#### The magnetic moment of the \(Z_c(3900)\) as an axialvector tetraquark state with QCD sum rules

Eur. Phys. J. C
The magnetic moment of the Zc(3900) as an axialvector tetraquark state with QCD sum rules
Zhi-Gang Wang 0
0 Department of Physics, North China Electric Power University , Baoding 071003 , People's Republic of China
In this article, we assign the Zc±(3900) to be the diquark-antidiquark type axialvector tetraquark state, study its magnetic moment with the QCD sum rules in the external weak electromagnetic field by carrying out the operator product expansion up to the vacuum condensates of dimension 8. We pay special attention to matching the hadron side with the QCD side of the correlation function to obtain solid duality, the routine can be applied to study other electromagnetic properties of the exotic particles.
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, and observed a structure Zc(3900) in the π ± J /ψ
invariant mass distribution [
1
]. Then the Zc(3900) was
confirmed by the Belle and CLEO collaborations [
2,3
]. Recently,
the BESIII collaboration determined the spin and parity of
the Zc±(3900) state to be J P = 1+ with a statistical
significance more than 7σ over other quantum numbers in a partial
wave analysis of the process e+e− → π +π − J /ψ [4]. If it
is really a resonance, the assignments of tetraquark state [
5–
11
] and molecular state [
12–19
] are robust according to the
non-zero electric charge. The newly observed exotic states
X , Y , Z , which are excellent candidates for the multiquark
states, provide a good platform for studying the
nonperturbative behavior of QCD and attract many interesting works
of the particle physicists.
The QCD sum rules is a powerful nonperturbative tool
in studying the ground state hadrons, and has given many
successful descriptions of the hadronic parameters on the
phenomenological side [
20–22
]. As far as the Zc(3900)
is concerned, the mass and decay width of the Zc(3900)
have been studied with the QCD sum rules in details
[
10,11,18,19,23,24
], while the magnetic moment of the
Zc(3900) is only studied with the light-cone QCD sum rules
[25]. The magnetic moment of the Zc(3900) is a
fundamental parameter as its mass and width, which also has
copious information about the underlying quark structure, can
be used to distinguish the preferred configuration from
various theoretical models and deepen our understanding of the
underlying dynamics.
For example, although the proton and neutron have
degenerated mass in the isospin limit, their electromagnetic
properties are quite different. If we take them as point particles,
the magnetic moments are μ p = 1 and μn = 0 in unit
of the nucleon magneton from Dirac’s theory of
relativistic fermions. In 1933, Otto Stern measured the magnetic
moment of the proton, which deviates from one
significatively and indicates that the proton has under-structure. The
neutron’s anomalous magnetic moment originates from the
Pauli form-factor. So it is interesting to study the interaction
of the Zc(3900) with the photon, which plays an important
role in understanding its nature. In fact, the works on the
magnetic moments of the exotic particles X , Y , Z are few,
only the magnetic moments of the Zc(3900), X (5568) and
Zb(10610) are studied with the light-cone QCD sum rules
[
25–27
].
In this article, we tentatively assign the Zc(3900) to be the
diquark-antidiquark type tetraquark state and study its
magnetic moment with the QCD sum rules in the external weak
electromagnetic field. In Ref. [
28
], Ioffe and Smilga (also in
Ref. [
29
], Balitsky and Yung) introduce a static
electromagnetic field which couples to the quarks and polarizes the QCD
vacuum, and extract the nucleon magnetic moments from
linear response to the external electromagnetic field with the
QCD sum rules. This approach has been applied successfully
to study the magnetic moments of the octet baryons and
decuplet baryons [
30–34
]. Now we extend this approach to study
the magnetic moment of the hidden-charm tetraquark state
Zc(3900).
The article is arranged as follows: we derive the QCD sum
rules for the magnetic moment μ of the Zc(3900) in Sect. 2;
in Sect. 3, we present the numerical results and discussions;
and Sect. 4 is reserved for our conclusion.
