Observation of pseudoscalar and tensor resonances in $J/\psi\rightarrow \gamma \phi \phi$

Based on a sample of $(1310.6 \pm 10.5) \times 10^{6}$ $J/\psi$ events collected with the BESIII detector operating at the BEPCII storage ring, a partial wave analysis of the decay $J/\psi\rightarrow \gamma \phi \phi$ is performed in order to study the intermediate states. Results of the partial wave analysis show that the structures are predominantly $0^{-+}$ states. The existence of the $\eta(2225)$ is confirmed, and its resonance parameters are measured. Two additional pseudoscalar states, the $\eta(2100)$ with a mass of $2050_{-24}^{+30}{}_{-26}^{+75}$~MeV/$c^{2}$ and a width of $250_{-30}^{+36}{}_{-164}^{+181}$~MeV/$c^{2}$ and the $X(2500)$ with a mass of $2470_{-19}^{+15}{}_{-23}^{+101}$~MeV/$c^{2}$ and a width of $230_{-35}^{+64}{}_{-33}^{+56}$~MeV/$c^{2}$, are observed. In addition to these three pseudoscalar states, the scalar state $f_0(2100)$, and three tensor states, the $f_2(2010)$, $f_2(2300)$ and $f_2(2340)$, are observed in the process $J/\psi\rightarrow \gamma \phi\phi$. The product branching fractions ${\cal B}$($J/\psi\rightarrow \gamma X)\times{\cal B}(X\rightarrow \phi \phi)$ are reported.

Based on a sample of (1310.6±10.5)×10 6 J/ψ events collected with the BESIII detector operating at the BEPCII storage ring, a partial wave analysis of the decay J/ψ → γφφ is performed in order to study the intermediate states. Results of the partial wave analysis show that the structures are predominantly 0 −+ states. The existence of the η(2225) is confirmed, and its resonance parameters are measured. Two additional pseudoscalar states, the η(2100) with a mass of 2050 +30

I. INTRODUCTION
In quantum chromodynamics (QCD)−gluons−the gauge bosons of the strong force, carry color charge and thus can form bound states called glueballs [1][2][3]. The search for glueballs is an important field of research in hadron physics. However, possible mixing of the pure glueball states with nearby qq nonet mesons makes the identification of glueballs difficult in both experiment and theory. The glueball spectrum has been predicted by Lattice QCD [4][5][6], where the lowest-lying glueballs are scalar (mass 1.5−1.7 GeV/c 2 ), tensor (mass 2.3−2.4 GeV/c 2 ), and pseudoscalar (mass 2.3−2.6 GeV/c 2 ). Radiative decays of the J/ψ meson provide a gluon-rich environment and are therefore regarded as one of the most promising hunting grounds for glueballs [7,8].
In this paper, we present a partial wave analysis (PWA) of J/ψ → γφφ, where both φ mesons are reconstructed from K + K − , based on a sample of (1310.6 ± 10.5) × 10 6 J/ψ events collected with the BESIII detector [18].

II. BESIII DETECTOR AND MONTE CARLO SIMULATION
The BESIII detector is a magnetic spectrometer operating at BEPCII, a double-ring e + e − collider with centerof-mass energies between 2.0 and 4.6 GeV. The cylindrical core of the BESIII detector consists of a heliumbased main drift chamber (MDC), a plastic scintillator time-of-flight system (TOF) and a CsI(Tl) electromagnetic calorimeter (EMC) that are all enclosed in a superconducting solenoidal magnet providing a magnetic field of 1.0 T (0.9 T in 2012, for about 1.09 × 10 9 J/ψ events). The solenoid is supported by an octagonal flux-return yoke with resistive plate counter muon identifier modules interleaved with steel. The acceptance for charged particles and photons is 93% of the 4π solid angle, and the charged-particle momentum resolution at p = 1 GeV/c is 0.5%. The EMC measures photon energies with a resolution of 2.5% (5%) at E γ = 1 GeV in the barrel (end caps).
A GEANT4-based [19] Monte Carlo (MC) simulation software package is used to optimize the event selection criteria, estimate backgrounds and determine the detection efficiency. We generate a large signal MC sample of J/ψ → γφφ, φ → K + K − uniformly in phase space.

