Measurement of the e + e − → K 0 S K 0 L π 0 cross sections from √ s = 2.000 to 3.080 GeV

: Based on e + e − collision data collected at center-of-mass energies from 2.000 to 3.080 GeV by the BESIII detector at the BEPCII collider, a partial wave analysis is performed for the process e + e − → K 0 S K 0 L π 0 . The results allow the Born cross sections of the process e + e − → K 0 S K 0 L π 0 , as well as its subprocesses e + e − → K ∗ (892) 0 ¯ K 0 and K ∗ 2 (1430) 0 ¯ K 0 to be measured. The Born cross sections for e + e − → K 0 S K 0 L π 0 are consistent with previous measurements by BaBar and SND, but with substantially improved precision. The Born cross section lineshape of the process e + e − → K ∗ (892) 0 ¯ K 0 is consistent with a vector meson state around 2.2 GeV with a statistical significance of 3.2 σ . A Breit-Wigner fit determines its mass as M Y = (2164 . 1 ± 9 . 6 ± 3 . 1) MeV /c 2 and its width as Γ Y = (32 . 4 ± 21 . 1 ± 1 . 5) MeV , where the first uncertainties are statistical and the second ones are systematic, respectively.

The e + e − → K 0 S K 0 L π 0 [34] and e + e − → K 0 S K ± π ∓ [28] processes have been investigated by the BaBar collaboration using the initial state radiation (ISR) technique.A Dalitz amplitude analysis was performed for e + e − → K 0 S K ± π ∓ , leading to the determination of the isoscalar and isovector cross sections for K * (892) K.A distinct asymmetry between neutral and charged channels is observed in the Dalitz plot for K 0 S π ∓ and K ± π ∓ within √ s ′ = 2 − 3 GeV.It may be related to a similar effect observed in the radiative decay rates of the neutral and charged K * 2 (1430) [28].The SND collaboration has studied e + e − → K 0 S K 0 L π 0 at √ s = 1.3 − 2.0 GeV, and the cross sections have been measured at a statistical uncertainty level of 10%-30% [35].
In this paper, we present a partial wave analysis (PWA) of the process e + e − → K 0 S K 0 L π 0 based on 19 data samples collected by the BESIII experiment, ranging from √ s = 2.000 to 3.080 GeV and corresponding to an integrated luminosity of 647 pb −1 [36,37].The Born cross section of the process e + e − → K 0 S K 0 L π 0 and its sub-processes e + e − → K * (892) 0 K0 and K * 2 (1430) 0 K0 are measured.Throughout the paper charge conjugated processes are also included by default.

BESIII detector and Monte Carlo simulation
The BESIII detector [38] records symmetric e + e − collisions provided by the BEPCII storage ring [39], which operates with a peak luminosity of 1 × 10 33 cm −2 s −1 in the range of √ s from 2.0 to 4.95 GeV.BESIII has collected large data samples in this energy region [40].The cylindrical core of the BESIII detector covers 93% of the full solid angle and consists of a helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which are all enclosed in a superconducting solenoidal magnet providing a 1.0 T (0.9 T in 2012) magnetic field.The solenoid is supported by an octagonal flux-return yoke with resistive plate counter muon identification modules interleaved with steel.The charged-particle momentum resolution at 1 GeV/c is 0.5%, and the dE/dx resolution is 6% for electrons from Bhabha scattering.The EMC measures photon energies with a resolution of 2.5% (5%) at 1 GeV in the barrel (end cap) region.The time resolution in the TOF barrel region is 68 ps, while that in the end cap region is 110 ps.
Simulated samples produced with GEANT4 based [41] Monte Carlo (MC) software, which includes the geometric description [42] of the BESIII detector and the detector response, are used to optimize the event selection criteria, estimate backgrounds, and determine the detection efficiency.The signal MC samples for the processes e + e − → K 0 S K 0 L π 0 , K * (892) 0 K0 and K * 2 (1430) 0 K0 are generated by ConExc [43] using an amplitude model with parameters fixed to the PWA results.For background studies, inclusive hadronic events are generated with a hybrid generator that includes ConExc, LUARLW [44] and PHOKHARA [45].

