# Non-trivial topological phases in transition metal rich half-Heusler Oxides

Bhautik R Dhori  Department of Physics, Faculty of Science,
The Maharaja Sayajirao University of Baroda, Vadodara, Gujarat-390 002, India
Raghottam M. Sattigeri  International Research Centre MagTop, Institute of
Physics,
Polish Academy of Sciences, Aleja Lotników 32/46, PL-02668 Warsaw, Poland
Prafulla K Jha<EMAIL_ADDRESS>Department of Physics, Faculty of Science,
The Maharaja Sayajirao University of Baroda, Vadodara, Gujarat-390 002, India

###### Abstract

Topological Insulators with gapless surface states and insulating bulk in non-
centrosymmetric cubic systems have been extensively explored following the
discovery of two-dimensional quantum spin hall effect in zincblende HgTe. In
such systems the negative band inversion strength EBIS ($=$
E${}_{\Gamma_{6}}-$ E${}_{\Gamma_{8}}<$ 0) governs the robustness of the non-
trivial topological states at ambient conditions. Hence, realizing large
negative values of EBIS has been a guiding motivation of several
investigations reported in literature. Here, we present a material design
approach which can be employed to realize large negative values of EBIS in
cubic materials such as half-Heusler (HH) oxides with 18 valence electron
configurations. We explore 27 HH oxides of the form ABO (A = Li, K, Rb; B =
Cu, Ag, Au) in $\alpha$-, $\beta$-, and $\gamma$-phase (by placing transition
metal atom at different Wyckoff positions) for their non-trivial topological
phase. Off these three phases, we found that, the $\alpha$-phase of nine HH
oxides (wherein the transition metal atoms occupy 4a Wyckoff positions in the
crystal structure) is the most promising with non-trivial topological phase
which is governed by the mass-darwin relativistic effects enhancing EBIS.
Whereas the other phases were found to be either trivial semiconductors or
semimetals or metals and most of them being dynamically unstable. We focus on
RbAuO in $\alpha$-phase with EBIS of $–$ 1.29 eV and the effect of strain
fields on the topological surface states of this compound. We conclude that
the $\alpha$-phase of HH oxide presented here can be synthesized
experimentally for diverse room temperature applications in spintronics and
nanoelectronics.

##  I Introduction

Topological quantum materials have generated tremendous excitement in the
condensed matter community over the recent years. Materials such as
topological insulators (TI) [1, 2], topological crystalline insulators [3, 4],
and kondo insulators [5, 6] have diverse and potential applications in
spintronics, quantum computations etc. Cubic systems like HgTe with inverted
band order are known to host non-trivial TI behaviour upon breaking the cubic
symmetry / on creating quantum well structures etc [7]. This was followed by
experimental efforts which eventually lead to the confirmation of non-trivial
topological phases and phase transitions in matter [8]. From theoretical
perspective such as, ab-initio studies, it is essential that the band order
predicted in cubic structures matches those observed experimentally. In this
regard, the standard generalised gradient approximations (GGA) is not accurate
and reliable [9]. Hence, computationally expensive methods such as, MBJGGA,
SCAN etc. have been implemented which successfully predict the inverted band
orders in tune with the experimental observations [9, 10, 11]. Standard GGA
approach typically fails to predict accurate inverted band orders due to band
inversion strengths (defined as, E${}_{BIS}=$ E${}_{\Gamma_{6}}-$
E${}_{\Gamma_{8}}$) of the order of meV in cubic systems. Hence, one can
expect reliable results from standard GGA if the EBIS is large enough i.e., of
the order of eV [12].

A ternary class of materials having ABO stoichiometric composition called
half-Heusler (HH) oxides are a prime candidate for TI owing to their excellent
thermoelectric [13, 14], superconducting [15, 16], and semimetallic properties
[13, 14, 15, 16, 17, 18, 19, 20, 21]. The non-centrosymmetric and face centred
cubic crystal structure of a typical HH compound PQR constitutes P+
substitution in a QR- zincblende sublattice [22]. Hence, ABO HH oxides can be
realised in three different phases ($\alpha$-, $\beta$-, and $\gamma$-) owing
to the presence of three non-equivalent atomic arrangements within the unit
cell [23]. These three distinct phases correspond to C1b structure [24]. Such
compounds are unique for example, the atomic arrangement of non-magnetic
elements can lead to a magnetic ordering in the compound [25]. This implies
that not only the stoichiometry but also the atomic arrangements play a vital
role as evident from the literature.[26, 27].

This motivated us to explore relativistic effects such as orbital inversion
and band splitting in HH oxides [28]. Using EBIS as a critical parameter Sawai
et.al., predicted a series of III-VIII-VA HH oxides with some of them hosting
non-trivial topological nature under ambient conditions [12]. It was proposed
that inclusion of transition metals and lighter anions such as O, S and Cl may
lead to large non-trivial energy gap in HH compounds [26]. Shi-Yuan Lin
et.al., predicted topological insulating phase in more than 2000 HH oxides
based on EBIS [26]. However, a systematic study of such HH oxides is scarce.
This motivated us to thoroughly investigate ABO (A= Li, K, Rb; B= Cu, Ag, Au)
HH oxides for SOC-induced topological phase.

In the present work, using ab-initio methods we have investigated the low-
energy electronic structure and topological insulating phase in transition
metal-rich HH oxides. We report 27 HH oxides with transition metal atoms
placed at different Wyckoff positions in the unit cell. Off these, the
$\alpha$-phase exhibits semimetallic electronic structure in non-relativistic
regime similar to HgTe [29]. These HH oxides are also analogous to well-
explored binary system HgTe [30] which is semimetallic in non-relativistic and
relativistic regime but hosts non-trivial topological properties under
dimensional confinement or uniaxial strain fields. However, unlike HgTe the
relativistic regime opens up large global gap in the HH oxides with potential
to host non-trivial topological phase at ambient conditions without
dimensional confinement or uniaxial strain fields.

The proposed materials have large EBIS which implies that these compounds are
excellent candidates facilitating experimental validation of non-trivial Dirac
dispersions at room temperature and ambient pressure. It also known that the
presence of oxygen intrinsically prevents bulk oxidations on the surface,
which makes these compounds robust in addition to topological protection [31].
In addition to this, the dynamical and thermodynamical stability give
directions on experimental realization of HH oxides in the $\alpha$-phase [29,
32]. Thus, indicating that ABO HH oxides are a potential candidate for room
temperature and ambient pressure applications in the field of nanoelectronics
and spintronics.

##  II Methodology

HH oxides of the form ABO were investigated using density functional theory
based first-principles calculations as implemented in Quantum ESPRESSO [33].
Norm conserving pseudopotential based on Martins-Troullier method with Perdew-
Burke-Ernzerhof type exchange-correlation functional under generalized
gradient approximation was used to obtain the ground state properties of the
materials [34]. The pseudopotentials account for the 4s1 orbitals of alkali
metals, 4s23d9 orbitals of transition metals, and 2s22p4 orbitals of O. For
SOC calculations we included the effects arising from the core valence
electrons using full relativistic pseudopotential based on the projector
augmented wave (PAW) method. Uniform optimized computational parameters were
used for all the calculations. The kinetic energy cut-off was 90 Ry with the
convergence threshold criteria of at least $<$10-8 Ry. A Monkhorst Pack [35]
grid with 10 $\times$ 10 $\times$ 10 k-points was used to map the momentum
space in all the calculations. The dynamic stability of the proposed oxides
are analysed in term of the phonon dispersion curves obtained using Density
Functional Perturbation Theory with a q-mesh of at least 3 $\times$ 3 $\times$
3 for all the systems. We performed ab-initio molecular dynamics (AIMD)
simulations on 2 $\times$ 2 $\times$ 2 super-cell of $\alpha$-phase for 3
picoseconds (3000 femtoseconds) time step with the temperature of thermostat
set to 300 K. For topological properties, we employed maximally localized
Wannier functions to build the exact tight binding Hamiltonian model for HH
oxides as implemented in the wannier90 code [36, 37]. We then used
WannierTools code to compute, $\mathbb{Z}_{2}$ invariants, ARPES-like surface
states, and slab band structures [38].

##  III Results and Discussion

###  III.1 Crystal structure and stability

Figure 1:  (a) Crystal structure of the proposed half-Heusler oxides in
$\alpha$-phase. (b) Irreducible Brillouin zone corresponding to the
F$\overline{4}$3m space group. (c) Ab-initio molecular dynamics simulation for
3 ps at 300 K thermostat temperature for $\alpha$-RbAuO system indicating
energetic stability of the structure at room temperature. (d) Phonon
dispersion curves of $\alpha$-RbAuO indicating absence of imaginary modes.

The ground state of the proposed ABO HH oxides exhibits face centered cubic
structure governed by F$\overline{4}$3m space group emerging from the
$\overline{4}$3m point group. The A, B, and O atoms occupy 4b, 4a, and 4c
Wyckoff positions respectively which corresponds to the $\alpha$-phase. In
this phase the elements at A site are alkali metal from group I(A) and
elements at B site are transition metals from group I(B). We have also
explored the $\beta$\- and $\gamma$-phase by exchanging the Wyckoff positions
of A, B and O sites in the ABO HH oxides (details in SUPPLEMENTARY MATERIAL
(SM)). Figure 1(a) and (b) presents the conventional crystal structure and
irreducible Brillouin zone of the proposed HH oxides. In such structures the
Pauling electronegativity of constituent elements plays a vital role as a
result, the electropositive element at A shares ‘n’electrons with the
electronegative element at B and O. This leads to HgTe-like zincblende lattice
is formed by B and O atom BOn- and an empty void within the zincblende
structure is stuffed with A atoms An+. The crystal structure of the proposed
HH oxides were obtained from literature [39] followed by optimization of the
lattice parameters by performing a proper convergence test. We have summarized
the optimized lattice parameters in Tab. 1 for $\alpha$-, $\beta$\- and the
$\gamma$-phase of HH oxides.

