Study of the process $e^{+}e^{-}\rightarrow\phi\eta$ at center-of-mass energies between 2.00 and 3.08 GeV

The process $e^{+}e^{-} \rightarrow \phi\eta$ is studied at 22 center-of-mass energy points ($\sqrt{s}$) between 2.00 and 3.08 GeV using 715 pb$^{-1}$ of data collected with the BESIII detector. The measured Born cross section of $e^{+}e^{-} \rightarrow \phi\eta$ is found to be consistent with {\textsl{BABAR}} measurements, but with improved precision. A resonant structure around 2.175 GeV is observed with a significance of 6.9$\sigma$ with mass ($2163.5\pm6.2\pm3.0$) MeV/$c^{2}$ and width ($31.1_{-11.6}^{+21.1}\pm1.1$) MeV, where the first uncertainties are statistical and the second are systematic.

The process e + e − → φη is studied at 22 center-of-mass energy points ( √ s) between 2.00 and 3.08 GeV using 715 pb −1 of data collected with the BESIII detector. The measured Born cross section of e + e − → φη is found to be consistent with BABAR measurements, but with improved precision. A resonant structure around 2.175 GeV is observed with a significance of 6.9σ with mass (2163.5 ± 6.2 ± 3.0) MeV/c 2 and width (31.1 +21.1 −11.6 ± 1.1) MeV, where the first uncertainties are statistical and the second are systematic.
The ratio between the φη and φη partial widths is an important observable to assess φ(2170) as a hybrid state. An ssg hybrid state is expected to have a stronger coupling to φη, with the partial width expected to be larger than that to φη by factors ranging from 3 up to 200 [13,14]. The resulting partial width in the study of e + e − → φη at BESIII is found to be B φ(2170) φη Γ φ(2170) e + e − = (7.1 ± 0.7 ± 0.7) eV [11]. The precision of the ratio of partial widths between the decays to φη and φη of provides a strong benchmark for theoretical models aiming to explain the nature of the φ(2170). Hence, a new measurement of the cross section of the process e + e − → φη is important in order to improve our understanding of the nature of the φ(2170) resonance.
In this paper, we present an improved measurement of the Born cross section (σ Born φη ) of the process e + e − → φη at 22 ( √ s) in the range between 2.00 and 3.08 GeV with a data sample corresponding to an integrated luminosity of 715 pb −1 collected with the BESIII experiment.

II. BESIII EXPERIMENT AND MONTE CARLO SIMULATION
The BESIII detector is a magnetic spectrometer [47] located at the Beijing Electron Positron Collider (BEPCII) [48]. The cylindrical core of the BESIII detector consists of a helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all enclosed in a superconducting solenoidal magnet providing a 1.0 T magnetic field. The solenoid is supported by an octagonal flux-return yoke instrumented with resistive plate counters interleaved with steel, which serve as muon identifiers. The acceptance of charged particles and photons is 93% of the full solid angle. The charged-particle momentum resolution at 1 GeV/c is 0.5%, and the dE/dx resolution is 6% for electrons from Bhabha scattering. The EMC measures photon energies with a resolution of 2.5% (5%) at 1 GeV in the barrel (end cap) region. The time resolution of the TOF barrel part is 68 ps, while that of the end cap part is 110 ps.
The detector response, including the interaction of secondary particles with the detector material, is simulated using geant4 [49] based Monte Carlo (MC) software. MC simulation samples of 2.5 million e + e − → φη events per energy point generated by using P -wave in the production process with the conexc [50] generator are used for the efficiency determination and to calculate the correction factors for radiative effects up to next-to-leading order (NLO), as well for the effect of vacuum polarization (VP). MC samples of inclusive hadronic events generated with conexc combined with luarlw [50] are used for background studies.

