#### The decay width of the \(Z_c(3900)\) as an axialvector tetraquark state in solid quark–hadron duality

Eur. Phys. J. C
The decay width of the Zc(3900) as an axialvector tetraquark state in solid quark-hadron duality
Zhi-Gang Wang 0
Jun-Xia Zhang 0
0 Department of Physics, North China Electric Power University , Baoding 071003 , People's Republic of China
In this article, we tentatively assign the Zc±(3900) to be the diquark-antidiquark type axialvector tetraquark state, study the hadronic coupling constants G Zc J/ψπ , G Zcηcρ , G Zc D D¯ ∗ with the QCD sum rules in details. We take into account both the connected and disconnected Feynman diagrams in carrying out the operator product expansion, as the connected Feynman diagrams alone cannot do the work. Special attentions are paid to matching the hadron side of the correlation functions with the QCD side of the correlation functions to obtain solid duality, the routine can be applied to study other hadronic couplings directly. We study the two-body strong decays Zc+(3900) → J /ψ π +, ηcρ+, D+ D¯ ∗0, D¯ 0 D∗+ and obtain the total width of the Zc±(3900). The numerical results support assigning the Zc±(3900) to be the diquark-antidiquark type axialvector tetraquark state, and assigning the Zc±(3885) to be the meson-meson type axialvector molecular state.
1 Introduction
In 2013, the BESIII Collaboration studied the process
e+e− → π +π − J /ψ at a center-of-mass energy of 4.260
GeV using a 525 pb−1 data sample collected with the BESIII
detector, and observed a structure Zc(3900) in the π ± J /ψ
mass spectrum [
1
]. Then the structure Zc(3900) was
confirmed by the Belle and CLEO Collaborations [
2,3
]. Also in
2013, the BESIII Collaboration studied the process e+e− →
π D D¯ ∗, and observed a distinct charged structure Zc(3885)
in the (D D¯ ∗)± mass spectrum [4]. The angular distribution
of the π Zc(3885) system favors a J P = 1+ assignment [
4
].
Furthermore, the BESIII Collaboration measured the ratio
Rex p [
4
],
Rex p =
(Zc(3885) → D D¯ ∗)
(Zc(3900) → J /ψ π ) = 6.2 ± 1.1 ± 2.7.
(1)
In 2015, the BESIII Collaboration observed the neutral
parter Zc0(3900) with a significance of 10.4 σ in the process
e+e− → π 0π 0 J /ψ [
5
]. Recently, the BESIII Collaboration
determined the spin and parity of the Zc±(3900) state to be
J P = 1+ with a statistical significance larger than 7σ over
other quantum numbers in a partial wave analysis of the
process e+e− → π +π − J /ψ [
6
].
Now we list out the mass and width from different
measurements.
Zc±(3900) : M = 3899.0 ± 3.6 ± 4.9 MeV,
= 46 ± 10 ± 20 MeV, BESIII [
1
],
Zc±(3900) : M = 3894.5 ± 6.6 ± 4.5 MeV,
Zc±(3900) : M = 3886 ± 4 ± 2 MeV,
= 63 ± 24 ± 26 MeV, Belle [
2
],
= 37 ± 4 ± 8 MeV, CLEO [
3
],
Zc±(3885) : M = 3883.9 ± 1.5 ± 4.2 MeV,
Zc0(3900) : M = 3894.8 ± 2.3 ± 3.2 MeV,
= 24.8 ± 3.3 ± 11.0 MeV, BESIII [
4
],
= 29.6 ± 8.2 ± 8.2 MeV, BESIII [
5
]. (2)
The values of the mass are consistent with each other from
different measurements, while the values of the width differ
from each other greatly. The Zc(3900) and Zc(3885) may be
the same particle according to the mass, spin and parity.
Faccini et al. tentatively assign the Zc(3900) to be the
negative charge conjunction partner of the X (3872) [
7
]. There
have been several possible assignments, such as tetraquark
state [
8–13
], molecular state [
14–20
], hadro-charmonium
[21], rescattering effect [22–24].
