# Contributions of negative-energy states to the E2-M1 polarizability of the
Sr clock

Fang-Fei Wu, Ting-Yun Shi, and Li-Yan Tang† 00footnotetext: †Email Address:
<EMAIL_ADDRESS>State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Innovation
Academy for Precision Measurement Science and Technology, Chinese Academy of
Sciences, Wuhan 430071, People’s Republic of China

###### Abstract

With the improvement of high-precision optical clock, the higher-order
multipolar interaction between atoms and light needs quantitative evaluation.
However for the Sr clock, the differential dynamic E2-M1 polarizability at the
magic wavelength has contradictions among available theoretical and
experimental results. Recently, the new experimental measurement of S.
Dörscher et al. [arXiv: 2210. 14727] is consistent with measurement of
Ushijima et al., which poses new challenges to theory and urgently calls for
theoretical explanations. In present work, we investigate contributions of
negative-energy states to the E2 and M1 polarizabilities. We find that for the
M1 polarizability, the contribution from negative-energy states is crucial and
dominant. Our new theoretical result for E2-M1 polarizability difference is
$-7.74(3.92)\times 10^{-5}$ a.u., which is in good agreement with the recent
experiment of S. Dörscher et al., so the inconsistency problem of E2-M1
polarizability in the Sr clock between theory and experiment is eliminated.

###### pacs:

31.15.ac, 31.15.ap, 34.20.Cf

##  I Introduction

The high-precision optical clock has extensive and important applications,
such as redefine the unit of time Bregolin et al. (2017); Yamanaka et al.
(2015), test variations of the fundamental constants Godun et al. (2014);
Huntemann et al. (2014); Safronova et al. (2018), probe dark matter and dark
energy Arvanitaki et al. (2015); Roberts et al. (2017), and search for new
physics Pihan-Le Bars et al. (2017); Shaniv et al. (2018); Kolkowitz et al.
(2016). Strontium as a typical representative of optical lattice clocks, the
systematical uncertainty has entered into $10^{-18}$ level of precision.
Aiming to develop and realize a new generation of higher-precision optical
clocks with uncertainty and stability beyond $10^{-18}$, the contribution of
the multipolar and higher-order Stark shifts need to be quantitatively
evaluated Ovsiannikov et al. (2013); Katori et al. (2015); Porsev et al.
(2018); Ushijima et al. (2018).

For the Sr clock, however, there is an obvious contradiction for the
differential dynamic multipolar E2-M1 polarizability of
$\Delta\alpha^{QM}(\omega)$ at the magic wavelength of 813.4280(5) nm between
recent theoretical and experimental studies. In 2011, P. G. Westergaard et al.
in SYRTE have performed the measurement of E2-M1 polarizability, which gives a
value of $0.0(2.6)\times 10^{-5}$ a.u. Westergaard et al. (2011). However, in
2013, V. D. Ovsiannikov et al. obtain the theoretical result of $-3.6\times
10^{-5}$ a.u. using the single-electron approximated method Ovsiannikov et al.
(2013), and in 2015, H. Katori et al. report their theoretical value of
$0.74\times 10^{-5}$ a.u. Katori et al. (2015), which has opposite sign to the
theoretical value of the single-electron approximation Ovsiannikov et al.
(2013). The complete contradiction between these two theoretical results
promotes the new-round of measurements and calculations. In 2018, S. G. Porsev
et al. have carried out the calculations of many-body perturbation theory
combined configuration interaction method (MBPT+CI) and report their result of
$2.80(36)\times 10^{-5}$ a.u. Porsev et al. (2018), and also the experiment
group of H. Katori has measured the E2-M1 polarizability of $-8.01(33)\times
10^{-5}$ a.u. Ushijima et al. (2018). In 2019, we also have performed our
calculations by using the combined method of the Dirac-Fock plus core
polarization (DFCP) and relativistic configuration interaction (RCI)
approaches and reported our value of $2.68(94)\times 10^{-5}$ a.u. Wu et al.
(2019), which gives an independently test for the theoretical work of Porsev
et al., but the contradiction of E2-M1 polarizability between theory and
experiment, even the difference in sign, is still pending. Recently, S.
Dörscher et al. has performed a new-round of measurement, their new
experimental value is $-987^{+174}_{-223}$ $\rm{\mu Hz}$ [equals to
$-8.22^{+1.45}_{-1.86}\times 10^{-5}$ a. u.] Dörscher et al. (2022), which
confirms the correctness of measurement in H. Katori’s group. The consistency
between two different experimental groups poses a new challenge to the
theoretical calculations, and a new theoretical explanation is urgently needed
for solving the current contradiction of the E2-M1 polarizability in the Sr
clock.

In present paper, we recalculate the dynamic multipolar polarizabilities of
the Sr clock by using the combined DFCP+RCI method. Different from our
previous paper, the negative-energy states of single electron are included to
construct the configuration interactions. In addition, when calculating the
multipolar polarizabilities using the method of sum over states, the summation
involves all the negative- and positive-energy states of the Sr atom. Compared
with recent measurement, our present results reveal the importance of the
negative-energy states and remove the contradiction between theory and
experiment for the differential dynamic multipolar polarizabilities of the Sr
clock.

##  II Theoretical Method

For an atom exposed under a linear polarized laser field with the laser
frequency $\omega$, at the magic wavelength, the leading-order of Stark shift
that related to the dynamic electric dipole (E1) polarizability can be
eliminated, but the multipolar Stark shifts that related to the dynamic
electric quadrupole (E2) and the magnetic dipole (M1) polarizabilities can’t
be cancelled. Generally, the dynamic M1 and E2 polarizabilities for the
initial state $|0\rangle\equiv|n_{0},J_{0}=0\rangle$ (where $n_{0}$ represents
all other quantum numbers) are written as Porsev et al. (2004)

$\displaystyle\alpha^{M1}(\omega)$ $\displaystyle=$
$\displaystyle\frac{2}{3}\sum_{n}\frac{\Delta E_{n0}|\langle
0\|M1\|nJ_{n}\rangle|^{2}}{\Delta E_{n0}^{2}-\omega^{2}}\,,$ (1)
$\displaystyle\alpha^{E2}(\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{30}(\alpha\omega)^{2}\sum_{n}\frac{\Delta
E_{n0}|\langle 0\|Q\|nJ_{n}\rangle|^{2}}{\Delta E_{n0}^{2}-\omega^{2}}\,,$ (2)

where $\alpha$ is the fine structure constant, $M1$ and $Q$ are, respectively,
the magnetic-dipole and electric-quadrupole transition operators. $\Delta
E_{n0}$ represents the transition energy between the initial state $|0\rangle$
and the intermediate state $|nJ_{n}\rangle$.

The wave functions and energies for the Sr atom are obtained by solving the
followed eigen equation,

$\displaystyle[\sum_{i}^{2}h_{\rm
DFCP}(\bm{r_{i}})+V(r_{ij})]|\Psi({\pi}JM)\rangle=E|\Psi({\pi}JM)\rangle\,,$
(3)

where the two-particle interaction potential $V(r_{ij})$ is

$\displaystyle V(r_{ij})=\frac{1}{r_{ij}}+V_{2}(r_{ij})\,,$ (4)

the first term is the Coulomb interaction between two valence electrons, the
second term is two-body core-polarization interaction Mitroy et al. (2010);
Mitroy and Bromley (2003), the detailed expression of $V_{2}(r_{ij})$ can be
referred to our previous paper Wu et al. (2019).

The wave function $|\Psi({\pi}JM)\rangle$ with parity $\pi$, angular momentum
$J$, and magnetic quantum number $M$ of the system is expanded as a linear
combination of the configuration-state wave functions
$|\Phi_{I}(\sigma{\pi}JM)\rangle$,

$\displaystyle|\Psi({\pi}JM)\rangle=\sum_{I}{C_{I}|\Phi_{I}(\sigma{\pi}JM)\rangle}\,,$
(5)

where $C_{I}$ and $\sigma$ are the expansion coefficients and the additional
quantum number that define each configuration state uniquely, respectively.
The configuration-state wave functions $|\Phi_{I}(\sigma{\pi}JM)\rangle$ are
constructed by the single-electron wave functions $\phi(\bm{r})$, which is
obtained by solving the followed DFCP equation,

$h_{\rm DFCP}(\bm{r})\phi(\bm{r})=\varepsilon\phi(\bm{r})\,,$ (6)

and $h_{\rm DFCP}(\bm{r})$ represents the DFCP Hamiltonian,

$h_{\rm
DFCP}(\bm{r})=c{{\bm{\alpha}}}\cdot{\mathbf{p}}+(\beta-1)c^{2}+V_{N}(r)+V_{\rm
core}(r)\,,$ (7)

where $\bm{\alpha}$ and $\beta$ are the $4\times 4$ Dirac matrices,
$\mathbf{p}$ is the momentum operator for the valence electron, $V_{N}(r)$ is
the Coulomb potential between a valence electron and nucleus, $V_{\rm
core}(r)$ is the interaction potential between core electrons and a valence
electron, which is the sum of a DF potential and a semi-empirical one body
core-polarization potential Mitroy and Norcross (1988),

$V_{\rm core}(r)=V_{\rm DF}(r)+V_{\rm 1}(r)\,.$ (8)

The DF potential $V_{\rm DF}(r)$ is constructed from the orbital functions
$\psi(\bm{r})$ by solving the Dirac-Fock (DF) calculation of frozen Sr2+ core.
The one body core-polarization potential $V_{\rm 1}(r)$ is kept the same as
our previous paper Wu et al. (2019).

It’s worth noting that when solving the Eq.(6), a series of energies for
univalent-electron system are obtained, including both positive- and negative-
energy states. In our previous works Wu et al. (2019, 2020), we only use the
single-electron wave functions corresponding to the positive-energy states to
construct the configuration-state wave function
$|\Phi_{I}(\sigma{\pi}JM)\rangle$. Different from our previous works Wu et al.
(2019, 2020), in present work, all the single-electron wave functions
corresponding to both of the positive- and negative-energy states are used to
construct configuration-state wave functions. Therefore, the inclusion of
negative-energy states makes the configuration scale increased greatly. In
addition, when solving the Eq. (3), we can obtain a series of positive- and
negative-energy states of divalent-electron atom, then the summation of the M1
and E2 polarizabilities in Eqs. (1) and (2) involves all the energy states.

In present work, we have tested the convergence for the multipolar
polarizabilities as the number of B-spline basis sets $N$ and the partial-wave
$\ell$ increased, we find that our results remain unchanged with at less 4
significant digits as $N$ and $\ell$ changed. So in the following section, we
only listed the results under the maximum basis set. The maximum number of
B-spline basis is 40, the maximum number of partial-wave is 5, and the number
of configuration has reached 128781.

