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

We study the processes $e^{+}{e}^{-}\to K_S^0{K}_L^0\gamma$, $K_S^0{K}_L^0{\pi }^{+}{\pi }^{-}\gamma$, $K_S^0{K}_S^0{\pi }^{+}{\pi }^{-}\gamma$, and $K_S^0{K}_S^0{K}^{+}{K}^{-}\gamma$, where the photon is radiated from the initial state, providing cross section measurements for the hadronic states over a continuum of center-of-mass energies. The results are based on $469{\text{fb}}^{-1}$ of data collected with the BABAR detector at SLAC. We observe the $\varphi (1020)$ resonance in the $K_S^0{K}_L^0$ final state and measure the product of its electronic width and branching fraction with about 3% uncertainty. We present a measurement of the $e^{+}{e}^{-}\to K_S^0{K}_L^0$ cross section in the energy range from 1.06 to 2.2 GeV and observe the production of a resonance at 1.67 GeV. We present the first measurements of the $e^{+}{e}^{-}\to K_S^0{K}_L^0{\pi }^{+}{\pi }^{-}$, $K_S^0{K}_S^0{\pi }^{+}{\pi }^{-}$, and $K_S^0{K}_S^0{K}^{+}{K}^{-}$ cross sections and study the intermediate resonance structures. We obtain the first observations of $J/\psi$ decay to the $K_S^0{K}_L^0{\pi }^{+}{\pi }^{-}$, $K_S^0{K}_S^0{\pi }^{+}{\pi }^{-}$, and $K_S^0{K}_S^0{K}^{+}{K}^{-}$ final states.

Figure 1

The ${\pi }^{+}{\pi }^{-}$ invariant mass distribution for the selected $K_S^0$ candidates for the data (points) and simulation (histogram). The vertical lines indicate the signal region.

Figure 2

The distribution of constrained recoil mass $m_{\text{rec}}^c$, obtained according to Eqs. (1) and (2), for selected $\gamma K_S^0$ candidates. The points represent the data, and the histogram an MC simulation of $e^{+}{e}^{-}\to \gamma \varphi \to \gamma K_S^0{K}_L^0$ events, normalized to the two most populated bins.

Figure 3

The experimental $m_{\text{rec}}^c$ distribution (points) compared with our estimated background contributions from (cumulatively): $K_S^0{K}_L^{0}2{\pi }^0\gamma$ (shaded area), $K_S^0{K}_L^0{\pi }^0\gamma$ (hatched), and $\varphi (K_S^0{K}_L^0)\eta \gamma$ (open histogram). The simulated signal distribution is shown as the dashed histogram and the sum of all simulated events as the solid histogram.

Figure 4

The EMC cluster energy vs the opening angle between the measured cluster direction and the predicted $K_L^0$ direction for all non-ISR clusters in the data.

Figure 5

The difference in azimuthal angle between the EMC cluster direction and the predicted $K_L^0$ direction for selected clusters in the data (points) and MC-simulated $K_S^0{K}_L^0(\varphi )\gamma$ events (histogram).

Figure 6

The three-constraint ${\chi }^2$ distribution for $K_S^0$-cluster-$\gamma$ events in the data (points) fitted under the $K_S^0{K}_L^0\gamma$ hypothesis. The open histogram represents the same distribution for the MC-simulated signal events, normalized to the data in the region ${\chi }^2<10$, and the shaded histogram represents the estimated background in the data.

Figure 7

The $K_S^0{K}_L^0$ invariant mass distribution in the data (points) and signal-MC simulation (histogram) for candidate events in the signal region of Fig. 6. The shaded histogram represents the estimated background, and the line is a smooth parametrization thereof.

Figure 8

The simulated distribution of differences between the reconstructed and generated $K_S^0{K}_L^0$ invariant mass for the 1 MeV bin of reconstructed mass at the $\varphi$ peak.

