1. Introduction
At present it is well known that all physical systems can be classified into several categories:
- The first corresponds to those, in which the symmetry broken at T = 0 is restored at
high temperature [1, 2]. Here high temperature means that T/M >> 1 for mass scale M of
the system. In addition, there is another alternative phenomenon, behavior associated with more
broken symmetry as temperature is increased. This is the so-called inverse symmetry breaking
(ISB).
- The second category deals with those cases which exhibit symmetry non-restoration
(SNR) at high temperature. This phenomenon emerges in a lot of systems and materials [3]. In
the context of quantum field theory, the high temperature SNR has been considered in [4, 5, 6].
The mechanisms of SNR/ISB have found a variety of applications. For instance, in
cosmology, where they have been implemented in realistic models, their consequences have been
explored in connection with high temperature phase transitions in the early stages of the Universe,
with applications covering problems involving CP violation and baryogenesis, topological defect
formation, inflation, etc. For example, the Kibble-Higgs sector of a SU(5) grand unified theory can
be mimicked by considering the case Nφ = 90 and Nψ = 24 and has been used to treat the monopole
problem. Setting Nφ = Nψ = 1 the model becomes invariant under the discrete transformation
Z2 × Z2. The latter version has been used in connection with the domain wall problem [7].

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2017-0031
Mathematical and Physical Sci., 2017, Vol. 62, Iss. 8, pp. 48-58
This paper is available online at
PHASE TRANSITION IN Z2 × Z2 MODEL BASED
ON THE CJT FORMALISM
Le Viet Hoa1, Nguyen Tuan Anh2 and Pham Van Dien3
1Faculty of Physics, Hanoi National University of Education
2Electric Power University
3Military Academy of Logistics
Abstract. Cornwall-Jackiw-Tomboulis (CJT) effective action at finite temperature is
applied to study the symmetry non-restoration (SNR) and inverse symmetry breaking
(ISB) at high temperature in the Z2 × Z2 model. A renormalization prescription is
developed for CJT effective action in the double bubble approximation. It is shown that
the triviality-related feature of the model does not show up. The phase transitions patterns
are considered in detail for a specified set of model parameters.
Keywords: Quantum condensation, two-component system, CJT effective action, SNR,
ISB, Z2 × Z2 model, double bubble approximation.
1. Introduction
At present it is well known that all physical systems can be classified into several categories:
- The first corresponds to those, in which the symmetry broken at T = 0 is restored at
high temperature [1, 2]. Here high temperature means that T/M >> 1 for mass scale M of
the system. In addition, there is another alternative phenomenon, behavior associated with more
broken symmetry as temperature is increased. This is the so-called inverse symmetry breaking
(ISB).
- The second category deals with those cases which exhibit symmetry non-restoration
(SNR) at high temperature. This phenomenon emerges in a lot of systems and materials [3]. In
the context of quantum field theory, the high temperature SNR has been considered in [4, 5, 6].
The mechanisms of SNR/ISB have found a variety of applications. For instance, in
cosmology, where they have been implemented in realistic models, their consequences have been
explored in connection with high temperature phase transitions in the early stages of the Universe,
with applications covering problems involving CP violation and baryogenesis, topological defect
formation, inflation, etc. For example, the Kibble-Higgs sector of a SU(5) grand unified theory can
be mimicked by considering the caseNφ = 90 andNψ = 24 and has been used to treat the monopole
problem. Setting Nφ = Nψ = 1 the model becomes invariant under the discrete transformation
Z2 × Z2. The latter version has been used in connection with the domain wall problem [7].
Received February 19, 2017. Accepted August 18, 2017.
Contact Le Viet Hoa, e-mail: dr.viethoa@gmail.com.
48
Phase transition in the Z2 × Z2 model based on CJT formalism
Domain walls with internal structure are common in interacting scalar field theories
involving at least two real scalar fields. They consist of one scalar field which generates the domain
wall and thus has a kink-like profile, and one or more scalar fields which interact with the kink-field
attaining a lump-like profile. One can think of these extra fields as being dynamically localized
to the wall which at the same time attain tachyonic masses in the interior of the wall and thus
non-zero vacuum expectation values. In general, if the back reactions of these fields on the kink
field are significant, these can affect the domain-wall solution, particularly its width.