III. EVENT SELECTION AND BACKGROUND ANALYSIS
To select e + e − → φη, the φ and η candidates are reconstructed through their decays to K + K − and γγ, respectively. Selected events must have exactly two charged tracks with opposite charge. Tracks are reconstructed using the MDC, and all track candidates have to be within the MDC acceptance of | cos θ| < 0.93, where θ is the polar angle with respect to the symmetry axis of the drift chamber. Additionally, both tracks are required to have their point of closest approach to the interaction point be within 10 cm along the beam direction and 1 cm in the transverse plane. The TOF and the dE/dx information are combined to calculate a particle identification (PID) likelihood for the π, K, and p hypotheses. For both tracks, it is required that the likelihood of a kaon assignment is larger than the two alternative hypotheses.
Photon candidates are selected from showers in the 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, while photon candidates reconstructed in the end caps must have a polar angle within 0.86 < | cos θ| < 0.92. In order to suppress the background from ISR processes, the energy of all photon candidates is required to be larger than 70 MeV. To suppress electronic noise and energy deposits unrelated to the event, the timing information from the EMC is required to be within 700 ns of the event start time for all photon candidates.
A four-constraint (4C) kinematic fit is applied using the hypothesis e + e − → K + K − γγ, constraining the measured four-momenta of all particles to the initial centerof-mass (c.m.) four-momentum. For events with more than two good photon candidates, the combination with the smallest χ 2 of the kinematic fit is retained for further study. Only events with χ 2 4C (K + K − γγ) < 100 are kept. In order to suppress background contributions from the reaction e + e − → γ ISR φ, φ → K + K − , where γ ISR is the detected ISR photon, a second kinematic fit is used testing the e + e − → K + K − γ hypothesis. Events are rejected if the χ 2 of the kinematic fit to the e + e − → K + K − γ hypothesis is smaller than the one for the signal hypothesis. The distribution of the invariant mass of the two kaons (M (K + K − )) versus the invariant mass of the two photons (M (γγ)) is shown in Fig. 1 for the data at a ( √ s) of 2.125 GeV using the above selection criteria. An enhancement at the η and φ meson masses is observed. Candidate events of e + e − → φη are required to be within the combined η and φ signal region, defined as |M (γγ) − m η | < 30 MeV/c 2 , 0.98 < M (K + K − ) < 1.08 GeV/c 2 , where m η is the mass of η meson as listed by the PDG [51], and 30 MeV/c 2 corresponds to three times the detector resolution at the η mass.
The main sources of the remaining background are processes of the type e + e − → K + K − η (where the K + K − pair does not originate from a φ decay), K + K − π 0 and e + e − → K + K − π 0 π 0 . Possible contaminations are estimated to be less than 1.0% from studies performed on inclusive hadronic MC samples. After selecting η candidates using the invariant mass M (γγ), background contributions are highly suppressed and do not exhibit narrow structures in M (K + K − ) so that they can be described by a smooth polynomial function.

A. Signal yields
The number of events of the process e + e − → φη is determined by an unbinned maximum likelihood fit to the K + K − invariant mass with a signal shape, which is parameterized by a P -wave relativistic Breit-Wigner function convolved with a Gaussian function. The background shape is described by a first order polynomial. According to Ref. [52], the P -wave relativistic Breit- where m φ is fixed to the mass of the φ meson [51], p K is the kaon momentum in the φ rest frame and p K is the kaon momentum at the nominal φ mass. The width Γ[M (K + K − )] is given by where Γ 0 is fixed to the nominal width of the φ meson [51] and B(p) is the P -wave Blatt-Weisskopf form chosen as the radius in the calculation of the centrifugal barrier factor. The amplitude squared convolved with the Gaussian function is added incoherently to the background polynomial. The parameters of the polynomial and the Gaussian function are free in the fit. The latter is used to compensate for absolute resolution and an offset of the mass scaling in data. The fit result for the data at √ s = 2.125 GeV is shown in Fig. 2, while the signal event yields N Signal for all energy points are summarized in Table I.

