Investigating the Υ ( 10753 ) → Υ ( 1 3 D J ) η transitions

In this work, we investigate the Υ (10753) → Υ (1 3 D J ) η ( J = 1 , 2 , 3) processes, where the Υ (10753) is assigned as a conventional bottomonium under the 4 S -3 D mixing scheme. Our result shows that the concerned processes have considerable branching ratios, i.e. , branching ratios B [ Υ (10753) → Υ (1 3 D 1 ) η ] and B [ Υ (10753) → Υ (1 3 D 2 ) η ] can reach up to the order of magnitude of 10 − 4 − 10 − 3 , while B [ Υ (10753) → Υ (1 3 D 3 ) η ] is around 10 − 6 − 10 − 5 . With the running of Belle II, it is a good opportunity for ﬁnding out the concerned hidden-bottom hadronic decays. 2 .


I. INTRODUCTION
As presented in the Belle II physics book [1], the designed luminosity of SuperKEKB can reach up to 8 × 10 35 cm −2 s −1 . Thus, the forthcoming Belle II experiment represents the precision frontier of particle physics, which is an ideal platform to perform the correlative study around heavy flavor physics. Obviously, some higher bottomonia can be accessible at Belle II, which may provide valuable hints to construct the bottomonium family.
Inspired by the 2S -1D mixing scheme for the charmonium ψ(3770), we have a reason to believe that the S -D mixing scheme should be considered when revealing the nature of the Υ(10753). Thus, the Lanzhou group introduced the 4S -3D mixing scheme to clarify the puzzling phenomenon of the Υ(10753) [12,13]. In Ref. [12], the Lanzhou group proposed the 4S -3D mixing scheme, which can solve the mass puzzle of the Υ(10753). A later result in Ref. [13] shows that the Υ(10753) under this mixing scheme has sizable dielectron decay width and the measured values R n = Γ e + e − × B[Υ(10753) → Υ(nS )π + π − ] (n = 1, 2, 3) by Belle [2] can be reproduced. In summary, the current measured data of the Υ(10753) [2], including its mass and R n values, can well be understood under the 4S -3D mixing scheme. Thus, the Υ(10753) can be still a good candidate of vector bottomonium. Along this line, several typical transitions of the Υ(10753) into other bottomonia with lower mass were explored in Refs. [12,13], which will be accessible at a future experiment like Belle II. In Fig. 1, we summarize the present status of the study of the transitions of the Υ(10753) into other bottomonia. Obviously, our knowledge of the transitions of the Υ(10753) into other bottomonia is still absent. A typical example is that the allowed Υ(10753) → Υ(1 3 D 2 )η is waiting to be explored, not only by theorists but also by experimentalists. This fact stimulates our interest in carrying out the investigation of Υ(10753) → Υ(1 3 D J )η (J = 1, 2, 3), where the Υ(1 3 D J ) denote three 1D bottomonium states. By checking the PDG values [3], we may find that only Υ(1 3 D 2 ) was observed. For the remaining 1D bottomonia, they are still missing in experiment. Thus, the present study of Υ(10753) → Υ(1 3 D J )η has a close relation to these two missing bottomonia Υ(1 3 D 1 ) and Υ(1 3 D 3 ).
For calculating the branching ratio of the Υ(10753) → Υ(1 3 D J )η transitions, the concrete phenomenological model should be involved. Borrowing the former experience of the decays of higher states of heavy quarkonium, the coupled channel effect should be considered here [12][13][14][15][16][17][18][19][20][21][22][23][24][25]. In this work, we adopt the hadronic loop mechanism to present the concrete calculation, which will be mentioned in the following section. We hope that our realistic investigation of the discussed processes may provide valuable information to experimentally search for Υ(10753) → Υ(1 3 D J )η, which will be an intriguing research task for Belle II.
