Study of the process $e^+ e^- \to K^0_{S}K^0_{L}$ in the center-of-mass energy range 1004--1060 MeV with the CMD-3 detector at the VEPP-2000 $e^+ e^-$ collider

The $e^+ e^- \to K^0_{S}K^0_{L}$ cross section has been measured in the center-of-mass energy range 1004--1060 MeV at 25 energy points using $6.1 \times 10^5$ events with $K^0_{S}\to \pi^+\pi^-$ decay. The analysis is based on 5.9 pb$^{-1}$ of an integrated luminosity collected with the CMD-3 detector at the VEPP-2000 $e^+ e^-$ collider. To obtain $\phi(1020)$ meson parameters the measured cross section is approximated according to the Vector Meson Dominance model as a sum of the $\rho, \omega, \phi$-like amplitudes and their excitations. This is the most precise measurement of the $e^+ e^- \to K^0_{S}K^0_{L}$ cross section with a 1.8\% systematic uncertainty.

The e + e − → K 0 S K 0 L cross section has been measured in the center-of-mass energy range 1004-1060 MeV at 25 energy points using 6.1 × 10 5 events with K 0 S → π + π − decay. The analysis is based on 5.9 pb −1 of an integrated luminosity collected with the CMD-3 detector at the VEPP-2000 e + e − collider. To obtain φ(1020) meson parameters the measured cross section is approximated according to the Vector Meson Dominance model as a sum of the ρ, ω, φ-like amplitudes and their excitations. This is the most precise measurement of the e + e − → K 0 S K 0 L cross section with a 1.8% systematic uncertainty.

Introduction
Investigation of e + e − annihilation into hadrons at low energy provides unique information about interactions of light quarks. High-precision studies of various hadronic cross sections are of great interest in connection with the problem of the muon anomalous magnetic moment [1] and constitute the main goal of experiments with the CMD-3 and SND detectors at the upgraded VEPP-2000 collider [2,3]. * Corresponding author.
In particular, e + e − → K 0 S K 0 L is one of the processes with a rather large cross section in the center-of-mass energy range from 1 to 2 GeV. A precise measurement of this cross section, dominated by the contribution of the φ(1020) and φ(1680) resonances, is required to improve our knowledge of the hadronic contributions to (g − 2) μ and α(M 2 Z ). Additional motivation for high-precision measurements of the e + e − → K 0 S K 0 L and e + e − → K + K − cross sections around the φ meson peak comes from a significant deviation of the ratio of the coupling constants g φ→K + K − g φ→K S K L from theoretical predictions [4].
The most precise previous studies of the process have been performed at the CMD-2 [5], SND [6] and BaBar [7]  In this paper we present results of the new measurement of the e + e − → K 0 S K 0 L cross section based on a high-statistics data sample collected at 25 energy points in the center-of-mass energy (c.m.) E c.m. range 1004-1060 MeV with the CMD-3 detector.

