$e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$, $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$, and $K^{+}{K}^{-}{K}^{+}{K}^{-}$ cross sections at center-of-mass energies 0.5–4.5 GeV measured with initial-state radiation

We study the process $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}\gamma$, with a hard photon radiated from the initial state. About 60 000 fully reconstructed events have been selected from $89{\mathrm{f}\mathrm{b}}^{-1}$ of BABAR data. The invariant mass of the hadronic final state defines the effective $e^{+}{e}^{-}$ $e^{+}{e}^{-}$ center-of-mass energy, so that these data can be compared with the corresponding direct $e^{+}{e}^{-}$ measurements. From the $4\pi$ -mass spectrum, the cross section for the process $e^{+}{e}^{-}$ $\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ is measured for center-of-mass energies from 0.6 to 4.5 $\mathrm{G}\mathrm{e}\mathrm{V}$. The uncertainty in the cross section measurement is typically 5%. We also measure the cross sections for the final states $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ and $K^{+}{K}^{-}{K}^{+}{K}^{-}$. We observe the $J/\psi$ in all three final states and measure the corresponding branching fractions. We search for $X(3872)$ in $J/\psi (\to {\mu }^{+}{\mu }^{-}){\pi }^{+}{\pi }^{-}$ and obtain an upper limit on the product of the $e^{+}{e}^{-}$ width of the $X(3872)$ and the branching fraction for $X(3872)\to J/\psi {\pi }^{+}{\pi }^{-}$.

Figure 1

The one-constraint ${\chi }^2$ distributions for data (upper histogram) and MC simulation (shaded histogram) four-charged-track events fitted to the four-pion hypothesis. The crosshatched histogram is the estimated background contribution from non-ISR events obtained from JETSET. The signal and control regions are indicated.

Figure 2

The four-pion invariant-mass distribution for the signal region of Fig. 1. The points indicate the background estimated from the difference between data and MC events for the control region of Fig. 1, normalized to the difference between data and MC events in the signal region of Fig. 1. The crosshatched histogram corresponds to the non-ISR background of Fig. 1.

Figure 3

(a) The four-pion invariant-mass distribution obtained from fits to three-charged-track events in which an observed photon is used to constrain the unobserved fourth pion; ISR and non-ISR background contributions are indicated by the points with error bars and the crosshatched histogram, respectively. (b) The four-pion mass dependence of the ratio between the three- and four-charged-track distributions for data (open circles) and MC simulation (solid squares).

Figure 4

(a) The four-pion mass distribution from MC simulation for the signal and control (shaded) regions of Fig. 1. (b) The mass dependence of the net reconstruction and selection efficiency obtained from simulation.

Figure 5

(a) The distribution in track-pair opening angle for the minimum of the six values possible for each event; (b) the distribution in polar angle; and (c) the transverse momentum distribution for all pions from all events. All quantities are in the laboratory frame; the points are for data and the histograms are obtained from MC simulation.

Figure 6

The angular distribution of the lowest-momentum pion (left) and of the three most energetic pions (right) in the four-pion rest frame with respect to the direction of the four-pion system in the laboratory frame for the five regions of four-pion mass indicated. The points are data, and the histograms are MC simulation.

Figure 7

The integrated ISR luminosity per $0.1\mathrm{G}\mathrm{e}\mathrm{V}$ in effective c.m. energy obtained from $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}\gamma$ events. The curve represents the fit used in the luminosity calculations. The point at the $\rho (770)$ mass position was excluded from the fit because of pion misidentification feedthrough from the ${\pi }^{+}{\pi }^{-}\gamma$ final state. The points at the $J/\psi$ mass position were also excluded because of ISR production of $J/\psi$ followed by decay to ${\mu }^{+}{\mu }^{-}$.

Figure 8

The $e^{+}{e}^{-}\mathrm{c}.\mathrm{m}.$ energy dependence of the ${\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ cross section measured with ISR data. The events due to $\psi (2S)\to {\pi }^{+}{\pi }^{-}J/\psi$ with $J/\psi \to {\mu }^{+}{\mu }^{-}$ (see Fig. 2) have been removed. Only statistical errors are shown.

Figure 9

The energy dependence of the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ cross section obtained with BABAR ISR data (black points) in comparison with that from direct $e^{+}{e}^{-}$ production measurements. Only statistical errors are shown.

Figure 10

A detailed view of the energy dependence of the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ cross section near threshold. All experimental points are indicated as in Fig. 9. Only statistical errors are shown.

Figure 11

The two-pion (bottom) and three-pion (top) vs four-pion invariant-mass distributions for data (left) and simulation (right); signals corresponding to $J/\psi$ and $\psi (2S)$ production are present in the data, but are not included in the simulation. Otherwise, agreement between data and MC simulation is quite good.

Figure 12

The three-pion (left) and ${\pi }^{+}{\pi }^{-}$ (center) mass distributions; the points represent data, the histograms simulation, and there are four entries per event. Five regions of four-pion mass are indicated in the center plots. For events having a ${\pi }^{+}{\pi }^{-}$ mass combination in the $\rho (770)$ region, the distribution of the highest mass other ${\pi }^{+}{\pi }^{-}$ pair is shown in the plots to the right.

Figure 13

The one-constraint ${\chi }^2$ distributions for the four-charged-track data events and $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ Monte Carlo events (shaded histogram) fitted to the $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ hypothesis. At least one kaon must be identified. The crosshatched histogram is the estimated background contribution from four-charged-pion ISR events. The signal and control regions are indicated.

