Interpretation of $Z_c(4025)$ as the Hidden Charm Tetraquark States via QCD Sum Rules

By using QCD Sum Rules, we found that the charged hidden charm tetraquark $[c u][\bar{c} \bar{d}]$ states with $ J^P = 1^-$ and $ 2^+$, which are possible quantum numbers of the newly observed charmonium-like resonance $Z_c(4025)$, have masses of $m_{1^-}^c = (4.54 \pm 0.20) \, \text{GeV}$ and $m_{2^+}^c = (4.04 \pm 0.19) \, \text{GeV}$. The contributions up to dimension eight in the Operator Product Expansion (OPE) were taken into account in the calculation. The tetraquark mass of $J^{P} = 2^{+}$ state was consistent with the experimental data of $Z_c(4025)$, suggesting the $Z_c(4025)$ state possessing the quantum number of $J^P = 2^+$. Extending to the b-quark sector, the corresponding tetraquark masses $m_{1^-}^b = (10.97 \pm 0.25) \, \text{GeV}$ and $m_{2^+}^b = (10.35 \pm 0.25) \, \text{GeV}$ were obtained, which are testable in future B-factories.


Introduction
A charged charmonium-like state Z c (4025) has been reported by BESIII in the process e + e − → (D * D * ) ± π ∓ [1]. Its mass and width are respectively (4026.3 ± 2.6 ± 3.7) MeV and (24.8 ± 5.7 ± 7.7) MeV. The mass and decay modes imply that it is a charged hidden charm state, which is similar to Z c (3900) [2,3,4]. In addition, the BESIII Collaboration recently announced another structure called Z c (3885) [5] in the invariant mass spectrum of (DD * ) ± . yielding significant conclusions. For J P = 1 − , an unstable mass sum rules was obtained [14], where the interpolating current was consistent with the 1 − current. However, stable results were extracted in Refs. [17,18]. This paper reanalyzes this case by adding several new ingredients [12] and performing moderate criteria [13], to determine the available threshold parameter √ s 0 and the Borel window M 2 B .
It should be noted that, very recently, the BESIII Collaboration has observed a charged charmonium-like resonance in the processes e + e − → (h c π ± )π ∓ , named Z c (4020) [19]. So far it is still too early to tell whether the Z c (4025) and the Z c (4020) are the same origin or not [20], though many theoretical investigations have already done [7,21,22,23,24,25,26]. argued that the Z c (4025) is a 2 + state, but being a D * D * bound state [23].
In Sec.II, various essential formulae are presented. Numerical analysis and mass extraction are shown in Sec.III, with conclusions given in Sec.IV.

Formalism
The QCD Sum Rules begin with the two-point correlation functions : The interpolating currents of 1 − and 2 + hidden charm tetraquark states are respectively constructed as: where, a, b, c, · · · , are color indices, and C represents the charge conjugation matrix.
For j 1 − µ (x), the correlation function has the following Lorentz covariance form: where the subscripts 1 and 0 respectively denote the quantum numbers of the spin 1 and 0 mesons.
On the phenomenological side, after separating out the ground state contribution from the pole term of the Π i (q 2 ), where i = 1 or 2, the correlation function is expressed as a dispersion integral over a physical regime, where m c i , λ c i and ρ h i (s) respectively represent the mass, decay constant, and spectral density of the tetraquark state. Here s 0 is the threshold of higher excited states and continuum states, and λ c i is defined in Refs. [12,27]. It is worth to note that the tetraquark state defined here couples to the four-quark interpolating field, which has a special structure differing from the D * D * configuration. And hence the D * D * threshold effect is neglected here.
On the OPE side of the Π i (q 2 ), the correlation function is expressed as a dispersion relation: where ρ OP E i (s) = Im[Π OP E i (s)]/π, and is expressed as: where the "· · · " stands for other higher dimension condensates omitted in our work.
To evaluate the spectral density of the OPE side, the "full" propagators S q ij (x) and S Q ij (p) of a light quark (q = u, d or s) and a heavy quark (Q = c or b) are respectively written with the vacuum condensates clearly displayed [9].
where the Lorentz indices κ ′ and λ ′ correspond to the indices of an outer gluon field from another propagator, and G ′ represents the outer gluon field [28].
Using the techniques of Refs. [13,12] the spectral density ρ OP E i (s) was calculated up to dimension eight at the leading order in α s .
The 1 − tetraquark state spectral density of the OPE side are given as: where M B is the Borel parameter introduced by the Borel transformation, s and the integration limits are given by For the 2 + tetraquark state: Matching the OPE side and the phenomenological side of the correlation function Π(q 2 ), i.e. the quark-hadron duality, and performing the Borel transformation, the sum rule for the mass of the hidden charm tetraquark state is determined to be: with where O 6 represents g 3 s G 3 or qq 2 for i = 1 or i = 2.

