Cross section measurement of $e^{+}e^{-} \to K_{S}^{0}K_{L}^{0}$ at $\sqrt{s}=2.00-3.08~{\rm GeV}$

The cross sections of the process $e^{+}e^{-} \to K_{S}^{0}K_{L}^{0}$ are measured at fifteen center-of-mass energies $\sqrt{s}$ from $2.00$ to $3.08~{\rm GeV}$ with the BESIII detector at the Beijing Electron Positron Collider (BEPCII). The results are found to be consistent with those obtained by BaBar. A resonant structure around $2.2~{\rm GeV}$ is observed, with a mass and width of $2273.7 \pm 5.7 \pm 19.3~{\rm MeV}/c^2$ and $86 \pm 44 \pm 51~{\rm MeV}$, respectively, where the first uncertainties are statistical and the second ones are systematic. The product of its radiative width ($\Gamma_{e^+e^-}$) with its branching fraction to $K_{S}^{0}K_{L}^{0}$ ($Br_{K_{S}^{0}K_{L}^{0}}$) is $0.9 \pm 0.6 \pm 0.7~{\rm eV}$.

The cross sections of the process e + e − → K 0 S K 0 L are measured at fifteen center-of-mass energies √ s from 2.00 to 3.08 GeV with the BESIII detector at the Beijing Electron Positron Collider (BEPCII). The results are found to be consistent with those obtained by BaBar. A resonant structure around 2.2 GeV is observed, with a mass and width of 2273.7 ± 5.7 ± 19.3 MeV/c 2 and 86 ± 44 ± 51 MeV, respectively, where the first uncertainties are statistical and the second ones are systematic. The product of its radiative width (Γ e + e − ) with its branching fraction to K 0 ) is 0.9 ± 0.6 ± 0.7 eV.
Recently, the cross sections for e + e − → K + K − were measured by the BESIII and BaBar collaborations [13,35]. A structure near 2.2 GeV was reported with a mass (width) differing from the world averaged parameters of the φ(2170) by 3σ (2σ). The structure is not supported by Babar based on the measurements of the process e + e − → K 0 S K 0 L [35], though the uncertainties are very large which are more than 100% in most of the energy intervals. On the other hand, a theoretically guided fit to the BESIII cross sections for e + e − → K + K − provided consistent results with respect to the φ(2170) parameters [36]. The structure observed in the cross section measurements can also be explained as an ω-like state [37]. In general, considering the interferences between resonance and non-resonance contributions, additional information from other processes, such as e + e − → K 0 S K 0 L , is needed. Although, this process has been investigated in the past by the DM1 [38], OLYa [39], CDM2 [40][41][42], SND [43,44] and BaBar [35,45] collaborations, these measurements mainly focused on the energy region below 2.0 GeV.
In this work, we present Born cross section measure-ments of the process e + e − → K 0 S K 0 L . The results obtained in the overlapping center-of-mass region from 2.00 − 2.54 GeV are compared to previous measurements by BaBar [35]. Moreover, we present, for the first time, Born cross section measurements taken in the interval from 2.54 to 3.08 GeV. A fit is applied to the cross section measurements of the e + e − → K 0 S K 0 L process, and the resonant structure result is compared with that found by BESIII [13] and BaBar [35] in e + e − → K + K − .

II. DETECTOR AND MONTE CARLO SIMULATION
The BESIII detector is a magnetic spectrometer [46] located at BEPCII [47]. The cylindrical core of the BESIII detector 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 magnetic field. The solenoid is supported by an octagonal flux-return yoke with resistive plate counter muon identifier modules interleaved with steel. The acceptance of charged particles and photons is 93% over 4π solid angle. The chargedparticle momentum resolution at 1 GeV/c is 0.5%, and the dE/dx resolution is 6% for the 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 of the TOF barrel part is 68 ps, while that of the end cap part is 110 ps.
The data samples used in this work are collected by the BESIII detector at fifteen center-of-mass (c.m.) energies between 2.00 and 3.08 GeV with an integrated luminosity of 583 pb −1 [48,49].
Monte Carlo (MC) samples simulated with a model of the complete detector are used to determine detection efficiency, optimize event selection criteria, and estimate backgrounds. Detector geometry, material description, propagation and interactions with the detector of the final-state particles are handled by GEANT4-based [50] simulation software, BESIII Object Oriented Simulation Tool [51].