Since these oxides have not yet been synthesized experimentally and due to
scare data on their thermodynamic stability, we performed AIMD simulations at
room temperature (300 K) for time period of 3 picoseconds (3000 femtoseconds).
The AIMD results for RbAuO in $\alpha$-phase are presented in Fig. 1(c). It is
evident that the structure thermalizes without structural deformations at room
temperature with mild variations in the total energy. AIMD for other compounds
in $\alpha$-phase are presented in SM Fig. 1. The AIMD results give insight
into the feasibility of material synthesis at room temperature. Apart from
structural stability established from the AIMD studies, we also report the
phonon dispersion curves of all the phases using DFPT (SM Fig. 2,22,24). In
Fig. 1(d) we present the phonon dispersion curves for RbAuO which indicates
the absence of imaginary modes implying dynamical stability of the structure
in $\alpha$-phase.

###  III.2 Electronic band structure

The 9 HH oxides in $\alpha$-phase host semimetallic electronic structure in
non-relativistic regime with degenerate valence and conduction bands at the
center of the BZ. However, in the relativistic regime the bands are non-
degenerate with inverted orbital character across the Fermi energy being
retained at high symmetry point $\Gamma$ while being absent at other time
reversal invariant momenta points in the BZ due to the lack of spatial
inversion symmetry.

Figure 2:  Electronic band structure of RbAuO in $\alpha$ phase (a) without
and (b) with spin-orbit coupling indicating the relativistic splitting due to
strong spin-orbit effects.

Hence, under ambient conditions the relativistic effects originating from the
strong SOC in these HH oxides gives rise to non-trivial topological character
without the influence of external strain/pressure. This eliminates the use of
techniques such as, breaking crystal symmetry or dimensional confinement which
is otherwise required to realise non-trivial topology in HgTe and other
similar systems. The band topology of these HH oxides is governed by the
$\Gamma_{7}$ (j = 1/2; 5/2) and $\Gamma_{8}$ (j = 3/2; 5/2) states in
conduction band minima and valence band maxima respectively (which are
composed of the -p, -d orbitals), and $\Gamma_{6}$ (j = 1/2; 5/2) states below
the valence band maxima (which is composed of -s, -d orbitals). The orbital
inversion mechanism involves $\Gamma_{7}$, $\Gamma_{8}$ and $\Gamma_{6}$
states with the band inversion strength defined as, E${}_{BIS}=$
E${}_{\Gamma_{6}}-$ E${}_{\Gamma_{8}}$. The electronegativity of lighter anion
and strong SOC of heavier transition metals arising from the core electrons
results in this intrinsic band inversion. For E${}_{BIS}<$ 0 the HH oxides
exhibit non-trivial topological phase, while a trivial topological phase is
observed when E${}_{BIS}>$ 0\. All HH oxides in $\alpha$-phase host inverted
band characteristic under equilibrium conditions with $\Gamma_{6}$ states
below the $\Gamma_{8}$ states. The calculated value of EBIS and the SOC
induced global gap for all these HH oxides are tabulated in Tab. 1, which
clearly demonstrates non-trivial topological character (further confirmed by
computing $\mathbb{Z}_{2}$ invariant). In all these 9 HH oxides in
$\alpha$-phase, we notice indistinguishable electronic band structures (as
shown in SM Fig. 3) and with topological band ordering (as shown in SM Fig.
4). Off the 9 HH oxides, 7 (LiCuO, LiAgO, LiAuO, KCuO, KAuO, RbCuO, RbAuo)
exhibit a Mexican hat-like band dispersion in their conduction band minima and
valence band maxima due to non-trivial orbital inversions resulting in higher
values of EBIS as compared to the global energy gap.

Figure 3:  (a-b) The degenerate electronic states at $\Gamma$ point in absence
of relativistic effects and the density of states indicating the semi-metallic
nature. (c-d) Non-trivial global gap evident in the electronic structure and
density of states indicating insulating characters. Figure 4:  Atomic and
orbital projected band structures for RbAuO. (a) Total density of states
projected onto the electronic band structures in low energy regime without SOC
alongside atomic and orbital projections. (b) Total density of states
projected onto the electronic band structures in low energy regime with SOC
alongside atomic and orbital projections.

Other HH oxides with the transition metals placed at different Wyckoff
positions leading to the $\beta$\- and $\gamma$-phase were investigated. We do
not discuss their topological properties since these compounds are either
trivial insulators, semimetals or metals with all of them being dynamically
unstable as evident from the phonon dispersions presented in Fig. 22 and 24 in
SM. Hence, in the following sections, we primarily focus only on the
dynamically stable compound RbAuO in $\alpha$-phase for its electronic and
topological properties owing to strong SOC and large EBIS.

###  III.3 The case of RbAuO

####  III.3.1 Electronic and Topological properties

With the thermodynamic and dynamical stability of RbAuO presented in Fig.
1(c,d) we report the electronic and topological properties under equilibrium
conditions presented in Fig. 3. The optimized lattice parameters for RbAuO are
presented in Tab. 1. The electronic band structures in absence and presence of
the relativistic effects are presented in Fig. 2 at equilibrium conditions. In
non-relativistic regime the doubly degenerate states at $\Gamma$ point are
protected by the cubic symmetry of the system. However, in the relativistic
regime the conduction and valence bands are well resolved with a global gap of
190 meV in the entire BZ with Fermi energy in between. Since the DOS of RbAuO,
without the inclusion of SOC, is zero at the Dirac points (as shown in Fig. 3
(b)). In the presence of SOC, the gap appears and the DOS are reducing near
the Fermi level (Fig. 3 (d)). The low energy excitations and orbital band
inversions are analyzed near the Fermi energy in terms of the atomic and
orbital projected electronic structures wherein for TIs without inversion
symmetry, the band inversion occurs at a specific time reversal invariant
momenta in the reciprocal space.

The atomic and orbital projected electronic structures in the non-relativistic
and relativistic regime are presented in Fig. 4. In the non-relativistic
regime the crystal field gives rise to intrinsically inverted band order with
semimetallic character being evident from the degenerate conduction and
valence bands. This degeneracy originates from the -p and -d orbitals of Au
and O atoms in the unit cell while the lighter -s orbitals are pushed deeper
below the Fermi energy which is a clear indication of the inverted bulk band
order making the system non-trivial.

However, in the relativistic regime, the strong SOC effects enhance the
inverted orbital order which already existed due to the crystal field. The
-p1/2, -p3/2, -d3/2, and -d5/2 orbitals originating from the Au and O atoms in
the unit cell dominate the conduction band minima and the valence band maxima.
While the -s1/2 orbitals originating from Rb, Au, and O get pushed even
farther from the Fermi energy (from -1.4 eV to -1.5 eV) as compared to the
non-relativistic regime with some contribution from the -d5/2 orbitals of Au
atoms. This inverted order is also enhanced due to the mass-darwin
relativistic effects in the $\alpha$-phase of HH oxides. This leads to high
E${}_{BIS}=-$1.29 eV with well resolved global energy gap of 190 meV in the
entire BZ which is quite high as compared to several HH compounds [40, 41].
Hence, the intrinsically inverted orbital order indicates that RbAuO is non-
trivial.

Figure 5:  (a) Comparison of electronic band structure from DFT and
wannierization (b) Computational surface states of the RbAuO oxides
demonstrate conducting surfaces (c) evolution of Wannier charge centers

This type of non-trivial inverted band orders involving -s, -p, and -d
orbitals has also been observed in other HH compounds such as ScPtBi, LuPdBi,
and other C1b type HH oxides [42, 43]. Such inverted band order is also
observed in HH oxides NaYO (Y = Ag, Au, and Cu) [42].

The non-trivial nature of the proposed HH oxides and the corresponding band
topologies are further classified using $\mathbb{Z}_{2}$ invariants. Since the
system does not posses inversion symmetry, we compute these invariants using
the Wilson loop method.

$\nu_{0}=(\mathbb{Z}_{2_{(k_{i}=0)}}+\mathbb{Z}_{2_{(k_{i}=0.5)}})\cdot mod2$
(1)

$\nu_{i}=\mathbb{Z}_{2_{(k_{i}=0.5)}}$ (2)

This is achieved by computing the maximally localized Wannier functions to
generate the exact tight-binding Hamiltonian for the system. Hence,
$\mathbb{Z}_{2}$ invariants along six time reversal invariant planes i.e., kx
= 0,$\pi$, ky = 0,$\pi$ and kz = 0,$\pi$ in the BZ are computed using equation
1 and 2 rather than computing the product of parity of the eigen functions
[44].

