Interpretation of $Z_b(10610)$ and $Z_b(10650)$ in the ISPE mechanism and the Charmonium Counterpart

The initial single pion emission (ISPE) mechanism is applied to the processes $\Upsilon(5S)\to \pi B^{(*)}\bar{B}^{(*)}$ whose details have been recently reported at ICHEP2012 and we obtain reasonable agreement with Bell's measurements, i.e., we succeed in reproducing the enhancement structures of $Z_b(10610)$ and $Z_b(10650)$. Inspired by this success, we predict the corresponding enhancement structures in higher charmonia open charm pion decay near the thresholds of $D^\ast \bar{D}$ and $D^\ast \bar{D}^\ast$.

Observation of these two structures has inspired theorists with the extensive interests. Various theoretical explanations were proposed after the Belle's observation. In the following, we will briefly review the research status of Z b (10610) and Z b (10650).
Considering that Z b (10610) and Z b (10650) are charged and close to BB * and B * B * thresholds, respectively, many theoretical efforts have been made to answer the question whether these newly observed structures are the real exotic states or not. Before the discovery of Z b (10610) and Z b (10650), the authors in Refs. [3,4] predicted the existence of loosely bound S-wave BB * molecular states. The heavy quark spin structure by Bondar et al. [5], study using the chiral constituent quark model in Ref. [6], the effective Lagrangian approach via the one-boson exchange in Ref. [7], and study on the line shape in the vicinity of B ( * )B( * ) thresholds as well as two-body decay rates using the effective field theory in Ref. [8], all showed that Z b (10610) and Z b (10650) can be the BB * and B * B * molecular states, respectively. The authors in Ref. [6] further showed that their quantum numbers are I(J PC ) = 1(1 +− ). The QCD sum rule (QSR) analysis by Zhang et al. [9] suggested † corresponding author * Electronic address: chendy@impcas.ac.cn ‡ Electronic address: xiangliu@lzu.edu.cn § Electronic address: matsuki@tokyo-kasei.ac.jp that Z b (10610) could be a BB * molecular state. Using the Bethe-Salpeter equation, the problem whether Z b (10610) is a BB * molecular state was studied in Ref. [10]. They claimed that BB * molecular state with isospin I = 1 cannot be formed when the contribution of σ exchange is small [10]. Apart from these studies of mass spectrum just mentioned above, there are some theoretical papers dedicated to the production and decay behavior of Z b (10610) and Z b (10650). Under the frameworks of BB * and B * B * molecular states, the radiative decay of Υ(5S ) into molecular bottomonium was calculated [11], and the processes of Z b (10610) and Z b (10650) decaying into bottomonium and pion were also investigated very recently [12]. In Ref. [13], the properties of Z b (10610) and Z b (10650) were studied assuming that Z b (10610) and Z b (10650) are the BB * and B * B * molecular states. Dong et al. [14] performed the calculation of molecular hadrons, Z b (10610) and Z b (10650), decaying into Υ(nS ) and π + by the effective Lagrangian approach.
In addition, tetraquark explanation for Z b (10610) and Z b (10650) was proposed. In Ref. [15], the masses of tetraquark states bubd and bdbū with J P = 1 + were obtained by the chromomagnetic interaction Hamiltonian, which are compatible with the corresponding masses of Z b (10610) and Z b (10650). Using the QSR approach, the authors in Ref. [16] calculated the mass of the tetraquark states with the configuration [bd][bū] and found that Z b (10610) and Z b (10650) can be described by tetraquark. Ali et al. also gave tetraquark interpretation for Z b (10610) and Z b (10650) and studied the decay of tetraquark state Y b (10890) into Z b (10610) ± π ∓ or Z b (10650) ± π ∓ , and the decays of Z b (10610)/Z b (10650) into π ± Υ(nS ) and π ± h b (mP) [17].
Although there have been many theoretical efforts to clarify Z b (10610) and Z b (10650), further study on these two Z b states is still an interesting research topic. For instance, it is crucial how to distinguish different explanations for Z b (10610) and Z b (10650). Very recently, the Belle Collaboration has reported new results on Z b (10610) and Z b (10650) at the ICHEP2012 conference that these Z b structures also exist in the BB * and B * B * invariant mass spectra of Υ(5S ) → πBB * , πB * B * decays [22]. This new experimental phenomenon of Z b (10610) and Z b (10650) can provide an important platform to test explanations for Z b (10610) and Z b (10650) proposed so far and this process also reminds us the ISPE mechanism.
In this work, we will explain why two charged structures Z b (10610) and Z b (10650) can appear in the BB * and B * B * invariant mass spectra of the Υ(5S ) → πBB * , πB * B * decays. We find that the ISPE mechanism proposed in Ref. [21] can be well applied to the Υ(5S ) → πBB * , πB * B * processes, which can further test this mechanism. Other than explaining the Belle's new observation, we will extend our study to the open-charm decays of higher charmonia with the emission of a single pion because of the similarity between bottomonium and charmonium [23]. As a result of our study, we will give the corresponding prediction of two charged charmonium-like structures close to the D * D and D * D * thresholds, which can be found in the invariant mass spectra m D * D and m D * D * of the open-charm decays of higher charmonia with the emission of a single pion.
This work is organized as follows. After introduction, we introduce the ISPE mechanism and its application to Υ(5S ) → πBB * , πB * B * decays in the next section. The relevant numerical results will be presented here. In Sec. III, we extend the ISPE mechanism to study the open-charm decays of higher charmonia with the emission of a single pion and give the corresponding prediction. The paper ends with summary in Sec. IV.

