Study of the Process e+ e- -->pi0 pi0 gamma in c.m. Energy Range 600--970 MeV at CMD-2

The cross section of the process e+ e- -->pi0 pi0 gamma has been measured in the c.m. energy range 600--970 MeV with the CMD-2 detector. The following branching ratios have been determined: B(rho -->pi0 pi0 gamma) =(5.2^{+1.5}_{-1.3} +- 0.6)x10^{-5} and B(omega -->pi0 pi0 gamma) =(6.4^{+2.4}_{-2.0} +- 0.8)x10^{-5}. Evidence for the rho -->f0(600) gamma decay has been obtained: B(rho -->f0(600) gamma) = (6.0^{+3.3}_{-2.7}\pm 0.9)x10^{-5}. From a search for the process e+ e- -->eta pi0 gamma the following upper limit has been obtained: B(omega -->eta pi0 gamma)<3.3 10^{-5} at 90% CL.

Information on the ρ(ω) decays to the ππγ(ηπ 0 γ) final states is rather scarce. Because of the large background from the initial state radiation in the process e + e − → π + π − , a search for ρ(ω) decays into the π + π − γ final state is difficult and among many such experiments [17] only one succeeded in the observation of the decay ρ 0 → π + π − γ [18]. A long search for the ω → π 0 π 0 γ decay (see [17] and references therein) finally proved successful for the GAMS Collaboration, which observed it in π − p collisions with the branching fraction of (7.4 ± 2.5) × 10 −5 [19]. Recently a high statistics study of the ρ(ω) energy range has been performed by the SND Collaboration [20,21]. They observe both ρ and ω decays into π 0 π 0 γ with a branching ratio higher than that predicted by vector dominance. While for the ρ meson the excess can be explained by the ρ 0 → f 0 (600)γ decay, first evidence for which is reported by SND, the reasons for the higher probability of the corresponding ω decay are not yet clear.
In our recent paper [22] we described a study of the process e + e − → π 0 π 0 γ in the c.m. energy range 920-1380 MeV, i.e. above the threshold of ωπ 0 production, using the CMD-2 detector at the VEPP-2M e + e − collider. In this work we report on the measurement of the cross section of the process e + e − → π 0 π 0 γ in the c.m. energy range 600-970 MeV with CMD-2. Also described is the first search for the process e + e − → ηπ 0 γ in this energy range.

Experiment
The general purpose detector CMD-2 has been described in detail elsewhere [23]. Its tracking system consists of a cylindrical drift chamber (DC) and doublelayer multiwire proportional Z-chamber, both also used for the trigger, and both inside a thin (0.38 X 0 ) superconducting solenoid with a field of 1 T. The barrel CsI calorimeter with a thickness of 8.1 X 0 placed outside the solenoid has energy resolution for photons of about 9% in the energy range from 100 to 700 MeV. The angular resolution is of the order of 0.02 radians. The end-cap BGO calorimeter with a thickness of 13.4 X 0 placed inside the solenoid has energy and angular resolution varying from 9% to 4% and from 0.03 to 0.02 radians, respectively, for the photon energy in the range 100 to 700 MeV. The barrel and end-cap calorimeter systems cover a solid angle of 0.92×4π radians. This analysis is based on a data sample corresponding to integrated luminosity of 7.7 pb −1 collected in 1998-2000 in the energy range 600-970 MeV. The step of the c.m. energy scan varied from 0.5 MeV near the ω peak to 5 MeV far from the resonance. The beam energy spread is about 4 × 10 −4 of the total energy. The luminosity is measured using events of Bhabha scattering at large angles [24].
We use Monte Carlo simulation (MC) to model the response of the detector and determine the efficiency. Due to the beam background additional ("fake") clasters can appear in the calorimeter. The corresponding probability as well as photon energy and angular spectra are obtained directly from the data using the process e + e − → π + π − π 0 . Then these photons are mixed with the detector response during simulation.

