Search for $\psi(4S)\to\eta J/\psi$ in $B^\pm \ra \eta J/\psi K^\pm$ and $e^+ e^- \to \eta J/\psi$ processes

We search for the $\psi(4S)$ state in $B^\pm \ra \eta J/\psi K^\pm$ and $e^+e^- \ra \eta J/\psi$ based on the Belle measurements with the assumed mass $M=(4230\pm8)$ MeV/$c^2$ and width $\Gamma=(38\pm12)$ MeV. No significant signal is observed in the $\eta J/\psi$ mass spectra. The 90\% confidence level upper limit on the product branching fraction $\BR(B^\pm \ra \psi(4S)K^\pm)\BR(\psi(4S) \ra \eta J/\psi)<6.8\times 10^{-6}$ is obtained in $B^\pm \ra \eta J/\psi K^\pm$ decays. By assuming the partial width of $\psi(4S) \to e^+e^-$ being 0.63 keV, the branching fraction limit $\BR(\psi(4S) \ra \eta J/\psi)<1.3\%$ is obtained at the 90\% confidence level in $e^+e^- \ra \eta J/\psi$, which is consistent with the theoretical prediction.


Introduction
Potential models predict some charmonium states above the DD threshold, but the number of observed states in experiments is more than the number predicted. The states which have been seen outwith the theoretical predictions are normally referred to as exotic states or XYZ particles. Many XYZ states have been announced in various processes, for example, the observation of X(3872) in B decays [1], the Y (4260) [2], Y (4360) [3] and Y (4660) [4] in e + e − annihilation, and the X(3915) [5] observed in the two-photon process.
On the other hand, there are still some charmonium states predicted by the potential models which have not yet been observed experimentally, especially in the mass region higher than 4 GeV/c 2 , such as η c (3S), η c (4S), ψ(4S) and ψ(5S). To some degree, some XYZ states are regarded as candidates for these unfound predicted states.
Searching for these missing predicted states is very helpful to test the potential models. When checking the mass spectra of the observed charmonia with spin-parity J P C = 1 −− and comparing them with those of the corresponding bottomonia, there might be a charmonium state ψ(4S) at about 4.2 GeV/c 2 compared to the Υ(4S) state [6]. The authors in Ref. [6] predicted that this missing charmonium state has a mass of 4.263 GeV/c 2 and a very narrow width. As a state with the same spin-parity 1 −− , the Y(4220) [7] may be a good candidate for the Recently, the BESIII Collaboration performed a study on the decay e + e − → ωχ cJ (J = 0, 1, 2) [7], where the Born cross sections at 9 energy points were measured. When using a Breit-Wigner (BW) function to fit the experimental data of e + e − → ωχ c0 , a resonant structure with mass M = (4230 ± 8) MeV/c 2 and width Γ = (38±12) MeV was observed with a statistical significance more than 9σ. However, for the remaining processes e + e − → ωχ c1 and e + e − → ωχ c2 , there were no significant signals.
The authors in Ref. [9] checked the thresholds of ωχ c0 , ωχ c1 and ωχ c2 , which are 4.197 GeV/c 2 , 4.293 GeV/c 2 and 4.338 GeV/c 2 , respectively. The central mass of ψ(4S) is just above the ωχ c0 threshold and below the ωχ c1,2 thresholds. Accordingly the newly observed structure in e + e − → ωχ c0 could be the missing charmonium ψ(4S) state, and the e + e − → ωχ c1,2 processes are kinematically forbidden to ψ(4S). Stimulated by this, the authors estimated the meson loop contribution to ψ(4S) → ωχ c0 and found the evaluation can overlap with the experimental data in a reasonable parameter range. As a typical transition accessible by experiment, the decay ψ(4S) → ηJ/ψ similar to ψ(4S) → ωχ c0 can occur.
As indicated in Ref. [9], the predicted upper limit of ψ(4S) → ηJ/ψ can be accessible at Belle and the forthcoming BelleII. We noticed that the Belle experiment previously measured the B ± → ηJ/ψK ± [11] and e + e − → ηJ/ψ [12] processes, where the ηJ/ψ invariant mass distributions were given. Hence, in this work we fit the ηJ/ψ mass spectra from the B ± → ηJ/ψK ± and e + e − → ηJ/ψ processes to search for the ψ(4S) state. The experimental measurements can be taken as a test of the theoretical calculation.
This work is organized as follows. We present the detailed fit results to the ηJ/ψ mass spectra from B ± → ηJ/ψK ± and e + e − → ηJ/ψ processes with the ψ(4S) state included in Sec. 2 and Sec. 3. If no clear ψ(4S) signal is observed, the branching fraction limits at the 90% confidence level (C.L.) will be given with the systematic errors included. The last section ends with the conclusion and discussion.