III. EVENT SELECTION
Charged tracks in the polar angle range | cos θ| < 0.93 are reconstructed from hits in the MDC. The combined information from the energy loss (dE/dx) measured in MDC and flight time in TOF is used to form particle identification confidence levels for the π, K and p hypotheses. Each track is assigned the particle type corresponding to the highest confidence level. Photon candidates are required to have an energy deposition above 25 MeV in the barrel EMC (| cos θ| < 0.80) and 50 MeV in the end cap EMC (0.86 < | cos θ| < 0.92). To exclude showers from charged particles, the angle between the shower position and the charged tracks extrapolated to the EMC must be greater than 10 degrees. A requirement on the EMC timing is used to suppress electronic noise and energy deposits unrelated to the event.
The study of the γK + K − K + K − final state is complicated by low momentum kaons significantly affecting the reconstruction efficiency, especially at low φφ masses. To improve the reconstruction efficiency, the J/ψ → γK + K − K + K − candidate decays are reconstructed with at least one photon and at least three charged tracks identified as kaons. A one-constraint (1C) kinematic fit under the hypothesis J/ψ → γK + K − K ± K ∓ miss is performed by constraining the mass of the missing particle to the kaon mass. The resulting χ 2 1C is required to be less than 5. If more than one combination of one photon and three kaon tracks meets this requirement, only the combination with the smallest χ 2 1C is accepted. To suppress possible background events with K + K − K + K − and π 0 K + K − K + K − final states, the χ 2 of a 1C kinematic fit under the hypothesis J/ψ → K + K − K ± K ∓ miss and the χ 2 of a 2C kinematic fit under the hypothesis J/ψ → π 0 K + K − K ± K ∓ miss , with an additional constraint on the invariant mass of the two photons to be equal to the π 0 mass, are both required to be larger than 10.
For the selected J/ψ → γK + K − K ± K ∓ miss candidates, one φ is reconstructed from the K + K − pair with an invariant mass closest to the nominal pole mass m φ , and the other φ is reconstructed from the remaining reconstructed kaon and the missing kaon. The scatter plot of Fig. 1(a), where a cluster of events corresponding to φφ production is evident. Because the processes J/ψ → φφ and J/ψ → π 0 φφ are forbidden by C-parity conservation, the presence of two φ mesons is a clear signal for the radiative decay J/ψ → γφφ. The φφ events are selected by requiring |M (K + K − ) − m φ | < 10 MeV/c 2 (referred to as φ 1 ) and |M (K ± K ∓ miss ) − m φ | < 15 MeV/c 2 (referred to as φ 2 ). Simulation studies show that only 0.2% of the selected J/ψ → γφφ events have a miscombination of kaons. The Dalitz plot and the invariant mass distributions of φφ for the selected γφφ candidate events are shown in Fig. 1  Possible backgrounds are studied with a MC sample of 1.2 × 10 9 J/ψ inclusive decays, in which the decays with known branching fractions are generated by EVTGEN [20] and the remaining J/ψ decays are generated according to the LUNDCHARM [21,22] model. The dominant backgrounds are found to be those with final states Non-φφ backgrounds are estimated using the φ sideband events from data. The two dimensional sidebands are illustrated by dashed boxes in Fig. 1(a), where the sideband regions are defined as 1.09 GeV/c 2 < M (K + K − ) < 1.11 GeV/c 2 and 1.10 GeV/c 2 < M (K ± K ∓ miss ) < 1.13 GeV/c 2 . The shaded histogram in Fig. 1(c) shows the background contribution estimated from the normalized sideband events, corresponding to a background level of 5.4%. The Dalitz plot for the estimated background events are shown in Fig. 1(d), where the accumulation of events in the left lower corner is mainly due to background events from J/ψ → φf 1 (1420).

