Study of the process e+e−→K+K−η with the CMD-3 detector at the VEPP-2000 collider

Article history: Received 19 June 2019 Received in revised form 12 September 2019 Accepted 12 September 2019 Available online 18 September 2019 Editor: L. Rolandi The process e+e−→K + K −η has been studied in the center-of-mass energy range from 1.59 to 2.007 GeV using the data sample of 59.5 pb−1, collected with the CMD-3 detector at the VEPP-2000 e+e− collider in 2011, 2012 and 2017. The K +K −η final state is found to be dominated by the contribution of the φ(1020)η intermediate state. The cross section of the process e+e−→φ(1020)η has been measured with a systematic uncertainty of 5.1% on the base of 3009 ± 67 selected events. The obtained cross section has been used to calculate the contribution to the anomalous magnetic moment of the muon: a μ (E < 1.8 GeV) = (0.321 ± 0.015stat ± 0.016syst) × 10−10, a μ (E < 2.0 GeV) = (0.440 ± 0.015stat ± 0.022syst) × 10−10. From the cross section approximation the φ(1680) meson parameters have been determined with better statistical precision, than in previous studies. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.


Introduction
A high-precision measurement of the cross section of e + e − → hadrons has numerous applications including, e.g., a cal- magnetic moment (g − 2) μ and running fine structure constant.
To confirm or deny the observed difference between the calculated (g − 2) μ value [1][2][3][4] and the measured one [5], more precise measurements of the exclusive channels of e + e − → hadrons are necessary.
The process e + e − →K + K − η has been previously studied by the BaBar collaboration at the center-of-mass energies (E c.m. ) from 1.56 to 3.48 GeV in the η→2γ decay mode [6] and from 1.56 to 2.64 GeV in the η→π + π − π 0 decay mode [7] (∼480 and ∼110 signal events were selected, respectively). Another study of this process in the E c.m. range from 1.56 to 2.0 GeV was performed by the SND collaboration [8] with ∼265 selected signal events. In the BaBar study [6] it was found that the dominant intermediate mechanism in this process is e + e − →φ(1680)→φ(1020)η (in what follows φ(1020)≡φ, φ(1680)≡φ and natural units h = c = 1 are used), so the total cross section σ (e + e − →K + K − η) was subdivided into two parts: σ (e + e − →φη)·B φ K + K − (for the invariant masses of kaons m inv, 2K < 1045 MeV) and σ NON−φ (e + e − →K + K − η) (for m inv, 2K > 1045 MeV). The latter was only 3-12% of the total cross section, and the data samples of BaBar were not sufficient to analyze the intermediate mechanisms in the NON − φ part of the reaction [6]. As the φ meson dominates in this process, its parameters can be extracted from the approximation of the e + e − →φη cross section.
In this paper we report the results of the study of the pro-