In Ref. [
12
], we study the masses and pole residues of the
J PC = 1+± hidden charm tetraquark states with the QCD
sum rules by calculating the contributions of the vacuum
6.2 ± 1.1 ± 2.7, if the Zc(3900) and Zc(3885) are the same
particle with the diquark–antidiquark type structure.
The article is arranged as follows: we derive the QCD sum
rules for the hadronic coupling constants G Zc J/ψπ , G Zcηcρ ,
G Zc D D¯ ∗ in Sect. 2; in Sect. 3, we present the numerical results
and discussions; and Sect. 4 is reserved for our conclusion.
2 The width of the Zc(3900) as an axialvector
tetraquark state
We study the two-body strong decays Zc+(3900) → J /ψ π +,
ηcρ+, D+ D¯ ∗0, D¯ 0 D∗+ with the following three-point
correlation functions 1μν ( p, q), 2μν ( p, q) and 3μν ( p, q),
respectively,
1μν ( p, q) = i2
2μν ( p, q) = i2
3μν ( p, q) = i2
d4xd4 yeipx eiqy 0|T {JμJ/ψ (x)J5π (y)Jν (0)}|0 , (3)
d4xd4 yeipx eiqy 0|T {J5ηc (x)Jμρ (y)Jν (0)}|0 ,
(4)
d4xd4 yeipx eiqy 0|T {JμD∗ (x)J5D(y)Jν (0)}|0 , (5)
where the currents
J J/ψ (x ) = c¯(x )γμc(x ),
μ
J5π (y) = u¯(y)i γ5d(y),
J5ηc (x ) = c¯(x )i γ5c(x ),
Jμρ (y) = u¯(y)γμd(y),
JμD∗ (x ) = u¯(x )γμc(x ),
condensates up to dimension-10 in a consistent way in the
operator product expansion, and explore the energy scale
dependence in details for the first time. The predicted masses
MX = 3.87+−00..0099 GeV and MZ = 3.91+−00..1019 GeV support
assigning the X (3872) and Zc(3900) to be the 1++ and 1+−
diquark–antidiquark type tetraquark states, respectively.
In Ref. [
20
], we study the axialvector hidden charm and
hidden bottom molecular states with the QCD sum rules by
calculating the vacuum condensates up to dimension-10 in
the operator product expansion, and explore the energy scale
dependence of the QCD sum rules for the heavy
molecular states in details. The numerical results support assigning
the X (3872), Zc(3900), Zb(10610) to be the color
singletsinglet type molecular states with J PC = 1++, 1+−, 1+−,
respectively.
We can reproduce the experimental value of the mass of
the Zc(3900) based on the QCD sum rules both in the
scenario of tetraquark states and in the scenario of molecule
states [
12,20
]. Additional theoretical works on the width are
still needed to identify the Zc(3900).
In Ref. [
13
], Dias et al identify the Zc±(3900) as the
charged partner of the X (3872) state, and study the
twobody strong decays Zc+(3900) → J /ψ π +, ηcρ+, D+ D¯ ∗0,
D0 D¯ ∗+ with the QCD sum rules by evaluating the
threepoint correlation functions and take into account only the
connected Feynman diagrams, and they obtain the width
Zc = 63.0 ± 18.1 MeV.
In Ref. [25], Agaev et al study the two-body strong decays
Zc+(3900) → J /ψ π +, ηcρ+ with the light-cone QCD sum
rules by taking into account both the connected and
disconnected Feynman diagrams, and obtain the width Zc =
(Zc+(3900) → J /ψ π +) + (Zc+(3900) → ηcρ+) =
65.7 ± 10.6 MeV.
It is interesting to know that the connected Feynman
diagrams alone or the connected plus disconnected
Feynman diagrams lead to the same result [
13,25
]. As far as the
X (5568) is concerned, if we take the scenario of tetraquark
states, the width can also be reproduced based on the
connected Feynman diagrams alone [26] or the connected plus
disconnected Feynman diagrams [27,28]. We should prove
that the contributions of the disconnected Feynman diagrams
can be neglected safely.
