Measurement of Branching Fractions of $\psi(3686)\rightarrow\phi\eta^\prime, \phi f_1(1285)$ and $\phi \eta(1405)$

Using a sample of $448.1\times10^6$ $\psi(3686)$ events collected with the BESIII detector, we perform a study of the decay $\psi(3686) \rightarrow \phi \pi^{+}\pi^{-}\eta$. The branching fraction of $\psi(3686)\rightarrow\phi\eta^\prime$ is determined to be $(1.51\pm0.16\pm0.12)\times 10^{-5}$, which is consistent with the previous measurement but with significantly improved precision. The resonances $f_{1}(1285)$ and $\eta(1405)$ are clearly observed in the $\pi^{+}\pi^{-}\eta$ mass spectrum with statistical significances of $18\sigma$ and $9.7\sigma$, respectively. The corresponding product branching fractions are measured to be $\mathcal{B}(\psi(3686)\rightarrow\phi f_{1}(1285),f_{1}(1285)\rightarrow\pi^{+}\pi^{-}\eta) =(1.03\pm0.10\pm0.09)\times 10^{-5}$ and $\mathcal{B}(\psi(3686)\rightarrow\phi\eta(1405),\eta(1405)\rightarrow\pi^{+}\pi^{-}\eta) =(8.46\pm1.37\pm0.92)\times 10^{-6}$.


I. INTRODUCTION
In the quark model, if the vector meson nonet is ideally mixed, then the φ contains only strange quarks and the ω contains only up and down quarks. This assumption can be used as a 'flavor filter' for the determination of the quark content of various resonant structures by observing their production in J/ψ or ψ(3686) decays in association with a φ or an ω.
In addition, it is expected in perturbative QCD that both J/ψ and ψ(3686) decays to hadrons are via three gluons or a photon, which provides the relation [4,5] This relation is referred to as the "12% rule" and it is expected to hold to a reasonably good degree for both inclusive and exclusive decays. By studying the ψ(3686) → φπ + π − η decay, we can check the "12% rule" in ψ → φη ′ , ψ → φf 1 (1285) and ψ → φη(1405) decays. A sample of 448.1 × 10 6 ψ(3686) events collected with the BESIII detector [6], about 30 times larger than that of the BES experiment, offers a unique opportunity to improve the precision of the BF of ψ(3686) → φη ′ and to investigate the π + π − η invariant mass spectrum above the η ′ mass.
In this paper, an improved measurement of the branching fraction B(ψ(3686) → φη ′ ) is presented. We also report the observation of the f 1 (1285) and η(1405) in the π + π − η mass spectrum and a measurement of the corresponding product branching fractions.

II. DETECTOR AND MONTE CARLO SIMULATION
BEPCII is a double-ring electron-positron collider with design peak luminosity of 10 33 cm −2 s −1 with a beam current of 0.93 A at √ s = 3.773 GeV. The cylindrical core of the BESIII detector consists of a helium-based main drift chamber (MDC), a plastic scintillator time-offlight (TOF) system, a CsI(Tl) electromagnetic calorimeter (EMC), a superconducting solenoidal magnet providing a 1.0 T magnetic field, and a muon system made of resistive plate chambers in the iron flux return yoke of the magnet. The acceptances for charged particles and photons are 93% and 92% of 4π, respectively. The charged particle momentum resolution is 0.5% at 1 GeV/c, and the barrel (end cap) photon energy resolution is 2.5% (5.0%) at 1 GeV.
The optimization of the event selection and the estimation of the physics background are performed using Monte Carlo (MC) simulated samples. The geant4based [7] simulation software boost [8] includes the geometry and material description of the BESIII detector, the detector response and digitization models, as well as a record of the detector running conditions and performance. To study the potential backgrounds, an inclusive MC sample of 506 × 10 6 ψ(3686) decays is generated, where the production of the ψ(3686) resonances are simulated by the MC event generator kkmc [9,10]. The known decay modes are generated by evtgen [11,12] with branching fractions set to the world average values [13], and by lundcharm [14] for the remaining unknown decays. Each MC-generated event is mixed with a randomly triggered event recorded during data taking in order to include effects of background contamination, such as beam-related background and cosmic rays, as well as electronic noise and hot wires. The analysis is performed in the framework of the BESIII offline software system which takes into account the detector calibration, event reconstruction and data storage.

