$D^*\bar D^*$ molecule interpretation of $Z_c(4025)$

We have used QCD sum rules to study the newly observed charged state $Z_c(4025)$ as a hidden-charm $D^*\bar D^*$ molecular state with the quantum numbers $I^G(J^{P})=1^+(1^{+})$. Using a $D^*\bar D^*$ molecular interpolating current, we have calculated the two-point correlation function and the spectral density up to dimension eight at leading order in $\alpha_s$. The extracted mass is $m_X=(4.04\pm0.24)$ GeV. This result is compatible with the observed mass of $Z_c(4025)$ within the errors, which implies a possible molecule interpretation of this new resonance. We also predict the mass of the corresponding hidden-bottom $B^*\bar B^*$ molecular state: $m_{Z_b}=(9.98\pm0.21)$ GeV.


I. INTRODUCTION
After the observation of charged charmonium-like resonance Z c (3900) [1], the BESIII Collaboration recently discovered another charged structure Z c (4025) in the process e + e − → (D * D * ) ± π ∓ [2]. This new resonance, which has a mass of M = (4026.3 ± 2.6 ± 3.7) MeV, lies very close to the D * D * threshold. Its width is Γ = (24.8 ± 5.6 ± 5.7) MeV [2]. To date, the experiment has not determined the quantum numbers of the Z c (4025) resonance. Since it was observed in both the D * D * and the h c π channels, the quantum numbers of the charged Z c (4025) was argued to be I G (J P ) = 1 + (1 + ) while its neutral partner carries negative C-parity [3].
Similar to the other charged charmonium-like states Z + (4050), Z + (4250) [4], Z + (4430) [5] and Z c (3900) [1], Z c (4025) cannot be a conventional cc state due to the charge it carries. Molecular and tetraquark configurations have recently been used to explore its underlying structure [3]. In Ref. [3], the authors have studied the mass spectrum of Z c (4025) and its pionic and radiative decays as a D * D * molecular state using the one-bosonexchange (OBE) model. They have also studied the Y (4260) → (D * D * ) − π + decay through the initial-singlepion-emission mechanism in Ref. [6]. Z c (4025) was also studied as a [cu][cd] tetraquark with the quantum numbers J P = 2 + using QCD sum rules [7]. In Ref. [8], the D * D * J P = 1 + molecular current with a derivative has been studied in QCD sum rules and the extracted mass coincides with Z c (4025).
There also exist other theoretical predictions of this * Electronic address: wec053@mail.usask.ca † Electronic address: tom.steele@usask.ca ‡ Electronic address: duml@pku.edu.cn § Electronic address: zhusl@pku.edu.cn new charged structure before its observation by BE-SIII [9][10][11]. Ref. [11] studied the I G (J P C ) = 1 + (1 +− ) charmonium-like tetraquark states, and predicted masses near the D * D * threshold and the possible decay patterns including the open-charm modes DD * , D * D * and other hidden-charm modes. Up to now, the BESIII Collaboration has not reported the DD * decay mode of Z c (4025). Right now, it seems that the D * D * molecule interpretation is slightly more natural.
At the hadronic level, the molecular states are commonly assumed to be bound states of two hadrons formed by the exchange of the color-singlet mesons. This configuration is very different from that of the tetraquark states, which are generally bound by the QCD force at the quark-gluon level. In this work, we study Z c (4025) as a D * D * molecular state using QCD sum rules approach [12][13][14].
Within the framework of the QCD sum rule, all the procedures such as the operator product expansion, the calculation of the Wilson coefficient and the Borel transform are very similar for the molecular-type current and tetraquark-type current. In principle, if we exhaust all the possible molecular-type currents and all the possible tetraquark-type of currents, we can rigorously show that these two sets of interpolating currents are equivalent by using a Fierz rearrangement [15,16].