##  III Results and Discussions

Using new configuration-state wavefunctions, which are constructed from both
positive- and negative-energy states of the Sr+ ion, we have performed the
new-round of calculations of energies, reduced matrix elements, E1, E2, and M1
polarizabilities of the Sr clock. The detailed comparisons of energies,
reduced matrix elements, and the static dipole polarizability between our new
results and our previous paper are also made. We find that after the inclusion
of the negative-energy states, the correction for energy is less than 3 ppm.
And for the main electric dipole, electric quadrupole, and magnetic dipole
transition matrix elements, even for the static dipole polarizability of the
clock states, there is no numerical change. That means the negative-energy
states have little influence on these atomic properties, these minor
corrections can’t be reflected under the current theoretical accuracy.
Therefore, we think for these atomic properties, it is unnecessary to report
the same data tables as the Ref. Wu et al. (2019).

Next, we can focus our attentions on the dynamic E2 and M1 polarizabilities of
the $5s^{2}\,{}^{1}S_{0}$ and $5s5p\,^{3}P_{0}^{o}$ clock states at the
813.4280(5) nm magic wavelength. Tables 1 and 2 list the itemized
contributions to the E2 and M1 polarizabilities, respectively. It’s seen that
for the E2 polarizability, the maximum contribution to the ground-state
$5s^{2}\,{}^{1}S_{0}$ comes from the positive-energy state of
$5s4d\,^{1}D_{2}$, which contributes about 75% to the total E2 polarizability.
The second largest contribution comes from the positive-energy state of
$5s5d\,^{1}D_{2}$, which contributes about 13% to the total E2 polarizability.
For the excited-state $5s5p\,^{3}P_{0}^{o}$, the main contribution comes from
the $5d5p\,^{3}F_{2}^{o}$, $5s6p\,^{3}P_{2}^{o}$ and $5s4f\,^{3}F_{2}^{o}$
states, these three items together contribute about 60% to the total E2
polarizability. From Table 1, it’s also clearly seen that for both of the
$5s^{2}\,{}^{1}S_{0}$ and $5s5p\,^{3}P_{0}^{o}$ clock states of the Sr atom,
the contribution of negative-energy states is less than $10^{-14}$, it can be
almost ignored, which means that the contribution of positive-energy states to
the total E2 polarizability is almost 100%.

Table 1:  Itemized contributions (Contr.) to the dynamic electric-quadrupole (E2) polarizability (in a.u.) for the $5s^{2}\,{}^{1}S_{0}$ and $5s5p\,^{3}P_{0}^{o}$ clock states at the 813.4280(5) nm magic wavelength. Tail represents the contribution from other positive-energy states, $\alpha^{E2+}$ and $\alpha^{E2-}$ represent the total contribution from positive- and negative-energy states, respectively. The numbers in the square brackets denote powers of ten. $5s^{2}\,{}^{1}S_{0}$ | $5s5p\,^{3}P_{0}^{o}$
---|---
Sub item | Contr. | Sub item | Contr.
$5s4d\,^{3}D_{2}$ | 1.258[-7] | $5s5p\,^{3}P_{2}^{o}$ |  $-$2.805[-6]
$5s4d\,^{1}D_{2}$ | 6.965[-5] | $5d5p\,^{3}F_{2}^{o}$ | 3.095[-5]
$5s5d\,^{1}D_{2}$ | 1.224[-5] | $5d5p\,^{1}D_{2}^{o}$ | 3.149[-6]
$5s5d\,^{3}D_{2}$ | 1.106[-8] | $5s6p\,^{3}P_{2}^{o}$ | 1.741[-5]
$5p^{2}\,{}^{3}P_{2}$ | 5.966[-8] | $4d5p\,^{3}D_{2}^{o}$ | 3.603[-6]
$5d^{2}\,{}^{1}D_{2}$ | 3.887[-8] | $5d5p\,^{3}P_{2}^{o}$ | 2.139[-6]
$5s6d\,^{3}D_{2}$ | 4.981[-10] | $5s4f\,^{3}F_{2}^{o}$ | 2.644[-5]
$5s6d\,^{1}D_{2}$ | 1.226[-7] | $5s7p\,^{3}P_{2}^{o}$ | 2.601[-6]
$5s7d\,^{1}D_{2}$ | 2.600[-6] | $5s5f\,^{3}F_{2}^{o}$ | 8.768[-6]
Tail | 7.950[-6] | Tail | 3.214[-5]
$\alpha^{E2+}$ | 9.28[-5] | $\alpha^{E2+}$ | 12.44[-5]
$\alpha^{E2-}$ |  $-$8.64[-16] | $\alpha^{E2-}$ |  $-$1.10[-15]
Total | 9.28[-5] | Total | 12.44[-5]
Table 2:  Itemized contributions (Contr.) to the dynamic magnetic-dipole (M1) polarizability (in a.u.) for the $5s^{2}\,{}^{1}S_{0}$ and $5s5p\,^{3}P_{0}^{o}$ clock states at the 813.4280(5) nm magic wavelength. Tail represents the contribution from other positive-energy states, $\alpha^{M1+}$ and $\alpha^{M1-}$ represent the total contribution from positive- and negative-energy states, respectively. The numbers in the square brackets denote powers of ten. $5s^{2}\,{}^{1}S_{0}$ | $5s5p\,^{3}P_{0}^{o}$
---|---
Sub item | Contr. | Sub item | Contr.
$5s4d\,^{3}D_{1}$ | 1.483[-15] | $5s5p\,^{3}P_{1}^{o}$ |  $-$4.811[-6]
$5s6s\,^{3}S_{1}$ | 4.098[-13] | $5s5p\,^{1}P_{1}^{o}$ |  $-$2.702[-7]
$5s5d\,^{3}D_{1}$ | 1.273[-12] | $5s6p\,^{3}P_{1}^{o}$ | 7.336[-10]
$5p^{2}\,{}^{3}P_{1}$ | 1.539[-9] | $5s6p\,^{1}P_{1}^{o}$ | 1.766[-8]
Tail | 5.81[-10] | Tail | 1.35[-8]
$\alpha^{M1+}$ | 2.17[-9] | $\alpha^{M1+}$ |  $-$5.05[-6]
$\alpha^{M1-}$ |  $-$3.84[-4] | $\alpha^{M1-}$ |  $-$4.88[-4]
Total |  $-$3.84[-4] | Total |  $-$4.93[-4]

However for the dynamic M1 polarizability, the influence of negative-energy
states is obvious. It can be seen from Table 2. For the $5s^{2}\,{}^{1}S_{0}$
state, if the negative-energy states are not considered, the largest
contribution comes from the $5p^{2}\,{}^{3}P_{1}$ state and the M1
polarizability at the 813.4280(5) nm magic wavelength is $2.17\times 10^{-9}$.
After considering the negative-energy states, the total value of the M1
polarizability at the 813.4280(5) nm magic wavelength is changed as
$-3.84\times 10^{-4}$, which is entirely from the contribution of the
negative-energy states, that because the contribution of the negative-energy
states is five orders of magnitude larger than that of the positive-energy
states, and directly changes the sign of the final M1 polarizability.
Similarly, for the $5s5p\,^{3}P_{0}^{o}$ state, the contribution of the
negative-energy states is two orders of magnitude larger than that of the
positive-energy states, accounting for 99% of the final M1 polarizability.

Table 3:  Summarized results of dynamic E2 and M1 polarizabilities (in a.u.) for the $5s^{2}\,{}^{1}S_{0}$ and $5s5p\,^{3}P_{0}^{o}$ clock states at the 813.4280(5) nm magic wavelength. $\Delta\alpha^{E2}(\omega)$ and $\Delta\alpha^{M1}(\omega)$ represent the difference for the clock states of the dynamic E2 and M1 polarizabilities, respectively. And $\Delta\alpha^{QM}(\omega)=\Delta\alpha^{M1}(\omega)+\Delta\alpha^{E2}(\omega)$. The numbers in parentheses are computational uncertainties. The numbers in the square brackets denote powers of ten. Polarizability | Present | Ref. Wu et al. (2019) | Ref. Porsev et al. (2018)
---|---|---|---
$\alpha^{E2}_{\,{}^{1}S_{0}}(\omega)$ | 9.28(57)[-5] | 9.26(56)[-5] | 8.87(26)[-5]
$\alpha^{E2}_{\,{}^{3}P_{0}^{o}}(\omega)$ | 12.44(76)[-5] | 12.44(76)[-5] | 12.2(25)[-5]
$\Delta\alpha^{E2}(\omega)$ | 3.16(95)[-5] | 3.18(94)[-5] | 3.31(36)[-5]
$\alpha^{M1}_{\,{}^{1}S_{0}}(\omega)$ |  $-$3.84(24)[-4] | 2.12(13)[-9] | 2.37[-9]
$\alpha^{M1}_{\,{}^{3}P_{0}^{o}}(\omega)$ |  $-$4.93(30)[-4] |  $-$5.05(31)[-6] |  $-$5.08[-6]
$\Delta\alpha^{M1}(\omega)$ |  $-$1.09(38)[-4] |  $-$5.05(31)[-6] |  $-$5.08[-6]
$\Delta\alpha^{QM}(\omega)$ |  $-$7.74(3.92)[-5] | 2.68(94)[-5] | 2.80(36)[-5]
Figure 1: (Color online) Comparison of the $\Delta\alpha^{QM}(\omega)$ (in
a.u.). The green line represents measurement results. The blue line represents
our present value, and the magenta line denotes other theoretical results.

Further, in order to conservatively evaluate the uncertainty of our
calculations, similar to our previous paper Wu et al. (2019), we can introduce
$\pm$3% fluctuation into all the reduced matrix elements to give the
uncertainties of present E2 and M1 polarizabilities. The final values are
summarized in Table 3, also a detailed comparison among present work, our
previous calculations Wu et al. (2019), and the CI+all order of perturbation
calculations of Porsev et al. Porsev et al. (2018) is given in this table. It
can be clearly seen that present dynamic E2 polarizabilities that with the
contribution of negative-energy states included for the $5s^{2}\,{}^{1}S_{0}$
and $5s5p\,^{3}P_{0}^{o}$ states at the 813.4280(5) nm magic wavelength are in
good agreement with the Refs. Porsev et al. (2018); Wu et al. (2019).