2 The magnetic moment of the Zc(3900) as an
axialvector tetraquark state
In the following, we write down the two-point correlation
function μν ( p, q) in the external electromagnetic field F ,
μν ( p, q) = i
d4x ei p·x 0|T Jμ(x ) Jν†(0) |0 F ,
Jμ(x ) =
εi jk εimn
√2
u j (x )C γ5ck (x )d¯m (x )γμC c¯n(x )
−u j (x )C γμck (x )d¯m (x )γ5C c¯n(x ) ,
the i , j , k, m, n are color indexes, the C is the charge
conjugation matrix.
The correlation function μν ( p, q) can be rewritten as
μν ( p, q) = i
d4x ei p·x 0 T Jμ(x ) Jν†(0) 0
(1)
(2)
(3)
+ i
×
d4x ei p·x 0 T
Jμ(x ) −i e
d4 y ηα(y) Aα(y) Jν†(0)
0 + · · · ,
where the ηα(y) is the electromagnetic current and the Aα(y)
is the electromagnetic field.
We can insert a complete set of intermediate hadronic
states with the same quantum numbers as the current operator
Jμ(x ) into the correlation function μν ( p, q) to obtain the
hadronic representation [
20–22
]. After isolating the ground
state and the first radial excited state contributions from the
pole terms, we obtain
μν ( p, q) = MZ2λ−2Z p2 ζμ( p)ζν∗( p) + M2λ2Z
Z − p2 ζμ( p)ζν∗( p)
1
− λ2Z e εα∗ (q) ζμ( p) ζν∗( p ) MZ2 − p2 Zc( p)|ηα(0)|Zc( p ) )
1
× MZ2 − p 2 − λZ λZ e εα∗ (q) ζμ( p) ζν∗( p )
1
Z − p2 Zc( p)|ηα(0)|Zc( p ) )
× M2
1
+
+
+
=
λ2Z
the εα and ζμ are the polarization vectors of the photon
and the axial-vector meson Zc( ), respectively. The hadronic
matrix element Zc( p)|ηα(0)|Zc( p ) can be parameterized
by three form-factors,
Zc( p)|ηα(0)|Zc( p ) = G1(Q2) ζ ∗( p) · ζ ( p ) pα + pα
−G3(Q2)
+G2(Q2) ζα( p )ζ ∗( p) · q − ζα∗( p)ζ ( p ) · q
1
ζ ∗( p) · q ζ ( p ) · q pα + pα ,
with Q2 = −q2, p = p + q [
35
]. The Lorentz-invariant
form-factors G1(Q2), G2(Q2) and G3(Q2) are related to
the charge, magnetic and quadrupole form-factors,
2
GC = G1 + 3 ηG Q ,
G M = −G2,
G Q = G1 + G2 + (1 + η)G3,
Q2
respectively, where η = 4M2 is a kinematic factor. At zero
Z
momentum transfer, these form-factors are proportional to
the usual static quantities of the charge e, magnetic moment
μZ and quadrupole moment Q1,
e GC (0) = e,
e G M (0) = 2MZ μZ ,
e G Q (0) = MZ2 Q1 .
The hadronic matrix elements Zc( p)|ηα(0)|Zc( p ) and
Zc( p)|ηα(0)|Zc( p ) are parameterized analogously by the
form-factors G1(Q2), G2(Q2) and G3(Q2). In this article,
we choose the tensor structure Fμν for analysis.
The current Jμ(x ) also has non-vanishing couplings with
the scattering states D D∗, J /ψ π , J /ψρ, etc [
36
]. In the
following, we study the contributions of the intermediate
meson-loops to the correlation function μν ( p, q),
μν ( p, q) = MZ2 +
+ MZ2 +
λ2Z
( p) − p2 −gμν
λ2Z
( p) − p2 −gμν
+
+
+
where the self-energy ( p) = D D∗ ( p) + J/ψπ ( p) +
J/ψρ ( p) + · · · . The λZ/Z and MZ/Z are bare quantities to
absorb the divergences in the self-energies ( p) and ( p ).