In this respect, there is growing interest in finding an adequate formalism which can provide
reliable results for SNR/ISB occurring at high temperature. As was pointed out in [4], the CJT
effective action is perhaps most suited for this purpose.
In this paper, based on the CJT effective action at finite temperature [7], we reconsider the
Z2 × Z2 model, which was used in [8, 9] for the domain wall problem and in other Refs. [10, 11].
2. Content
2.1. Renormalized CJT effective action at finite T
For simplicity let us start from the system described by the simple Lagrangian which is
invariant under transformations of symmetry group Z2 × Z2:
£ =
1
2
(∂µφ)
2 +
µ21
2
φ2 +
λ1
4!
φ4 +
1
2
(∂µψ)
2 +
µ22
2
ψ2 +
λ2
4!
ψ4 +
λ
4
φ2ψ2 +∆£. (2.1)
The counter-terms are chosen as
∆L =
δµ21
2
φ2 +
δλ1
4!
φ4 +
δµ22
2
ψ2 +
δλ2
4!
ψ4 +
δλ
4
φ2ψ2.
The boundedness of the potential appearing in (2.1) requires
λ1 > 0, λ2 > 0 and λ1λ2 > 9λ
2. (2.2)
Shifting {φ,ψ} → {φ+ φ0, ψ + ψ0} leads to the interaction Lagrangian
£int =
λ1 + δλ1
24
φ4 +
λ1 + δλ1
6
φ0φ
3 +
λ2 + δλ2
24
ψ4 +
λ2 + δλ2
6
ψ0ψ
3
+
λ+ δλ
4
φ2ψ2 +
λ+ δλ
2
φ0φψ
2 +
λ+ δλ
2
ψ0ψφ
2
and the tree-level propagators in the momentum space
D−10 (k;φ0, ψ0) = k
2+µ1
2+δµ1
2+
λ1+δλ1
2
φ20+
λ+δλ
2
ψ20 ,
G−10 (k;φ0, ψ0) = k
2+µ22+δµ
2
2+
λ2+δλ2
2
ψ20+
λ+δλ
2
φ20.
Next the double bubble approximation is used to yield the truncated expression for CJT
49
Le Viet Hoa, Nguyen Tuan Anh and Pham Van Dien
effective potential V CJTβ [φ0, ψ0,D,G] at finite temperature
V CJTβ [φ0,ψ0,D,G]= δΩ+
µ21+δµ
2
1
2
φ20+
λ1+δλ1
24
φ40+
µ22+δµ
2
2
2
ψ20 +
λ2+δλ2
24
ψ40+
λ+δλ
4
φ20ψ
2
0
+
1
2
∫
β
{
lnD−1(k) + lnG−1(k) +D−10 (k;φ0, ψ0)D +G
−1
0 (k;φ0, ψ0)G− 2
}
+
λ1 + δλ1
8
[ ∫
β
D(k)
]2
+
λ2+δλ2
8
[ ∫
β
G(k)
]2
+
λ+ δλ
4
[ ∫
β
D(k)
][ ∫
β
G(k)
]
. (2.3)
Here the usual notation is introduced
∫
β
≡ T
∞∑
−∞
∫
d3~k
(2π)3
.
Following [4] we introduce the temperature dependent effective masses M1 and M2
D−1(k) = k2 +M21 ; G
−1(k) = k2 +M22 ,
where
M21 = µ
2
1 + δµ
2
1 +
λ1 + δλ1
2
[
φ20 + P (M1)
]
+
λ+ δλ
2
[ψ20 + P (M2)],
M22 = µ
2
2 + δµ
2
2 +
λ2 + δλ2
2
[ψ20 + P (M2)] +
λ+ δλ
2
[φ20 + P (M1)]. (2.4)
in which
P (M) =
∫
β
1
k2 +M2
.
Inserting (2.4) into (2.3) we get
V CJTβ [φ0,ψ0,M1,M2]=
µ21+δµ
2
1
2
φ20+
λ1+δλ1
24
φ40+
µ22+δµ
2
2
2
ψ20+
λ2+δλ2
24
ψ40+
λ+δλ
4
φ20ψ
2
0
+ Q(M1) +Q(M2) +
1
2
(
µ21 + δµ
2
1 +
λ1 + δλ1
2
φ20 +
λ+ δλ
2
ψ20 −M
2
1
)
P (M1)
+
1
2
(
µ22 + δµ
2
2 +
λ2 + δλ2
2
ψ20 +
λ+ δλ
2
φ20 −M
2
2
)
P (M2)
+
λ1 + δλ1
8
[P (M1)]
2 +
λ2 + δλ2
8
[P (M2)]
2 +
λ+ δλ
4
P (M1)P (M2), (2.5)
where
Q(M) =
1
2
∫
β
ln(k2 +M2).