However, there exists an important difference between one single molecular-type current and one single tetraquark-type current. By Fierz rearrangement, every single tetraquark-type current can be expressed as a linear combination of several (sometimes up to five) independent molecular-type currents. We can decompose the tetraquark interpolating current into these explicit molecular-type operators. In the single molecular-type QSR, the color flow of the correlation function is quite simple and forms two closed loops. In the tetraquark correlator, there exist additional contributions from the non-diagonal correlator besides the many diagonal corre-lators as in the molecular-type QSR. Now the color flow is complicated, which is the interference and transition between different molecular structures [17]. In this respect, one well-known example is the light scalar-isoscalar sigma meson. The tetraquark-type current (or their combination/mixing) leads to a better mass prediction than the simple pion-pion molecular current [15].
The paper is organized as follows. In Sect. II, we calculate the correlation function and spectral density using the D * D * molecule current. In Sect. III, we perform a numerical analysis and extract the mass of Z c (4025). The last section is a brief summary.

II. QCD SUM RULE AND SPECTRAL DENSITY
The starting point of QCD sum rules is the two-point correlation function where J µ (x) is the D * D * molecular interpolating current with I G (J P ) = 1 + (1 + ) (2) in which a, b are color indices and q denotes an up or down quark. In principle, the anti-symmetric tensor operator q a σ αµ γ 5 c a can couple to both J P = 1 + (q a σ 0i γ 5 c a components) and J P = 1 − (q a σ ij γ 5 c a components) channels. However, we can pick out the 1 − piece by multiplication with the vector operatorc b γ α q b so that the molecular operator (q a σ αµ γ 5 c a )(c b γ α q b ) carries the quantum numbers J P = 1 + after contracting the Lorentz index. The molecule current in Eq.(2) contains both the charged components with (ūc)(cd) and (dc)(cu) pieces and the neutral component with (ūc)(cu) and (dc)(cd) pieces. For the neutral component, it carries negative C-parity and the quantum numbers should be I G (J P C ) = 1 + (1 +− ). However, we do not differentiate between u and d quarks in our analysis, so the charged component and the neutral component are the same in QCD sum rules due to isospin symmetry. The correlation function in Eq. (1) can be written as two independent Lorentz structures since J µ is not a conserved current: in which the invariant functions Π 1 (q 2 ) and Π 0 (q 2 ) are related to the spin-1 and spin-0 mesons, respectively. We focus on Π 1 (q 2 ) to study the 1 + channel in this work. The correlation function in Eq. (1) can be obtained at both the hadron level and the quark-gluon level. To determine the correlation function at the hadron level, we use the dispersion relation where b n is the unknown subtraction constant which can be removed by taking the Borel transform. The lower limit of integration is the square of the sum of the masses of all current quarks (omitting the light quark mass). ρ(s) is the spectral function Here we adopt the pole plus continuum parametrization of the hadronic spectral density. The intermediate states |n must have the same quantum numbers as the interpolating currents J µ . |X is the lowest lying resonance with mass m X and it couples to the current J µ via the coupling parameter f X where ǫ µ is the polarization vector (ǫ · q = 0). At the quark-gluon level, the correlation function can be calculated in terms of quark and gluon fields via the operator product expansion (OPE) method. We evaluate the correlation function up to dimension-eight condensate contributions at leading order in α s using the same technique as in Refs. [11,[18][19][20]. The spectral density is then obtained: ρ(s) = 1 π ImΠ(q 2 ). Sum rules for the hadron parameters are established by equating the correlation functions obtained at both the hadron level and quark-gluon level via quark-hadron duality. The Borel transform is applied to the correlation functions at both levels to remove the unknown constants in Eq. (4), suppress the continuum contribution, and improve the convergence of the OPE series. Using the spectral