Event selection and background analysis
The signal process e + e − → K 0 S K 0 L π 0 is reconstructed with K 0 S → π + π − , π 0 → γγ, and K 0 L treated as a missing particle.Signal candidates are required to have two charged pions with zero net charge and at least two photons.
Charged tracks detected in the MDC are required to be within a polar angle (θ) range of |cos θ| < 0.93.Here, θ is defined with respect to the z-axis, which is the symmetry axis of the MDC.Each K 0 S candidate is reconstructed from two oppositely charged tracks satisfying that the distance of closest approach to the interaction point (IP) must be less than 20 cm along the z-axis.The two charged tracks are assigned as π + π − without imposing further particle identification criteria.They are constrained to originate from a common vertex and are required to have an invariant mass within |M (π is the invariant mass of π + π − pair with kinematics updated by the vertex fit and m K 0 S is the K 0 S nominal mass [1].The decay length of the K 0 S candidate is required to be greater than twice the vertex resolution away from the IP. Photon candidates are identified using showers in the EMC.The deposited energy of each shower must be more than 25 MeV in the barrel region (|cos θ| < 0.80) and more than 50 MeV in the end cap region (0.86 < |cos θ| < 0.92).To exclude showers that originate from charged tracks, the angle subtended by the EMC shower and the position of the closest charged track at the EMC must be greater than 10 • as measured from the IP.To suppress electronic noise and showers unrelated to the event, the difference between the event start time and the EMC time of the photon candidate is required to be within [0, 700] ns.
To suppress background and improve the kinematic resolution, a one-constraint (1C) kinematic fit imposing energy-momentum conservation is carried out under the K 0 S K 0 L γγ hypothesis with K 0 L treated as a missing particle.If there are more than two photons in an event, the combination with the minimum χ 2 1C is retained for further analysis, and candidate events are required to satisfy χ 2 1C < 30.To suppress the contamination from the process e + e − → γ ISR K 0 S K 0 L , an additional 1C kinematic fit is performed under the hypothesis of γK 0 S K 0 L , and only events which satisfy χ 2 1C < χ 2 1C (γK 0 S K 0 L ) are retained.To remove K 0 L showers in the EMC that could be mistaken as photons, the angles between the candidate EMC shower and the K 0 L momentum after the kinematic fit are required to be greater than 20 • .Each signal candidate is required to have the invariant mass of the two photons within the π 0 mass region (|M (γγ) − m π 0 | < 0.015 GeV/c 2 ).
Potential background sources are studied by analyzing inclusive e + e − → hadrons and exclusive e + e − → π + π − π 0 π 0 , K 0 S K ± π ∓ π 0 and K 0 S K 0 L π 0 π 0 MC samples after applying the same event selection criteria.The dominant background process is e + e − → π + π − π 0 π 0 .Exclusive e + e − → π + π − π 0 π 0 events are generated by PHOKHARA [45] based on the results of the BaBar collaboration [46].The e + e − → K 0 S K ± π ∓ π 0 and e + e − → K 0 S K 0 L π 0 π 0 events are generated by ConExc based on the dressed cross sections for e + e − → K 0 S K ± π ∓ π 0 and e + e − → K 0 S K 0 L π 0 π 0 from the BaBar experiment [34, 47] with a phase space model and re-weighted to improve the agreement with BESIII data using a multidimensional gradient-boosting algorithm (HEPML) [48], respectively.The exclusive e + e − → π + π − π 0 π 0 , K 0 S K ± π ∓ π 0 and K 0 S K 0 L π 0 π 0 samples, which have been normalized to the experimental integrated luminosity, are used to evaluate the numbers of background events.The contribution of K 0 S peaking background events from e + e − → K 0 S K ± π ∓ π 0 and K 0 S K 0 L π 0 π 0 is at a level of 0.1% − 0.4% for different energy points, which is negligible in the following fit.The background levels are summarized in table 1. Figure 1 shows distributions of the invariant masses of π + π − , M (π + π − ) and γγ, M (γγ) without the K 0 S and π 0 mass window requirements, respectively.Non-K 0 S events are characterized by a flat shape in M (π + π − ) and are estimated with the events in the K 0 S sideband, which is defined by The signal yields of the e + e − → K 0 S K 0 L π 0 process are obtained by performing an unbinned maximum likelihood fit to the M (π + π − ) spectrum.The signal component is described by the signal MC-simulated shape convolved with a Gaussian function which describes the difference between data and MC simulation.The mean value and width of the Gaussian function are separately floated parameters at different energy points.The background function is parameterized by a first-order polynomial function.The corresponding fit results for data taken at √ s = 2.125 and 2.900 GeV are shown in figure 2. The same event selection criteria and fit procedure are applied for all data samples at the nineteen c.m. energies.
In order to improve the resolution of kinematic variables, the remaining e + e − → K 0 S K 0 L π 0 events are subjected to a three-constraint (3C) kinematic fit, which, in addition to imposing energy and momentum conservation, further constrains the π 0 and K 0 S masses to their The region between red arrows is the signal region, and the regions between the green arrows are the sideband regions.
125 and 2.900 GeV, where the black dots with error bars are data, the blue solid curve is the total fit result, the green dashed curve indicates the fitted background shape, and the red dashed curve is the fitted signal shape.(d) , where the (black) dots with error bars are data, and the shaded histograms are non-K 0 S events estimated by the PDG values [1].After all above criteria, the invariant mass spectra of K 0 S K 0 L , K 0 S π 0 , K 0 L π 0 and the invariant masses squared of K 0 S π 0 versus K 0 L π 0 are shown in figure 3, where the K * (892) 0 structure is clear.For the invariant mass spectra of K 0 S K 0 L , K 0 S π 0 and K 0 L π 0 , the contributions of background events which are obtained by the K 0 S sideband are smooth and confirm that there is no peaking structure.Those non-K 0 S events are used to estimate the background contributions and those K 0 S peaking backgrounds are negligible in the following amplitude analysis.

