Charged bottomonium-like states and and the decay
Inspired by the newly observed two charged bottomonium-like states, we consider the possible contribution from the intermediate and states to the decay process, which naturally explains Belle’s previous observation of the anomalous production near the peak of at GeV [K.F. Chen et al. (Belle Collaboration), Phys. Rev. Lett. 100, 112001 (2008)]. The resulting and distributions agree with Belle’s measurement after inclusion of these states. This formalism also reproduces the Belle observation of the double-peak structure and its reflection in the invariant mass spectrum of the decay.
pacs:14.40.Pq, 13.25.Gv, 13.66.Bc
Very recently, the Belle Collaboration announced the first observation of two charged bottomonium-like states and in the hidden-bottom decay channels () and () of Collaboration:2011gj (). The measured parameters of and are
The analysis of the angular distribution indicates that the quantum numbers of both and are . Both and are charged hidden-bottom states. Moreover they are very close to the thresholds of and Nakamura:2010zzi (), respectively. Thus, and are ideal candidates of the and S-wave molecular states, which were studied extensively in Refs. Liu:2008fh (); Liu:2008tn ().
On the other hand, a new puzzle arises in the theoretical study Chen:2011qx (); Ali:2009es () of the dipion invariant mass distribution and the distribution of the anomalous production near the peak of Abe:2007tk (). While all the other calculations are well in accord with the Belle data, the predicted differential width disagrees with the Belle measurement Chen:2011qx (). In this work, we will illustrate that the inclusion of these two states in the decays explains the puzzling line shape of very naturally.
In general, there exist three mechanisms for the hidden-bottom decays with the dipion emission
The first one is the direct production by decay (see Fig. 1 (a)). The so-called direct production of denotes that there does not exist the contribution from the intermediate mesons (such as , hadronic loop constructed by or mesons, ) to . Thus, the direct production of provides the background contribution.
The QCD Multipole Expansion (QME) method Kuang:1981se () is generally applied to deal with the dipion transition between heavy quarkonia. So far, there exist many theoretical efforts study the dipion transitions between the bottomonia Yan:1980uh (); Kuang:1981se (); Zhou:1990ik (); Anisovich:1995zu (); Guo:2004dt (); Guo:2006ai (); Simonov:2008qy (); Simonov:2008sw () (see Refs. Voloshin:1987rp (); Besson:1993mm (); Kuang:2006me () for a detailed review). In this work, we do not intend to calculate the contribution from the direct transition under the framework of the QME method, but alternatively follow the effective Lagrangian approach to describe transitions. The transition amplitude of the direct production of can be written as
which was suggested by Novikov and Shifman in the study of the decay Novikov:1980fa (), where the subscripts S-wave and D-wave denote the S-wave and D-wave contributions respectively. is the mass difference between and . denotes the invariant mass of , while is the angle between and in the rest frame. The pion decay constant and mass are taken as MeV and MeV, respectively. In Eq. (LABEL:direct), and are free parameters to be determined when fitting the experimental data.
Different from the other low-lying bottomonia with , is above the thresholds and predominantly decays into pair, which may render the coupled channel effect quite important Meng:2007tk (); Meng:2008dd (); Simonov:2008ci (). When exploring the hidden-bottom decay, the coupled channel effect has to be taken into account. In other words, there also exists the second mechanism contributing to the transitions as shown in Fig. 1 (b), where the intermediate and hadronic loop is the bridge to connect the initial state and final state . Furthermore, can be approximately expressed as a sequential decay process. first transits into and the scalar meson . Then couples with the dipion. Choosing as the intermediate state contribution to the process is not only consistent with the Belle data Collaboration:2011gj (); Abe:2007tk () but also allowed by the phase space of the decay channel.
If comparing the dipion invariant mass spectrum of in Refs. Collaboration:2011gj (); Abe:2007tk (), the data in Ref. Collaboration:2011gj () at the higher end of are qualitatively different from those in Ref. Abe:2007tk (), where the total events in Ref. Abe:2007tk () are at least one order of magnitude less than those in Ref. Collaboration:2011gj (). Such a large accumulation of events at MeV Collaboration:2011gj () might be due to the contribution from the tail of the intermediate . We did not include the contribution when we analyzed the data in Abe:2007tk (). Considering the situation of the new data of the dipion invariant mass spectrum of Collaboration:2011gj (), we also include contribution to the analysis of in the following.
The effective Lagrangians relevant to the Fig. 1 (b) include
where and . There are 4 diagrams. Thus, the concrete expressions of decay amplitudes are written as
The amplitude indicates that the initial dissolves into intermediate , which transit into the final and scalar meson by exchanging meson . In the above expressions, the form factor is introduced by . And is the mass of the exchanged meson in the transitions shown in Fig. 1 (b) and with MeV. As indicated in Ref. Chen:2011qx (), we can parameterize the decay amplitude of corresponding to Fig. 1 (b) as
if only considering the S-wave contribution. Here, we introduce as the fitting parameter.
Similar to Eq. (10), the parameterized decay amplitude of with as the intermediate state can be expressed as
which corresponds to Fig. 1 (b) with the replacement .
