$\mathbf{\Upsilon(10753)\to\Upsilon(nS)\pi^+\pi^-}$ decays induced by hadronic loop mechanism

In this work, we investigate the $\Upsilon(10753)\to\Upsilon(nS)\pi^+\pi^-$ ($n=1,2,3$) processes by considering the hadronic loop mechanism, where $\Upsilon(10753)$ is assigned to a conventional bottomonium in the $4S$-$3D$ mixing scheme. Our results of the concerned processes own considerable branching ratios, which can reach up to the order of magnitude of $10^{-4}-10^{-3}$. We should indicate that the measured $\Gamma_{e^+e^-}\times\mathcal{B}[\Upsilon(10753)\to\Upsilon(nS)\pi^+\pi^-]$ values given by Belle can be reproduced well. This fact supports the former bottomonium assignment to the $\Upsilon(10753)$ in the $4S$-$3D$ mixing scheme. Obviously, it is a good opportunity for the ongoing Belle II experiment if the predicted result in this work can be tested further.


I. INTRODUCTION
Heavy quarkonium spectroscopy, especially with the observation of the higher states above the open heavy-flavor thresholds, provides a unique platform to deepen our understanding of the nonperturbative behavior of quantum chromodynamics (QCD) and hints for investigating how quarks form different types of hadrons.As a typical example, there were abundant charmonium and charmoniumlike XYZ states above the D ( * ) D( * ) thresholds (see more details in Refs.[1][2][3][4][5][6]) reported by experiments in the last two decades [7], which greatly enhance our knowledge of hadron physics.However, up to now, only a few members in the bottomonium family have been observed [7] and we should still pay more attention to the construction of the bottomonium family, which has become one of the intriguing topics in the study of hadron spectroscopy.
Facing this anomaly of the Υ(10753), the Lanzhou group proposed the 4S -3D mixing scheme for the Υ(10753) inspired by the research experience of charmonium [22][23][24], where the Υ(10753) can be a mixture of the Υ(4S ) and Υ(3D) states [25].Under this mixing scheme, the mass problem of the Υ(10753) can be understood, and the dielectron width of the Υ(10753) have a significant enhancement due to the mixing of the Υ(4S ) component [25]. 1 Along this line, the hiddenbottom hadronic decays of the Υ(10753) with a η or ω emission was studied [25], which can be used in future experiments.
When facing the new theoretical progress as mentioned above, the story on the Υ(10753) should continue.In this work, we investigate the scalar meson contributions to the hidden-bottom hadronic transitions Υ(10753) → Υ(nS )π + π − (n = 1, 2, 3), by treating the Υ(10753) as a traditional bottomonium state in the 4S -3D mixing scheme, to test whether the experimental results can be reproduced or not.According to the previous experience, the hadronic transitions between low-lying heavy-quarkonium systems can be estimated by the QCD multipole expansion (QCDME) [14,27].However, when solving the problem of higher states of the heavyquarkonium systems whose masses are above the corresponding open flavor thresholds, the coupled-channel effect may play an important role.There are anomalous decay behaviors of higher bottomonia that have been announced in Refs.[8,[28][29][30][31].Besides, enhancement of the decay rate for some spin-flipped transitions, which are forbidden by the heavy quark spin symmetry [32,33], was observed [29,34].
This paper is organized as follows: In Sec.II, we illustrate the detailed calculation of Υ(10753) → Υ(nS )π + π − (n = 1, 2, 3) with the effective Lagrangian approach.Then, we present numerical results in Sec.III.Finally, the paper concludes with a summary.In this section we introduce the hadronic loop mechanism and present the detailed formula of the calculation for Υ(10753) → Υ(nS )π + π − (n = 1, 2, 3) when the Υ(10753) is treated as a conventional bottomonium state in the 4S -3D mixing scheme.Under the framework of the hadronic loop mechanism, the Υ(10753) firstly decays into a bottom meson pair, and then the bottom meson pair is converted into the final state of the Υ(nS ) and a light scalar meson by exchanging a bottom meson.Finally, the intermediate light scalar meson decays into π + π − .The concrete diagrams are shown in Fig. 1.It is worth noting that the contribution from the B s Bs loop is not included in this work due to the weak coupling of the Υ(10753) with B s Bs [54].
In Eq. (2.1), a monopole form factor F (q 2 , m 2 E ) is introduced to compensate the off shell effect of the exchanged B ( * ) meson and represent the structure effect of the interaction vertices [35,[58][59][60], i.e., is adopted with m E and q representing the mass and momentum of the exchanged bottom meson, respectively.Here, Λ, the cutoff parameter, can be parametrized as Λ = m E + α Λ Λ QCD with Λ QCD = 220 MeV [36][37][38], and α Λ is expected to be of the order of unity to ensure that the cutoff Λ does not deviate from the physical mass of the exchanged meson [35].
In the framework of the 4S -3D mixing scheme, for the Υ(10753), the decay amplitude is expressed as where θ 33 • [25] is the mixing angle, and the factor 4 comes from the charge conjugation and the isospin transformation on the bridged B ( * ) meson.In this work, we take both σ and f 0 (980) contributions into account.If taking approximation of ignoring the interference between the σ and f 0 (980) contributions 3 , the total amplitude is given by Finally, the differential decay width can be obtained by ) where the overbar denotes summation over the polarizations of the Υ(nS ), and the coefficient 1/3 comes from an average over the polarizations of the initial state.p Υ(nS ) is the threemomentum of Υ(nS ) in the initial state rest frame, and p * π is the three-momentum of a π meson in the center-of-mass frame of a di-pion system.m ππ is the π + π − invariant mass.Besides, Ω Υ(nS ) and Ω π are the solid angles of p Υ(nS ) and p * π , respectively.