In this article, we assign the Zc(3900) to be the diquark–
antidiquark type tetraquark state with J PC = 1+−, study
the hadronic coupling constants G Zc J/ψπ , G Zcηcρ , G Zc D D¯ ∗
with the three-point QCD sum rules by including both
the connected and disconnected Feynman diagrams,
special attentions are paid to the hadronic spectral densities
of the three-point correlation functions, then calculate the
partial decay widths of the strong decays Zc+(3900) →
J /ψ π +, ηcρ+, D+ D¯ ∗0, D0 D¯ ∗+, and diagnose the nature
of the Zc±(3900) based on the width and the ratio Rex p =
(6)
(7)
(8)
(9)
J5D(y) = c¯(y)i γ5d(y),
εi jk εimn
Jν (0) = √2
−cn (0)C γ5um (0)c¯k (0)γν C d¯ j (0)},
{cn(0)C γν um (0)c¯k (0)γ5C d¯ j (0)
interpolate the mesons J /ψ, π, ηc, ρ , D∗, D and Zc(3900),
respectively.
We insert a complete set of intermediate hadronic states
with the same quantum numbers as the current operators into
the three-point correlation functions 1μν ( p, q), 2μν ( p, q)
and 3μν ( p, q) [29–31], and isolate the ground state
contributions to obtain the following results,
1μν ( p, q) = fπ Mπ2 f J/ψ MJ/ψ λZc G Zc J/ψπ
mu + md
3μν ( p, q) =
=
=
=
=
−i
+ (MZ2c − p 2)(MJ2/ψ − p2) sπ0
∞ dt ρZcπ ( p 2, p2, t)
t − q2
−i
+ (MZ2c − p 2)(Mπ2 − q2) s0J/ψ
−i
+ (MJ2/ψ − p2)(Mπ2 − q2)
∞ dt ρZcψ ( p 2, t, q2)
t − p2
∞ dt ρZc J/ψ (t, p2, q2) + ρZcπ (t, p2, q2)
× sZ0c t − p 2
× (gμν + · · · ) + · · ·
1( p 2, p2, q2) gμν + · · · ,
−i
× (MZ2c − p 2)(MD2∗ − p2)(MD2 − q2)
−i
+ (MZ2c − p 2)(MD2 − q2) s0D∗
−i ∞
+ (MD2∗ − p2)(MD2 − q2) sZ0c
−i ∞ dt ρZc D ( p 2, p2, t)
+ (MZ2c − p 2)(MD2∗ − p2) s0D t − q2
∞ dt ρZc D∗ ( p 2, t, q2)
t − p2
dt
×
ρZc D∗ (t, p2, q2) + ρZc D(t, p2, q2)
t − p 2
,
0| J5D(0)|D(q) =
mc
Zc( p )| Jν (0)|0 = λZc ζν∗
J /ψ ( p)π(q)|Zc( p ) = ξ ∗( p) · ζ ( p ) G Zc J/ψπ ,
ηc( p)ρ(q)|Zc( p ) = ε∗(q) · ζ ( p )G Zcηcρ ,
D∗( p)D(q)|Zc( p ) = ς ∗( p) · ζ ( p ) G Zc D D¯ ∗ ,
the ξ, ε, ς and ζ are polarization vectors of the J /ψ , ρ , D∗
and Zc(3900), respectively. The sπ0 , sJ/ψ , sZ0c , sη0c , sρ0, sD0∗
0
and sD0 are the continuum threshold parameters. The 12
unknown functions ρZcπ ( p 2, p2, t ), ρZcψ ( p 2, t, q2), ρZcπ
(t, p2, q2), ρZc J/ψ (t, p2, q2), ρZcρ ( p 2, p2, t ), ρZcηc ( p 2, t,
q2), ρZcρ (t, p2, q2), ρZcηc (t, p2, q2), ρZc D∗ ( p 2, t, q2),
ρZc D ( p 2, p2, t ), ρZc D∗ (t, p2, q2), ρZc D(t, p2, q2) have
complex dependence on the transitions between the ground
states and the high resonances or the continuum states.