Regularizing the divergent integrals P (M) and Q(M), appearing in (2.5) we make use of the
three-dimensional momentum cut-off scheme. Then each divergent integral is written to be the
50
Phase transition in the Z2 × Z2 model based on CJT formalism
sum of divergent and finite parts, namely
Q(M) = DivQ(M) +Qf (M), DivQ(M) = −
M4
4
I2 +
M2
2
I1,
Qf (M) =
M4
64π2
(
ln
M2
µ2
−
1
2
)
+T
∫
d3~k
(2π)3
ln
(
1−e−
E(~k)
T
)
,
P (M) = DivP (M) + Pf (M), DivP (M) = I1 −M
2I2,
Pf (M) =
M2
16π2
ln
M2
µ2
−
∫
d3~k
(2π)3
[
E(~k)
(
1− e
E(~k)
T
)]
−1
,
I1 =
Λ2
8π2
, I2=
1
16π2
ln
Λ2
µ2
, E(~k)=(~k2 +M2)1/2. (2.6)
Now let us develop the renormalization carried out in [10] requiring that all divergent terms have
to be absorbed into counter-terms, corresponding to renormalizing masses and coupling constants.
To this end, the renormalized masses and coupling constants are defined as
δ2V
δφ2
∣∣∣∣
φ=φ0;ψ=ψ0
= µ21R,
δ2V
δψ2
∣∣∣∣
φ=φ0;ψ=ψ0
= µ22R,
δ4V
δφ4
∣∣∣∣
φ=φ0;ψ=ψ0
= λ1R,
δ4V
δψ4
∣∣∣∣
φ=φ0;ψ=ψ0
= λ2R,
δ4V
δφ2δψ2
∣∣∣∣
φ=φ0;ψ=ψ0
= λR, (2.7)
which lead to two equations for δλ1, δλ2 and δλ
δλ1
[
φ20 + P (M1)
]
+ δλ[ψ20 + P (M2)] + λ1
[
φ20 + P (M1)
]
+ λ[ψ20 + P (M2)] = 0,
δλ2P (M2) + δλ
[
φ20 + P (M1)
]
+ λ2[ψ
2
0 + P (M2)] + λ
[
φ20 + P (M1)
]
= 0. (2.8)
Besides, another equation is imposed additionally,
δµ21DivP (M1) + δµ
2
2DivP (M2)+
δλ1
4
[
P 2(M1)+2φ
2
0DivP (M1)
]
+
δλ2
4
[P 2(M2)+2ψ
2
0DivP (M2)] +
δλ
4
[
P (M1)P (M2)+φ
2
0DivP (M2)+ψ
2
0DivP (M1)
]
+
(
µ21 +
λ1
2
φ20 +
λ
2
ψ20 −M
2
1
)
DivP (M1) +
(
µ22 +
λ
2
φ20 +
λ2
2
ψ20 −M
2
2
)
DivP (M2)
+2 [DivQ(M1) + DivQ(M2)] = 0. (2.9)
Altogether, we have a system of three linear equations for five unknown quantities δµ21, δµ
2
2,
δλ1, δλ2 and δλ. The existence of nontrivial roots ensures that only finite terms would be present
in the renormalized effective potential
V CJTβ [φ0, ψ0,M
2
1R,M
2
2R] =
µ21R
2
φ20 +
λ1R
24
φ40 +
µ22R
2
ψ20 +
λ2R
24
ψ40 +
λR
4
φ20ψ
2
0
+ Qf (M1R) +Qf (M2R) +
1
2
(
µ21R+
λ1R
2
φ20+
λR
2
ψ20−M
2
1R
)
Pf (M1R)
+
1
2
(
µ22R+
λ2R
2
ψ20+
λR
2
φ20−M
2
2R
)
Pf (M2R) +
λ1R
8
[Pf (M1R)]
2
+
λ2R
8
[Pf (M2R)]
2 +
λR
4
Pf (M1R)Pf (M2R). (2.10)
51
Le Viet Hoa, Nguyen Tuan Anh and Pham Van Dien
From (2.10) the renormalized gap equations are obtained
[
µ21R +
λ1R
6
φ20+
λR
2
ψ20+
λ1R
2
Pf (M1R)+
λR
2
Pf (M2R)
]
φ0=0,[
µ22R +
λ2R
6
ψ20+
λR
2
φ20+
λ2R
2
Pf (M2R)+
λR
2
Pf (M1R)
]
ψ0=0. (2.11)
and
M21R = µ
2
1R+
λ1R
2
[
φ20+Pf (M1R)
]
+
λR
2
[ψ20+Pf (M2R)],
M22R = µ
2
2R+
λ2R
2
[ψ20+Pf (M2R)]+
λR
2
[φ20+Pf (M1R)]. (2.12)
Substituting (2.12) into (2.10) we arrive at
Vβ[φ0,ψ0] =
µ21R
2
φ20 +
