#### $$Z_c(3900)/Z_c(3885)$$ as a virtual state from $$\pi J/\psi -\bar{D}^*D$$πJ/ψ-D¯∗D interaction

Eur. Phys. J. C
Zc(3900)/Zc(3885) as a virtual state from π J/ψ − D¯ ∗ D interaction
Jun He 1
Dian-Yong Chen 0
0 School of Physics, Southeast University , Nanjing 210094 , China
1 Department of Physics and Institute of Theoretical Physics, Nanjing Normal University , Nanjing 210097 , China
In this work, we study the πJ/ψ and D¯ ∗ D invariant mass spectra of the Y (4260) decay to find out the origin of the Zc(3900) and Zc(3885) structures. The πJ/ψ − D¯ ∗ D interaction is studied in a coupled-channel quasipotential Bethe-Saltpeter equation approach, and embedded to the Y (4260) decay process to reproduce both π −J/ψ and D∗− D0 invariant mass spectra observed at BESIII simultaneously. It is found out that a virtual state at energy about 3870 MeV is produced from the interaction when both invariant mass spectra are comparable with the experiment. The results support that both Zc(3900) and Zc(3885) have the same origin, that is, a virtual state from πJ/ψ − D¯ ∗ D interaction, in which the D¯ ∗ D interaction is more important and the coupling between D¯ ∗ D and πJ/ψ channels plays a minor role.
1 Introduction
In recent years, many exotic resonance structures were
observed near the threshold of two hadrons, which are
difficult to put into the conventional quark model. The exotic
resonance structures near the D¯ ∗ D threshold (in this work
we will remark hidden charmed system with a vector D∗/D¯ ∗
meson and a pseudoscalar D¯ /D meson as D¯ ∗ D if the explicit
is not necessary) are good examples of such phenomena. The
first X Y Z particle, X (3872) is almost on the D¯ ∗ D threshold,
which was interpreted as a D¯ ∗ D hadronic molecular state
immediately after its observation [
1,2
]. Later, an isovector
resonant structure named Zc±(3900) was observed in 2013
at BESIII and Bell in the π ±J/ψ invariant mass spectrum
of e+e− → π +π − J /ψ at √s = 4.26 GeV [
3,4
], and
further confirmed at CLEO-c in the same channel at √s = 4.17
GeV [
5
]. The observed Zc(3900) is also near the D¯ ∗ D
threshold, thus, it is natural to explain it as an isovector partner of
X (3872) in the D¯ ∗ D molecular scenario and expected to be
observed in the (D¯ ∗ D)I =1 channel. It was confirmed by the
observation of Zc(3885) in the D D¯ ∗ invariant mass
spectrum of Y (4260) decay in process e+e− → π ±(D D¯ ∗)∓ [
6
].
Recently, the neutral partners of Zc±(3900) and Zc±(3885)
were also observed at BESIII [
7,8
]. Besides, the spin parity
of these state has been determined as J P = 1+ by a partial
wave analysis [9].
Thanks to the experiments at BESIII, Belle and
CLEOc, the isovector Zc(3900)/Zc(3885) has been established.
Since the Zc±(3900)/Zc±(3885) carries a charge, it cannot
be explained as a cc¯ state, which must be neutral. After the
observation at BESIII, many interpretations of the origin of
Zc(3900)/Zc(3885) have been proposed, which includes the
hadronic molecular state [
10–14
], tetraquark state [
15–17
],
initial-single-pion-emission mechanism [
18,19
], cusp effect
from triangle singularity [20]. And its decays were also
studied though intermediate meson loop [
21–23
]. Due to its
closeness to the D¯ ∗ D threshold, the hadronic molecular state is an
important picture to explain the Zc(3900)/Zc(3885)
structure. In Refs. [
24,25
], the D¯ ∗ D interaction as well as the B¯ ∗ B
interaction was studied in a one-boson-exchange model. No
bound state was found with the light-meson exchange. In
the chiral unitary approach, the contribution of exchange
of heavy meson was included into the D¯ ∗ D interaction,
which provides attraction strong enough to produce a bound
state [26]. The importance of the heavy-meson exchange
was confirmed by a further study in the one-boson-exchange
model combined with a quasipotential Bethe–Salpeter
equation approach [
27
].