Amplitude analysis
Based on the GPUPWA framework [49], a PWA is performed on the surviving candidate events to identify the intermediate processes present in e + e − → K 0 S K 0 L π 0 .The quasi-two-body decay amplitudes in the process e + e − → K 0 considered and constructed using the covariant tensor amplitude formalism [50], where R 1 and R 2 are the intermediate states that can decay to K 0 S K 0 L and K 0 S (K 0 L )π 0 , respectively.
According to Ref. [50], the general form for the decay amplitude of a 1 − state (Y ) is where Y µ (m) is the polarization vector of Y , m is the spin projection of Y , and U µ i is the i-th partial-wave amplitude with coupling strength determined by a complex parameter Λ i .The amplitude U µ i is constructed with the spin factor, Blatt-Weisskopf barrier factors and propagators of resonances under the assumption of isobar model [50].The differential cross-section can be written as The spin factor is constructed with the covariant Zemach (Rarita-Schwinger) tensor formalism [50][51][52][53] by combining pure-orbital-angular-momentum covariant tensors t(L) and the momenta of parent particles together with Minkowski metric g µν and Levi-Civita symbol ϵ µνλσ .For a process a → bc, the covariant tensors t(L) µ 1 ...µ L for the final states of pure orbital angular momentum L are where r = p b − p c , P (p a ) is the spin projection operator of the particle a, Q abc is the magnitude of p b or p c in the rest system of a.The Blatt-Weisskopf barrier factors B L (Q abc ), are derived by assuming a square well interaction potential as Here Q 0 = 0.197321/R GeV/c is a hadron "scale" parameter, where R is the radius of the centrifugal barrier in fm.In this paper, the radius R is taken to be 0.7 fm.The propagator of intermediate resonance is parameterized by a relativistic Breit-Wigner (BW) function with an invariant mass dependent width [54] where s is the invariant mass squared of the daughter particle, m and Γ 0 are the mass and width of the intermediate resonance, respectively, l is the orbital angular momentum for a daughter particle, and p(s) or p(m 2 ) is the momentum of a daughter particle in the rest frame of the resonance with mass √ s or m.To include the resolution effect for the narrow ϕ resonance, the BW function is convolved with a Gaussian function.
The relative magnitudes and phases of the individual intermediate processes are determined by performing an unbinned maximum likelihood fit using MINUIT [55], where the magnitude and phase of the reference amplitude e + e − → K * (892) 0 K0 are fixed to 1 and 0, respectively, while those of other amplitudes are free parameters of the fit.
The negative log-likelihood function for observing N events in the data sample is expressed as where ω i is the decay-amplitude squared evaluated from the four-momenta of final particles for the i-th event, ϵ i is the detection efficiency and Φ is the standard element of phase space.
The contribution of background events to the N LL is canceled out by evaluating the signal model on K 0 S sideband events injected into the data sample with negative weights.Conservation of J P C for intermediate states, in the process e + e − → R 1 π 0 → K 0 S K 0 L π 0 , allows both P and F wave contributions both in e + e − → R 1 π 0 and R , the contributions of P, D and F waves are all allowed both in the primary and secondary processes.The PWA fit procedure starts by including the K * (892) 0 K0 and K * 2 (1430) 0 K0 as the initial baseline solutions, and then adds one at a time other possible intermediate states which can decay to K 0 S (K 0 L )π 0 or K 0 S K 0 L .The masses and widths of possible intermediate resonances are fixed to their PDG values [1].Intermediate states are included in the solution if the statistical significance is greater than 5σ, where the statistical significance is evaluated from the changes in likelihood and degrees of freedom with and without the corresponding amplitude included in the PWA fit.The direct decay process without an intermediate resonance is treated as a phase space distribution without a propagator [12].The procedure is repeated until a best solution is obtained.
The above strategy is implemented individually on the experimental data sets collected at √ s = 2.125, 2.396 and 2.900 GeV, which have the largest luminosities and yields among the nineteen data sets.As the c.m. energy increases, we test the significance of the process with higher threshold at √ s = 2.900 GeV, such as e + e − → K * 3 (1780) 0 K0 , which cannot be produced at √ s = 2.125 GeV.But the significances of these processes are less than 5σ and they are not retained in the final best solution.The invariant mass spectra, angular distributions and fit results for √ s = 2.125 GeV are shown in figure 4.