Regarding the contribution of these two newly observed states to the process, we introduce the third mechanism depicted in Fig. 1 (c) and (d), where s are the intermediate states and interact with and . The general expressions of the amplitudes of Fig. 1 (c) and (d) are
respectively, where we define and . Since Fig. 1 (c) and (d) are related to each other by charge-conjugation, thus .
Thus, the total decay amplitude of the decay is
where we have introduced the phase angles , , and .
As a three body decay, the differential decay width for read as,
with and . The relevant resonance parameters are listed in Table. 1.
|State||Mass (GeV)||State||Mass (GeV)||Width (GeV)|
If considering only the contributions from Fig. 1 (a) and (b) in our present scenario, we have four free parameters as listed in Table 2, where the contribution is included to fit the Belle data Abe:2007tk (). With the help of the MINUIT package, we perform the global fit to the experimental data of the dipion invariant mass spectrum distribution and the distribution of the production near the peak of Abe:2007tk (). The best fit to the dipion invariant mass spectrum distribution is shown in the left-panel in Fig. 2. Unfortunately the corresponding distribution of the production strongly deviates from the Belle data as shown in the right-panel of Fig. 2. The values of the obtained fitting parameters are presented in Table 2. Such discrepancy between theoretical and experimental results stimulates a New Puzzle first indicated in Ref. Chen:2011qx (). At present, solving these new puzzle becomes an important and intriguing research topic, which will be helpful to underlying mechanism behind the decay.
In contrast, we consider the contribution from and in the following and discuss the dependence of and of on and respectively. Under this scheme, we refit the Belle data Collaboration:2011gj () with Eq. (14). There are 10 fitting parameters as listed in Table 3. In Fig. 3, we present a comparison between the Belle data (dots with error bars) and our best fit (histograms) to the Belle data Collaboration:2011gj (), which indicates that the line shapes of the invariant mass spectra of and for describe the Belle data Collaboration:2011gj () well. The double-peak structure around GeV and its reflection around 10.25 GeV are reproduced by our model well. With the central values of these parameters in Table 3, we obtain the partial decay width of MeV, which is consistent with the Belle measurement MeV Abe:2007tk (). Thus, the contribution from these charged resonances provides a possible solution to the puzzle why the decay width is abnormally large Abe:2007tk ().
From Table 3, we notice that the uncertainty of is one order-of-magnitude larger than that of , which means the fit is less sensitive to the than to the . Using Eq. (14), we reanalyze the new Belle data in Ref. Collaboration:2011gj () with the obtained fitting parameters in Table 4, where we do not include the contribution. The comparison between our fitting result and the experimental data are given in Fig. 4. By the scenario in Eq. (14), we reproduce the Belle data well, which confirms that the intermediate contribution to is small. If comparing the obtained values of the fitting parameter in Tables 3 and 4, we notice that the eight common parameters do not change much in the two schemes. With the parameters listed in Tables 3 and 4, we also present the distribution with and without the intermediate contribution. The experimental measurement of the distribution for Abe:2007tk () can be described well with the scenarios in this work. This fact indicates that the two structures play important role in the understanding of the Belle data, especially the distribution of .
In summary, the Belle Collaboration announced an exciting observation of two charged bottomonium-like states and . These states are good candidates of exotic states, which calls for theoretical efforts in revealing their underlying structures. Carrying out the phenomenological study relevant to and is one of the important and valuable issues of heavy quarkonium physics, which is full of challenges and opportunities Brambilla:2004wf (); Brambilla:2010cs ().
The and states are related to the anomalous phenomena of production near previously reported by Belle Abe:2007tk (). Comparing the fitting results without and with the contributions from the newly observed states, we notice that the intermediate and play a crucial role in the behavior of . The inclusion of the and contribution to provides a unique mechanism of understand the puzzling distribution of production near Abe:2007tk (). The double-peak structure and its reflection in the invariant mass spectrum of Collaboration:2011gj () are also reproduced by this mechanism. In this work, the values of the fitting parameters in our scenario are obtained by fitting Belle data Collaboration:2011gj (); Abe:2007tk (). To some extent, the interpretation of the values of these parameters is related to the understanding of background, the structures of two states etc, which is an interesting research topic.
Besides finding the signals of and in decay channel, Belle’s analysis of its remaining four hidden-bottom decay channels () and () also indicate the observation of and Collaboration:2011gj (). The present formalism can be extended to study the dipion invariant mass distribution and the distribution of and decay.
Additionally, Belle’s measurement favors the and molecular explanation of the and resonances respectively. The possible S-wave and molecular states were investigated extensively in Refs. Liu:2008fh (); Liu:2008tn (). Very recently, the authors in Ref. Bondar:2011ev () discussed the special decay behaviour of the J=1 S-wave and molecular states based on the heavy quark symmetry. Future dynamical study of the mass and decay pattern of the S-wave and molecular states are very desirable.
This project is supported by the National Natural Science Foundation of China under Grants No. 11175073, No. 11005129, No. 11035006, No. 11047606, No. 11075004, No. 11021092; the Ministry of Education of China (FANEDD under Grant No. 200924, DPFIHE under Grant No. 20090211120029, NCET under Grant No. NCET-10-0442, the Fundamental Research Funds for the Central Universities); and the West Doctoral Project of the Chinese Academy of Sciences.
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