The obvious difference between present results and other theoretical values in
Table 3 is from the M1 polarizability. Present values of $-3.84(24)\times
10^{-4}$ and $-4.93(30)\times 10^{-4}$ a.u. show that the M1 polarizabilities
for the $5s^{2}\,{}^{1}S_{0}$ and $5s5p\,^{3}P_{0}^{o}$ clock states at the
813.4280(5) nm magic wavelength are equivalent in magnitude and the sign are
the same, which is totally different from the values of Refs. Porsev et al.
(2018); Wu et al. (2019), where the $\alpha^{M1}_{{}^{3}P_{0}^{o}}(\omega)$
polarizability is three orders of magnitude larger than
$\alpha^{M1}_{{}^{1}S_{0}}(\omega)$ polarizability. The reason for this large
difference is that in present work we have taken account of the contributions
from negative-energy states. The final value for the differential $M1$
polarizability $\Delta\alpha^{M1}(\omega)$ is $-1.09(38)\times 10^{-4}$.

When adding the two values of $\Delta\alpha^{E2}(\omega)$ and
$\Delta\alpha^{M1}(\omega)$, we can get the final value of the differential
dynamic multipolar polarizability $\Delta\alpha^{QM}(\omega)$, it is
$-7.74(3.92)\times 10^{-5}$ a.u. The large uncertainty in present value of
$-7.74(3.92)\times 10^{-5}$ a.u. mainly comes from the error bar of
$\Delta\alpha^{M1}(\omega)$, since the differential M1 polarizability
$\Delta\alpha^{M1}(\omega)$ is an order of magnitude larger than the
differential E2 polarizability $\Delta\alpha^{E2}(\omega)$. To further reduce
the theoretical uncertainty in the future, it is necessary to develop high-
accuracy theoretical methods for calculations of multi-electron atomic
structure.

Fig. 1 plots the detailed comparison for the differential E2-M1 polarizability
between theory and experiment. It is seen that for
$\Delta\alpha^{QM}(\omega)$, the recent measurement in PTB Dörscher et al.
(2022) is coincident with the earlier measurement of Ref. Ushijima et al.
(2018), but disagrees with the theoretical values Katori et al. (2015);
Ovsiannikov et al. (2013); Porsev et al. (2018); Wu et al. (2019), even the
sign of $\Delta\alpha^{QM}(\omega)$ between theory and experiment is opposite,
which indicates that there might be problems with theoretical results. Our
present value of $-7.74(3.92)\times 10^{-5}$ a.u., which includes the
contribution of negative-energy states, agrees well with the experimental
results of $-8.01(33)\times 10^{-5}$ Ushijima et al. (2018) and
$-8.22^{+1.45}_{-1.86}\times 10^{-5}$ a.u. Dörscher et al. (2022). This
illustrates that the negative-energy states are crucial to the calculation of
multipolar polarizabilities, especially for the magnetic polarizability.

##  IV Summary

The dynamic magnetic-dipole and electric-quadrupole polarizabilities at the
magic wavelength for the $5s^{2}\,{}^{1}S_{0}$ and $5s5p\,^{3}P_{0}^{o}$
states of the Sr clock have been calculated by using the combined DFCP+RCI
method, where the negative-energy states have been taken into account to
construct the configurations. For the final differential multipolar
polarizability of $\Delta\alpha^{QM}(\omega)$, our result of
$-$7.74(3.92)$\times$10-5 a.u. is in good agreement with the recent
measurement result of Ref. Dörscher et al. (2022), also the difference between
present result of $-$7.74(3.92)$\times$10-5 a.u. and the CI+all order value of
2.80(36)$\times$10-5 a.u. Porsev et al. (2018) verifies the importance of the
negative-energy states. Our calculations have solved the discrepancy of
$\Delta\alpha^{QM}(\omega)$ between theory and experiment in the Sr clock. In
addition, the contribution of the negative-energy states to the magnetic
multipolar polarizabilities can be extended to other optical clocks for study
of multipolar optical frequency shift. Especially, for the accurate
calculations of g-factor and electric quadrupole moment, the contribution of
negative-energy states also need to be taken into account.

###### Acknowledgements.

We thank Yong-Hui Zhang for helpful discussions on the negative-energy states.
This work was supported by the National Natural Science Foundation of China
under Grant Nos. 12174402 and 12004124, and by the Nature Science Foundation
of Hubei Province Nos.2019CFA058 and 2022CFA013.

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Analysis of the $Z_{cs}(3985)$ as the axialvector tetraquark state

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

###### Abstract

In this paper, we choose the scalar and axialvector diquark operators in the
color antitriplet as the fundamental building blocks to construct the four-
quark currents and investigate the diquark-antidiquark type axialvector
tetraquark states $c\bar{c}u\bar{s}$ in the framework of the QCD sum rules.
The predicted tetraquark mass $M_{Z}=3.99\pm 0.09\,\rm{GeV}$ is in excellent
agreement with the experimental value $3985.2^{+2.1}_{-2.0}\pm 1.7\,\rm{MeV}$
from the BESIII collaboration, which supports identifying the $Z_{cs}(3985)$
as the cousin of the $Z_{c}(3900)$ with the quantum numbers $J^{PC}=1^{+-}$.
We take into account the light flavor $SU(3)$ mass-breaking effect to estimate
the mass spectrum of the diquark-antidiquark type hidden-charm tetraquark
states having the strangeness according to the previous works.

PACS number: 12.39.Mk, 12.38.Lg

Key words: Tetraquark state, QCD sum rules

##  1 Introduction

Very recently, the BESIII collaboration observed an excess over the known
contributions of the conventional charmed mesons near the $D_{s}^{-}D^{*0}$
and $D^{*-}_{s}D^{0}$ mass thresholds in the $K^{+}$ recoil-mass spectrum with
the significance of 5.3 $\sigma$ in the processes of the $e^{+}e^{-}\to
K^{+}(D_{s}^{-}D^{*0}+D^{*-}_{s}D^{0})$ [1]. It is the first candidate of the
charged hidden-charm tetraquark state with strangeness. The Breit-Wigner mass
and width of the new structure, $Z_{cs}(3985)$, were determined to be
$3985.2^{+2.1}_{-2.0}\pm 1.7\,\rm{MeV}$ and $13.8^{+8.1}_{-5.2}\pm
4.9\,\rm{MeV}$, respectively. According to the production mode, it is nature
to draw the conclusion that the $Z_{cs}(3985)$ should be a cousin of the well-
known $Z_{c}(3885/3900)$ with strangeness, the $Z_{c}(3885)$ was observed in
the precess $e^{+}e^{-}\to(D\bar{D}^{*})^{-}\pi^{+}$, which means that the
$Z_{c}(3885)$ and $Z_{cs}(3985)$ are governed by a similar production
mechanism and have a similar quark structure $c\bar{c}q\bar{u}$ with the $q=d$
or $s$. The discovery of the $Z_{cs}(3985)$ could provide some unique hints to
uncover the secrets of the charged exotic $Z_{c}$ structures.

The existence of an exotic $Z_{cs}$ state with a mass, which lies near the
$D_{s}^{-}D^{*0}$ and $D^{*-}_{s}D^{0}$ thresholds $3975.2\,\rm{MeV}$ and
$3977.0\,\rm{MeV}$ respectively, has been predicted in several theoretical
models, including the diquark-antidiquark type tetraquark model [3, 4], the
$D_{s}\bar{D}^{*}$ molecule model [5, 6], the hadro-quarkonium model [4, 7],
and the initial-single-chiral-particle-emission mechanism [8]. Just like the
$Z_{c}$ states, the decay rate of the $Z_{cs}$ to open-charm final states is
expected to be larger than the decay rate to charmonium final states.

After the discovery of the $Z_{cs}(3985)$, several possible explanations for
its nature were proposed, such as the tetraquark state [9, 10],
$D^{*}\bar{D}_{s}-D\bar{D}_{s}^{*}$ hadronic molecular state or dynamically
generated resonance with the coupled-channel effects [9, 11, 12, 13, 14, 15,
16], re-scattering effects [17, 18], and we can study its nature with the
photo-production [19].

In Ref.[20], we investigate the spin-parity $J^{P}=1^{+}$ hidden-charm
tetraquark states without strangeness in the framework of the QCD sum rules,
and examine the dependence of the masses and pole residues on the energy
scales $\mu$ of the spectral densities at the quark-gluon level at length for
the first time. Our reliable calculations support identifying the $X(3872)$
and $Z_{c}(3900)$ as the diquark-antidiquark type tetraquark states having
hidden-charm with the quantum numbers $J^{PC}=1^{++}$ and $1^{+-}$,
respectively. In Ref.[21], we take the diquark and antidiquark operators in
the color antitriplet and color triplet respectively as the fundamental
building blocks to construct the scalar, axialvector and tensor four-quark
local currents to investigate the mass spectrum of the ground state tetraquark
states with hidden-charm but without strangeness in the framework of the QCD
sum rules comprehensively, and revisit the possible identifications of the
existing tetraquark candidates with hidden-charm, such as the $X(3860)$,
$X(3872)$, $X(3915)$, $X(3940)$, $X(4160)$, $Z_{c}(3900)$, $Z_{c}(4020)$,
$Z_{c}(4050)$, $Z_{c}(4055)$, $Z_{c}(4100)$, $Z_{c}(4200)$, $Z_{c}(4250)$,
$Z_{c}(4430)$, $Z_{c}(4600)$, etc.

