Exploring the $\Upsilon(6S)\to \chi_{bJ}\phi$ and $\Upsilon(6S)\to \chi_{bJ}\omega$ hidden-bottom hadronic transitions

In this work, we investigate the hadronic loop contributions to the $\Upsilon(6S) \to \chi_{bJ} \phi~(J=0,1,2)$ along with $\Upsilon(6S) \to \chi_{bJ} \omega~(J=0,1,2)$ transitions. We predict that the branching ratios of $\Upsilon(6S) \to \chi_{b0} \phi$, $\Upsilon(6S) \to \chi_{b1} \phi$ and $\Upsilon(6S) \to \chi_{b2} \phi$ are $(0.68 \sim 4.62) \times 10^{-6}$, $(0.50 \sim 3.43) \times 10^{-6}$ and $(2.22 \sim 15.18) \times 10^{-6}$, respectively and those of $\Upsilon(6S) \to \chi_{b0} \omega$, $\Upsilon(6S) \to \chi_{b1} \omega$ and $\Upsilon(6S) \to \chi_{b2} \omega$ are $(0.15 \sim 2.81) \times 10^{-3}$, $(0.63 \sim 11.68) \times 10^{-3}$ and $(1.08 \sim 20.02) \times 10^{-3}$, respectively. Especially, some typical ratios, which reflect the relative magnitudes of the predicted branching ratios, are given, i.e., for $\Upsilon(6S)\to \chi_{bJ}\phi$ transitions, $\mathcal{R}^\phi_{10}={\mathcal{B}[\Upsilon(6S) \to \chi_{b1} \phi]}/{\mathcal{B}[\Upsilon(6S) \to \chi_{b0} \phi]} \approx 0.74$, $\mathcal{R}^\phi_{20}= {\mathcal{B}[\Upsilon(6S) \to \chi_{b2} \phi]}/{\mathcal{B}[\Upsilon(6S) \to \chi_{b0} \phi]} \approx 3.28$, and $\mathcal{R}^\phi_{21} = {\mathcal{B}[\Upsilon(6S) \to \chi_{b2} \phi]}/{\mathcal{B}[\Upsilon(6S) \to \chi_{b1} \phi]} \approx 4.43$, and for $\Upsilon(6S)\to \chi_{bJ}\omega$ transitions, $\mathcal{R}^\omega_{10}={\mathcal{B}[\Upsilon(6S) \to \chi_{b1} \omega]}/{\mathcal{B}[\Upsilon(6S) \to \chi_{b0} \omega]} \approx 4.11$, $\mathcal{R}^\omega_{20}= {\mathcal{B}[\Upsilon(6S) \to \chi_{b2} \omega]}/{\mathcal{B}[\Upsilon(6S) \to \chi_{b0} \omega]} \approx 7.06$, and $\mathcal{R}^\omega_{21} = {\mathcal{B}[\Upsilon(6S) \to \chi_{b2} \omega]}/{\mathcal{B}[\Upsilon(6S) \to \chi_{b1} \omega]} \approx 1.72$. With the running of BelleII in the near future, experimental measurement of these two kinds of transitions will be a potential research issue.


I. INTRODUCTION
As an interesting research issue, experimental studies of the hadronic transitions of Υ(5S ) have been focused on by the Belle Collaboration in the past decade. When surveying the reported hadronic transitions of Υ(5S ), we found their general property, i.e., their observed hadronic transitions have large branching ratios. For example, Belle observed anomalous decay widths of the Υ(5S ) → Υ(nS )π + π − [1], and Υ(5S ) → χ bJ ω (J = 0, 1, 2) transitions [2]. In addition, two bottomonium-like states Z b (10610) and Z b (10650) were observed in Υ(5S ) → Υ(nS )π + π − [3]. As indicated in a serial of theoretical studies [4][5][6][7][8][9][10], the puzzling phenomena happening on Υ(5S ) transitions reflect an underlying mechanism mediated by a coupled channel effect since Υ(5S ) is above the thresholds of B ( * ) (s)B ( * ) (s) [11]. In the bottomonium family, the Υ(6S ) has the similar situation to that of Υ(5S ). We have a reason to believe that the coupled channel effect is still important to the hadronic transitions of Υ(6S ), whose exploration is, thus, an intriguing topic. This theme can provide us a valuable information of the coupled-channel effect on these decays.
to B ( * ) andB ( * ) and the diagrams change simultaneously as in Figs. 4-6. To calculate these diagrams at the hadron level, we adopt the effective Lagrangian approach, in which we first introduce the Lagrangians relevant to our calculation.
For the interactions between a heavy quarkonium and two heavy-light mesons, the Lagrangians are constructed based on the heavy quark effective theory. In the heavy quark limit, the light degrees of freedom s is a good quantum number. Thus, each value of s is assigned to a doublet formed by the states with a total angular momentum J = s ± 1/2, while for the heavy quarkonium, since the degeneracy is expected under the rotations of two heavy quark spins, there is a multiplet formed by heavy quarkonia with the same angular momentum . Therefore, under the framework of heavy quark symmetry, general forms of couplings between an S-wave or P-wave heavy quarkonium and two heavy-light mesons can be constructed as [18] L s = igTr R (QQ)H(Qq) γ µ ↔ ∂ µH (Qq) + H.c., in which R (QQ) and P (QQ) denote multiplets formed by bottomonia with = 0 and = 1, and their detailed expressions, as in Ref. [21], can be written as respectively. H (Qq) represents a doublet formed by heavy-light pseudoscalar and vector mesons [18][19][20][21]  s ) T as in Ref. [13]. H (Qq) corresponds to a doublet formed by heavy-light anti-mesons, which can be obtained by applying the charge conjugation operation to H (Qq) .
For the interaction between a light vector meson and two heavy-light mesons, the general form of the Lagrangian reads as [18,[22][23][24][25][26] where and a vector octet V has the form By expanding the Lagrangians in Eqs. (1) and (5), the following concrete expressions are obtained With the above effective Lagrangians, we can write out the amplitudes of hadronic loop contributions to Υ(6S ) → χ bJ φ (J = 0, 1, 2). For the Υ(6S ) → χ b0 φ transition, the amplitudes corresponding to Fig. 1 are where p 1 , p 2 and p 3 are momenta of Υ(6S ), φ/ω and χ bJ , and k 1 , k 2 , and q are momenta of internal B ( * ) (s) and exchanged B ( * ) (s) , respectively. In these expressions of the decay amplitudes, the monopole form factor is introduced, by which the inner structure of interaction vertices is reflected and the off-shell effect of the exchanged bottom-strange mesons is compensated. Here, the adopted form factor is taken as , with m E being the mass of the exchanged boson and the cutoff Λ being parameterized as Λ = m E + α Λ Λ QCD with Λ QCD = 0.22 GeV as in Refs. [14][15][16]. We need to specify that the monopole behavior of the adopted form factor was suggested by the QCD sum rule studied in Ref. [17]. In a serial of published papers (see Refs. [4-6, 8-10, 12-16, 21, 29]), the monopole form factor was adopted to study the transitions of charmonia and bottomonia, and B decays. Thus, this approach has been tested by these successful studies.
In a similar way, we can further write out the decay amplitudes of Υ(6S ) → χ b1 φ and Υ(6S ) → χ b2 φ, which are collected in Appendix. By considering the isospin symmetry, a general expression of the total amplitude of Υ(6S ) → χ bJ φ with J = 0, 1, 2 is written as Then, the partial decay width reads where the overline indicates the sum over polarizations of Υ(6S ), φ, and χ b1 (or χ b2 ) and the factor 1 3 denotes the average over the polarization of initial Υ(6S ).
In the case of Υ(6S ) → χ bJ ω, the expression of the partial decay width is given by with a general expression of the total amplitude of by considering the isospin and charge symmetry. The detailed expressions of A (J− j) are collected in Appendix.