Data analysis
At the initial stage, we select events which have no tracks in the DC, five photons, the total energy deposition 1.7 < E tot /E beam < 2.2, the total momentum P tot /E beam < 0.3 and at least three photons detected in the CsI calorimeter. The minimum photon energy is 30 MeV for the CsI and 40 MeV for the BGO calorimeter. Then a kinematic fit requiring energy-momentum conservation is performed with an additional reconstruction of two π 0 mesons. We require good reconstruction quality (χ 2 < 5) and the ratio of the reconstructed to measured energy to be 0.75 < ω i /E i < 1.8 for each photon. After this stage 219 events remain in the whole energy range.
The dominant background comes from the processes e + e − → 3γ, 4γ.
Events from the process (1) can imitate signal events if two soft photons are lost. The processes (2,3) can meet the selection criteria with one ore two additional ("fake") photons coming from shower splitting, "noisy" electronic channels in the calorimeter and beam background.
To determine the background contribution, the following procedure is used. All above listed processes are simulated using the Monte Carlo. To decrease a statistical uncertainty, ten times more events than is expected from the background cross section and luminosity are used at each energy. Then the same selection criteria as for the data are applied. The obtained number of selected events is divided by ten and subtracted from experimental data at each energy. In total, about 29 background events are expected after this procedure. Figure 1 shows the E tot and χ 2 distribution for the data, signal and background MC. We use the χ 2 distributions to estimate the accuracy of the background estimation. The experimental distribution was fitted by a sum of MC and background. The ratio N exp bg /N MC bg = 1.2 ± 0.2 was obtained. We conclude that the background level is estimated well and its systematic error does not exceed 40%. This results in a 6% systematic uncertainty in the cross section.

Approximation of the cross sections
At each energy point the cross section of the process σ is calculated from the observed number of events and background MC expectation using the following formula: where N exp is the number of observed events, N bg is the expected number of background events from MC, L is the integrated luminosity, ε is the detection efficiency and (1 + δ) is the radiative correction at the corresponding energy.
To calculate the detection efficiency, we use Monte Carlo simulation taking into account the neutral trigger (NT) efficiency. NT is based on the information from the CsI calorimeter and its efficiency depends on the number of clusters and total energy deposition. The NT efficiency is estimated using events of the process e + e − → π + π − π 0 . We require the charged trigger signal and three or more clusters in the CsI calorimeter, and study the NT efficiency as a function of the energy deposition in CsI. The NT efficiency varies from 85% at c.m. energy 600 MeV to about 95% at 980 MeV.
The radiative corrections are calculated according to [27]. The dependence of the detection efficiency on the energy of the emitted photon is determined from simulation.
The obtained Born cross section of the process e + e − → π 0 π 0 γ is shown in Fig. 2 while Table 1 lists detailed information on the analysis of this reaction. No events were selected in the energy range 600 to 690 MeV, therefore our results are presented as upper limits at 90% C.L. The Feldman-Cousins procedure [25] was used to calculate errors (upper limits) at each energy. This cross section (the "dressed" one from the column VII) was used in the approximation of the energy dependence with resonances.
Meanwhile, for applications to various dispersion integrals like that for the leading order hadronic contribution to the muon anomalous magnetic mo- Table 1 The c.m. energy, integrated luminosity, detection efficiency, number of observed events, background expectation, radiative correction, Born cross section σ, vacuum polarization correction and "bare" cross sectionσ of the process e + e − → π 0 π 0 γ. ment, the "bare" cross section should be used. Following the procedure in Ref. [26], the latter is obtained from the "dressed" one by multiplying it by the vacuum polarization correction |1 − Π(s)| 2 , where Π(s) is the photon polarization operator calculated taking into account the effects of both leptonic and hadronic vacuum polarization. The values of the correction and the "bare" cross sectionσ are presented in two last columns of Table 1.
The maximum likelihood method is applied to fit the experimental data obtained from the relation (4). We parameterize the amplitude of the process by a sum of the ρ and ω contributions. The former contains the ρ → ωπ 0 transition plus one more mechanism beyond the vector dominance model which is chosen to be the ρ 0 → f 0 (600)γ one. Because of the small width of the ω meson, the ρ → ωπ 0 amplitude is rapidly falling below the ωπ 0 threshold, making thereby the ρ 0 → ωπ 0 and ρ 0 → f 0 (600)γ transitions distinguishable. On the contrary, the small width of the ω meson prevents from distinguishing various mechanisms possibly existing in the ω → π 0 π 0 γ decay. For this reason we parameterize the ω meson amplitude by the ω → ρπ transition only.
The Born cross section of the process is written as: where dΦ is the final state phase space and Here the first term describes the amplitude of the e + e − → ρ, ρ′ → ωπ 0 transition, while the second and third ones are the e + e − → ρ → f 0 (600)γ and e + e − → ω → ρπ 0 amplitudes. m V is the mass and D V is the propagator of the vector meson V given by D V (s) = s − m 2 V + i √ sΓ V (s), Γ V (s) is the corresponding energy dependent width. The real parameter α = g ρ′ωπ /g ρωπ is the ratio of the coupling constants for the ρ and ρ′ mesons. The A ρ→ωπ 0 amplitude, proportional to the coupling constant g ρωπ of the ρ → ωπ transition, is written as in our previous analysis above 1 GeV [22].

