Observation of a structure in $e^{+}e^{-} \to \phi \eta^{\prime}$ at $\sqrt{s}$ from 2.05 to 3.08 GeV

The process $e^{+}e^{-} \to \phi \eta^{\prime}$ has been studied for the first time in detail using data sample collected with the BESIII detector at the BEPCII collider at center of mass energies from 2.05 to 3.08 GeV. A resonance with quantum numbers $J^{PC}=1^{--}$ is observed with mass $M$ = (2177.5 $\pm$ 4.8 (stat) $\pm$ 19.5 (syst)) MeV/${ \it{c}^{\mathrm{2}}}$ and width $\Gamma$ = (149.0 $\pm$ 15.6 (stat) $\pm$ 8.9 (syst)) MeV with a statistical significance larger than 10$\sigma$. The observed structure could be identified with the $\phi(2170)$, then the ratio of partial width between the $\phi \eta^{\prime}$ by BESIII and $\phi \eta$ by BABAR is ($\mathcal{B}^{R}_{\phi \eta}\Gamma^{R}_{ee})/{(\mathcal{B}^{R}_{\phi \eta^{\prime}}\Gamma^{R}_{ee})}$ = 0.23 $\pm$ 0.10 (stat) $\pm$ 0.18 (syst), which is smaller than the prediction of the $s\bar{s}g$ hybrid models by several orders of magnitude.

Peking University, Beijing 100871, People's Republic of China l School of Physics and Electronics, Hunan University, Changsha 410082, China (Dated: March 31, 2020) The process e + e − → φη ′ has been studied for the first time in detail using data sample collected with the BESIII detector at the BEPCII collider at center of mass energies from 2.05 to 3.08 GeV. A resonance with quantum numbers J P C = 1 −− is observed with mass M = (2177.5 ± 4.8 (stat) ± 19.5 (syst)) MeV/c 2 and width Γ = (149.0 ± 15.6 (stat) ± 8.9 (syst)) MeV with a statistical significance larger than 10σ. The observed structure could be identified with the φ(2170), then the ratio of partial width between the φη ′ by BESIII and φη by BABAR is (B R φη Γ R ee )/(B R φη ′ Γ R ee ) = 0.23 ± 0.10 (stat) ± 0.18 (syst), which is smaller than the prediction of the ssg hybrid models by several orders of magnitude.
PACS numbers: 13.25.Gv,12.38.Qk,14.20.Gk,14.40.Cs One of the most challenging questions in contemporary physics is how quarks and gluons form hadrons. Quantum Chromodynamics (QCD) allows for any color neutral combinations, however, a striking majority of all observed hadronic states are consistent with either a quarkantiquark pair, i.e. mesons, or triplet quark systems, i.e. baryons. Although it has been difficult to unambiguously identify so-called exotic hadrons, such as glueballs, hybrids and multiquarks, remarkable progress has been made in the charm sector during the last decade. Some of those newly observed charmonium-like states are good candidates for exotics [1,2]. The strangeonium family maybe have states similar to those found in heavier quarkonia, and the experimental information for the spectrum of strangeonium is terra incognita.
The φ(2170) was first discovered in the process e + e − → φf 0 (980) by BABAR [3,4] via the initial-state-radiation (ISR) technique, and was later confirmed later by Belle [5], BES [6] and BESIII [7,8]. There are several interpretations of the φ(2170), including a regular ss meson in a 2 3 D 1 [9] or 3 3 S 1 configuration [10], an ssg hybrid [11,12], a tetraquark state [13][14][15][16], a ΛΛ bound state [17][18][19], an S-wave threshold effect [20], or a threemeson system φK + K − [21]. The conventional ss meson is predicted to decay with significant fraction into the ss-signature modes φη and φη ′ [10]. According to the Okubo-Zweig-Iizuka (OZI) rule [22] and taking isospin effect into account, the contributions of ω-like and ρ-like states are suppressed in the φη and φη ′ modes. These two decay modes are useful to measure the mass and width of φ-like states. On the other hand, the ssg hybrid state is expected to have a stronger coupling to φη, whose partial width is expected to be larger than that of φη ′ by a factor of 3-200 [11,12]. The ratio between the φη and φη ′ decay widths is therefore an important observable to test φ(2170) as a hybrid state.
