$\bar{D}^{(*)}_{s}D^{(*)}$ molecular state with $J^{P}=1^{+}$

In this paper, we construct $\bar{D}^{(*)}_{s}D^{(*)}$-molecule-type interpolating currents $J_{(\pm)\mu}(x)$ with $J^{P}=1^{+}$, calculate the corresponding mass and magnetic moment using the QCD sum rule method and its extension in the weak electromagnetic field, and study the processes of $Z_{(\pm)cs}$ to $\eta_{c}K^{*}$, $J/\psi K$, $\bar{D}D^{*}_{s}$, and $\bar{D}^{*}D_{s}$ via three-point sum rules. The numerical values are $m_{Z_{(\pm)cs}}=3.99^{+0.17}_{-0.14}~\mbox{GeV}$, and $\lambda_{Z_{(\pm)cs}}=2.07^{+0.28}_{-0.16}\times10^{-2}~\mbox{GeV}^{5}$, $\mu_{Z_{(\pm)cs}}=0.18^{+0.16}_{-0.09}~\mu_{N}$ with $\mu_{N}$ the nucleon magneton, $\Gamma_{Z_{(+)cs}}=17.47^{+12.70}_{-8.08}$, and $\Gamma_{Z_{(-)cs}}=13.86^{+10.37}_{-6.51}$. The masses are in agreement with the recently measured value of $Z_{cs}(3985)$ by the BESIII Collaboration, $m^{exp}_{Z_{cs}}=(3982.5^{+1.8}_{-2.6}\pm2.1)~\mbox{MeV}$. The widths are compatible with the experimental value, $\Gamma^{exp}_{Z_{cs}}=(12.8^{+5.3}_{-4.4}\pm3.0)~\mbox{MeV}$. The magnetic moment and the various decay modes can help us to determine the inner structure of $Z_{cs}(3985)$ when being confronted with experimental data in the future.

In the present work, we constructD ( * ) s D ( * ) -molecule-type interpolating currents J (±)µ (x) with J P = 1 + , calculate the corresponding mass and magnetic moment using the QCD sum rule method and its extension in the weak electromagnetic field, and study their decay properties via three-point sum rule. The QCD sum rule method [35,36] is a nonperturbative analytic formalism firmly entrenched in QCD with minimal modeling and has been successfully applied in almost every aspect of strong interaction physics. Its extension in the weak electromagnetic field can be used to calculate the magnetic moment of ground hadron states [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. The electromagnetic multipole moments of the hadron encode the spatial distributions of charge and magnetization in the hadron and provide important information about the quark configurations of the hadron and the underlying dynamics. So it is interesting to study the electromagnetic multipole moments of the hadron.
The rest of the paper is organized as follows. In Sec.II, the relevant sum rules are derived. Section III is devoted to the numerical analysis, and a short summary is given in Sec.IV. In Appendix B, the spectral densities are shown.