Figure 14

(a) The $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ mass distribution from simulation for the signal and control (shaded) regions of Fig. 13; (b) the mass dependence of the net reconstruction and selection efficiency obtained from simulation.

Figure 15

The $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ invariant-mass distribution for the signal region of Fig. 13. The points indicate the background estimated from the difference between data and MC simulation for the control region of Fig. 13 normalized to the difference between the number of data and MC events in the signal region of Fig. 13. The shaded histogram corresponds to the non-ISR background estimated using JETSET.

Figure 16

The c.m. energy dependence of the $e^{+}{e}^{-}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ cross section obtained from ISR events at BABAR compared with the only direct $e^{+}{e}^{-}$ measurement, by DM1. Only statistical errors are shown.

Figure 17

For the $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ data sample: (a) the scatter plot of the $K^{+}{\pi }^{-}$ and $K^{-}{\pi }^{+}$ invariant-mass values; (b) the scatter plot of the $K^{+}{\pi }^{+}$ and $K^{-}{\pi }^{-}$ invariant-mass values; (c) the $K^{+}{\pi }^{-}$ or $K^{-}{\pi }^{+}$ mass projection of (a); (d) the mass distribution for other $K\pi$ combinations for events in the $K^{*0}(892)$ bands of (a).

Figure 18

For the $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ data sample, after removing events contributing to the $K^{*0}(892)$ mass regions indicated in Fig. 17: (a) the scatter plot of ${\pi }^{+}{\pi }^{-}$ and $K^{+}{K}^{-}$ mass values; (b) the ${\pi }^{+}{\pi }^{-}$ mass projection of (a); (c) the $K^{+}{K}^{-}$ mass projection of (a); (d) the ${\pi }^{+}{\pi }^{-}$ mass projection of (a) for events in the $\varphi (1020)$ mass peak.

Figure 19

The $2K2\pi$ events from the $K^{*0}(892)$ bands in Fig. 17: (a) the $K^{*0}(892)K^{\pm }$ vs $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ invariant-mass distribution scatter plot; (b) the $K^{*0}(892)K^{\pm }$ mass projection; (c) the $K^{*0}(892){\pi }^{\pm }$ mass distribution; (d) the $m(K^{*0}{\pi }^{\mp })$ vs $m(K^{*0}{K}^{\pm })$ scatter plot.

Figure 20

The $2K2\pi$ events with the $K^{*0}(892)$ bands in Fig. 17 excluded: (a) the $K^{+}{K}^{-}{\pi }^{\pm }$ mass distribution with the $K^{+}{K}^{-}$ mass in the $\varphi$ region; (b) the $K^{+}{K}^{-}{\pi }^{\pm }$ mass distribution with $K^{+}{K}^{-}$ in the $\varphi$ mass region excluded; (c) the ${\pi }^{+}{\pi }^{-}$ vs $K^{\pm }{\pi }^{+}{\pi }^{-}$ scatter plot, and (d) the $K^{\pm }{\pi }^{+}{\pi }^{-}$ projection, for the events from (b).

Figure 21

The one-constraint ${\chi }^2$ distributions for four-charged-track data events and $K^{+}{K}^{-}{K}^{+}{K}^{-}$ Monte Carlo events (shaded histogram) fitted to the $K^{+}{K}^{-}{K}^{+}{K}^{-}$ hypothesis. At least three kaons must be identified. The crosshatched histogram is the estimated background contribution from $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ ISR events. The signal and control regions are indicated.

Figure 22

(a) The $K^{+}{K}^{-}{K}^{+}{K}^{-}$ mass distribution from simulation for the signal and control (shaded) regions of Fig. 21; (b) the mass dependence of the net reconstruction and selection efficiency obtained from simulation.

Figure 23

The $K^{+}{K}^{-}{K}^{+}{K}^{-}$ invariant-mass distribution for the signal region of Fig. 21. The points indicate the background estimated from the difference between data and MC simulation for the control region of Fig. 21, normalized to the difference between the number of data and MC events in the signal region of Fig. 21. The shaded histogram corresponds to the non-ISR background estimated from JETSET.

Figure 24

The c.m. energy dependence of the $e^{+}{e}^{-}\to K^{+}{K}^{-}{K}^{+}{K}^{-}$ cross section obtained from ISR events at BABAR. Only statistical errors are shown.

Figure 25

(a) The $K^{+}{K}^{-}$ and (b) the $K^{\pm }{K}^{+}{K}^{-}$ invariant-mass distributions for events from the $K^{+}{K}^{-}{K}^{+}{K}^{-}$ sample.

Figure 26

The ${\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ mass distribution for ISR-produced $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ events in the $J/\psi –\psi (2S)$ region; there are clear signals at the $J/\psi$ and $\psi (2S)$ mass positions. The shaded region at the latter corresponds to $\psi (2S)\to J/\psi {\pi }^{+}{\pi }^{-}$, with $J/\psi \to {\mu }^{+}{\mu }^{-}$ and the muons are treated as pions.

Figure 27

The $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ (top) and $K^{+}{K}^{-}{K}^{+}{K}^{-}$ (bottom) mass distributions for ISR-produced events in the $J/\psi$ region. A clear signal is observed at the $J/\psi$ mass.

Figure 28

The $m(2\mu 2\pi )-m(2\mu )$ difference for events with $m(2\mu )$ near the $J/\psi$ mass and both particles identified as muons. The lines show the $\psi (2S)$ and $X(3872)$ selection regions.