Figure 5(a) presents the exact match between the electronic structure
calculated using DFT and the electronic structure extracted from the exact
tight-binding Hamiltonian generated using wannierization. The $\mathbb{Z}_{2}$
invariants are obtained by tracing the evolution of wannier charge centres in
the momentum space presented in Fig. 5 (c) which is obtained by using the
Wilson loop method. A non-trivial topology is characterized by the presence of
Kramer’s doublet exchange occuring during the evolution of wannier charge
centers. For non-trivial systems, the evolution of wannier charge centres
presented in Fig. 5(c) odd(even) number of crossings for $k_{i=0}$ and
$k_{i=0.5}$ respectively. Thus the $\mathbb{Z}_{2}$ invariants are ($\nu_{0}$,
$\nu_{1}$ $\nu_{2}$ $\nu_{3}$) = (1, 000) quantifying the non-trivial nature
of RbAuO. Since $\nu_{0}\neq 0$ it implies that the proposed system RbAuO is a
strong TI. Furthermore, using Green’s function method, we calculate ARPES-like
surface state spectra which is presented in Fig. 5(b). It is evident that, the
bulk global gap in the momentum space hosts a robust Dirac dispersion on the
surface projected along [111] direction. Hence, RbAuO can host dissipationless
transport of Fermions across the surface originating from the robust inverted
band order at room temperatures owing to high EBIS. We summarise the
structural properties of other HH oxides in $\beta$\- and $\gamma$-phase in
Tab. 1.

####  III.3.2 Strain effects

Figure 6:  Electronic band structures with SOC at (a) a $=$ a${}_{0}-$ 6% a0
and (b) a $=$ a${}_{0}+$ 6% a0 indicating band inversion at high symmetric
point $\Gamma$. ARPES-like surface state spectra for (c) compressive and (d)
tensile strains at $\pm$ 6%. Table 1: Summary of atomic Wyckoff positions and
lattice parameters of the proposed transition metal rich HH oxides in
$\alpha$-, $\beta$\- and $\gamma$-phase. Along with the energy band gap (Eg),
band inversion strength (EBIS) and the $\mathbb{Z}_{2}$ invariant for the
$\alpha$-phase. Note that, the symbols ∗, † and, # correspond to atoms
presented in the respective columns above.

Compounds $\xrightarrow{}$ | Li∗Cu†O# | Li∗Ag†O# | Li∗Au†O# | K∗Cu†O# | K∗Ag†O# | K∗Au†O# | Rb∗Cu†O# | Rb∗Ag†O# | Rb∗Au†O#
---|---|---|---|---|---|---|---|---|---
$\alpha$-phase | 4b∗4a†4c# | 4b∗4a†4c# | 4b∗4a†4c# | 4b∗4a†4c# | 4b∗4a†4c# | 4b∗4a†4c# | 4b∗4a†4c# | 4b∗4a†4c# | 4b∗4a†4c#
$\beta$-phase | 4a∗4c†4b# | 4a∗4c†4b# | 4a∗4c†4b# | 4a∗4c†4b# | 4a∗4c†4b# | 4a∗4c†4b# | 4a∗4c†4b# | 4a∗4c†4b# | 4a∗4c†4b#
$\gamma$-phase | 4c∗4b†4a# | 4c∗4b†4a# | 4c∗4b†4a# | 4c∗4b†4a# | 4c∗4b†4a# | 4c∗4b†4a# | 4c∗4b†4a# | 4c∗4b†4a# | 4c∗4b†4a#
$\alpha$-phase, a (Å) | 4.95 | 5.45 | 5.50 | 5.74 | 5.99 | 6.04 | 5.98 | 6.19 | 6.24
$\beta$-phase, a (Å) | 5.02 | 5.53 | 5.59 | 5.75 | 6.17 | 6.34 | 5.88 | 6.33 | 6.55
$\gamma$-phase, a (Å) | 4.91 | 5.30 | 5.39 | 5.97 | 6.24 | 6.41 | 6.18 | 6.52 | 6.72
$\alpha$-phase, Eg (meV) | 65.3 | 110.0 | 84.5 | 76.6 | 76.8 | 186.0 | 65.9 | 53.6 | 190.0
$\alpha$-phase, EBIS (eV) |  $-$1.25 |  $-$0.97 |  $-$2.37 |  $-$0.72 |  $-$0.19 |  $-$1.47 |  $-$0.59 |  $-$0.08 |  $-$1.29
$\alpha$-phase, $\mathbb{Z}_{2}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1

Since RbAuO exhibits high EBIS, we further explore the robustness of the non-
trivial topological nature by subjecting the system to hydrostatic strain of
compressive and tensile nature retaining the cubic symmetry. We find that the
electronic and topological properties of RbAuO has drast variations in the
wide range of strain $\pm$6%. In the tensile regime, the system global gap
reduces with the the Dirac cone being annihilated on the surface, making the
system weak TI. Whereas in the compressive regime, the Dirac cone is retained
and the global gap in the momentum space gets enhanced reaching a maximum
value of 230 meV. This is evident from electronic structure presented in Fig.
6(a,b). Although, the inverted orbital order is retained in both; the tensile
and compressive regimes, the stark difference is due to weaker orbital
hybridization’s in the tensile regime as compared to the compressive regime.
Hence, we show that the, non-trivial nature and Dirac dispersions along the
surface would persist under compressive strains and annihilated under tensile
strains. The compressive strains can be realised in experimental conditions
subjecting the system to substrate effects hence indicating towards device
applications for spintronics and nanoelectronics at room temperature.

##  IV Conclusion

To summarise, we propose a design approach wherein transition metals are
placed at different Wyckoff positions leading to 27 transition metal rich HH
oxides in $\alpha$-, $\beta$\- and $\gamma$-phase. We show that, off the 27
predicted materials, 18 are dynamically unstable in the $\beta$\- and
$\gamma$-phase which makes $\alpha$-phase the most promising configuration of
the HH oxides. However, off the 9 HH oxides in $\alpha$-phase, we have 4
systems which are dynamically unstable with thermodynamic stability being
observed at room temperature. This left us with only 5 dynamically stable
configurations in the $\alpha$-phase. Off these, RbAuO was found to be the
most promising HH oxide owing to its high E${}_{BIS}=-$ 1.29 eV which is quite
high compared to other compounds with a global energy gap in the entire BZ of
190 meV under equilibrium conditions. With such strong inverted band order
originating from the mass-darwin relativistic effects, we further report the
non-trivial topological nature in terms of the $\mathbb{Z}_{2}$ invariant and
ARPES-like surface states. To assess the practicality in device applications
of the proposed compound, we subjected the system to tensile and compressive
strains of $\pm$6%. The results show that, under compressive regime of strain
the electronic bands under go blue shift in energy with increment in the
global energy gap to maximum of 230 meV. With the non-trivial inverted band
order being retained which is evident from the $\mathbb{Z}_{2}$ invariant and
the Dirac cone on the projected surface states. Whereas, in the tensile
strain, the Dirac cone is anihilated with the system exhibiting weak TI
nature. Thus we propose transition metal rich HH oxides in $\alpha$-phase,
especially RbAuO for spintronic and nanoelectronic applications under ambient
conditions and at room temperature.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgement

BRD highly appreciates the Department of Science and Technology (DST),
Government of India, for the Inspire Fellowship award (No. DST/INSPIRE
fellowship/2020/IF200265). RMS acknowledges the support from Foundation for
Polish Science through the International Research Agendas program co-financed
by the European Union within the Smart Growth Operational Programme (Grant No.
MAB/2017/1) and access to the computing facilities of the Poznan
Supercomputing and Networking Center Grant No. 609.

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The ground states and first radial excitations of the vector tetraquark states
with explicit P-waves via the QCD sum rules

Zhi-Gang Wang 111E-mail<EMAIL_ADDRESS>
Department of Physics, North China Electric Power University, Baoding 071003,
P. R. China

###### Abstract

In this work, we choose the diquark-antidiquark type four-quark currents with
an explicit P-wave between the diquark and antidiquark pairs to study the
ground states and first radial excitations of the hidden-charm tetraquark
states with the quantum numbers $J^{PC}=1^{--}$. And we obtain the lowest
vector tetraquark masses and make possible assignments of the existing $Y$
states. There indeed exists a hidden-charm tetraquark state with the
$J^{PC}=1^{--}$ at the energy about $4.75\,\rm{GeV}$ as the first radial
excitation to account for the BESIII data.

PACS number: 12.39.Mk, 12.38.Lg

Key words: Tetraquark state, QCD sum rules

##  1 Introduction

There have been observed a number of $Y$ states, such as the $Y(4008)$
observed in the $J/\psi\pi^{+}\pi^{-}$ channel [1], the $Y(4220)$ and
$Y(4320)$ observed in the $J/\psi\pi^{+}\pi^{-}$ channel [2], the $Y(4230)$
observed in the $\omega\chi_{c0}$ channel [3], the $Y(4260)$ observed in the
$J/\psi\pi^{+}\pi^{-}$ channel [4, 5, 6], the $Y(4320)$ observed in the
$\psi^{\prime}\pi^{+}\pi^{-}$ channel [7], the $Y(4360)$ and $Y(4660)$
observed in the $\psi^{\prime}\pi^{+}\pi^{-}$ channel [8, 9], the $Y(4390)$
observed in the $\pi^{+}\pi^{-}h_{c}$ channel [10], the $Y(4469)$ observed in
the $D^{*0}D^{*-}\pi^{+}$ channel [11], the $Y(4484)$ observed in the
$K^{+}K^{-}J/\psi$ channel [12], the $Y(4544)$ observed in the
$\omega\chi_{c1}$ channel [13], the $Y(4630)$ observed in the
$\Lambda_{c}^{+}\Lambda_{c}^{-}$ channel [14], the $Y(4710)$ observed in the
$K^{+}K^{-}J/\psi$ channel [15], the $Y(4790)$ observed in the
$D_{s}^{\ast+}D_{s}^{\ast-}$ channel [16].

