Partial wave analysis of J/\psi \to \gamma \phi \phi

Using $5.8 \times 10^7 J/\psi$ events collected in the BESII detector, the radiative decay $J/\psi \to \gamma \phi \phi \to \gamma K^+ K^- K^0_S K^0_L$ is studied. The $\phi\phi$ invariant mass distribution exhibits a near-threshold enhancement that peaks around 2.24 GeV/$c^{2}$. A partial wave analysis shows that the structure is dominated by a $0^{-+}$ state ($\eta(2225)$) with a mass of $2.24^{+0.03}_{-0.02}{}^{+0.03}_{-0.02}$ GeV/$c^{2}$ and a width of $0.19 \pm 0.03^{+0.06}_{-0.04}$ GeV/$c^{2}$. The product branching fraction is: $Br(J/\psi \to \gamma \eta(2225))\cdot Br(\eta(2225)\to \phi\phi) = (4.4 \pm 0.4 \pm 0.8)\times 10^{-4}$.


Introduction
In addition to the spectrum of ordinary light q q meson states, QCD-motivated models predict a rich spectrum of gg glueballs, q qg hybrids and qq q q four quark states [1].Radiative J/ψ decays provide an excellent laboratory for testing these predictions, and systems of two vector particles have been intensively examined for signatures of gluonic bound states [2].Pseudoscalar enhancements in ρρ and ωω final states have been seen in radiative J/ψ decays [3][4][5][6][7].Recently, a near-threshold scalar, the f 0 (1810) or X(1810) was reported in the ωφ invariant mass distribution from the doubly OZI suppressed decays of J/ψ → γωφ [8], thereby adding an additional puzzle to the already confusing spectrum of low-lying scalar mesons [9,10].
In this letter we present results from a high statistics study of J/ψ → γφφ in the γK + K − K 0 S K 0 L final state, using a sample of 5.8 × 10 7 J/ψ events collected with the BESII detector at the Beijing Electron-Positron Collider (BEPC).The presence of a signal around 2.24 GeV/c 2 and its pseudoscalar character are confirmed, and the mass, width, and branching fraction are determined by a partial wave analysis (PWA).

BES detector and Monte Carlo simulation
BESII is a large solid-angle magnetic spectrometer that is described in detail in Ref. [15].Charged particle momenta are determined with a resolution of σ p /p = 1.78% √ 1 + p 2 (with p in GeV/c) in a 40-layer cylindrical main drift chamber (MDC).Particle identification is accomplished using specific ionization (dE/dx) measurements in the MDC and time-of-flight (TOF) measurements in a barrel-like array of 48 scintillation counters.The dE/dx resolution is σ dE/dx = 8.0%; the TOF resolution is σ T OF = 180 ps for the Bhabha events.Outside of the TOF counters is a 12-radiation-length barrel shower counter (BSC) comprised of gas tubes interleaved with lead sheets.The BSC measures the energies and directions of photons with resolutions of σ E /E ≃ 21%/ √ E (with E in GeV), σ φ = 7.9 mrad, and σ z = 2.3 cm.The iron flux return of the magnet is instrumented with three double layers of counters that are used to identify muons.
In this analysis, a GEANT3-based Monte Carlo (MC) simulation program (SIMBES) [16], which includes detailed consideration of the actual detector responses (such as dead electronic channels), is used.The consistency between data and Monte Carlo has been checked with many high-purity physics channels [16].