Figure 9

The simulated detection efficiency $\epsilon (s)$ vs the generated $K_S^0{K}_L^0$ invariant mass, calculated by dividing the number of events in the signal region of Fig. 6 by the number generated in each bin.

Figure 10

The $K_S^0{K}_L^0$ invariant mass distribution in the $\varphi (1020)$ region. Only statistical uncertainties are shown. The curve represents the result of the fit described in the text.

Figure 11

Two-dimensional plot of the higher cluster energy in a photon-candidate pair vs the corresponding diphoton mass $m_{\gamma \gamma }$ for all pairs of EMC clusters, containing neither the ISR photon nor the $K_L^0$ candidate.

Figure 12

The 3C ${\chi }^2$ distributions for $K_S^0{K}_L^0\gamma$ candidate events in the data (points) and signal simulation (open histogram), fitted under the $K_S^0{K}_L^0$ hypothesis. The shaded, cross-hatched, and hatched areas represent the simulated contributions from the ISR $\varphi \eta$, $K_S^0{K}_L^0{\pi }^0$, and $K_S^0{K}_L^0{\pi }^0{\pi }^0$ channels, respectively.

Figure 13

The $K_S^0{K}_L^0$ invariant mass distribution for data (circles) in the ${\chi }^2$ control (a) and signal (b) regions (see Fig. 12). The histogram in (a) represents MC-simulated signal events in the control region, and that in (b) represents the simulated background from $\varphi \eta$, $K_S^0{K}_L^0{\pi }^0$, and $K_S^0{K}_L^{0}2{\pi }^0$ events; The squares show the total estimated background, obtained as the difference between the data and simulated distributions in (a), normalized to the data in the signal region. (c) The $K_S^0{K}_L^0$ invariant mass distribution above $1.0\mathrm{GeV}/c^2$ for the data after background subtraction.

Figure 14

(a) The $K_S^0{K}_L^0$ mass distributions from the MC simulation for the signal (unshaded) and control (shaded) regions of Fig. 12. (b) The mass dependence of the simulated net reconstruction and selection efficiency for the $K_S^0{K}_L^0$ final state in the $m(K_S^0{K}_L^0)>1.06\mathrm{GeV}/c^2$ region.

Figure 15

The $e^{+}{e}^{-}\to K_S^0{K}_L^0$ cross section compared with all available data in this $E_{\text{c.m.}}$ region.

Figure 16

The $e^{+}{e}^{-}\to K_S^0{K}_L^0$ cross section (points) compared with the results of the fits described in the text with the nonresonant amplitude fixed to zero (solid lines) and floating (dashed lines).

Figure 17

Comparison of the $e^{+}{e}^{-}\to K_S^0{K}_L^0$ cross section (points) with that for $e^{+}{e}^{-}\to K^{+}{K}^{-}$ [8] (crosses).

Figure 18

(a) The three-constraint ${\chi }^2$ distributions for data (points) and MC-simulated $K_S^0{K}_L^0{\pi }^{+}{\pi }^{-}\gamma$ events (open histogram). The shaded, cross-hatched, and hatched histograms represent the simulated contributions from ISR $\varphi \eta$, ISR $K_S^{0}K\pi {\pi }^0$, and non-ISR $q\overline{q}$ events, respectively. (b) The $K_S^0{K}_L^0{\pi }^{+}{\pi }^{-}$ invariant mass distribution for data events in the signal region of (a) (points). The shaded and cross-hatched histograms represent the simulated contributions from ISR $\varphi \eta$+$K_S^{0}K\pi {\pi }^0$ and non-ISR $q\overline{q}$ events, respectively, and the hatched area represents that estimated from the control region. The curve shows the empirical fit used for background subtraction. (c) The $K_S^0{K}_L^0{\pi }^{+}{\pi }^{-}$ invariant mass distribution after background subtraction.

