Exploring the $\Upsilon(4S,5S,6S) \to h_b(1P)\eta$ hidden-bottom hadronic transitions

Recently, Belle Collaboration has reported the measurement of the spin-flipping transition $\Upsilon(4S) \to h_b(1P)\eta$ with an unexpectedly large branching ratio: $\mathcal{B}(\Upsilon(4S) \to h_b(1P)\eta) =(2.18\pm 0.11\pm 0.18)\times 10^{-3}$. Such a large branching fraction contradicts with the anticipated suppression for the spin flip. In this work, we examine the effects induced by intermediate bottomed meson loops and point out that these effects are significantly important. Using the effective Lagrangian approach (ELA), we find the experimental data on $\Upsilon(4S) \to h_b(1P)\eta$ can be accommodated with the reasonable inputs. We then explore the decays $\Upsilon(5S,6S)\to h_b(1P)\eta$ and find that these two channels also have sizable branching fractions. We also calculate these these processes in the framework of nonrelativistic effective field theory (NREFT). For the decays $\Upsilon(4S) \to h_b(1P) \eta$, the NREFT results are at the same order of magnitude but smaller than the ELA results by a factor of $2$ to $5$. For the decays $\Upsilon(5S, 6S) \to h_b(1P) \eta$ the NREFT results are smaller than the ELA results by approximately one order of magnitude. We suggest future experiment Belle-II to search for the $\Upsilon(5S, 6S)\to h_b(1P) \eta$ decays which will be helpful to understand the transition mechanism.

fractions. We also calculate these these processes in the framework of nonrelativistic effective field theory (NREFT). For the decays Υ(4S) → h b (1P )η, the NREFT results are at the same order of magnitude but smaller than the ELA results by a factor of 2 to 5. For the decays Υ(5S, 6S) → h b (1P )η the NREFT results are smaller than the ELA results by approximately one order of magnitude. We suggest future experiment Belle-II to search for the Υ(5S, 6S) → h b (1P )η decays which will be helpful to understand the transition mechanism.
A low-lying heavy quarkonium system is expected to be compact and nonrelativistic, so the QCD multipole expansion (QCDME) [8][9][10] can be applied to explore the hadronic transitions.
For the excited states that lie above open flavor thresholds, QCDME might be problematic due to the coupled channel effects. Several possible new mechanisms have been proposed in order to explain the anomalous decay widths of Υ(4S) → h b (1P )η. For instance a nonrelativistic effective field theory (NREFT) is used in Ref. [11], where the branching ratio can reach the order of 10 −3 .
It has been noticed for a long time that the intermediate meson loop (IML) is one prominent nonperturbative mechanism in hadronic transitions [12][13][14]. In recent years, this mechanism has been successfully applied to study the production and decays of ordinary and exotic states , and a global agreement with experimental data is found. This approach has also been extensively used to study the Υ(4S, 5S, 6S) hidden bottomonium decays [46][47][48][49][50][51][52]. In this work, we will investigate the process Υ(4S, 5S, 6S) → h b (1P )η via IML model. As we will show in the following the experimental data on Υ(4S) → h b (1P )η can be accommodated in this approach. We then predict the branching ratios of the decays Υ(5S, 6S) → h b (1P )η and find that they are measurable in future.
The rest of this paper is organized as follows. We will first introduce the effective Lagrangian for our calculation in Sec. II and calculate the IML contributions to decay widths. Then, we will present our numerical results in Sec. III. A brief summary will be given in Sec. IV. In reality, we only pick up the leading order contributions as a reasonable approximation due to the breakdown of the local quark-hadron duality [12,53]. In this work, we consider the IML illustrated in Fig. 1 as the leading order contributions of Υ(4S, 5S, 6S) → h b (1P )η. To calculate these diagrams, we need the effective Lagrangians to derive the couplings. Based on the heavy quark symmetry and chiral symmetry [54,55], the Lagrangian for the S-and P-wave bottomonia at leading order is given as