Event selection
Since the K 0 L is difficult to identify in BESII, its detection is not required in the selection of J/ψ → γK + K − K 0 S K 0 L events.J/ψ → γK + K − π + π − (K 0 L ) miss candidates are selected from events with four charged tracks with net charge zero in the MDC and with one or two isolated photons in the BSC.Charged tracks are required to be well fitted to a helix, be within the polar angle region | cos θ| < 0.8, and have a transverse momentum larger than 50 MeV/c.For each track, the time-of-flight (TOF) and specific ionization (dE/dx) measurements in the MDC are combined to form a particle identification confidence level for the π, K and p hypotheses; the particle type with the highest confidence level is assigned to each track.The four selected charged tracks are required to consist of an unambiguously identified K + , K − , π + and π − combination.Each candidate photon is required to have an energy deposit in the BSC greater than 60 MeV, to be isolated from charged tracks by more than 20 • , have an angle between the cluster development direction in the BSC and the photon emission direction less than 30 • , and have its first hit in the BSC within the first six radiation lengths.A one-constraint(1C) kinematic fit is performed with the J/ψ → γK + K − π + π − (K 0 L ) miss hypothesis.For events with two isolated photons, the one that gives the smallest χ 2 (γK The K 0 S K 0 L versus K + K − invariant masses are plotted in 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-invariance, the presence of two φ's is a clear signal for the radiative decay J/ψ → γφφ.The histogram in Fig. 1 (b) shows the K + K − invariant mass distribution after the requirement that the invariant mass distribution for events where the K + K − and K 0 S K 0 L invariant masses lie within the φφ mass region (|M(K There are a total of 508 events, which survive the above-listed criteria (optimized for low φφ masses), with a prominent structure around 2.24 GeV/c 2 .The phase space invariant mass distribution and the acceptance versus φφ invariant mass are also shown in Fig. 2 (a) as the dashed histogram and dotted curve, respectively.The peak is also evident as a diagonal band along the upper right-hand edge of the Dalitz plot, shown in Fig. 2 (b).The asymmetry in the dalitz plot of data is caused by detection efficiency.
Non-φφ backgrounds are studied using events in the φ sideband regions shown in Fig. 1 (a).Figure 3 (b) shows the 2 ), and Fig. 3 (c) shows the corresponding spectrum of events within the φ → K 3 (d) shows the events in the corner region, which is defined as (1.069 GeV/c 2 < M(K + K − ) < 1.094 GeV/c 2 and 1.082 GeV/c 2 < M(K 0 S K 0 L ) < 1.132 GeV/c 2 ).The background, estimated by summing up the  The backgrounds in the selected event sample are studied with Monte Carlo simulations of a number of potential background decay channels listed in the PDG tables [17].The main background originates from J/ψ → φK * K, K * → Kπ 0 (+ c.c.).Using the PDG's world average branching fraction for this mode, we estimate that about 40 events from this channel are in the φφ invariant mass signal region.However, the simulation also shows that they do not peak at low φφ masses.Using a Monte Carlo sample of 35.1M inclusive J/ψ decay events generated with the LUND-charm model [18], we searched for other possible background channels.None of the simulated channels produce a peak at low φφ invariant masses.The dotted histogram in Fig. 3 (a) shows the φφ invariant mass distribution and the normalized background estimated with inclusive J/ψ Monte Carlo samples where the events of J/ψ → γφφ and J/ψ → γη c (η c → φφ) are removed.

Partial wave analysis
A partial wave analysis (PWA) of the events with M(φφ) < 2.7 GeV/c 2 was performed.The two-body decay amplitudes in the sequential decay process J/ψ → γX, X → φφ, φ → K + K − and φ → K 0 S K 0 L are constructed using the covariant helicity coupling amplitude method [19].The intermediate resonance X is described with the normal Breit-Wigner propagator BW = 1/(M 2 − s − iMΓ), where s is the φφ invariant mass-squared and M and Γ are the resonance's mass and width.The amplitude for the sequential decay process is the product of all decay amplitudes and the Breit-Wigner propagator.The total differential cross section dσ/dΦ is where A(J P C ) is the total amplitude for all resonances whose spin-parity are J P C , and BG denotes the background contribution, which is described by a non-interfering phase space term.
The relative magnitudes and phases of the amplitudes are determined by an unbinned maximum likelihood fit.The basis of likelihood fitting is the calculation of the probability that a hypothesized probability distribution function would produce the data set under consideration.The probability to observe the event characterized by the measurement ξ is where ω(ξ) ≡ dσ dΦ and ǫ(ξ) is the detection efficiency.The normalization integral dξω(ξ)ǫ(ξ) is done with a weighted phase space MC sample; the details are described in Ref. [20].The joint probability density for observing the N events in the data sample is For technical reasons, rather than maximizing L, S = −lnL is minimized, i.e.
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.So, for the fitting, lnL, defined as: is used.The free parameters are optimized by MINUIT [21].In the minimization procedure, a change in log likelihood of 0.5 represents a one standard deviation effect for the one parameter case.
For the production of a pseudoscalar, only P waves are allowed in both the radiative decay J/ψ → γX and the hadronic decay X → φφ.For the case of a scalar, both S and D waves are possible in both the radiative and hadronic decays; only the S wave is considered in the fit.For the production of a 2 + resonance, there are five possible amplitudes for both the radiative and the hadronic decays, one S wave, three D waves and one (L = 4) wave.In this case, only S and D waves in both decays, corresponding to the lower overall spin of the φφ system, are considered.(The (L = 4) wave is ignored in this analysis.) When J/ψ → γX, X → φφ is fitted with both the φφ and γX systems in a P wave, which corresponds to a X = 0 −+ pseudoscalar state, the fit gives 195.5 +18.6 −18.7 events with mass M = 2.24 +0.03 −0.02 GeV/c 2 , width Γ = 0.19 ± 0.03 GeV/c 2 , and a statistical significance larger than 10 σ.The errors are statistical only.Using a selection efficiency of 3.29%, which is determined from the Monte-Carlo simulation using the magnitudes and phases of the partial amplitudes from the PWA, we obtain a product branching fraction of: Details of the fitting procedure and the detection efficiency determination can be found in Ref. [20]. Figure 4   We also tried to fit the resonance with 0 ++ and 2 ++ spin-parity hypotheses using all possible combinations of orbital angular momenta in the φφ and γ(φφ) systems.The log-likelihood values of the best fits are worse than that of the 0 −+ assignment by 95 and 27 for the 0 ++ and 2 ++ assignments, respectively.We therefore conclude that the J P C of the resonance strongly favors 0 −+ (η(2225)).
Because of the possible existence of a coherent J/ψ → γφφ phase space amplitude, we also fitted with an interfering phase space (0 − ) term included.The log-likelihood value S improves by 0.4, which suggests that the contribution from 0 −+ phase space is negligible.
If an additional resonance is included in the fit, the significance of the additional resonance after reoptimization is 0.8 σ, 2.1 σ and 3.3 σ for the 0 −+ , 0 ++ and 2 ++ assignments of the additional resonance, respectively.The differences between the results including and not including the additional 0 ++ or 2 ++ are included in the systematic errors; the systematic uncertainty contributions for the mass, width, and branching fraction of the η(2225) are +0.01GeV/c 2 , +0.04 GeV/c 2 , and +2.6%, respectively.