CMD-3 detector and data set
The VEPP-2000 e + e − collider [9][10][11][12] at the Budker Institute of Nuclear Physics covers the E c.m. range from 0.32 to 2.01 GeV and uses a technique of round beams to reach an instantaneous luminosity of 10 32 cm −2 s −1 at E c.m. =2.0 GeV. The Cryogenic Magnetic Detector (CMD-3) described in [13] is installed in one of the two interaction regions of the collider. The detector tracking system consists of the cylindrical drift chamber (DC) [14] and doublelayer cylindrical multiwire proportional Z-chamber, installed inside a thin (0.085 X 0 ) superconducting solenoid with 1.0-1.3 T magnetic field. Both subsystems are also used to provide the trigger signals. DC contains 1218 hexagonal cells in 18 layers and allows one to measure charged particle momentum with 1.5-4.5% accuracy in the 40-1000 MeV range, and the polar (θ ) and azimuthal (ϕ) angles with 20 mrad and 3.5-8.0 mrad accuracy, respectively. Amplitude information from the DC signal wires is used to measure ionization losses (dE/dx) of charged particles. The barrel electromagnetic calorimeters based on liquid xenon (LXe) [15] (5.4 X 0 ) and CsI crystals (8.1 X 0 ) are placed outside the solenoid [16]. The total amount of material in front of the barrel calorimeter is 0.13 X 0 that includes the solenoid as well as the radiation shield and vacuum vessel walls. The endcap calorimeter is made of 680 BGO crystals of 13.4 X 0 thickness [16]. The magnetic flux-return yoke is surrounded by scintillation counters which are used to tag cosmic events.
To study a detector response and determine a detection efficiency, we have developed a code for Monte Carlo (MC) simulation of our detector based on the GEANT4 [17] package so that all simulated events are subjected to the same reconstruction and selection procedures as the data.
The energy range E c.m. = 1.0-2.007 GeV was scanned in the runs of 2011, 2012 and 2017. The integrated luminosity at each energy point was determined using events of the processes e + e − →e + e − and e + e − →2γ [18]. The beam energy was monitored by measuring the current in the dipole magnets of the main ring (in 2011 and 2012), and by using the Back-Scattering-Laser-Light system (in 2017) [19,20]. In the runs of 2011 and 2012 we use the measured average momentum of electrons and positrons in events of Bhabha scattering, as well as the average momentum of proton-antiproton pairs from the process e + e − →pp process [21] to determine the actual E c.m. values for each nominal beam energy with about 6 and 2 MeV accuracy, respectively.

Event selection
In order to measure the cross section of K + K − η production, one needs to determine the detection efficiency for these events. The detection efficiency strongly depends on the intermediate mechanisms of the process and to reveal those mechanisms K + K − η events are selected in the η→2γ decay mode resulting in a sample of almost background-free events.

Selection of "good" tracks
Candidates for K + K − η events are required to have exactly two "good" tracks in the DC with the following "good" track definition: 1) a track transverse momentum p ⊥ is larger than 60 MeV; 2) a distance of the closest track approach (PCA) to the beam axis in the transversal plane (ρ PCA ) is less than 0.5 cm; 3) a distance from the PCA to the center of the interaction region along the beam axis (z PCA ) is less than 12 cm; 4) a polar angle θ of the track is in the range from θ cut ≡0.9 to π − θ cut radians; 5) for positively charged particles ionization losses dE/dx of the track are smaller than ionization losses typical of a proton with the same momentum.

Selection of kaons
To select events with two oppositely charged kaons, we use the functions f K /π (p, dE/dx) representing the probability density for charged kaon/pion with the momentum p to produce the energy losses dE/dx in the DC. These functions were obtained at each E c.m. in the analysis of the process e + e − →K + K − π + π − with the CMD-3 detector [22], and we use them to simulate dE/dx of the kaons and pions. Further, the log-likelihood function (LLF) for the hypothesis that for i = 1, 2 two oppositely charged particles with the momenta p i and energy losses (dE/dx) i are kaons is defined as see its distribution in Fig. 1. We apply the cut L 2K > −0.3 to select events with kaons.

Kinematic fit
To select K + K − η events in the η→2γ mode, we select events with two or more photons with energies larger than 40 MeV and polar angles θ γ in the range from 0.5 to π − 0.5 radians. Then we perform a kinematic fit (assuming energy-momentum conservation) of a K + K − pair with each pair of selected photons, searching for the combination that gives the minimal χ 2 4C . We apply a requirement on the χ 2 4C < 75 value to select signal events, see Fig. 2 (unless otherwise stated, in what follows the simulated histograms are normalized to the expected number of events according to the cross sections measured in [6,7,22,23]; the simulation of signal and background processes includes the emission of photon jets by initial electrons and positrons according to [24]). The resulting distributions of dE/dx vs particle momentum, m inv,2γ and m inv,2K are shown in Figs. 3-5. It is seen that the φη→K + K − η mechanism dominates in the process. Furthermore, events with m inv, 2K > 1075 MeV show no peaking structure near m inv,2γ = m η (Fig. 6), thus mainly coming from the background (the expected contribution of φη→K + K − 2γ is about 30 events).
Thus, we do not observe a contribution of any intermediate states in the process e + e − →K + K − η other than φη. In what follows we measure the cross section of the process e + e − →φη using   the recoil to an η meson. Such an inclusive approach allows us to avoid the loss of statistics due to selection of specific η decay modes, but in turn it increases the amount of background.