In this article, we choose the tensor gμν to study the
hadronic coupling constants G Zc J/ψπ , G Zcηcρ and G Zc D∗ D
to avoid the contaminations from the corresponding scalar
and pseudoscalar mesons, as the following current-meson
couplings are non-vanishing,
0| JμJ/ψ (0)|χc0( p) = fχc0 pμ,
0| Jμρ (0)|a0(q) = fa0 qμ,
0| JμD∗ (0)|D0∗( p) = f D0∗ pμ,
Zc0( p )| Jν (0)|0 = −i λZc0 pν ,
where the fχc0 , fa0 , f D0∗ , λZc0 are the decay constants of the
χc0(3414), a0(980), D0∗(2400) and Zc( J P = 0−),
respectively. The terms proportional to pμ pν in the 1μν ( p, q)
and 3μν ( p, q) and the terms proportional to qμ pν in the
2μν ( p, q) have contaminations from the hadronic coupling
constants G Zcχc0π , G Zc D0∗ D and G Zcηca0 , respectively.
(12)
(13)
(14)
(15)
(16)
(17)
We introduce the notations CZcπ , CZcψ , CZcπ , CZc J/ψ ,
CZcρ , CZcηc , CZcρ , CZcηc , CZc D∗ , CZc D , CZc D∗ and CZc D
to parameterize the net effects,
CZcπ =
CZcψ =
CZcπ =
CZc J/ψ =
CZcρ =
CZcηc =
CZcρ =
CZcηc =
CZc D∗ =
CZc D =
CZc D∗ =
CZc D =
∞ dt ρZcπ ( p 2, p2, t )
t − q2
s0
π
∞ dt ρZcψ ( p 2, t, q2)
s0J/ψ t − p2
,
,
ρZcπ (t, p2, q2)
t − p 2
,
ρZc J/ψ (t, p2, q2)
t − p 2
0
sZc
∞ dt ρZcρ ( p 2, p2, t )
t − q2
ρZcηc ( p 2, t, q2)
t − p2
ρZcρ (t, p2, q2)
t − p 2
ρZcηc (t, p2, q2)
,
,
t − p 2
0
sZc
∞ dt ρZc D∗ ( p 2, t, q2)
t − p2
0
sD∗
∞ dt ρZc D ( p 2, p2, t )
t − q2
∞
0
sZc
.
,
fπ Mπ2 f J/ψ MJ/ψ λZc G Zc J/ψπ
mu + md
−i
× (MZ2c − p 2)(Mη2c − p2)(Mρ2 − q2)
−i CZcρ
−i CZcηc
mc
+ (MZ2c − p 2)(Mη2c − p2)
+ (MZ2c − p 2)(Mρ2 − q2)
−i CZcηc − i CZcρ
+ (Mη2c − p2)(Mρ2 − q2) + · · · ,
the CZcπ , CZcψ , CZcπ , CZc J/ψ , CZcρ , CZcηc , CZcρ , CZcηc ,
CZc D∗ , CZc D , CZc D∗ and CZc D on the momentums p 2,
p2, q2, and take them as free parameters, and choose the
suitable values to eliminate the contaminations from the high
resonances and continuum states to obtain the stable QCD
sum rules with the variations of the Borel parameters.
We carry out the operator product expansion up to the
vacuum condensates of dimension 5 and neglect the tiny
contributions of the gluon condensate. On the QCD side, the
correlation functions 1( p 2, p2, q2) and 2( p 2, p2, q2) can
be written as
i ∞
1( p 2, p2, q2) = 32√2π 4 4mc2 ds
1
s − p2 0
∞
1
× u − q2 u s + 2mc2
i mc q¯q ∞ ds
+ 4√2π 2 4mc2
4mc2
ds
1 p 2 − s − q2
× s − p2 q4
i ∞
2( p 2, p2, q2) = − 32√2π 4 4mc2 ds
1
where the last two terms originate from the Feynman
diagrams where a quark pair q¯q absorbs a gluon emitted from
other quark line. The term
|m A→mc ,
(26)
in above equations comes from the connected Feynman
diagrams, if we set p 2 = p2, then it reduces to
∂
∂
s
∂
∞
∞
∞
It has no contribution after performing the double Borel
transformation with respect to the variables P2 = − p2
4mc2
ds
4mc2
s
∞
s
ds
(27)
and Q2 = −q2. It is more reasonable to performing the
Borel transformation than taking the limit q2 → 0, as we
carry out the operator product expansion at the large
spacelike region Q2 = −q2 → ∞. So the connected Feynman
diagrams have no contributions in the correlation functions
1/2( p 2, p2, q2), which are in contrary to Refs. [
13,26
],
where only the connected Feynman diagrams have
contributions and the limit Q2 → 0 is taken.