II. THE ISPE MECHANISM AND THE
The ISPE mechanism has been first proposed to study the Υ(5S ) → Υ(nS )π + π − (n = 1, 2, 3) and Υ(5S ) → h b (mP)π + π − (m = 1, 2) decays [21], and it explains why two Z b structures can be observed in these processes. Via the ISPE mechanism, the hidden-bottom dipion decays of Υ(5S ) can occur through two steps. First, Υ(5S ) decays into the B ( * )B( * ) plus one pion, where most of the kinematical energy is carried out by the emitted pion and is continuously distributed. Secondly, the B ( * ) andB ( * ) mesons with low momentum can easily interact with each other to convert into the final state Υ(nS )π or h b (mP)π via the B ( * ) meson exchange [21].
In this paper, we would like to apply the ISPE mechanism to the open-bottom decays of Υ(5S ) with the emission of a single pion. In Figs. 1 and 2, we present the typical diagrams describing Υ(5S ) → πB ( * )B( * ) via the ISPE mechanism, where the intermediate B ( * ) andB ( * ) meson can convert into BB * or B * B * final state by exchanging light mesons such as π and ρ.  In the following, we will write out the decay amplitudes corresponding to the diagrams listed in Figs. 1 and 2. The effective Lagrangians relevant to our study are given by where V and P are 3 × 3 matrices corresponding to the pseudoscalar and vector octets, which satisfy with ω 8 = ω cos θ+φ sin θ and sin θ = −0.761. These effective Lagrangians are constructed by considering heavy quark limit and chiral symmetry. The coupling constants in the above Lagrangians can be defined as where g V = m ρ / f π , β = 0.9, λ = 0.56 GeV −1 and f π = 132 MeV.
Using the above Lagrangians, we obtain the decay amplitudes for Υ(5S ) → B * 0 B − π + corresponding to five diagrams shown in Fig. 1 as Similarly, one also gets the amplitudes corresponding to two diagrams listed in Fig. 2 as In these expressions for decay amplitudes, the dipole form factor (FF) The total decay amplitudes are expressed as where A 1 and A 2 correspond to Υ(5S ) → B * 0 B − π + with the intermediate BB * + h.c. and B * B * , respectively, while A 3 to Υ(5S ) → B * 0 B * − π + with the intermediate B * B * . The general differential decay width for Υ(5S )(p 0 ) → π(p 3 )B ( * ) (p 4 )B * (p 5 ) is with m 2 B * B ( * ) = (p 4 + p 5 ) 2 and m 2 B * π = (p 3 + p 5 ) 2 , where the overline indicates the sum over the polarization of Υ(5S ) in the initial state and the polarizations of B * orB * meson in the final state.
Since we mainly concentrate on the lineshapes of the BB * and B * B * invariant mass spectrum distributions of Υ(5S ) → B * 0 B − π + and Υ(5S ) → B * 0 B * − π + decays, the interference effects between A 1 and A 2 are not considered in this work. Calculating the distributions of Eq. (9), we can see whether there exist the enhancement structures close to the BB * and B * B * thresholds steming from the ISPE mechanism. As one can see from Fig. (3), peaks of our theoretical curves nicely match with those of the experimental enhancement structures. Here, the maximum of the theoretical line shape is normalized to be 1 and the typical value of α = 1 is taken in our calculation. Diagrams (a) and (b) correspond to Υ(5S ) → B * 0 B − π + via the intermediates BB * + h.c. and B * B * , respectively, by the ISPE mechanism. The diagram (c) reflects the distribution dΓ(Υ(5S ) → B * 0 B * − π + )/dm B * B * of Υ(5S ) → B * 0 B * − π + . To compare our theoretical results with the experimental data, we also show Belle's data (the blue dots with error) of the Υ(5S ) → BB * π (left) and Υ(5S ) → B * B * π (right) [22]. The thresholds of BB * and B * B * are marked by the dotted lines.

