Hidden-bottom hadronic transitions of $\Upsilon(10753)$

Assuming that the $\Upsilon(10753)$ is a $4S$-$3D$ mixed state, we investigated the hidden-bottom hadronic decays of the $\Upsilon(10753) \to \eta_b(1S)\omega(\eta^{(\prime)})$ via the intermediate meson loops. In a commonly accepted range of the model parameter $\alpha$ in the form factor, the predicted branching ratios may reach to the order of $10^{-3}$--$10^{-2}$. The relative ratio of the partial decay widths of the $\Upsilon(10753)\to\eta_b\eta^{(\prime)}$ to $\Upsilon(10753)\to \eta_b\omega$ is found to be dependent on the $\eta$-$\eta'$ mixing angle. In addition, we also calculated the ratios of the partial decays widths of the $\Upsilon(10753) \to \eta_b \omega$ to $\Upsilon(10753)\to \Upsilon(nS)\pi^+\pi^-$ ($n=1\,,2$), which are found to be around 0.4 and 0.2 for $n=1$ and $n=2$, respectively. These values are in accordance with the preliminary experimental results. The calculations presented here tend to favor the $\Upsilon(10753)$ as the $4S$-$3D$ mixture. We hope these predictions could be verified by the future BelleII experiments.


I. INTRODUCTION
Decays of the heavy quarkonia can provide us insight of the dynamics of quark interactions and the formation of hadrons [1][2][3].By means of comparing the theoretical predictions based on quantum chromodynamics (QCD) to the experimental data, we can test the validity of the model and improve our understanding of the strong interaction in the low energy region.Therefore, searching in experiments more new states in the cc and b b sectors and predicting their properties from different theoretical models are of particular importance.As summarized in the Review of Particle Data Group (PDG) [4], huge data samples have been accumulated.Among these samples, there are quite a few complex structures that cannot be interpreted by the conventional quark model, which are usually referred to as exotic or XYZ states, e.g., the celebrated X(3872) [5] and Z b (10610)/Z b (10650) [6] (For review, see Ref. [7][8][9][10][11][12][13]).
The rest of the paper is organized as follows.In Sec.II, we present the theoretical framework used in this work.Then in Sec.III the numerical results are presented, and a brief summary is given in Sec.IV.
Another way to estimate the mixing angle θ is to use the mass-mixing formula [4].In terms of the quadratic mass mixing scheme, the mixing angle θ is governed by A search in the literature yields that the mass of the pure Υ(4S) were theoretically predicted to range from 10607 MeV to 10635 MeV [17][18][19][20][21]47].This mass range leads to the mixing angle between 23.4 • and 36.1 • , agreeing with the results obtained by the foregoing fitting procedure.Moreover, such mass range of the Υ(4S) would require the mass of the pure Υ 1 (3 3 D 1 ) to be 10698-10725 MeV, coinciding also with the early theoretically predicted masses from 10653 to 10717 MeV [17][18][19][20][21]47].These estimations provide ostensible feasibility of the formation of the Υ(10753) by the 4S-3D mixing, which needs verification by comparing various predictions of this mixing scheme to the experimental results.Furthermore, it also indicates that the Υ 1 (3 3 D 1 ) is the dominant component to form the Υ(10753).
In the following, we shall, based on the 4S-3D mixing scenario, investigate the decays of the Υ(10753) to the η b (1S) with emission of the ω, η, and η ′ .

A. The Effective Lagrangians
According to the Heavy Quark Effective Theory (HQET), the interactions of the S-wave bottomonium Υ(nS) and η b (nS) with the bottom and anti-bottom mesons are described by the Lagrangian [32,33,48,49] Here  involved mesons, i.e., The value of g 1 could be determined by the experimental or theoretical branching ratios of the open b-flavored strong decays.For those below the B B threshold, the g 1 could be calculated by [40,45] with the decay constant f Υ , which could be extracted from the dielectron width Γ ee using Eq. ( 3).
The coupling of the D-wave bottomonium Υ 1 (3 3 D 1 ) to a pair of bottom and anti-bottom mesons is [32,33,48] where the coupling constants are governed by another global factor g 2 in the following form Based on the heavy quark limit and chiral symmetry, the interactions of the light vector and pseudoscalar mesons with the heavy bottom mesons read [35,50,51] where the V and P are, respectively, the nonet vector and pseudoscalar mesons in the matrix form The coupling constants g B ( * ) B ( * ) V and g B ( * ) B ( * ) P could be determined using the following relations [50] Here β = 0.9 and g V = m ρ /f π with the pion decay constant f π = 132 MeV [48] and the ρ meson mass m ρ = 775.26MeV [4].Moreover, λ = 0.56 GeV −1 and g = 0.59 based on the matching of the form factors obtained from the light cone sum rule and from the lattice QCD calculations [56].