The early studies in the hadronic molecular picture focus
on how to produce a bound state corresponding to the
Zc(3900)/Zc(3885) from the D¯ ∗ D interaction. It is
interesting to study if the bound state obtained in those studies
can reproduce the original experimental data of the invariant
mass spectra. In Ref. [
26
], the invariant mass spectra were
studied in a chiral unitary approach while the Breit–Wigner
form with mass and width obtained from the interaction were
adopted. In Ref. [
19
] the invariant mass spectra were
studied in an initial-single-pion-emission mechanism, where the
D¯ ∗ D rescattering were not considered. Besides, when this
study was done, only π J /ψ invariant mass spectrum was
available. In Refs. [
28–31
], the mass invariant mass spectra
was explicitly studied and fitted, and the poles
corresponding to the Zc(3900)/Zc(3885) were extracted, especially
in Ref. [28] the analysis suggested the Zc(3885)/Zc(3900)
maybe originate from a virtual state. However, in theses
studies, all coupling constants of the interactions were chosen
as free parameters. A recent lattice work suggests that the
off-diagonal π J /ψ − D¯ D∗ coupling is more important than
the D¯ D∗ interaction, and a semiphenomenological
analysis was adopted to study the invariant mass spectrum [
32
].
In their comparison with data, two general free
parameters were adopted to π J /ψ and D¯ D∗ channel, respectively,
which smeared an important experiment results about the
relative magnitudes of decays in these two channels. BESIII
reported that the decay width of Zc(3885) in D D¯ ∗ is still
much larger than that of Zc(3900) in J /ψ π channel with a
ratio 6.2 ± 1.1 ± 2.7, though which is much smaller than
conventional charmonium states above the open charm
threshold [
33
].
In this work, we try to reproduce both line shapes and
relative magnitudes of the π J /ψ and D¯ ∗ D invariant mass
spectra, simultaneously. It is performed by studying the Y (4260)
decay with reacattering of π J /ψ − D¯ ∗ D, which is
calculated in a quasipotential Bethe–Salpeter equation approach.
In the calculation, the interaction is constructed with the
Lagrangians from the heavy-quark effective theory. And in
this work, we only consider the system with negative charge,
the positive and neutral cases are analogous due to the SU(3)
symmetry.
In the next section, the formalism adopted to calculate
the three-body decay of the Y (4260) 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. 4. A brief summary is given in the last section.
2 Formalism of three-body decay
The Zc(3900)/Zc(3885) resonance structure were observed
at the e+e− → Y (4260) → π Zc → π(π J /ψ/D¯ ∗ D)
process at BESIII. The internal structure of the Y (4260)
is still in the debate. To avoid the complexity, we adopt a
phenomenological vertex Y (4260) → π(D¯ ∗ D). The effect
of the e−e− → Y (4260) is also absorbed into this
vertex. Hence, to study the invariant mass spectrum, we
consider the three-body-decay diagram in Fig. 1. In this work,
we focus on the invariant mass spectrum near the D¯ D∗
threshold and the πJ/ψ interaction is suppressed by the
OZI rule, The explicit calculation in our model also
suggests that the intermediate π J /ψ channel will be suppressed
Y (4260)
(a)
π+
D−(π−)
D∗0(J/ψ)
Y (4260)
D−(D∗−)
D∗0(D0)
T
(b)
π+
D−(π−)
D∗0(J/ψ)
seriously. Hence, we only consider the intermediate D− D∗0
and D∗− D0 channels in the loop between direct decay vertex
and the rescattering from beginning, which was also adopted
in Ref. [
31
]. When calculating the rescattering amplitude T ,
the π − J /ψ − D− D∗0 − D∗− D0 interaction is considered.
