Search for the $\eta_{c2}(1D)$ in $e^+e^-\to\gamma\eta_{c2}(1D)$ at $\sqrt{s}$ near 10.6 GeV at Belle

For the first time we search for the $\eta_{c2}(1D)$ in $e^+e^-\to\gamma\eta_{c2}(1D)$ at $\sqrt{s}$ = 10.52, 10.58, and 10.867 GeV with data samples of 89.5 fb$^{-1}$, 711 fb$^{-1}$, and 121.4 fb$^{-1}$, respectively, accumulated with the Belle detector at the KEKB asymmetric energy electron-positron collider. No significant $\eta_{c2}(1D)$ signal is observed in the mass range between 3.8 and 3.88 GeV/$c^2$. The upper limit at 90\% confidence level on the product of the Born cross section for $e^+e^- \to \gamma\eta_{c2}(1D)$ and branching fraction for $\eta_{c2}(1D)\to \gamma h_c(1P)$ is determined to be $\sigma(e^+e^- \to \gamma\eta_{c2}(1D)){\cal B}(\eta_{c2}(1D)\to \gamma h_c(1P))$ $<$ 4.9 fb at $\sqrt{s}$ near 10.6 GeV.

The contemporary interest into heavy quarkonia largely stems from the fact that many experimental results are poorly understood on the theoretical side. Discoveries of new states or new processes, and improved precisions for known states in experiments help to verify various QCD models. Among the charmonium states, a D-wave charmonium state ψ 2 (1D) decaying into γχ c1 was observed in B decays and e + e − annihilations by Belle [1] and BESIII [2], respectively. The mass of ψ 2 (1D) is (3822.2 ± 1.2) MeV/c 2 [1,2], and the upper limit at 90% confidence level (C.L.) on the width is 16 MeV [2]. Using proton-proton collision data, LHCb reported the observation of X(3842) in D 0D0 and D + D − mass spectra [3]. The observed mass and width are (3842.71 ± 0.16 ± 0.12) MeV/c 2 and (2.79 ± 0.51 ± 0. 35) MeV, which suggest the interpretation of the new state as the spin-3 charmonium ψ 3 (1D) state [3]. Only for the spin-singlet low-lying D-wave state η c2 (1D) there has been no experimental clue yet.
The data used in this analysis correspond to 89.5 fb −1 of integrated luminosity at 10.52 GeV, 711 fb −1 at 10.58 GeV, and 121.4 fb −1 at 10.867 GeV, respectively, which were recorded by the Belle detector [18,19] at the KEKB asymmetric-energy e + e − collider [20,21]. The Belle detector was a large solid-angle magnetic spectrometer that consists of a silicon vertex detector, a 50-layer cen-tral drift chamber (CDC), an array of aerogel threshold Cherenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter comprised of CsI(TI) crystals (ECL) located inside a superconducting solenoid coil that provides a 1.5T magnetic field. The ECL is divided into three regions spanning θ, the angle of inclination in the laboratory frame with respect to the direction opposite the e + beam. The ECL backward end cap, barrel, and forward end cap cover ranges of −0.91 < cos θ < −0.65, −0.63 < cos θ < 0.85, and 0.85 < cos θ < 0.98, respectively. An iron flux-return yoke instrumented with resistive plate chambers located outside the coil was used to detect K 0 L mesons and identify muons. Monte Carlo (MC) samples are generated with evtgen [22] to determine detection efficiency and optimize event selection criteria. The simulated events are processed with a detector simulation based on geant3 [23]. No definite model exists for the distribution of polar angle θ γ in the e + e − C.M. system for e + e − → γη c2 (1D) because the combination of tensor-meson production and γ emission is theoretically complicated and requires experimental input. So we model the production as evenly distributed in phase space and account for differences from (1 ± cos 2 θ γ ) distributions as systematic uncertainties. Correction of initial state radiation (ISR) is taken into account in the studied mode, where we assume the Born cross section σ(e + e − → γη c2 (1D)) ∼ 1/s n (n = 2) in the calculation of the radiative correction factor, where s is the e + e − C.M. energy squared. Changing the s dependence of the cross section from n = 2 to n = 1 or n = 4, the maximum difference of the radiative correction factor is considered as the systematic uncertainty. Generic MC samples, i.e. B = B + , B 0 , or B ( * ) s decays and e + e − → qq (q = u, d, s, c) at √ s = 10.52, 10.58, and 10.867 GeV, normalized to the same integrated luminosity as real data, are used to check for possible peaking backgrounds [24].