The renormalized self-energies contribute a finite imaginary
part to modify the dispersion relation,
+
+
+
the physical width of the ground state (MZ2 ) = Zc(3900) =
(28.1 ± 2.6) MeV is small enough [
36
], while the effect of
the large width of the first radial excited state (MZ2 ) =
Z(4430) = 181 ± 31 MeV can be absorbed into the
continuum states (the Z (4430) can be tentatively assigned to
be the first radial excitation of the Zc(3900) based on the
QCD sum rules [
37
]), the zero width approximation in the
hadronic spectral density works, the contaminations of the
intermediate meson-loops are expected to be small and can
be neglected safely [
38
]. In fact, even the effect of the large
width of the Zc(4200) ( Zc(4200) = 370 ± 70+−71032 MeV) can
be safely absorbed into the pole residue [
39
]. For detailed
discussions about this subject, one can consult Refs. [
10,39
].
In the following, we briefly outline the operator product
expansion for the correlation function μν ( p, q) in
perturbative QCD. We contract the quark fields in the correlation
function μν ( p, q) with Wick theorem, obtain the result,
μν ( p, q) = −
i εi jk εimnεi j k εi m n
2
×
d4x ei p·x Tr γ5Ckk (x )γ5CU j j T (x )C
× Tr γν Cn n(−x )γμC Dm mT (−x )C
+ Tr γμCkk (x )γν CU j j T (x )C
× Tr γ5Cn n(−x )γ5C Dm mT (−x )C
−
−
+
+
×
×
+
σ αβ x 2 − 2xλx β σ λα
δi j q¯q χ σ αβ eeq Fαβ
24
δi j q¯q eeq Fαβ
288
δi j q¯q eeq Fαβ
576
i
Ci j (x ) = (2π )4
d4ke−ik·x
σ αβ x 2 (κ + ξ ) − 2xλx β σ λα
ξ
κ − 2
+ · · · , (12)
δi j
k − mc −
gs Gnαβ tinj σ αβ (k + mc) + (k + mc)σ αβ
4 (k2 − mc2)2
δi j eeq Fαβ σ αβ (k + mc) + (k + mc)σ αβ
4 (k2 − mc2)2
+ · · · ,
and t n = λ2n , the λn is the Gell-Mann matrix [
10,22,28,
31,33,34,40
]. We retain the term q¯ j σμν qi originates from
Fierz re-ordering of the qi q¯ j to absorb the gluons emitted
from other quark lines to form q¯ j gs Gaαβ tma nσμν qi to extract
the mixed condensate q¯ gs σ Gq [
10
]. The new condensates
or vacuum susceptibilities χ , κ and ξ induced by the external
electromagnetic field are defined by
0|q¯ σαβ q|0 F = e eq χ q¯q Fαβ ,
0|q¯ gs Gαβ q|0 F = e eq κ q¯q Fαβ ,
εαβλτ 0|q¯ gs Gλτ γ5q|0 F = i e eq ξ q¯q Fαβ ,
with the convention ε0123 = −ε0123 = 1 [
28,31,33,34
].
Then we compute the integrals both in the coordinate
and momentum spaces, and obtain the correlation function
( p 2, p2) Fμν therefore the spectral density at the level of
quark-gluon degrees of freedom. Furthermore, we take into
(13)
(14)
account contributions of the new condensates originate from
the quark interacting with the external electromagnetic field
as well as the vacuum gluons,
0|qi (x )q¯ j (0)Gnαβ (x )|0 F
= −
qq
¯
16
tinj eeq
i
κ Fαβ − 4 ξ γ5 εαβλτ F λτ
,
see the Feynman diagrams shown in Figs. 1 and 2.