A. Analysis method
Using the GPUPWA framework [23], a PWA is performed on 45,852 events in the region M (φφ) < 2.7 GeV/c 2 in order to disentangle the structures present in the light mesons. Due to the detector resolution not being included in the PWA fit, events in the η c signal region are excluded. The quasi two-body decay amplitudes in the sequential decay process J/ψ → γX, X → φφ, φ → K + K − are constructed using the covariant tensor amplitudes described in Ref. [24]. J/ψ → φf 1 (1285), f 1 (1285) → γφ is ignored due to its low branching fraction [25]. For the radiative J/ψ decay to mesons, the general form for the decay amplitude is where ψ µ (m 1 ) is the J/ψ polarization four-vector, e ν (m 2 ) is the polarization vector of the photon and U µν i is the partial wave amplitude with coupling strength determined by a complex parameter Λ i . The partial wave amplitudes U i used in the analysis are constructed with the four-momenta of the particles in the final state, and their specific expressions are given in Ref. [24].
In this analysis, we use Breit-Wigner (BW) as an approximation to describe the leading singularity since no model is available yet for the high energy region with many channels opened. Each resonance X is parametrized by a constant-width, relativistic BW propagator, where s is the invariant mass-squared of φφ, and M and Γ are the mass and width of the intermediate resonance.
The complex coefficients of the amplitudes and resonance parameters are determined by an unbinned maximum likelihood fit with the likelihood function constructed as in Ref. [26].
The probability to observe the ith event characterized by the measurement ξ i , i.e., the measured four-momenta of the particles in the final state, is: where ǫ(ξ i ) is the detection efficiency and ω(ξ i ) ≡ ( dσ dΦ ) i is the differential cross section, and dΦ is the standard element of phase space. The full differential cross section is: where A(J P C ) is the full amplitude for all possible resonances whose spin-parity are J P C . dξω(ξ)ǫ(ξ) ≡ σ ′ is the measured total cross section. The joint probability density for observing the N events in the data sample is: For technical reasons, rather than maximizing L, S = − ln L is minimized, with for a given data set. The second term is a constant and has no impact on the determination of the parameters of the amplitudes or on the relative changes of S values. In the fitting, − ln L is defined as: The free parameters are optimized by MINUIT [27]. The measured total cross section σ ′ is evaluated using MC techniques. An MC sample of N gen is generated with signal events that are distributed uniformly in phase space.
These events are subjected to the selection criteria and yield a sample of N acc accepted events. The normalization integral is computed as: Since data contains the contribution of signal and background, the contribution of non-φφ background events is taken into account by subtracting the negative loglikelihood (NLL) value obtained for events in the φφ sidebands from the NLL value obtained for events in the φφ signal region, i.e., − ln L sig = −(ln L data − ln L bkg ).
The number of the fitted events N X for an intermediate resonance X, which has N WX independent partial wave amplitudes A i , is defined as where N ′ is the number of selected events after background subtraction, and is the measured cross section of the resonance X and is calculated with the same MC sample as the measured total cross section σ ′ . The branching fraction of J/ψ → γX, X → φφ is calculated as: where the detection efficiency ε X is obtained by the partial wave amplitude weighted MC sample, N J/ψ is the total number of J/ψ events, and B φ→K + K − = (48.9 ± 0.5)% is the branching fraction of φ → K + K − taken from Ref. [25].

B. PWA results
In this analysis, all possible combinations of J P C = 0 −+ , 0 ++ and 2 ++ resonances [28] listed in the PDG [25] are evaluated. Given the small phase space of J/ψ → γφφ, J ≥ 4 states should be suppressed. The changes in the NLL value and the number of free parameters in the fit with and without a resonance are used to evaluate its statistical significance. In the baseline solution, there are three 0 −+ resonances (η(2225), η(2100), and X(2500)), one 0 ++ resonance (f 0 (2100)), three 2 ++ resonances (f 2 (2010), f 2 (2300), and f 2 (2340)), and the direct decay of J/ψ → γφφ, which is modeled by a 0 −+ phase space distribution (0 −+ PHSP) of the φφ system. The statistical significance of each component in the baseline solution is larger than 5 σ. The masses and widths of the three 0 −+ resonances are free parameters in the fit. The resonance parameters of the 0 ++ and 2 ++ resonances are fixed to the PDG [25] values due to limited statistics. The masses and widths of the resonances, product branching fractions of J/ψ → γX, X → φφ, and the statistical significances are summarized in Table I, where the first errors are statistical, and the second ones are systematic. The fit fraction of each component and their interference fractions are shown in Table II. Figure 2(a) shows a comparison of the data and the PWA fit projection (weighted by MC efficiencies) of the invariant mass distributions of φφ for the fitted parameters. The comparisons of the projected data and MC angular distributions for the events with φφ invariant mass less than 2.7 GeV/c 2 are shown in Fig. 2(b)−2(e). The χ 2 /n bin value is displayed on each figure to demonstrate the goodness of fit, where n bin is the number of bins of each figure and χ 2 is defined as: where n i and ν i are the number of events for the data and the fit projections with the baseline solution in the ith bin of each figure, respectively.  [29] or η(2320) [30] worsens the NLL values by 21.2 and 33.0, respectively. The spin-parity assignment J P C of the X(2500) as 0 −+ is significantly better than the 0 ++ hypothesis, with the NLL value improving by 44.1 units. Changing the spin-parity assignment of the X(2500) to 2 ++ , resulting in 10 additional free parameters, worsens the NLL value by 0.5, instead. Therefore, the preferred assignment for the X(2500) is pseudoscalar. If we replace the two tensor states f 2 (2300) and f 2 (2340) by a single one with free resonance parameters in the fit, the NLL value is worsened by 14.7. In this case, a statistical significance test of the f 2 (2340) yields a value of 6.1 σ. The narrow f J (2220) (alternatively known as the ξ(2230)), which was seen in J/ψ → γK + K − at MarkIII [31] and BES [32], but not seen in J/ψ → γK 0 S K 0 S at CLEO [33], is also studied. When included in the PWA, the statistical significance of the f J (2220) is found to be 0.8 σ. The upper limit on the branching fraction ratio B(ξ(2230) → φφ)/B(ξ(2230) → K + K − ) at the 90% C.L. is estimated to be 1.91 × 10 −2 . For the description of the nonresonant contribution, the statistical significance of additional non-resonant contributions with J P C = 0 ++ or 2 ++ is less than 5 σ. Additional resonances listed in Ref. [25] as well as two extra states, the X(2120) and X(2370) from Ref. [34], are tested with all possible J P C assignments. None of them has a statistical significance larger than 5 σ, as shown in Table III. The existence of possible additional resonances is further studied by performing scans for extra resonances (J P C = 0 −+ , 0 ++ , 1 ++ , 2 −+ , 2 ++ and 4 ++ ) with different masses and widths. The scan results yield no evidence for extra intermediate states. The reliability of the fit procedure is tested by an input-output check, as follows: An MC sample is generated with given components. After the fitting procedure described above, the properties of the components (mass, width, branching fraction, and the effect of interference terms) are compared with the input values. The output values agree with the input around ±1 σ, confirming the reliability of the fitting procedure.
In addition to the PWA fit with resonances described by BW functions, a model-independent fit where the intermediate states are parameterized by a separate complex constant for each of 35 bins of 20 MeV/c 2 width is performed in the region M (φφ) < 2.7 GeV/c 2 to extract the contribution of components with each J P C using the method described in Ref. [35]. The fit results are shown in Fig. 2(f). The 0 −+ contribution is dominant, and a strong 2 ++ component at 2.3 GeV/c 2 is observed. In general, the model-independent fit gives similar features to those of the model-dependent fit, and the results of these two fits are consistent with each other.

