Cross sections for the reactions $e^+ e^-\to K^0_S K^0_L\pi^0$, $K^0_S K^0_L\eta$, and $K^0_S K^0_L\pi^0\pi^0$ from events with initial-state radiation

TL;DR

This paper measures cross sections for specific hadronic final states in electron-positron collisions with initial-state radiation, revealing resonance structures and first observations of certain decays with data from the BaBar detector.

Contribution

First measurements of $e^+ e^- o K^0_S K^0_L\pi^0$, $K^0_S K^0_L\eta$, and $K^0_S K^0_L\pi^0\pi^0$ cross sections up to 4 GeV, including resonance analysis and $J/\psi$ decay observations.

Findings

First cross section measurements for these final states.

Observation of $J/\psi$ decays to these states for the first time.

Identification of intermediate resonance structures.

Abstract

We study the processes $e^+ e^-\to K^0_S K^0_L\pi^0\gamma$, $K^0_S K^0_L\eta\gamma$, and $K^0_S K^0_L\pi^0\pi^0\gamma$, where the photon is radiated from the initial state, providing cross section measurements for the hadronic final states over a continuum of center-of-mass energies. The results are based on 469~$fb^{-1}$ of data collected at or near the $\Upsilon(4S)$~ resonance with the BaBar detector at SLAC. We present the first measurements of the $e^+ e^-\to K^0_S K^0_L\pi^0$, $K^0_S K^0_L\eta$, and $K^0_S K^0_L\pi^0\pi^0$ cross sections up to a center-of-mass energy of 4~GeV, and study their intermediate resonance structures. We observe $J/\psi$ decays to all of these final states for the first time, present measurements of their $J/\psi$ branching fractions, and search for $\psi(2S)$ decays.

Tables

Table 1. Table 1: Summary of the e + ​ e − → K S ​ K L ​ π 0 → superscript 𝑒 superscript 𝑒 subscript 𝐾 𝑆 subscript 𝐾 𝐿 superscript 𝜋 0 e^{+}e^{-}\rightarrow K_{S}K_{L}\pi^{0} cross section measurement. Uncertainties are statistical only.

$E_{c.m.}$ (GeV)	$\sigma$ (nb)	$E_{c.m.}$ (GeV)	$\sigma$ (nb)	$E_{c.m.}$ (GeV)	$\sigma$ (nb)	$E_{c.m.}$ (GeV)	$\sigma$ (nb)
1.425	0.28 $\pm$ 0.07	2.075	0.55 $\pm$ 0.11	2.725	0.12 $\pm$ 0.05	3.375	0.03 $\pm$ 0.04
1.475	0.94 $\pm$ 0.15	2.125	0.44 $\pm$ 0.09	2.775	0.12 $\pm$ 0.08	3.425	0.02 $\pm$ 0.01
1.525	1.68 $\pm$ 0.22	2.175	0.42 $\pm$ 0.09	2.825	0.10 $\pm$ 0.06	3.475	0.05 $\pm$ 0.03
1.575	2.60 $\pm$ 0.28	2.225	0.36 $\pm$ 0.08	2.875	0.15 $\pm$ 0.10	3.525	0.01 $\pm$ 0.03
1.625	2.89 $\pm$ 0.26	2.275	0.48 $\pm$ 0.09	2.925	0.04 $\pm$ 0.03	3.575	0.04 $\pm$ 0.03
1.675	3.15 $\pm$ 0.30	2.325	0.31 $\pm$ 0.07	2.975	0.10 $\pm$ 0.10	3.625	0.03 $\pm$ 0.03
1.725	2.79 $\pm$ 0.29	2.375	0.18 $\pm$ 0.07	3.025	0.06 $\pm$ 0.03	3.675	0.00 $\pm$ 0.01
1.775	1.96 $\pm$ 0.23	2.425	0.26 $\pm$ 0.07	3.075	0.31 $\pm$ 0.09	3.725	0.04 $\pm$ 0.04
1.825	1.30 $\pm$ 0.18	2.475	0.25 $\pm$ 0.08	3.125	0.24 $\pm$ 0.12	3.775	0.01 $\pm$ 0.01
1.875	1.12 $\pm$ 0.18	2.525	0.19 $\pm$ 0.06	3.175	0.05 $\pm$ 0.04	3.825	0.03 $\pm$ 0.02
1.925	0.79 $\pm$ 0.12	2.575	0.09 $\pm$ 0.04	3.225	0.02 $\pm$ 0.03	3.875	0.00 $\pm$ 0.01
1.975	0.55 $\pm$ 0.10	2.625	0.14 $\pm$ 0.05	3.275	0.04 $\pm$ 0.03	3.925	0.02 $\pm$ 0.02
2.025	0.58 $\pm$ 0.11	2.675	0.07 $\pm$ 0.03	3.325	0.05 $\pm$ 0.04	3.975	0.01 $\pm$ 0.01

Table 2. Table 2: Summary of the e + ​ e − → ϕ ​ π 0 → superscript 𝑒 superscript 𝑒 italic-ϕ superscript 𝜋 0 e^{+}e^{-}\rightarrow\phi\pi^{0} cross section measurement. Uncertainties are statistical only.

$E_{c.m.}$ (GeV)	$\sigma$ (nb)	$E_{c.m.}$ (GeV)	$\sigma$ (nb)
1.25	0.00 $\pm$ 0.02	2.15	0.00 $\pm$ 0.01
1.35	0.00 $\pm$ 0.02	2.25	0.01 $\pm$ 0.01
1.45	0.06 $\pm$ 0.06	2.35	0.02 $\pm$ 0.02
1.55	0.17 $\pm$ 0.06	2.45	0.02 $\pm$ 0.01
1.65	0.04 $\pm$ 0.03	2.55	0.00 $\pm$ 0.01
1.75	0.05 $\pm$ 0.03	2.65	0.01 $\pm$ 0.01
1.85	0.02 $\pm$ 0.02	2.75	0.00 $\pm$ 0.01
1.95	0.00 $\pm$ 0.01	2.85	0.00 $\pm$ 0.01
2.05	0.03 $\pm$ 0.02

Table 3. Table 3: Summary of the e + ​ e − → K S ​ K L ​ η → superscript 𝑒 superscript 𝑒 subscript 𝐾 𝑆 subscript 𝐾 𝐿 𝜂 e^{+}e^{-}\rightarrow K_{S}K_{L}\eta cross section measurement. Uncertainties are statistical only.