Born cross sections measurement
The Born cross section for e + e − → K 0 S K 0 L π 0 is obtained at each c.m. energy using where N sig is the number of signal events, L is the integrated luminosity, ϵ is the efficiency obtained by weighting MC simulation according to the PWA results, B is the product of BFs in the full decay chain [56], and 1 + δ is the ISR correction factor, which is obtained by a QED calculation [57].Both ϵ and 1 + δ depend on the line shape of cross sections and are determined by an iterative procedure [11,58].The Born cross section for an intermediate process, e + e − → K * (892) 0 K0 or K * 2 (1430) 0 K0 , at each energy is obtained with the same approach, where N sig is replaced with the product of the total number of surviving events and the corresponding fraction relative to the total obtained according to the PWA results, and B is replaced with the product of the BFs of the decays K 0 S → π + π − , π 0 → γγ and that of the intermediate state (K * (892) 0 → K 0 π 0 = 33.23%,K * 2 (1430) 0 → K 0 π 0 = 16.60%) from the PDG [1], respectively.The Born cross sections are listed in tables 4, 5 and 6, separately for the processes e + e − → K 0 S K 0 L π 0 , K * (892) 0 K0 and K * 2 (1430) 0 K0 , respectively.The previous BESIII measurement [12] with the charged channel e + e − → K + K − π 0 shows that K * 2 (1430) + K − is the dominant component, with the fraction of K * (892) + K − at the 2-10% level.However, in this study with the neutral channel e + e − → K 0 S K 0 L π 0 , K * (892) 0 K0 is dominant, while K * 2 (1430) 0 K0 is at the 5% level in the BESIII c.m. energy region.The asymmetry is also observed by BaBar [28] in the production of K * (892) 0 K0 , K * (892) + K − , K * 2 (1430) 0 K0 and K * 2 (1430) + K − .To quantify the effect, we define relative  ratios of the Born cross sections: The corresponding results for the relative ratio are summarized in figure 5.