A. Event selection
To select candidate events of the process ψ(3686) → φπ + π − η with φ → K + K − and η → γγ, the following criteria are imposed on the data and MC samples. We select charged tracks in the MDC within the polar angle range |cosθ| < 0.93 and require that the points of closest approach to the beam line be within ±10 cm of the interaction point in the beam direction and within 1.0 cm in the plane perpendicular to the beam. The TOF and the specific energy loss, dE/dx, of a particle measured in the MDC are combined to calculate particle identification (PID) probabilities for pion, kaon and proton hypotheses. The particle type with the highest probability is assigned to each track. In this analysis, two kaon and two pion tracks with opposite charges are required.
Photon candidates are reconstructed from isolated clusters of energy deposits in the EMC. The energy deposited in nearby TOF counters is included to improve the photon reconstruction efficiency and energy resolution. At least two photon candidates are required, with a minimum energy of 25 MeV for barrel showers (|cosθ| < 0.80) and 50 MeV for end-cap showers (0.86 < |cosθ| < 0.92). To exclude showers due to charged particles, the angle between the nearest charged track and the shower in EMC must be greater than 10 • . An EMC shower timing requirement, 0 ≤ t ≤ 700 ns, is applied to suppress electronic noise and energy depositions unrelated to the event.
A four-constraint (4C) kinematic fit using fourmomentum conservation is performed under the ψ(3686) → K + K − π + π − γγ hypothesis. In events with more than two photon candidates, all pairs are tried and the combination with the smallest χ 2 4C value is retained. An event is rejected if χ 2 4C > 80. The resulting distribution of the invariant mass M γγ versus M K + K − is illustrated in Fig. 1(a); the area indicated by the solid box corresponds to the ψ(3686) → φπ + π − η signal region. The distributions of M γγ and M K + K − are shown in Figs. 1(b) and 1(c), respectively, where the η and φ peaks are clearly observed. The φ and η signal regions are defined as where m φ and m η are the world average values of the φ and η masses [13].

B. Background study
To investigate the background events, we apply the same selection to the inclusive MC sample of 506 × 10 6 ψ(3686) events. The π + π − η invariant mass distribution of the selected events is displayed in Fig. 2. The BF of ψ(3686) → φη ′ in the MC was adjusted to provide agreement in the number of events with the data. For the mass region of M π + π − η > 1.1 GeV/c 2 , the contribution from the inclusive MC events is reasonably smooth, which indicates that the f 1 (1285) and η(1405) peaks observed in data are not from the known decays of ψ(3686) decays listed by the Particle Data Group (PDG) [13].
For the background events from the continuum process e + e − → φπ + π − η, we perform a study with the sample of (2.93 ± 0.01) fb −1 [15] taken at √ s = 3.773 GeV. After the same event selection as described above, clear η ′ , f 1 (1285) and η(1405) peaks are seen in the π + π − η mass spectrum recoiling against the φ. An unbinned maximum likelihood fit, analogous to the one in Sec. III C, yields 221 ± 15, 26.8 ± 7.1, and 87 ± 16 events for η ′ , f 1 (1285) and η(1405), respectively. We assume that the observed signals come directly from e + e − annihilations and not ψ(3770) decays. A scale factor f is defined as the ratio of the observed number N in ψ(3686) data to that in the ψ(3770) data, where N , L, σ and ε refer to the observed number events, integrated luminosity of data samples, cross section and detection efficiency at the two c.m. energies. The details on the cross section can be found in Ref. [16]. The detection efficiency ratios ε ψ(3686) ε ψ(3770) can be determined by Monte Carlo simulations. The scale factors are calculated to be 0.232, 0.242, and 0.236 and the normalized numbers of continuum events for e + e − → φη ′ , e + e − → φf 1 (1285) and e + e − → φη(1405) at 3.686 GeV are determined to be 51.3 ± 3.5, 6.5 ± 1.7, and 20.5 ± 3.8, respectively. Due to identical event topology, these background events are indistinguishable from signal events and are subtracted directly from our nominal yields. Since the η ′ signal is well isolated from the f 1 (1285) and η(1405) peaks, we perform an extended unbinned maximum likelihood fit to the π + π − η invariant mass in the range of [0.85, 1.10] GeV/c 2 to obtain the signal yields of η ′ . In the fit, the total probability density function consists of a signal and a background contribution. The signal component is modeled from the MCsimulated signal shape using a nonparametric method [17], convolved with a Gaussian function to account for different mass resolutions in data and MC simulation. The background contribution is described by the twodimensional η -φ sideband and the background from ψ(3686) → π + π − J/ψ. The fit, shown in Fig. 3(a), yields 201 ± 15 φη ′ events.