Search for ψ(4S) in B Decays
Using 772×10 6 BB pairs collected with the Belle detector, the decays B ± → ηJ/ψK ± were studied to search for a new narrow charmonium(-like) state X in the ηJ/ψ mass spectrum, where the J/ψ and η mesons were reconstructed by a lepton-pair ℓ + ℓ − (ℓ = e, µ) and two photons [11]. Except for the known ψ ′ → ηJ/ψ decay, no significant narrow excess was found in the ηJ/ψ mass spectrum. Figure 1 shows the ηJ/ψ mass distribution of interest after all the event selection requirements are applied. A binned maximum likelihood fit to the ηJ/ψ mass distribution is performed to extract the signal and background yields. A BW function (mass and width fixed at 4.23 GeV/c 2 and 38 MeV [9]) is convolved with a Gaussian function (the mass resolution is about 11 MeV/c 2 ) as the ψ(4S) signal shape and a second polynomial function is taken as the background shape. The fit range and results to the ηJ/ψ mass spectrum are shown in Fig. 1.
From the fit, we obtain 5.9±5.5 signal events, with a statistical significance of 0.9σ, from the difference of the logarithmic likelihoods, −2 ln(L 0 /L max ), taking the difference in the number of degrees of freedom (∆ndf = 1) in the fits into account, where L 0 and L max are the likelihoods of the fits without and with a resonance component, respectively.
We determine a Bayesian 90% C.L. upper limit on the number of ψ(4S) signal events (N sig ) by finding the value where N sig is the number of ψ(4S) signal events and L is the value of the likelihood as a function of N sig . To take into account the systematic uncertainty, the above likelihood is convolved with a Gaussian function whose width equals the total systematic uncertainty described below. The upper limit on the number of ψ(4S) signal events is 22.7 at 90% C.L. There are several sources of systematic error for the branching fraction measurement. Most of the systematic errors are the same as those in Ref. [11], except that the dominant uncertainty associated with the fitting procedure is different, which is estimated by changing the order of the background polynomial, the range of the fit, the ψ(4S) mass and width by ±1σ. Finally, the uncertainty due to the fitting procedure is 11%. Assuming all the sources are independent and adding them in quadrature, the final total systematic uncertainties are summarized in Table 1. The 90% C.L. upper limit is set on the product branching fraction B(B ± → ψ(4S)K ± )B(ψ(4S) → ηJ/ψ) using where N UP sig , N BB , ǫ, B(J/ψ → ℓ + ℓ − ) and B(η → γγ) are the upper limit on the number of ψ(4S) signal events at 90% C.L., the number of BB pairs, the corrected detection efficiency of 9.23% at 4.23 GeV/c 2 obtained from the fitted efficiency curve using the efficiencies at ψ ′ , ψ(4040) and ψ(4160) points [11], the branching fractions of J/ψ to lepton pair and η to two photons [13], respectively. Finally, the 90% C.L. upper limit on the product branching fraction B(B ± → ψ(4S)K ± )B(ψ(4S) → ηJ/ψ) is found to be 6.8 × 10 −6 .
To obtain the transition rates of ψ(4040) and ψ(4160) to the ηJ/ψ final state, an unbinned maximum likelihood fit was performed to the ηJ/ψ mass spectra from the signal candidate events and the η and J/ψ sideband events simultaneously [12]. The fit to the signal events includes two coherent P -wave BW functions convolved by the effective luminosity and efficiency curve for ψ(4040) and ψ(4160) signals and an incoherent second-order polynomial background; the fit to the sideband events includes the same background function only. Due to the low statistics, the masses and widths of the ψ(4040) and ψ(4160) were fixed [14] and the effects of mass resolution were small and therefore were neglected [12].
Similarly here, to obtain the transition rate of ψ(4S) to ηJ/ψ final state, a binned maximum likelihood fit with three coherent P -wave BW functions for ψ(4040), ψ(4160) and ψ(4S) is applied to the e + e − → ηJ/ψ cross sections directly, as shown in Fig. 2. In the fits, besides the masses and widths of ψ(4040) and ψ(4160) are fixed [14], the ψ(4S) parameters are also fixed [9]. Figure 2 and Table 2 show the fit results. There are four solutions with equally good fit quality. The results of BΓ e + e − for ψ(4040) and ψ(4160) are consistent with the published results within errors [12]. The significance of the ψ(4S) is estimated by comparing the likelihood of fits with and without ψ(4S) included. We obtain a statistical significance of 2.6σ. The most conservative upper limit with the systematic errors included on B(ψ(4S) → ηJ/ψ)Γ ψ(4S) e + e − is obtained to be 8.2 eV at 90% C.L., which corresponds to Solution III in Fig. 2(c).
The expected integrated luminosity at the BelleII experiment is 50 ab −1 in 2024, which is about 50 times the current total integrated luminosity at Belle. With this huge data sample, the expected upper limit on B(ψ(4S) → ηJ/ψ) will be 1.9×10 −3 if it scales as 1/ √ L, where L is the integrated luminosity, and can therefore reach the theoretical prediction level.