Systematic uncertainties
Two categories of systematic uncertainties are considered in the measurement of the Born cross sections.The first category includes systematic sources not associated with the PWA fit that are evaluated as follows: 1.The uncertainty associated with the integrated luminosity is 1% and estimated by using large angle Bhabha events [36].
2. The uncertainty concerning K 0 S reconstruction is studied with control samples of J/ψ → K 0 S K ± π ∓ and J/ψ → ϕK 0 S K ± π ∓ .The result shows that the difference in efficiency between data and MC simulation is 1% per K 0 S [59].
3. The uncertainty of the requirement on the number of charged tracks (N charge ) is estimated with a control sample of J/ψ → K 0 S K 0 L π 0 .The difference in efficiency between data and MC simulation with and without this requirement is taken as the uncertainty.

The uncertainty concerning photon detection efficiency is studied with a control sam-
ple of e + e − → K + K − π + π − π 0 [60].The result shows that the difference in detection efficiency between data and MC simulation is 1% per photon.
5. The uncertainty related to the kinematic fit is studied with a control sample of J/ψ → K 0 S K 0 L π 0 .The difference in efficiency between data and MC simulation with and without the kinematic fit is taken as the uncertainty.6.The uncertainty of the VP and ISR correction factors (Rad) is obtained with the accuracy of the radiation function, which is about 0.5% [56], and has an additional contribution from the cross section line shape, which is estimated by varying the model parameters of the fit to the cross section.All parameters are randomly varied within their uncertainties, and the resulting parametrization of the line shape is used to recalculate (1+δ)ϵ and the corresponding cross section.This procedure is repeated one thousand times, and the standard deviation of the resulting cross sections is taken as the systematic uncertainty.The systematic uncertainty associated with the VP and ISR correction factor is evaluated as the quadratic sum of contributions from the QED theory and line shape parametrization [10].
8. The uncertainty caused by the M (π + π − ) fit (Fit) includes the descriptions of signal shape and background shape.The nominal MC-simulated shape convolved with a Gaussian function is replaced by a MC-simulated shape convolved with a Crystal Ball function, and the nominal background shape is replaced by a second-order polynomial function, and the differences with the nominal results are taken as the uncertainties.The uncertainties from above sources are added in quadrature and taken as the total uncertainty from the M (π + π − ) fit.The second category of uncertainties includes those associated with the PWA fit that are evaluated as follows: 1.The uncertainty from the fit parameters (FPar) is estimated by the standard deviation of re-calculated efficiencies derived from one thousand groups of randomly generated fit parameters using a correlated multi-variable Gaussian function.
2. The uncertainty related to the resonance parameters (Par) is estimated by performing alternative fits shifting the world-average parameter value by its error from the PDG [1].
3. The uncertainty concerning the extra additional resonances (Extra) is estimated by performing alternative fits with all components whose significances are greater than 3σ.In the alternative fit K * (1410) 0 → K 0 π 0 is added for the data sample at √ s = 2.125 GeV, K * (1410) 0 → K 0 π 0 and ρ(1450) → K 0 K0 are added at √ s = 2.396 GeV, ρ(1450)/ρ(1700) → K 0 K0 and K * (1410 4. The uncertainty of the background estimation in the PWA fit (Bkg) is estimated by using only the lower or higher sideband.
5. The uncertainty from the Blatt-Weisskopf barrier factor (BWf) is estimated by varying the radius of the centrifugal barrier from 0.7 to 1.0 fm.
Assuming all the sources of systematic uncertainties as independent, the total systematic uncertainty is obtained by adding them in quadrature.The 100% correlated uncertainties for the Born cross sections of e + e − → K 0 S K 0 L π 0 , K * (892) 0 K0 and K * 2 (1430) 0 K0 are listed in table 7. The other uncorrelated and total systematic uncertainties are listed in tables 8-10.