With the decay amplitude M the invariant of the mass
spectrum can be obtained from the differential decay width
of Y (4260) as
(1)
(2)
1
dΓ = 2M |M|2dΦ,
where the M is the mass of the Y (4260) and phase space can
be written as
Here cm means the center of mass frame of particles 2 and
3. The explicit deduction is given in Appendix A.
The key to study the decay amplitude is to write
rescattering amplitude T of the J /ψ − − D− D∗0 − D∗− D0
interaction, which can be obtained with the help of the Bethe–
Salpeter equation as shown in Fig. 2.
To avoid difficulty of solving a four-dimensional
equation in the Minkowski space, with quasipotential
approximation, the Bethe–Salpeter equation is often reduced to a
three-dimensional equation, which can be further reduced
to a one-dimensional equation by partial wave
decomposition. In this work, the OBE interaction will be adopted. The
off-shellness of two constituent hadrons should be kept to
avoid the unphysical singularity below the threshold. The
covariant spectator theory, in which the heavier constituent
is put on shell [
34–38
], will be adopted in our study of the
π −J/ψ − D− D∗0 − D∗− D0 interaction. Such treatment was
explained explicitly in the appendices of Ref. [
27
] and has
been applied to studied the X(3250), the Zc(3900) and the
LHCb pentaquarks and its strange partners [
13,39–41
]. The
partial-wave Bethe–Salpeter equation with fixed spin parity
J P of system is written as [27]
i TλJ2Pλ3,λ2λ3 (p , p) = i Vλ2λ3,λ2λ3 (p , p) +
J P
λ2λ3≥0
with the reduced propagator written down in the
center-ofmass frame with P = (M, 0) as
Here the heavier particle (remarked with h) is put on shell,
which has ph0 = Eh (p ) = mh2 + p 2. The pl 0 for the
lighter particle (remarked as l) is then W − Eh (p ). Here
and hereafter we will adopt a definition p = | p|. And the
momentum of particle 2 p2 = − p and the momentum of
particle 3 p3 = p . The potential kernel VλJ2Pλ3λ2λ3 with
spin-parity J P is defined as
J P
i Vλ2λ3λ2λ3 (p , p)
= 2π
J
d cos θ [dλ32λ32 (θ )i Vλ2λ3λ2λ3 ( p , p)
+ηd−J λ32λ32 (θ )i Vλ2λ3−λ2−λ3 ( p , p)],
where λ32 = λ3 − λ2 and η = P P2 P3(−1)J −J2−J3 with
J(2,3) and P(2,3) being the spin and parity of constituent
2 or 3. Without loss of generality the initial and final
relative momenta are chosen as p = (0, 0, p) and p =
(p sin θ , 0, p cos θ ), and the dλJλ (θ ) is the Wigner d-matrix.
In most cases, the integral in Eq. (3) is non-convergent.
In this work an exponential regularization is introduced by a
replacement of the propagator as
G0(p) → G0(p) e− pl 2−ml2 2/Λ4 2 .
We would like to remind that the regularization of heavier
particle vanishes because it is put onshell in the
quasipotential approximation adopted. With the regularization, the
contributions at large momentum p is suppressed heavily at
the energies higher than 2 GeV [
42
], which guarantees the
convergence of the integral. if we multiply exponential factor
on both sides of the Eq. (3), it can be found that the
regularization factor can be seen as a form factor introduced due to
the off-shell effect of particle 1 in a form of e−(k2−m2)2/Λ4 .
The interested reader is referred to Ref. [
27
] for further
information about the regularization.
(5)
(6)
dΓ
d M23 =
1
6M
2 1
|Mλ2,λ3;λ| (2π )5
p˘1pc3m dΩ1dΩ3cm
M
λ2,λ3;λ
1 1 p˘1pc3m
= 6M (2π )5 M
λ2,λ3;λ;J
| MˆλJ2,λ3;λ(M23)|2.