III. EVENT SELECTION AND BACKGROUND ANALYSIS
The momentum of the K 0 S meson is reconstructed from its K 0 S → π + π − decay. Events containing the recon-structed K 0 S candidates are retained for further analysis. The K 0 L meson is not detected directly; because of the two-body decay, its presence is inferred by a requirement on the K 0 S candidate momentum. To select signal candidates, the following criteria are applied: • Exactly two oppositely-charged tracks are required without any requirement on neutral tracks. The distance of closest approach of the track with respect to the interaction point is required to be less than 20 cm along the beam direction (z−axis of the BESIII coordinate system), while no requirement is made with respect to the transverse direction. Tracks are required to be within the acceptance of the MDC, i.e. | cos θ| < 0.93, where θ is the polar angle between the track and the z−axis.
A vertex fit is applied to constrain the two tracks to a common vertex, and subsequently a secondary vertex fit is performed to determine the flight distance L and corresponding uncertainty δL, where L corresponds to the separation between the secondary vertex and the interaction point. We require L/δL to be larger than 2, as illustrated by the green vertical line in Figure 1. The invariant mass of the two tracks (m π + π − ), where the tracks are treated as π + and π − candidates, is required to is the mass of K 0 S taken from the Particle Data Group (PDG) [56]. The signal yields are determined from fits to the invariant-mass distributions, as discussed in Section IV.
• To reject backgrounds from the e + e − → e + e − and e + e − → γγ processes, we require the ratio E/cp between the deposited energy in the EMC (E) and the momentum measured by the MDC (p) to be less than 0.8.
• |p π + π − − p K 0 S | < σ p must be satisfied to suppress backgrounds from three (or more) body decays, where p π + π − is the momentum reconstructed from the π + π − system, S momentum, and σ p = 15 MeV/c is the momentum resolution of the reconstructed K 0 S determined using the signal MC. The p π + π − distribution is shown in Figure 2.
MC studies indicate that the non-hadronic background and two-photon process contribute less than 5% in the region |m π + π − − m K 0 S | < 3σ K 0 S at low c.m. energies (< 2.396 GeV), where σ K 0 S = 4 MeV/c 2 is the mass resolution of the pion pair determined by fitting the signal MC shape, and it dominates at 3.080 GeV with a contribution of less than 20%. No peaking backgrounds were found from non-hadronic processes after applying the previously described criteria at all c.m. energies. Detailed event type analysis with a generic tool, TopoAna [57], shows that the dominant hadronic background channels Dots refer to data and the shaded area corresponds to simulated signal events normalized to the integrated luminosity of the data. The (green) vertical line indicates the requirement that is applied to select signal candidate events. are e + e − → K 0 S K 0 L π 0 , e + e − → π + π − π + π − , e + e − → π + π − π 0 and e + e − → (γ)π + π − . A study using exclusive hadronic MC samples showed that only at 3.080 GeV a peaking background can be expected from the process e + e − → K 0 S K 0 L π 0 . This will be further discussed in Section V.

IV. CROSS SECTION
Born cross sections (σ B ) are obtained at each energy point by: where N sig is the signal yield, ǫ is the detection efficiency, 1 + δ is the correction factor including vacuum polarization (VP) and initial-state radiation (ISR) effects, and L is the integrated luminosity measured using large-angle Bhabha scattering events with the method elucidated in Ref. [48]. The branching ratio of K 0 S → π + π − has been incorporated into ǫ.
The signal yields are determined with an unbinned maximum-likelilood fit to the invariant-mass distribution of π + π − pairs of the selected events obtained for each c.m. energy point, where the signal shape is described by a Gaussian function and the background is represented with a zero-order Chebychev polynomial. The fit range is taken with a window of more than 8σ K 0 S around the signal K 0 S . The mass and the width of the Gaussian function is fixed to m K 0 S and σ K 0 S for all the c.m. energy points, except for the two energies with the highest statistics (2.000 and 2.125 GeV). The signal and background yields are set free for all c.m. energies. As an example, Figure 3 illustrates the m π + π − distribution together with the corresponding fit result for data taken at √ s = 2.125 GeV.
Both ǫ and 1 + δ depend on the line shape of the cross sections and are determined via an iterative procedure. In the first iteration, the cross sections from 2.00 GeV to 3.08 GeV are obtained and taken as initial inputs. The cross sections below 2.00 GeV are provided by previous experiments [38][39][40][41][42][43][44][45] and fitted together with our measurements above 2.00 GeV. The parameters ǫ and 1 + δ are calculated according to the fit curve at each c.m. energy and are taken as input for the next iteration. The procedure is repeated until the measured Born cross sections converge.
Results are summarized in Table I with both statistical and systematic uncertainties given in the last column. The systematic uncertainties are discussed in Section V. I: Born cross sections of the e + e − → K 0 S K 0 L process. The columns Nsig and N bkg show the numbers of signal and background events determined by fitting the m π + π − distribution. The detection efficiency ǫ, ISR and VP correction factor 1 + δ, and the integrated luminosity L are summarized in the 4 th , 5 th , and 6 th column, respectively. The values presented in the column labeled with σB correspond to the measured Born cross section, where the first uncertainty is statistical and the second one is systematic.  The solid curve denotes the best fit through the data of the complete model, whereby the dash-dotted and dashed lines are the corresponding signal and background components, respectively.