λ1R
24
φ40+
µ22R
2
ψ20 +
λ2R
24
ψ40+
λR
4
φ20ψ
2
0 +Qf (M1R) +Qf (M2R)
−
λ1R
8
[Pf (M1R)]
2 −
λ2R
8
[Pf (M2R)]
2 −
λR
4
Pf (M1R)Pf (M2R), (2.13)
For convenience the subscript R will drop out in what follows.
It is worth emphasizing that the present renormalization prescription leads to two
important results:
- The expression (2.10) for the renormalized V CJTβ does not contain any cut-off
dependent term.
- The so-called triviality-related features of the model under consideration does not
show up.
They are our major successes in comparison with [4].
2.2. High Temperature SNR/ISB
Considering high temperature SNR/ISB let us assume that µ21 < 0 and µ
2
2 > 0. As
a consequence,
φ0 6= 0 and ψ0 = 0, (2.14)
which implies that at T = 0 symmetry of the system is spontaneously broken in the φ sector and
unbroken in the ψ sector.
At high temperature the necessary condition for symmetry restoration (SR) in sector φ
is that
M21 (T )
∣∣
T=T1
= 0 (2.15)
at some value T = T1.
In the vicinity of T1, T
2/M21 ≫ 1, and the high temperature behavior of M
2
1 (T ) looks like
M21 (T ) ≃ µ
2
1 +
T 2
24
(λ1 + λ), (2.16)
52
Phase transition in the Z2 × Z2 model based on CJT formalism
which gives immediately
T 21 ≃ −
24µ21
λ1 + λ
. (2.17)
It is clear that T1 is real only if λ1 + λ > 0, then the symmetry gets restored at T = T1. Inversely,
the symmetry non-restoration occurs in φ sector if
λ1 + λ < 0 (2.18)
or
λ < 0 and λ1 < |λ|. (2.19)
Analogously, the symmetry in the ψ sector is spontaneously broken at high temperature T = Tc2
only if
M22 (T2) = 0. (2.20)
In the vicinity of T2 we obtain the high temperature expansion of M
2
2 (T )
M22 (T ) ≃ µ
2
2 +
T 2
24
(λ2 + λ) (2.21)
which leads to
T 22 ≃ −
24µ22
λ2 + λ
. (2.22)
Therefore the condition for ISB is
λ < 0, λ2 < |λ|. (2.23)
In resuming, the parameters are constrained by
λ1 > 0, λ2 > 0, µ
2
1 < 0, µ
2
2 > 0,
λ1λ2 > 9λ
2, λ λ1, λ2, (2.24)
for the present model, in which both SNR/ISB simultaneously take place at high temperatures in
corresponding sector.
2.3. Phase Transition Patterns For Specified Values of Parameters
In order to gain an insight into the model it is very interesting to consider the phase
transitions for specified values of the model parameters. As is easily seen, there is no value of
λ which fulfils both conditions
λ1λ2 > 9λ
2, |λ| > λ1,λ2. (2.25)
53
Le Viet Hoa, Nguyen Tuan Anh and Pham Van Dien
Fig. 1. The T dependence of M1 and M2, corresponding to the region in wich the broken symmetry in φ-sector
is restored (see Fig. 3). The phase transition happens in the interval [T1, Tc1].
