Eur. Phys. J. C (2013) 73:2661 DOI 10.1140/epjc/s10052-013-2661-x

Regular Article - Theoretical Physics

Could Zc (4025) be a J P = 1+ D ∗ D¯ ∗ molecular state? Chun-Yu Cui1,a , Yong-Lu Liu2 , Ming-Qiu Huang2 1 2

Department of Physics, School of Biomedical Engineering, Third Military Medical University, Chongqing 400038, China College of Science, National University of Defense Technology, Hunan 410073, China

Received: 13 October 2013 / Revised: 12 November 2013 / Published online: 3 December 2013 © Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica 2013

Abstract We investigate whether the newly observed narrow resonance Zc (4025) can be described as a D ∗ D¯ ∗ molecular state with quantum numbers J P = 1+ . Using QCD sum rules, we consider contributions up to dimension six in the operator product expansion and work at leading order of αs . The mass obtained for this state is (4.05 ± 0.28) GeV. It is concluded that the D ∗ D¯ ∗ molecular state is a possible candidate for Zc (4025).

1 Introduction The existence of charged states containing cc¯ represents an indisputable signal of the exotic states. There are three charged states, Z + (4050), Z + (4250), and Z + (4430), which were discovered by the Belle collaboration in B-decays [1, 2]. This discovery has triggered many theoretical investigations on the nature of this structure [3–9]. However, Babar did not confirm the existence of these three charged states [10, 11]. Recall that two charged bottomonium-like resonances Zb (10610) and Zb (10650) were observed by Belle Collaboration [12], most of theoretical investigations support the B ∗ B¯ (∗) molecular structure with J P = 1+ [13– 23]. That implies that there exist similar structures in charmonium-like energy regions. As anticipated, a charged charmonium-like structure, Zc (3900), was discovered by the BESIII experiment in the π ± J /ψ mass spectrum in the process e+ e− → π + π − J /ψ [24], and subsequently confirmed by the Belle experiment [25]. Based on heavy quark spin symmetry and heavy flavor symmetry, Guo et al. predict Zc (3900) as the D ∗ D¯ partner of the Zb (10610) [26]. Recently, the BECIII Collaboration reported a new enhancement structure Zc (4025) in the e+ e− → (D ∗ D¯ ∗ )± π ∓ √ at s = 4.26 GeV [27]. The mass and width of this state is M = (4026.3 ± 2.6 ± 3.7) MeV/c2 and Γ = (24.8 ± a e-mail:

[email protected]

5.6 ± 7.7) MeV/c2 . It attracts many attempts to investigate its possible configurations with various models [28– 33]. In Ref. [28], they give an explanation of Y (4260) → (D ∗ D¯ ∗ )− π + decay via the ISPE mechanism. In Ref. [29], the authors have studied the loosely bound D ∗ D¯ ∗ system with one-pion exchange model, which indicates that Zc (4025) is the ideal candidate of the D ∗ D¯ ∗ molecular state with J P = 1+ . We also notice that the charmoniumlike charged tetraquark state with J P = 1+ was studied using QCD sum rules. The mass obtained is about (4.05 ± 0.05) GeV [34]. Due to the asymptotic property of the QCD, studies of the hadron physics have to concern themselves with the nonperturbative effect which is difficult in quantum field theory. There are many methods to estimate the mass of a hadron, among which the QCD sum rule (QCDSR) [35–43] is a fairly reliable one. In Ref. [44], by assuming Zc (3900) as a D D¯ ∗ molecular state with J P = 1+ , we investigate the mass of this possible molecular configuration within the framework of QCDSR. Although the J P quantum numbers of Zc (4025) remain to be determined, it is still preferred to have spin parity J P = 1+ experimentally [24]. Following the opinion in Refs. [24, 29], we propose that Zc (4025) is a D ∗ D¯ ∗ molecular candidate with J P = 1+ , which is the Zb (10650)’s corresponding particle in the charmonium sector. It is difficult to construct a suitable axial-vector style molecular current interpolating the state Zc (4025) using both D ∗ and D¯ ∗ fields. A possible interpolating field is supposed to describe this kind of state D ∗ D¯ ∗ :     ¯ ¯ j μ (x) = ε μναβ u(x)iγ (1) ν c(x) Dα c(x)γ β d(x) . Performing the parity transformation to the current, it satisfies the condition (j μ )p = j μ . The rest of the paper is organized as follows. The QCDSR for the Zc (4025) state is derived in Sect. 2, with contributions up to dimension six in the operator product expansion (OPE). The numerical analysis is presented to extract the hadronic mass in Sect. 3, where a short summary and our conclusion are also presented.