Here the explicit form of AλJ2,λ3;λ(p3cm , Ω1) with J = 0, 1
in Eq. (9) is applied.
The distribution can be further rewritten with the partial
wave amplitudes withe J P as
dΓ 1 1
d M23 = 6M (2π )5
p˘1pc3m
M
1
i≥0; j≥0;J P N 2 | MˆiJ;Pj (M23)|2,
J
(12)
The write the amplitude of three-body decay of the
Y (4260), we adopt an effective Lagrangian for the Y →
π D D∗ as,
LY →π D D∗ = gY →π D D∗ Y μ Dτ · π D¯ μ∗ + Dμ∗τ · π D¯ ,
The partial-wave amplitudes with spin parity J P is
J P
Aλ2,λ3;λ(pcm )
=
dΩ3cm [Aλ2,λ3;λ( P, p2cm , p3cm )DλJR∗,λ32 (Ω3cm )
× ηA−λ2,−λ3;λ( P, p2cm , p3cm )DλJR∗,−λ32 (Ω3cm ).
With Lagrangian we adopted, only the J P = 0+ and 1−
partial wave survive as
(7)
(8)
(9)
(10)
(11)
P0cm
M δλ0 Dλ1λ(Ω1).
The total three-body decay with the rescattering is written
A1λ+2,λ3;λ = N212 δλ3± + Emcm δλ30)(δλ± +
0− 2 pcm Pcm
Aλ2,λ3;λ = N 2 δλ30 m δλ0 Mcm .
0
as
with
bk,J P (M23) +
Mˆ iJ;iP (M23) = Aˆ j;i
× i Tˆ jJ;kP (p3cm , M23)G0(p3cm ) Aˆk;i (p3cm , M23)
J P
(13)
where i and j denote the independent λ2,3 and λ, and the
factors fi=0 = 1/√2 and fi =0 = 1 are inserted. In this work
we introduce parameterized background contribution with
k
dp3cm p3cm2
(2π )3
Aˆ bjk;i,J P (M23)
J P
= c(M23 − Mmin)a (Mmax − M23)b Aˆ j;i (M23).
(14)
The parameters will be determined by comparing with
experiment.
3 Lagrangians and π − J/ψ − D− D∗0/ D∗− D0
interaction
Now we need to construct the potential V of the π − J /ψ −
D− D∗0 − D∗− D0 interaction to provide the rescattering
amplitude T . In this work, we adopt the Lagrangians from the
heavy quark effective theory. The effective Lagrangian of the
pseudoscalar mesons with heavy flavor mesons reads [
43,44
]
LD∗ DP = −i 2g√mfπDm D∗ −Db Da∗λ† + Db∗λ Da† ∂λPba
+ i 2g√mfπDm D∗ −D˜ a∗λ† D˜ b + D˜ a† D˜ b∗λ ∂λPab,
LD∗ D∗P = fg αμνλ Db∗μ←→∂α Da∗λ†∂νPab − gf αμνλ D˜ a∗μ†←→∂α Db∗λ∂νDba,
π
LD∗ DV = √2λgV ελαβμ −Da∗μ†←→∂λ Db + Da†←→∂λ Db∗μ ∂αVβ ba
+ √2λgV ελαβμ −D˜ a∗μ†←→∂λ D˜ b + D˜ a†←→∂λ D˜ b∗μ ∂αVβ ab ,
(15)
with the octet pseudoscalar and nonet vector meson matrices
as
P = ⎜⎜
⎝
V = ⎜⎜
⎝
⎛ √π02 + √η6
⎛ ρ0 ω
√2 + √2
π −
K −
ρ−
K ∗−
π +
π0 η
− √2 + √6
K 0
¯
ρ+
ρ0 ω
− √2 + √2
K¯ ∗0
K + ⎞
K 0 ⎟⎟ ,
2η ⎠
− √6
K ∗+ ⎞
K ∗0 ⎟⎟ .
φ ⎠
(16)
G j>0 =
w(p j )p j2 G0(p j ),
(2π )3
Here, D and D˜ correspond to (D0, D+, Ds+) and (D¯ 0, D−,
Ds−).