V. SYSTEMATIC UNCERTAINTIES AND LINE SHAPE A. Systematic uncertainties of the Born cross sections
Several sources of the systematic uncertainties are estimated at each c.m. energy point, including uncertainties in the determination of the K 0 S selection efficiency, in applying the E/cp requirement, in the ISR and VP correction factors, in the integrated luminosity, and in the fit procedure that was used to determine the signal yield. The uncertainty in the K 0 S → π + π − branching ratio is only 0.07% [56], which is considered to be negligible in this study.
The systematic uncertainty of the K 0 S selection efficiency is obtained using the control samples J/ψ → K * (892) ∓ K ± , K * (892) ∓ → K 0 S π ∓ and J/ψ → φK 0 S K ∓ π ± , and the uncertainties are between 2.2% and 4.8% depending on the reconstructed K 0 S momentum [58]. The uncertainty from the E/cp requirement is estimated by changing the momentum p of each track to its value before applying the secondary-vertex fit. The uncertainties of the signal model, background model and fit range determine the uncertainties of the signal yields. The uncertainty from the signal model is estimated by changing the signal model to the shape predicted by the MC data. The uncertainty due to the background model is determined by replacing the background function with a first-order Chebychev polynomial. The uncertainty associated to the fit range is estimated by enlarging or reducing the fit range with an amount corresponding to the mass resolution.
The systematic uncertainty of ǫ × (1 + δ) is obtained by fluctuating randomly all the fit parameters within the iteration procedure by one σ and taking into account the correlations among the parameters. The distribution of the randomly produced ǫ × (1 + δ) is fitted by a Gaussian function, and the width of the fitted parameter is defined as the systematic uncertainty of ǫ × (1 + δ). The uncertainty due to the luminosity is estimated using large-angle Bhabha scattering events, which is about 0.9% [48,49].
A MC study shows a peaking background from the process K 0 S K 0 L π 0 at a center-of-mass energy of 3.080 GeV. However, the contribution normalized according to the integrated data luminosity is expected to be only 2.6 events. To compensate for a possible incomplete simulation, such as an incorrect angular distribution, the systematic uncertainty from the possible K 0 S K 0 L π 0 background is increased to 3.1% assuming the background level might be higher by 50%.
All the systematic uncertainties are listed in Table II. The total systematic uncertainty is obtained by summing the individual contributions in quadrature.