BESIII measured the processes e + e − → K + K − [23] and e + e − → ΛΛ [24] to test the prediction of φ(2170) as the ΛΛ bound state. An enhancement at √ s = 2.232 GeV in the process e + e − → φK + K − [25] is difficult to be interpreted by the Faddeev calculation for the three-meson system φKK. Assuming that the observed structure in the process e + e − → K + (1460)K − is φ(2170), it implies that the theoretical expectation for the hybrid state is not in agreement with the experimental results [26]. BABAR observed evidence of φ(2170) in a study of the process e + e − → γ ISR φη [27] and a small signal in e + e − → γ ISR φη ′ [28]. The tail of the φ(1680) contributes to the φη mode. However, φ(1680) decays into φη ′ are highly suppressed [29]. In a BESIII study of J/ψ → ηφη ′ , evidence was found of a structure in the φη ′ mass spectrum in the 2.0 − 2.1 GeV/c 2 region under the assumption of J P = 1 − [30]. The e + e − → φη ′ process provides therefore important input for the understanding of the φ(2170).
In this Letter, we present a measurement of the Born cross sections of e + e − → φη ′ as a function of center-ofmass (c.m.) energies from 2.05 to 3.08 GeV based on 20 data samples corresponding to an integrated luminosity of 640 pb −1 collected at the Beijing spectrometer (BE-SIII).
The BESIII detector is a magnetic spectrometer located at the Beijing Electron Position Collider (BEPCII) [31]. A detailed description of the detector and its performance can be found in Ref. [32]. A geant4based [33] Monte Carlo (MC) simulation software is used to generate simulated data samples. The software implementation includes geometric and material description of the BESIII detector, the detector response and digitization models. It also accounts for the variation in detector running conditions and performance.
To study backgrounds, a generic MC sample for the process e + e − → qq with q = u, d, s is generated with conexc [34], while the hadronization processes are generated by evtgen [35,36] for known modes with branching fractions set to Particle Data Group (PDG) world average values [37] and by luarlw [38] for the remaining unknown decays. The signal MC sample for e + e − → φη ′ is also generated by conexc, taking radiative corrections, the angular distributions of the final state and the amplitude of η ′ → γπ + π − [39] into account at each c.m. energy point.
The MC simulations are used to optimize the selection criteria, the determination of detection efficiencies and estimation of the background. Taking the branching fractions of intermediate states and the efficiency of photon detection into consideration for the process of e + e − → φη ′ , the φ candidate is identified from a K + K − pair and the η ′ from π + π − γ combinations. To improve the detection efficiency, candidate events are required to have three or four good charged tracks, corresponding to two detected pions and one or two detected kaons, and at least one good photon. Tracks are reconstructed from hits in the multilayer drift chamber (MDC) within | cos θ| < 0.93, where θ is the polar angle with respect to the magnetic field direction. The tracks are required to pass the interaction point within 10 cm along the beam direction and within 1 cm in the transverse direction to the beam. For each charged track, the time-of-flight (TOF) and the energy loss measurement (dE/dx) information are combined to form particle identification confidence levels (C.L.) for the π, K, and p hypotheses. The particle type with the highest C.L. is assigned to each track. Two pions and at least one kaon are required per event.