In Refs.[20, 21], we choose the local axialvector four-quark currents
$\eta^{N}_{\mu}(x)$ and $\eta^{P}_{\mu}(x)$,

$\displaystyle\eta^{N}_{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon^{ijk}\varepsilon^{imn}}{\sqrt{2}}\Big{\\{}u^{T}_{j}(x)C\gamma_{5}c_{k}(x)\bar{d}_{m}(x)\gamma_{\mu}C\bar{c}^{T}_{n}(x)-u^{T}_{j}(x)C\gamma_{\mu}c_{k}(x)\bar{d}_{m}(x)\gamma_{5}C\bar{c}^{T}_{n}(x)\Big{\\}}\,,$
$\displaystyle\eta^{P}_{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon^{ijk}\varepsilon^{imn}}{\sqrt{2}}\Big{\\{}u^{T}_{j}(x)C\gamma_{5}c_{k}(x)\bar{d}_{m}(x)\gamma_{\mu}C\bar{c}^{T}_{n}(x)+u^{T}_{j}(x)C\gamma_{\mu}c_{k}(x)\bar{d}_{m}(x)\gamma_{5}C\bar{c}^{T}_{n}(x)\Big{\\}}\,,$
(1)

to investigate the lowest tetraquark states with the quantum numbers
$J^{PC}=1^{+-}$ and $1^{++}$, respectively, where the $i$, $j$, $k$, $m$, $n$
are color indices. The four-quark current $\eta^{N}_{\mu}(x)$ has the quantum
numbers $J^{PC}=1^{+-}$ and couples potentially to the $Z_{c}^{+}(3900)$, in
fact, the $Z_{c}^{\pm}(3900)$ have non-zero electric charge, and do not have
definite charge conjugation, or they are not eigenstates of the charge
conjugation, only the electric neutral state $Z_{c}^{0}(3900)$ has definite
charge conjugation. If the $Z_{cs}^{-}(3985)$ is really a tetraquark state,
irrespective of whether it is the diquark-antidiquark type or meson-meson type
tetraquark state, its valence quarks are $c\bar{c}s\bar{u}$, and has no
definite electric conjugation indeed (to be more precisely), we suppose that
it has definite conjugation, just like its cousins $c\bar{c}q\bar{q}$ and
$c\bar{c}s\bar{s}$. In the present work, we tentatively identify the
$Z_{cs}^{-}(3985)$ as the diquark-antidiquark type axialvector tetraquark
state with the four valence quarks $c\bar{c}s\bar{u}$ and examine its mass in
the framework of the QCD sum rules at length. Then we take into account the
light flavor $SU(3)$ mass-breaking effect to explore the mass spectrum of the
diquark-antidiquark type tetraquark states with hidden-charm and with
strangeness according to our previous works.

The article is arranged as follows: in Sect.2, we get the QCD sum rules for
the masses and pole residues of the $J^{P}=1^{+}$ tetraquark states with
hidden-charm and with strangeness; in Sect.3, we obtain numerical results and
give discussions; and Sect.4 is aimed to get a conclusion.

##  2 The QCD sum rules for the axialvector tetraquark states with strangeness

If we choose the favorable diquark configurations, the scalar ($S$) and
axialvector ($A$) diquark states in the color antitriplet, as the fundamental
building blocks to construct the diquark-antidiquark type tetraquark states,
we can obtain two nonets with the symbolic structures,

$\displaystyle I=1$ $\displaystyle:$
$\displaystyle[uc]_{S}[\bar{d}\bar{c}]_{A}-[uc]_{A}[\bar{d}\bar{c}]_{S}\,,\,[dc]_{S}[\bar{u}\bar{c}]_{A}-[dc]_{A}[\bar{u}\bar{c}]_{S}\,,\,$
$\displaystyle\frac{[uc]_{S}[\bar{u}\bar{c}]_{A}-[uc]_{A}[\bar{u}\bar{c}]_{S}-[dc]_{S}[\bar{d}\bar{c}]_{A}+[dc]_{A}[\bar{d}\bar{c}]_{S}}{\sqrt{2}}\,;$
$\displaystyle I=0$ $\displaystyle:$
$\displaystyle\frac{[uc]_{S}[\bar{u}\bar{c}]_{A}-[uc]_{A}[\bar{u}\bar{c}]_{S}+[dc]_{S}[\bar{d}\bar{c}]_{A}-[dc]_{A}[\bar{d}\bar{c}]_{S}}{\sqrt{2}}\,,$
$\displaystyle[sc]_{S}[\bar{s}\bar{c}]_{A}-[sc]_{A}[\bar{s}\bar{c}]_{S}\,;$
$\displaystyle I=\frac{1}{2}$ $\displaystyle:$
$\displaystyle[qc]_{S}[\bar{s}\bar{c}]_{A}-[qc]_{A}[\bar{s}\bar{c}]_{S}\,,\,[sc]_{S}[\bar{q}\bar{c}]_{A}-[sc]_{A}[\bar{q}\bar{c}]_{S}\,,$
(2)

and

$\displaystyle I=1$ $\displaystyle:$
$\displaystyle[uc]_{S}[\bar{d}\bar{c}]_{A}+[uc]_{A}[\bar{d}\bar{c}]_{S}\,,\,[dc]_{S}[\bar{u}\bar{c}]_{A}+[dc]_{A}[\bar{u}\bar{c}]_{S}\,,\,$
$\displaystyle\frac{[uc]_{S}[\bar{u}\bar{c}]_{A}+[uc]_{A}[\bar{u}\bar{c}]_{S}-[dc]_{S}[\bar{d}\bar{c}]_{A}-[dc]_{A}[\bar{d}\bar{c}]_{S}}{\sqrt{2}}\,;$
$\displaystyle I=0$ $\displaystyle:$
$\displaystyle\frac{[uc]_{S}[\bar{u}\bar{c}]_{A}+[uc]_{A}[\bar{u}\bar{c}]_{S}+[dc]_{S}[\bar{d}\bar{c}]_{A}+[dc]_{A}[\bar{d}\bar{c}]_{S}}{\sqrt{2}}\,,$
$\displaystyle[sc]_{S}[\bar{s}\bar{c}]_{A}+[sc]_{A}[\bar{s}\bar{c}]_{S}\,;$
$\displaystyle I=\frac{1}{2}$ $\displaystyle:$
$\displaystyle[qc]_{S}[\bar{s}\bar{c}]_{A}+[qc]_{A}[\bar{s}\bar{c}]_{S}\,,\,[sc]_{S}[\bar{q}\bar{c}]_{A}+[sc]_{A}[\bar{q}\bar{c}]_{S}\,,$
(3)

where the $I=1$, $0$, $\frac{1}{2}$ are the isospins of the hidden-charm
tetraquark states, $q=u$, $d$.

In the first nonet, only the charge-neutral tetraquark states
$\frac{1}{\sqrt{2}}\Big{(}[uc]_{S}[\bar{u}\bar{c}]_{A}-[uc]_{A}[\bar{u}\bar{c}]_{S}-[dc]_{S}[\bar{d}\bar{c}]_{A}+[dc]_{A}[\bar{d}\bar{c}]_{S}\Big{)}$,
$\frac{1}{\sqrt{2}}\Big{(}[uc]_{S}[\bar{u}\bar{c}]_{A}-[uc]_{A}[\bar{u}\bar{c}]_{S}+[dc]_{S}[\bar{d}\bar{c}]_{A}-[dc]_{A}[\bar{d}\bar{c}]_{S}\Big{)}$
and $[sc]_{S}[\bar{s}\bar{c}]_{A}-[sc]_{A}[\bar{s}\bar{c}]_{S}$ are
eigenstates of the charge conjugation operators, and have the definite charge
conjugation $C=-$.

In the second nonet, only the charge-neutral tetraquark states
$\frac{1}{\sqrt{2}}\Big{(}[uc]_{S}[\bar{u}\bar{c}]_{A}+[uc]_{A}[\bar{u}\bar{c}]_{S}-[dc]_{S}[\bar{d}\bar{c}]_{A}-[dc]_{A}[\bar{d}\bar{c}]_{S}\Big{)}$,
$\frac{1}{\sqrt{2}}\Big{(}[uc]_{S}[\bar{u}\bar{c}]_{A}+[uc]_{A}[\bar{u}\bar{c}]_{S}+[dc]_{S}[\bar{d}\bar{c}]_{A}+[dc]_{A}[\bar{d}\bar{c}]_{S}\Big{)}$
and $[sc]_{S}[\bar{s}\bar{c}]_{A}+[sc]_{A}[\bar{s}\bar{c}]_{S}$ are
eigenstates of the charge conjugation operators, and have the definite charge
conjugation $C=+$.

Thereafter, we will assume that the first nonet and second nonet have the
negative and positive charge conjugations respectively so as to distinguish
the two nonets, and we should bear in mind that the charge conjugation is not
a good quantum number.

Routinely, let us write down the two-point vacuum Green’s functions
$\Pi_{\mu\nu}(p)$,

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

where the interpolating currents $J_{\mu}(x)=J^{N}_{\mu}(x)$ and
$J^{P}_{\mu}(x)$,

$\displaystyle J^{N}_{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon^{ijk}\varepsilon^{imn}}{\sqrt{2}}\Big{\\{}u^{T}_{j}(x)C\gamma_{5}c_{k}(x)\bar{s}_{m}(x)\gamma_{\mu}C\bar{c}^{T}_{n}(x)-u^{T}_{j}(x)C\gamma_{\mu}c_{k}(x)\bar{s}_{m}(x)\gamma_{5}C\bar{c}^{T}_{n}(x)\Big{\\}}\,,$
$\displaystyle J^{P}_{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{\varepsilon^{ijk}\varepsilon^{imn}}{\sqrt{2}}\Big{\\{}u^{T}_{j}(x)C\gamma_{5}c_{k}(x)\bar{s}_{m}(x)\gamma_{\mu}C\bar{c}^{T}_{n}(x)+u^{T}_{j}(x)C\gamma_{\mu}c_{k}(x)\bar{s}_{m}(x)\gamma_{5}C\bar{c}^{T}_{n}(x)\Big{\\}}\,,$
(5)

the superscripts $N$ and $P$ stand for the negative and positive charge
conjugations, respectively, the $i$, $j$, $k$, $m$, $n$ are color indices. We
suppose that the interpolating four-quark currents $J^{N}_{\mu}(x)$ and
$J_{\mu}^{P}(x)$ have the negative and positive charge conjugation
respectively in the sense of the limit $J_{\mu}(x)|_{s\to u}$ or
$J_{\mu}(x)|_{u\to s}$, which should be understood in the same way as in the
two tetraquark nonets in Eqs.(2)-(2). In Refs.[20, 21], we choose the four-
quark currents $\eta^{N}_{\mu}(x)$ and $\eta^{P}_{\mu}(x)$ to explore the
axialvector tetraquark states $uc\bar{d}\bar{c}$, and observe that the
$uc\bar{d}\bar{c}$ states with the quantum numbers $J^{PC}=1^{+-}$ and
$1^{++}$ have almost degenerated masses. The currents $J^{N}_{\mu}(x)$ and
$J^{P}_{\mu}(x)$ are the $SU(3)$ partners of the currents $\eta^{N}_{\mu}(x)$
and $\eta^{P}_{\mu}(x)$, respectively with the simple relation
$d\leftrightarrow s$. Now we also expect that the four-quark currents
$J^{N}_{\mu}(x)$ and $J_{\mu}^{P}(x)$ couple potentially to the
$J^{PC}=1^{+-}$ and $1^{++}$ tetraquark states with almost the same masses.