$E_{c.m.}$ (GeV)	$\sigma$ (nb)	$E_{c.m.}$ (GeV)	$\sigma$ (nb)	$E_{c.m.}$ (GeV)	$\sigma$ (nb)	$E_{c.m.}$ (GeV)	$\sigma$ (nb)
		2.075	0.41 $\pm$ 0.12	2.725	0.15 $\pm$ 0.06	3.375	0.01 $\pm$ 0.02
		2.125	0.45 $\pm$ 0.12	2.775	0.10 $\pm$ 0.05	3.425	0.02 $\pm$ 0.03
		2.175	0.44 $\pm$ 0.11	2.825	0.08 $\pm$ 0.05	3.475	0.01 $\pm$ 0.02
1.575	0.12 $\pm$ 0.14	2.225	0.49 $\pm$ 0.12	2.875	0.14 $\pm$ 0.06	3.525	0.02 $\pm$ 0.02
1.625	0.52 $\pm$ 0.14	2.275	0.30 $\pm$ 0.11	2.925	0.05 $\pm$ 0.04	3.575	-0.01 $\pm$ 0.02
1.675	0.78 $\pm$ 0.18	2.325	0.31 $\pm$ 0.09	2.975	0.05 $\pm$ 0.06	3.625	-0.01 $\pm$ 0.01
1.725	1.01 $\pm$ 0.23	2.375	0.30 $\pm$ 0.10	3.025	0.14 $\pm$ 0.06	3.675	0.07 $\pm$ 0.03
1.775	0.50 $\pm$ 0.14	2.425	0.32 $\pm$ 0.10	3.075	0.22 $\pm$ 0.06	3.725	0.04 $\pm$ 0.03
1.825	0.53 $\pm$ 0.14	2.475	0.19 $\pm$ 0.07	3.125	0.09 $\pm$ 0.05	3.775	0.07 $\pm$ 0.04
1.875	0.41 $\pm$ 0.12	2.525	0.12 $\pm$ 0.06	3.175	0.03 $\pm$ 0.04	3.825	0.01 $\pm$ 0.02
1.925	0.36 $\pm$ 0.12	2.575	0.29 $\pm$ 0.09	3.225	-0.01 $\pm$ 0.03	3.875	0.01 $\pm$ 0.02
1.975	0.28 $\pm$ 0.09	2.625	0.11 $\pm$ 0.06	3.275	0.04 $\pm$ 0.04	3.925	0.01 $\pm$ 0.02
2.025	0.46 $\pm$ 0.11	2.675	0.19 $\pm$ 0.08	3.325	0.05 $\pm$ 0.03	3.975	0.02 $\pm$ 0.02

Table 4. Table 4: Summary of the e + ​ e − → K S 0 ​ K L 0 ​ π 0 ​ π 0 → superscript 𝑒 superscript 𝑒 subscript superscript 𝐾 0 𝑆 subscript superscript 𝐾 0 𝐿 superscript 𝜋 0 superscript 𝜋 0 e^{+}e^{-}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0} cross section measurement. Uncertainties are statistical only.

$E_{c.m.}$ (GeV)	$\sigma$ (nb)	$E_{c.m.}$ (GeV)	$\sigma$ (nb)
1.3500	0.018 $\pm$ 0.029	2.7500	0.029 $\pm$ 0.059
1.4500	0.059 $\pm$ 0.070	2.8500	0.028 $\pm$ 0.054
1.5500	0.334 $\pm$ 0.108	2.9500	0.051 $\pm$ 0.083
1.6500	0.350 $\pm$ 0.115	3.0500	0.213 $\pm$ 0.081
1.7500	0.549 $\pm$ 0.126	3.1500	-0.002 $\pm$ 0.071
1.8500	0.290 $\pm$ 0.113	3.2500	0.002 $\pm$ 0.033
1.9500	0.113 $\pm$ 0.103	3.3500	-0.002 $\pm$ 0.029
2.0500	0.185 $\pm$ 0.104	3.4500	0.031 $\pm$ 0.029
2.1500	0.141 $\pm$ 0.098	3.5500	0.015 $\pm$ 0.025
2.2500	0.038 $\pm$ 0.088	3.6500	-0.015 $\pm$ 0.020
2.3500	0.008 $\pm$ 0.081	3.7500	0.052 $\pm$ 0.024
2.4500	-0.022 $\pm$ 0.073	3.8500	0.002 $\pm$ 0.018
2.5500	0.080 $\pm$ 0.074	3.9500	-0.016 $\pm$ 0.016
2.6500	-0.019 $\pm$ 0.062

Table 5. Table 5: Summary of the J / ψ 𝐽 𝜓 {J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu} and ψ ​ ( 2 ​ S ) 𝜓 2 𝑆 \psi{(2S)} branching fractions obtained in this analysis.

Measured		Measured			Calculated Branching Fractions (10-3)
Quantity		Value ($\mathrm{eV}$)			This work			Previous
${\Gamma }_{ee}^{J/\psi }$ $\cdot$	${ℬ}_{J/\psi \to K_S^0{K}_L^0{\pi }^0}$	$11.4$±	1.3 $\pm$	0.6	$2.06$±	0.24 $\pm$	0.10	–
${\Gamma }_{ee}^{J/\psi }$ $\cdot$	${ℬ}_{J/\psi \to K_S^0{K}_L^0\eta }$	$8.0$±	1.8 $\pm$	0.4	$1.45$±	0.32 $\pm$	0.08	–
${\Gamma }_{ee}^{J/\psi }$ $\cdot$	${ℬ}_{J/\psi \to K_S^0{K}_L^0{\pi }^0{\pi }^0}$	$10.3$±	2.3 $\pm$	0.5	$1.86$±	0.43 $\pm$	0.10	–
${\Gamma }_{ee}^{J/\psi }$ $\cdot$	${ℬ}_{J/\psi \to K^{\ast }{\left(892\right)}^0\overline{K}{}_{}{}^0+c.c.}\cdot {ℬ}_{K^{\ast }{\left(892\right)}^0\to K^0{\pi }^0}$	$6.7$±	0.9 $\pm$	0.4	$1.20$±	0.15 $\pm$	0.06	–
${\Gamma }_{ee}^{J/\psi }$ $\cdot$	${ℬ}_{J/\psi \to K_2^{\ast }{\left(1430\right)}^0\overline{K}{}_{}{}^0+c.c.}\cdot {ℬ}_{K_2^{\ast }\left(1430\right)\to K^0{\pi }^0}$	$2.4$±	0.7 $\pm$	0.1	$0.43$±	0.12 $\pm$	0.02		PDG
${\Gamma }_{ee}^{\psi \left(2S\right)}$ $\cdot$	${ℬ}_{\psi \left(2S\right)\to K_S^0{K}_L^0{\pi }^0}$							–
${\Gamma }_{ee}^{\psi \left(2S\right)}$ $\cdot$	${ℬ}_{\psi \left(2S\right)\to K_S^0{K}_L^0\eta }$	$3.14$±	1.08 $\pm$	0.16	$1.33$±	0.46 $\pm$	0.07	–
${\Gamma }_{ee}^{\psi \left(2S\right)}$ $\cdot$	${ℬ}_{\psi \left(2S\right)\to K_S^0{K}_L^0{\pi }^0{\pi }^0}$	$2.92$±	1.27 $\pm$	0.15	$1.24$±	0.54 $\pm$	0.06	–

Equations

$$\sigma (K_S^0{K}_L^0{\pi }^0)(E_{c.m.})=\frac{d{N}_{K_S^0{K}_L^0{\pi }^0\gamma }(E_{c.m.})}{d\mathcal{L}(E_{c.m.})\cdot ϵ(E_{c.m.})\cdot R},$$

Full text

I Introduction

Electron-positron annihilation events with initial-state radiation (ISR) can be used to study processes over a wide range of energies below the nominal $e^{+}e^{-}$ center-of-mass (c.m.) energy ($E_{\rm c.m.}$), as demonstrated in Ref. baier . The possibility of exploiting ISR to make precise measurements of low-energy cross sections at high-luminosity $\phi$ and $B$ factories is discussed in Refs. arbus ; kuehn ; ivanch , and motivates the studies described in this paper. Such measurements are of particular current interest because of a three-standard-deviation discrepancy between the measured value of the muon anomalous magnetic moment ($g_{\mu}-2$) and that computed in the Standard Model dehz , where the hadronic loop contributions require experimental $e^{+}e^{-}$ annihilation cross sections as input. The calculation is most sensitive to the low-energy region, where the inclusive hadronic cross section cannot be measured reliably, and a sum of exclusive states must be used. Not all accessible states have been measured yet, and new measurements will improve the reliability of the calculation. In addition, studies of ISR events at $B$ factories provide information on resonance spectroscopy for masses up through the charmonium region.