Recently, the BESIII collaboration studied the processes $e^{+}e^{-}\to\omega
X(3872)$ and $\gamma X(3872)$ using data samples with an integrated luminosity
of $4.5~{}\text{fb}^{-1}$ at center-of-mass energies from 4.66 to 4.95 GeV,
and observed that the relatively large cross section for the
$e^{+}e^{-}\to\omega X(3872)$ process is mainly attributed to the enhancement
around 4.75 GeV, which maybe indicate a potential structure in the
$e^{+}e^{-}\to\omega X(3872)$ cross section [17]. If the enhancement is
confirmed in the future by enough experimental data, there maybe exist another
$Y$ state, the $Y(4750)$.

Even if the $Y(4220)$, $Y(4230)$ and $Y(4260)$ are the same particle, the
$Y(4320)$, $Y(4360)$ and $Y(4390)$ are the same particle, the $Y(4469)$,
$Y(4484)$, and $Y(4544)$ are the same particle $Y(4500)$, the $Y(4630)$ and
$Y(4660)$ are the same particle, the $Y(4710)$ and $Y(4750)$ are the same
particle, the $Y$ states are beyond the compatibility of the traditional quark
models, we have to introduce tetraquark states, molecular states, hybrid
states, etc, to make reasonable assignments [18, 19, 20, 21, 22, 23, 24, 25,
26, 27].

In Refs.[28, 29], we take the scalar, pseudoscalar, axialvector, vector and
tensor (anti)diquarks as the elementary constituents to construct the four-
quark currents without introducing explicit P-waves, and investigate the
hidden-charm and hidden-charm-hidden-strange tetraquark states with the
quantum numbers $J^{PC}=1^{--}$ and $1^{-+}$ in a comprehensive and consistent
way via the QCD sum rules, and revisit the assignments of the $X/Y$ states in
the hidden-charm tetraquark scenario, see Tables 1-2, where the subscripts
$S$, $P$, $V$ ($\widetilde{V}$) and $A$ ($\tilde{A}$) stand for the scalar,
pseudoscalar, vector and axialvector (anti)diquarks, respectively.

$Y_{c}$ | $J^{PC}$ | $M_{Y}(\rm{GeV})$ | Assignments
---|---|---|---
$[uc]_{P}[\overline{dc}]_{A}-[uc]_{A}[\overline{dc}]_{P}$ | $1^{--}$ | $4.66\pm 0.07$ | ? $Y(4660)$
$[uc]_{P}[\overline{dc}]_{A}+[uc]_{A}[\overline{dc}]_{P}$ | $1^{-+}$ | $4.61\pm 0.07$ |
$[uc]_{S}[\overline{dc}]_{V}+[uc]_{V}[\overline{dc}]_{S}$ | $1^{--}$ | $4.35\pm 0.08$ | ? $Y(4360/4390)$
$[uc]_{S}[\overline{dc}]_{V}-[uc]_{V}[\overline{dc}]_{S}$ | $1^{-+}$ | $4.66\pm 0.09$ |
$[uc]_{\tilde{V}}[\overline{dc}]_{A}-[uc]_{A}[\overline{dc}]_{\tilde{V}}$ | $1^{--}$ | $4.53\pm 0.07$ | ? $Y(4500)$
$[uc]_{\tilde{V}}[\overline{dc}]_{A}+[uc]_{A}[\overline{dc}]_{\tilde{V}}$ | $1^{-+}$ | $4.65\pm 0.08$ |
$[uc]_{\tilde{A}}[\overline{dc}]_{V}+[uc]_{V}[\overline{dc}]_{\tilde{A}}$ | $1^{--}$ | $4.48\pm 0.08$ | ? $Y(4500)$
$[uc]_{\tilde{A}}[\overline{dc}]_{V}-[uc]_{V}[\overline{dc}]_{\tilde{A}}$ | $1^{-+}$ | $4.55\pm 0.07$ |
$[uc]_{S}[\overline{dc}]_{\tilde{V}}-[uc]_{\tilde{V}}[\overline{dc}]_{S}$ | $1^{--}$ | $4.50\pm 0.09$ | ? $Y(4500)$
$[uc]_{S}[\overline{dc}]_{\tilde{V}}+[uc]_{\tilde{V}}[\overline{dc}]_{S}$ | $1^{-+}$ | $4.50\pm 0.09$ |
$[uc]_{P}[\overline{dc}]_{\tilde{A}}-[uc]_{\tilde{A}}[\overline{dc}]_{P}$ | $1^{--}$ | $4.60\pm 0.07$ |
$[uc]_{P}[\overline{dc}]_{\tilde{A}}+[uc]_{\tilde{A}}[\overline{dc}]_{P}$ | $1^{-+}$ | $4.61\pm 0.08$ |
$[uc]_{A}[\overline{dc}]_{A}$ | $1^{--}$ | $4.69\pm 0.08$ | ? $Y(4660)$
Table 1:  The possible assignments of the hidden-charm tetraquark states, the isospin limit is implied [28].  $Y_{c}$ | $J^{PC}$ | $M_{Y}(\rm{GeV})$ | Assignments
---|---|---|---
$[sc]_{P}[\overline{sc}]_{A}-[sc]_{A}[\overline{sc}]_{P}$ | $1^{--}$ | $4.80\pm 0.08$ | ? $Y(4790)$
$[sc]_{P}[\overline{sc}]_{A}+[sc]_{A}[\overline{sc}]_{P}$ | $1^{-+}$ | $4.75\pm 0.08$ |
$[sc]_{S}[\overline{sc}]_{V}+[sc]_{V}[\overline{sc}]_{S}$ | $1^{--}$ | $4.53\pm 0.08$ |
$[sc]_{S}[\overline{sc}]_{V}-[sc]_{V}[\overline{sc}]_{S}$ | $1^{-+}$ | $4.83\pm 0.09$ |
$[sc]_{\tilde{V}}[\overline{sc}]_{A}-[sc]_{A}[\overline{sc}]_{\tilde{V}}$ | $1^{--}$ | $4.70\pm 0.08$ | ? $Y(4710)$
$[sc]_{\tilde{V}}[\overline{sc}]_{A}+[sc]_{A}[\overline{sc}]_{\tilde{V}}$ | $1^{-+}$ | $4.81\pm 0.09$ |
$[sc]_{\tilde{A}}[\overline{sc}]_{V}+[sc]_{V}[\overline{sc}]_{\tilde{A}}$ | $1^{--}$ | $4.65\pm 0.08$ | ? $Y(4660)$
$[sc]_{\tilde{A}}[\overline{sc}]_{V}-[sc]_{V}[\overline{sc}]_{\tilde{A}}$ | $1^{-+}$ | $4.71\pm 0.08$ |
$[sc]_{S}[\overline{sc}]_{\tilde{V}}-[sc]_{\tilde{V}}[\overline{sc}]_{S}$ | $1^{--}$ | $4.68\pm 0.09$ | ? $Y(4660)$
$[sc]_{S}[\overline{sc}]_{\tilde{V}}+[sc]_{\tilde{V}}[\overline{sc}]_{S}$ | $1^{-+}$ | $4.68\pm 0.09$ | ?? $X(4630)$
$[sc]_{P}[\overline{sc}]_{\tilde{A}}-[sc]_{\tilde{A}}[\overline{sc}]_{P}$ | $1^{--}$ | $4.75\pm 0.08$ |
$[sc]_{P}[\overline{sc}]_{\tilde{A}}+[sc]_{\tilde{A}}[\overline{sc}]_{P}$ | $1^{-+}$ | $4.75\pm 0.08$ |
$[sc]_{A}[\overline{sc}]_{A}$ | $1^{--}$ | $4.85\pm 0.09$ |
Table 2:  The possible assignments of the hidden-charm-hidden-strange
tetraquark states [29].

In Refs.[30, 31], we introduce an explicit P-wave between the diquark and
antidiquark pairs to construct the hidden-charm four-quark currents, and
explore the vector tetraquark states systematically via the QCD sum rules, and
obtain the lowest vector tetraquark masses up to today, and revisit the
assignments of the $Y$ states in the hidden-charm tetraquark scenario, see
Table 3, where the angular momenta
$\vec{S}=\vec{S}_{qc}+\vec{S}_{\bar{q}\bar{c}}$ and $\vec{J}=\vec{L}+\vec{S}$.

$|S_{qc},S_{\bar{q}\bar{c}};S,L;J\rangle$ | $M_{Y}(\rm{GeV})$ | Assignments
---|---|---
$|0,0;0,1;1\rangle$ | $4.24\pm 0.10$ | $Y(4220)$
$|1,1;0,1;1\rangle$ | $4.28\pm 0.10$ | $Y(4220/4320)$
$\frac{1}{\sqrt{2}}\left(|1,0;1,1;1\rangle+|0,1;1,1;1\rangle\right)$ | $4.31\pm 0.10$ | $Y(4320/4390)$
$|1,1;2,1;1\rangle$ | $4.33\pm 0.10$ | $Y(4320/4390)$
Table 3:  The possible assignments of the hidden-charm tetraquark states with
explicit P-waves, the isospin limit is implied [31].