Systematic error
The systematic uncertainties are estimated by considering the following: the uncertainties in the modeling of the background, different Breit-Wigner parameterizations, the possible presence of additional resonances, the simulation of the MDC wire resolution, as well as possible biases in the fitting procedure.
The uncertainties in the background include the uncertainty in the treatment of the background in the fitting.We also tried to subtract the backgrounds determined from the sidebands in the fit, and the differences are taken as systematic errors.Since the enhancement is in the near-threshold region of the φφ invariant mass spectrum, a fit using a Breit-Wigner with a momentum dependent width [22] is also performed, and the differences between the fit with a constant width Breit-Wigner are included as systematic errors.Possible fitting biases are estimated from the differences obtained between input and output masses and widths from Monte Carlo samples, which are generated as J/ψ → γ0 −+ , 0 −+ → φφ using the fitted parameters.The total systematic errors are obtained by adding the individual errors in quadrature.The total systematic errors on the mass and width are determined to be +0.03−0.02 GeV/c 2 and +0.06 −0.04 GeV/c 2 , respectively.
For the systematic error on the branching fraction measurement, the systematic uncertainties of the photon detection efficiency and the particle identification efficiency, as well as the φ and K 0 S decay branching fractions, the mass and width uncertainties of η(2225), and the total number of J/ψ events [23] are also included.The total relative systematic error on the product branching fraction is +17.7 −18.5 %.

Fig. 1 .
Fig. 1.(a) The K 0S K 0 L versus K + K − invariant mass distribution.(b) The K + K − invariant mass distribution when the K 0 S K 0 L invariant mass is in the φ → K 0 S K 0

Fig. 2 .
Fig. 2. (a) The K + K − K 0 S K 0 L invariant mass distribution for J/ψ → γφφ candidate events.The dashed histogram is the phase space invariant mass distribution, and the dotted curve indicates how the acceptance varies with the φφ invariant mass.(b) Dalitz plot for J/ψ → γφφ candidate events.scaledevent yields in Figs.3(b) and (c) and subtracting that in Fig.3 (d), is shown as the dashed histogram in Fig.3 (a).No sign of an enhancement near the φφ mass threshold is evident in the non-φφ background events.

Fig. 3 .
Fig. 3.The K + K − K 0 S K 0 L invariant mass distributions for (a) events in the φφ signal region; (b) events in the φ → K + K − sideband region; (c) events in the φ → K 0 S K 0 L sideband region; (d) events in the corner region, as described in the text.The dashed histogram in (a) shows the background distribution obtained from the sideband evaluation.The dotted histogram in (a) shows the φφ invariant mass distribution of inclusive J/ψ Monte Carlo samples (a) shows a comparison of the data and MC projections of the φφ invariant mass distribution for the fitted parameters.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 Figs. 4 (b)-(f).

Fig. 4 .
Fig. 4. Comparisons between data and MC projections for the φφ invariant mass distribution and the angular distributions for events with φφ invariant mass less than 2.7 GeV/c 2 using the fitted η(2225) resonance parameters.The points with error bars are data, the solid histograms are the MC projections, and the dashed lines are the background contributions.(a) The φφ invariant mass distribution; (b) the polar angle of the radiative photon (θ γ ); (c) the polar angle of the φ in the φφ rest system (θ φ ); (d) the polar angle of the kaon in the φ rest system (θ K ); (e) the azimuthal angle of the φ in the φφ rest system (φ φ ); and (f) the χ distribution, where χ is the azimuthal angle between the normal directions of the two decay planes in the φφ rest frame.