The effective Lagrangian of the vector mesons with heavy
flavor mesons reads
LDDV = −i √2 βg2V D˜ a†←→∂μ D˜ bVaμb,
βgV Da†←→∂μ DbVbμa + i √
βgV Da∗†←→∂μ Db∗Vbμa
LD∗D∗V = i √2
− i 2√2λgV m D∗ Db∗μ Da∗ν†(∂μVν − ∂ν Vμ)ba
βgV D˜ a∗†←→∂μ D˜ b∗Vaμb
− i √2
√
− i 2 2λgV m D∗ D˜ a∗μ† D˜ b∗ν (∂μVν − ∂ν Vμ)ab,
LDDσ = −2gσ m D Da† Daσ − 2gσ m D D˜ a† D˜ aσ,
LD∗D∗σ = 2gσ m D∗ Da∗† Da∗σ + 2gσ m D∗ D˜ a∗† D˜ a∗σ.
Here the parameters are determined as g = 0.59, β = 0.9,
λ = 0.56 GeV−1, gV = 5.8 and gσ = gπ /(2√6) with
gπ = 3.73 [
45,46
].
The couplings of heavy-light charmed mesons to J /ψ
follow form,
LD∗ D¯ ∗ J/ψ = −igD∗ D∗ψ ψ · D¯ ∗←→∂ · D∗
−ψμ D¯ ∗ · ←→∂μ D∗ + ψμ D¯ ∗ · ←→∂D∗μ) ,
LD∗ D¯ J/ψ = −gD∗ Dψ βματ ∂β ψμ(D¯ ←→∂τ D∗α + D¯ ∗α←→∂τ D),
LDD¯ J/ψ = igDDψ ψ · D←→∂D¯ .
The three couplings in (18) are related to the single parameter
g2 as gDm∗DD∗∗ψ = gmDDDψ = gD∗ Dψ = 2g2√mψ and g2 =
2√mmDψfψ with fψ = 405 MeV.
With above Lagrangians, the potential for the
interactions can be constructed, which is presented explicitly
in Appendix B.
4 The numerical results
The amplitude T for the π − J /ψ − D− D∗0 − D∗− D0
interaction can be obtained by discretizing the momenta p, p , and
p in the integral equation (3) by the Gauss quadrature with
a weight w(pi ). After such treatment, the integral equation
can be transformed to a matrix equation [
27
]
N
j=0
Tik = Vik +
Vi j G j T jk .
The propagator G is a diagonal matrix as
i po
G j=0 = − 32π 2W +
j
w(p j ) po2
(2π )3 2W (p j2 − po2)
(17)
(18)
(19)
(20)
with on-shell momentum
1
po = 2W
[W 2 − (M1 + M2)2][W 2 − (M1 − M2)2].
(21)
The rescattering amplitude T can be solved as T = (1 −
V G)−1V . The pole of rescattering amplitude can be found
at |1 − V G| = 0 after analytic continuation total energy W
into the complex plane as z. The amplitude for the Y (4260)
decay M can be written as M = Abk +T G A with the on-shell
element being chosen.
