies higher than
2 GeV as shown in Fig. 1 of Ref. [18], and convergence
of the integral is guaranteed. By multiplying the
exponential factor on both sides of the Eq. (2), it is easy to found
that the exponential factor can also be seen as a form
factor to reflect the off-shell effect of particle 1 in a form of
e−(k2−m2)2/ 4 . It is also the reason why a square of the
exponential factor is introduced in Eq. (7). The interested reader
is referred to Ref. [17] for further information as regards
the regularization. A sharp cutoff of the momentum of p
at certain value p max, namely cutoff regularization, is also
often adopted in the literature [31]. The exponential
regularization can be seen as a soft version of the cutoff
regularization. A comparison of the exponential regularization
and the cutoff regularization as adopted in the chiral
unitary approach [31] was made in Ref. [28] and it was found
that the different treatments do not affect the conclusion.
Because the current treatment guarantees the convergence
of the integration, we do not introduce the form factor for
the exchanged meson, which is redundant and its effect can
be absorbed into variation of the cutoffs as discussed in
Ref. [32].
The integral equation (2) can be solved by discretizing
the momenta p, p , and p by the Gauss quadrature with a
weight w(pi ). After such treatment, the integral equation can
be transformed to a matrix equation [17]
j=0
Mik = Vik +
The propagator G is a diagonal matrix with
G j>0 =
with on-shell momentum
[W 2 − (M1 + M2)2][W 2 − (M1 − M2)2].
The scattering amplitude M can be solved as M = (1 −
V G)−1V . Obviously, the pole of scattering amplitude we
wanted can be found at |1 − V G| = 0 after analytic
continuation total energy W into the complex plane as z. In the
current work, the pole is searched by scanning the value of
|1 − V (z)G(z)| by variation of real and imaginary parts of
z, Re(z) and I m(z), in complex plane to find position of z
with |1 − V (z)G(z)| = 0.
0−(1−−)
Fig. 2 The log |1 − V (z)G(z)| for the D∗ D¯ 1 interaction. The results
for the bound state with 0−(1− −) (left panel) and the resonance state
with 1+(1+) (right panel) are drawn to the same scale
3 The numerical results
With potential in Eq. (3), the pole from the scattering
amplitude can be found at |1 − V (z)G(z)| = 0 at complex plane
by a continuation of the real center-of-mass energy W to a
complex z. In this work, only free parameter is the
regularization cutoff . By varying the cutoff, we try to found a bound
state with 0−(1− −) and a resonance state with 1+(1+), which
correspond to Y (4390) and Z (4430), respectively, with the
same cutoff. In Fig. 2, the log |1 − V (z)G(z)| is plotted with
variations of Re(z) and Im(z). It is found that with cutoff
= 1.4 GeV two states expected can be produced from the
D∗ D¯ 1 interaction.
Under the D∗ D¯ 1 threshold, a bound state with quantum
number 0−(1− −) can be found at z = 4384 MeV, which can
be obviously related to Y (4390) with a mass of 4391 MeV
observed at BESIII. Since only the D∗ D¯ 1 interaction is
considered in this work, no width is produced and the pole is
at real axis. This state has a negative parity, so can be
produced from the D∗ D¯ 1 interaction in S wave. For the state with
1+(1+), the P wave should be introduced to produce its
positive parity. As discussed in Ref. [18], the P-wave interaction
is usually weaker than the S-wave interaction. Furthermore,
for the D∗ D¯ 1 interaction considered in this work, the flavor
factor for the isoscalar sector is three times larger than that
for the isovector sector, which makes the isovector
interaction weaker. Hence, one can expect that the 1+(1+) state is
considerably higher than the 0−(1− −) state. The result in
Fig. 2 confirms such surmise. The expected 1+(1+) state is
found at z = 4461 + i 39, which is much higher than the
0−(1− −) state, even above the D∗ D¯ 1 threshold. Obviously,
this pole can be related to the charged charmonium-like state
Z (4430) whose mass is about 4475 MeV as suggested by the
new LHCb experiment. Though only one-channel is included
in this work, the resonance state carries a width as suggested
by the scattering theory.