A. Mass and magnetic moment
First, we write down the molecule-type interpolating currents with J P = 1 + : which can couple to theD ( * ) s D ( * ) molecular state (labeled as Z (±)cs ), and the coupling strength can be parameterized as follows: 0|J (±)µ (0)|Z (±)cs (p) = λ Z (±)cs ǫ (±)µ (p) (2) with λ Z (±)cs and ǫ (±)µ (p) being the pole residue and polarization vector of Z (±)cs state, respectively. The charge, magnetic, and quadrupole form factors of the Z (±)cs state are related to three functions-G 1 (Q 2 ), G 2 (Q 2 ), and G 3 (Q 2 ): with η = Q 2 4m 2 Z (±)cs and the functions G (±)1 (Q 2 ), G (±)2 (Q 2 ), and G (±)3 (Q 2 ) defined as with q = p ′ − p and Q 2 = −q 2 . At zero momentum transfer, these form factors are proportional to the usual static quantities of the charge e, magnetic moment µ Z (±)cs , and quadrupole moment Q (±)1 : To derive the needed sum rules, we begin with the time-ordered correlation function in the QCD vacuum in the presence of a constant background electromagnetic field F µν : where J (±)µ (x) is the interpolating current of Z (±)cs state (1). The Π (0) (±)µν (p) term is the correlation function without an external electromagnetic field, and gives rise to the mass and pole residue of Z (±)cs . The magnetic moment will be extracted from the linear response term Π (1) (±)µναβ (p)F αβ . Following the method stated in Ref. [50,51], we express physically the correlation function (6) as where the constant a parametrizes the contributions from the pole-continuum transitions. In Eq. (6), we substitute J (±)µ (x) with Eq. (1), contract the relevant quark fields via Wick's theorem, and obtain where S (c) (x) = 0|T [c(x)c(0)]|0 and S (q) (x) = 0|T [q(x)q(0)]|0 , q = u, s are the full charm-and up (strange)-quark propagators, respectively, whose expressions are given in Appendix A, T r denotes the trace of the Dirac spinor indices, and a, b, c, and d are color indices. Through dispersion relation, Π OP E (±)µν (p) can be written as where ρ i (±) (s) = 1 π ImΠ OP E (±)i (s), i = 0, 1 are the spectral densities, and m c and m s are the masses of the charm and strange quark, respectively. We find that ρ i (+) (s) = ρ i (−) (s), i = 0, 1, and do not distinguish the subscripts ± in the rest of this subsection. The expressions of ρ i (s), i = 0, 1, are given in Appendix B.
Finally, matching the phenomenological side (7) and the QCD representation (9), we obtain for the Lorentz structure (−g µν + pµpν p 2 ), and for the Lorentz structure iF µν . According to quark-hadron duality, the excited and continuum states' spectral density can be approximated by the QCD spectral density above some effective threshold s 0 Zcs , whose value will be determined in Sec.III: Subtracting the contributions of the excited and continuum states, one gets In order to improve the convergence of the operator production expansion (OPE) series and suppress the contributions from the excited and continuum states, it is necessary to make a Borel transform. As a result, we have Zcs e −m 2 where M 2 B is the Borel parameter and A = a λ 2

Zcs
. Taking the derivative of the first equation in (14) with respect to − 1 and dividing it by the original expression, one has

B. Strong decay form factors
In this subsection, we calculate the strong decay form factors of Z (±)cs to η c K * , J/ψK, DD * s , andD * D s . To this end, we start with the following three-point functions: We will take Γ 1 (+)µν (p, p 1 , p 2 ) as an example to illustrate the steps involved in our calculation. Inserting complete sets of hadronic states into the three-point correlation function, we have where we make use of Eq.(2) and the following matrix elements: The constant A represents the transition between ground state and excited states, which cannot be ignored.
On the other hand, we can calculate Γ 1 (+)µν (p, p 1 , p 2 ) theoretically. Substituting the interpolating currents in Γ 1 (+)µν (p, p 1 , p 2 ) with their explicit expressions and contracting the quark fields via Wick's theorem, we obtain Substituting the quark propagators with their explicit expressions and carrying out the integrals, we obtain the theoretical side of the correlation function: where the coefficient Γ 1OP E (+) (p 2 , p 2 1 , p 2 2 ) can be written as with ρ 1 3(+) (p 2 , t 1 , t 2 ) being the QCD special density. The explicit expressions of ρ 1 3(+) (p 2 , t 1 , t 2 ) are given in Appendix B.
It is time to match the physical representation (18) and theoretical representation (21) of Γ 1 (+)µν (p, p 1 , p 2 ). For the Lorentz structure g µν , we have, by using the quark-hadron duality, where t 0 1 and t 0 2 are the threshold parameters in the channels of η c and K * , respectively. Setting p 2 = p 2 1 and performing Borel transforms p 2 1 → M 2 B1 and p 2 2 → M 2 B2 , one gets Acting on the above equation with operator m 2 ηc − We can study similarly other three-point correlation functions, and the corresponding spectral densities are given in Appendix B.

