Research Article | Open Access
Production of in Radiative Decays
We investigate the production of in the process , where is assumed to be the counterpart of in the bottomonium sector as molecular state. We use the effective Lagrangian based on the heavy quark symmetry to explore the rescattering mechanism and calculate their production ratios. Our results have shown that the production ratios for are orders of with reasonable cutoff parameter range . The sizeable production ratios may be accessible at the future experiments like forthcoming BelleII, which will provide important clues to the inner structures of the exotic state .
In the past decades, many so-called have been observed by the Belle, BaBar, CDF, D0, CMS, LHCb, and BESIII collaborations . Some of them cannot fit into the conventional heavy quarkonium in the quark model [2–5]. Up to now, many studies on the production and decay of these states have been carried out in order to understand its nature (for a recent review, see [6–8]).
In 2003, the Belle collaboration discovered an exotic candidate in the process  which was subsequently confirmed by the BaBar collaboration  in the same channel. It was also discovered in proton-proton/antiproton collisions at the Tevatron [11, 12] and LHC [13, 14]. is a particularly intriguing state because on the one hand its total width MeV  is tiny compared to typical hadronic widths and on the other hand the closeness of its mass to threshold ( MeV) and its prominent decays to  suggest that it may be an meson-meson molecular state [15, 16].
Many theoretical works have been carried out in order to understand the nature of since the first observation of . It is also natural to look for the counterpart with (denoted as hereafter) in the bottom sector. These two states are related by heavy quark symmetry which should have some universal properties. The search for may provide us with important information on the discrimination of a compact multiquark configuration and a loosely bound hadronic molecule configuration. Since the mass of may be very heavy and its is , it is less likely for a direct discovery at the current electron-positron collision facilities, though the Super KEKB may provide an opportunity in radiative decays . In , a search for in the final states has been presented and no significant signal is observed for such a state.
The production of at the LHC and the Tevatron [19, 20] and other exotic states at hadron colliders [21–26] has been extensively investigated. In the bottomonium system, the isospin is almost perfectly conserved, which may explain the escape of in the recent CMS search . As a result, the radiative decays and isospin conserving decays will be of high priority in searching for [28–30]. In , we have studied the radiative decays of (), with being a candidate for molecular state, and found that the partial widths into are about 1 keV. In , we studied the rescattering mechanism of the isospin conserving decays , and our results show that the partial width for is about tens of keVs.
In this work, we will further investigate production in with being molecule candidate. To investigate this process, we calculate the intermediate meson loop (IML) contributions. As well know, IML transitions have been one of the important nonperturbative transition mechanisms being noticed for a long time [31–33]. Recently, this mechanism has been used to study the production and decays of ordinary and exotic states [34–60] and decays [61–68], and a global agreement with experimental data was obtained. Thus this approach may be suitable for the process .
The paper is organized as follows. In Section 2, we present the effective Lagrangians for our calculation. Then in Section 3, we present our numerical results. Finally we give the summary in Section 4.
2. Effective Lagrangians
Based on the heavy quark symmetry, we can write out the relevant effective Lagrangian for [68, 69]:where and correspond to the bottom meson isodoublets. is the antisymmetric Levi-Civita tensor and . Since is above the threshold of , the coupling constants between and can be determined via experimental data for . The experimental branching ratios and the corresponding coupling constants are listed in Table 1. Since there is no experimental information on , we choose the coupling constants between and , the same values as that of .
In order to calculate the process depicted in Figure 1, we also need the photonic coupling to the bottomed mesons. The magnetic coupling of the photon to heavy bottom meson is described by the Lagrangian [70, 71]withwhere is an unknown constant, is the light quark charge matrix, and is the heavy quark electric charge (in units of ). is determined in the nonrelativistic constituent quark model and has been adopted in the study of radiative decays . In and systems, value is the same due to heavy quark symmetry . In (2), the first term is the magnetic moment coupling of the light quarks, while the second one is the magnetic moment coupling of the heavy quark and hence is suppressed by .
At last, assume that is -wave molecule with given by the superposition of . and . hadronic configurations asAs a result, we can parameterize the coupling of to the bottomed mesons in terms of the following Lagrangian:where denotes the coupling constant. Since is slightly below -wave threshold, the effective coupling of this state is related to the probability of finding component in the physical wave function of the bound states and the binding energy, [36, 72, 73]:where and is the reduced mass. Here, we should also notice that the coupling constant in (6) is based on the assumption that is a shallow bound state where the potential binding the mesons is short-ranged.
Based on the relevant Lagrangians given above, the decay amplitudes in Figure 1 can be generally expressed as follows:where and are the vertex functions and the denominators of the intermediate meson propagators. For example, in Figure 1(a), are the vertex functions for the initial , final , and photon, respectively. are the denominators for the intermediate , , and propagators, respectively.
Since the intermediate exchanged bottom mesons in the triangle diagram in Figure 1 are off-shell, in order to compensate these off-shell effects arising from the intermediate exchanged particle and also the nonlocal effects of the vertex functions [74–76], we adopt the following form factors:where corresponds to monopole and dipole form factor, respectively. and the QCD energy scale MeV. This form factor is supposed and many phenomenological studies have suggested . These two form factors can help us explore the dependence of our results on the form factor.