We have to be cautious in matching the QCD side of the
correlation function ( p 2, p2) with the hadron side of the
correlation function ( p 2, p2), as there appears the
variable p 2 = ( p + q)2. We rewrite the correlation function
H ( p 2, p2) on the hadron side into the following form
through dispersion relation,
H ( p 2, p2) =
s0Z
4mc2
ds
sZ0
4mc2
ds ρH (s , s)
(s − p 2)(s − p2) + · · · ,
(15)
(16)
where the ρH (s , s) is the hadronic spectral density,
ρH (s , s) = lim lim
2→0 1→0
Ims Ims
H (s + i 2, s + i 1)
π 2
= λ2Z G2(0) δ s − M Z2 δ s − M Z2
+λZ λZ G2(0) δ s − M Z2
δ s − M Z2
+λZ λZ G2(0) δ s − M Z2 δ s − M Z2
+ · · · ,
(17)
the sZ0 is the continuum threshold parameter, we add the
subscript H to denote the hadron side. However, on the QCD
side, the QCD spectral density ρQC D (s , s) does not exist,
Ims Ims
QC D (s + i 2, s + i 1)
π 2
(18)
(19)
(20)
(21)
ρQC D (s , s) = lim lim
2→0 1→0
= 0,
because
lim
2→0
Ims
QC D (s + i 2, p2)
π
= 0,
we add the subscript QC D to denote the QCD side.
On the QCD side, the correlation function QC D ( p2)
(the QC D ( p 2, p2) is reduced to QC D ( p2) due to
lacking dependence on the variable p 2) can be written into the
following form through dispersion relation,
QC D ( p2) = an p2n +
s0Z
4mc2
ds ρQC D (s)
s − p2
+ · · · ,
where the ρQC D (s) is the QCD spectral density,
ρQC D (s) =
lim
1→0
Ims
QC D (s + i 1) ,
π
which is independent on the variable p 2, the coefficients an
with n = 0, 1, 2, . . . are some constants, the terms an p2n
disappear after performing the Borel transform with respect
to the variable P 2 = − p2, we can neglect the terms an p2n
safely.
We math the hadron side with the QCD side of the
correlation function, and carry out the integral over ds firstly to
obtain the solid duality,
sZ0
4mc2
ds ρQC D (s)
s − p2
=
=
=
sZ0
4mc2
ds
1
s − p2
λ2Z G2(0)
+
MZ2 − p2
sZ0
ds
4mc2
1
s − p2
λ2Z G2(0)
where we introduce the unknown parameter C to denote the
transition between the ground state and the excited states,
λ2Z C =
sZ0
4mc2
ds
∞
s0Z
ds ρH (s , s)
s − p 2 =
λZ λZ G2(0)
M 2
Z − p 2
In numerical calculation, we smear dependency of the C on
the momentum p 2 and take it as a free parameter, and choose
the suitable value to eliminate the contaminations from the
high resonances and continuum states to obtain the stable
QCD sum rule with respect to the variation of the Borel
parameter T 2.
Now we set p 2 = p2 and perform the Borel transform
with respect to the variable P2 = − p2 to obtain the following
QCD sum rule:
y f
yi
1−y
d y
y f
zi
d y
yi
2s − mc2
zi
y f
mc q¯q
ρ3(s) = − 96π 4
ρQC D(s) = ρ0(s) + ρ3(s) + ρ5(s) + ρ6(s) + ρ8(s), (25)
1−y
zi
dz yz (1 − y − z)
× s − mc2 2 4s − mc2 ,
(1 − y − z) 3s − 2mc2
(1 − y − z) − (y + z) s − mc2 ,
mc q¯ gsσ Gq
256π 4
−
−
mc q¯ gsσ Gq
1536π 4
5mc q¯ gsσ Gq
1536π 4
y f
yi
yi
× 1 + s δ s − mc2
d y 1 + s δ s − mc2
y f
yi
d y
y f
d y
dz yz + yz
−
−
+
y f
yi
mc2 = (y+yzz)mc2 , mc2 = y(1m−c2 y) , yyi f d y → 01 d y, z1i−y d z →
01−y d z when the δ functions δ s − mc2 and δ s − mc2
appear, we neglect the small contributions of the gluon
condensate.