B. Transition Amplitudes
According to the mixed wave function of the Υ(10753) in Eq. ( 1), the amplitude of the decays we consider is written as where M S and M D are the amplitudes due to the pure Υ(4S) and Υ 1 (3 3 D 1 ) contributions, respectively, of which the proportion is described by the mixing angle θ.
Selecting the neutral loop in Fig. 1(a) as an example, the tensor structures N S(D) µν Additionally, the vector structures N

S(D) µ
for the case of Υ(p) Here D and D µν , respectively, represent the propagators for the scalar B and vector B * in the following form Moreover, the F (q , m B 0 ) is a form factor to account for the off-shell effect as well as the inner structure of the exchanged mesons [57][58][59][60][61]. Here, we adopt a dipole form factor [51] where q and m stand for the momentum and mass of the exchanged meson, respectively; Λ = m + αΛ QCD with Λ QCD = 0.22 GeV [61], in which the α is usually regarded as an undetermined parameter in the vicinity of unity.
In present calculations, we take the α ranging from 0.2 to 1.0.To obtain the charged and strange terms C µ(ν) and S µ in Eq.( 14), we only need to replace the neutral B's by charged and strange ones.
) where a summation over the polarizations of initial and final vector mesons is included.p is the three momentum of the final light mesons.

III. NUMERICAL RESULTS AND DISCUSSION
In Ref. [20], the total width of the pure Υ(4S) was predicted to be 24.7 MeV, of which the branching fraction for the Υ(4S) → B B approaches unity.In view of the experimental measurement that the branching fractions for the Υ(10580) → B 0 B0 and Υ(10580) → B + B − are nearly equal and their sum is larger than 0.96 [4], we therefore assume that the total width of the pure Υ(4S) are saturated by the two processes Υ(4S) → B 0 B0 and Υ(4S) → B + B − with equal proportion.According to the two-body decay model together with the interactions in Eq. ( 5), we get the coupling constant of the Υ(4S) to B B is about 13.343, and g 1 = 0.388 GeV −3/2 .Then, the other relevant constants can be obtained using Eq. ( 6).
To determine the coupling constants for the interactions of the pure Υ 1 (3 3 D 1 ) with the B ( * ) (s)
In order to estimate the coupling constant of the η b and the possible bottom meson pairs, we need to calculate the decay constant of the Υ(1S) using Eq.(3).In terms of Γ ee = 1.340 keV for the Υ(1S) [4], f Υ is about 0.715 GeV, and thereby leading to g 1 = 0.407 GeV −3/2 , which is similar to the result in Ref. [40].For easy reference, we summarized the global factors g 1 and g 2 obtained above in Table I.
Figure 2 shows the branching fractions for the decays of Υ(10753) → η b ω, η b η, and η b η ′ as a function of the TABLE I. Global parameters g1 and g2 (Units: GeV −3/2 ) we employed in the calculations.Their estimations are based on the theoretical and experimental data in Refs.[4,20].model parameter α introduced in the form factor (see Eq. ( 18)).The calculation were performed using the 4S-3D mixing angle of 33 • and the η-η ′ mixing angle of −19.1 • that was determined by DM2 Collaboration [62].
The parameter α is varied from 0.2 to 1.0.It is seen that the results are strongly sensitive to the parameter α, changing from about 10 −5 to 10 −2 .Explicitly, In the absence of relevant experimental data, it seems difficult to narrow down the α range.However, it is recalled that the Υ(10753) have the same quantum numbers J P C = 1 −− with Υ(10580) and Υ(10860), and its mass is between m Υ(10580) and m Υ(10860) .Within the S-D mixing framework, it is, hence, plausible to expect that the Υ(10753) has comparable decay modes to the Υ(10580) and Υ(10860).In view of the fact that the branching fraction for the Υ(10580) → η b ω was measured to be less than 1.8 × 10 −4 and for the Υ(10860) → η b ω it was smaller than 1.3 × 10 −3 [4], we could limit the α value to 0.3 ∼ 0.5.Such small α's or even smaller ones were selected in previous work [33,35].In this range of the α, the predicted partial widths for all the decays we consider are between 1 and 50 keV: It should be noted that the partial decay widths of the Υ(10753) → η b ω we obtained using the S-D mixing scenario are much smaller than those predicted when assigning the Υ(10753) as a tetraquark state, under which the predicted width is (2.46 +4.70 −1.60 ) MeV [22].This great difference is quite favorable for us to distinguish the internal structure of the Υ(10753) when we have relevant experimental data in the future.That is to say, when the future experiments give smaller widths on the order of keV for the Υ(10753) → η b ω, the Υ(10753) appears to favor the 4S-3D mixed state.
In Fig. 3 the branching fractions of the decays Υ(10753) → η b η and η b η ′ are plotted for different η-η ′ mixing angles.These calculations were performed at the fixed model parameter α = 0.5.As seen, with increasing the η-η ′ mixing angle θ P , the branching fraction for the Υ(10753) → η b η decreases distinctly, while for the Υ(10753) → η b η ′ the branching fraction exhibits only a slight increase.In the vicinity of θ P = −18 • , the branching fractions for these two decays are more likely to be equal.The strong α dependence is usually weakened when considering the relative ratios between the branching fractions of different processes.We here define the fol-lowing ratios The calculated relative ratios for two η-η ′ mixing angles of θ P = −19.1 • and θ P = −14.4• are shown in Fig. 4.
It is seen that although the α dependence exists, it is actually weakened strongly in comparison to the absolute branching fractions shown in Fig. 2 that cover 3 orders of magnitude.On the other hand, the ratio R η/ω changes strongly with the η-η ′ mixing angle, while the R η ′ /ω varies slightly.This finding is straightforward in view of the results shown in Fig. 3.Moreover, we calculated the ratio of The calculation procedure for the Υ(10753) → Υ(nS)π + π − are similar with that by Bai et al in Ref. [34] so that the related details are not repeated.The calculated results are shown in Fig. 5.For comparison, we also show the upper limits of the experimental data at 90% confidence level as the points, which are extracted from Refs.[14,64].It is clearly seen that the ratios R η b ω/Υ(nS)ππ are nearly independent of the model parameter α.The theoretical values of R η b ω/Υ(1S)ππ and R η b ω/Υ(2S)ππ are, respectively, around 0.4 and 0.2, which are in line with the experimental measurements with the upper limits of 2.5 ± 1.3 and 0.83 ± 0.28 for the cases of Υ(10753) → Υ(1S)ππ and Υ(2S)ππ, respectively.This finding, to some extent, supports the interpretation of the Υ(10753) as a 4S-3D mixture.