Photon candidates are selected from showers in the electromagnetic calorimeter (EMC) that are not associated with charged tracks. Good photon candidates reconstructed in the barrel part of the EMC must have a polar angle within | cos θ| < 0.8 and a minimum deposited energy of 25 MeV. To be reconstructed in the end caps, the photon candidates must have a polar angle within 0.86 < | cos θ| < 0.92 and a minimum energy deposit of 50 MeV. Timing information in the EMC is used to suppress electronic noise and energy deposits unrelated to the event. In order to suppress the background from ISR processes, the energy of photon candidates is required to be larger than 70 MeV. The tracks and photon candidates are then combined and subject to further analysis. The interaction vertex of the event is reconstructed by two pions and one kaon. A one-constraint (1C) kinematic fit is performed under the hypothesis that the Kπ + π − γ missing mass corresponds to the kaon mass [37]. If both kaons are identified in an event, the combination with the smallest χ 2 of the 1C kinematic fit is retained. The corresponding χ 2 , denoted as χ 2 1C (π + π − KK miss γ), is required to be smaller than 20. The candidate event of e + e − → φη ′ is required to be within the φ signal region, defined as |M (KK miss ) − m φ | < 0.015 GeV/c 2 . The sideband region, defined as 1.050 GeV/c 2 < M (KK miss ) <1.130 GeV/c 2 , is used to estimated the non-φ background contributions.
A study of the e + e − → qq MC sample shows that the dominant background processes are e + e − → φπ + π − , e + e − → KK * π and e + e − → K + K − ρ(770). No peaking background is observed in the signal region of the π + π − γ invariant-mass distribution. The signal region is defined as |M (π + π − γ) − m η ′ | < 3σ, where m η ′ is the nominal η ′ mass from PDG and σ is the η ′ width convolved with detector resolution.
The e + e − → φη ′ signal yield is determined by performing an unbinned maximum likelihood fit to the π + π − γ invariant-mass distribution. The signal is described by the line-shape obtained from the signal MC simulation convolved with a Gaussian function that accounts for the difference in resolution between data and MC simulation. The shape of the background is parameterized by a second-order polynomial function. The corresponding fit result is shown in Fig. 1   The same event selection criteria and fit procedure are applied to the other 19 data samples taken at different c.m. energies. The numbers of siganl events for these samples are listed in Table I. The Born cross section is calculated using where N obs is the number of signal events, L the integrated luminosity measured with the method described in Ref. [40], B the product of the branching fractions of the decays φ → K + K − and η ′ → π + π − γ [37], ǫ the detection efficiency and (1 + δ) is the correction factor due to ISR and vacuum polarization (VP). Both ǫ and (1 + δ) are obtained from MC simulations of the signal reaction at the individual c.m. energies [41,42]. The detection efficiency and ISR factor depend on the input Born cross section, where the iterations are performed until the measured Born cross section does not change by more than 0.5%. The resulting Born cross sections and related variables are listed in Table I. The following sources of systematic uncertainties are considered in the measurement of the Born cross sections in addition to the common uncertainties including the integrated luminosity, the tracking efficiency, photon detection, PID and branching fractions of intermediate state decays for each energy point, systematic uncertainties arise from the kinematic fit, the fit procedure, mass window requirement of φ, ISR correction factor, as well as MC statistics. The uncertainty of the integrated luminosity is 1% at each energy point [40]. The uncertainty of the efficiency for each charged track and PID are estimated to be 1% [23]. The uncertainty due to photon detection is 1% [43]. The uncertainty of the branching fractions of intermediate states is taken from the PDG [37], it is 2.2 %. The uncertainty related to the kinematic fit is estimated by correcting the helix parameters of the simulated charged tracks to match the resolution in data [44]. The difference in (1 + δ)ǫ between the last two iterations of the cross section measurement is taken as the uncertainty related to the ISR correction factor. The φ line shape in simulated data is smeared to better match the data line shape. The difference in the detection efficiency before and after smearing are assigned as systematic uncertainties for the φ mass window requirement. The difference in the signal yield between fits in a range of (0.8, 1.10) GeV/c 2 compared to the nominal fit is treated as the systematic uncertainty from the fit range. The