Except for the charged tracks from the relatively long-lived K 0 S decaying into π + π − , impact parameters with respect to the interaction point are required to be less than 0.5 cm perpendicular to and 2 cm along the beam axis, respectively. For e + e − → γη c2 (1D), we require the numbers of charged tracks, N trk , to be exactly 6 for η c (1S) → 3(π + π − ) and 4 for η c (1S) , also with a zero net charge. For the particle identification (PID) of a well-reconstructed charged track, information from different detector subsystems, including specific ionization in the CDC, time measurement in the TOF, and the response of the ACC, is combined to form a likelihood L i [25] for particle species i. Tracks with R K = L K /(L K + L π ) < 0.4 are identified as pions with an efficiency of 93%, while 6% of kaons are misidentified as pions; tracks with R K > 0.6 are identified as kaons with an efficiency of 94%, while 7% of pions are misidentified as kaons.
Using a multivariate analysis with a neural network [26] based on two sets of input variables [27], a K 0 S candidate is reconstructed from a pair of oppositely charged tracks that are treated as pions. The invariant mass of the K 0 S candidates is required to be within 10 MeV/c 2 (∼ 2.5σ) of the nominal K 0 S mass [28]. The η c (1S) candidates are required to satisfy |M (η c (1S)) − m ηc(1S) | < 60 MeV/c 2 (see below), where m ηc(1S) is the nominal mass of η c (1S) [28].
An ECL cluster is treated as a photon candidate if it is isolated from the projected path of charged tracks in the CDC. The energy of photons is required to be larger than 50 MeV in the barrel and larger than 100 MeV in the endcaps. The ratio of energy deposited in the 3 × 3 array of crystals, with the center being the one with the largest energy deposit, to that in the 5 × 5 array sharing the same center is required to be greater than 0.8. The number of photons in an event is required to be at least 3. The most energetic photon in the C.M. frame (γ max ) is regarded as the primary photon in the process e + e − → γη c2 (1D). Among the remaining photons, all combinations of two photons are considered for the decay chain of η c2 (1D). We call the photon that appears directly from η c2 (1D) decay as γ 1 , and the one from h c (1P ) as [28]. Note that we do not distinguish γ 1 and γ 2 , i.e. rank them by their energies, to increase efficiency and avoid possible bias. In this case, the incorrect attribution rate of γ 1 η c (1S)/γ 2 η c (1S) for h c (1P ) signal is only 0.8% according to signal MC simulations after applying a five-constraint (5C) kinematic fit (see below) for final states from η c (1S), γ max , γ 1 , and γ 2 . In both signal MC and the data, no events have multiple entries. Therefore, the entire decay chain can be written as e + e − → γ max η c2 (1D), A 5C kinematic fit constraining the four-momenta of the final-state particles to the initial e + e − collision system and the invariant mass of γ 2 η c (1S) to the h c (1P ) nominal mass [28] is performed; the fit has five degrees of freedom. The fit is required to satisfy χ 2 5C < 20 to improve the resolutions of the momenta of charged tracks and the energies of photons, and to suppress backgrounds with more than three photons, such as ISR processes. The requirement of χ 2 5C is optimized by maximizing the Punzi parameter S/(3/2 + √ B) [29], where S is the number of signal events in signal MC samples assuming σ(e + e − → γη c2 (1D)) = 1.5 fb [17] and B(η c2 (1D) → γh c (1P )) = 0.5 [7,9,14,15], and B is the number of background events from h c (1P ) mass sidebands in data (see below).
We veto γ from π 0 and η having |M (γ i γ j ) − m π 0 | < 15 MeV/c 2 and |M (γ i γ j ) − m η | < 30 MeV/c 2 , where γ i is either γ 1 or γ 2 and γ j is any photon in the event other than γ i . These veto ranges correspond to about 2.5σ in resolution. The signal efficiencies are decreased by 1.5% and 10.6% after vetoing π 0 and η backgrounds.
After applying the above requirements, the invariant mass distribution for the selected η c (1S) candidates from signal MC samples is shown in Fig. 1(a). The η c (1S) signal shape is described by a Breit-Wigner (BW) convolved with a Gaussian function. The fitted result is shown by the solid curve in Fig. 1(a). Here, we require η c (1S) candidates to satisfy |M (η c (1S)) − m ηc(1S) | < 60 MeV/c 2 , where m ηc(1S) is the nominal mass of η c (1S) [28]. In this region, more than 90% signal events are retained. This relatively tight mass requirement is applied to avoid possible ISR backgrounds with J/ψ candidates.
The invariant mass spectrum, before the application of the 5C kinematic fit, for h c (1P ) candidates from signal MC samples is shown in Fig. 1(b). We fit the M (η c (1S)γ 2 ) distribution by modelling the h c (1P ) signal with the convolution of a Crystal Ball (CB) [30] and Gaussian function due to the asymmetric energy resolution of γ 2 . The h c (1P ) signal is required to satisfy |M (η c (1S)γ 2 ) − m hc(1P ) | < 30 MeV/c 2 (∼ 2.5σ), where m hc(1P ) is the nominal mass of h c (1P ) [28]. h c (1P ) signals can be seen. A few events are around the J/ψ mass point [28], where three candidates are expected from ISR production of ψ(2S), with ψ(2S) → γχ c1 (→ γJ/ψ). The red dashed curves show the required signal regions as above. The blue dashed curves in Fig. 2(b) indicate the h c (1P ) mass sidebands on each side of the signal region. Each sideband is separated from the signal region with a 30 MeV/c 2 gap and has the same width as the signal region. To determine the normalization factor of h c (1P ) mass sidebands, a second-order polynomial function is used to describe the γ 2 η c (1S) invariant mass spectrum, as shown by the blue solid curve in Fig. 2(b).