For the correlation function 3( p 2, p2, q2), only the
connected Feynman diagrams have contributions, we can
set p 2 = 4 p2 according to the relation MZc(3900) ≈
2MD∗ , the complex expression of the correlation function
3( p 2, p2, q2) can be reduced to a more simple form,
3(4 p2, p2, q2) = i mc q¯ gs σ Gq
96√2π 2
∞
mc2
ρ → 0, M D2 → 0 and mc2 → 0,
In the limit Mπ2 → 0, M 2
we maybe expect to choose Q2 = −q2 off-shell, and match
the terms proportional to Q12 in the limit Q2 → 0 on
the hadron side with the ones on the QCD side to obtain
QCD sum rules for the momentum dependent hadronic
coupling constants G Zc J/ψπ (Q2), G Zcηcρ (Q2), G Zc D D¯ ∗ (Q2),
then extract the values to the mass-shell Q2 = −Mπ2 , −Mρ2
or −M D2 to obtain the physical values [
13
]. However, the
approximations Mρ2 → 0, M D2 → 0 and mc2 → 0 are rather
crude, and we carry out the operator product expansion at
the large space-like region Q2 = −q2 → ∞. We
prefer taking the imaginary parts of the correlation functions
1/2/3( p 2, p2, q2) with respect to q2 + i through
dispersion relation and obtain the physical hadronic spectral
densities, then take the Borel transform with respect to the Q2 to
obtain the QCD sum rules for the physical hadronic coupling
constants.
We have to be cautious in matching the QCD side with the
hadron side of the correlation functions 1/2/3( p 2, p2, q2),
as there appears the variable p 2 = ( p + q)2. We rewrite the
correlation functions 1H/2/3( p 2, p2, q2) on the hadron side
into the following form through dispersion relation,
1H ( p 2, p2, q2) =
2H ( p 2, p2, q2) =
0
sZc
(MJ/ψ +Mπ )2
ds
0
sJ/ψ ds
4mc2
0
u0π
du
ρ1H (s , s, u)
× (s − p 2)(s − p2)(u − q2) + · · · , (29)
s0Zc sη0c u0ρ
(Mηc +Mρ )2
ds
4mc2
ds
du
0
3H ( p 2, p2, q2) =
ρ2H (s , s, u)
× (s − p 2)(s − p2)(u − q2) + · · · , (30)
sZ0c ds s0D∗ ds u0D du
mc2 mc2
(MD∗ +MD)2
ρ3H (s , s, u)
× (s − p 2)(s − p2)(u − q2) + · · · , (31)
ρ1Q/C2/D3 ( p 2, s, u) = lim lim
2→0 1→0
×
Ims Imu
1Q/C2/D3( p 2, s + i 2, u + i 1)
π 2
,
We math the hadron side of the correlation functions with the
QCD side of the correlation functions,
where the ρ1H/2/3(s , s, u) are the hadronic spectral densities,
H lim lim
ρ1/2/3(s , s, u) = 3li→m0 2→0 1→0
we add the superscript H to denote the hadron side. However,
on the QCD side, the QCD spectral densities ρQ1/C2/D3(s , s, u)
do not exist,
ρQ1/C2/D3(s , s, u) = lim lim lim
3→0 2→0 1→0
Ims Ims Imu
1Q/C2/D3(s + i 3, s + i 2, u + i 1)
π 3
×
because
lim
3→0
Ims
1Q/C2/D3(s + i 3, p2, q2)
π
= 0,
we add the superscript QC D to denote the QCD side.