II. RADIATIVE DECAYS
The S-wave bottomonium doublet and P-wave bottomonium multiplet states are expressed as where Υ and η b are the S-wave bottomonium fields. The h b and χ bJ (J=0,1,2) are the P-wave bottomonium fields. The v µ is the 4-velocity of these bottomonium states.
The bottomed and anti-bottomed meson triplet read where B and B * denote the pseudoscalar and vector bottomed meson fields, respectively, i.e. B ( * ) = . v µ is the 4-velocity of the bottomed mesons. ε µναβ is the antisymmetric Levi-Civita tensor and ε 0123 = +1.
Consequently, the relevant effective Lagrangian for S-wave Υ(nS) and P-wave h b (1P ) read where the coupling constants will be determined later.
The effective Lagrangian for a light pseudoscalar meson coupled to bottomed mesons pair can be constructed using the heavy quark symmetry and chiral symmetry [54][55][56] where P is a 3×3 matrix for the pseudoscalar octet. The physical states η is the linear combinations of nn = (uū + dd)/ √ 2 and ss with the mixing scheme: The mixing angle is given as α P ≃ θ P + arctan √ 2, where the empirical value for the θ P should be in the range −24.6 • ∼ −11.5 • [57]. In this work, we will take With the above Lagrangians, we can derive the transition amplitudes for Υ(nS)(p 1 ) → where p 1 , p 2 and p 3 are the four momenta of the initial state Υ(nS), final state h b (1P ) and η, respectively. ε 1 and ε 2 are the polarization vector of Υ(nS) and h b (1P ), respectively. q 1 , q 3 and q 2 are the four momenta of the bottomed meson connecting Υ(nS) and η, the bottomed meson connecting Υ(nS) and h b (1P ), and the exchanged bottomed meson, respectively.
In the triangle diagrams of Fig. 1, the exchanged bottomed mesons are off shell. To compensate the offshell effects and regularize the ultraviolet divergence [59][60][61], we introduce the monopole form factor, where q 2 and m 2 are the momentum and mass of the exchanged bottomed meson, respectively.
The parameter Λ ≡ m 2 + αΛ QCD and the QCD energy scale Λ QCD = 220MeV. The dimensionless parameter α, which is usually of order 1, depends on the specific process.

III. NUMERICAL RESULTS
With the experimental data on the decay width of Υ(4S) → BB [57], the coupling constant g Υ(4S)BB is determined as g Υ(4S)BB = 24.2 which is comparable to the estimation in the vector For the coupling constants between Υ(5S) and B ( * )B( * ) , we use the experimental data on the decay width of Υ(5S) → B ( * )B( * ) [57]. The measured branching ratios and the corresponding coupling constants are given in Table I. One can see that the values determined from the Υ(5S) data in Table I are very small. This is partly due to the fact that as a high-excited bb state, the wave function of Υ(5S) has a complicated node structure, and the coupling constants will be small The coupling constants between h b (1P ) and B ( * )B * in Eq. (9) are determined as where g 1 = − m χ b0 /3/f χ b0 . m χ b0 and f χ b0 are the mass and decay constant of χ b0 (1P ), respectively [62], i.e. f χ b0 = 175 ± 55 MeV [63].
In the chiral and heavy quark limits, the couplings between bottomed meson pair and light pseudoscalar mesons have the following relationships [55], where f π = 132 MeV is the pion decay constant, and g = 0.59 [64]. For the bottom meson loop contributions in Fig. 1, the decay amplitude scales as follows, In Fig. 2 (a), we plot the branching ratios for Υ(4S) → h b (1P )η in terms of the cutoff parameter α with the monopole form factor. We also zoom into details of the figure with a narrow range α = 0.1−0.2 in order to show the best fit of the α parameter. As shown in Fig. 2 (a), the branching ratios are not drastically sensitive to the cutoff parameter α. Our calculated branching ratios can reproduce the experimental data [57] at about α = 0.12. In Fig. 2  Here, we should notice several uncertainties may influence our numerical results, such as the coupling constants and off-shell effects arising from the exchanged particles of the loops, and the cutoff parameter can also be different in decay channels.
In order to illustrate the impact of the η-η ′ mixing angle, in Fig. 3, we present the branching ratios in terms of the η-η ′ mixing angle with α = 0.15 (solid line) and 0.25 (dashed line), respectively. As shown in this figure, when the η-η ′ mixing angle α P increases, the branching ratios of Υ(4S) → h b (1P )η decrease, while the branching ratios of Υ(5S, 6S) → h b (1P )η increase. This behaviour suggests how the η-η ′ mixing angle influences our calculated results to some extent.
As a comparison, in Fig. 3, we also give the results using the NREFT approach denoted as dotted lines. The NREFT approach provides a systematic tools to control the uncertainties [11,34,65].