B. Efficiency and radiative corrections
With N Signal determined, the Born cross section σ Born φη (s) of the process e + e − → φη at the c.m. energy squared s can be determined using where L is the integrated luminosity measured with large angle Bhabha scattering [53], is the reconstruction efficiency, (1 + δ) is the radiative correction factor and 1 (1−Π) 2 is the VP correction factor. The explicit ( √ s) dependence of those variables is omitted here. The total branching fraction B is the product of the branching fractions for the decays contained in the full decay chain B = B(φ → K + K + ) · B(η → γγ) = (19.39 ± 0.22)% [51]. The product (1 + δ) is determined in an iterative procedure. At each step of the iteration, a set of 1000 MC samples is produced taking into account the fit to the Born cross section introduced in Sec. V. The MC samples are produced by sampling the model parameters according to a Gaussian distribution whose width is set equal to the uncertainties of the model parameters as obtained in the fit. Each of the 1000 MC samples gives a new value of (1 + δ). The mean of those 1000 values is used to recalculate the Born cross section. This process is repeated until the resulting Born cross section converges. After two iterations, the observed change in the Born cross section σ Born φη (s) is smaller than the uncertainty of the generator, which is 0.5%. The efficiency, the radiative, VP correction factors and the results for σ Born φη are summarized in Table I.

C. Systematic uncertainties
Several sources of systematic uncertainties are considered in the determination of σ Born φη . The uncertainties associated with the knowledge of the tracking efficiency of the two charged tracks as well as from the PID efficiency are studied with a e + e − → K + K − π + π − control sample. The difference of the efficiency measured in data and MC is assigned as the uncertainty, and it is found to be 1.0% per track for both tracking and PID efficiency [8]. The uncertainty due to photon reconstruction efficiency is 1.0% per photon [54]. The uncertainty of the luminosity measurement is smaller than 1.0% [53]. The uncertainty associated with the kinematic fit is estimated by not using the correction of the helix parameters of the charged tracks described in detail in Ref. [55] and taking the difference to the nominal result as the uncertainty. In order to estimate the contribution from the η selection, the mass window is varied from |M (γγ) − m η | < 3σ to 2.5σ and 3.5σ, and the larger difference to the nomimal result is taken as the uncertainty. The systematic uncertainty of the signal yield is estimated by varying the fit range from (0.98, 1.08) GeV/c 2 to (0.99, 1.09) GeV/c 2 , where the difference to the nominal result is the uncertainty. The uncertainty related to the signal shape is estimated with an alternative fit using the φ MC shape convolved with a Gaussian function. The uncertainty due to background shape is estimated with an alternative fit using an Argus function [56] instead of a polynomial. The uncertainty due to φ peaking background is estimated by investigating the η sideband which is defined as 40 MeV/c 2 < |M (γγ) − m η | < 150 MeV/c 2 . We take the difference of the normalized number of the events from a signal fit to the sideband region and the number of events estimated from MC as systematic uncertainties. The uncertainty in (1 + δ) arises from the accuracy of the radiator function, which is about 0.5% [57], and has an additional contribution from the parametrization of the σ Born φη (s) line shape, which is taken as the standard deviation of the fit to the sampled parameters. The two contributions are summed in quadrature. The uncertainties of the branching fractions of intermediate states are 1.1% [51]. Assuming that these contributions to the systematic uncertainties are uncorrelated, the total systematic uncertainties are obtained by adding the individual uncertainties in quadrature. The resulting values for all √ s are shown in Table II. The fluctuations of some relative uncertainties among the different energies origin from the influence of statistics, and have negligible effect on the total absolute uncertainties and the final results. The total systematic uncertainties on σ Born φη range from 3.9% to 8.3%.