From Tables 1-3, we can see explicitly that there are no rooms for the
$Y(4008)$ and $Y(4750)$ in the vector hidden-charm tetraquark scenario. If we
take the $X(3872)$ and $Z_{c}(3900)$ as the lowest tetraquark states with the
$J^{PC}=1^{++}$ and $1^{--}$, respectively [32, 33, 34, 35, 36], the $Y(4008)$
cannot be assigned as a vector tetraquark state due to the small mass
splitting $\delta M\approx 100\,\rm{MeV}$. If the $Y(4220/4230/4260)$ can be
assigned as the ground state vector tetraquark state, the lowest vector
hidden-charm tetraquark state can be obtained by the QCD sum rules, the
$Y(4750)$ can be assigned as its first radial excitation according to the mass
gap $M_{Y(4750)}-M_{Y(4260)}=0.51\,\rm{GeV}$, which happens to be our naive
expectation of the mass gap between the ground state and first radial
excitation.

If the $Y(4750)$ can be assigned as the first radial excitation of the
$Y(4220/4230/4260)$, there maybe exist a spectrum for the first radial excited
states, which lie at about $4.8\,\rm{GeV}$, it is interesting to explore such
a possibility.

As we know, the heavy-light diquarks $\varepsilon^{ijk}q^{T}_{j}C\Gamma Q_{k}$
have five structures, where the $i$, $j$ and $k$ are color indexes,
$C\Gamma=C\gamma_{5}$, $C$, $C\gamma_{\mu}\gamma_{5}$, $C\gamma_{\mu}$ and
$C\sigma_{\mu\nu}$ for the scalar, pseudoscalar, vector, axialvector and
tensor diquarks, respectively, the P-wave is implicitly embodied in the
negative parity of the diquarks. We can also introduce an explicit P-wave
inside the heavy-light diquarks to obtain
$\varepsilon^{ijk}q^{T}_{j}C\Gamma\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\mu}Q_{k}$,
where the derivative
$\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\mu}=\stackrel{{\scriptstyle\rightarrow}}{{\partial}}_{\mu}-\stackrel{{\scriptstyle\leftarrow}}{{\partial}}_{\mu}$
embodies the explicit P-wave, then take the diquarks
$\varepsilon^{ijk}q^{T}_{j}C\Gamma\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\mu}Q_{k}$
as the basic building blocks to construct the four-quark currents to study the
tetraquark states with the $J^{PC}=1^{--}$, and we would explore such a
possibility in our next work.

In Refs.[28, 29, 30, 31], we use the (modified) energy scale formula to obtain
the suitable energy scales of the QCD spectral densities to enhance the pole
contributions and improve the convergent behaviors of the operator product
expansion [37], it is the unique feature of our works. In this direction, we
have also explored the hidden-charm tetraquark states with the
$J^{PC}=0^{++}$, $0^{-+}$, $0^{--}$, $1^{+-}$, $2^{++}$ [38, 39], hidden-
bottom tetraquark states with the $J^{PC}=0^{++}$, $1^{+-}$, $2^{++}$ [40],
hidden-charm molecular states with the $J^{PC}=0^{++}$, $1^{+-}$, $2^{++}$
[41], doubly-charm tetraquark (molecular) states with the $J^{P}=0^{+}$,
$1^{+}$, $2^{+}$ [42] ([43]), hidden-charm pentaquark (molecular) states
[44]([45]), and assign the existing exotic states consistently.

In the isospin limit, the vector tetraquark states with the symbol valence
quarks

$\displaystyle I=1$ $\displaystyle:$ $\displaystyle
c\bar{c}u\bar{d}\,,\,\,\,c\bar{c}\frac{u\bar{u}-d\bar{d}}{\sqrt{2}}\,,\,\,\,c\bar{c}d\bar{u}\,,$
$\displaystyle I=0$ $\displaystyle:$ $\displaystyle
c\bar{c}\frac{u\bar{u}+d\bar{d}}{\sqrt{2}}\,,$ (1)

have degenerated masses, and we would explore the $c\bar{c}u\bar{d}$
tetraquark states for simplicity. We update the analysis of our previous works
[30, 31], extend our previous works to explore the first radial citations of
the vector hidden-charm tetraquark states with the QCD sum rules
systematically, and take the modified energy scale formula to acquire the
suitable energy scales of the QCD spectral densities, and make possible
assignments of the existing $Y$ states and make predictions for the mass
spectrum of the first radial excitations at the energy about $4.8\,\rm{GeV}$.

The article is arranged as follows: we derive the QCD sum rules for the vector
tetraquark states in section 2; in section 3, we present the numerical results
and discussions; section 4 is reserved for our conclusion.

##  2 QCD sum rules for the vector tetraquark states

We write down the two-point correlation functions $\Pi_{\mu\nu}(p)$ and
$\Pi_{\mu\nu\alpha\beta}(p)$ firstly,

$\displaystyle\Pi_{\mu\nu}(p)$ $\displaystyle=$ $\displaystyle i\int
d^{4}xe^{ip\cdot x}\langle
0|T\left\\{J_{\mu}(x)J_{\nu}^{\dagger}(0)\right\\}|0\rangle\,,$ (2)
$\displaystyle\Pi_{\mu\nu\alpha\beta}(p)$ $\displaystyle=$ $\displaystyle
i\int d^{4}xe^{ip\cdot x}\langle
0|T\left\\{J_{\mu\nu}(x)J_{\alpha\beta}^{\dagger}(0)\right\\}|0\rangle\,,$ (3)

where $J_{\mu}(x)=J_{\mu}^{1}(x)$, $J_{\mu}^{2}(x)$ and $J_{\mu}^{3}(x)$,

$\displaystyle J^{1}_{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon^{ijk}\varepsilon^{imn}}{\sqrt{2}}u^{Tj}(x)C\gamma_{5}c^{k}(x)\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\mu}\bar{d}^{m}(x)\gamma_{5}C\bar{c}^{Tn}(x)\,,$
(4) $\displaystyle J^{2}_{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon^{ijk}\varepsilon^{imn}}{\sqrt{2}}u^{Tj}(x)C\gamma_{\alpha}c^{k}(x)\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\mu}\bar{d}^{m}(x)\gamma^{\alpha}C\bar{c}^{Tn}(x)\,,$
(5) $\displaystyle J^{3}_{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon^{ijk}\varepsilon^{imn}}{2}\left[u^{Tj}(x)C\gamma_{\mu}c^{k}(x)\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\alpha}\bar{d}^{m}(x)\gamma^{\alpha}C\bar{c}^{Tn}(x)\right.$
(6)
$\displaystyle\left.+u^{Tj}(x)C\gamma^{\alpha}c^{k}(x)\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\alpha}\bar{d}^{m}(x)\gamma_{\mu}C\bar{c}^{Tn}(x)\right]\,,$
$\displaystyle J_{\mu\nu}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon^{ijk}\varepsilon^{imn}}{2\sqrt{2}}\left[u^{Tj}(x)C\gamma_{5}c^{k}(x)\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\mu}\bar{d}^{m}(x)\gamma_{\nu}C\bar{c}^{Tn}(x)\right.$
(7)
$\displaystyle+u^{Tj}(x)C\gamma_{\nu}c^{k}(x)\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\mu}\bar{d}^{m}(x)\gamma_{5}C\bar{c}^{Tn}(x)$
$\displaystyle-u^{Tj}(x)C\gamma_{5}c^{k}(x)\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\nu}\bar{d}^{m}(x)\gamma_{\mu}C\bar{c}^{Tn}(x)$
$\displaystyle\left.-u^{Tj}(x)C\gamma_{\mu}c^{k}(x)\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\nu}\bar{d}^{m}(x)\gamma_{5}C\bar{c}^{Tn}(x)\right]\,.$

Under charge conjugation transform $\widehat{C}$, the currents $J_{\mu}(x)$
and $J_{\mu\nu}(x)$ have the properties,

$\displaystyle\widehat{C}J_{\mu}(x)\widehat{C}^{-1}$ $\displaystyle=$
$\displaystyle-J_{\mu}(x)\,,$
$\displaystyle\widehat{C}J_{\mu\nu}(x)\widehat{C}^{-1}$ $\displaystyle=$
$\displaystyle-J_{\mu\nu}(x)\,,$ (8)

in other words, they have negative conjugation.

