Proposal of searching for the $\Upsilon(6S)$ hadronic decays into $\Upsilon(nS)$ plus $\eta^{(\prime)}$

In this work, we propose a possible experimental research topic on the $\Upsilon(6S)\to \Upsilon(nS)\eta^{(\prime)}$ ($n=1,2,3$) transitions. Considering the hadronic loop effect from the intermediate $S$-wave $B_{(s)}$ and $B_{(s)}^*$ mesons, we estimate the partial decay widths and the corresponding branching ratios of the $\Upsilon(6S)\to \Upsilon(nS)\eta^{(\prime)}$ hadronic decays, which become considerably large. With the running of the Belle II experiment, it becomes possible to explore these $\Upsilon(6S)\to \Upsilon(nS)\eta^{(\prime)}$ transitions.

Along this line, in this work we continue to focus on the specific hadronic transitions Υ(6S ) → Υ(nS )η ( ) (n = 1, 2, 3). Using the hadronic loop mechanism, we need to further estimate their partial decay widths and the corresponding branching ratios, which are crucial information for experimental analysis of these decays at Belle II. In addition, we predict several typical ratios of these discussed hadronic decays, which can be tested in future experiment.
Frankly speaking, another motivation inspiring us to carry out such a study is that the present work is one part of the whole physics around the higher bottomonium Υ(6S ). We hope that the entire aspect of the Υ(6S ) can be described by our step-by-step effort, which will be valuable to in determining these potential research topics at Belle II when more and more data will be accumulated in the near future.
This work is organized as follows. After the Introduction, we spend the next section illustrating the detailed calculation of the Υ(6S ) → Υ(nS )η ( ) transitions (see Sec. II). Then, the numerical results including the partial decay widths, the branching ratios, and several ratios will be presented in Sec. III. This paper ends with conclusions and a discussion. All the diagrams shown in Fig. 2 can be abstracted into one general diagram presented in Fig. 3, in which the bottom or bottom-strange mesons construct the hadronic loop and connect the initial Υ(6S ) with the final Υ(nS ) and η ( ) so that the general expression of the amplitude, according to can be written as where V i (i = 1, 2, 3) are interaction vertices and 1/P j ( j = 1, 2) and 1/P E correspond to the propagators. In Eq. (1), monopole form factor F (q 2 , m E ) = (m 2 E − Λ 2 )/(q 2 − Λ 2 ) is adopted as suggested by a QCD sum rule study in Ref. [19] , in which we have introduced m E as the mass of the exchanged bottom meson and the cutoff Λ = m E + α Λ Λ QCD with Λ QCD = 220 MeV [20][21][22]. This included form factor is applied to mainly describe the structure effect of each vertex. In addition, this form factor also plays a role similar to the Pauli-Villas renormalization scheme which is often used to avoid ultraviolet divergence in the loop integrals [23,24].
After expanding the Lagrangians in Eqs.
(2) and (3), the interaction vertices V i in Eq. (1) can be explicitly expressed by Finally, the decay widths of the processes Υ(6S ) → Υ(nS )η ( ) can be evaluated by in which the average over polarizations of the initial Υ(6S ) and the sum over those of the Υ(nS ) have been done, and In Eq. (12), 6 i=1 means that all the diagrams shown in Fig. 2 will give contributions to this loop-described transition Υ(6S ) → Υ(nS )η ( ) , M q i and M s i denote that in the amplitudes, the loops are composed of bottom (B ( * ) ) and bottomstrange (B ( * ) s ) mesons, respectively, and the factors 4 and 2 come from the charge conjugation and the isospin transformations on the bridged B ( * ) (s) .

III. NUMERICAL RESULTS
Before presenting our numerical results, we want to illustrate how to determine the relevant coupling constants. For g ΥB ( * ) B ( * ) , we use the corresponding partial decay widths calculated via the potential model in Ref. [37] to fit them. The results are collected in Table I. For g ΥB ( * ) B ( * ) , we need to consider the internal symmetry implied by the heavy quark effective theory, under which all the coupling constants are related to each other through a global constant g 1 in Eq. (2). With the help of the vector meson dominance ansatz [38][39][40][41], a relation among these couplings can be given as with m Υ and f Υ being the mass and decay constant of Υ [38][39][40][41]. The decay constant of Υ(nS ), i.e., f Υ , can be obtained by fitting to the leptonic decay width Γ[Υ(nS ) → e + e − ] given in Ref. [5] by where α is the fine-structure constant and C V = 1/9 for the Υ(nS ) meson. We estimate the different values of g 1 and list them in Table II. Similarly, g B ( * ) B ( * ) η ( ) can also be connected by a coupling constant g 2 = g π / f π with f π =131 MeV and g π =0.569 shown in Eq. (3), i.e., in which α, β, γ, δ have been defined in Eq. (8).
Besides these fixed coupling constants, there is still a free parameter α Λ , which is included in the form factor F (q 2 , m E ) in the amplitudes. According to the experiences in Refs. [7][8][9], α Λ = 3 is taken in our calculation.
Our calculation gives the partial decay widths for the processes Υ(6S ) → Υ(nS )η ( ) with α Λ = 3 as The corresponding branching ratios are extracted as The above results indicate that most of the hidden-bottom η ( ) transitions of the Υ(6S ) have branching ratios around 10 −3 . Because of these predicted considerable branching ratios, we are confident that all these transitions can be observed in future experiments like Belle II. For the process Υ(6S ) → Υ(2S )η ( ) , it is easy to see that although its branching ratio is 3 orders of magnitude lower than others, we note here that this smallness is mainly due to the restriction of phase space together with the η − η mixing depicted in Eqs. (7) and (8), which gives opposite contributions to M q i and M s i in Eq. (12). Although this also happens in Υ(6S ) → Υ(1S )η ( ) , its branching ratio can still hold at 10 −3 due to the larger phase space.
We also find that the ratios of these calculated branching ratios are insensitive to the α Λ values. When scanning α Λ in a range [2.7, 3.3], we present the ratios, which reflects this insensitivity well. In the following equations, we list the typical ratios and which, in our view, can be tested by a future experiment.

IV. CONCLUSIONS AND DISCUSSION
Two very recent experimental observations of the hadronic transitions Υ(5S ) → Υ(1 3 D J )η [15] and Υ(6S ) → χ bJ π + π − π 0 [16] again enforced our ambition to explore the hiddenbottom decays of higher bottomonia, since these Belle measurements are consistent with our predictions made in Refs. [14,17]. Here, the hadronic loop mechanism may play a crucial role in mediating these hidden-bottom transitions of higher bottomonia to the final states.
In this work, we have applied the hadronic loop mechanism to investigate the Υ(6S ) decays into Υ(nS ) plus η or η , which are still missing in an experiment. Just illustrated in the former section, these discussed transitions have considerable branching ratios, which make the experimental exploration of them accessible, especially for the Belle II experiment.
At present, the knowledge of the Υ(11020) is still not enough due to the absence of the relevant experimental information [5]. Since the present work is one part of the whole study on the Υ(11020), we hope that these predictions given in this paper can stimulate the experimentalist's interest in carrying out the search for the discussed transitions. This will make the information of the Υ(11020) more and more abundant. Surely, if these predicted branching ratios and partial widths can be confirmed in a future experiment, the importance of the hadronic loop mechanism to the hidden-bottom transitions of higher bottomonia can be further realized. To some extent, it is also helpful to deepen our understanding of the nonperturbative behavior of strong interaction.
In addition, the involved propagators are given by