At the hadron side, we insert a complete set of intermediate tetraquark states
with hidden-charm, strangeness and other quantum numbers, such as the spin,
parity, charge conjugation, as the four-quark current operators $J_{\mu}(x)$,
into the Green’s functions $\Pi_{\mu\nu}(p)$ to obtain the hadron spectral
representation [22, 23], and separate the lowest tetraquark states $Z_{N/P}$
with hidden-charm and with strangeness (or the lowest pole terms), and obtain
the results:

$\displaystyle\Pi_{\mu\nu}(p)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{Z}^{2}}{M^{2}_{Z}-p^{2}}\left(-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{p^{2}}\right)+\cdots\,,$
(6) $\displaystyle=$
$\displaystyle\Pi(p^{2})\left(-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{p^{2}}\right)+\cdots\,,$

where the pole residues $\lambda_{Z}$ (in other works, the decay constants)
are defined by $\langle
0|J_{\mu}(0)|Z_{N/P}(p)\rangle=\lambda_{Z}\varepsilon_{\mu}$, the
$\varepsilon_{\mu}$ are the polarization vectors of the axialvector tetraquark
states $Z_{N/P}$. On the other hand, there are two-particle scattering state
contributions from the two-meson pairs $KJ/\psi$, $\eta_{c}K^{*}$,
$D_{s}\bar{D}^{*}$, $D^{*}\bar{D}_{s}$, $\cdots$, as the quantum field theory
does allow non-vanishing couplings between the four-quark currents and two-
particle scattering states in case that they share the same quantum numbers.
In Ref.[24], we investigate the $Z_{c}(3900)$ as an axialvector tetraquark
state with the quantum numbers $J^{PC}=1^{+-}$ in the framework of the QCD sum
rules at length by considering the two-particle scattering state contributions
and the nonlocal effects between the two colored constituents (diquark and
antidiquark) in the four-quark current operator, and obtain the conclusion
that the contribution of the $Z_{c}(3900)$ as a pole term plays a non-
substitutable role, we can saturate the QCD sum rules at the hadron side no
matter with or without the two-meson scattering state contributions. The net
effects of the two-particle scattering states of the intermediate meson pairs
can be taken into account effectively by adding an energy-dependent finite
width to the pole term. In the present case, the energy-dependent Breit-Wigner
width $13.8^{+8.1}_{-5.2}\pm 4.9\,\rm{MeV}$ of the $Z_{cs}(3985)$ is really
small enough so as to be neglected safely.

In the QCD side, we carry out the operator product expansion up to the vacuum
condensates of dimension 10 in a consistent way, just as what we did in our
previous works [20, 21], in the deep Euclidean region $P^{2}=-p^{2}\to\infty$
or $\gg\Lambda^{2}_{QCD}$, which corresponds to the small spatial distance and
time interval $\vec{x}\sim t\sim\frac{1}{\sqrt{P^{2}}}$. We take into account
or calculate the vacuum condensates $\langle\bar{q}q\rangle$,
$\langle\bar{s}s\rangle$, $\langle\bar{q}g_{s}\sigma Gq\rangle$,
$\langle\bar{s}g_{s}\sigma Gs\rangle$,
$\langle\frac{\alpha_{s}GG}{\pi}\rangle$,
$\langle\bar{q}q\rangle\langle\bar{s}s\rangle$,
$\langle\bar{q}q\rangle\langle\frac{\alpha_{s}GG}{\pi}\rangle$,
$\langle\bar{s}s\rangle\langle\frac{\alpha_{s}GG}{\pi}\rangle$,
$\langle\bar{q}q\rangle\langle\bar{s}g_{s}\sigma Gs\rangle$,
$\langle\bar{s}s\rangle\langle\bar{q}g_{s}\sigma Gq\rangle$,
$\langle\bar{q}q\rangle\langle\bar{s}s\rangle\langle\frac{\alpha_{s}GG}{\pi}\rangle$
and $\langle\bar{q}g_{s}\sigma Gq\rangle\langle\bar{s}g_{s}\sigma Gs\rangle$,
which are the vacuum expectation values of the quark-gluon operators of the
orders $\mathcal{O}(\alpha_{s}^{k})$ with the restriction $k\leq 1$ as
$\alpha_{s}=\frac{g_{s}^{2}}{4\pi}$. The vacuum condensates $\langle
g_{s}^{3}f_{abc}G^{a}G^{b}G^{c}\rangle$,
$\langle\frac{\alpha_{s}GG}{\pi}\rangle^{2}$,
$\langle\frac{\alpha_{s}GG}{\pi}\rangle\langle\bar{q}g_{s}\sigma Gq\rangle$
and $\langle\bar{q}q\rangle\langle g_{s}^{3}f_{abc}G^{a}G^{b}G^{c}\rangle$ are
of the dimensions $6$, $8$, $9$ and $9$, respectively, and are vacuum
expectation values of the quark-gluon operators of the orders
$\mathcal{O}(\alpha_{s}^{\frac{3}{2}})$, $\mathcal{O}(\alpha_{s}^{2})$,
$\mathcal{O}(\alpha_{s}^{\frac{3}{2}})$ and
$\mathcal{O}(\alpha_{s}^{\frac{3}{2}})$, respectively, and are neglected in
the present work, just like in our previous works [21, 25, 26, 27], direct
calculations indicate that those contributions are tiny indeed [28]. In
calculations, we have assumed vacuum saturation for the sake of factorizing
the higher dimensional vacuum condensates into the lowest ones to reduce the
numbers of the fundamental parameters, which works very well indeed in the
large color numbers limit.

Once we get the analytical expressions of the correlation (or Green’s)
functions $\Pi(p^{2})$ at the degrees of freedom of the quarks and gluons,
then we resort to dispersion relation to get the spectral densities at the
quark level straightforwardly, and match the two sides of the correlation
(Green’s) functions $\Pi(p^{2})$ (i.e. the hadron side and QCD side),
accomplish the quark-hadron duality (in other words, the current-hadron
duality) below the thresholds $s_{0}$ of the continuum states or higher
resonances, complete the Borel transform in regard to the variable or
parameter $P^{2}=-p^{2}$ and acquire the QCD sum rules:

$\displaystyle\lambda_{Z}^{2}\exp\left(-\frac{M_{Z}^{2}}{T^{2}}\right)$
$\displaystyle=$
$\displaystyle\int_{4m_{c}^{2}}^{s_{0}}ds\,\rho_{QCD}(s)\,\exp\left(-\frac{s}{T^{2}}\right)\,,$
(7)

where the explicit expressions of the spectral densities $\rho_{QCD}(s)$ at
the quark level are neglected for simplicity.

We differentiate both sides of the above equation in regard to the parameter
$\frac{1}{T^{2}}$, then eliminate the pole residues $\lambda_{Z}$ by
introducing a fraction, and reach the QCD sum rules for the masses of the
hidden-charm axialvector tetraquark states with strangeness,

$\displaystyle M^{2}_{Z}$ $\displaystyle=$
$\displaystyle\frac{-\int_{4m_{c}^{2}}^{s_{0}}ds\frac{d}{d(1/T^{2})}\,\rho_{QCD}(s)\,\exp\left(-\frac{s}{T^{2}}\right)}{\int_{4m_{c}^{2}}^{s_{0}}ds\,\rho_{QCD}(s)\,\exp\left(-\frac{s}{T^{2}}\right)}\,.$
(8)

##  3 Numerical results and discussions

We adopt the standard values or conventional values of all the vacuum
condensates $\langle\bar{q}q\rangle=-(0.24\pm 0.01\,\rm{GeV})^{3}$,
$\langle\bar{q}g_{s}\sigma Gq\rangle=m_{0}^{2}\langle\bar{q}q\rangle$,
$\langle\bar{s}s\rangle=(0.8\pm 0.1)\langle\bar{q}q\rangle$,
$\langle\bar{s}g_{s}\sigma Gs\rangle=m_{0}^{2}\langle\bar{s}s\rangle$,
$m_{0}^{2}=(0.8\pm 0.1)\,\rm{GeV}^{2}$,
$\langle\frac{\alpha_{s}GG}{\pi}\rangle=0.012\pm 0.004\,\rm{GeV}^{4}$ at the
typical energy scale $\mu=1\,\rm{GeV}$ [22, 23, 29, 30, 31], and prefer the
$\overline{MS}$ masses of the charm and strange quarks,
$m_{c}(m_{c})=(1.275\pm 0.025)\,\rm{GeV}$ and
$m_{s}(\mu=2\,\rm{GeV})=(0.095\pm 0.005)\,\rm{GeV}$, from the Particle Data
Group [32], just like in our previous works [20, 21]. In addition, we take
into account the energy-scale dependence of all the input parameters at the
quark level, such as the quark condensates $\langle\bar{q}q\rangle$,
$\langle\bar{s}s\rangle$, the mixed quark condensates
$\langle\bar{q}g_{s}\sigma Gq\rangle$, $\langle\bar{s}g_{s}\sigma Gs\rangle$
and the $\overline{MS}$ masses $m_{c}(\mu)$, $m_{s}(\mu)$ in the light of
renormalization group equation [33],

$\displaystyle\langle\bar{q}q\rangle(\mu)$ $\displaystyle=$
$\displaystyle\langle\bar{q}q\rangle({\rm 1GeV})\left[\frac{\alpha_{s}({\rm
1GeV})}{\alpha_{s}(\mu)}\right]^{\frac{12}{33-2n_{f}}}\,,$
$\displaystyle\langle\bar{s}s\rangle(\mu)$ $\displaystyle=$
$\displaystyle\langle\bar{s}s\rangle({\rm 1GeV})\left[\frac{\alpha_{s}({\rm
1GeV})}{\alpha_{s}(\mu)}\right]^{\frac{12}{33-2n_{f}}}\,,$
$\displaystyle\langle\bar{q}g_{s}\sigma Gq\rangle(\mu)$ $\displaystyle=$
$\displaystyle\langle\bar{q}g_{s}\sigma Gq\rangle({\rm
1GeV})\left[\frac{\alpha_{s}({\rm
1GeV})}{\alpha_{s}(\mu)}\right]^{\frac{2}{33-2n_{f}}}\,,$
$\displaystyle\langle\bar{s}g_{s}\sigma Gs\rangle(\mu)$ $\displaystyle=$
$\displaystyle\langle\bar{s}g_{s}\sigma Gs\rangle({\rm
1GeV})\left[\frac{\alpha_{s}({\rm
1GeV})}{\alpha_{s}(\mu)}\right]^{\frac{2}{33-2n_{f}}}\,,$ $\displaystyle
m_{c}(\mu)$ $\displaystyle=$ $\displaystyle
m_{c}(m_{c})\left[\frac{\alpha_{s}(\mu)}{\alpha_{s}(m_{c})}\right]^{\frac{12}{33-2n_{f}}}\,,$
$\displaystyle m_{s}(\mu)$ $\displaystyle=$ $\displaystyle m_{s}({\rm
2GeV})\left[\frac{\alpha_{s}(\mu)}{\alpha_{s}({\rm
2GeV})}\right]^{\frac{12}{33-2n_{f}}}\,,$ $\displaystyle\alpha_{s}(\mu)$
$\displaystyle=$
$\displaystyle\frac{1}{b_{0}t}\left[1-\frac{b_{1}}{b_{0}^{2}}\frac{\log
t}{t}+\frac{b_{1}^{2}(\log^{2}{t}-\log{t}-1)+b_{0}b_{2}}{b_{0}^{4}t^{2}}\right]\,,$
(9)