Numerical Analysis
In the numerical calculation, the values of the condensates and the quark masses are used as [13,29,30]: In the QCD Sum Rules, to select the appropriate threshold s 0 and the Borel parameter M 2 B , there are two criteria [8,9,11]. As the convergence of the OPE must be retained in order to determine their convergence, it is essential to compare the relative contributions of each term to the total contributions on the OPE side.
The second criterion to constrain the M 2 B is that the pole contribution (PC) must be larger than the continuum contribution. Thus, for various values of the M 2 B , it is necessary to analyze the relative pole contribution, defined as the pole contribution divided by the total contribution, i.e. pole plus continuum. To safely eliminate the contributions of the higher excited and continuum states, the PC is generally greater than 50% [11,13], which is slightly different from the constraint used in [14].
To find a proper value for √ s 0 , we carry out a similar analysis as in Refs. [11,31]. Since the continuum threshold is connected to the mass of the studied state by the relation √ s 0 ∼ m c i + 0.5 GeV, various √ s 0 satisfying this constraint are taken into account. Among these values, one needs then to find out the proper one which has an optimal window for Borel parameter M 2 B . That is, within this window, the physical quantity, here the tetraquark mass m c i , is independent of the Borel parameter M 2 B as much as possible. Through the above procedure one obtains the central value of √ s 0 . However, in practice, in the QCD Sum Rules calculation, it is normally acceptable to vary the √ s 0 by 0.1 GeV [31], which gives the lower and upper bounds and hence the uncertainties of √ s 0 .

1 − Hidden Charm Tetraquark State
The OPE convergence of the 1 − hidden charm tetraquark state is shown in Fig.1, which reflects a strong OPE convergence for M 2 B ≥ 2.1 GeV 2 , making it possible to determine the lower limit constraint of the M 2 B .
The result of the PC is shown in Fig.2, which indicates the upper limit constraint of the To determine an appropriate value of the s 0 , a similar analysis is utilized as was applied in Ref. [13,18]. Thus, for the s 0 , √ s 0 = 5.0 GeV, the M 2 B ≤ 3.4 GeV 2 .   The mass of the 1 − hidden charm tetraquark state was determined to be: where M 2 B was 3.4 GeV 2 , and the errors stemmed from the uncertainties of the charm quark mass, the condensates and the threshold parameter √ s 0 .

2 + Hidden Charm Tetraquark State
For the 2 + sector, the OPE convergence is shown in Fig.4, which reflects a strong OPE The dependence of m c 2 + is drawn on the parameter τ in Fig.6, where τ = 1/M 2 B , and the continuum thresholds √ s 0 are respectively taken as 3.9, 4.2, 4.5, 4.8, and 5.1 GeV, from down to up. In Fig.6, the optimal mass curve of the 2 + hidden charm tetraquark state is shown with √ s 0 = 4.5 GeV, where both the aforementioned criteria are satisfied.
The mass of the 2 + hidden charm tetraquark was determined to be:

Conclusions
In this paper, we estimated the masses of the hidden charm tetraquark states with J P = 1 − and 2 + , which are possible quantum numbers possessed by the charmonium-like resonance Z c (4025). In our calculations contributions up to dimension eight in the OPE were taken into account. Noticeably, with the 1 − current, the mass was found to be m c 1 − = (4.54±0.20) GeV, so we deduced that such a tetraquark structure was not the candidate for Z c (4025). As in the discussions in Ref. [14], it may correspond to the charged partner of the charmonium-like state Y (4360) or Y (4660), within the uncertainties. However, in the case of 2 + , we found that m c 2 + = (4.04 ± 0.19) GeV, which is consistent within the errors with the experimental data of the Z + c (4025) resonance.
As was mentioned in the introduction, the existing analyses favor Z c (3900) having quantum number of J P = 1 + , and these calculations can not discriminate Z c (3900) with the Z c (3885). Possibly they may have the same origin. In this work, we calculate the masses of J P = 2 + tetraquark states in the framework of QCD Sum Rules. Our result suggests that the mass of hidden charm tetraquark state is a bit more than 4 GeV, which in certain degree agrees with the recent observations of Z c (4025) or Z c (4020) by BESIII Collaboration. For Addentum: during the finalization of this work, two reports about Z c (4025) appear, wherein the molecular picture [32] and initial-single-pion-emission mechanism [33] were employed.