In our model, the parameters in the Lagrangians are
determined by the heavy quark symmetry. The free parameters are
the cutoff Λ and the a, b and c for the background. The cutoffs
in the regularization and in the form factor for the exchanged
meson have the same value for simplification. In this work,
we try to reproduce the line shapes and relative magnitudes
of the π −J/ψ and D∗− D0 invariant mass spectra by varying
the parameters. When comparing the theoretical results and
the experimental data, we should be careful about the
number of the events of two channels which were obtained with
different efficiencies in experiments. Fortunately, in
original report of BESIII [
33
], both the cross sections and
corresponding numbers of the events for the Zc(3900) in π − J /ψ
channel and Zc(3885) in D∗− D channel were presented as
13.5 pb with 307 events and 83.5 pb with 502 events,
respectively. Here we adopt the Mmax (π ± J /ψ ) distribution as the
π −J/ψ invariant mass spectrum to avoid the reflection peak
because the background contribution is parametrized in this
work. The theoretical results for events can be obtained by
multiplied the efficiencies on theoretical decay distribution
for π − J /ψ and D∗− D0, respectively. Besides, the different
bin sizes adopted in two channels are also considered in the
calculation. After such treatment and a general normalization
to the experimental data, the comparison between the
theoretical and experimental results can be carried out. It is found
that with a cutoff Λ = 1.85 GeV the invariant mass spectra
can be reproduced as shown in Fig. 3, and the corresponding
parameters for the background are (a, b, c) = (0.5, 1.2, 3.6)
and (1.0, 0.05, 8.5) for the π − J /ψ and D∗− D0 invariant
mass spectra, respectively.
At low energies, the π − J /ψ invariant mass spectrum is
mainly from the background contribution, which decreases
with increase of the energies near and higher than the D∗− D0
threshold. A sharp peak arises near the threshold due to the
π J /ψ − D¯ ∗ D rescattering, which effect decreases a little
slower at energies above the threshold than at energies below
the threshold. The full model can reproduce the π − J /ψ
invariant mass spectrum generally. The peak seems too sharp
compared with the experiment, which may be from the
contributions neglected in this work. As in our previous work
in Ref. [
27
] where only D∗ D¯ scattering were considered,
the D∗− D0 invariant mass spectrum of the Y (4260) decay
π−J/ψ
D*−D0
10−2
10−4
can be reproduced. At low energies, the peak near
threshold is almost from the π J /ψ − D¯ ∗ D rescattering and the
background contribution becomes important at higher
energies. Combined the results of both invariant mass spectra,
the Zc(3900) in π − J /ψ invariant mass spectrum and the
Zc(3885) in D∗− D0 invariant mass spectrum can be
reproduced simultaneously from the π − J /ψ − D∗− D0 − D− D∗0
rescattering.
Though peaks can be produced in the invariant mass
spectra, we still need to find out that the resonance structures are
from a pole or just cusps. In the literatures [
47,48
], the
category of the pole from the two-body interaction has been
studied. In the order of the attraction of interaction from strong
to weak, there exist four types of poles, bound state which
is below threshold and usually called molecular state,
virtual state which is also below the threshold but in the second
Riemann surface, virtual state with width which is below the
threshold but has an imaginary part, and resonance which
is beyond the threshold and has an imaginary part. Hence,
we adjust the cutoff, with which the strength of the
interaction has positive correlation. The poles produced form the
π − J /ψ − D∗− D0 − D− D∗0 interaction with typical cutoffs
are listed in Table 1.
First, we list both the results for the π − J /ψ − D∗− D0 −
D− D∗0 interaction and these after turning off the π − J /ψ
channel. From the results, one can find that the interaction
is dominant with the D∗ D¯ interaction. If the cutoff larger
than about 2.1 GeV, a pole below the D¯ ∗ D threshold is
produced from the interaction. Compared with the results
withFull model
Λ
out π − J /ψ channel, the imaginary part of the pole is
obviously from the coupled-channel effect. By varying the values
if cutoff a little, the results in full model and these without
π − J /ψ channel are almost same. Hence, it is a bound state
mainly from the D¯ ∗ D interaction. With the decrease of the
cutoff, the interaction becomes weaker and the pole is
running to the threshold. When the cutoff is smaller than about
2.1 GeV, a pole will appear in the second Riemann surface of
the D¯ ∗ D interaction. This pole is leaving the threshold with
the decrease of the cutoff and will merge with the lower pole
as shown in Fig. 3. If the cutoff decreases further, the pole
dies away and no virtual state with width is produced, though
a peak still can be produced as a cusp near the D¯ ∗ D
threshold in the invariant mass spectra which is much wider than
the experimental Zc(3885). Combined with results in Fig. 3
and in Table 1, one can find both Zc(3900) and Zc(3885)
are from a virtual bound state mainly from the D¯ ∗ D
interaction. The invariant mass spectra can not be reproduced with
a bound state or cusp effect without pole.