According to Eqs. (22, 23), the non-diagonal transitions
between the ground state and the excited states can be written
as
λ2Z C
MZ2 − p2 =
If we set p 2 = p2 and perform the Borel transform with
respect to the variable P2 = − p2, we can obtain
λZ λZ G2(0)
M 2
Z − MZ2
exp
MZ2
− T 2
− exp
M 2
Z
− T 2
In Ref. [
37
], we observe that the Z (4430) can be tentatively
assigned to be the first radial excitation of the Zc(3900) based
on the QCD sum rules. If we set Z = Z (4430), then
exp
exp
M2
Z
− T 2
M2
Z
− T 2
= 0.11 ∼ 0.17,
(30)
ysy−mmc2 c2 ,
(31)
(32)
(33)
y f
for the Borel parameter T 2 = (2.2 − 2.8) GeV2. So the terms
exp − MTZ22 , exp − MTZ22 , . . . can be neglected
approximately, the non-diagonal transitions can be approximated as
λ2Z C exp − MT2Z2 .
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 =
0 = (0.8 ± 0.1) GeV2, at the energy scale μ =
m20 q¯q , m2
1 GeV [
20–22,31,32,41
], and mc(mc) = (1.28 ± 0.03) GeV
from the Particle Data Group [36]. Furthermore, we set
mu = md = 0 due to the small current quark masses. For
the new condensates or vacuum susceptibilities χ , κ and ξ
induced by the external electromagnetic field, we take two
set parameters:
(I) the old values χ = −3 GeV−2, κ = −0.75, ξ = 1.5 at
the energy scale μ = 1 GeV fitted in the QCD sum rules
for the magnetic moments of the p, n and [
31,32
];
(II) the new values χ = −(3.15 ± 0.30) GeV−2, κ = −0.2,
ξ = 0.4 at the energy scale μ = 1 GeV determined in the
detailed QCD sum rules analysis of the photon light-cone
distribution amplitudes [42].
The existing values of the vacuum susceptibilities χ , κ and
ξ are quite different from different determinations, the old
values χ (1 GeV) = −(4.4 ± 0.4) GeV−2 [
44
] or −(5.7 ±
0.6) GeV−2 [
45
] determined in the QCD sum rules combined
with the vector meson dominance, while the most recent
value χ (1 GeV) = −(2.85 ± 0.50) GeV−2 determined in
the light-cone QCD sum rules for the radiative heavy meson
decays [
43
]. The old values χ = −6 GeV−2, κ = −0.75,
ξ = 1.5 at the energy scale μ = 1 GeV determined in the
QCD sum rules for the octet and decuplet baryon magnetic
moments [
33,34
] are also different from the parameters (I)
determined in analysis of the magnetic moments of the p, n
and [
31,32
]. The most popularly used values are the
parameters (II) [
42
], which are consistent with the most recent
values determined in Ref. [
43
]. In the parameters (I), we
choose the value χ (1 GeV) = −3 GeV−2 [
31,32
] instead
of χ (1 GeV) = −6 GeV−2 [
33,34
] according to the most
recent value χ (1 GeV) = −(2.85 ± 0.50) GeV−2 [
43
].
We take into account the energy-scale dependence of the
input parameters from the renormalization group equation,
αs (Q) 225
αs (μ)
q¯ gs σ Gq (μ) = q¯ gs σ Gq (Q)
12
αs (Q) 25
αs (μ)
,
χ (μ) = χ (Q)
κ(μ) = κ(Q)
ξ(μ) = ξ(Q)
mc(μ) = mc(mc)
1
αs (μ) = b0t
16
αs (μ) 25
αs (Q)
36
αs (μ) 25
αs (Q)
36
αs (μ) 25
αs (Q)
,
,
+
b12(log2 t − log t − 1) + b0b2
b4t2
0
2
where t = log μ2 , b0 = 331−22πn f , b1 = 1532−4π192n f , b2 =
2857− 50933 n f + 32275 n2f
128π3 , = 210 MeV, 292 MeV and 332 MeV
for the flavors n f = 5, 4 and 3, respectively [
36,42,46,47
],
and evolve all the input parameters to the optimal energy scale
μ = 1.4 GeV to extract the form-factor G2(0) [
10,24,48
].