III. THE OPEN-CHARM DECAYS OF HIGHER CHARMONIA WITH A SINGLE PION EMISSION
Being inspired by the success of the former section, we would like to apply the ISPE mechanism to the open charm decays of higher charmonia with a single pion emission, for example, to the processes ψ(4415) → π + D * 0 D ( * )− and ψ(4160) → π + D * 0 D − .
What we need in this section is to replace Υ(5S ), B, and B * in Figs. 1 and 2 with ψ(4415)/ψ(4160), D, and D * , re-spectively. We also need to replace the corresponding fields in the effective Lagrangians in Eqs. (1)(2)(3). The parameters are of course new definitions. The resultant curves are shown in Fig. (4). Similarly to Υ(5S ) → πB * B( * ) , there are two significant enhancement structures near the thresholds of D * D and D * D * in the invariant mass spectra m D * D and m D * D * of ψ(4415) → πD * D( * ) . For ψ(4160) → πD * D , only one enhancement near D * D threshold is predicted and the threshold of D * D * is out of the range of the invariant mass spectra m D * D of this process.

IV. SUMMARY
Very recently, the Belle Collaboration has reported new results on Z b (10610) and Z b (10650) at the ICHEP2012 conference that these Z b structures also exist in the BB * and B * B * invariant mass spectra of Υ(5S ) → πBB * , πB * B * decays [22]. This motivates us to apply the ISPE mechanism because these are the typical processes for this mechanism to be applied. Using the effective Lagrangian approach among hadrons as well as chiral particles, we have computed the theoretical curves of the invariant mass spectra of B * B( * ) for the above processes, which are shown in Figs. 3 and have successful agreement with experimental enhancement structures of Z b (10610) and Z b (10650).
This success has further driven us to apply the ISPE mechanism to the open-charm decays of higher charmonia with a single pion emission, ψ(4415) → π + D * 0 D − and ψ(4160) → π + D * 0 D − . Similar procedures to those in Sec. II have led us to depict the theoretical curves of the invariant mass spectra as shown in Fig. 4. Figure 4 shows two clear peaks for the invariant mass spectra m D * D and m D * D * of the decay ψ(4415) → π + D * 0 D − and one peak for m D * D of ψ(4160) → π + D * 0 D − . These predictions can be easily tested by Bell, BaBar, forthcoming BellII and SuperB.