Studies of the ISR processes $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}\gamma$ Druzhinin1 ; isr2pi and $e^{+}e^{-}\rightarrow X_{h}\gamma$, where $X_{h}$ represents any of several exclusive multihadron final states, using data of the BABAR experiment at SLAC, have been reported previously. The $X_{h}$ studied so far include: charged hadron pairs $\pi^{+}\pi^{-}$ isr2pi , $K^{+}K^{-}$ isr2k , and $p\overline{p}$ isr2p ; four or six charged mesons isr4pi ; isr2k2pi ; isr6pi ; charged mesons plus one or two $\pi^{0}$ mesons isr2k2pi ; isr6pi ; isr3pi ; isr5pi ; a $K^{0}_{\scriptscriptstyle S}$ plus charged and neutral mesons isrkkpi ; and the first ISR measurement from BABAR that includes $K^{0}_{\scriptscriptstyle L}$ mesons isrkskl . Together, they demonstrate good detector efficiency for events of this kind, and well understood tracking, particle identification, and $\pi^{0}$, $K^{0}_{\scriptscriptstyle S}$ and $K^{0}_{\scriptscriptstyle L}$ reconstruction.

In this paper we report measurements of the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$, $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$, and $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$ final states, produced in conjunction with a hard photon that is assumed to result from ISR. Candidate $K^{0}_{\scriptscriptstyle S}$ mesons are reconstructed in the $\pi^{+}\pi^{-}$ decay mode, candidate $\pi^{0}$ and $\eta$ mesons are reconstructed in the $\gamma\gamma$ decay mode, and $K^{0}_{\scriptscriptstyle L}$ mesons are detected via their nuclear interactions in the electromagnetic calorimeter. For these final states, we measure cross sections from threshold to $E_{\rm c.m.}=4$ $\mathrm{\,Ge\kern-1.00006ptV}$, study their internal structure, perform the first measurements of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ branching fractions, and search for $\psi{(2S)}$ decays. We also search for the $e^{+}e^{-}\rightarrow\gamma K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{0}$ and $e^{+}e^{-}\rightarrow\gamma K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{0}\pi^{0}$ processes, which are forbidden by $C$-parity conservation, and we see no indication of them at the level of single background events. Together with our previous measurements isr2k ; isr2k2pi ; isrkskl , these results provide a much more complete understanding of the $K\kern 1.99997pt\overline{\kern-1.99997ptK}{}\pi$, $K\kern 1.99997pt\overline{\kern-1.99997ptK}{}\eta$ and $K\kern 1.99997pt\overline{\kern-1.99997ptK}{}\pi\pi$ final states in $e^{+}e^{-}$ annihilation.

II The BABAR detector and data set

The data used in this analysis were collected with the BABAR detector at the PEP-II asymmetric-energy $e^{+}e^{-}$ storage ring. The total integrated luminosity used is 468.6 $\mbox{\,fb}^{-1}$ lumi , which includes data collected at the $\mathchar 28935\relax(4S)$ resonance (424.7 $\mbox{\,fb}^{-1}$) and at a c.m. energy 40 $\mathrm{\,Me\kern-1.00006ptV}$ below this resonance (43.9 $\mbox{\,fb}^{-1}$).

The BABAR detector is described in detail elsewhere babar . Charged particles are reconstructed using the BABAR tracking system, which comprises the silicon vertex tracker (SVT) and the drift chamber (DCH) inside the 1.5 T solenoid. Separation of pions and kaons is accomplished by means of the detector of internally reflected Cherenkov light (DIRC) and energy-loss measurements in the SVT and DCH. The hard ISR photon, photons from $\pi^{0}$ and $\eta$ decays, and $K^{0}_{\scriptscriptstyle L}$ are detected in the electromagnetic calorimeter (EMC). Muon identification, provided by the instrumented flux return, is used to select the $\mu^{+}\mu^{-}\gamma$ final state.

To study the detector acceptance and efficiency, we have developed a special package of simulation programs for radiative processes based on the approach suggested by Kühn and Czyż kuehn2 . Multiple collinear soft-photon emission from the initial $e^{+}e^{-}$ state is implemented with the structure-function technique kuraev ; strfun , while additional photon radiation from the final-state particles is simulated using the PHOTOS package PHOTOS . The precision of the radiative simulation contributes less than 1% to the uncertainty of the measured hadronic cross sections.

In addition to the signal channels $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$, $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$, and $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$, we simulate ISR processes which result in high backgrounds, $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}$, $K^{0}_{\scriptscriptstyle S}K^{\pm}\pi^{\mp}$, and $K^{0}_{\scriptscriptstyle S}K^{\pm}\pi^{\mp}\pi^{0}$, with cross sections and mass dependences based on our previous measurements and isospin relations. The $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}$ and $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$ channels are dominated by $e^{+}e^{-}\rightarrow\gamma\phi$ and $\gamma\phi\eta$, respectively. Samples of three to five times the number of expected events are generated for each final state and processed through the detector response simulation GEANT4 . These events are then reconstructed using the same software chain as the data.

We also simulate several non-ISR backgrounds, including $e^{+}e^{-}\rightarrow q\overline{q}$ $(q=u,d,s,c)$ events using the Jetset 7.4 jetset generator, and $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ events using the KORALB koralb generator. Variations in detector and background conditions are taken into account.

III Event Selection and Reconstruction

We begin with events containing at least two charged particles and at least four clusters of energy deposits in contiguous crystals in the EMC. We then consider the cluster in the event with the highest energy in the $e^{+}e^{-}$ c.m. frame as the ISR photon candidate, and require $E^{\gamma}_{\rm c.m.}>3~{}\mathrm{\,Ge\kern-1.00006ptV}$. Since the ISR photons are produced mostly along the beamline, this accepts only about 10% of the signal events, but in the selected events, the hadronic system is fully contained and can be studied reliably.

In these events, we reconstruct candidate $K^{0}_{\scriptscriptstyle S}$ decays to two charged pions from pairs of oppositely charged tracks not identified as electrons. They must have a well reconstructed vertex between 0.1 and 40.0 cm in radial distance from the beam axis, and their total momentum must be consistent with the assumption that they originate from the interaction region. The $m(\pi^{+}\pi^{-})$ invariant mass distribution for these $K^{0}_{\scriptscriptstyle S}$ candidates is shown in Fig. 1 for both data (points) and the $e^{+}e^{-}\rightarrow\gamma\phi\rightarrow\gamma K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}$ simulation (histogram). The signal is very clean, and requiring $482<m(\pi^{+}\pi^{-})<512~{}{\mathrm{\,Me\kern-1.00006ptV\!/}c^{2}}$ (vertical lines in Fig. 1) accepts 98% of the signal events. We use the sidebands 472–482 and 512–522 ${\mathrm{\,Me\kern-1.00006ptV\!/}c^{2}}$ to estimate the contributions from non-$K^{0}_{\scriptscriptstyle S}$ backgrounds, which are found to be negligible in all cases after final selection.