The above results suggest that the π − J /ψ channel plays
a minor role in the π − J /ψ − D∗− D0 − D− D∗0
interaction, which also leads to a relatively small decay width in
π J /ψ channel compared with that in D¯ ∗ D channel reported
at BESIII [
33
]. It is interesting to give the results only with
coupling between π −J/ψ channel and D∗− D0 − D− D∗0
channel. From the results in Fig. 4, if we increase the
cutoff to a value about 3 GeV, the peak in the π −J/ψ
invariant mass spectrum can be reproduced. However, the peak in
the D∗− D0 invariant mass spectrum is much wider than the
experiment. No pole is produced from the interaction, and
the peaks are from the cusp effect. Hence, in our model, with
only the coupling of π −J/ψ and D∗− D0 − D− D∗0 channels,
the D∗− D0 invariant mass spectrum can not be explained.
This result supports two structures are from the virtual state
mainly from the D¯ ∗ D interaction.
5 Summary
In this work, the π J /ψ and D¯ ∗ D invariant mass spectra of
the Y (4260) decay is studied with rescattering of π J /ψ −
D¯ ∗ D, which is calculated in a quasipotential Bethe–Salpeter
equation approach. The theoretical invariant mass spectra
are compared with BESIII experiment to determine the pole
structure of π J /ψ − D¯ ∗ D interaction.
The peaks in both invariant mass spectra are reproduced
from the π J /ψ − D¯ ∗ D rescattering in the Y (4260) decay.
When the experimental data at BESIII is reproduced, the
π J /ψ − D¯ ∗ D interaction produce a virtual state at energy
of about 3870 MeV. The D¯ ∗ D channel plays important role
to produce the virtual state and the coupling between π J /ψ
and D¯ ∗ D is relatively small, which is consistent with
experimentally observed larger cross section of Zc(3855) in D¯ ∗ D
channel than that of Zc(3900) in the π J /ψ channel. After
turning off the D¯ ∗ D interaction and keeping only the
coupling between π J /ψ and D¯ ∗ D, the cusp effect still can give
peaks near the D¯ ∗ D threshold. However, the peak in the D¯ ∗ D
invariant mass spectrum is quite broad, which conflicts with
the BESIII experiment.
Acknowledgements Authors thank Dr. Bin Zhong for useful
discussion. This project is supported by the National Natural Science
Foundation of China (Grants no. 11675228, no. 11375240, and no. 11775050),
and the Major State Basic Research Development Program in China
under Grant 2014CB845405.
Open Access This article is distributed under the terms of the Creative
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Funded by SCOAP3.
Appendix A: Phase space in the center of mass frame
After partial-wave decomposition, the amplitude is
To study the invariant mass spectrum of particles 2 and 3, it is
convenient to rewrite the Lorentz-invariant phase space dΦ
in the center-of-mass frame of particles 2 and 3. With such
treatment, the results of the Bethe–Salpeter equation in the
center of mass frame also can be embedded directly. Thus,
we first rewrite the phase factor as [
49
]
dΦ = (2π )4δ4( P − p1 − p2 − p3)d3 p˜1d3 p˜2 p˜3
= (2π )4δ(E2cm + E3cm − W23)δ3
× ( pc2m + pc3m )d3 p˜1d3 p˜2cm d3 pcm
˜3
where d p˜ = d3 p/[(2π )32E ] and W223 = (M − E1)2 − |p1|2.
Here the Lorentz invariance of the d3 p˜ and δ4( P − p1 − p2 −
p3) is used.
Owing to the three-momentum δ function, the integral
over pcm can be eliminated. The momentum of the particle
2
3 has a relation pc3m = 2M123 λ(M223, m32, m22) with invariant
mass of the 23 system M23 = E2cm + E3cm . Now the quantity
d3 p3cm can be converted to d M23 by the relation,
d3 p3cm =
M23
E2cm E3cm pcm
3 d M23dΩ3cm .