In this respect, let us proceed to the phase transitions study for the case, in which broken
symmetry gets restored in the φ sector and ISB takes place in the ψ sector. Accordingly, the
parameters are constrained as follows.
λ1 > 0, λ2 > 0, µ
2
1 < 0, µ
2
2 > 0,
λ 9λ
2. (2.26)
It is clear from the expressions for the effective couplings given above, that for perturbative
values for the tree-level coupling parameters the predicted results for SNR/ISB are very stable
even for very large temperatures (in units of the regularization scale M1) which is due to the slow
logarithmic change with the temperature. So, in order to illustrate, let us choose at random some
specified values for µ21, µ
2
2, λ1, λ2 and λ, which obey the inequalities (2.26):
µ21 = −(4 MeV)
2, µ22 = (2 MeV)
2,
λ1 = 24, λ2 = 1.8 and λ = −2. (2.27)
They are the inputs for numerical computations. We first remark that, in addition
to the model parameters, the renormalization introduced another parameter µ, which is the
renormalization scale. Then we must determine a suitable value µ20 of µ
2 , which is defined as
the real root of the following equation
φ0(µ
2, 0)
∣∣
µ2=µ20
= 2 MeV, (2.28)
where φ0(µ
2, 0) is a solution of the system of Eqs. (2.11) and (2.12) at T = 0. The numerical
computation gives µ0 = 5.657 MeV.
54
Phase transition in the Z2 × Z2 model based on CJT formalism
Fig. 2. The T dependence of M1 and M2, corresponding to the region that the symmetry in ψ-sector
is broken (see Fig. 5). The phase transition happens in the interval [Tc2, T2].
Fig. 3. The T evolution of the order parameter φ, in which the broken symmetry in φ-sector is restored. The phase
transition happens in the interval [T1, Tc1]. At T1, the value φ0 = 0 is a maximum of V (φ0, T ), while the value
φ0 = 1.7 MeV is at a minimum. In the interval T1 < T < Tc1, the value φ0 = 0 is at a minimum at V (φ0, T ) = 0,
value φ02 is maximal, and φ01 is at a minimum. At Tc1, there is an inflexion point of V (φ0, T ) at φ0 = 0.988 MeV.
(see Fig. 4).
Let us consider three sectors. In the sector {φ0 6= 0, ψ0 = 0}, after eliminating φ0 from
(2.11) and (2.12) it leads to
M21 (T ) = −2µ
2
1 − λ1Pf (M1)− λPf (M2),
M22 (T ) = µ
2
2 +
3λ
λ1
µ21 − λPf (M1)+
(
λ2
2
−
6λ2
4λ1
)
Pf (M2). (2.29a)
55
Le Viet Hoa, Nguyen Tuan Anh and Pham Van Dien
Fig. 4. The evolution of the V (φ0, T ) as a function of the order parameter φ0 for several temperature steps is:
T = 4.11, 4.5, 4.7, 4.878, 5.MeV from bottom to top. At T1, the value φ0 = 0 is a maximum of V (φ0, T ),
while the value φ0 = 1.7 MeV is at a minimum. In the interval T1 < T < Tc1, the value φ0 = 0
is at a minimum at V (φ0, T ) = 0, the value φ02 is maximal, and φ01 in minimal (see Fig. 4). At Tc1,
there is an inflexion point of V (φ0, T ) at φ0 = 0.988 MeV.
In the sector {φ0 = 0, ψ0 = 0}, we have
M21 (T ) = µ
2
1 +
λ1
2
Pf (M1) +
λ
2
Pf (M2),
M22 (T ) = µ
2
2 +
λ2
2
Pf (M2) +
λ
2
Pf (M1). (2.29b)
And in the sector {φ0 = 0, ψ0 6= 0}, after eliminating ψ0 from (2.11) and (2.12) it reads
M21 (T ) = µ
2
1 +
3λ
λ2
µ22 − λPf (M2)+
(
λ1
2
−
6λ2
4λ2
)
Pf (M1).
M22 (T ) = −2µ
2
2 − λ2Pf (M2)− λPf (M1). (2.29c)
Inserting µ2 = µ20 into (2.29) and then solving numerically this system of equations we obtain the
solutions M1 and M2 presented in Fig. 1 and Fig. 2.