A few thousand events (about 1% of the total) have more than one selected $K^{0}_{\scriptscriptstyle S}$ candidate, and we use only the candidate with $m(\pi^{+}\pi^{-})$ closest to the nominal PDG $K^{0}_{\scriptscriptstyle S}$ mass. We also require the event to contain no other tracks that extrapolate within 2 cm of the beam axis and 3 cm along the axis from the nominal interaction point.

Any number of additional tracks and EMC clusters is allowed. We consider all clusters with reconstructed energy above 0.1 $\mathrm{\,Ge\kern-1.00006ptV}$ as photon candidates, and calculate the invariant mass of each pair. Every pair with a mass within 30 (50) ${\mathrm{\,Me\kern-1.00006ptV\!/}c^{2}}$ of the nominal $\pi^{0}$ ($\eta$) mass is considered a $\pi^{0}$ ($\eta$) candidate. The efficiency of $\pi^{0}$ and $\eta$ reconstruction in these events is about 97%.

The decay length of the $K^{0}_{\scriptscriptstyle L}$ meson is large, and the probability to detect a $K^{0}_{\scriptscriptstyle L}$ decay in the DCH is low. Instead, we look for a cluster in the EMC resulting from the interaction of a $K^{0}_{\scriptscriptstyle L}$ with a nucleus in the EMC material. Such clusters are indistinguishable from photon-induced clusters, and give poor resolution on the $K^{0}_{\scriptscriptstyle L}$ energy. The characteristics of these clusters were studied in detail in our previous publication isrkskl , where it was shown using $e^{+}e^{-}\rightarrow\phi\gamma$ events that $K^{0}_{\scriptscriptstyle L}$ clusters are detected with high efficiency and good angular resolution. Background from low-energy clusters is high, and the requirement of at least 0.2 $\mathrm{\,Ge\kern-1.00006ptV}$ in cluster energy yields a clean sample with 48% efficiency. Here, we apply the same energy requirement, and use the efficiency and angular resolution measured as a function of polar and azimuthal angles in Ref. isrkskl .

IV The kinematic fit procedure

Each event selected as described in Sec. III is subjected to a set of constrained kinematic fits, in which the four-momenta and covariance matrices of the initial $e^{+}e^{-}$, the ISR photon, the best $K^{0}_{\scriptscriptstyle S}$ candidate, and zero, two or four relevant photon candidates are taken into account. The direction and angular resolution, but not the energy, of the $K^{0}_{\scriptscriptstyle L}$ candidate is also used, and the $K^{0}_{\scriptscriptstyle L}$ momentum is determined in the fit. The three-momentum vectors for all other particles, including the photons, are also determined with better accuracy from the fits, and the fitted values are used in further calculations.

For every event, we first perform kinematic fits under the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\gamma$ hypothesis, considering the ISR photon and $K^{0}_{\scriptscriptstyle S}$ candidates, along with each cluster with energy over 0.2 $\mathrm{\,Ge\kern-1.00006ptV}$ in turn as the $K^{0}_{\scriptscriptstyle L}$ candidate. Each fit has 3 constraints, and we consider the combination with the best $\chi^{2}$ value, denoted $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L})$. This variable is useful in suppressing the large background arising from combinations of background photons with a mass near the $\pi^{0}$ or $\eta$ mass.

Next, we consider each $\pi^{0}$ and $\eta$ candidate, and perform a set of fits to the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\gamma$ and $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta\gamma$ hypothesis, including the ISR photon, $K^{0}_{\scriptscriptstyle S}$, and two photon candidates, along with each $K^{0}_{\scriptscriptstyle L}$ candidate not included in the $\pi^{0}$ or $\eta$ candidate. These fits have four constraints, including one on the $\pi^{0}$ or $\eta$ candidate mass. We retain the $\pi^{0}K^{0}_{\scriptscriptstyle L}$ and $\eta K^{0}_{\scriptscriptstyle L}$ candidate combinations yielding the best values of $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0})$ and $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta)$, respectively.

Similarly, for events with six or more clusters, we consider each pair of non-overlapping $\pi^{0}$ candidates, and perform a set of five-constraint fits under the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}\gamma$ hypothesis. Both $\pi^{0}$ masses are constrained, and we retain the $\pi^{0}\pi^{0}K^{0}_{\scriptscriptstyle L}$ candidate combination yielding the lowest value of $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0})$.

Finally, we perform similar fits for all the other simulated signal and background processes discussed in Sec. II, giving us additional $\chi^{2}$ variables that can be used to select (or suppress) these processes.

V The $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ final state

V.1 Additional selection criteria

For the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ final state, a few additional selection criteria are applied. Considering all pairs of EMC clusters not assigned to the ISR photon, $\pi^{0}$, or $K^{0}_{\scriptscriptstyle L}$ candidates, we observe a large signal from extra $\pi^{0}$’s, shown in Fig. 2. It is especially strong when one of the clusters has high energy, so we require $\rm E_{\gamma}^{max}<0.5~{}\mathrm{\,Ge\kern-1.00006ptV}$. This reduces backgrounds from several sources with a loss of 3% in simulated signal efficiency. However, many ISR $\phi\gamma$ events with a false $\pi^{0}$, formed by accidential photons, remain. To reduce this background, we require $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L})>15$ if the fitted $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}$ invariant mass $m(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L})$ is smaller than 1.04 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$.

The 4C $\chi^{2}$ distribution for the remaining events under the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\gamma$ hypothesis is shown as the points in Fig. 3(a), with the corresponding distribution for MC-simulated pure $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\gamma$ events shown as the open histogram. Both distributions are broader than typical 4C $\chi^{2}$ distributions due to higher-order ISR, which is present in both data and simulation, but not taken into account in the fit. The reliability of the simulated distribution has been demonstrated in our previous studies, and is discussed below. In the figure, the simulated signal distribution is normalized to the data in the region $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0})<3$, where the contribution of higher-order ISR is small and the background contamination is lowest, but still amounts to about 5% of the signal. The difference between the two distributions at high values gives an indication of the level of background.

We define a signal region $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0})<25$ and a control region $25<\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0})<50$ (vertical lines in Fig. 3(a)), from which we estimate backgrounds in the signal region. The signal region contains 5441 data and 3402 signal-MC events, while the control region contains 2733 and 632 events, respectively.

V.2 Background subtraction

We estimate known ISR backgrounds from simulation, and normalize the simulated non-ISR background using the $\pi^{0}$ peak, as described in Ref. isrkskl . The largest backgrounds we can evaluate in this way are shown in Fig. 3(a): the shaded, cross-hatched, and hatched areas represent the simulated backgrounds from non-ISR $q\bar{q}$, ISR $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}(\phi)$, and ISR $\phi\eta$ events, respectively. The shapes of these three distributions are consistent with each other and quite distinct from that expected for signal events. However, these backgrounds account for less than half of the observed difference between data and simulation at large $\chi^{2}$ values. We assume the remaining background is from other ISR processes, with a $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0})$ distribution similar in shape to those shown.