This paper is organized as follows. After the introduction, we illustrate the detailed calculation of Υ(10753) → Υ(1 3 D J )η (J = 1, 2, 3) with the hadronic loop mechanism in Sec. II. And then, the numerical results are presented in Sec. III. Finally, we end the paper with a discussion and conclusion. Before studying the decays, we need to briefly introduce the 4S -3D mixing scheme. If assigning the Υ(10753) as the pure Υ(3D) state, the predicted mass of pure Υ(3D) state ranges from 10653 MeV to 10717 MeV [4-6, 26, 27]. Thus, there exists difference between the theoretical result and current measurement of the Υ(10753). Furthermore, the dielectron width of the Υ(3D) was estimated to be just a few eV [4,6,26,27], which is lower than the corresponding dielectron widths of the Υ(4S ) and Υ(5S ) states. Thus, it is difficult to find pure Υ(3D) state via the electron-positron annihilation process. However, the Υ(10753) signal was observed in the e + e − → Υ(nS )π + π − processes by Belle [2], which is puzzling for us. As proposed in Refs. [12,13], the 4S -3D mixing scheme for the Υ(10753) was introduced where θ denotes the mixing angle, and Υ 4S −3D and Υ 4S −3D are physical states. Here, the Υ 4S −3D state corresponds to the observed Υ(10753). Obviously, the puzzle on mass can be solved as shown in Fig. 1 of Ref. [12], and the dielectron decay width of the Υ(10753) is sizable. Thus, the Υ(10753) still can be as a good candidate of vector bottomonium. Based on hadronic loop mechanism, the initial Υ(10753) can be converted into final low-lying D-wave bottomonium Υ(1 3 D J ) through the triangle loops composed of bottom mesons. The concerned diagrams are displayed in Fig. 2, where the contributions from the B ( * ) s meson loops can be ignored due to the weak coupling between the Υ(10753) and the B ( * ) sB ( * ) s pair [28]. For the diagrams shown in Fig. 2, the general expression of their amplitude mediated by the hadronic loop mechanism reads as where V i (i = 1, 2, 3) are interaction vertices, and 1/P 1,2,E denote the corresponding propagators of intermediate bottom mesons. In addition, the form factor F (q 2 , m 2 E ) should be introduced to compensate the off shell effect of the exchanged B ( * ) meson and depict the structure effect of interaction vertices [29][30][31]. In our calculation, the monopole form factor [32] emphasized by QCD sum rules is adopted with m E and q denoting the mass and four-momentum of the exchanged intermediate meson, respectively. Here, we take Λ QCD = 220 MeV [33][34][35], and α Λ is a phenomenological dimensionless parameter.
The effective Lagrangian approach is used to give the concrete expressions of the decay amplitudes defined in Eq. (2.2). Due to the requirement from the heavy quark limit and the chiral symmetry, the concerned effective Lagrangians include [36][37][38][39] Here, the abbreviations S (bb) and D (bb) µν represent the S -wave and D-wave multiplets of bottomonium, respectively, i.e., (2.6) Additionally, the (0 − , 1 − ) doublets of bottom and antibottom mesons is abbreviated as H (bq) and H (bq) , respectively, which can be expressed as where the normalization factor √ m B ( * ) is neglected here.
With the above preparation, we expand the compact Lagrangians in Eq. (2.4) to get the following effective Lagrangians (2.10)  Fig. 2 as an example, the expression of its amplitude is based on the Cutkosky cutting rule. And then, the remaining amplitudes can be obtained similarly. Under the 4S -3D mixing scheme, the total amplitude is 3D cos θ, (2.15) where the superscript i(j) denotes the i(j)-th amplitudes from the bottom meson loops in the above diagrams, the index J denotes differential final D-wave bottomonium states Υ(1 3 D J ), and the subscripts 4S and 3D is applied to distinguish the contributions from the Υ(4S ) and Υ(3D) components, respectively. The mixing angle θ ≈ 33 • is suggested in Refs. [12,13]. In addition, the charge conjugation transformation (B ( * ) ↔B ( * ) ) and the isospin transformations on the bridged B ( * ) mesons (B ( * )0 ↔ B ( * )+ andB ( * )0 ↔ B ( * )− ) require a fourfold factor. Finally, the decay widths of the transitions of the Υ(10753) into a low-lying D-wave bottomonium by emitting a light pseudoscalar meson η can be evaluated by where the overbar above amplitude denotes the sum over the polarizations of the Υ(1 3 D J ). The coefficient 1/3 comes from averaging over spins of the initial state. Besides, m is the mass of the Υ(10753), and p η is the three-momentum of η meson in the rest frame of the initial Υ(10753).