IV. SUMMARY
Calculations of partial decay widths of the Υ(10753) → η b ω, η b η and η b η ′ through the intermediate meson loop mechanism have been performed using an effective Lagrangian approach.In the calculations, we assumed that the Υ(10753) is a 4S-3D mixed state with a mixing angle of 33 • .The branching fractions of these decay processes are predicted to be 10 −4 -10 −3 when the model parameter α is between 0.3 and 0.5, which correspond to partial widths of 1-50 keV.For the decays of Υ(10753) → η b η and η b η ′ , their branching fractions depend on the η-η ′ mixing angle.
Moreover, the relative ratios of the process Υ(10753) → η b ω to Υ(10753) → Υ(nS)π + π − (n = 1 , 2) are found to be in accordance with the experimental results.Our calculated results tend to support the interpretation of the Υ(10753) as a 4S-3D mixture.It is hoped that the present calculated results could be verified by the experiments in BelleII.

FIG. 1 .
FIG. 1. Feynman diagrams for the decay Υ(10753) → η b ω(η (′) ) in the neutral bottom meson loop mechanism.The corresponding charged and strange bottom meson loops are not shown but included in the calculations.The symbol Υ in the diagrams stands for the Υ(4S) and Υ1(3 3 D1).

FIG. 2 .
FIG. 2. (Color online) Branching fractions of the processes Υ(10753) → η b ω, η b η, and η b η ′ .The 4S-3D mixing angle is fixed to be 33 • and the η-η ′ mixing angle is taken to be the widely used value of −19.1 • that was determined by DM2 Collaboration [62].The light-colored bands indicate the margin of error resulting mainly from the errors of the mass and width for the Υ(10753) and of the 4S-3D mixing angle.