A. Fit of the line shape
To study a possible resonant behavior in e + e − → φη, a least χ 2 fit taking into account the correlation between systematic uncertainties for different ( √ s) is performed to the measured values of the Born cross section σ Born φη . Previous results from the BABAR collaborations [3] are also included to be able to describe the low-energy behavior of the cross section. Following Ref. [3] and assuming that the reaction proceeds mostly through the decay of the two resonances φ(1680) and φ(2170), the line shape is fitted using a coherent sum of two phase-space factor modified Breit-Wigner functions and a non-resonant term: where P φη (s) is the phase space factor of the φη system, A n.r. φη (s) describes the non-resonant contribution, For the phase space for the φη system, we use A reasonable description of the non-resonant contribution is given by the power-law dependence A n.r. φη (s) = a 0 /s a1 . We describe the φ(1680) resonance using Breit-Wigner amplitude e + e − , mass m φ(1680) , width Γ φ(1680) . The width Γ φ(1680) on the denominator use an energy dependent width [3] Γ φ(1680) (s) =Γ φ(1680) P φη (s) P φη (m 2 φ(1680) ) B φ(1680) φη ) .   In the fit, the previous results from the BABAR collaboration [3] are included to be able to also describe the low-energy behavior (<2 GeV) of the cross section where we have no data. The parameters of the φ(1680), such as mass (m φ(1680) = 1709 MeV/c 2 ), width (Γ φ(1680) = 369 MeV) and branching fraction (Γ φ(1680) e + e − B φ(1680) φη = 138 eV), are fixed to the results from Ref. [3]. The φ(2170) is described using Breit-Wigner amplitude The fit has two solutions with an identical mass and width of the resonance. The fit quality is estimated by the χ 2 , where the best fit gives a χ 2 /n.d.f = 0.9, with n.d.f = 97 being the number of degrees of freedom. The mass and width of φ(2170) are determined to be m φ(2170) = (2163.5 ± 6.2) MeV/c 2 and Γ φ(2170) = (31.1 +21.1 −11.6 ) MeV from our fit, illustrated in Fig. 3. The significance of the φ(2170) resonance is determined to be 6.9σ. This is obtained by comparing the change of ∆χ 2 = 59.05 with (blue solid line in Fig. 3 (a)∼(d)) and without (grey dotted line in Fig. 3 (a)∼(d)) the resonance in the fit and taking the change of number of degree of freedom ∆n.d.f = 4 into account. The cross section and the fit results are summarized in Table III and shown in Fig. 3. Fig. 3 (a) and (c) are solution I; Fig. 3 (b) and (d) are solution II. Fig. 3 (b) shows the same data subtracting the fit result that is obtained without inclusion of the φ(2170). It is obvious that an additional resonant structure around 2.175 GeV is needed.

B. Systematic uncertainties
For the systematic uncertainties of the resonance parameters, we examine effects from the choice of the model for the non-resonant contribution and of the fit range. For the model dependence of the non-resonant contribution, a A n.r. φη (s) = a 0 /s is used instead, resulting in differences of 3.0 MeV/c 2 and 0.1 MeV for mass and width, respectively. For the dependence on the fit range, we set aside the energy point 2.00 and 3.08 GeV, resulting in differences of 0.3 MeV/c 2 and 1.1 MeV for mass and width, respectively. The total systematic uncertainties are thus 3.0 MeV/c 2 , 1.1 MeV for mass and width, respectively.

VI. SUMMARY AND DISCUSSION
This paper presents the most accurate measurement of the Born cross section of e + e − → φη, at 22 c.m. energies in the interval 2.00 to 3.08 GeV. A resonant structure is observed in the σ Born φη line shape. We determine the parameters of this resonance to be m φ(2170) = (2163.5 ± 6.2 ± 3.0) MeV/c 2 and Γ φ(2170) = (31.1 +21.1 −11.6 ± 1.1) MeV. Here, the first uncertainties are statistical and the second ones are systematic. The significance is larger than 6.9σ. With the input of the partial width of B R φη Γ R ee = (7.1 ± 0.7) eV [11], the ratio of partial widths between the decay modes φη and φη would be (0.03 +0.02 −0.01 ) or (1.42 +0. 56 −0.46 ). Compared to a previous measurement by the BABAR experiment [3], the mass value reported here is significantly larger. While similar resonances have been observed in many different channels, the observed decay widths vary significantly.
The fitted result is compared with the parameters of the φ(2170) state measured by previous experiments via various processes as shown in Fig. 4. The results obtained in this paper are consistent with the world average parameters of the φ(2170). However, differences to other individual measurements in different channels can be sizable. Among the existing measurements, the result of this measurement yields the smallest width of the φ(2170) resonance observed so far.