III. NUMERICAL RESULT
Before displaying the numerical results, we need to introduce how to fix the values of these involved parameters, which include the masses and the related coupling constants. For the mass and width of the Υ(10753), the measured central values from the Belle Collaboration, m Υ(10753) = 10.753 GeV and Γ Υ(10753) = 35.5 MeV [2], are adopted in our calculation. For the mass of the Υ(1 3 D 2 ), we take its experimental result m Υ(1 3 D 2 ) = 10.164 GeV [40]. For the masses of the Υ(1 3 D 1 ) and Υ(1 3 D 3 ) still missing in experiment, the theoretical results 10.153 and 10.170 GeV predicted in Ref. [6] are taken in our calculation, respectively. Moreover, the PDG values [3] are used for other involved bottom mesons and η meson in this work.
In the following, we should determine the relevant coupling constants. The coupling constants g Υ 1 B ( * ) B ( * ) depicting the coupling between the Υ(3D) and a pair of bottom mesons are extracted from the corresponding decay widths given in Ref. [6]. The corresponding coupling constants are listed in Table I. And then, the g Υ BB value is determined by the corresponding partial decay width given in Ref. [6], while the g Υ BB * value can be fixed by g Υ BB and the relations shown in Eq. (3.1) which is from the heavy quark symmetry. For convenience of reader, the values of these coupling constants are also collected into Table I. The coupling constants defined in Eq. (2.10), Eq. (2.11), and Eq. (2.12) read as where g D = 9.83 GeV −3/2 [21,25]. According to the SU(3) quark model, the observed η and η are mixing of the singlet η 1 and octet η 8 , Thus, the coupling constant g B ( * ) B ( * ) η can be expressed by the coupling constant g H , i.e., where g H = 0.569 and f π = 131 MeV [21,24,25]. The mixing angle θ = −19.1 • had been fixed by the DM2 Collaboration [41]. We also collect the involved coupling constants in Table I. Until now, we have obtained all coupling constants involving in our calculation. However, there exists the phenomenological parameter α Λ introduced in Eq. (2.3) to parametrize the cutoff Λ. Since the cutoff Λ should not deviate from the physical mass of the exchanged meson, α Λ is restricted to be of the order of unity [31]. Since there does not exist direct experimental data to constrain the α Λ value, we have to borrow the experience of the Υ(10860) transition into Υ J (1D)η [42] 1 , where the branching ratio of Υ(10860) → Υ J (1D)η is of the order of magnitude of 10 −3 . To reach up to this order of magnitude, we should take the range 0.2 ≤ α Λ ≤ 0.4, which satisfies the requirement of α Λ [31]. For the Υ(10753) → Υ(1 3 D J )η (J = 1, 2, 3) processes, the α Λ dependence of the discussed branching ratios is displayed in left panel of Fig. 3. From Fig. 3, we can summarize the behavior of the obtained branching ratios by which the corresponding decay widths can be further presented as Additionally, we also notice that the ratios Fig. 3) act weakly dependence of α Λ , i.e., The sizable branching ratios of the Υ(10753) → Υ(1 3 D 1 )η and Υ(10753) → Υ(1 3 D 2 )η decays indicate the probability of finding out them in Belle II. Thus, experimental search for them will be an interesting task for future experiment like Belle II.
With running of Belle II, the physics relevant to the Υ(10753) should be paid more attention. Searching for different decay modes of the Υ(10753) is crucial step of establishing the Υ(10753) as bottomonium. We hope that the present work may provide valuable information to future experimental exploration. (1.18)