The invariant mass distributions for η c2 (1D) candidates from √ s = 10.52, 10.58, and 10.867 GeV data samples are shown in Fig. 3. No clear signals can be seen,  and only a few candidates survived in the γ 1 h c (1P ) mass spectra. The green cross-hatched histograms are from the normalized h c (1P ) mass sidebands.
Since the number of selected signal candidate events is small, we obtain the 90% C.L. upper limit of the signal yield (N UL ) at each η c2 (1D) mass point by using the frequentist approach [31] implemented in the POLE (Poissonian limit estimator) program [32], where each mass region is selected to contain 95% of the signal according to MC simulations, the number of signal candidate events is counted directly, and the number of expected background events is estimated from the normalized h c (1P ) mass sidebands. The scan mass region is from 3.8 to 3.88 GeV/c 2 in steps of 4 MeV/c 2 , which corresponds to the half mass resolution of M (γ 1 h c (1P )). The systematic uncertainties discussed below have been taken into account in the POLE program [32].
The upper limit on the product of the Born cross section for e + e − → γη c2 (1D) and branching fraction for η c2 (1D) → γh c (1P ) is calculated by where N UL is the upper limit on the signal events in data, |1 − Π| 2 is the vacuum polarization factor [33,34], L is the integrated luminosity, (1 + δ) ISR is the radiative correction factor [35], and Σ i B i ε i is the sum over decay modes of the product of branching fractions and reconstruction efficiencies. The values of these variables (except N UL ) are summarized in Table I. The values of σ UL (e + e − → γη c2 (1D))B(η c2 (1D) → γh c (1P )) at √ s = 10.52, 10.58, 10.867 GeV, and from the combined data sample are shown in Fig. 4. The upper limit at 90% C.L. on the product of the Born cross section for e + e − → γη c2 (1D) and branching fraction for η c2 (1D) → γh c (1P ) is 4.9 fb at √ s near 10.6 GeV.
There are several sources of systematic uncertainty in the Born cross section measurements, including the detection efficiency, the uncertainty on the estimated signal efficiency due to limited MC statistics, the distribution of polar angle θ γ for e + e − → γη c2 (1D), intermediate state branching fractions, the energy dependence of the cross sections, and the integrated luminosity, which are listed in Table II. The systematic uncertainty in detection efficiency is a final-state-dependent combined uncertainty for all the different types of particles detected, including tracking efficiency, PID, K 0 S selection, and photon reconstruction.
Based on a study of D * + → π + D 0 , D 0 → K 0 S π + π − , the uncertainty in tracking efficiency is taken to be 0.35% per track. The uncertainties in PID for charged kaons and pions are determined to be 1.1% and 1.0%, respectively, based on a low-background sample of D * + decays. The ratio of the efficiencies in K 0 S selection between data and MC simulation is found to be (97.9 ± 0.4 ± 0.6)% in the decay chain of D * + → π + D 0 , D 0 → K 0 S π + π − . We take   2.2% as the systematic uncertainty due to K 0 S selection. The uncertainty in the photon reconstruction is 2.0% per photon, according to a study of radiative Bhabha events. The above individual uncertainties from different η c (1S) reconstructed modes are added linearly, weighted by the product of the detection efficiency and all secondary branching fractions (ε i B i ). Assuming these uncertainties are independent and adding them in quadrature, the final uncertainty related to the reconstruction efficiency is 7.0%.
The statistical uncertainty in the determination of efficiency from signal MC samples is 1.0%. For e + e − → γη c2 (1D), signal events are generated isotropically by default. Alternative angular distributions (1 ± cos 2 θ γ ) are also generated, and the maximum difference in the detection efficiency between the alternatives and the default sample, 13.7%, is taken as the systematic uncertainty. Uncertainties from the h c (1P ) and η c (1S) decay branching fractions are taken from the PDG [28]; the final uncertainties on intermediate state branching fractions are summed in quadrature over all the η c (1S) decay modes weighted by the product of the detection efficiency and secondary branching fractions (ε i B i ).
Changing the s dependence of the cross section from n = 2 to n = 1 or n = 4, the differences of the radiative correction factor are very small (< 1%). Thus, this uncertainty is neglected. The total luminosity is determined to 1.4% precision using wide-angle Bhabha scattering events. All the uncertainties are summarized in Table II and, assuming all the sources are independent, summed in quadrature to give the total systematic uncertainty.