Another fit to the π + π − η invariant mass in the range of [1.1, 2.2] GeV/c 2 is performed to obtain the signal yields of f 1 (1285) and η(1405) with an assumption of no interference between them. In this case, the total probability density function can be described with , which are free parameters in the fit. The Gaussian function G represents the mass resolution, and the corresponding parameters, σ 1 and σ 2 , are taken from the MC simulations. The detection efficiency ε(M ππη ) as a function of the π + π − η invariant mass is obtained from the MC simulation. BKG refers to the following background components: 1) a smoothed shape from the two-dimensional η -φ sidebands with fixed normalization, and 2) a linear polynomial describing the remaining background events.
The fit, shown in Fig. 3(b), yields 234 ± 22 φf 1 (1285) events and 195 ± 28 φη(1405) events. The corresponding statistical significances for f 1 (1285) and η(1405) are 18σ and 9.7σ, respectively. They are determined from differences of the likelihood values and the degrees of freedom between the fits with and without the resonance. The  [13], but with larger uncertainties due to our limited statistics.

IV. SYSTEMATIC UNCERTAINTIES
The sources of systematic uncertainties include the efficiency difference between data and MC simulation for charged track reconstruction, photon detection, PID requirements and kinematic fit as well as input branching fractions, the number of ψ(3686) events, and yield fitting procedures. The corresponding contributions to the measurement of the branching fractions are discussed in detail below.
(a) MDC tracking efficiency: The charged tracking efficiency has been investigated with the clean control channels J/ψ → π + π − pp and J/ψ → ρπ [18]. It is found that the MC simulation agrees with data within 1% for each charged track. Therefore, 4% is taken as the systematic uncertainty from the four charged tracks in the final state.
1.6 · · · · · · Number of ψ(3686) events 0.6 0.6 0.6 φ mass window (b) Photon detection efficiency: The photon detection efficiency has been studied using a control sample of J/ψ → ρπ [18]. The results indicate that the difference between the detection efficiencies of data and MC is around 1% per photon. Thus, 2% is taken as the total systematic uncertainty for the detection of the two photons in this analysis.
(c) PID efficiency: To evaluate the PID efficiency uncertainty, we have studied the kaon and pion PID efficiencies using the clean control samples of J/ψ → K * ± K ∓ and J/ψ → ρπ [18], respectively. We find that the difference in the PID efficiency between data and MC is 1% for each kaon or pion. Hence, 4% is taken as the total systematic uncertainty from the PID efficiency.
(d) Kinematic fit: The uncertainty associated with the 4C kinematic fit comes from the inconsistency between data and MC simulation of the fit; this difference is reduced by correcting the track helix parameters of the MC simulation. Following the method described in Ref. [19], we obtain the systematic uncertainties for the 4C kinematic fit as 0.3%, 0.2% and 0.2% for the branching fractions of ψ(3686) → φη ′ , ψ(3686) → φf 1 (1285) and ψ(3686) → φη(1405), respectively.
(f) Number of ψ(3686) events: The number of ψ(3686) events is determined from an analysis of inclusive hadronic ψ(3686) decays. The uncertainty of the number of ψ(3686) events, 0.6% [20], is taken as the systematic uncertainty in the calculation of the BFs.
(g) φ mass window: In Ref. [21], a control sample of J/ψ → φη ′ , φ → K + K − , η ′ → γπ + π − is used to study the uncertainty due to the φ mass window requirement. We adopt the resulting uncertainty of 1.2% from that study.