On the QCD side, the correlation functions 1Q/C2/D3
( p 2, p2, q2) can be written into the following form through
dispersion relation,
1QC D( p 2, p2, q2) =
2QC D( p 2, p2, q2) =
3QC D( p 2, p2, q2) =
0
sJ/ψ
4mc2
ds
du
0
u0π
4mc2
0
ρ1QC D( p 2, s, u)
× (s − p2)(u − q2) + · · · ,
sη0c u0ρ
ds
du
ρ2QC D( p 2, s, u)
× (s − p2)(u − q2) + · · · ,
s0D∗ u0D
ds
du
mc2
mc2
ρ3QC D( p 2, s, u)
× (s − p2)(u − q2) + · · · ,
where the ρ1Q/C2/D3 ( p 2, s, u) are the QCD spectral densities,
(38)
(39)
(40)
(41)
(32)
= 0,
(33)
(34)
(35)
(36)
(37)
=
0
sηc
4mc2
=
=
mc2
=
=
0
sJ/ψ
4mc2
=
ds
∞
0
u0π
du
ds
ρ1QC D( p 2, s, u)
(s − p2)(u − q2)
s0J/ψ u0π
ds
du
(MJ/ψ +Mπ )2 4mc2 0
ρ1H (s , s, u)
× (s − p 2)(s − p2)(u − q2)
where the integrals over ds are carried out firstly to obtain
the solid duality,
s0
2
s
the s2 and 2u denote the thresholds 4mc2, mc2, 0, the
2
denotes the thresholds (MJ/ψ + Mπ )2, (Mηc + Mρ )2 and
(MD∗ + MD)2. No approximation is needed, the continuum
threshold parameter sZ0c in the s channel is also not needed.
The present routine can be applied to study other hadronic
rceoluaTpthiloiennngsfwudneicretsiceottnlysp. 2 1=/2( pp22, apn2d, qp2)2 an=d 4 p32( pin2,tph2e, qco2r)-,
respectively, and perform the double Borel transformations
with respect to the variables P2 = − p2 and Q2 = −q2,
respectively to obtain the following QCD sum rules,
(43)
(44)
where the s0J/ψ , u0π , sη0c , u0ρ , sD0∗ and u0D are the continuum
threshold parameters, the T 2 and T22 are the Borel
parameters.
In the three QCD sum rules, the terms depend on T22 can
be factorized out explicitly,
(46)
MD2∗
+ CZc D∗ + CZc D exp − T 2
the dependence on the Borel parameter T22 is trivial,
exp − u−T2M2ρ2 , exp − u−TM22D2 ,
u−Mπ2 , exp
− T22
exp − mc2−T22MD2 , which differ from the QCD sum rules
for the three-meson hadronic coupling constants greatly
[
32
]. It is difficult to obtain T22 independent regions in the
present three QCD sum rules, as no other terms to
stabilize the QCD sum rules. We can take the local limit
T22 → ∞, which is so called local-duality limit (the
local QCD sum rules are reproduced from the original
QCD sum rules in infinite Borel parameter limit) [
33–
35
], then exp − Tu22 = exp − mT22c2 = exp − MT22π2 =
exp − MT22ρ2 = exp − MT22D2 = 1, the three QCD sum rules
are greatly simplified.