Figure 19

(a) The $K_S^0{K}_L^0{\pi }^{+}{\pi }^{-}$ invariant mass distribution for MC-simulated signal events in the signal (open histogram) and control region (shaded) of Fig. 18. (b) The net reconstruction efficiency from the simulation.

Figure 20

The $e^{+}{e}^{-}\to K_S^0{K}_L^0{\pi }^{+}{\pi }^{-}$ cross section. The error bars are statistical only.

Figure 21

The $K_L^0{\pi }^{\pm }$ invariant mass vs the $K_S^0{\pi }^{\mp }$ invariant mass (two entries per event).

Figure 22

The (a) $K_S^0{\pi }^{\pm }$ and (b) $K_L^0{\pi }^{\pm }$ mass projections of Fig. 21. The curves represent the results of the fits described in the text, with the hatched areas representing the nonresonant components.

Figure 23

The number of $K^{*}(892{)}^{\pm }$ events obtained from fits to the $K_S^0{\pi }^{\pm }$ invariant mass distribution in each $0.04\mathrm{GeV}/c^2$ interval of $K_L^0{\pi }^{\mp }$ mass. The curve represents the result of the fit described in the text, with the hatched areas representing the nonresonant component.

Figure 24

(a) The $K_S^0{K}_L^0$ invariant mass distribution for the selected $K_S^0{K}_L^0{\pi }^{+}{\pi }^{-}$ events. The solid and dashed lines represent the result of the fit described in the text and its non-$\varphi$ component, respectively. (b) The ${\pi }^{+}{\pi }^{-}$ invariant mass distribution for events in the $\varphi$ peak (see the text).

Figure 25

The four-constraint ${\chi }^2$ distributions for $K_S^0{K}_S^0{\pi }^{+}{\pi }^{-}\gamma$ candidate events selected in the data (points) and signal-MC simulation (open histogram) fitted under the $K_S^0{K}_S^0{\pi }^{+}{\pi }^{-}$ hypothesis. The cross-hatched histogram represents the simulated background contribution from non-ISR $q\overline{q}$ events.

Figure 26

Scatter plot of the ${\pi }^{+}{\pi }^{-}$ invariant mass of one $K_S^0$ candidate vs that of the other $K_S^0$ candidate calculated using the results of the constrained fit.

Figure 27

The $K_S^0{K}_S^0{\pi }^{+}{\pi }^{-}$ invariant mass distribution (points) for events in the signal region of Fig. 25. The cross-hatched and hatched histograms represent the backgrounds from non-ISR $q\overline{q}$ events and others estimated from the ${\chi }^2$ control region of Fig. 25, respectively. The curve represents the smooth empirical fit to the total background used for subtraction.

Figure 28

(a) The $K_S^0{K}_S^0{\pi }^{+}{\pi }^{-}$ invariant mass distribution for the MC-simulated signal events in the signal and control (shaded) regions of Fig. 25. (b) The net reconstruction and selection efficiency from the simulation.

Figure 29

The $e^{+}{e}^{-}\to K_S^0{K}_S^0{\pi }^{+}{\pi }^{-}$ cross section.

Figure 30

The $K_S^0{\pi }^{-}$ invariant mass vs the $K_S^0{\pi }^{+}$ invariant mass (two entries per event).

Figure 31

The (a) $m(K_S^0{\pi }^{+})$ and (b) $m(K_S^0{\pi }^{-})$ projections of Fig. 30. The lines and hatched areas represent the results of the fits described in the text and their non-$K^{*}$ components, respectively.

Figure 32

The fitted number of $K^{*}(892{)}^{+}$ events in each $0.04\mathrm{GeV}/c^2$ interval of the $K_S^0{\pi }^{-}$ mass. The curve represents the result of the fit described in the text, with the hatched area representing the nonresonant component.