To obtain any distribution of the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ signal events, we use the control region to estimate the sum of all backgrounds, following the procedure described in detail in Ref. isrkskl . In each bin of the distribution in question, the background contribution is estimated as the difference between the numbers of data and signal-MC events in the control region (see Fig. 3(a)), normalized to the corresponding difference in the signal region.

The $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ invariant mass distribution of events in the signal region is shown in Fig. 3(b) as the points. The shaded, cross-hatched, and hatched areas represent the same simulated backgrounds as in Fig. 3(a). The sum of all backgrounds, estimated from the control region, is shown as the open histogram in Fig. 3(b), and the extracted mass distribution for $e^{+}e^{-}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ signal events is shown as the filled points in Fig. 3(c). We observe 3669 events in the mass range from threshold to 4.0 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$. In addition to a main peak around 1.8 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$, a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ signal is visible.

This procedure relies on good agreement between data and simulation in the $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0})$ distribution. Considering our previous studies of $\chi^{2}$ distributions isrkskl , along with simulation and normalization statistics, we estimate the relative systematic uncertainty on the background to be 30%. This results in an uncertainty on the background-subtracted signal of about 10% for $m(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0})<2.2$ ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$, increasing roughly linearly with mass to about 40% at 3.2 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$ and above.

V.3 Detection efficiency

The selection procedures applied to the data are also applied to the MC-simulated event sample. The resulting distribution of the reconstructed $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ invariant mass is shown in Fig. 4(a) for events with $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0})$ in the signal (open histogram) and control (hatched) regions. The reconstruction efficiency as a function of mass is obtained by dividing the number of reconstructed MC events in each 50 ${\mathrm{\,Me\kern-1.00006ptV\!/}c^{2}}$ mass interval by the number generated in that interval, and is shown in Fig. 4(b). The effects of detector resolution, about 25 ${\mathrm{\,Me\kern-1.00006ptV\!/}c^{2}}$, are included in this efficiency. Below 1.5 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$ the efficiency becomes very large, due to the rapidly changing cross section near threshold. Since the resolution is measured from the data and the shape of the threshold rise is well simulated, we apply no correction. Nevertheless, the backgrounds and the resolution-effect uncertainties are high in this region, so we do not quote a cross section measurement below 1.4 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$, and we assign an additional 50% (30%) relative systematic uncertainty for the mass bin at 1.425(1.475) ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$.

This efficiency is corrected for the data-MC differences evaluated in our previous studies. The ISR photon detection efficiency has been studied using $\mu^{+}\mu^{-}\gamma$ events isr2pi , and we apply a polar-angle-dependent correction of typically $-$1.5$\pm$0.5% to the simulated efficiency. The $K^{0}_{\scriptscriptstyle S}$ detection efficiency has been studied very carefully at BABAR, with data-MC differences in the efficiency determined as a function of the $K^{0}_{\scriptscriptstyle S}$ direction and momentum. We apply a correction event by event, which introduces an overall correction of $+$1.1$\pm$1.0% to the efficiency. The $\pi^{0}$ reconstruction efficiency has been studied in BABAR using $\omega\gamma$ and $\omega\pi^{0}\gamma$ events, and the correction is found to be ($-$3$\pm$1)%. The $K^{0}_{\scriptscriptstyle L}$ detection requires a ($-$6.1$\pm$0.6)% correction isrkskl . In total, there is a ($-$9.5$\pm$1.6)% correction; this systematic uncertainty is small compared with that due to the backgrounds, described above.

V.4 The $e^{+}e^{-}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ cross section

The cross section for $e^{+}e^{-}$ annihilation into $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ is calculated from

[TABLE]

where $E_{\rm c.m.}\equiv m(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0})$; $dN_{K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\gamma}$ is the number of selected, background-subtracted $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ events in the interval $dE_{\rm c.m.}$; $\epsilon(E_{\rm c.m.})=\epsilon^{\rm MC}(E_{\rm c.m.})\cdot(1+\delta_{\rm corr})$ is the simulated detection efficiency corrected for data-MC differences, as described above. The radiative correction $R$ is unity within 1%, with an estimated precision of about 1%. The differential ISR luminosity $d{\cal L}(E_{\rm c.m.})$ associated with the interval $dE_{\rm c.m.}$ centered at an effective collision energy of $E_{\rm c.m.}$ is calculated using the leading order formula (see, for example, Ref. isr3pi ), and the systematic uncertainty associated with the luminosity determination is estimated to be 0.5%.

The cross section is shown as a function of energy in Fig. 5, and listed in Table 1. There are no previous measurements for this final state. We do not quote the cross section from threshold (1.13 $\mathrm{\,Ge\kern-1.00006ptV}$) to 1.4 $\mathrm{\,Ge\kern-1.00006ptV}$, where it shows a sharp rise to a maximum value of about 3 nb near 1.7 $\mathrm{\,Ge\kern-1.00006ptV}$, presumably dominated by the $\phi(1680)$ resonance, and a slow decrease toward higher energies, perturbed by the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ signal. Only statistical uncertainties are shown. The systematic uncertainty is dominated by background contributions, and amounts to about 10% near the peak of the cross section (1.7 $\mathrm{\,Ge\kern-1.00006ptV}$), increasing roughly lineary with decreasing cross section to about 30% in the 2.5–3 $\mathrm{\,Ge\kern-1.00006ptV}$ region, always similar in size to the statistical uncertainty. Above the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass, statistics dominate the $\sim$40% systematic uncertainty.

V.5 The $K^{*}(892)^{0}$ and $K_{2}^{*}(1430)^{0}$

contributions

Figure 6 shows the distributions of the fitted $K^{0}_{\scriptscriptstyle S}\pi^{0}$ and $K^{0}_{\scriptscriptstyle L}\pi^{0}$ invariant masses in the selected $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ events, after background subtraction. Clear signals corresponding to the $K^{*}(892)^{0}$ resonance are visible, as well as indications of $K_{2}^{*}(1430)^{0}$ production.

We fit these distributions with a sum of two incoherent Breit-Wigner functions and a function describing the non-resonant contribution, yielding $1750\pm 84$ $K^{*}(892)^{0}\rightarrow K^{0}_{\scriptscriptstyle S}\pi^{0}$ decays, $1795\pm 56$ $K^{*}(892)^{0}\rightarrow K^{0}_{\scriptscriptstyle L}\pi^{0}$ decays, and a total of $145\pm 54$ $K_{2}^{*}(1430)^{0}$ decays. The sum of these $K^{*0}$ decays is consistent with the total number of $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ events, indicating that the process is dominated by $K^{*0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}+c.c.$, and mostly $K^{*}(892)^{0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}+c.c.$, production.

Indeed, if we perform fits similar to those shown in Fig. 6 for events in each 0.05 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$ interval of the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ invariant mass, and sum the $K^{*}(892)^{0}K^{0}_{\scriptscriptstyle S}$ and $K^{*}(892)^{0}K^{0}_{\scriptscriptstyle L}$ yields, we obtain the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ mass distribution shown in Fig. 3(c) by the open circles. The errors are statistical only, and the difference between the number of $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ events and the $K^{*}(892)^{0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}$ contribution in each bin is less than the systematic uncertainty due to the background subtraction procedure.