Table 1 The bound states from the D∗ D¯ 1 interaction with typical
cutoffs . The cutoff and energy W are in units of GeV, and MeV,
respectively
Table 2 The bound states from the D∗ D¯ 1 interaction with typical
cutoffs . The cutoff and energy W are in units of GeV, and MeV,
respectively
0+(0−+)
0−(1+−)
0−(2+−)
0+(1−+)
0−(1− −)
0+(2−+)
1+(0−)
1+(2−)
The above results show that the experimentally observed
Y (4390) and Z (4430) can be reproduced from the D∗ D¯ 1
interaction with the same cutoff = 1.4 GeV. In the rest part
of this section, we will study whether there exist other
possible states produced from this interaction. Here, we only
consider the D∗ D¯ 1 interaction with spin parties 0±, 1±, 2±, and
3−. Other partial waves are not considered because their spin
parities cannot be constructed with S and P waves. Because a
coupled-channel effect is not included in this work, we allow
the regularization cutoff to deviate from the value above,
1.4 GeV, by 0.5 GeV, i.e. from 0.9 to 1.9 GeV. Only poles in
an energy range 4.35 < Re(z) < 4.50 GeV are searched
for in the calculation. The isovector states from the D∗ D¯ 1
interaction with typical cutoffs are listed in Table 1.
In the isovector sector, besides the 1+(1+) state
corresponding to Z (4430), there exist other two possible states
with 1+(0−) and 1+(2−) produced from the D∗ D¯ 1
interaction. With the decrease of the regularization cutoff, the
interaction gradually weaken. As a result, the poles of these
states will run to and cross the threshold at certain cutoff, and
then the bound state becomes a resonance state. If we fix the
cutoff at 1.4 GeV as in the case of reproducing Y (4390) and
Y (4430), 1+(0−) is a bound state around 4.4 GeV. 1+(2−)
is a resonance state much higher the=an Z (4430). A
dependence of the results on the cutoff can be found in Table 1,
which is from neglecting of the coupled-channel effect and
other approximations adopted in our approaches. It is also
the reason why we will vary the cutoff in the calculation, that
is, the effects of the approximations can be absorbed into the
variation of the cutoff.
The results of the isoscalar sector is listed in Table 2.
Nine states with 0+(0±+), 0±(1+±), 0±(1−±), 0±(2+±)
and 0+(2−+) are produced in this sector, which are much
more than three states in the isoscalar sector. It is
reasonable because the flavor factor for the isoscalar sector is three
times larger than that for the isovector sector, which means
stronger interaction in this sector. Generally speaking, the
spin-negative states are more binding than the positive states
which reflects the P-wave interaction is usually weaker than
the S-wave state. In our calculation, more than one state is
found in some cases, which can be seen as excited state. As in
the study of the hydrogen energy level and the hadron
spectrum in the constituent quark model, it is natural to find radial
excited states besides the ground state. For the Y (4390) and
Z (4430), which we focused on in this work, only one state
was found in a considerable large range of the Re(z), and it
is not so meaningful to present the results of excited states
for other states even which ground state has not yet been
observed in the experiment. So, in Tables 1 and 2, only the
results of the ground state are presented.
4 Summary
In this work, the D∗ D¯ 1 interaction is investigated in
a quasipotential Bethe–Salpeter equation approach, and
bound and resonance states are searched for to interpret
Y (4390) observed recently at BESIII and the first observed
charged charmonium-like state Z (4430). A bound state with
0−(1− −) at 4384 GeV and a resonance state with 1+(1+)
at 4461 + i 39 MeV are produced from the D∗ D¯ 1 interaction
which can be related to Y (4390) and Z (4430), respectively.
Hence, Y (4390) is an isoscalar partner of Z (4430) and a
partner of Y (4260) by replacing the D meson by the D∗ meson
in the hadronic-molecular state picture.
Acknowledgements This project is supported by the National Natural
Science Foundation of China (Grants No. 11675228 and No. 11375240),
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