The energy-conserving δ function is substituted as,
δ(M23 − W23) =
W23
|M p1/E1|
δ(p˘1 − p1)
where the p˘1 satisfies M223 = (M − E˘1)2 − p˘12. Performing
the integral over p1, we obtain the final expression of phase
space factor,
Now, we treat the three-body amplitude of the decay of
Y (4260) with rescattering which is written as
With the Lorentz invariance, the amplitude can be rewritten
in the center-of-mass frame of particles 2 and 3 as
Z
Mλ2,λ3;λ( p1, p2, p3)
= d(42πp3)c4m Tλ2,λ3 ( p2cm , p3cm ; p2cm , p3cm )
· G( p3cm )Aλ( Pcm , p2cm , p3cm ).
(A.1)
(A.2)
(A.4)
(A.5)
(A.6)
Z
Mλ2,λ3;λ( p1, p2, p3)
=
J λR
N J DλJR∗,λ32 (Ω3cm )
λ2λ3
Here, we present the explicit form of the one-boson-exchange
potential. The potentials for the D− D∗0 → D− D∗0
interaction with vector V, J /ψ and σ meson exchanges are
i IVβ2g2
V
VV = 2 q2 − m2V (k2 + k2) · (k1 + k1) 2 · 2,
(A.7)
(A.8)
(B.9)
(A.3)
VJ/ψ =
[ 2 · (k1 + k1) 2 · (k2 + k2)
−i gD∗ D∗ J/ψ gD D J/ψ
q2 − m2J/ψ
+ 2 · (k2 + k2) 2 · (k1 + k1)
− (k2 + k2) · (k1 + k1) 2 · 2],
Vσ =
i 4gσ2 m P m P∗
q2 − m2
σ
The potential for the D∗− D0 → D∗− D0 interaction can be
obtained from these for the D− D∗0 → D− D∗0 interaction
by alternating particle 1 and particle 2.
The potentials for the D∗− D0 → D− D∗0 interaction with
V, J /ψ and P meson exchanges are
For vector meson exchange flavor factor Iρ = −Iω = 1/2,
and for the pseudoscalar meson Iπ = −3Iη = −1/2.
For the coupling of the D− D∗0 → π − J /ψ interaction,
there exist two type of potentials, t and u, as
VD∗,t =
VD,u =
fπ q2 − m2D∗
− i2g√m Dm D∗ gD∗ D∗ J/ψ kμ
1 −gμν + qμqν /m2D∗
− i4g√m Dm D∗ gDD J/ψ k1 · 2 k1 · 2,
fπ q2 − m2D
· 2 · 2(k2 − q)ν − 2 · (k2 − q) 2ν + 2ν 2 · (k2 − q) ,
− 4iggJ/ψ D∗ D
VD∗,u = f q2 − m2D∗
αβρμ 2β qαk1ρ αβντ k2α 2β qτ .
(B.11)
Here, qt = k2 − k2 and qu = k1 − k2 = k1 − k2. The potential
of the D∗− D0 → π − J /ψ interaction can be obtained from
these of the D− D∗0 → π − J /ψ interaction by alternating
initial particles 1 and 2.
A form factor is introduced to compensate the off-shell
effect of exchanged meson [
50,51
]
f (q2) =
Λ4 + qt2 − m2 2 /4
Λ4 + q2 − qt2 + m2 /2
where qt2 denotes the value of q2 at the kinematical
threshold. The kinematical regime between the threshold and the
on-shell point of the exchange particle is stressed and t
channel contributions at threshold are directly given by their
couplings. The form factor is only function of the Lorentz
invariant q2, pole free on the real q2 axis, normalized to 1
for q2 = m2 and q2 = qt2, but does not have its maximum
at q2 = m2. In the propagator of the meson exchange we
make a replacement q2 → −|q2| to remove the singularities
as Ref. [
52
].
(B.12)
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