Figure 33

The (a) ${\pi }^{+}{\pi }^{-}$ and (b) $K_S^0{K}_S^0$ invariant mass distributions for the selected $K_S^0{K}_S^0{\pi }^{+}{\pi }^{-}$ events with $K^{*}(892{)}^{+}{K}^{*}(892{)}^{-}$ events excluded (see the text).

Figure 34

The four-constraint ${\chi }^2$ distributions for $K_S^0{K}_S^0{K}^{+}{K}^{-}\gamma$ candidate events in the data (points) and signal MC simulation (open histogram) fitted under the $K_S^0{K}_S^0{K}^{+}{K}^{-}$ hypothesis. The cross-hatched histogram represents the simulated contribution from non-ISR $q\overline{q}$ events.

Figure 35

The $K_S^0{K}_S^0{K}^{+}{K}^{-}$ invariant mass distribution for data events in the signal region, ${\chi }^2(K_S^0{K}_S^0{K}^{+}{K}^{-})<40$ (open histogram). The subset of events with $m(K^{+}{K}^{-})<1.04\mathrm{GeV}/c^2$, predominantly $K_S^0{K}_S^0\varphi (1020)$ events, is shown as the shaded histogram.

Figure 36

(a) The $K_S^0{K}_S^0{K}^{+}{K}^{-}$ invariant mass distribution for the MC-simulated signal events in the signal and control (shaded) regions of Fig. 34. (b) The net reconstruction and selection efficiency from the simulation.

Figure 37

The $e^{+}{e}^{-}\to K_S^0{K}_S^0{K}^{+}{K}^{-}$ cross section. Events with invariant mass within $0.05\mathrm{GeV}/c^2$ of the $J/\psi$ mass are excluded.

Figure 38

(a) The $K^{+}{K}^{-}$ vs $K_S^0{K}_S^0$ invariant mass for all selected $K_S^0{K}_S^0{K}^{+}{K}^{-}$ events in the data. (b) The $m(K_S^0{K}_S^0)$ projection of (a). (c) An expanded view of (b) in which the line represents the result of the fit described in the text.

Figure 39

Expanded views of the invariant mass distributions near the $J/\psi$ mass for the (a) $K_S^0{K}_L^0{\pi }^{+}{\pi }^{-}$, (b) $K_S^0{K}_S^0{\pi }^{+}{\pi }^{-}$, and (c) $K_S^0{K}_S^0{K}^{+}{K}^{-}$ final states. The lines represent the results of the fits described in the text.

Figure 40

(a) The $K_S^0{\pi }^{\pm }$ invariant mass distribution (four entries per event) for the $K_S^0{K}_S^0{\pi }^{+}{\pi }^{-}$ events under the $J/\psi$ peak with the non-$J/\psi$ contribution subtracted (see the text). (b) The distribution of the other $m(K_S^0{\pi }^{\mp })$ for those events with one $m(K_S^0{\pi }^{\pm })$ value within $0.15\mathrm{GeV}/c^2$ of the $K^{*}(892{)}^{\pm }$ mass (up to two entries per event, pairings in the overlap region taken once). The lines represent the results of the fits described in the text, with the shaded areas representing the combinatorial components.

Figure 41

(a) The $m({\pi }^{+}{\pi }^{-})$ distribution for the $J/\psi \to K_S^0{K}_S^0{\pi }^{+}{\pi }^{-}$ events. (b) The $m(K_S^0{K}_S^0)$ distribution for those events in the $\rho (770)$ region, $0.6<m({\pi }^{+}{\pi }^{-})<1.0\mathrm{GeV}/c^2$.

Figure 42

(a) The $K^{+}{K}^{-}$ vs $K_S^0{K}_S^0$ invariant mass for the $K_S^0{K}_S^0{K}^{+}{K}^{-}$ events under the $J/\psi$ peak (see the text). (b) The $K_S^0{K}_S^0$ invariant mass distribution for the events in (a) with $m(K^{+}{K}^{-})<1.04\mathrm{GeV}/c^2$. The line represents the result of the fit described in the text.