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Eur. Phys. J. C (2013) 73:2661

2 QCD sum rules for Zc (4025)

light-quark masses mu and md . After some tedious calculations, the concrete forms of spectral densities can be derived:

In order to get the mass of the particle in the QCDSR approach, we start by considering the following two-point correlation function:      Π μν q 2 = i d 4 x eiq·x 0|T j μ (x)j ν+ (0) |0. (2) In consideration of the Lorentz covariance, the above correlation function can be generally decomposed as   μ ν     q μ q ν (0)  2  q q μν 2 μν (1) 2 Π q = Π q + 2 Π q . − g q2 q (3) We select the term proportional to gμν to extract the mass sum rule, since it only gets contributions from the 1+ state. The QCD sum rule method attempts to link the hadron phenomenology with the interactions of quarks and gluons. Three main ingredients are contained in the process when using this method: an approximate description of the correlation function in terms of intermediate states through the dispersion relation, an evaluation of the same correlation function in terms of QCD degrees of freedom via the operator product expansion (OPE), and a procedure for matching these two descriptions and extracting the parameters that characterize the hadronic state of interest. We can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators j μ into the correlation function to obtain the phenomenological expression. The coupling of the current with the state can be defined by the decay constant as follows: 0|jμ |Z = λ μ .

(4)

Therefore the invariant function Π (1) (q 2 ) can be expressed as    λ2 1 ∞ Im Π (1)phen (s) (1) 2 Π q = 2 + ds , (5) s − q2 MZ − q 2 π s 0 where MZ denotes the mass of the molecular state, and s0 is the threshold parameter. In the OPE side, Π (1) (q 2 ) can be written as  ∞   ρ OPE (s) Π (1) q 2 = ds , (6) s − q2 4m2c where the spectral density is ρ OPE (s) = π1 Im Π (1) (s). Applying the quark–hadron duality hypothesis with the Borel transformation, one obtains the following sum rule:  s0 2 2 2 λ2 e−MZ /M = ds ρ OPE (s)e−s/M , (7) 4m2c

with M 2 being the Borel parameter. Technically, we work at the leading order of αs and consider vacuum condensates up to dimension six, in use of the similar techniques in Refs. [45–47]. Considering the isospin breaking effect, we keep the terms which are linear in the

¯ ρ OPE (s) = ρ pert (s) + ρ qq (s) + ρ g

+ρ

qq ¯ 2

with ρ

pert

m2 (s) = 12 c 6 2 π



(s) + ρ αmax

αmin

dα α4

g 3 G3 



2 G2 

¯ ·Gq (s) + ρ g qσ (s)

(8)

(s),

1−α

βmin

dβ (1 − α − β)2 β3

× (7α + 4β + 8)r(mc , s)4  αmax  dα 1−α dβ 1 − 5 × 212 π 6 αmin α 4 βmin β 4  × 29α 3 + 60α 2 β + 39αβ 2 + 8β 3 − 6α 2  − 12αβ − 27α − 24β + 16 r(mc , s)5 ,  αmax  2 3 dα qq ¯ ¯ ρ qq mc − α(1 − α)s (s) = 6 4 mc 2 2 π αmin α(1 − α)  αmax  dα 1−α dβ 3qq ¯ 3 + 7 4 mc 2 2 2 π αmin α βmin β  2  × α + 3αβ + 2β 2 − 4α − 3β + 1 r(mc , s)2  αmax  dα 1−α dβ qq ¯ − 7 4 mc 2 3 2 π βmin β αmin α  2  2 × 4α + 8αβ + 3β − 10α − 4β + 2 × r(mc , s)3 , ρ

g 2 G2 

g 2 G2  (s) = 12 6 m4c 2 π



αmax

αmin 2

dα α4



1−α

dβ βmin

(9) × (1 − α − β) (2α + β + 1)r(mc , s)   αmax dα 1−α dβ g 2 G2  − 13 6 m2c 4 2 π βmin β αmin α  3 2 × 14α + 27α β + 16αβ 2 + 3β 3 + 14α 2  + 4αβ − 2β 2 − 20α − 5β + 4 r(mc , s)2 ,  αmax 3g qσ ¯ · Gq ¯ ·Gq ρ g qσ (s) = m dα αs c 27 π 4 αmin   × m2c − α(1 − α)s  αmax 3g qσ ¯ · Gq dα − m c 2 210 π 4 αmin α(1 − α)  2  2 2 × 7α − 25α + 10 mc − α(1 − α)s   3g s¯ σ · Gs 3 αmax dα 1−α dβ − m c α βmin β 28 π 4 αmin × (2 − 2α − β)r(mc , s)  αmax  dα 1−α dβ 3g s¯ σ · Gs − m c 2 2 29 π 4 αmin α βmin β × (α + β)r(mc , s)2 ,    qq ¯ 2 2 αmax qq ¯ 2 ρ (s) = − 4 2 mc dα m2c − α(1 − α)s . 2 π αmin