uncertainty related to the signal shape is estimated with an alternative fit using the same function for the sig-nal shape, but fixing the width of the Gaussian function to the value obtained in the nominal fit plus one standard deviation. The background shape is described as a second-order polynomial function. A fit with a thirdorder polynomial function for the background shape is used to estimate the uncertainty. The uncertainty due to MC statistics is estimated by the number of the generated events. Assuming that all of the above systematic uncertainties are uncorrelated, the total systematic uncertainties are obtained by adding the individual uncertainties in quadrature, shown in Table II. TABLE II: Relative systematic uncertainties (in %) in the Born cross sections of e + e − → φη ′ . In addition to the common uncertainties, these represent the uncertainties in the estimated effects of the kinematic fit (KinFit), signal and background shape (Signal and Background), fit range (Range), the initial state radiation factor (ISR), φ mass window (m φ ), MC statistics (MC). The total uncertainty is obtained by summing the individual contributions in quadrature. The measured Born cross sections are shown in Fig. 2  (a), where a clear structure is observed around 2.2 GeV. To study a possible resonant behavior, a χ 2 fit incorporating the correlated and uncorrelated uncertainties is performed to the measured Born cross sections. Assuming that the final states φη ′ come from a resonance decay, we fit the line shape using a coherent sum of a phase-space modified Breit-Wigner (BW) function and a phase-space term. The probability density function (PDF) is defined as where the BW function is written as where M R is the mass of the resonance, Γ R tot the total width, Γ R e + e − the e + e − partial width, B R (φη ′ ) the branching fraction of the resonance decay to φη ′ , ϕ the phase angle between the resonance and the phase-space contribution and Φ( √ s) the phase space factor for a Pwave two-body system.
The fit has two solutions with an identical mass and width of the resonance. The product Γ R e + e − B R (φη ′ ) is also the same in the two solutions, while the phases are different. The fit quality is estimated by inspecting the χ 2 , which gives a χ 2 /ndf = 28.30/15, where ndf is the number of degrees of freedom. The parameters of the structure are determined to be M = (2177.5 ± 4.8) MeV/c 2 and Γ = (149.0 ± 15.6) MeV, where the uncertainty is statistical only. Figure 2 shows the fit result, and the parameters of the resonance are summarized in Table III. The significance of the resonance is determined to be larger than 10σ, including systematic uncertainties. This is obtained by comparing the change of ∆(χ 2 ) with and without the resonance in the fit and taking the change in the number of degrees of freedom ∆ndf = 4 into account. The systematic uncertainties of the resonance parameters are mainly due to the signal model. To assess this systematic uncertainty, a modified BW function with mass-dependent width is used for the fit, resulting in differences of 19.5 MeV/c 2 and 8.9 MeV for the mass and width, respectively. The dependence on the c.m. energy determination and the fit procedure were also investigated and found to be negligible. The uncertainties (statistical and systematic) of the measured Born cross sections have been considered in the fit. The figure of the Supplemental Material shows the comparison of the parameters of the φ(2170) state measured by experiments via various processes.
In summary, we present a precise measurement of the cross section line-shape for e + e − → φη ′ based on data samples collected with the BESIII detector at the BEPCII collider at 20 different c.m. energies from 2.050 to 3.080 GeV. A clear structure is observed in the line shape of the measured Born cross sections. Assuming that the φη ′ comes from a single resonance, we detemine the mass and width of this resonance to be (2177.5 ± 5.1 (stat) ± 18.6 (syst)) MeV/c 2 and (149.0 ± 15.6 (stat) ± 8.9 (syst)) MeV, respectively. Here, the first uncertainties are statistical and the second ones are systematic. The statistical significance of the resonance is estimated to be larger than 10 σ. The J P C of the resonance should be 1 −− since it is produced in formation via e + e − collisions. The mass of the resonance is compatible with the φ(2170). Assuming that the observed resonance is the φ(2170), the measured B R φη Γ R ee = 1.7 ± 0.7 (stat) ± 1.3 (syst) by BABAR [27] is smaller than that of φη ′ mode. The ratio (B R φη Γ R ee )/(B R φη ′ Γ R ee ) is estimated to be 0.23 ± 0.10 (stat) ± 0.18 (syst). This is smaller than the prediction of the ssg hybrid models by several orders of magnitude [11,12] and casts severe doubt on the validity of these models.