V. SYSTEMATIC UNCERTAINTIES
The sources of systematic uncertainty are divided into two categories. The first includes the systematic uncertainties from the number of J/ψ events (0.8% [36,37]), MDC tracking (1.0% each for three charged tracks [38]), kaon PID (1.0% each for three kaons [38]), photon detec-  tion efficiency (1.0% [38]), kinematic fit (2.5%), φ mass resolution (0.3%) and B φ→K + K − (2.0%). These systematic uncertainties are applicable to all the branching fraction measurements. The total systematic uncertainty from these sources is 5.5%. The second source concerns the PWA fit procedure, where the systematic uncertainties are applicable to measurements of the branching fractions and resonance parameters. These sources of systematic uncertainties are described below.
(i) BW parametrization. Uncertainties from the BW parametrization are estimated by the changes in the fit results caused by replacing the fixed width Γ 0 of the BW for the threshold states η(2100) and η(2225) with a mass-dependent width form Γ(m) [39].
(ii) Uncertainty from resonance parameters. In the nominal fit, the resonance parameters of the 0 ++ and 2 ++ states are fixed. An alternative fit is per-  [25], and the changes in the results are taken as systematic uncertainties.
(iii) Background uncertainty. To estimate the background uncertainty, alternative fits are performed with background events from different φ sideband regions and different normalization factors, and the changes in the results are assigned as the systematic uncertainties.
(iv) Uncertainty from additional resonances. Uncertainties from possible additional resonances are estimated by adding the f 0 (2020) and the f 2 (2150), which are the two most significant additional resonances, into the baseline configuration individually, the changes of the measurements caused by them are assigned as the systematic uncertainties.
For each alternative fit performed to estimate the systematic uncertainties from the PWA fit procedure, the changes of the measurements are taken as the one-sided systematic uncertainties. For each measurement, the individual uncertainties are assumed to be independent and are added in quadrature to obtain the total systematic uncertainty on the negative and positive side, respectively. The sources of systematic uncertainties applicable to the measurements of masses and widths of η(2225), η(2100) and X(2500), and their contributions are summarized in Table IV. The relative systematic uncertainties relevant for the branching fraction measurements are summarized in Table V, where the last row is the to-tal relative systematic uncertainty from fitting irrelevant sources.

VI. SUMMARY
In summary, a PWA on J/ψ → γφφ has been performed based on (1310.6 ± 10.5) ×10 6 J/ψ events collected with the BESIII detector. The most remarkable feature of the PWA results is that 0 −+ states are dominant. The existence of the η(2225) is confirmed and two additional pseudoscalar states, η(2100) with a mass 2050 +30

+56
−33 MeV/c 2 , are observed. The new experimental results are helpful for mapping out pseudoscalar excitations and searching for a 0 −+ glueball. The three tensors f 2 (2010), f 2 (2300) and f 2 (2340) observed in π − p → φφn [9] are also observed in J/ψ → γφφ. Recently, the production rate of the pure gauge tensor glueball in J/ψ radiative decays has been predicted by Lattice QCD [40], which is compatible with the large production rate of the f 2 (2340) in J/ψ → γφφ and J/ψ → γηη [26].