At the hadron side, we insert a complete set of intermediate hadronic states
with the same quantum numbers as the interpolating currents $J_{\mu}(x)$ and
$J_{\mu\nu}(x)$ into the correlation functions $\Pi_{\mu\nu}(p)$ and
$\Pi_{\mu\nu\alpha\beta}(p)$ respectively to acquire the hadronic
representation [46, 47, 48]. Then we isolate the ground states and obtain the
expressions,

$\displaystyle\Pi_{\mu\nu}(p)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{Y}^{2}}{M_{Y}^{2}-p^{2}}\left(-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{p^{2}}\right)+\cdots\,\,,$
(9) $\displaystyle=$
$\displaystyle\Pi_{Y}(p^{2})\left(-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{p^{2}}\right)+\cdots\,,$
$\displaystyle\Pi_{\mu\nu\alpha\beta}(p)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{Y}^{2}}{M_{Y}^{2}\left(M_{Y}^{2}-p^{2}\right)}\left(p^{2}g_{\mu\alpha}g_{\nu\beta}-p^{2}g_{\mu\beta}g_{\nu\alpha}-g_{\mu\alpha}p_{\nu}p_{\beta}-g_{\nu\beta}p_{\mu}p_{\alpha}+g_{\mu\beta}p_{\nu}p_{\alpha}+g_{\nu\alpha}p_{\mu}p_{\beta}\right)$
(10)
$\displaystyle+\frac{\lambda_{Z}^{2}}{M_{Z}^{2}\left(M_{Z}^{2}-p^{2}\right)}\left(-g_{\mu\alpha}p_{\nu}p_{\beta}-g_{\nu\beta}p_{\mu}p_{\alpha}+g_{\mu\beta}p_{\nu}p_{\alpha}+g_{\nu\alpha}p_{\mu}p_{\beta}\right)+\cdots\,,$
$\displaystyle=$
$\displaystyle\widetilde{\Pi}_{Y}(p^{2})\left(p^{2}g_{\mu\alpha}g_{\nu\beta}-p^{2}g_{\mu\beta}g_{\nu\alpha}-g_{\mu\alpha}p_{\nu}p_{\beta}-g_{\nu\beta}p_{\mu}p_{\alpha}+g_{\mu\beta}p_{\nu}p_{\alpha}+g_{\nu\alpha}p_{\mu}p_{\beta}\right)$
$\displaystyle+\widetilde{\Pi}_{Z}(p^{2})\left(-g_{\mu\alpha}p_{\nu}p_{\beta}-g_{\nu\beta}p_{\mu}p_{\alpha}+g_{\mu\beta}p_{\nu}p_{\alpha}+g_{\nu\alpha}p_{\mu}p_{\beta}\right)\,,$

where we take the definitions for the pole residues $\lambda_{Y}$ and
$\lambda_{Z}$,

$\displaystyle\langle 0|J_{\mu}(0)|Y(p)\rangle$ $\displaystyle=$
$\displaystyle\lambda_{Y}\,\varepsilon_{\mu}\,,$ $\displaystyle\langle
0|J_{\mu\nu}(0)|Y(p)\rangle$ $\displaystyle=$
$\displaystyle\frac{\lambda_{Y}}{M_{Y}}\,\varepsilon_{\mu\nu\alpha\beta}\,\varepsilon^{\alpha}p^{\beta}\,,$
$\displaystyle\langle 0|J_{\mu\nu}(0)|Z(p)\rangle$ $\displaystyle=$
$\displaystyle\frac{\lambda_{Z}}{M_{Z}}\left(\varepsilon_{\mu}p_{\nu}-\varepsilon_{\nu}p_{\mu}\right)\,,$
(11)

the $\varepsilon_{\mu}$ are the polarization vectors of the tetraquark states
$Y$ and $Z$ with the quantum numbers $J^{PC}=1^{--}$ and $1^{+-}$,
respectively. Now we project out the components $\Pi_{Y}(p^{2})$ and
$\Pi_{Z}(p^{2})$ explicitly with the projectors $P_{Y}^{\mu\nu\alpha\beta}$
and $P_{Z}^{\mu\nu\alpha\beta}$,

$\displaystyle\Pi_{Y}(p^{2})$ $\displaystyle=$ $\displaystyle
p^{2}\widetilde{\Pi}_{Y}(p^{2})=P_{Y}^{\mu\nu\alpha\beta}\Pi_{\mu\nu\alpha\beta}(p)\,,$
$\displaystyle\Pi_{Z}(p^{2})$ $\displaystyle=$ $\displaystyle
p^{2}\widetilde{\Pi}_{Z}(p^{2})=P_{Z}^{\mu\nu\alpha\beta}\Pi_{\mu\nu\alpha\beta}(p)\,,$
(12)

where

$\displaystyle P_{Y}^{\mu\nu\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left(g^{\mu\alpha}-\frac{p^{\mu}p^{\alpha}}{p^{2}}\right)\left(g^{\nu\beta}-\frac{p^{\nu}p^{\beta}}{p^{2}}\right)\,,$
$\displaystyle P_{Z}^{\mu\nu\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left(g^{\mu\alpha}-\frac{p^{\mu}p^{\alpha}}{p^{2}}\right)\left(g^{\nu\beta}-\frac{p^{\nu}p^{\beta}}{p^{2}}\right)-\frac{1}{6}g^{\mu\alpha}g^{\nu\beta}\,,$
(13)

we take the components $\Pi_{Y}(p^{2})$ as we explore the hidden-charm
tetraquark states with the $J^{PC}=1^{--}$.

At the QCD side, we accomplish the operator product expansion up to the vacuum
condensates of dimension-10, and consider the vacuum condensates which are
vacuum expectations of the operators of the orders
$\mathcal{O}(\alpha_{s}^{k})$ with $k\leq 1$ consistently, i.e. we consider
the $\langle\bar{q}q\rangle$, $\langle\frac{\alpha_{s}GG}{\pi}\rangle$,
$\langle\bar{q}g_{s}\sigma Gq\rangle$, $\langle\bar{q}q\rangle^{2}$,
$\langle\bar{q}q\rangle\langle\frac{\alpha_{s}GG}{\pi}\rangle$,
$\langle\bar{q}q\rangle\langle\bar{q}g_{s}\sigma Gq\rangle$,
$\langle\bar{q}g_{s}\sigma Gq\rangle^{2}$ and
$\langle\bar{q}q\rangle^{2}\langle\frac{\alpha_{s}GG}{\pi}\rangle$. The
interested readers can consult Refs.[30, 31] for more details.

Now we adopt quark-hadron duality below the continuum thresholds $s_{0}$ and
$s_{0}^{\prime}$, respectively, and accomplish Borel transform with respect to
$P^{2}=-p^{2}$ to acquire two QCD sum rules:

$\displaystyle\lambda^{2}_{Y}\,\exp\left(-\frac{M^{2}_{Y}}{T^{2}}\right)$
$\displaystyle=$
$\displaystyle\int_{4m_{c}^{2}}^{s_{0}}ds\,\rho_{QCD}(s)\exp\left(-\frac{s}{T^{2}}\right)\,,$
(14)
$\displaystyle\lambda^{2}_{Y}\,\exp\left(-\frac{M^{2}_{Y}}{T^{2}}\right)+\lambda^{2}_{Y^{\prime}}\,\exp\left(-\frac{M^{2}_{Y^{\prime}}}{T^{2}}\right)$
$\displaystyle=$
$\displaystyle\int_{4m_{c}^{2}}^{s^{\prime}_{0}}ds\,\rho_{QCD}(s)\exp\left(-\frac{s}{T^{2}}\right)\,,$
(15)

where the $\rho_{QCD}(s)$ are the QCD spectral densities obtained through
dispersion relation, the $s_{0}$ and $s_{0}^{\prime}$ correspond to the ground
states $Y$ and first radial excitations $Y^{\prime}$, respectively.

We adopt the notations $\tau=\frac{1}{T^{2}}$,
$D^{n}=\left(-\frac{d}{d\tau}\right)^{n}$, and use the subscripts $1$ and $2$
to denote the $Y$ and $Y^{\prime}$ respectively to simplify the expressions.
Now we rewrite the two QCD sum rules in Eqs.(14)-(15) as

$\displaystyle\lambda_{1}^{2}\exp\left(-\tau M_{1}^{2}\right)$
$\displaystyle=$ $\displaystyle\Pi_{QCD}(\tau)\,,$ (16)
$\displaystyle\lambda_{1}^{2}\exp\left(-\tau
M_{1}^{2}\right)+\lambda_{2}^{2}\exp\left(-\tau M_{2}^{2}\right)$
$\displaystyle=$ $\displaystyle\Pi^{\prime}_{QCD}(\tau)\,,$ (17)

where the $\Pi_{QCD}(\tau)$ and $\Pi^{\prime}_{QCD}(\tau)$ represent the
correlation functions below the continuum thresholds $s_{0}$ and
$s_{0}^{\prime}$, respectively. We derive the QCD sum rules in Eq.(16) with
respect to $\tau$ to get the ground states masses,

$\displaystyle M_{1}^{2}$ $\displaystyle=$
$\displaystyle\frac{D\Pi_{QCD}(\tau)}{\Pi_{QCD}(\tau)}\,,$ (18)

then it is straightforward to get the ground state masses and pole residues
with the two coupled QCD sum rules, see Eq.(16) and Eq.(18) [30, 31].