Fit to the lineshape
The Born cross sections for the process e + e − → K 0 S K 0 L π 0 are shown in figure 6  Table 8.Systematic uncertainties (%) for the Born cross section of e + e − → K 0 S K 0 L π 0 at each c.m. energy associated with the ISR and VP correction factors (Rad), the M(π + π − ) fit (Fit), the fit parameters in PWA (FPar), the resonance parameters (Par), the extra additional resonances (Extra), the background estimation (Bkg) and the Blatt-Weisskopf barrier factor (BWf).
A χ 2 fit, incorporating the correlated and uncorrelated uncertainties among different energy points, is performed to determine the resonance parameters for the Born cross sections of e + e − → K * (892) 0 K0 .The fit probability density function is a coherent sum of a continuum component and a resonant component.where M Y and Γ Y are the mass and width of the resonance; ϕ Y is the relative phase between the continuum component and the resonance; Γ e + e − Y is its partial width to e + e − ; B is the BF of Y → K * (892) 0 K0 ; and c 1 and c 2 are additional parameters of the fit.Γ( √ s) is defined as m)| 2 dΦ 3 is the phase-space factor for the relative orbital angular momentum L = 1 of the process e + e − → K * (892) 0 K0 → K 0 S K 0 L π 0 and Φ 3 is three-body phase space [1].The amplitude A is the partial wave amplitude in the covariant Rarita-Schwinger tensor formalism [50] and is described as: where T is the covariant tensor, f is the Breit-Wigner propagator, ϵ µνλσ is the Levi-Civita symbol, and the other operators can be found in Ref. [50].In total, there are six free parameters in the fit:  11.Rectangles with error bars are BESIII data, where errors include both statistical and systematic uncertainties.The solid black curves represent the total fit result, the dashed blue curves for the resonance and the dashed green curves for the continuum component, and the dash-dotted pink curves for the interference between the resonance and continuum components.
significance of the resonance is determined to be 3.2σ by comparing the change of χ 2 (∆χ 2 = 18.27) and the change of ndf (∆ndf = 4) between the nominal fit and the fit without the resonance.As both solutions are mathematically equivalent, we do not prefer one over the other, but list them in order of increasing interference fraction.Figure 7(b) shows a large interference between resonance and continuum components.The uncertainties (statistical and systematic) of the measured Born cross sections have been included when fitting the line shape, determining the resonance parameters and estimating the significance of the resonance.
Besides the uncertainties of individual cross-section measurements, the fit to the lineshape is also affected by the uncertainty of the BEPCII c.m. energy and the description of the continuum.The uncertainty of the c.m. energy calibration is estimated as 0.1% and is ignored in the determination of resonance parameters [36].To evaluate the systematic uncertainty associated with the lineshape model, the continuum term c 1 √ P ( √ s) s c 2 is replaced with an exponential function of the form c 0 • e −p 0 ( √ s−M th ) , where c 0 and p 0 are free parameters and M th = m K * (892) 0 + m K0 is the mass threshold for K * (892) 0 K0 production [1,12].The difference of the parameters from the nominal results are taken as the systematic uncertainties.

Summary
In summary, a partial wave analysis of the process e + e − → K 0 S K 0 L π 0 is performed for nineteen data samples collected in the BESIII experiment with center-of-mass energies ranging from 2.000 to 3.080 GeV corresponding to a total integrated luminosity of 647 pb −1 .The Born cross sections of the process e + e − → K 0 S K 0 L π 0 , as well as those for the intermediate processes e + e − → K * (892) 0 K0 and K * Table 11.Result of the fit to the e + e − → K * (892) 0 K0 Born cross sections, where the first uncertainties originate from the cross section measurement and the second from the line shape fit methodology, respectively.
wave analysis on each data sample individually, where the charge conjugated processes are also included.The measured Born cross sections of the process e + e − → K 0 S K 0 L π 0 are consistent with earlier results by BaBar [34], while the precision is significantly improved.The Born cross section lineshape of the process e + e − → K * (892) 0 K0 hints at a resonant structure around 2.2 GeV with a significance of 3.2σ.A Breit-Wigner fit yields its mass M Y = (2164.7 ± 9.1 ± 3.1) MeV/c 2 and width Γ Y = (32.4± 21.0 ± 1.8) MeV.The resonance parameters, especially the very narrow width, are very close to the BESIII results measured through the ϕη channel [7] of the ϕ(2170) meson [1].
The ratio of Born cross section measurements of the process e + e − → K * (892) + K − to the process e + e − → K * (892) 0 K0 is less than 0.2 and that for K * 2 (1430) is in the region of 0-40, where the statistical and systematic uncertainties are included.If we apply the isospin decomposition for the decay from isospin vector (ρ * ) or isospin scalar (ω * , ϕ * ) state to the final state K * K, the ratio of the yields in the neutral and charged K * K should be 1.On the other hand, the electromagnetic interaction also contributes to the production of e + e − → K * K, and it does not require isospin conservation.Future experimental and theoretical studies are needed to understand the observed phenomenon.