where $t=\log\frac{\mu^{2}}{\Lambda^{2}}$, $b_{0}=\frac{33-2n_{f}}{12\pi}$,
$b_{1}=\frac{153-19n_{f}}{24\pi^{2}}$,
$b_{2}=\frac{2857-\frac{5033}{9}n_{f}+\frac{325}{27}n_{f}^{2}}{128\pi^{3}}$,
$\Lambda=213\,\rm{MeV}$, $296\,\rm{MeV}$ and $339\,\rm{MeV}$ for the quark
flavor numbers $n_{f}=5$, $4$ and $3$, respectively [32]. In the present work,
we investigate the hidden-charm tetraquark states $c\bar{c}u\bar{s}$ with
strangeness, it is better to adopt the quark flavor numbers $n_{f}=4$, then
evolve all the input parameters at the quark level to a typical energy scale
$\mu$, which satisfies the restriction of the energy scale formula
$\mu=\sqrt{M_{Z}-(2{\mathbb{M}}_{c})^{2}}$ with the updated effective charm
quark mass ${\mathbb{M}}_{c}=1.82\,\rm{GeV}$ [21, 25, 26, 27]. If we take the
$Z_{cs}(3985)$ as the ground state tetraquark candidate for the $Z_{N/P}$ with
hidden-charm, with strangeness, and with $J^{P}=1^{+}$, the best energy scales
(or our preferred energy scales) of the spectral densities at the quark level
are $\mu=1.6\,\rm{GeV}$.

Now let us take a short digression to discuss the energy scale dependence of
the QCD sum rules. In preforming the operator product expansion, we can choose
any energy scale if the perturbative calculations are feasible at this special
energy scale, and the physical quantities extracted from the QCD sum rules
should be independent on selections of the energy scales. In this sense, the
correlation functions $\Pi(p^{2})$ are independent on the energy scales,
$\frac{d}{d\mu}\Pi(p^{2})=0$.

On the other hand, the two-quark (three-quark, four-quark, $\cdots$) currents
$J(x)$ are operators and are renormalized at special energy scales, thus they
are defined at special energy scales and are energy scale dependent
quantities,

$\displaystyle J(x,\mu)$ $\displaystyle=$ $\displaystyle
L^{\gamma_{J}}J(x,\mu_{0})\,,$ (10)

where $L=\frac{\alpha_{s}(\mu_{0})}{\alpha_{s}(\mu)}$, and the $\gamma_{J}$
are the anomalous dimensions of the currents $J(x)$. We usually neglect
renormalization of the hadron states, and set the anomalous dimensions of the
pole residues (or decay constants) to be the anomalous dimensions of the
current operators,

$\displaystyle\langle 0|J(0,\mu)|H(p)\rangle$ $\displaystyle=$
$\displaystyle\lambda_{H}(\mu)\,,$ (11) $\displaystyle=$ $\displaystyle
L^{\gamma_{J}}\langle 0|J(0,\mu_{0})|H(p)\rangle$ $\displaystyle=$
$\displaystyle L^{\gamma_{J}}\lambda_{H}(\mu_{0})\,.$

In fact, even in the QCD sum rules for the conventional $D$ meson, where the
radiative corrections of the perturbative terms have been calculated up to the
order $\mathcal{O}(\alpha_{s}^{2})$ and the radiative corrections of the quark
condensate have been calculated up to the order $\mathcal{O}(\alpha_{s}^{1})$
[34, 35, 36], the relation for the decay constant
$f_{D}(\mu)=L^{\frac{12}{25}}f_{D}(\mu_{0})$ cannot take account of the energy
scale dependence of the QCD sum rules in a consistent way.

On one hand, the non-local operators $J(x)J(0)$ have their own anomalous
dimensions $\gamma_{JJ}$, $\gamma_{JJ}\neq 2\gamma_{J}$, just like re-
normalization of the quark fields $q(x)$ alone is not enough for the
conventional current operators $J(x)=\bar{q}(x)\Gamma q(x)$, where the
$\Gamma$ stand for some Dirac $\gamma$-matrixes. On the other hand, we usually
neglect the radiative corrections due to the cumbersome calculations and
factorize the higher dimensional vacuum condensates to the lower dimension
vacuum condensates in performing the operator product expansion, the energy
scale dependence of the QCD sum rules is modified. Furthermore, we introduce
the continuum threshold parameters $s_{0}$ to exclude the contaminations of
the higher resonances and continuum states, the correlation between the
thresholds and continuum thresholds is not clear.

The energy scale dependence cannot be absorbed into the pole residues alone,
we cannot obtain energy scale independent QCD sum rules for the masses,
selections of the energy scales of the QCD spectral densities influence the
masses extracted from the QCD sum rules. We can rewrite the energy scale
formula in another form,

$\displaystyle M^{2}_{X/Y/Z}$ $\displaystyle=$ $\displaystyle\mu^{2}+C\,,$
(12)

where the constants $C=4{\mathbb{M}}_{c}^{2}$, then explain the energy scale
formula in another way. We conjecture that the predicted tetraquark masses and
pertinent energy scales of the QCD spectral densities have a Regge-trajectory-
like relation, see Eq.(12), where the $C$ are free parameters and fitted by
the QCD sum rules. Direct calculations have proven that the $C$ have universal
values and work well for all the tetraquark (molecular) states.

The standard values of the quark condensates and mixed condensates determined
in the original works still survive [22, 29, 30, 31], while the value of the
gluon condensate was updated from time to time in the literatures, however,
the standard value determined in the original works [22] is still feasible
[31]. The gluon condensate is the vacuum expectation value of the gluon
operator of the order $\mathcal{O}(\alpha_{s}^{1})$, and plays a minor
important role in the present work, the standard value and updated value do
not make much difference. The most important parameter is the $c$-quark mass
$m_{c}(m_{c})$. In 2006, R. D. Matheus et al investigated the $X(3872)$ as the
diquark-antidiquark type tetraquark state with the QCD sum rules by carrying
out the operator product expansion up to the vacuum condensates of dimension 8
[37]. Thereafter the QCD sum rules became a powerful theoretical approach in
studying the exotic $X$, $Y$ and $Z$ states. In Ref.[37], the $\overline{MS}$
mass $m_{c}(m_{c})=1.23\pm 0.05\rm{GeV}$ was chosen, thereafter the value was
adopted without or with slightly modified uncertainty [38]. Only in recent
years, new values $m_{c}(m_{c})=1266\pm 6\,\rm{MeV}$ [39] and $1.275\pm
0.025\,\rm{GeV}$ [9] and $1.275^{+0.025}_{-0.035}\,\rm{GeV}$ [16] were chosen.
The values of the $\overline{MS}$ mass of the $c$-quark listed in The Review
of Particle Physics in 2012, 2014, 2016, 2018 and 2020 were $1.275\pm
0.025\,\rm{GeV}$, $1.275\pm 0.025\,\rm{GeV}$, $1.27\pm 0.03\,\rm{GeV}$,
$1.275^{+0.025}_{-0.035}\,\rm{GeV}$ and $1.27\pm 0.02\,\rm{GeV}$,
respectively. We fitted the constants $C$ with the value $1.275\pm
0.025\,\rm{GeV}$ from The Review of Particle Physics (2012) [25, 26], and
adopted the value eversince.

The energy gaps between the ground states (or 1S) and the first radial excited
states (or 2S) of the hidden-charm tetraquark states without strangeness or
with hidden-strangeness are about $0.6\,\rm{GeV}$ [21, 40, 41, 42], we adopt
the continuum threshold parameters as $\sqrt{s_{0}}=M_{Z}+0.55\pm
0.10\,\rm{GeV}=4.55\pm 10\,\rm{GeV}$ tentatively, and vary the Borel
parameters $T^{2}$ to satisfy the two requirements that the pole contributions
dominate the QCD sum rules at the hadron side and the operator product
expansions converge rather quickly at the QCD side via trial and error. In
doing so, we should bear in mind that if our numerical results do no support
identifying the $Z_{cs}(3985)$ to be the axialvector tetraquark state, we
should refit the best continuum threshold parameters and the best energy
scales of the spectral densities at the quark level to obtain the real ground
state tetraquark masses.

In the end, we acquire the Borel windows (or working Borel parameters) and
pole contributions in the two QCD sum rules for the currents $J_{\mu}^{N}(x)$
and $J^{P}_{\mu}(x)$, and show them plainly in the Table 1. From the table, we
can see clearly that the contributions of the lowest pole terms are about
$(42-62)\%$, while the central values exceed $50\%$, so we can say confidently
that the contributions of the pole terms dominate the QCD sum rules at the
hadron side, one of the fundamental criterions is satisfied very well. In
Fig.1, we plot the absolute values of the contributions of the vacuum
condensates with the centroids of the values of all the input parameters under
the condition that the total contributions are normalized to be 1. From the
figure, we can see clearly that the largest contributions come from the quark
condensates $\langle\bar{q}q\rangle$ and $\langle\bar{s}s\rangle$, the vacuum
condensates of the dimensions $7$, $8$ and $10$ play a tiny role, the operator
product expansion converges very good. Now we can acquire the conclusion
confidently that it is reliable and reasonable to extract the tetraquark
masses in the Borel windows.

Figure 1:  The absolute values of the contributions of the vacuum condensates in the condition of central values of the input parameters, where the $N$ and $P$ stand for the negative and positive charge conjugation, respectively.  $J^{PC}$ | $T^{2}(\rm{GeV}^{2})$ | $\sqrt{s_{0}}(\rm{GeV})$ | $\mu(\rm{GeV})$ | pole | $M(\rm{GeV})$ | $\lambda(\rm{GeV}^{5})$
---|---|---|---|---|---|---
$1^{+-}$ | $3.0-3.4$ | $4.55\pm 0.10$ | 1.6 | $(42-63)\%$ | $3.99\pm 0.09$ | $(2.85\pm 0.45)\times 10^{-2}$
$1^{++}$ | $3.0-3.4$ | $4.55\pm 0.10$ | 1.6 | $(41-62)\%$ | $3.99\pm 0.09$ | $(2.85\pm 0.45)\times 10^{-2}$
Table 1:  The Borel windows, continuum threshold parameters, ideal energy
scales, pole contributions, masses and pole residues for the axialvector
tetraquark states with strangeness.