Then we derive the QCD sum rules in Eq.(17) with respect to $\tau$ to get

$\displaystyle\lambda_{1}^{2}M_{1}^{2}\exp\left(-\tau
M_{1}^{2}\right)+\lambda_{2}^{2}M_{2}^{2}\exp\left(-\tau M_{2}^{2}\right)$
$\displaystyle=$ $\displaystyle D\Pi^{\prime}_{QCD}(\tau)\,.$ (19)

From Eq.(17) and Eq.(19), we get the QCD sum rules,

$\displaystyle\lambda_{i}^{2}\exp\left(-\tau M_{i}^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{\left(D-M_{j}^{2}\right)\Pi^{\prime}_{QCD}(\tau)}{M_{i}^{2}-M_{j}^{2}}\,,$
(20)

where $i\neq j$. Then we derive the QCD sum rules in Eq.(20) with respect to
$\tau$ to acquire

$\displaystyle M_{i}^{2}$ $\displaystyle=$
$\displaystyle\frac{\left(D^{2}-M_{j}^{2}D\right)\Pi_{QCD}^{\prime}(\tau)}{\left(D-M_{j}^{2}\right)\Pi_{QCD}^{\prime}(\tau)}\,,$
$\displaystyle M_{i}^{4}$ $\displaystyle=$
$\displaystyle\frac{\left(D^{3}-M_{j}^{2}D^{2}\right)\Pi_{QCD}^{\prime}(\tau)}{\left(D-M_{j}^{2}\right)\Pi_{QCD}^{\prime}(\tau)}\,.$
(21)

The squared masses $M_{i}^{2}$ obey the equation,

$\displaystyle M_{i}^{4}-bM_{i}^{2}+c$ $\displaystyle=$ $\displaystyle 0\,,$
(22)

where

$\displaystyle b$ $\displaystyle=$ $\displaystyle\frac{D^{3}\otimes
D^{0}-D^{2}\otimes D}{D^{2}\otimes D^{0}-D\otimes D}\,,$ $\displaystyle c$
$\displaystyle=$ $\displaystyle\frac{D^{3}\otimes D-D^{2}\otimes
D^{2}}{D^{2}\otimes D^{0}-D\otimes D}\,,$ $\displaystyle D^{j}\otimes D^{k}$
$\displaystyle=$ $\displaystyle
D^{j}\Pi^{\prime}_{QCD}(\tau)\,D^{k}\Pi^{\prime}_{QCD}(\tau)\,,$ (23)

the subscripts $i=1,2$ and the superscripts $j,k=0,1,2,3$. At last, we solve
the simple equation and obtain two solutions, i.e. the masses of the ground
states and first radial excitations [36, 49, 50],

$\displaystyle M_{1}^{2}$ $\displaystyle=$
$\displaystyle\frac{b-\sqrt{b^{2}-4c}}{2}\,,$ (24) $\displaystyle M_{2}^{2}$
$\displaystyle=$ $\displaystyle\frac{b+\sqrt{b^{2}-4c}}{2}\,.$ (25)

We can acquire the ground state masses either from the QCD sum rules in
Eq.(18) or in Eq.(24), but we prefer the QCD sum rules in Eq.(18), because
there are larger ground state contributions and less uncertainties from the
continuum threshold parameters. We acquire the masses and pole residues of the
first radial excitations from the two coupled QCD sum rules in Eq.(20) and in
Eq.(25).

##  3 Numerical results and discussions

We adopt the traditional vacuum condensates $\langle\bar{q}q\rangle=-(0.24\pm
0.01\,\rm{GeV})^{3}$, $\langle\bar{q}g_{s}\sigma
Gq\rangle=m_{0}^{2}\langle\bar{q}q\rangle$, $m_{0}^{2}=(0.8\pm
0.1)\,\rm{GeV}^{2}$, $\langle\frac{\alpha_{s}GG}{\pi}\rangle=0.012\pm
0.003\,\rm{GeV}^{4}$ at the energy scale $\mu=1\,\rm{GeV}$ [46, 47, 48, 51],
and choose the modified minimum subtracted mass $m_{c}(m_{c})=(1.275\pm
0.025)\,\rm{GeV}$ from the Particle Data Group [52], and set $m_{u}=m_{d}=0$.
Moreover, we consider the energy scale dependence of the input parameters,

$\displaystyle\langle\bar{q}q\rangle(\mu)$ $\displaystyle=$
$\displaystyle\langle\bar{q}q\rangle({\rm 1GeV})\left[\frac{\alpha_{s}({\rm
1GeV})}{\alpha_{s}(\mu)}\right]^{\frac{12}{33-2n_{f}}}\,,$
$\displaystyle\langle\bar{q}g_{s}\sigma Gq\rangle(\mu)$ $\displaystyle=$
$\displaystyle\langle\bar{q}g_{s}\sigma Gq\rangle({\rm
1GeV})\left[\frac{\alpha_{s}({\rm
1GeV})}{\alpha_{s}(\mu)}\right]^{\frac{2}{33-2n_{f}}}\,,$ $\displaystyle
m_{c}(\mu)$ $\displaystyle=$ $\displaystyle
m_{c}(m_{c})\left[\frac{\alpha_{s}(\mu)}{\alpha_{s}(m_{c})}\right]^{\frac{12}{33-2n_{f}}}\,,$
$\displaystyle\alpha_{s}(\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{b_{0}t}\left[1-\frac{b_{1}}{b_{0}^{2}}\frac{\log
t}{t}+\frac{b_{1}^{2}(\log^{2}{t}-\log{t}-1)+b_{0}b_{2}}{b_{0}^{4}t^{2}}\right]\,,$
(26)

where $t=\log\frac{\mu^{2}}{\Lambda_{QCD}^{2}}$,
$b_{0}=\frac{33-2n_{f}}{12\pi}$, $b_{1}=\frac{153-19n_{f}}{24\pi^{2}}$,
$b_{2}=\frac{2857-\frac{5033}{9}n_{f}+\frac{325}{27}n_{f}^{2}}{128\pi^{3}}$,
$\Lambda_{QCD}=210\,\rm{MeV}$, $292\,\rm{MeV}$ and $332\,\rm{MeV}$ for the
flavors $n_{f}=5$, $4$ and $3$, respectively [52, 53]. We choose the flavor
$n_{f}=4$ because we explore the tetraquark states consist of the valence
quarks $u$, $d$ and $c$. We evolve all the input parameters to the suitable
energy scales $\mu$ to extract the masses of the hidden-charm tetraquark
states, which satisfy the modified energy scale formula
$\mu=\sqrt{M^{2}_{X/Y/Z}-(2{\mathbb{M}}_{c}+0.5\,\rm{GeV})^{2}}=\sqrt{M^{2}_{X/Y/Z}-(4.1\,\rm{GeV})^{2}}$,
where the $\mathbb{M}$ is the effective charm quark mass [30, 31].

In the scenario of tetraquark states, we can tentatively assign the $X(3915)$
and $X(4500)$ to be the 1S and 2S states with the $J^{PC}=0^{++}$ [54, 55],
assign the $Z_{c}(3900)$ and $Z_{c}(4430)$ to be the 1S and 2S states with the
$J^{PC}=1^{+-}$, respectively [32, 34, 36], assign the $Z_{c}(4020)$ and
$Z_{c}(4600)$ to be the 1S and 2S states with the $J^{PC}=1^{+-}$,
respectively [50, 56], and assign the $X(4140)$ and $X(4685)$ to be the 1S and
2S states with the $J^{PC}=1^{++}$, respectively [57, 58], where the energy
gaps between the 1S and 2S states are about $0.57\sim 0.59\,\rm{GeV}$.

In Refs.[30, 31], we choose the continuum threshold parameters as
$\sqrt{s_{0}}=M_{Y}+0.55\sim 0.60\pm 0.10\,\rm{GeV}$ for the hidden-charm
tetraquark states with the $J^{PC}=1^{--}$, the ground state contribution can
be as large as $(49-81)\%$. Compared with the usually chosen pole
contributions $(40-60)\%$ [38, 39, 40, 41, 42, 43, 44, 45], the ground state
contributions $(49-81)\%$ in the old analysis are too large in the QCD sum
rules for the multiquark states, which maybe suffer from contaminations from
the first radial excitations, so in the present calculations, we choose
slightly smaller continuum threshold parameters $\sqrt{s_{0}}=M_{Y}+0.50\sim
0.55\pm 0.10\,\rm{GeV}$ to reduce the ground state contributions, and perform
a consistent and detailed analysis. Furthermore, the continuum threshold
parameters $s_{0}^{\prime}$ are set to be
$\sqrt{s^{\prime}_{0}}=M_{Y^{\prime}}+0.40\pm 0.10\,\rm{GeV}$ according to the
mass-gap of the $\psi^{\prime}$ and $\psi^{\prime\prime}$ from the Particle
Data Group [52].

We search for the suitable Borel parameters and continuum threshold parameters
via trial and error. The pole contributions (PC) and the vacuum condensate
contributions ($D(n)$) are defined by

$\displaystyle{\rm{PC}}$ $\displaystyle=$
$\displaystyle\frac{\int_{4m_{c}^{2}}^{s_{0}/s_{0}^{\prime}}ds\rho_{QCD}\left(s\right)\exp\left(-\frac{s}{T^{2}}\right)}{\int_{4m_{c}^{2}}^{\infty}ds\rho_{QCD}\left(s\right)\exp\left(-\frac{s}{T^{2}}\right)}\,,$
(27)

and

$\displaystyle D(n)$ $\displaystyle=$
$\displaystyle\frac{\int_{4m_{c}^{2}}^{s_{0}/s_{0}^{\prime}}ds\rho_{QCD,n}(s)\exp\left(-\frac{s}{T^{2}}\right)}{\int_{4m_{c}^{2}}^{s_{0}/s_{0}^{\prime}}ds\rho_{QCD}\left(s\right)\exp\left(-\frac{s}{T^{2}}\right)}\,,$
(28)

respectively.