(h) η mass window: To estimate the uncertainty from the η mass requirement, we select a clean sample of J/ψ → φη without this requirement. Events with two oppositely charged tracks and two good photons are selected. The charged tracks must be identified as kaons. A 4C kinematic fit is performed with the J/ψ → K + K − γγ hypothesis and the χ 2 4C is required to be less than 40. The K + K − invariant mass is required to be in the φ mass region, We perform a fit to the mass spectrum of γγ, where a Crystal Ball function [22] is used to describe the η signal and a first-order Chebyshev polynomial describes the background. Requiring |M γγ − m η | < 0.04 GeV/c 2 , we regard the difference of the η selection efficiencies between data and MC samples, 0.3%, as the systematic uncertainty.
(i) J/ψ veto: To remove the background events from ψ(3686) → π + π − J/ψ, we have applied a requirement of |M K + K − γγ − M J/ψ | > 0.05 GeV/c 2 . In order to estimate the systematic uncertainty, this requirement is varied by ± 0.01 GeV/c 2 for both ψ(3686) and continuum data. The maximum changes to the nominal results, 2.7% and 3.2%, respectively, for f 1 (1285) and η(1405) are taken as the systematic uncertainties.
(j) Fit range: We perform alternative fits for ψ(3686) and continuum data by varying the fit ranges, and assign the maximum change of the results, 2.7%, 1.3% and 6.9%, as the systematic uncertainties.
(k) Signal shape: To obtain the number of φη ′ events in the fit to M π + π − η , the MC shape of the η ′ convolved with a Gaussian function is used to describe the signal shape. In order to estimate the systematic uncertainty due to this shape, alternative fits are performed to determine the yields of signal and peaking background events, replacing the MC shape with a Breit-Wigner function. The change of the result, 2.7%, is taken as the systematic uncertainty. The uncertainties from the signal shape of the f 1 (1285) and η(1405) are estimated by varying the mass resolutions by ±10%, in both ψ(3686) and continuum data, to account for the difference between data and MC simulation. We take the changes of the signal yields of φf 1 (1285) and φη(1405) events, 1.8% and 1.7%, as the systematic uncertainties.
(l) Background shape from Chebyshev polynomial: To estimate the uncertainty of background shape in the fit to M π + π − η , we performed alternative fits by replacing the first-order Chebyshev polynomial with a second-order Chebyshev polynomial for both ψ(3686) and continuum data. The changes of 0.0% and 4.2% are taken as systematic uncertainties.
(m) Sideband: The uncertainty on the φη ′ yield caused by the sideband regions is estimated by changing those regions to 0.447 < M γγ < 0.487, 0.609 < M γγ < 0.649, and 1.046 < M K + K − < 1.076 GeV/c 2 . The change of the yields, 1.7%, is regarded as the systematic uncertainty for that mode. For the determination of the f 1 (1285) and η(1405) signal yields, the background events estimated from the two-dimensional η -φ sidebands have only a smooth contribution under the f 1 (1285) and η(1405) peaks.
To estimate the uncertainty associated with the sidebands, we performed an alternative fit by removing the constraint on the number of the background events estimated from the two-dimensional η -φ sidebands. The changes to the nominal results, 1.7% and 1.2% for f 1 (1285) and η(1405), respectively, are considered as systematic uncertainties.
(n) Parameters of φf 1 (1285) generation: Due to the limited statistics, and because ψ(3686) → φf 1 (1285) is expected to be similar to the process of J/ψ → φf 1 (1285), we use the same model for J/ψ → φf 1 (1285) [2] to generate the signal MC sample of ψ(3686) → φf 1 (1285) to determine the detection efficiency, and the angular distribution of this decay can be expressed as dN dcosθ = 1 + αcos 2 θ. The parameter α used in the MC sample generation is taken from the angular distribution of φ in the rest frame of J/ψ found in real data. Following the method described in Ref. [2], the impact of the uncertainty of these parameters on the efficiency, 4.7%, is taken as a source of systematic uncertainty on the branching fraction.
A summary of the systematic errors is shown in Table II. By assuming that all of them are independent, the total systematic uncertainty is obtained by adding the individual contributions in quadrature.