Signal/background separation
We use the requirement L 2K > −0.3 to select events with two oppositely charged kaons and then the requirement m inv, 2K < 1050 MeV to select events from the φ-meson region, see Fig. 7.
Simulation shows that the major background final states are   We perform the signal/background separation using the distribution of the E parameter ( Fig. 8), which is defined as and represents the "energy disbalance" of the event assuming the η to be the recoil particle for the K + K − pair.
We approximate the distribution of E in the range from −150 to 100 MeV at each E c.m. , see Fig. 9. The linear function is used to describe the background shape. The shape of the signal is determined at each E c.m. by fitting the simulated signal E distribution by the sum of three Gaussians: In the fit of the experimental E distribution we fix the parameters a 1,2 , μ i and σ i characterising the signal shape at the values obtained from the fit of the simulation. The signal amplitude a 0 , the possible shift δx and smearing δσ of the signal distribution are taken as floating parameters: In total, 3009 ± 67 of signal events were selected.   In the study of the process e + e − →K + K − π + π − with the CMD-3 detector [22] it was found that the average detection efficiencies for kaons in experiment, (ε K exp ), and simulation, (ε K MC ), agree with the accuracy of 1% (the 0.85 < θ < π − 0.85 range was considered). Thus we estimate the systematic uncertainty of the kaon detection efficiency for the "good" polar angle range 1.0 < θ < π − 1.0 as less than 1%. At polar angles θ < 1.0 and θ > π − 1.0 the kaon detection efficiency decreases in a different way in data and simulation, leading to the difference of the experimental and simulated kaon polar angle spectra. From that difference one can obtain the correction to the selection efficiency for the K + K − η final state. To do this we select events from the signal peak region −40 MeV < E < 20 MeV with at least one kaon having the polar angle in the range 1.1 to π − 1.1 (we assume ε K exp to be equal to ε K MC in this range). Fig. 11 shows the comparison of the |π /2 − θ| distributions for the second kaon in data and simulation. The approximation of the ratio of spectra in simulation and data by the function 1 + exp(p 0 (p 1 − θ)) provides a correction for the kaon selection efficiency (1 + δ K eff )(θ) as a function of θ , see Fig. 12. The uncertainty of this function is obtained by the multifold variation of the points in the histogram, shown in Fig. 12, and it's subsequent refitting.

Efficiencies
The correction (1 + δ eff ) for the kaons selection efficiency in with the polar angle distributions of the kaons reconstructed in simulation: The values of this correction at different energies are shown in Fig. 13. The systematic uncertainty of these values is derived from the uncertainty of (1 + δ K eff )(θ) function and is estimated to be 1.5%.
To test the validity of the obtained correction, we use the value of the estimated total number of signal events N sig.tot , actually produced at the collider during the experimental runs: where N i sig.events is the number of selected signal events at the i-th energy, ε i -the corrected detection efficiency at that energy.
Application of the efficiency correction makes N sig.tot almost independent of θ cut , as one can see from Fig. 14.
Next, since the ε MC value does not include the trigger efficiency ε trig , the latter should be found separately from the experimental data. The trigger of the CMD-3 detector consists of two subsystems, so-called "neutral" trigger (NT) and "charged" trigger (CT), connected into the OR scheme, and the overall trigger efficiency equals where the efficiencies of NT and CT are expressed in terms of the number of events in the experiment, in which only NT (N NT ), only CT (N CT ) or both subsystems (N NT&CT ) were triggered:

Cross section calculation and approximation
The Born cross section of the process e + e − →φη at each E c.m. is calculated by dividing the visible cross section σ vis by the radiative where N sig.events is the number of selected events of the signal pro- . (11) We perform the calculation iteratively, using for the first iteration the approximation of the cross section measured by BaBar [6], in the E c.m. range from 1.58 to 2.0 and from 2.3 to 3.5 GeV (excluding the region from 2.0 to 2.3 GeV to avoid the fitting of the φ(2170) resonance). For the cross section approximation we use the formula where | is the momentum of the kaon in the φ→K + K − decay in the φ rest frame, θ normal is the polar angle of the normal to the plane formed by the p K + and p K − vectors, d K + K − η is the element of three-body phase space. We neglect the OZIsuppressed [25] contribution of ω(1650), but consider the possible contribution of the resonance below reaction threshold, describing it via the amplitude a n.r. e i n.r. /s. The factor F (s) represents the "dynamic" part of the squared matrix element averaged over the three-body K + K − η phase space.
The quantity φ (s) is given by the following expression (see [23]): where σ designates the f 0 (500) meson, the P K * (892)K and P φη functions represent the phase spaces of quasi-two-body final states in φ →K * (892)K and φ →φη decays. According to [23] we take where V = K * , φ, P = K , η. The P φσ function in (14) represents the phase space of the quasi-two-body final state in φ →φσ decay and is calculated as: is the quantity proportional to the width of the φ →φσ decay with the σ mass equal to m. Integration in the formula 16 is performed in the range available for m = m inv,2π ∈ (2m π ; √ s − m φ ). Similarly to φ (s) the φ (s) is calculated taking into account the K + K − , K S K L and π + π − π 0 modes of φ-meson decay.
It should be noted that in the work of BaBar [6] for the e + e − →φη cross section fit the quasi-two-body formula was used. The normalized difference (σ 3body φη /σ 2body φη − 1) of the three-body and quasi-two-body cross section parametrizations is shown in Fig. 16. At the current level of a systematic uncertainty (see Section 3.5) it becomes important for us to use a more precise three-body formula.
After the first iteration we use CMD-3 data along with the BaBar data in the range from 2.3 to 3.5 GeV, which is necessary to fix the asymptotic behavior of the cross section. Four iterations are sufficient for the radiative corrections to converge with the accuracy of 0.5%. Fig. 17 shows the values of the radiative correction at the last iteration. The uncertainties of the radiative corrections caused by the cross section shape are calculated by the multifold variation of the visible cross sections and subsequent recalculation of the radiative corrections and were found to be <1.5%.  The obtained e + e − →φη Born cross section (see Tables 1-3) along with that of BaBar [6] and SND [8] is shown in Fig. 18. The fit of the cross section asymptotics is shown in Fig. 19. The obtained Born cross section exhibits a hint to the wavelike deviation from the fit near E c.m. ≈1.9 GeV, see Fig. 20. This may be due to the uncertainties of the branching fractions of φ decay modes or due to the decay modes, that were not taken into account in our cross section parameterization. However, at the current level of statistics we are not sensitive to these effects.
The φ parameters, obtained from the approximation of the CMD-3 cross section are shown in Table 4. Along with the cross section parametrization using  [6] and other previous measurements, but have better statistical precision. The estimation of the systematic uncertainties of φ parameters is described in Section 3.5.