V.6 The $\phi(1020)\pi^{0}$ contribution

Figure 7 shows the distribution of the fitted $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}$ invariant mass for the selected $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ events before background subtraction (dots), along with the background (histogram) estimated from the $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0})$ control region. A $\phi(1020)$ signal is visible in the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ signal, but not in the background. A fit with a Gaussian plus polynomial function yields a total of $29\pm 9$ $\phi\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}$ decays in $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ events.

Fitting the $m(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L})$ distribution in 0.1 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$ bins of the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ mass, we obtain a $\phi\pi^{0}$ invariant mass spectrum for the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ final state. Using Eq. 1 and the $\phi\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}$ branching fraction PDG , we calculate a cross section for this intermediate state, shown in Fig. 8 (dots) and listed in Table 2. Only statistical uncertainties are shown. The systematic uncertainties of 10–30% relative in this mass range are smaller than the statistical uncertainties.

The results are consistent with those observed in our previous study of the $K^{+}K^{-}\pi^{0}$ final state isrkkpi , also shown in Fig. 8 (circles). Together, our measurements suggest a possible resonant structure near 1.6 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$ with isospin $I=1$. The low cross section is expected, as the $\phi\pi^{0}$ channels are suppressed by the OZI-rule.

VI The $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$ final state

VI.1 Final selection and backgrounds

We apply the same requirements on extra $\pi^{0}$’s and $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L})$ as for the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ final state (see Sec. V), and consider the $\eta$-$K^{0}_{\scriptscriptstyle L}$ combination in each event with the best $\chi^{2}$ under the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta\gamma$ hypothesis. Figure 9(a) shows the $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta)$ distribution of the remaining events in the data (dots) compared with that of the signal simulation (open histogram). The simulated distribution is normalized to the data in the region $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta)<3$, where the contribution of higher-order ISR is small and the background contamination is lowest, but still amounts to about 10% of the signal. The cross-hatched and hatched areas represent the simulated contributions from non-ISR $q\bar{q}$ events and the sum of ISR $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$, ISR $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}$ , and ISR $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$ events, respectively; together, they account for about half of the excess of data over signal events at high values of $\chi^{2}$.

We define a signal region $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta)<20$ and a control region $20<\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta)<40$ (vertical lines in Fig. 9(a)), containing 1829 data and 2518 signal-MC events, and 1473 data and 495 signal-MC events, respectively. The $m(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta)$ distribution for the events in the signal region is shown in Fig. 9(b) as points, along with the sum of the simulated background processes as the cross-hatched and hatched areas. Using events from the control region (see Sec. V.2) we calculate the total background contribution, assumed to be dominated by ISR channels, and show it as the open histogram in Fig. 9(b).

We fit the total background with a smooth function to reduce fluctuations, and use the result (curve in Fig. 9(b)) for the background subtraction. This yields a total of $864\pm 43$ signal events with masses between threshold and 4.0 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$, with the mass distribution shown in Fig. 9(c). Again, we estimate the relative systematic uncertainty on the background as 30%, corresponding to an uncertainty on the cross section of about 15% for $m(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta)<2.2$ ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$, increasing roughly lineary to 30% at 3.0 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$, and over 100% above 3.2 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$.

VI.2 Cross section for $e^{+}e^{-}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$

We calculate the $e^{+}e^{-}\!\!\rightarrow\!K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$ cross section as a function of the effective c.m. energy using Eq. 1. The simulated efficiency is 1.6% and shows no dependence on the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$ invariant mass. All efficiency corrections discussed in Sec. V.4 are applied, in particular, the same correction is applied to the $\eta$ reconstruction efficiency as for the $\pi^{0}$.

The fully corrected cross section is shown in Fig. 10 and listed in Table 3, with statistical uncertainties only. There are no other measurements for this final state. The cross section shows a steep rise from threshold at 1.6 $\mathrm{\,Ge\kern-1.00006ptV}$, a maximum value of about 1 nb near 1.7 $\mathrm{\,Ge\kern-1.00006ptV}$, and a decrease with increasing energy, punctuated by a clear ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ signal (discussed in Sec. VIII). The relative systematic uncertainty is dominated by the uncertainty of the backgrounds, totals 15% at the peak of the cross section, increases roughly lineary to about 30% at 3 $\mathrm{\,Ge\kern-1.00006ptV}$, and exceeds 100% at higher energies.

VI.3 The $\phi(1020)\eta$ contribution

Figure 11 shows the background-subtracted $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}$ invariant mass distribution in $e^{+}e^{-}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$ events (dots), compared with that of simulated ISR $\phi\eta$ events (histogram). The two distributions are consistent at low mass values, and we simply take the number of events with $m(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L})<1.05$ ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$ as an estimate of the $\phi\eta$ contribution. It totals $386\pm 20$ events, with the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$ invariant mass distribution shown in Fig. 9(c) as the open circles. The $\phi\eta$ channel dominates $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$ production for masses below about 2 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$, but its contribution decreases rapidly for higher masses, and shows no significant ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ signal.

Using these events, we calculate the $e^{+}e^{-}\rightarrow\phi\eta$ cross section, which is shown in Fig. 12 as the points. It is consistent with our previous measurement isrkkpi in the $K^{+}K^{-}\eta$ final state (circles). Again, only statistical uncertainties are shown, and they are larger than the 15–30% systematic uncertainties. We observe no significant structures in the $K^{0}_{\scriptscriptstyle S}\eta$ or in the $K^{0}_{\scriptscriptstyle L}\eta$ invariant mass distributions.

VII The $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$ final state

VII.1 Final selection and backgrounds

From all events with a $K^{0}_{\scriptscriptstyle S}$, a $K^{0}_{\scriptscriptstyle L}$ and at least two non-overlapping $\pi^{0}$ candidates, we consider the combination with the best value of $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0})$, as described in Sec. IV. Since background candidates are not well suppressed using additional photon or $\pi^{0}$, no additional requirements are imposed. Figure 13(a) shows the $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0})$ distribution of the data (dots), compared with that of the signal simulation (open histogram). The simulated distribution is normalized to the data in the region $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0})<5$, where the contribution of higher-order ISR is small and the background contamination is lowest, but still amounts to about 20% of the signal. The cross-hatched and hatched areas represent the simulated backgrounds from non-ISR $q\bar{q}$ and the sum of ISR $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}$, $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$, and $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$ events, respectively, where the latter two are normalized to our measurements reported above. Together, these account for a substantial fraction of the entries at high $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0})$ values.

We define a signal region $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0})<30$ and a control region $30<\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0})<60$ (vertical lines in Fig. 13(a)), containing 1748 data and 2465 signal-MC signal, and 990 data and 517 signal-MC events, respectively. The $m(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0})$ distribution for the events in the signal region is shown in Fig. 13(b) as the points. The cross-hatched and hatched areas show the simulated contributions from non-ISR $q\bar{q}$ events, and the sum of the ISR $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}$, $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$, and $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$ events, respectively.