In particular, since the σ dominantly decays into dipion, the Γ σ [7] can be used to determine the coupling constant g σππ .
For the f 0 (980), it dominantly decays into a pair of pions or kaons, so the ratio Apart from the fixed coupling constants, there still exists a free parameter α Λ introduced in the form factor.Thus, in Fig. 3, we present the α Λ dependence of the obtained branching ratios.
Next, we turn to the experimental status.Till now, there have not been any measurements on the corresponding widths.But some other concerned data, e.g.[8], were presented by the Belle Collaboration as R 1 = 0.295 ± 0.175 eV, R 2 = 0.875 ± 0.345 eV, R 3 = 0.235 ± 0.025 eV, 0 .7 0 .9 1 . 1 1 .3 1 .5 1 .7 0 .5 Here, the red solid lines are our predicted values by the hadronic loop mechanism, while the LT Gray bands with the blue dotted lines represent the extracted ones with errors.We should indicate that the common α Λ range is fixed as 0.5 < α Λ < 1.8 for the discussed transitions since α Λ is of order 1 as suggested in Ref. [35].which can shed light on some features about the partial decay widths.In other words, once the dielectron decay width is fixed, the branching rates can be extracted.The dielectron width can be determined in the following steps.In the framework of the 4S -3D mixing scheme, the dielectron decay width of the Υ(10753) is [22] Here, M is the mass of the Υ(10753), e b = −1/3 is the charge of the b quark, α is the fine structure constant, and α s = 0.18 [10].Besides, R 4S and R 3D are the radial parts of the Υ(4S ) spatial wave function and the second derivative of the radial part of Υ(3D) spatial wave function, respectively.By sub-stituting R 4S (0) and R 3D (0) extracted from Ref. [10], and the mixing angle θ = (33 ± 4) • fixed by Ref. [25] into Eq.(3.3), the dielectron decay width of the Υ(10753) is obtained as (0.159 ± 0.030) keV.Finally, the concerned branching ratios are estimated as where the large uncertainness mainly come from the poor accuracies of R n .The α dependence of the calculated branching ratios is given in Fig. 2, where the common α Λ range is fixed as 0.5 < α Λ < 1.8 since α Λ is of order 1 as suggested in Ref. [35].As shown in Fig. 3, we give the α Λ range after matching the calculated numerical branching ratios B[Υ(10753) → Υ(nS )π + π − ] (n = 1, 2, 3) with the extracted ones.In the following, we should discuss the reasonable values of α Λ .For the Υ(10753) → Υ(nS )π + π − ] (n = 1, 2), there exists a common α Λ range around 1.2, where the extracted B[Υ(10753) → Υ(nS )π + π − ] (n = 1, 2) can be well reproduced.This fact may reflect the similarity between Υ(10753) → Υ(1S )π + π − and Υ(10753) → Υ(2S )π + π − .What is more important is that this α Λ range is consistent with the requirement of determining α Λ value as suggested in Ref. [35], where α Λ is expected to be of order unity [35].For the discussed Υ(10753) → Υ(3S )π + π − , only if α Λ is reduced to about 50% of 1.2, the extracted B[Υ(10753) → Υ(3S )π + π − ] can be reproduced.Thus, in a reasonable region of α Λ , the extracted branching ratios B[Υ(10753) → Υ(nS )π + π − ] can be reproduced by introducing the hadronic loop mechanism.In other words, the calculated results are comparable with the measured R n = Γ e + e − × B[Υ(10753) → Υ(nS )π + π − ] values [8] by treating the Υ(10753) as a mixture of Υ(3D) and Υ(4S ) states.We should indicate that a direct measurement of branching ratios of these three discussed decays is still lacking.The ongoing Belle II experiment on measuring the absolute branching rates is necessary for helping us to make further constraints on the parameter α Λ .Besides, we also present the di-pion invariant mass spectrum distributions dΓ[Υ(10753) → Υ(nS )π + π − ]/dm π + π − in Fig. 4, where the maxima of the theoretical line shapes are all normalized to 1. Judging from the current node in the experiment, the predicament that the experiment lacks the direct measurement on the partial decay widths makes it difficult to make a firm judgment.Thus, we expect further measurements on the partial decay widths, as well as the di-pion invariant mass spectrum distributions, from the running Belle II experiment.They will play essential roles both in enriching our knowledge about these transitions and further identifying the coupledchannel effect.
In this work, we have investigated the scalar meson contributions to Υ(10753) → Υ(nS )π + π − (n = 1, 2, 3) processes in the same hypothesis with the effective Lagrangian approach.By taking the hadronic-loop mechanism into account, the corresponding transitions acquire considerably large branching ratios and can reach up to 10 −4 − 10 −3 .Additionally, our results can reproduce the Γ e + e − × B[Υ(10753) → Υ(nS )π + π − ] measured by Belle [8] well with a reasonable cutoff parameter α Λ , which strongly supports our assumption of Υ(10753) that it is the 4S -3D mixture.In addition, the line shape of the di-pion invariant mass spectrum distributions dΓ[Υ(10753) → Υ(nS )π + π − ]/dm π + π − are also presented, which should be used to identify the coupled-channel effects of B meson loops by the future experiments by Belle II.
In conclusion, the precise measurement on the resonance parameters, e.g. the decay modes, the di-pion invariant mass spectrum distributions of Υ(10753) would help us further confirm its nature.We suggest that the experimentalists pay more continuous attention on this issue.With joint efforts of theorists and experimentalists, the nature of Υ(10753) will be fully understood in future.Meanwhile, we expect to see more and more bottomonium and bottomonium-like states in the ongoing and forthcoming experiments, especially the Belle II experiment, which would lead us to a new era of hadron physics.