The hadronic parameters are taken as !sZ0 = 4.4 GeV,
MZ = 3.899 GeV, λZ = 2.1 × 10−2 GeV5 [
10,48
]. 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 [37]. We choose
the value !sZ0 = 4.4 GeV to avoid contamination of the
Z (4430) and reproduce the experimental value of the mass
MZ = 3.899 GeV from the QCD sum rules. The unknown
parameter is fitted to be C = 0.97 GeV−2 and 0.94 GeV−2
for the parameters (I) and (II) respectively to obtain platforms
in the Borel window T 2 = (2.2 − 2.8) GeV2.
We take into account uncertainties of the input
parameters, and obtain the value of the form-factor G2(0) therefore
the magnetic moment μZ of the Zc(3900), which is shown
explicitly in Fig. 3,
|μZ | = G2(0)
e
2MZ
e
= 1.46+0.98
−0.80 2MZ
= 0.35+−00..2149 μN for parameters, (I),
e
= 1.96+1.14
−0.91 2MZ
= 0.47+−00..2272 μN for parameters (II),
where the μN is the nucleon magneton. The present
prediction can be confronted to the experimental data in the future.
As the vacuum susceptibilities κ and ξ are less well
studied compared to the vacuum susceptibility χ , in Fig. 4, we
plot the value of the G2(0) with variation of the
parameter y = −κ(1 GeV) = ξ(1 GeV)/2 for the parameters
T 2 = 2.5 GeV2, χ (1 GeV) = −3.15 GeV−2 and C =
(35)
(36)
Fig. 4 The form-factor G2(0) with variation of the parameter y =
−κ(1 GeV) = ξ(1 GeV)/2
0.94 GeV−2. From the figure, we can see that the value of
the G2(0) decreases monotonously with increase of the
absolute values of the κ and ξ . Once the value of the magnetic
moment μZ is precisely measured, we can obtain a powerful
constraint on the values of the κ and ξ .
In Ref. [
25
], Ozdem and Azizi obtain the magnetic
moment μZ = 0.67 ± 0.32 μN from the light-cone QCD
sum rules, where the μZ decreases monotonously with
increase of the Borel parameter, the Borel window is not
flat enough. Compared to the QCD sum rules in the
external electromagnetic field, the light-cone QCD sum rules
have more parameters with limited precision. Furthermore,
there exists no experimental data on the electromagnetic
multipole moments of the exotic particles X , Y and Z , so
theoretical studies play an important role, more theoretical
works are still needed. For example, the magnetic moment
of the Zc(3900) as a molecular state is still not studied
with the QCD sum rules or light-cone QCD sum rules, a
comparison of the two possible assignments is not
possible at the present time. We can diagnose the nature of the
Zc(3900) as a genuine resonance or anomalous triangle
singularity in the photoproduction process [
49
].
Experimentally, the COMPASS collaboration observed the X (3872)
in the subprocess γ ∗ N → N X (3872)π ± in the process
μ+ N → μ+ N X (3872)π ± → μ+ N π +π −π ± [
50
], but
have not observed the Zc±(3900) in the subprocess γ ∗ N →
N Zc(3900)± in the process μ+ N → μ+ N Zc(3900)± →
μ+ N J /ψ π ± yet [
51
]. We can study the photon associated
production γ ∗ N → N Zc(3900)±γ or other processes with
the final states Zc±(3900)γ to study the electromagnetic
multipole moments of the Zc±(3900), although it is a difficult
work.
4 Conclusion
In this article, we tentatively assign the Zc±(3900) to be the
diquark-antidiquark type axialvector tetraquark state, study
its magnetic moment with the QCD sum rules in the external
weak electromagnetic field by carrying out the operator
product expansion up to the vacuum condensates of dimension 8,
and neglect the tiny contributions of the gluon condensate.
We pay special attention to matching the hadron side of the
correlation function with the QCD side of the correlation
function to obtain solid duality, the routine can be applied
to study other electromagnetic properties of the exotic
particles X , Y and Z directly. Finally, we obtain the magnetic
moment of the Zc(3900), which can be confronted to the
experimental data in the future and shed light on the nature
of the Zc(3900).
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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