Now we write down the simplified QCD sum rules
explicitly,
+
3 Numerical results and discussions
The input parameters on the QCD side are taken to be the
standard values q¯q = −(0.24 ± 0.01 GeV)3, q¯ gs σ Gq =
m20 q¯q , m02 = (0.8 ± 0.1) GeV2 at the energy scale μ =
1 GeV [
29–31,36
], mc(mc) = (1.28 ± 0.03) GeV from the
Particle Data Group [
37
]. Furthermore, we set mu = md = 0
due to the small current quark masses. We take into account
the energy-scale dependence of the input parameters from
the renormalization group equation,
q¯ gs σ Gq (μ) = q¯ gs σ Gq (Q)
2
αs (Q) 25
,
αs (μ)
12
αs (μ) 25
αs (mc)
,
mc(μ) = mc(mc)
1
αs (μ) = b0t
b1 log t
1 − b2 t
0
(49)
+
b12(log2 t − log t − 1) + b0b2
b4t 2
0
, (52)
where t = log μ22 , b0 = 331−22πn f , b1 = 1532−4π192n f , b2 =
2857− 51093238nπf3+ 32275 n2f , = 210, 292 and 332 MeV for the
flavors n f = 5, 4 and 3, respectively [
37
], and evolve all the
input parameters to the optimal energy scale μ = 1.4 GeV
to extract hadronic coupling constants [
12,38
].
The hadronic parameters are taken as Mπ = 0.13957 GeV,
Mρ = 0.77526 GeV, MJ/ψ = 3.0969 GeV, Mηc = 2.9834
GeV [
37
], fπ = 0.130 GeV, fρ = 0.215 GeV, s0
π =
0.85 GeV, sρ0 = 1.3 GeV [
36
], MD = 1.87 GeV, f D =
208 MeV, u0D = 6.2 GeV2, MD∗ = 2.01 GeV, f D∗ =
263 MeV, sD0∗ = 6.4 GeV2 [
39,40
], f J/ψ = 0.418 GeV,
fηc = 0.387 GeV [41], s0J/ψ = 3.6 GeV, sη0c =
3.5 GeV, MZc = 3.899 GeV, λZc = 2.1 × 10−2 GeV5
[
12,38
], fπ Mπ2 /(mu + md ) = −2 q¯q / fπ from the Gell–
Mann–Oakes–Renner relation.
In the scenario of tetraquark states, the QCD sum rules
indicate that the Zc(3900) and Z (4430) can be
tentatively assigned to be the ground state and the first radial
excited state of the axialvector tetraquark states, respectively
[
42
], the coupling of the current Jν (0) to the excited state
Z (4430) is rather large, so the unknown parameters cannot be
neglected. The unknown parameters are fitted to be CZc J/ψ +
CZcπ = 0.001 GeV8, CZcηc + CZcρ = 0.0046 GeV8 and
CZc D∗ + CZc D = 0.00013 GeV8 to obtain platforms in the
Borel windows T 2 = (1.9 − 2.6) GeV2, (1.9 − 2.5) GeV2
and (1.5 − 2.1) GeV2 for the hadronic coupling constants
G Zc J/ψπ , G Zcηcρ , G Zc D D¯ ∗ , respectively.
Then it is easy to obtain the values of the hadronic coupling
constants,
|G Zc J/ψπ | = 3.63 ± 0.70 GeV,
G Zcηcρ = 4.38 ± 1.86 GeV,
|G Zc D D¯ ∗ | = 0.62 ± 0.09 GeV,
(53)
which are shown explicitly in Fig. 1
We choose the masses Mπ = 0.13957 GeV, Mρ =
0.77526 GeV, MJ/ψ = 3.0969 GeV, Mηc = 2.9834 GeV,
MD+ = 1.8695 GeV, MD∗0 = 2.00685 GeV, MD0 =
1.86484 GeV, MD∗+ = 2.01026 GeV [
37
], MZc = 3.899 GeV
[
1
], and obtain the partial decay widths,
(Zc+(3900) → J /ψ π +) = 25.8 ± 9.6 MeV,
(Zc+(3900) → ηcρ+) = 27.9 ± 20.1 MeV,
(Zc+(3900) → D+ D¯ ∗0) = 0.22 ± 0.07 MeV,
(Zc+(3900) → D¯ 0 D∗+) = 0.23 ± 0.07 MeV,
and the total width,
which is consistent with the experimental data
considering the uncertainties [
1–3,5
]. If we take the central
values of the hadronic coupling constants |G Zc J/ψπ | =
3.63 GeV, G Zcηcρ = 4.38 GeV, |G Zc D D¯ ∗ | = 0.62 GeV, we
can obtain the total width Zc(3900) = 48.9 MeV, which
hap(54)
(55)
pens to coincide with the central value of the experimental
dada = 46 ± 10 ± 20 MeV from the BESIII Collaboration
[1], while the predicted ratio
(56)
R =
=
(Zc(3900) → D D¯ ∗)
(Zc(3900) → J /ψ π ) = 0.02
Rex p
(Zc(3885) → D D¯ ∗)
(Zc(3900) → J /ψ π ) = 6.2 ± 1.1 ± 2.7,
from the BESIII Collaboration [
4
]. It is difficult to assign the
Zc(3900) and Zc(3885) to be the same diquark–antidiquark
type axialvector tetraquark state. We can assign the Zc(3900)
to be the diquark–antidiquark type axialvector tetraquark
state, and assign the Zc+(3885) to be the molecular state
D+ D¯ ∗0 + D∗+ D¯ 0 according to the predicted mass 3.89 ±
0.09 GeV from the QCD sum rules [
20
]. If the Zc(3885) is
the D+ D¯ ∗0 + D∗+ D¯ 0 molecular state, the decays to D+ D¯ ∗0
and D∗+ D¯ 0 take place through its component directly, it is
easy to account for the large ratio Rex p.