III. NUMERICAL RESULTS
With the formulas listed in Sec. II and Appendix, we estimate the hadronic loop contributions to the Υ(6S ) → χ bJ φ together with Υ(6S ) → χ bJ ω (J = 0, 1, 2) transitions. Besides the masses taken from the Particle Data Book [11], all the other input parameters we need are the coupling constants. Since the Υ(6S ) is above the threshold of B ( * )  Table I we list the relevant partial decay widths given in Ref. [27] as well as the corresponding extracted coupling constants. The coupling constants relevant to the interactions between χ bJ and B ( * ) (s)B ( * ) (s) in the heavy quark limit are related to one gauge coupling g 1 given in Eq. (1), i.e., [21] and f χ b0 = 175 ± 55 MeV is the decay constant of χ b0 [28].
With the above preparation, we can evaluate the branching ratios of the Υ(6S ) → χ bJ φ and Υ(6S ) → χ bJ ω transitions. However, in our model, there still exists a free parameter α Λ , which is introduced to parameterize the cutoff Λ. Since the cutoff Λ should not be too far from the physical mass of the exchanged mesons [29], in this work we set the range 0.65 ≤ α Λ ≤ 1.15 for Υ(6S ) → χ bJ φ transitions and set 0.45 ≤ α Λ ≤ 1.15 for Υ(6S ) → χ bJ ω transitions to present the numerical results.
The main reason to introduce the hadronic loop mechanism is that Υ(5S ) is the second observed bottomonium above the BB threshold, where the coupled-channel effect may become important, which was tested by the studies in Refs. [4-10, 12, 13]. It is obvious that this is not the end of the whole story.
If the hadronic loop mechanism is a universal mechanism existing in higher bottomonium transitions, we have a reason to believe that this mechanism also plays an important role in higher bottomonium transitions. Considering the similarity between Υ(6S ) and Υ(5S ), where Υ(6S ) is the third bottomomium with open-bottom channels, we have focused on Υ(6S ) → χ bJ φ and Υ(6S ) → χ bJ ω hadronic decays. Using the hadronic loop mechanism, we have estimated the branching ratios of Υ(6S ) → χ bJ φ and Υ(6S ) → χ bJ ω, which can reach up to 10 −6 and 10 −3 , respectively. In the near future, BelleII will be running near the energy range of Υ(6S ), which makes BelleII have a great opportunity to find the χ bJ φ and χ bJ ω decay modes of Υ(6S ). If these rare decays are observed, the hadronic loop effects can be further tested.
In this work, we have especially obtained six almost stable ratios R φ 10 , R φ 20 and R φ 21 in addition to R ω 10 , R ω 20 and R ω 21 reflecting the relative magnitudes of the Υ(6S ) → χ bJ φ and Υ(6S ) → χ bJ ω decays, which are weakly dependent on our model parameter α Λ . Thus, these obtained ratios are important observable quantities. We have also suggested their experimental measurement, which is also a crucial test of our model. We notice the recent talk of the status of SuperKEKB and the future plan of taking data at the BelleII experiment [30]. Since the collision data on Υ(6S ) will be taken, we need to explore the possible interesting research issues about Υ(6S ). Our present work is only one step toward the long march.