Systematic uncertainties
We estimate a systematic uncertainty related to some selection criterion as a relative variation of the N sig.tot (see Section 3.3) with the variation (or switching on/off) of this criterion. The limits for the variation of the cuts are chosen as wide as possible with two requirements: 1) restriction does not seriously cut the signal;   2) the background shape is reasonably described by the contribution of K + K − π 0 π 0 and K + K − π + π − final states. The following sources of systematic uncertainties were considered: • The requirements on ρ PCA , z PCA , p ⊥ and dE/dx < (dE/dx) protons for positively charged particles applied in the "good" track selection procedure, give the uncertainties of 1.0, 0.5, 0.3 and 0.4%, respectively. The values are estimated by switching on/off these requirements.
• The cut on L 2K used for the kaon selection was varied from −0.6 to −0.1. The corresponding uncertainty was 0.8%. 0.62 ± 0.14 Table 3 Center-of-mass energy E c.m. , integrated luminosity L, number of signal events N sig.events , corrected detection efficiency ε, • The cut on m inv,2K , used for the φ-meson region selection, was varied from 1050 to 1100 MeV. The corresponding uncertainty was 0.7%.
• The lower limit of the E distribution fit was varied from −180 to −100 MeV. The corresponding uncertainty was 1%. • The upper limit of the E distribution fit was varied from 50 to 150 MeV. The corresponding uncertainty was 1%.
• The signal peak position can be fixed from simulation (δx ≡ 0) or released in the fit of the experimental E distribution, the corresponding uncertainty is 2%.
• The signal width can be fixed from the simulation (δσ ≡ 0) or released, the corresponding uncertainty is 2.5%.
• The background shape in the fit of the experimental E distribution can be taken as linear with floating parameters, or it can be fixed from the fit of the simulated background distribution. The corresponding uncertainty is 2.3%.
• The uncertainty of the single kaon detection efficiency is estimated to be 1%, for the pair of kaons -1.5%. The uncertainty of the correction to the K + K − η selection efficiency related to the angular dependence of the kaon detection efficiency (see Section 3.3), was estimated to be 1.5%.
• The systematic uncertainty of the luminosity measurement is 1% [18].
• The uncertainty of the B φ K + K − is about 1%. Table 5 shows a summary of the analyzed systematic uncertainties of the cross section measurement. The overall systematic uncertainty is obtained by a quadratic summation of the individual uncertainties and is estimated to be 5.1%.
The following contributions to the systematic uncertainties of the φ parameters were analyzed: • The systematic uncertainty of cross section measurement in- • The uncertainty of the branching fractions of φ -meson decay channels causes the uncertainty of φ shape. According to [26]  • The contribution of the uncertainty of nonresonant amplitude energy dependence was studied by performing the fit with different non-φ amplitudes: 0, a n.r. , a n.r. /s 3/2 , a n.r. /s, a n.r. / √ s, a n.r. · √ s, a n.r. ·s (a n.r. is constant). The resulting φ uncertainties are 14 eV for The overall systematic uncertainties of the φ parameters, shown in Table 4, are obtained by a quadratic summation of the listed individual uncertainties.

Contribution to (g − 2) μ
Using the result obtained for the e + e − → φη cross section we calculate the corresponding leading-order hadronic contribution to the anomalous magnetic moment of muon a μ . According to Ref. [27] this contribution for the E c.m. range from E min ≡ 2m K + + m η to E max is expressed as Here for a φη μ (E < 2.0 GeV) the total uncertainty is shown. Note, that the work [2] uses the quadratic interpolation between data points, while in the work [4] the trapezoidal rule is used. It is seen that our values for a φη μ for E c.m. < 1.8 GeV are about 1σ lower than previous results.

Conclusion
The process e + e − →K + K − η has been studied in the center-ofmass energy range from 1.59 to 2.01 GeV using the data sample of 59.5 pb −1 collected with the CMD-3 detector. In the production of the K + K − η final state we observed the contribution of the φ(1020)η intermediate state only. On the base of 3009 ± 67 selected signal events the cross section of e + e − →φ(1020)η has been measured with the systematic uncertainty of 5.1%. The obtained cross section has been used to calculate the contribution to the anomalous magnetic moment of the muon: a φη μ (E < 1.8 GeV) = (0.321 ± 0.015 stat ± 0.016 syst ) × 10 −10 , a φη μ (E < 2.0 GeV) = (0.440 ± 0.015 stat ± 0.022 syst ) × 10 −10 . From the e + e − →φ(1020)η cross section approximation the φ(1680) meson parameters have been determined with precision comparable or better than in previous measurements.