We take into account all the uncertainties of the input parameters to
accomplish the error analysis, and acquire the values of the masses and pole
residues of the axialvector tetraquark states with hidden-charm and with the
strangeness $S=1$, which are presented explicitly in Table 1 and Fig.2. From
Table 1, we can see clearly that the energy scale $\mu=1.6\,\rm{GeV}$ is
consistent with the mass $3.99\,\rm{GeV}$ inferred from the energy scale
formula $\mu=\sqrt{M_{Z}-(2{\mathbb{M}}_{c})^{2}}$ or the relation between the
tetraquark masses and the energy scales of the QCD spectral densities
$M_{Z}^{2}=\mu^{2}+4\mathbb{M}_{c}^{2}$. In Fig.2, we plot the predicted
tetraquark masses with respect to variations of the Borel parameters at much
larger regions than the Borel windows, furthermore, we also show the
experimental value of the mass of the $Z_{cs}(3985)$ from the BESIII
collaboration for the sake of comparing [1]. From the figure, we can see
clearly that there really appear very flat platforms in the Borel windows. In
the Borel windows, the mass of the $Z_{cs}(3985)$ overlaps with the central
values of the masses of the tetraquark states with strangeness and sharing the
quantum numbers $J^{PC}=1^{+\pm}$. From the Table 1, we can see plainly that
the axialvector tetraquark states with the negative and positive charge
conjugation have degenerated masses. In fact, the central values of the masses
of the axialvector tetraquark states with the negative and positive charge
conjugation are $3.98669\,\rm{GeV}$ and $3.99303\,\rm{GeV}$ respectively, the
axialvector tetraquark states having the $J^{PC}=1^{++}$ have slightly larger
masses than the corresponding states having the $J^{PC}=1^{+-}$ [21]. In all
the calculations including the present work, we observe that if we choose the
same values of the input parameters, such as the quark masses, vacuum
condensates, continuum threshold parameters, Borel parameters, etc, the
predicted masses of the $J^{PC}=1^{++}$ tetraquark states are slightly larger
than that of the $J^{PC}=1^{+-}$ tetraquark states. However, we should bear in
mind that the tiny mass difference cannot be quantified considering the
uncertainties of the QCD sum rules.

Figure 2:  The masses of the axialvector tetraquark states with variations of
the Borel parameter $T^{2}$, where the $N$ and $P$ stand for the negative and
positive charge conjugation, respectively, the regions between the two
vertical lines are the Borel windows, the Expt stands for the experimental
value of the mass of the $Z_{cs}(3985)$.

In Fig.3, we plot the predicted masses of the $J^{PC}=1^{+-}$ and $1^{++}$
tetraquark states with variations of the energy scales of the QCD spectral
densities in the condition of central values of the input parameters. From the
figure, we can see clearly that the predicted masses decrease monotonously
with the increase of the energy scales. If we set
${\mathbb{M}}_{c}=1.82\,\rm{GeV}$ [21, 25, 26, 27], we can obtain the dash-
dotted line $M_{Z}=\sqrt{\mu^{2}+4\times(1.82\,\rm{GeV})^{2}}$, which
intersects with the lines of the masses of the $J^{PC}=1^{+-}$ and $1^{++}$
tetraquark states at the energy scales about $\mu=1.6\,\rm{GeV}$. In this way,
we choose the energy scales of the QCD spectral densities in a consistent way.
As a byproduct, we can see clearly, if we choose the same input parameters,
the predicted masses of the $J^{PC}=1^{++}$ tetraquark states are slightly
larger than that of the $J^{PC}=1^{+-}$ tetraquark states.

Figure 3:  The masses of the axialvector tetraquark states with variations of
the energy scales $\mu$ in the condition of central values of the input
parameters, where the $N$ and $P$ stand for the negative and positive charge
conjugation, respectively, the ESF denotes the formula
$M=\sqrt{\mu^{2}+4\times(1.82\,\rm{GeV})^{2}}$.

When we contract the quark fields in the correlation functions
$\Pi_{\mu\nu}(p)$ with Wick theorem, we obtain the formula,

$\displaystyle\Pi_{\mu\nu}(p)$ $\displaystyle=$
$\displaystyle-\frac{i\varepsilon^{ijk}\varepsilon^{imn}\varepsilon^{i^{\prime}j^{\prime}k^{\prime}}\varepsilon^{i^{\prime}m^{\prime}n^{\prime}}}{2}\int
d^{4}xe^{ip\cdot x}$ (13) $\displaystyle\left\\{{\rm
Tr}\left[\gamma_{5}C^{kk^{\prime}}(x)\gamma_{5}CU^{jj^{\prime}T}(x)C\right]{\rm
Tr}\left[\gamma_{\nu}C^{n^{\prime}n}(-x)\gamma_{\mu}CS^{m^{\prime}mT}(-x)C\right]\right.$
$\displaystyle+{\rm
Tr}\left[\gamma_{\mu}C^{kk^{\prime}}(x)\gamma_{\nu}CU^{jj^{\prime}T}(x)C\right]{\rm
Tr}\left[\gamma_{5}C^{n^{\prime}n}(-x)\gamma_{5}CS^{m^{\prime}mT}(-x)C\right]$
$\displaystyle\mp{\rm
Tr}\left[\gamma_{\mu}C^{kk^{\prime}}(x)\gamma_{5}CU^{jj^{\prime}T}(x)C\right]{\rm
Tr}\left[\gamma_{\nu}C^{n^{\prime}n}(-x)\gamma_{5}CS^{m^{\prime}mT}(-x)C\right]$
$\displaystyle\left.\mp{\rm
Tr}\left[\gamma_{5}C^{kk^{\prime}}(x)\gamma_{\nu}CU^{jj^{\prime}T}(x)C\right]{\rm
Tr}\left[\gamma_{5}C^{n^{\prime}n}(-x)\gamma_{\mu}CS^{m^{\prime}mT}(-x)C\right]\right\\}\,,$

where the $\mp$ correspond the positive and negative charge conjugations of
the currents, respectively, the $U_{ij}(x)$, $S_{ij}(x)$ and $C_{ij}(x)$ are
the full $u$, $s$ and $c$ quark propagators, respectively. In carrying out the
operator product expansion, we observe that the dominant contributions come
from the first two terms in the bracket, the contributions come from the last
two terms in the bracket play a tiny role. It is natural that we obtain the
almost degenerated tetraquark masses.

From the Table 1, we can see clearly that the predicted tetraquark masses
$M_{Z}=3.99\pm 0.09\,\rm{GeV}$ are in excellent agreement with the
experimental value $3985.2^{+2.1}_{-2.0}\pm 1.7\,\rm{MeV}$ from the BESIII
collaboration [1], and support identifying the $Z_{cs}(3985)$ to be the
tetraquark state with hidden-charm, with strangeness and with the
$J^{PC}=1^{+-}$. We prefer the quantum numbers $J^{PC}=1^{+-}$ to the quantum
numbers $J^{PC}=1^{++}$ as the $Z_{cs}(3985)$ was observed in the
$D_{s}^{-}D^{*0}$ and $D^{*-}_{s}D^{0}$ mass spectrum [1], just like the
$Z_{c}(3900)$, which was observed in the $D\bar{D}^{*}$ mass spectrum [2].
However, the assignment $J^{PC}=1^{++}$ cannot be excluded as the charge
conjugation is not a good quantum number.

In Ref.[21], we introduce the four-vector $t^{\mu}=(1,\vec{0})$ to project the
axialvector and vector components of the tensor diquark operators, and take
the color-antitriplet diquark operators
$\varepsilon^{ijk}q^{T}_{j}C\gamma_{5}q^{\prime}_{k}$ ($S$),
$\varepsilon^{ijk}q^{T}_{j}Cq^{\prime}_{k}$ ($P$),
$\varepsilon^{ijk}q^{T}_{j}C\gamma_{\mu}q^{\prime}_{k}$ ($A$),
$\varepsilon^{ijk}q^{T}_{j}C\sigma^{v}_{\mu\nu}q^{\prime}_{k}$
($\widetilde{A}$),
$\varepsilon^{ijk}q^{T}_{j}C\gamma_{\mu}\gamma_{5}q^{\prime}_{k}$ ($V$) and
$\varepsilon^{ijk}q^{T}_{j}C\sigma^{t}_{\mu\nu}q^{\prime}_{k}$
($\widetilde{V}$) as the fundamental building blocks to construct the four-
quark currents to investigate the mass-spectrum of the tetraquark states with
hidden-charm but without strangeness in a comprehensive way, where the $S$,
$P$, $A/\widetilde{A}$ and $V/\widetilde{V}$ stand for the scalar,
pseudoscalar, axialvector and vector diquark operators, respectively,
$\sigma^{t}_{\mu\nu}=\frac{i}{2}\Big{[}\gamma^{t}_{\mu},\gamma^{t}_{\nu}\Big{]}$,
$\sigma^{v}_{\mu\nu}=\frac{i}{2}\Big{[}\gamma^{v}_{\mu},\gamma^{t}_{\nu}\Big{]}$,
$\gamma^{v}_{\mu}=\gamma\cdot tt_{\mu}$,
$\gamma^{t}_{\mu}=\gamma_{\mu}-\gamma\cdot tt_{\mu}$. In Table 2, we show
plainly the mass-spectrum of the hidden-charm tetraquark states
$c\bar{c}u\bar{d}$ obtained via the QCD sum rules in Ref.[21] with the
possible assignments.

If we assign the $Z_{cs}(3985)$ to be the cousin of the $Z_{c}(3900)$ with
strangeness, the mass gap or the light flavor $SU(3)$ mass-breaking effect
$M_{Z_{cs}(3985)}-M_{Z_{c}(3900)}=94\,\rm{MeV}$, then we take the light flavor
$SU(3)$ mass-breaking effect $m_{s}=0.09\,\rm{GeV}$, and estimate the mass
spectrum of the tetraquark states with hidden-charm and with strangeness based
on our previous work [21], which are shown explicitly in Table 3. We should
bear in mind that the charge conjugation is not a very good quantum number in
the present case. We cannot exclude that the $Z_{cs}(3985)$ can be identified
as the axialvector hidden-charm tetraquark state with the quantum numbers
$J^{PC}=1^{++}$.