function defined in Eq. (5), the sum rules can be written as where s 0 is the continuum threshold parameter and M B is the Borel mass. Then m X can be extracted by the ratio In the following, we study the lowest lying hadron mass m X in Eq. (8) as function of the continuum threshold s 0 and Borel mass M B . We calculate the spectral density at the quark-gluon level including the perturbative term, quark condensate qq , gluon condensate g 2 s GG , quark-gluon mixed condensate qg s σ · Gq , four quark condensate qq 2 and the dimension eight condensate qq qg s σ · Gq : ρ(s) = ρ pert (s) + ρ qq (s) + ρ GG (s) + ρ qq 2 (s) +ρ qGq (s) + ρ qq qGq (s), where αs−m 2 c , β max = 1 − α, m c is the charm quark mass, and H(α) is the Heaviside step function. As is evident from the above expressions, our calculations are of leading order in α s . Both the quark condensate qq and the quark-gluon mixed condensate qg s σ · Gq are proportional to the charm quark mass m c . They give important power corrections to the correlation functions. We ignore the chirally suppressed terms proportional to the light quark mass. Based on Ref. [18] the contribution of the three gluon condensate g 3 s f GGG expected to be numerically small and has not been included in this work. The dimension-eight condensate qq qg s σ · Gq contains the delta function δ s − m 2 c α(1−α) and its derivative. These terms compensate for the singular behavior of the spectral densities at the s = 4m 2 c threshold.

III. NUMERICAL ANALYSIS
The following QCD parameters are used in our analysis [21][22][23][24][25]: where the charm and bottom quark masses are the running mass in the M S scheme. As mentioned earlier, we set the light quark masses m q = 0 in the analysis. The convention for the mixed condensate is consistent with Refs. [11,[18][19][20], which have a sign difference from some other QCD sum rule studies because of the definition of the coupling constant g s . We define the pole contribution (PC) using the sum rules established in Eq. (7), which is the function of the continuum threshold s 0 and the Borel mass M B . PC represents the lowest lying resonance contribution to the correlation function, which also includes the continuum and higher state contributions with s 0 → ∞. We begin with the analysis by determining the Borel window. A good mass sum rule requires a suitable working region of the Borel scale M B . To obtain the lower bound on M 2 B , we let s 0 → ∞ and then study the OPE convergence in Fig. 1. One notes that the quark condensate qq contribution is much bigger than other condensates and is therefore the dominant power correction. Besides the quark condensate, the quark-gluon mixed condensate qg s σ·Gq also gives a significant contribution to the correlation function. From the expression for the spectral density in Eq. (11), the quark condensate and quark-gluon mixed condensate are proportional to the charm quark mass. The gluon condensate g 2 s GG , four quark condensate qq 2 and dimension-eight condensate qq qg s σ · Gq are smaller. However, they also give important corrections to the correlation function and stabilize the mass sum rules. Requiring the quark condensate contribution be less than one third of the perturbative term contribution, while the quark-gluon mixed condensate contribution be less than one third of the quark condensate contribution, we obtain the lower bound on the Borel window M 2 min = 4.3 GeV 2 . One may notice from Fig. 1 that the power corrections are small enough in the parameter region M 2 B ≥ 4.3 GeV 2 so that the OPE convergence is very good.  The continuum threshold s 0 is also an important parameter in QCD sum rules. An optimized choice of s 0 is the value minimizing the variation of the extracted hadron mass m X with the Borel mass M 2 B . This is achieved by studying the variation of m X with s 0 in Fig.  2 by varying the value of Borel mass from its lower bound M 2 min . One notes that these curves with a different value of M 2 B intersect at s 0 = 19 GeV 2 , around which the variation of m X with M 2 B is minimum. Then the upper bound on the Borel mass can be determined by studying the pole contribution defined in Eq. (12). We require that the pole contribution be larger than 10%, which results in the upper bound on the Borel mass