Figure 1 .
Figure 1.Distributions of (a),(c) M (π + π − ) with π 0 mass window requirement and (b),(d) M (γγ) at √ s = 2.125 and 2.900GeV, where the (black) dots with error bars are data, and the shaded histogram are the stacked MC samples of the signal process, π + π − π 0 π 0 , K 0 S K 0 L π 0 π 0 and K 0 S K ± π ∓ π 0 .The region between red arrows is the signal region, and the regions between the green arrows are the sideband regions.

Figure
Figure 2. Fit to the M (π + π − ) distribution at √ s = 2.125 and 2.900GeV, where the black dots with error bars are data, the blue solid curve is the total fit result, the green dashed curve indicates the fitted background shape, and the red dashed curve is the fitted signal shape.

Figure 4 .
Figure 4. Superposition of data and the PWA fit projections for invariant mass distributions of (a) K 0 S K 0 L , (b) K 0 S π 0 and (c) K 0 L π 0 , and the cos θ distributions of (d) K 0 S in e + e − c.m. frame, (e) K 0 S in K 0 S π 0 rest frame and (f) K 0 S in K 0 S K 0 L rest frame at √ s = 2.125 GeV.The pull projection of the residuals is shown beneath each distribution correspondingly.Different styles of the curves denote different components.
(a).The results are consistent with the previous results from BaBar.The Born cross sections for the intermediate process e + e − → K * 2 (1430) 0 K0 and e + e − → K * (892) 0 K0 are shown in figures 6(b) and 7, respectively.

Figure 6 .
Figure 6.The Born cross sections for (a) the process e + e − → 0 S K 0 L π 0 and (b) the process e + e − → K * 2 (1430) 0 K0 .The red dots are the measured results from BESIII, where errors include both statistical and systematic uncertainties.The green triangles and brown squares are the results from BaBar and SND, respectively.

Figure 7 .
Figure 7.The Born cross section and fit curves for e + e − → K * (892) 0 K0 , (a) and (b), corresponding to the two solutions in table11.Rectangles with error bars are BESIII data, where errors include both statistical and systematic uncertainties.The solid black curves represent the total fit result, the dashed blue curves for the resonance and the dashed green curves for the continuum component, and the dash-dotted pink curves for the interference between the resonance and continuum components.

Table 1 .
Summary of the background level for each √ s.

Table 2 .
The statistical significances of the intermediate states and fit fractions for √ s = 2.125, 2.396 and 2.900 GeV are listed in table 2 and table 3, respectively.For the other sixteen data samples with lower luminosities and limited statistics, the intermediate components are assumed to be the same as those of the nearby c.m. energies with higher statistics.The intermediate component candidates of Statistical significances of the intermediate states for data at √ s = 2.125, 2.396 and 2.900 GeV.

Table 3 .
Fit fractions of the intermediate states for data at √ s = 2.125, 2.396 and 2.900 GeV.

Table 4 .
The measured Born cross sections for e + e − → K 0 S K 0 L π 0 , where the first uncertainties are statistical, the second ones are systematics from table 7 and the third ones are model uncertainties.

Table 5 .
The measured Born cross sections for e + e − → K * (892) 0 K0 , where the first uncertainties are statistical, the second ones are systematics from table 7 and the third ones are model uncertainties.

Table 6 .
The measured Born cross sections for e + e − → K * 2 (1430) 0 K0 , where the first uncertainties are statistical, the second ones are systematics from table 7 and the third ones are model uncertainties.

Table 7 .
The 100% correlated systematic uncertainties for the Born cross section of e + e − → The cross section is modeled as:

Table 10 .
Systematic uncertainties (%) for the Born cross section of e + e − → K * 2