CMD-3 detector and data set
The Cryogenic Magnetic Detector (CMD-3) described elsewhere [8] is installed in one of the two interaction regions of the VEPP-2000 e + e − collider [9]. The detector tracking system consists of the cylindrical drift chamber (DC) and double-layer cylindrical multiwire proportional Z-chamber, both installed inside a thin (0.085 X 0 ) superconducting solenoid with 1.3 T magnetic field. DC contains 1218 hexagonal cells and provides a measurement of charged particle momentum and of the polar (θ ) and azimuthal (φ) angles. An amplitude information from the DC wires is used to measure the ionization losses dE/dx of charged particles with σ dE/dx ≈ 11-14% accuracy for minimum ionization particles (m.i.p.). A barrel electromagnetic calorimeter placed outside the solenoid consists of two subsystems: an inner liquid xenon (LXe) calorimeter (5.4 X 0 thick) surrounded by a scintillation CsI crystal calorimeter (8.1 X 0 thick) [10]. BGO crystals with 13.4 X 0 are used as an endcap calorimeter. The detector has two triggers: neutral and charged. A signal for neutral one is generated by the information from calorimeters, while the charged trigger comes from the tracking system. The return yoke of the detector is surrounded by scintillation counters which veto cosmic events.
To obtain a detection efficiency, Monte Carlo (MC) simulation of the detector based on the GEANT4 [11] package has been developed. Simulated events are subject to the same reconstruction and selection procedures as the data. MC simulation includes photon jet radiation by initial electrons calculated according to Refs. [12,13]. Background was estimated using a multihadronic Monte Carlo generator [14] based on experimental data for all measured processes in the energy range up to 2 GeV.
The analysis uses 5.9 pb −1 of an integrated luminosity collected in two scans of the φ(1020) resonance region at 25 energy points in the E c.m. = 1004-1060 MeV range. The beam energy E beam has been monitored by using the Back-Scattering-Laser-Light system [15,16] which determines E c.m. at each energy point with about 0.06 MeV accuracy.

Event selection
Signal identification is based on detection of two pions from the K 0 S → π + π − decay. For each pair of oppositely charged tracks a constrained fit to a common vertex is performed to determine track parameters. Assuming tracks to be pions, the pair with the best χ 2 from the vertex fit and with the invariant mass in the range 420-580 MeV/c 2 is selected as a K 0 S candidate. The following requirements are applied to events with a found K 0 S candidate: • The longitudinal distance and the transverse coordinate of the vertex should have |Z K 0 S | < 10 cm and |ρ K 0 S | < 6 cm, respectively; • Pions from K 0 S decay are required to have polar angles 1 < θ π + ,π − < π − 1 radians; • Each track has momentum 130 MeV/c < P π ± < 320 MeV/c corresponding to the kinematically allowed region for pions from the K 0 S decay and its ionization losses in DC are within three standard deviations from the average value, expected for pions. The last requirement rejects charged kaons and back-  ground protons, as shown in Fig. 1 for positive (a) and negative (b) tracks, respectively, at E beam = 505 MeV; • The momentum of the K 0 S candidate, P K 0 S = | P π + + P π − |, is required to be not larger than five standard deviations from the nominal momentum shown by the arrows in Fig. 2(a); • The cosine of the angle ψ between the tracks should be smaller than the cosine of the minimal angle between two pions originating from the two-body decay of the K 0 S meson, shifted by five standard deviations, as shown by the arrow in Fig. 2 The reconstructed polar angle of the K 0 S meson and the transverse distance of the K 0 S decay vertex from the e + e − interaction point are shown in Fig. 3 after above selections for data (points) and MC-simulation (shaded histogram). The dark shaded histograms show a sum of the background contributions from the MC-simulated hadronic processes (predominantly e + e − → π + π − 2π 0 ) and a contribution from cosmic muons estimated using events from the |Z K 0 We determine the number of signal events for data and simulation from a binned maximum likelihood fit of two-pion invariant mass shown in Fig. 4. The signal shape is described by a sum of four Gaussian functions with parameters fixed from the simulation and with additional Gaussian smearing to account for the difference in data-MC detector responses. The background in data, described by a second-order polynomial function, constitutes about 30% outside the φ meson peak and 0.5% under it. By toy MC experiments with fixed signal and background profiles as well as by varying the background shape and approximation range used we estimate an uncertainty on the number of extracted signal events as less than 1.1%. The number of obtained signal events, N exp , for each energy is listed in Table 3.