We use a two-step procedure to subtract backgrounds in this final state. From all experimental distributions, we first subtract the normalized MC-simulated events just discussed. In the case of the $m(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0})$ distribution, we fit an empirical function to the sum of these, shown as the solid line in Fig. 13(b), and use that for a bin-by-bin subtraction. A similar procedure is applied to all other distributions, including that of $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0})$ (see Fig. 13(a)).

We then use events from the $\chi^{2}(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0})$ control region, after subtraction of the backgrounds just described, to calculate the remaining background in each bin of each distribution, as described in Sec. V.2. We show this contribution to the $m(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0})$ distribution by the open squares in Fig. 13(b). We fit a smooth function to reduce fluctuations, and use the results (dotted curve in Fig. 13(b)) to subtract the remaining background.

After subtraction of all backgrounds, we obtain 392$\pm$55 signal events with masses between threshold and 4.0 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$, distributed as shown in Fig. 13(c). We estimate the systematic uncertainty due to backgrounds to be about 25% of the signal for $m(K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0})<2.2$ ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$, increasing roughly linearly to 100% at 3.0 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$, and everywhere smaller than the statistical uncertainty. There is no significant signal above about 2 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$, apart from an indication of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ signals in the 0.1 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$ wide bins.

VII.2 Cross section for $e^{+}e^{-}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$

We calculate the $e^{+}e^{-}\!\!\rightarrow\!K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$ cross section as a function of the effective c.m. energy $E_{\rm c.m.}$ using Eq. 1. The simulated efficiency is 1.5% and shows no dependence on the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$ invariant mass. All corrections discussed in Sec. V.4 are applied, plus an additional 3% for the detection of the second $\pi^{0}$. The fully corrected cross section is shown in Fig. 14 and listed in Table 4, with statistical uncertainties only. There are no other measurements for this final state.

The cross section shows a rise from a threshold at 1.4 $\mathrm{\,Ge\kern-1.00006ptV}$, a maximum value of about 0.5 nb near 1.8 $\mathrm{\,Ge\kern-1.00006ptV}$, and a decrease with increasing energy. Apart from a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and possibly a $\psi{(2S)}$ signal (discussed below), the cross section is statistically consistent with zero above 2.2 $\mathrm{\,Ge\kern-1.00006ptV}$. The relative systematic uncertainty is dominated by the uncertainty of the backgrounds, and totals 25% at the peak of the cross section, increasing linearly to about 60% at 2 $\mathrm{\,Ge\kern-1.00006ptV}$, and 100% at higher energies.

VII.3 The $K^{*}(892)^{0}$ and $\phi$ contributions

Figure 15 shows the $K^{0}_{\scriptscriptstyle S}\pi^{0}$ and the $K^{0}_{\scriptscriptstyle L}\pi^{0}$ invariant mass distributions for the selected $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$ events after background subtraction (two entries per event). Signals corresponding to the $K^{*}(892)^{0}$ resonance are evident, but the statistics are not sufficient to study them in detail.

As an exercise, we fit these distributions with a sum of a Breit-Wigner function and a smooth function describing the non-resonant contribution, yielding $190\pm 44$ $K^{*}(892)^{0}\rightarrow K^{0}_{\scriptscriptstyle S}\pi^{0}$ decays, and $171\pm 32$ $K^{*}(892)^{0}\rightarrow K^{0}_{\scriptscriptstyle L}\pi^{0}$ decays. There is no indication from the scatter plot (not shown) of any contribution from the $e^{+}e^{-}\rightarrow K^{*}(892)^{0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*}(892)^{0}$ reaction, and the sum of the two $K^{*}(892)$ yields is less than the total number of $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$ events, limiting any such contribution to half the signal events. This pattern is consistent with the dominance of the $e^{+}e^{-}\rightarrow K^{*}(892)^{0}K^{-}\pi^{+}+c.c.$ processes seen in our previous measurement of the $e^{+}e^{-}\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}$ reaction isr2k2pi .

Fitting the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}$ invariant mass distribution (not shown), we observe $71\pm 16$ events from the $e^{+}e^{-}\rightarrow\phi(1020)\pi^{0}\pi^{0}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$ process. This is consistent with expectations from our measurement in the $K^{+}K^{-}\pi^{0}\pi^{0}$ mode isr2k2pi , but with substantially lower precision.

VIII The charmonium region

Figure 16 shows expanded views of the mass distributions in Figs. 3(b), 9(b), and 13(b), respectively, in the 2.5–4.0 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$ mass region without any background subtraction. There are clear ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ signals in all three distributions, and indications of $\psi{(2S)}$ signals in (b) and (c). Fitting the distribution in Fig. 16(a) with the sum of the simulated ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ signal shape and a second-order polynomial function yields $182\pm 21$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ decays. No signal from the $\psi{(2S)}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ decay is observed ( $<$ 8 events at 90% C.L.). Fitting the other two distributions with the sum of simulated ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ signal shapes and a second-order polynomial function yields $45\pm 10$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$ decays, $47\pm 11$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$ decays, $16\pm 5$ $\psi{(2S)}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$ decays, and $14\pm 6$ $\psi{(2S)}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$ decays.

Using the corrected simulated efficiencies described above and the differential luminosity, we calculate the products of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ ($\psi(2S)$) electronic width and branching fractions to these modes, and list them in Table 5. Using the PDG value of $\Gamma_{ee}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu})=5.55$ $\mathrm{\,ke\kern-1.00006ptV}$ ($\Gamma_{ee}(\psi(2S))=2.35$ $\mathrm{\,ke\kern-1.00006ptV}$) PDG , we obtain the corresponding branching fractions, also presented in Table 5. Systematic uncertainties of typically 5% arise from the corrections to the simulated efficiencies, discussed in Sec. V.3, and variations of the signal shape.

These are the first observations of these three ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay modes. Our $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ branching fraction can be compared with existing measurements of similar modes ${\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow K^{+}K^{-}\pi^{0})=(2.8\pm 0.8)\times 10^{-3}$ franklin and ${\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow K^{\pm}K^{0}_{\scriptscriptstyle S}\pi^{\mp})=(2.6\pm 0.7)\times 10^{-3}$ vannucci . The data are consistent with the expectation from isospin conservation that they are equal. Our other two measured branching fractions are consistent with existing results for the corresponding modes involving charged kaons PDG , of ${\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow K^{+}K^{-}\eta)=(0.85\pm 0.14)\times 10^{-3}$ and ${\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow K^{+}K^{-}\pi^{0}\pi^{0})=(2.35\pm 0.41)\times 10^{-3}$, respectively.

There are no previous observations of $\psi{(2S)}$ decays into any of these modes. Our measurements indicate the presence of the $\psi{(2S)}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$ and $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$ decay modes at just over three and two standard deviations, respectively, and we give in Table  5 an upper limit at the 90% C.L on the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ mode.