III. NUMERICAL ANALYSIS
The input parameters needed in numerical analysis are qq = −(0. 24 24 GeV, and f Ds = 0.24 GeV, which can be found in Ref. [21]. For the vacuum susceptibilities χ, κ, and ξ, we take the values χ = −(3.15 ± 0.30) GeV −2 , κ = −0.2, and ξ = 0.4 determined in the detailed QCD sum rules analysis of the photon light-cone distribution amplitudes [52]. Besides these parameters, we should determine the working intervals of the threshold parameters and the Borel mass in which the mass, the pole residue, the magnetic moment, and strong decay form factors vary weakly. The continuum threshold is related to the square of the first excited states having the same quantum number as the interpolating field, while the Borel parameter is determined by demanding that both the contributions of the higher states and continuum are sufficiently suppressed and the contributions coming from higher-dimensional operators are small.
We define two quantities-the ratio of the pole contribution to the total contribution (RP) and the ratio of the highest-dimensional term in the OPE series to the total OPE series (RH) as follows: with i = 0, 1. Similar quantities can be defined for three-point correlation functions. In Fig.1(a), we compare the various terms in the OPE series as functions of M 2 B with s 0 Zcs = 4.5 GeV. From it one can see that, except the quark condensate qq , other vacuum condensates are much smaller than the perturbative term. So the OPE series are under control. Figure 1(b) shows RP 0 and RH 0 varying with M 2 B at s 0 Zcs = 4.5 GeV. The figure shows that the requirement RP 0 > 50% gives M 2 B ≤ 3.05 GeV 2 and |RH 0 | < 10% when M 2 B ≥ 1.80 GeV 2 , which is the lower limit of M 2 B .
where µ N is the nucleon magneton. This result can be confronted with experimental data in the future and give important information about the inner structure of the Z cs (3985) state.
We summarize our results about the strong decays of Z (±)cs to η c K * , J/ψK,DD * s , and D * D s in Table I

IV. CONCLUSION
In this paper, we constructD ( * ) s D ( * ) -molecule-type interpolating currents J (±)µ (x) with J P = 1 + , calculate the corresponding mass and magnetic moment using the QCD sum rule method and its extension in the weak electromagnetic field, and study the processes of Z (±)cs to η c K * , J/ψK,DD * s , andD * D s via three-point sum rules. Starting with the two-point correlation function in the external electromagnetic field and expanding it in power of the electromagnetic interaction Hamiltonian, we extract the masses and pole residues of Z (±)cs states from the leading term in the expansion and the magnetic moments from the linear response to the external electromagnetic field. The numerical values are m Z (±)cs = 3.99 +0. 17 −0.14 GeV, λ Z (±)cs = 2.07 +0.28 for light quarks, and for heavy quarks. In these expressions, t a = λ a 2 and λ a are the Gell-Mann matrix, g s is the strong interaction coupling constant, i and j are color indices, e Q(q) is the charge of the heavy (light) quark, and F µν is the external electromagnetic field. On the QCD side, we carry out the OPE up to dimension 8 for the spectral densities ρ (0) (s) and ρ (1) (s). The explicit expressions of the spectral densities are given below: In the above equations, a max =

The spectral densities of the three-point correlation function
We will give the explicit expressions of the three-point correlation functions in this subsection.
For Γ 1 (+)µν (p, p 1 , p 2 ), the spectral density is For Γ 1 (−)µν (p, p 1 , p 2 ), we find that the perturbative part and quark-condensation part contain only Lorentz structure p 1ν p 2µ . As a result, we choose this structure to obtain our sum rule for the strong decay form factor g 1(−) . The corresponding spectral density is .
The spectral densities are proportional to either m s , qq − ss or g sq σGq − g ss σGs .