Now we compare the present work with the work in Ref.
[
13
] in details. In the two works, the same currents are
chosen except for the currents to interpolate the π meson, the
operator product expansion is carried out at the large
spacelike regions P2 = − p2 → ∞ and Q2 = −q2 → ∞. In
the present work, we take into account both the connected
and disconnected Feynman diagrams, and obtain the solid
quark–hadron duality by getting the physical spectral
densities through dispersion relation, then perform double Borel
transforms with respect to the variables P2 and Q2 to obtain
the QCD sum rules for the physical hadronic coupling
constants directly. We pay special attention to the hadron spectral
spectral densities, and present detailed discussions and
subtract the continuum contaminations in a solid foundation. In
Ref. [
13
], Dias et al take into account only the connected
Feynman diagrams, and obtain the quark–hadron duality by
taking the limit Q2 → 0, Mπ2 → 0, Mρ2 → 0, M D2 → 0 and
mc2 → 0 and choosing special tensor structures, then
perform single Borel transform with respect to the variable P2
to obtain the QCD sum rules for the momentum dependent
hadronic coupling constants. They subtract the continuum
contaminations by hand, then parameterize the momentum
dependent hadronic coupling constants by some
exponential functions with arbitrariness to extract the values to the
mass-shell Q2 = −Mπ2 , −Mρ2 or −M D2 to obtain the
physical hadronic coupling constants. Although the values of the
width of the Zc(3900) obtained in the present work and in
Ref. [
13
] are both compatible with the experimental data, the
present predictions have much less theoretical uncertainties.
4 Conclusion
In this article, we tentatively assign the Zc±(3900) to be the
diquark–antidiquark type axialvector tetraquark state, study
the hadronic coupling constants G Zc J/ψπ , G Zcηcρ , G Zc D D¯ ∗
with the QCD sum rules in details. We introduce the
threepoint correlation functions, and carry out the operator
product expansion up to the vacuum condensates of dimension-5,
and neglect the tiny contributions of the gluon condensate.
In calculations, we take into account both the connected and
disconnected Feynman diagrams, as the connected Feynman
diagrams alone cannot do the work. Special attentions are
paid to matching the hadron side of the correlation functions
with the QCD side of the correlation functions to obtain solid
duality, the routine can be applied to study other hadronic
couplings directly. We study the two-body strong decays
Zc+(3900) → J /ψ π +, ηcρ+, D+ D¯ ∗0, D¯ 0 D∗+ and obtain
the total width of the Zc±(3900), which is consistent with the
experimental data. The numerical results support assigning
the Zc±(3900) to be the diquark–antidiquark type
axialvector tetraquark state, and assigning the Zc±(3885) to be the
meson–meson type axialvector molecular state.
Acknowledgements This work is supported by National Natural
Science Foundation, Grant number 11775079.
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.
Funded by SCOAP3.
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