Cross section of e
The Born cross section of the process e + e − → K 0 S K 0 L is calculated at each energy from the expression: where reg is a detection efficiency, trig is a trigger efficiency, L is an integrated luminosity, 1 + δ rad. is a radiative correction, and 1 + δ en.spr. represents a correction due to the spread of the collision energy.
The detection efficiency reg is obtained by dividing the number of MC simulated events after reconstruction and selection described above by the total number of generated K 0 S K 0 L pairs taking into account the branching fraction  0.05)% [17]. Fig. 5 shows the obtained detection efficiency (triangles) vs c.m. energy in comparison with the expected geometrical efficiency (squares). The geometrical efficiency is calculated as the probability of pions to be in the polar angle range 1 < θ π + ,π − < π − 1 radians at the generator level.
The trigger efficiency is studied using responses of two independent triggers, charged and neutral, for selected signal events, and is found to be close to unity, trig = 0.998 ± 0.001.
The integrated luminosity L is determined using events of the processes e + e − → e + e − (Bhabha events) with about 1% [18] systematic accuracy.
The initial-state radiative correction 1 + δ rad. , shown by squares in Fig. 6, is calculated using the structure function method with an accuracy better than 0.1% [19].
The spread of collision energy is about 350 keV, that is significant in comparison with the φ meson width, and we introduce the correction of the cross section, shown by points in Fig. 6, which has a maximum value of 1.028 ± 0.004 at the peak of the φ resonance.
The resulting cross section is listed in Table 3 for each energy and shown in Fig. 8

Systematic uncertainties
MC simulation may not exactly reproduce all detector responses, so an additional study was performed to obtain corrections for data-MC difference in the detection efficiency.
The data-MC difference in the charged pion detection by DC is studied using the process e + e − → φ → π + π − π 0 . Three-pion events can be fully reconstructed from one detected charged track and two detected photons from the π 0 decay, and a probability to detect another charged track can be determined. For the polar angle requirement 1 < θ π + ,π − < π − 1 radians, the average detection inefficiency is about 1% per track for high momentum, and decreases with pion momentum, as shown in Fig. 7. The rise of efficiency vs momentum is explained by the decreasing number of pions that decayed or interacted in DC. Good data-MC agreement is observed for charged pion detection, so no efficiency correction is introduced and the uncertainty in the detection is estimated as 0.5%. DC calibration is checked using signals of the Bhabha events [18] in the DC and Z-chamber, and for pions from the K 0 S decay the uncertainty due to the polar angle selection in the range of polar angles chosen is estimated as 0.4%.
By variation of corresponding selection criteria we estimate the uncertainty due to the data-MC difference in the angular and momentum resolutions as 0.5%, while other selection criteria contribute another 0.6%.
The total uncertainty of the detection efficiency is calculated as a quadratic sum of uncertainties from the different sources and is estimated to be 1.0%.
The systematic uncertainties of the e + e − → K 0 S K 0 L cross section discussed above are summarized in Table 1 giving 1.8% in total.

S K 0 L cross section
To obtain φ(1020) parameters we approximate the energy dependence of the cross section according to the vector meson dom- Table 1 Summary of systematic uncertainties in the e + e − → K 0 S K 0 L cross section measurement.
The coupling constants of the intermediate vector meson V with initial and final states can be presented as: where Vee and B VKK are electronic width and branching fraction of the V meson decay to a pair of kaons.
In our approximation we use the world-average values of mass, total width and electronic width of the ρ(770) and ω(782): ρ→ee = 7.04 ± 0.06 keV, ω→ee = 0.60 ± 0.02 keV [17]. The branching fractions of the ρ(770) and ω(782) to a kaon pair are unknown, and we use the relation g ωK 0  Table 2 and the fit result is shown in Fig. 8(a). Fig. 8(b) shows the relative difference between the obtained data  Table 3 The c.m. energy E c.m. , number of selected signal events N, detection efficiency MC , radiative-correction factor 1 + δ rad. , integrated luminosity L, and Born cross section σ of the process e + e − → K 0    Fig. 9 by the dotted lines, while the long-dashed line shows a contribution from higher excitations. The first uncertainties presented in Table 2 are statistical, and the second are the  Table 2. The obtained values agree with results of other measurements and some are more precise. ing that the destructive interference with these states dominates in the shown energy region.