The T dependence of the order parameter φ0 is given in Fig. 3. It is observed in these figures
that for 0 < T < T1 ≈ 4.11 MeV the system is in symmetrical breaking phase for the φ-sector.
A first order phase transition emerges in the interval T1 ≤ T ≤ Tc1 and the symmetry for the
φ-sector is restored completly since T > Tc1 ≈ 4.878 MeV, at which
dφ0(T )
dT
∣∣∣∣
T=Tc1
=∞, (2.30)
where Tc1 may be considered the critical temperature. This phenomenon is highlighted by
means of the numerical computation performed for Vβ[φ0, ψ0 = 0], as a function of φ0 at several
values of T . It is easily proved that the curve, corresponding to T = Tc1 = 4.878 MeV in Fig.
4, has an inflexion point at φ0(Tc1) = 0.998 MeV and V [φ0(Tc1)] = 0.227 MeV. The broken
symmetry is then restored at Tc1.
56
Phase transition in the Z2 × Z2 model based on CJT formalism
Fig. 5. The T evolution is of the order parameter ψ, in which the symmetry in ψ-sector is broken. The phase
transition happens in the interval [Tc2, T2]. At T2, the value ψ0 = 0 is at a maximum of V (ψ0, T ), while the value
ψ0 = 33.1 MeV is at a minimum. In the interval Tc2 < T < T2, the value ψ0 = 0 is at a minimum at
V (ψ0, T ) = 0, value ψ02 is at a maximum, and ψ01 at a minimum. At Tc2, there is an inflexion point of V (ψ0, T )
at ψ0 = 15.2 MeV. (see Fig. 6).
In order to consider the high temperature ISB in the ψ sector the T dependence of ψ0(T )
for large T are plotted in Fig. 5.
It is evident that the symmetry is broken for T = Tc2 = 212.253 MeV, at which
dψ0(T )
dT
∣∣∣∣
T=Tc2
=∞. (2.31)
Fig. 6. The evolution of the V (ψ0, T ) is a function of the order parameter ψ0 for several temperature steps:
T = 200, 212.253, 218, 228, 238.232 MeV from top to bottom. At T2, the value ψ0 = 0 is at a maximum of V (ψ0, T ),
while the value ψ0 = 33.1 MeV is at a minimum. In the interval Tc2 < T < T2, the value ψ0 = 0 is at a minimum
at V (ψ0, T ) = 0, value ψ02 is at a maximum, and ψ01 at a minimum (see Fig. 5). At Tc2, there is an inflexion point
of V (ψ0, T ) at ψ0 = 15.2 MeV.
57
Le Viet Hoa, Nguyen Tuan Anh and Pham Van Dien
Tc2 is considered as the critical temperature. It is the temperature for ISB to take place in
the ψ sector. The evolution of Vβ[φ0 = 0, ψ0] against ψ0 for different temperatures is shown in Fig.
6. It is properly asserted that the inflection point of the curve T = Tc2 = 212.253 MeV possesses
coordinates ψ0(Tc2) = 15.230 MeV and Vβ[ψ0(Tc2)] = 789.02 MeV.
3. Conclusions
In this paper the phase transitions were considered for Z2×Z2 model by means of the finite
temperature CJT effective action. It was obtained that the renormalization prescription carried out
for the T dependent CJT effective potential in the double bubble approximation remedied two
shortcomings that arose in [9], the cut-off dependence of the renormalized effective potential and
the triviality-related feature of the model. The latter, perhaps, is an artifact of the renormalization
method under consideration.
It was indicated that the SNR or ISB occurs if
λ λ1 or λ λ2. (3.1)
To better understand critical phenomena appearing in the model we investigated in detail phase
transitions for a set of parameters chosen at random. The numerical solutions for the gap equations
and the shape of the effective potential, as a function of order parameters at different temperatures,
exhibit the existence of first order phase transitions for SR in the φ sector and ISB in the ψ sector.
Although the model studied earlier is too simple, all those we observed in the preceding section
are extremely interesting and their main features do not depend on a chosen set of parameters,
provided the latter obeys (2.26), of course. Our present study, in some sense, could be considered
complementary to those obtained in [8, 9].
Acknowledgment. The authors would like to thank HNUE for its financial support.
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