Results of the fits
In all the following fits the ρ, ω and ρ′ meson masses and widths are fixed at the world average values [17]. The f 0 (600) mass and width are badly known [17]. Therefore, we use a wide range of these parameters: M f 0 (600) = 400-800 MeV, Γ f 0 (600) = 300-600 MeV. For the ρ and ρ′ resonances the energy dependence of the total width was described similarly to Ref. [22] while for the ω meson the total width was assumed to be energy independent. We perform three main fits: with g ρωπ equal to the value (16.7 ± 0.4 ± 0.6) GeV −1 obtained in our analysis above 1 GeV [22] (fit I), with free g ρωπ (fit II) and without a contribution from the ρ 0 → f 0 (600)γ decay (fit III). The results of the fits are shown in Table 2 and in Fig. 2 by the curves.
The value of the coupling constant g ρωπ = 16.2 ± 1.4 obtained in the second fit is in good agreement with the one from our measurement of the e + e − → ωπ 0 cross section above 1 GeV [22].
• The value of the coupling constant g ρωπ = 18.6 ± 1.1 obtained in the fit III is above that from our analysis above 1 GeV [22] by almost two standard deviations. • The recent analysis of the process e + e − → π 0 π 0 γ by SND [21] also showed evidence for the ρ 0 → f 0 (600)γ decay.
Thus, we choose the first model as our final result. The ρ 0 → f 0 (600)γ branching fraction is calculated from the results listed in Table 2 taking into account that B(f 0 (600) → π 0 π 0 ) = 1/3. Figure 3 shows the spectrum of a π 0 π 0 invariant mass for experimental events from the ω meson energy range (770-800 MeV). The experimental distribution agrees well with the ω → ρπ 0 decay model, however, the contribution from the ω → f 0 (600)γ decay is also acceptable. The existing statistics is not enough to distinguish these contributions, therefore we obtain only the total branching fraction of the ω → π 0 π 0 γ decay.

Search for the decay ω → ηπ 0 γ
For a search of events of the process e + e − → ηπ 0 γ we first apply the same criteria as for the preliminary selection of e + e − → π 0 π 0 γ events. After that a kinematic fit requiring energy-momentum conservation is performed with the additional reconstruction of one soft π 0 meson and a good reconstruction quality, χ 2 < 6, is required. To reject the dominant background from the process e + e − → π 0 π 0 γ, we perform an additional kinematic fit with the π 0 π 0 γ hypothesis and reject events that are consistent with it, χ 2 π 0 π 0 γ < 6. Then we look for a possible η signal in the invariant mass of two hard photons of the remaining three, M γγ . The M γγ distribution is approximated with a Gaussian for the signal and a polynomial function for the background. The Gaussian mean value and width are fixed from the MC simulation of the signal events. The background shape is obtained using the π 0 π 0 γ MC. In all energy ranges the resulting ηπ 0 γ signal is consistent with zero. Figure 4 shows the M γγ distribution for events from the ω resonance region: 381 MeV < E beam < 401 MeV. The 90% CL upper limit for the number of ηπ 0 γ events is obtained: N ηπ 0 γ < 2.4. Using the detection efficiency of 1.3%, we set the following upper limit for the e + e − → ηπ 0 γ cross section: σ(e + e − → ηπ 0 γ) < 57 pb , and for the branching fraction of the ω meson:

Systematic errors
The main sources of systematic uncertainties in the cross section determination are listed in Table 3. The systematic error due to selection criteria is obtained by varying the photon energy threshold, total energy deposition, total momentum, and χ 2 . The model uncertainty corresponds to different detection efficiencies for the ωπ 0 , ρπ 0 and f 0 (600)γ intermediate states. It also includes dependence on the f 0 (600) mass and width. The uncertainty in the determination of the integrated luminosity comes from the selection criteria of Bhabha events, radiative corrections and calibrations of DC and CsI. The error of the NT efficiency was estimated by trying various fitting functions for the energy dependence and variations of the cluster threshold. The uncertainty of the radiative corrections comes from the dependence on the emitted photon energy and the accuracy of the theoretical formulae. The resulting systematic uncertainty of the cross section in Table 1 as well as of the branching fractions in Table 2 is 12%.

Mode
Branching fraction with the standard value of the coupling constant: ∼ 1 × 10 −5 and ∼ 3 × 10 −5 [4], respectively. An attempt to explain the obtained branching ratios results in a high value of g ρωπ contradicting the other observations like, e.g., the experimental values of the ω → π + π − π 0 and ω → π 0 γ widths.
Predictions of these models differ rather strongly from each other reflecting various approaches applied by their authors [10]. While most of the recent papers agree that the observed value of the branching fraction for the ρ 0 → π 0 π 0 γ decay can be ascribed to the intermediate f 0 (600) state, the situation with the corresponding ω decay remains controversial. The corresponding ranges of the predicted values of branching fractions are summarized in Table 4. Note that from the upper limits for the non-ωπ π 0 π 0 γ cross section obtained by us at higher energy in Ref. [22] a significant contribution from the f 0 (980)γ or f 2 (1270)γ mechanisms appears not very likely.
Much higher data samples of the ρ and ω decays expected in experiments at the upgraded collider VEPP-2000 in Novosibirsk [28] will help to significantly improve our understanding of their radiative decays.
From the obtained results on the cross section of the radiative processes e + e − → π 0 π 0 γ, ηπ 0 γ one can estimate a possible contribution of the previously unstudied radiative processes to the leading order hadronic correction to the muon anomalous magnetic moment. To this end we first calculate the contribution of the process e + e − → π 0 π 0 γ using the "bare" cross section,σ, from Table 1 in the energy range below 920 MeV. The result contains a piece coming from the ω → π 0 π 0 γ decay already taken into account in Ref. [29] in the whole ω meson contribution, a ω µ = (37.96 ± 1.07) · 10 −10 . This ω meson contribution is subtracted from the value above using the branching ratio B(ω → π 0 π 0 γ) with the result (6.08 ± 0.82) · 10 −12 . A possible contribution from the process e + e − → π + π − γ is twice that of e + e − → π 0 π 0 γ, so that a LO,ππγ µ (600 MeV − 920 MeV) = (18.2 ± 2.5) · 10 −12 .

Conclusions
The following results are obtained in this work: • Using a data sample corresponding to integrated luminosity of 7.7 pb −1 , the cross section of the process e + e − → π 0 π 0 γ has been measured in the c.m. energy range 600-970 MeV. The values of the cross section are consistent with those obtained by the SND detector [21] and have the similar accuracy.
• A first search for the process e + e − → ηπ 0 γ was performed allowing to set the 90% CL upper limits: σ(e + e − → ηπ 0 γ) < 57 pb in the c.m. energy range 685-920 MeV and B(ω → ηπ 0 γ) < 3.3 × 10 −5 . • A possible contribution of the studied radiative processes to the muon anomalous magnetic moment was estimated to be negligible.