Eur. Phys. J. C (2013) 73:2661

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3 Numerical analysis and summary For numerical analysis of Eq. (7), we first specify the input parameters. The quark masses are chosen as mu = 2.3 MeV, md = 6.4 MeV, and mc = (1.23±0.06) GeV [48]. ¯ = qq The condensates are uu ¯ = dd ¯ = −(0.23 ± 3 2 3 0.03) GeV , g qσ ¯ · Gq = m0 qq, ¯ m20 = 0.8 GeV2 , 4 2 2 3 3 g G  = 0.88 GeV , and g G  = 0.045 GeV6 [40]. Complying with the standard procedure of the QCDSR, the threshold s0 and Borel parameter M 2 are varied to obtain the optimal stability window. There are two criteria (pole dominance and convergence of the OPE) for choosing the Borel parameter M 2 and threshold s0 . The contributions from the high dimension vacuum condensates in the OPE are shown in Fig. 1, as a function √ of M 2 . We have used s0 ≥ 4.5 GeV. From this figure it can be seen that for M 2 ≥ 2.0 GeV2 , the contribution of the dimension-6 condensate is less than 3 % of the total contribution and the contribution of the dimension-5 condensate is less than 13 % of the total contribution, which indicate the starting point for a good Borel convergence. Therefore, we fix the uniform lower value of M 2 in the sum rule win2 = 2.0 GeV2 . The upper limit of M 2 is dedow as Mmin termined by imposing that the pole contribution should be larger than continuum contribution. In Fig. 2, we show the M 2 dependence of the contributions from the pole terms. Ta2 for several values of √s . In ble 1 shows the values of Mmax 0 Fig. 3, we show the molecular state mass, for different val√ ues of s0 , in the relevant sum rule window. It can be seen that the mass is stable in the Borel window with the corre√ sponding threshold s0 . The final estimate of the J P = 1+ molecular state is obtained as MZ = (4.05 ± 0.28) GeV.

Fig. 1 The OPE convergence for the molecular state with contributions from different terms by varying the Borel parameter M 2 . The (A) and (B) correspond to contributions from the D = 6 term and the D = 5 term, respectively. The notations α, β, γ , λ and ρ corre√ spond to the threshold parameters s0 = 4.5 GeV, 4.6 GeV, 4.7 GeV, 4.8 GeV and 4.9 GeV, respectively

(10)

In summary, we construct a possible interpolator to describe the Zc (4025) as an axial-vector D ∗ D¯ ∗ molecular state. The QCD sum rule approach has been applied to calculate the mass of the resonance. Our numerical result is MD ∗ D¯ ∗ = (4.05 ± 0.28) GeV which is compatible with the experimental data of Zc (4025) by BECIII Collaboration. Thus it is concluded that Zc (4025) may be a D ∗ D¯ ∗ molecular state with quantum number J P = 1+ , which is supposed to be the charmonium-like partner of Zb (10650). Table 1 Upper limits in the Borel window for the J P = 1+ D ∗ D¯ ∗ current obtained from the sum rule √ for different values of s0

√ s0 (GeV)

2 Mmax (GeV2 )

4.5

2.62

4.6

2.75

4.7

2.91

4.8

3.12

4.9

3.22

Fig. 2 Contributions from the pole terms with variation of the Borel parameter M 2 . The notations α, β, γ , λ and ρ correspond to the thresh√ old s0 = 4.5 GeV, 4.6 GeV, 4.7 GeV, 4.8 GeV and 4.9 GeV, respectively

Page 4 of 4

Fig. 3 The mass of the molecular state as a function of M 2 from sum rule (7). The notations α, β, γ , λ and ρ correspond to the threshold pa√ rameters s0 = 4.5 GeV, 4.6 GeV, 4.7 GeV, 4.8 GeV, and 4.9 GeV, respectively

Acknowledgements This work was supported in part by the National Natural Science Foundation of China under Contract Nos. 11275268 and 11105222.

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