The explicit expression of transition amplitudes can be found in Appendix (A.) in , where radiative decays of charmonium are studied extensively based on effective Lagrangian approach.
3. Numerical Results
Before proceeding the numerical results, we first briefly review the predictions on mass of . The existence of is predicted in both the tetraquark model  and those involving a molecular interpretation [79–81]. In , the mass of the lowest-lying tetraquark is predicated to be 10504 MeV, while the mass of molecular state is predicated to be a few tens of MeV higher [79–81]. For example, in , the mass was predicted to be 10562 MeV, which corresponds to a binding energy to be 42 MeV, while the mass was predicted to be MeV, which corresponds to a binding energy MeV in . As can be seen from the theoretical predictions, it might be a good approximation and might be applicable if the binding energy is less than 50 MeV. In order to cover the range of the previous molecular and tetraquark predictions on [78–81], we present our results up to a binding energy of 100 MeV, and we will choose several illustrative values: MeV.
In Table 2, we list the predicted branching ratios by choosing the monopole and dipole form factors and three values for the cutoff parameter in the form factor. As a comparison, we also list the predicted branching ratios in NREFT approach. From this table, we can see that the branching ratios for are orders of . The results are not sensitive to both the form factors and the cutoff parameter we choose.
In Figure 2(a), we plot the branching ratios for in terms of the binding energy with the monopole form factors (solid line), 2.5 (dashed line), and (dotted line), respectively. The coupling constant of in (6) and the threshold effects can simultaneously influence the binding energy dependence of the branching ratios. With the increasing of the binding energy , the coupling strength of increases, and the threshold effects decrease. Both the coupling strength of and the threshold effects vary quickly in the small region and slowly in the large region. As a result, the behavior of the branching ratios is relatively sensitive at small , while it becomes smooth at large . Results with the dipole form factors , 2.5, and 3.0 are shown in Figure 2(b) as solid, dash, and dotted curves, respectively. The behavior is similar to that of Figure 2(a).
We also predict the branching ratios of and present the relevant numerical results in Table 3 and Figure 3 with the monopole and dipole form factors. At the same cutoff parameter , the predicted rates for are a factor of 2-3 smaller than the corresponding rates for . It indicates that the intermediate -meson loop contribution to the process is smaller than that to . This is understandable since the mass of is more far away from the thresholds of than . But their branching ratios are also about orders of with a reasonable cutoff parameter .
In , authors introduced a nonrelativistic effective field theory method to study the meson loop effects of . Meanwhile they proposed a power counting scheme to estimate the contribution of the loop effects, which is used to judge the impact of the coupled-channel effects. For the diagrams in Figure 1, the vertex involving the initial bottomonium is in -wave. The momentum in this vertex is contracted with the final photon momentum and thus should be counted as . The decay amplitude scales as follows:where is understood as the average velocity of the intermediate bottomed mesons.
As a cross-check, we also present the branching ratios of the decays in the framework of NREFT. The relevant transition amplitudes are similar to that given in  with only different masses and coupling constants. The obtained numerical results for and in terms of the binding energy are listed in the last column of Tables 2 and 3, respectively. As shown in Table 2, except for the largest binding energy MeV, the NREFT predictions of are about 1 order of magnitude smaller than the ELA results at the commonly accepted range. For shown in Table 3, the NREFT predictions are several times smaller than the ELA results in small binding energy range, while the predictions of these two methods are comparable at large binding energy. These differences may give some sense of the theoretical uncertainties for the predicted rates and indicate the viability of our model to some extent.
Here we should notice, for the isoscalar , the pion exchanges might be nonperturbative and produce sizeable effects [81–83]. In , their calculations show that the relative errors of are about 20% for . Even if we take into account this effect, the estimated order of the magnitude for the branching ratio may also be sizeable, which may be measured in the forthcoming BelleII experiments.
In this work, we have investigated the production of in the radiative decays of . Based on molecular state picture, we considered its production through the mechanism with intermediate bottom meson loops. Our results have shown that the production ratios for are about orders of with a commonly accepted cutoff range . As a cross-check, we also calculated the branching ratios of the decays in the framework of NREFT. Except for the large binding energy, the NREFT predictions of are about 1 order of magnitude smaller than the ELA results. The NREFT predictions of are several times smaller than the ELA results in small binding energy range, while the predictions of these two methods are comparable at large binding energy. In [28, 29], we have studied the radiative decays and the hidden bottomonium decays of . If we consider that the branching ratios of the isospin conserving process are relatively large, a search for may be possible for the updated BelleII experiments. These studies may help us investigate deeply. The experimental observation of will provide us with further insight into the spectroscopy of exotic states and is helpful to probe the structure of the states connected by the heavy quark symmetry.
The authors declare that they have no competing interests.
This work is supported in part by the National Natural Science Foundation of China (Grants nos. 11275113, 11575100, and 11505104) and the Natural Science Foundation of Shandong Province (Grant no. ZR2015JL001).
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