M 2 max = 4.9 GeV 2 . We obtain the Borel window 4.3 GeV 2 ≤ M 2 B ≤ 4.9 GeV 2 with the threshold value s 0 = 19 GeV 2 . Now we can perform the QCD sum rule analysis in the Borel window 4.3 GeV 2 ≤ M 2 B ≤ 4.9 GeV 2 . In Fig. 3, we show the variation of the extracted mass m X with the Borel mass M 2 B using continuum thresholds s 0 = 17 GeV 2 , 19 GeV 2 and 21 GeV 2 respectively. The mass curves are very stable in the Borel window around these threshold values. Finally, we extract the hadron mass: which is very well compatible with the mass of Z c (4025). This implies the possible D * D * molecule interpretation of this new resonance. Using this value of the hadron mass, we can also calculate the coupling parameter defined in Eq. (6), This parameter represents the strength of the coupling of the current J µ in Eq. (2) to the Z c (4025) resonance. The errors of our numerical results in Eq. (13) and Eq. (14) involve the uncertainties in the heavy quark masses and the values of the quark condensate, quark-gluon mixed condensate, and gluon condensate in Eq. (12). Other possible error sources such as truncation of the OPE series, the uncertainty of the continuum threshold value s 0 and the variation of the Borel mass M B are not taken into account. We can extend the analysis to the hidden-bottom Z b system, where Z b represents a B * B * molecular state with I G (J P ) = 1 + (1 + ). Using the same interpolating current in Eq. (2), we repeat all the above analysis procedures with the replacement m c → m b . To find a suitable working region of the Borel scale, we use the same criteria as in the D * D * system to study the OPE convergence and pole contribution. We find a Borel window 9.3 GeV 2 ≤ M 2 B ≤ 11.6 GeV 2 for the continuum threshold value s 0 = 107 GeV 2 . We show the Borel curves of the extracted Z b mass with s 0 and M 2 B in Figs. 4 and 5, respectively. In Fig. 4, the optimized value of the continuum threshold is chosen as s 0 = 107GeV 2 , which minimize the variation of the Z b mass m Z b with the Borel parameter M 2 B . This result is also shown in Fig. 5, in which the mass curve is very stable as a function of M 2 B in the obtained Borel window. Considering the same error sources as the D * D * system, we predict the mass and the coupling parameter of the Z b state to be m Z b = (9.98 ± 0.21) GeV, (15)

IV. SUMMARY
The BESIII Collaboration has discovered Z c (4025) in the process e + e − → (D * D * ) ± π ∓ near the D * D * threshold. This new structure is a charged resonance and thus cannot be a conventional charmonium state. It is thus a candidate for an exotic hadron state.
In Ref. [8], a D * D * molecular interpolating current with a derivative operator has been used to investigate the structure of Z c (4025) in QCD sum rules. In this paper, we use a different hidden-charm D * D * current with the quantum numbers I G (J P ) = 1 + (1 + ). We have calculated the correlation function and the spectral density up to dimension eight at leading order in α s , including the perturbative term, quark condensate qq , quark-gluon mixed condensate qg s σ · Gq , gluon condensate g 2 s GG , and the dimension-eight condensate qq qg s σ · Gq contributions. The quark condensate and the quark-gluon mixed condensate are proportional to the charm quark mass and are a larger contribution than the other condensates. The quark condensate is the dominant power correction to the correlation function. Other condensates are also important because they can improve the OPE convergence and stabilize the mass sum rules.
After performing the QCD sum rule analysis, we extract the hadron mass m X = (4.04±0.24) GeV consistent with BESIII's result of the mass of the Z c (4025). Our result supports Z c (4025) resonance as an axial-vector D * D * molecular state. In principle, our result also contains the neutral partner of Z c (4025) with the quantum numbers J P C = 1 +− . However, it has the same mass with the charged state in QCD sum rules due to isospin symmetry. We have also studied the corresponding hidden-bottom B * B * molecular state and predicted the mass m Z b = (9.98 ± 0.21) GeV. Hopefully our investigation will be useful for the understanding of the structure of the newly observed charged state Z c (4025) and the future search of its neutral partner.