VIII.1 Internal structure of the

${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$, $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$, and $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$ decays

We consider $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ events in the charmonium region with a $K^{0}_{\scriptscriptstyle S}\pi^{0}$ or $K^{0}_{\scriptscriptstyle L}\pi^{0}$ invariant mass within 0.15 ${\mathrm{\,Ge\kern-1.00006ptV\!/}c^{2}}$ of the nominal $K^{*}(892)^{0}$ or $K^{*}_{2}(1430)^{0}$ mass, and show their $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ invariant mass distributions in Figs. 17(a) and 17(b), respectively. Fits using simulated ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ signal shapes and polynomial backgrounds yield $106\pm 13$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow(K^{*}(892)^{0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}+c.c.)\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ events and $37\pm 11$ ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow(K^{*}_{2}(1430)^{0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}+c.c.)\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ events. For each of these intermediate states we calculate the product of its ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ branching fraction, $\Gamma^{J/\psi}_{ee}$, and the relevant branching fractions for the intermediate resonances, and list the values in Table 5. Using $\Gamma^{J/\psi}_{ee}=5.55$ $\mathrm{\,e\kern-1.00006ptV}$ PDG we calculate the corresponding products of branching fractions.

This first measurement of ${\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow(K^{*}_{2}(1430)^{0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}+c.c.)\rightarrow K^{0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}\pi^{0})$ is consistent with the existing upper limit of $4\times 10^{-3}$ PDG . According to isospin relations, the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow(K^{*}(892)^{0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}+c.c.)\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ decay rate should be the same as the existing world average of $1.97\pm 0.20\times 10^{-3}$ PDG for the charged-kaon decay chain ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow(K^{*}(892)^{+}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{-}+c.c.)\rightarrow K^{+}K^{-}\pi^{0}$, and a factor of two lower than the $3.2\pm 0.4\times 10^{-3}$ PDG observed in the $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}K^{+}\pi^{-}+c.c.$ final state. Our result is consistent with the latter expectation, and 2.5 standard deviations below the former. In Sec. VI.3 we noted that the $\phi\eta$ contribution to the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ signal in the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$ mode is very small. We estimate $5\pm 3$ events, corresponding to ${\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\phi\eta)=(0.52\pm 0.32)\times 10^{-3}$, which is consistent with the PDG PDG value of $(0.75\pm 0.08)\times 10^{-3}$. The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ signal in the $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$ mode has high background (see Fig. 16(c)), and we are unable to quantify the contributions from the $K^{*}(892)^{0}K^{0}\pi^{0}$ and $\phi\pi^{0}\pi^{0}$ intermediate states with reasonable accuracy. The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\phi\pi^{0}\pi^{0}$ decay rate is relatively well measured PDG , dominated by our previous measurement in the $K^{+}K^{-}\pi^{0}\pi^{0}$ final state.

IX Summary

We have presented studies of the processes $e^{+}e^{-}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$, $e^{+}e^{-}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$, and $e^{+}e^{-}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$ at center-of-mass energies below 4 $\mathrm{\,Ge\kern-1.00006ptV}$, using events with initial-state radiation collected with the BABAR detector. The cross sections for all three processes are measured for the first time, over the energy range from threshold to 4 $\mathrm{\,Ge\kern-1.00006ptV}$, and their resonant structure is studied.

The $e^{+}e^{-}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$ cross section is measured with 10–30% systematic uncertainty below 3 $\mathrm{\,Ge\kern-1.00006ptV}$, and is similar in shape to the $e^{+}e^{-}\rightarrow K^{+}K^{-}\pi^{0}$ cross section isrkkpi . It is dominated by resonant, quasi-two body intermediate states. The $K^{*}(892)^{0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}+c.c.$ processes account for about 90% of the cross section, and there are few-percent contributions from the $K^{*}(1430)^{0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}+c.c.$ and $\phi\pi^{0}$ processes. The cross section for the latter is consistent with that measured previously in the $K^{+}K^{-}\pi^{0}$ final state.

The $e^{+}e^{-}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$ cross section is measured with 15–30% systematic uncertainty below 3 $\mathrm{\,Ge\kern-1.00006ptV}$, and is similar to the $e^{+}e^{-}\rightarrow K^{+}K^{-}\eta$ cross section isrkkpi . The $\phi\eta$ intermediate state dominates below 2.0 $\mathrm{\,Ge\kern-1.00006ptV}$ and contributes up to 3.0 $\mathrm{\,Ge\kern-1.00006ptV}$, and its cross section is consistent with that measured previously in the $K^{+}K^{-}\eta$ final state. No other intermediate states are observed, and non-resonant $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$ production is substantial in the 2.2–3.0 $\mathrm{\,Ge\kern-1.00006ptV}$ range.

The $e^{+}e^{-}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$ cross section is measured with 25–60% systematic uncertainty below 3 $\mathrm{\,Ge\kern-1.00006ptV}$, and is consistent with the $e^{+}e^{-}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle S}\pi^{+}\pi^{-}$ cross section isrkskl . Its $E_{\rm c.m.}$ behavior is similar in shape to the $e^{+}e^{-}\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}$, $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{+}\pi^{-}$, and $K^{+}K^{-}\pi^{0}\pi^{0}$ cross sections, but factors of about 8, 2, and 1.5 smaller, respectively. There are substantial, but not dominant, contributions from the $K^{*}(892)^{0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}\pi^{0}$, $K^{*}(1430)^{0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}\pi^{0}$ and $\phi(1020)\pi^{0}\pi^{0}$ intermediate states. These are hard to quantify, but are consistent with expectations from our previous studies of the $e^{+}e^{-}\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}$ and $K^{+}K^{-}\pi^{0}\pi^{0}$ processes. We see no evidence for any $K^{*0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ intermediate states, also consistent with the low rates we have observed in final states involving charged kaons.

We observe the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}$, $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$, and $K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$ decays for the first time, and measure the product of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ electronic width and branching fraction to each of these modes. We study the resonant structure of these decays, and obtain measurements of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow K^{*}(892)^{0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}+c.c.$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow K_{2}^{*}(1430)^{0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{0}+c.c.$ branching fractions times $\Gamma^{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}_{ee}$. In addition, we observe the $\psi{(2S)}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\eta$ and $\psi{(2S)}\rightarrow K^{0}_{\scriptscriptstyle S}K^{0}_{\scriptscriptstyle L}\pi^{0}\pi^{0}$ decays for the first time, and measure the products of the $\psi{(2S)}$ electronic width and the corresponding branching fractions.

Acknowledgements

We are grateful for the extraordinary contributions of our PEP-II colleagues in achieving the excellent luminosity and machine conditions that have made this work possible. The success of this project also relies critically on the expertise and dedication of the computing organizations that support BABAR. The collaborating institutions wish to thank SLAC for its support and the kind hospitality extended to them. This work is supported by the US Department of Energy and National Science Foundation, the Natural Sciences and Engineering Research Council (Canada), the Commissariat à l’Energie Atomique and Institut National de Physique Nucléaire et de Physique des Particules (France), the Bundesministerium für Bildung und Forschung and Deutsche Forschungsgemeinschaft (Germany), the Istituto Nazionale di Fisica Nucleare (Italy), the Foundation for Fundamental Research on Matter (The Netherlands), the Research Council of Norway, the Ministry of Education and Science of the Russian Federation, Ministerio de Economía y Competitividad (Spain), the Science and Technology Facilities Council (United Kingdom), and the Binational Science Foundation (U.S.-Israel). Individuals have received support from the Marie-Curie IEF program (European Union) and the A. P. Sloan Foundation (USA).