Potential observation of the $\Upsilon(6S) \to \Upsilon(1^3D_J) \eta$ transitions at Belle II

We perform the investigation of two-body hidden-bottom transitions of the $\Upsilon(6S)$, which include $\Upsilon(6S) \to \Upsilon(1^3D_J) \eta~~(J=1,2,3)$ decays. For estimating the branching ratios of these processes, we consider contributions from the triangle hadronic loops composed of S-wave $B_{(s)}$ and $B_{(s)}^*$ mesons, which are a bridge to connect the $\Upsilon(6S)$ and final states. Our results show that both of the branching ratios of these decays can reach $10^{-3}$. Due to such considerable potential to observe these two-body hidden-bottom transitions of the $\Upsilon(6S)$, we suggest the forthcoming Belle II experiment to explore them.


I. INTRODUCTION
As an updated accelerator with luminosity 8×10 35 cm −2 s −1 , SuperKEKB is currently being constructed. The forthcoming Belle II experiment will accumulate 50 times more data than the previous Belle experiment. It is a good time to explore the potential physical issues close to the Belle II.
In Particle Data Group (PDG) [11], there is the Υ(11020) above the Υ(10860). Usually, the Υ(11020) is treated as the n 2S +1 L J = 6 3 S 1 state. Thus, in the following discussions, the Υ(11020) is abbreviated as the Υ(6S ) for convenience. Here, we want to emphasize that the Υ(6S ) has situation similar to * Electronic address: xiangliu@lzu.edu.cn † Electronic address: matsuki@tokyo-kasei.ac.jp that of the Υ(10860) since the Υ(6S ) is also above the B ( * ) thresholds. This fact gives us a reason to believe that the coupled-channel effect cannot be ignored when carrying out the studies around the Υ(6S ).
In this work, using the hadronic loop mechanism, we calculate the branching ratios of the discussed Υ(6S ) → Υ(1 3 D J )η decays, by which the potential observation of these two-body hidden-bottom transitions at Belle II can be suggested. We want to emphasize that the present study must become an important part of the physics around the Υ(6S ), and further push experimental exploration of these decays. This paper is organized as follows. After introduction, we present the detailed deductions of Υ(6S ) → Υ(1 3 D J )η via the hadronic loop mechanism in Sec. II. Then, various parameters are determined in Sec. III. In Sec. IV, numerical results are presented. Finally the paper will end with a short summary. In this work, we adpot the hadronic loop mechanism to study the Υ(6S ) → Υ(1 3 D J )η transitions.
Finally, the total sum of amplitudes reads as s ↔B ( * ) s . With the total sum of amplitudes for each process, the general expression of the partial decay widths is which averages over the polarization of initial Υ(6S ) and sums over the polarizations of the Υ(1 3 D J ).

III. INPUT PARAMETERS
Before displaying our results, we have to illustrate various parameters including masses and coupling constants used in this work. First, we need the input of the masses. For the masses of the bottomonia Υ(1 3 D 1 ) and Υ(1 3 D 3 ), 10.153 GeV [13,29] and 10.174 GeV [13,29] are adopted, respectively, whereas PDG values [11] are used for other bottomonia involved in this work.
Utilizing the partial decay widths of Υ(6S ) → B ( * )  Table I we list the calculated partial decay widths in Ref. [30] as well as the extracted coupling constants [31]. are related to one coupling g 2 of Eq. (1) under the heavy quark symmetry: We now have to determine the last remaining coupling constant g 2 . For g 2 , we fix it as follows [13]: we first define the decay constant of the vector meson Υ(1 3 D 1 ) [32] 0|Qγ with M V and µ V being the mass and the polarization vector of Υ(1 3 D 1 ), respectively. Then, using the relation g V BB M V / f V under the vector meson dominance ansatz [33][34][35], we can get g Υ(1 3 D 1 )BB , and g 2 eventually.
Our calculation shows that the branching ratios of Υ(6S ) → Υ(1 3 D J )η shown in Fig. 2 can reach up to 10 −3 , i.e., These significant values of the branching ratios show that the Υ(6S ) → Υ(1 3 D J )η decays can be accessible at Belle II.
We also obtain three ratios as shown in Figs. 3, where these ratios weakly depend on α Λ . The predicted ratios include which can be tested further in future experiment like Belle and Belle II.
Besides these observations, Belle discovered the Υ(5S ) decays into Υ(1 3 D J )η very recently, which has large branching ratios [12]. In fact, this Belle measurement confirmed the prediction made in Ref. [13], which again shows the important role of the hadronic loop effect to the Υ(5S ) hidden-bottom decays. What is more important is that this fact also inspires our ambition to further explore the hidden-bottom decays of the Υ(6S ).
In this work, we have selected the Υ(6S ) → Υ(1 3 D J )η, and have studied the potential discovery of these decays in a future experiment. Our calculation has shown that the branching ratios of the Υ(6S ) → Υ(1 3 D J )η can reach up to 10 −3 when the hadronic loop effect is introduced. It is evident that these significant branching ratios could arouse experimentalist's interest in finding them.
With the running of Belle II, we expect the observation of these anomalous Υ(6S ) → Υ(1 3 D J )η transitions, which will make our understanding of higher bottomonia more in-depth and thorough. Notably, the present study should become a part of the whole research around the Υ(6S ).