At last, we obtain the Borel windows, continuum threshold parameters, suitable
energy scales and pole contributions, which are shown explicitly in Tables
4-5. From the tables, we can see explicitly that the pole contributions of the
ground states (the ground states plus first radial excited states) are about
$(40-60)\%$ ($(67-85)\%$), just like in our previous works on other tetraquark
states [38, 39, 40, 41, 42, 43, 44, 45], the pole dominance criterion is
satisfied very good. On the other hand, the contributions from the highest
dimensional condensates play a minor important role, $|D(10)|<3\%$ or $\ll
1\%$ ($<1\%$ or $\ll 1\%$) for the ground states (the ground states plus first
radial excited states), the operator product expansion converges very good and
better than that in our previous work [31]. So we have confidence to extract
reliable tetraquark masses and pole residues.

From Tables 4-7, we can see explicitly that the modified energy scale formula
can be well satisfied, and the relations $\sqrt{s_{0}}=M_{Y}+0.50\sim 0.55\pm
0.10\,\rm{GeV}$ and $\sqrt{s^{\prime}_{0}}=M_{Y^{\prime}}+0.40\pm
0.10\,\rm{GeV}$ are hold, our analysis is consistent.

In Fig.1, we plot the masses of the ground states and first radial excitations
of the hidden-charm tetraquark states with the quantum numbers
$J^{PC}=1^{--}$. From the figure, we can see explicitly that there appear flat
platforms in the Borel windows, the uncertainties come from the Borel
parameters are rather small.

$|S_{qc},S_{\bar{q}\bar{c}};S,L;J\rangle$ | $\mu(\rm{GeV})$ | $T^{2}(\rm{GeV}^{2})$ | $\sqrt{s_{0}}(\rm{GeV})$ | pole | $D(10)$
---|---|---|---|---|---
$|0,0;0,1;1\rangle$ | $1.1$ | $2.6-3.0$ | $4.75\pm 0.10$ | $(40-65)\%$ | $<1\%$
$|1,1;0,1;1\rangle$ | $1.2$ | $2.5-2.9$ | $4.80\pm 0.10$ | $(39-64)\%$ | $<3\%$
$\frac{1}{\sqrt{2}}\left(|1,0;1,1;1\rangle+|0,1;1,1;1\rangle\right)$ | $1.3$ | $3.0-3.4$ | $4.85\pm 0.10$ | $(38-60)\%$ | $\ll 1\%$
$|1,1;2,1;1\rangle$ | $1.3$ | $2.7-3.1$ | $4.85\pm 0.10$ | $(39-63)\%$ | $<1\%$
Table 4:  The Borel windows $T^{2}$, continuum threshold parameters $s_{0}$, energy scales of the QCD spectral densities, contributions of the ground states, and values of the $D(10)$.  $|S_{qc},S_{\bar{q}\bar{c}};S,L;J\rangle$ | $\mu(\rm{GeV})$ | $T^{2}(\rm{GeV}^{2})$ | $\sqrt{s^{\prime}_{0}}(\rm{GeV})$ | pole | $D(10)$
---|---|---|---|---|---
$|0,0;0,1;1\rangle$ | $2.4$ | $2.8-3.2$ | $5.15\pm 0.10$ | $(67-85)\%$ | $\ll 1\%$
$|1,1;0,1;1\rangle$ | $2.5$ | $2.6-3.0$ | $5.20\pm 0.10$ | $(67-86)\%$ | $<1\%$
$\frac{1}{\sqrt{2}}\left(|1,0;1,1;1\rangle+|0,1;1,1;1\rangle\right)$ | $2.6$ | $3.0-3.4$ | $5.25\pm 0.10$ | $(67-84)\%$ | $\ll 1\%$
$|1,1;2,1;1\rangle$ | $2.6$ | $2.7-3.1$ | $5.25\pm 0.10$ | $(68-87)\%$ | $\ll 1\%$
Table 5:  The Borel windows $T^{2}$, continuum threshold parameters $s_{0}^{\prime}$, energy scales of the QCD spectral densities, contributions of the ground states plus first radial excitations, and values of the $D(10)$.  $|S_{qc},S_{\bar{q}\bar{c}};S,L;J\rangle$ | $M_{Y}(\rm{GeV})$ | $\lambda_{Y}(10^{-2}\rm{GeV}^{6})$
---|---|---
$|0,0;0,1;1\rangle$ | $4.24\pm 0.09$ | $2.28\pm 0.42$
$|1,1;0,1;1\rangle$ | $4.28\pm 0.09$ | $4.80\pm 0.95$
$\frac{1}{\sqrt{2}}\left(|1,0;1,1;1\rangle+|0,1;1,1;1\rangle\right)$ | $4.31\pm 0.09$ | $2.94\pm 0.50$
$|1,1;2,1;1\rangle$ | $4.33\pm 0.09$ | $6.55\pm 1.19$
Table 6:  The masses and pole residues of the ground states.  $|S_{qc},S_{\bar{q}\bar{c}};S,L;J\rangle$ | $M_{Y}(\rm{GeV})$ | $\lambda_{Y}(10^{-2}\rm{GeV}^{6})$
---|---|---
$|0,0;0,1;1\rangle$ | $4.75\pm 0.10$ | $8.19\pm 1.23$
$|1,1;0,1;1\rangle$ | $4.81\pm 0.10$ | $18.3\pm 3.0$
$\frac{1}{\sqrt{2}}\left(|1,0;1,1;1\rangle+|0,1;1,1;1\rangle\right)$ | $4.85\pm 0.09$ | $8.63\pm 1.22$
$|1,1;2,1;1\rangle$ | $4.86\pm 0.10$ | $21.7\pm 3.4$
Table 7:  The masses and pole residues of the first radial excited states.  $|S_{qc},S_{\bar{q}\bar{c}};S,L;J\rangle$ | $M_{Y}(\rm{GeV})$ | Assignments
---|---|---
$|0,0;0,1;1\rangle$ (1P) | $4.24\pm 0.09$ | $Y(4220/4260)$
$|0,0;0,1;1\rangle$ (2P) | $4.75\pm 0.10$ | $Y(4750)$
$|1,1;0,1;1\rangle$ (1P) | $4.28\pm 0.09$ | $Y(4220/4320)$
$|1,1;0,1;1\rangle$ (2P) | $4.81\pm 0.10$ |
$\frac{1}{\sqrt{2}}\left(|1,0;1,1;1\rangle+|0,1;1,1;1\rangle\right)$ (1P) | $4.31\pm 0.09$ | $Y(4320/4390)$
$\frac{1}{\sqrt{2}}\left(|1,0;1,1;1\rangle+|0,1;1,1;1\rangle\right)$ (2P) | $4.85\pm 0.09$ |
$|1,1;2,1;1\rangle$ (1P) | $4.33\pm 0.09$ | $Y(4320/4390)$
$|1,1;2,1;1\rangle$ (2P) | $4.86\pm 0.10$ |
Table 8:  The masses of the vector tetraquark states and possible assignments,
where the 1P and 2P denote the ground states and first radial excitations,
respectively.

Figure 1:  The masses of the vector tetraquark states with variations of the
Borel parameters $T^{2}$, where the $A$, $B$, $C$ and $D$ stand for the
$|0,0;0,1;1\rangle$, $|1,1;0,1;1\rangle$,
$\frac{1}{\sqrt{2}}\left(|1,0;1,1;1\rangle+|0,1;1,1;1\rangle\right)$ and
$|1,1;2,1;1\rangle$ states, respectively; the lower lines and upper lines
stand for the ground states and first radial excitations, respectively.

In Table 8, we present the possible assignments of the vector tetraquark
states based on the QCD sum rules. From the table, we can see explicitly that
there is a room to accommodate the $Y(4750)$, i.e. the $Y(4220/4260)$ and
$Y(4750)$ can be assigned as the ground state and first radial excited state
of the
$C\gamma_{5}\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\mu}\gamma_{5}C$
type tetraquark states with the $J^{PC}=1^{--}$, respectively. We cannot
identify a particle unambiguously with the mass alone, we have to study the
decays of those vector tetraquark candidates with the QCD sum rules to testify
the assignments, it is our next work.

##  4 Conclusion

In the present work, we choose the diquark-antidiquark type four-quark
currents with an explicit P-wave between the diquark and antidiquark pairs to
investigate the ground states and first radial excitations of the hidden-charm
tetraquark states with the quantum numbers $J^{PC}=1^{--}$ via the QCD sum
rules. Firstly, we take account of the ground states at the hadronic side
only, and update the old analysis by refitting the continuum threshold
parameters and Borel parameters. Comparing with the old calculations, we
obtain better convergent behaviors in the operator product expansion at the
QCD side and uniform pole contributions $(40-60)\%$ at the hadronic side.
Secondly, we take account of both the ground states and first radial
excitations, and focus on the first radial excitations, and obtain new
predictions. In calculations (also old calculations), we use the modified
energy scale formula $\mu=\sqrt{M^{2}_{X/Y/Z}-(4.1\,\rm{GeV})^{2}}$ to select
the suitable energy scales of the QCD spectral densities so as to improve the
convergent behavior of the operator product expansion and enhance the pole
contributions. All in all, we explore and obtain the masses and pole residues
of those 1P and 2P vector tetraquark states in a systematic and consistent
way. We obtain the lowest vector tetraquark masses and make possible
assignments of the existing $Y$ states, and observe that there indeed exists a
hidden-charm tetraquark states with the $J^{PC}=1^{--}$ at the energy about
$4.75\,\rm{GeV}$, which can account for the BESIII data.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
with Grant No.12175068.

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