$Z_{c}$($X_{c}$) | $J^{PC}$ | $M_{Z}(\rm{GeV})$ | Assignments |  $Z_{c}^{\prime}$ ($X_{c}^{\prime}$)
---|---|---|---|---
$[uc]_{S}[\overline{dc}]_{S}$ | $0^{++}$ | $3.88\pm 0.09$ | ? $X(3860)$ |
$[uc]_{A}[\overline{dc}]_{A}$ | $0^{++}$ | $3.95\pm 0.09$ | ? $X(3915)$ |
$[uc]_{\tilde{A}}[\overline{dc}]_{\tilde{A}}$ | $0^{++}$ | $3.98\pm 0.08$ |  |
$[uc]_{V}[\overline{dc}]_{V}$ | $0^{++}$ | $4.65\pm 0.09$ |  |
$[uc]_{\tilde{V}}[\overline{dc}]_{\tilde{V}}$ | $0^{++}$ | $5.35\pm 0.09$ |  |
$[uc]_{P}[\overline{dc}]_{P}$ | $0^{++}$ | $5.49\pm 0.09$ |  |
$[uc]_{S}[\overline{dc}]_{A}-[uc]_{A}[\overline{dc}]_{S}$ | $1^{+-}$ | $3.90\pm 0.08$ | ? $Z_{c}(3900)$ | ? $Z_{c}(4430)$
$[uc]_{A}[\overline{dc}]_{A}$ | $1^{+-}$ | $4.02\pm 0.09$ | ? $Z_{c}(4020/4055)$ | ? $Z_{c}(4600)$
$[uc]_{S}[\overline{dc}]_{\widetilde{A}}-[uc]_{\widetilde{A}}[\overline{dc}]_{S}$ | $1^{+-}$ | $4.01\pm 0.09$ | ? $Z_{c}(4020/4055)$ | ? $Z_{c}(4600)$
$[uc]_{\widetilde{A}}[\overline{dc}]_{A}-[uc]_{A}[\overline{dc}]_{\widetilde{A}}$ | $1^{+-}$ | $4.02\pm 0.09$ | ? $Z_{c}(4020/4055)$ | ? $Z_{c}(4600)$
$[uc]_{\widetilde{V}}[\overline{dc}]_{V}+[uc]_{V}[\overline{dc}]_{\widetilde{V}}$ | $1^{+-}$ | $4.66\pm 0.10$ | ? $Z_{c}(4600)$ |
$[uc]_{V}[\overline{dc}]_{V}$ | $1^{+-}$ | $5.46\pm 0.09$ |  |
$[uc]_{P}[\overline{dc}]_{V}+[uc]_{V}[\overline{dc}]_{P}$ | $1^{+-}$ | $5.45\pm 0.09$ |  |
$[uc]_{S}[\overline{dc}]_{A}+[uc]_{A}[\overline{dc}]_{S}$ | $1^{++}$ | $3.91\pm 0.08$ | ? $X(3872)$ |
$[uc]_{S}[\overline{dc}]_{\widetilde{A}}+[uc]_{\widetilde{A}}[\overline{dc}]_{S}$ | $1^{++}$ | $4.02\pm 0.09$ | ? $Z_{c}(4050)$ |
$[uc]_{\widetilde{V}}[\overline{dc}]_{V}-[uc]_{V}[\overline{dc}]_{\widetilde{V}}$ | $1^{++}$ | $4.08\pm 0.09$ | ? $Z_{c}(4050)$ |
$[uc]_{\widetilde{A}}[\overline{dc}]_{A}+[uc]_{A}[\overline{dc}]_{\widetilde{A}}$ | $1^{++}$ | $5.19\pm 0.09$ |  |
$[uc]_{P}[\overline{dc}]_{V}-[uc]_{V}[\overline{dc}]_{P}$ | $1^{++}$ | $5.46\pm 0.09$ |  |
$[uc]_{A}[\overline{dc}]_{A}$ | $2^{++}$ | $4.08\pm 0.09$ | ? $Z_{c}(4050)$ |
$[uc]_{V}[\overline{dc}]_{V}$ | $2^{++}$ | $5.40\pm 0.09$ |  |
Table 2:  The possible assignments of the ground state hidden-charm tetraquark states, where the isospin limit is implied [21].  $Z_{c}$($X_{c}$) | $J^{PC}$ | $M_{Z}(\rm{GeV})$ | Assignments
---|---|---|---
$[uc]_{S}[\overline{sc}]_{S}$ | $0^{++}$ | $3.97\pm 0.09$ |
$[uc]_{A}[\overline{sc}]_{A}$ | $0^{++}$ | $4.04\pm 0.09$ |
$[uc]_{\tilde{A}}[\overline{sc}]_{\tilde{A}}$ | $0^{++}$ | $4.07\pm 0.08$ |
$[uc]_{V}[\overline{sc}]_{V}$ | $0^{++}$ | $4.74\pm 0.09$ |
$[uc]_{\tilde{V}}[\overline{sc}]_{\tilde{V}}$ | $0^{++}$ | $5.44\pm 0.09$ |
$[uc]_{P}[\overline{sc}]_{P}$ | $0^{++}$ | $5.58\pm 0.09$ |
$[uc]_{S}[\overline{sc}]_{A}-[uc]_{A}[\overline{sc}]_{S}$ | $1^{+-}$ | $3.99\pm 0.09$ | ? $Z_{cs}(3985)$
$[uc]_{A}[\overline{sc}]_{A}$ | $1^{+-}$ | $4.11\pm 0.09$ |
$[uc]_{S}[\overline{sc}]_{\widetilde{A}}-[uc]_{\widetilde{A}}[\overline{sc}]_{S}$ | $1^{+-}$ | $4.10\pm 0.09$ |
$[uc]_{\widetilde{A}}[\overline{sc}]_{A}-[uc]_{A}[\overline{sc}]_{\widetilde{A}}$ | $1^{+-}$ | $4.11\pm 0.09$ |
$[uc]_{\widetilde{V}}[\overline{sc}]_{V}+[uc]_{V}[\overline{sc}]_{\widetilde{V}}$ | $1^{+-}$ | $4.75\pm 0.10$ |
$[uc]_{V}[\overline{sc}]_{V}$ | $1^{+-}$ | $5.55\pm 0.09$ |
$[uc]_{P}[\overline{sc}]_{V}+[uc]_{V}[\overline{sc}]_{P}$ | $1^{+-}$ | $5.54\pm 0.09$ |
$[uc]_{S}[\overline{sc}]_{A}+[uc]_{A}[\overline{sc}]_{S}$ | $1^{++}$ | $3.99\pm 0.09$ | ? $Z_{cs}(3985)$
$[uc]_{S}[\overline{sc}]_{\widetilde{A}}+[uc]_{\widetilde{A}}[\overline{sc}]_{S}$ | $1^{++}$ | $4.11\pm 0.09$ |
$[uc]_{\widetilde{V}}[\overline{sc}]_{V}-[uc]_{V}[\overline{sc}]_{\widetilde{V}}$ | $1^{++}$ | $4.17\pm 0.09$ |
$[uc]_{\widetilde{A}}[\overline{sc}]_{A}+[uc]_{A}[\overline{sc}]_{\widetilde{A}}$ | $1^{++}$ | $5.28\pm 0.09$ |
$[uc]_{P}[\overline{sc}]_{V}-[uc]_{V}[\overline{sc}]_{P}$ | $1^{++}$ | $5.55\pm 0.09$ |
$[uc]_{A}[\overline{dc}]_{A}$ | $2^{++}$ | $4.17\pm 0.09$ |
$[uc]_{V}[\overline{dc}]_{V}$ | $2^{++}$ | $5.49\pm 0.09$ |
Table 3:  The possible assignments of the ground state hidden-charm tetraquark
states with strangeness.

##  4 Conclusion

In present paper, we choose the scalar and axialvector diquark operators (in
color antitriplet) and antidiquark (in color triplet) operators as the
fundamental building blocks to construct the four-quark currents and
investigate the diquark-antidiquark type axialvector tetraquark states
$c\bar{c}u\bar{s}$ with the QCD sum rules in the condition that we accomplish
the operator product expansion up to the vacuum condensates of dimension $10$
consistently based on our reasonable analysis and successful experience, and
apply the energy scale formula $\mu=\sqrt{M_{Z}-(2{\mathbb{M}}_{c})^{2}}$
using the effective charmed quark mass ${\mathbb{M}}_{c}=1.82\,\rm{GeV}$ to
fix the best energy scales of the spectral densities at the quark level. The
predicted tetraquark mass $M_{Z}=3.99\pm 0.09\,\rm{GeV}$ is in excellent
agreement with the experimental value $3985.2^{+2.1}_{-2.0}\pm 1.7\,\rm{MeV}$
from the BESIII collaboration, and supports identifying the $Z_{cs}(3985)$ as
the cousin of the $Z_{c}(3900)$ with the quantum numbers $J^{PC}=1^{+-}$.
Furthermore, we obtain the mass of the corresponding tetraquark state
$c\bar{c}u\bar{s}$ with the quantum numbers $J^{PC}=1^{++}$, which can be
compared to the international high energy experimental data in the future. We
take into account the light flavor $SU(3)$ mass-breaking effect about
$90\,\rm{MeV}$, and make crude estimations of the mass spectrum of the
diquark-antidiquark type tetraquark states with hidden-charm and with
strangeness.
Note added:
After the manuscript was submitted to https://arxiv.org/, the LHCb
collaboration reported two new exotic states with the valence quarks
$c\bar{c}u\bar{s}$ in the $J/\psi K^{+}$ mass spectrum in the decays $B^{+}\to
J/\psi\phi K^{+}$ [43]. The most significant state, $Z_{cs}(4000)$, has a mass
of $4003\pm 6{}^{+4}_{-14}\,\rm{MeV}$, a width of $131\pm 15\pm 26\,\rm{MeV}$,
and the spin-parity $J^{P}=1^{+}$, while the broader state, $Z_{cs}(4220)$,
has a mass of $4216\pm 24{}^{+43}_{-30}\,\rm{MeV}$, a width of $233\pm
52{}^{+97}_{-73}\,\rm{MeV}$, and the spin-parity $J^{P}=1^{+}$ or $1^{-}$,
with a $2\sigma$ difference in favor of the first hypothesis [43]. As there
exist two tetraquark nonets, see Eqs.(2)-(2), and there maybe exist mixings
between the two tetraquark nonets, so there are enough rooms to accommodate
the $Z_{cs}(3985)$ and $Z_{cs}(4000)$ as the diquark-antidiquark type
axialvector tetraquark states with strangeness.

## Acknowledgements

This work is supported by National Natural Science Foundation, Grant Number
11775079.

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