B. Line shape
The line shape of the Born cross section of e + e − → K 0 S K 0 L , obtained from the results given in Table I, is displayed in Figure 4. A resonance structure R around 2.2 GeV is observed. The cross section data are fitted by where β(s) = 1 − 4m 2 is a Breit-Wigner function describing the resonance; M , Γ and σ are the mass, width and peak cross section of the resonance, respectively; P (s) = c p0 +c p1 √ s+c p2 s is a secondorder polynomial function that is used to describe the nonresonant contribution, c pi corresponds to the coefficient of the i th -degree polynomial function, and φ is the relative phase between nonresonant and resonant amplitudes.
The least-squares (χ 2 ) method is used to perform the fit with both statistical and systematic uncertainties taken into account. The χ 2 is obtained via a matrix (see Eq. (1) in Ref. [59] and Eq. (2) in Ref. [60]) in which correlation effects of the various terms are included. Uncertainties from the K 0 S -selection efficiency, 1+δ, luminosity and ǫ are considered to be correlated, while the remaining ones are treated as uncorrelated. The line shape and the individual contributions obtained from the fit are shown in Figure 4.
The mass and width of the structure determined by the fit are M = 2273.7 ± 5.7 MeV/c 2 and Γ = 86 ± 44 MeV, respectively, where the uncertainties are statistical. The goodness of the fit is χ 2 /N DF = 4.6/8, and the statistical significance of the structure is 7.5σ.
Various sources of systematic uncertainties of the observed structure are considered including those associated with the choice of the model used to describe the nonresonant component, the description of its width and the chosen fit range. To estimate the systematic uncertainties, we changed the description of the nonresonant component to a coherent sum of a second-order polynomial and continuum functions where P ′ (s) and φ are the same as those defined in Eq.
(2) but only used in the fit when √ s < c p2 , c c is the coefficient of the continuum function and φ c is the relative phase between continuum and resonant amplitudes. The differences in the values of the peak cross section, mass, and width with respect to the nominal ones are ∆σ = 0.0150 nb, ∆m = 17.7 MeV/c 2 , and ∆Γ = 8.4 MeV, respectively. By replacing the description of the width with an energy dependent one (Γ(s, m) = Γ× s (2), the peak cross section, mass, and width change by an amount of ∆σ = 0.0001 nb, ∆m = 2.2 MeV/c 2 , and ∆Γ = 0.3 MeV, respectively. Uncertainties from the fit range are estimated by excluding the point at the c.m. energy of 2.00 GeV or the one at 3.08 GeV. ∆σ 1 and ∆σ 2 (∆m 1 and ∆m 2 , ∆Γ 1 and ∆Γ 2 ) denote the dif- The relative systematic uncertainties (in %) from the K 0 S selection (ǫ(K 0 S )), E/cp, the ISR and VP correction factor (1 + δ), the luminosity (L) and the fit on the invariant mass of π + π − pair (Fit). The column peak denotes the source from the peaking background and it has been estimated only at the c.m. energy of 3.080 GeV as elucidated in the text. The total systematic uncertainty (syst.) is calculated by summing the individual contributions in quadrature. The relative statistical uncertainty (stat.) is shown in the last column. ferences of the peak cross sections (masses and widths) obtained by fitting all energy points with a fit excluding those two energy points. Systematic uncertainties associated with the fit range on the mass and width are subsequently estimated by (∆σ 1 ) 2 + (∆σ 2 ) 2 = 0.0030 nb, (∆m 1 ) 2 + (∆m 2 ) 2 = 7.5 MeV/c 2 , and (∆Γ 1 ) 2 + (∆Γ 2 ) 2 = 50.2 MeV. Total systematic uncertainties are obtained by taking the quadratic sum of all the differences, which amount to 0.0153 nb, 19.3 MeV/c 2 , and 50.9 MeV on the peak cross section, mass, and width, respectively. Only the statistic uncertainty on φ is considered.
Γ e + e − Br K 0 S K 0 L of the resonance R is calculated from the peak cross section by making use of σ R = 12πCΓ e + e − Br K 0 [45], where σ R represents the peak cross section obtained through Eq. 2, Br K 0 is partial width of R → e + e − , M and Γ are the mass and width of the resonance, and C = 0.3894 × 10 12 nb MeV 2 /c 4 [56]. Γ e + e − Br K 0 S K 0 L for the process is obtained from the fit results and listed in Eq. 4. The χ 2 obtained by the earlier-described matrix may cause a bias in the fit [59][60][61][62]. To estimate the bias effect, an unbiased χ 2 definition (Eq. (7) in Ref. [62]) is used to fit the line shape. The differences between the two cases are negligible in this analysis.
The parameters of the resonance around 2.2 GeV are where the quoted uncertainties are statistical and systematic, respectively. The mass and width are consistent within 2σ with measurements of the mass and width of a similar structure observed in e + e − → K + K − at BE-SIII [13], which gave M = 2239.2 ± 7.1 ± 11.3 MeV/c 2 and Γ = 139.8 ± 12.3 ± 20.6 MeV.

VI. SUMMARY
We report a measurement of the Born cross sections in e + e − → K 0 S K 0 L from √ s = 2.00 to 3.08 GeV obtained at fifteen energy points with BESIII. The data are consistent within 2σ with previous measurements by the BaBar collaboration [35] in the overlap region from 2.00 to 2.54 GeV, but with a significantly improved precision as demonstrated in Figure 4. Moreover, the Born cross sections from 2.54 to 3.08 GeV are reported for the first time. A structure is observed around 2.2 GeV, which is similar to the one observed earlier in e + e − → K + K − [13]. The results of both processes taken with BESIII and BaBar are shown in Figure 5 for comparison.
A fit is applied to the data, where the mass and width of the resonance are determined to be M = 2273.7 ± 5.7 ± 19.3 MeV/c 2 and Γ = 86 ± 44 ± 51 MeV, re-spectively. In addition, Γ e + e − Br K 0 S K 0 L is found to be 0.9 ± 0.6 ± 0.7 eV. The first uncertainties in the parameters are statistical and the second ones are systematic. The mass and width are consistent within 2σ and 1σ, respectively, with the resonance parameters obtained by fitting the cross sections for the process e + e − → K + K − (M = 2239.2 ± 7.1 ± 11.3 MeV/c 2 and Γ = 139.8 ± 12.3 ± 20.6 MeV) [13].
Comparing to one of the 1 −− candidate of the structure φ(2170) by looking up the PDG [56], the mass parameter obtained in this paper differs from the world average for more than 4σ. The width parameter is consistent within 1σ compared to the world averaged width of φ(2170). The uncertainty on the width in this paper is large. For another 1 −− candidate of the structure ρ(2150) [56], the mass parameter in this paper is more than 5σ different from the world average and the wold averaged width is not given in the PDG [56]. But the mass and width are consistent with the individual measurement of the process e + e − → γπ + π − by Babar [63]. The conclusions support the discussions in the e + e − → K + K − study by BESIII [13]. Due to limit of the statistics, especially the cross section measurements above 2.4 GeV, it is difficult to discuss deeper on the structure found in this paper. More precise and fine interval measurements are needed. computing center of USTC for their strong support. This