$e^{+}{e}^{-}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$, $K^{+}{K}^{-}{\pi }^0{\pi }^0$ and $K^{+}{K}^{-}{K}^{+}{K}^{-}$ cross sections measured with initial-state radiation

We study the processes $e^{+}{e}^{-}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}\gamma$, $K^{+}{K}^{-}{\pi }^0{\pi }^0\gamma$ and $K^{+}{K}^{-}{K}^{+}{K}^{-}\gamma$, where the photon is radiated from the initial state. About 34 600, 4400 and 2300 fully reconstructed events, respectively, are selected from $232{\mathrm{fb}}^{-1}$ of BABAR data. The invariant mass of the hadronic final state defines the effective $e^{+}{e}^{-}$ center-of-mass energy, so that the $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}\gamma$ data can be compared with direct measurements of the $e^{+}{e}^{-}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ reaction; no direct measurements exist for the $e^{+}{e}^{-}\to K^{+}{K}^{-}{\pi }^0{\pi }^0$ or $e^{+}{e}^{-}\to K^{+}{K}^{-}{K}^{+}{K}^{-}$ reactions. Studying the structure of these events, we find contributions from a number of intermediate states, and we extract their cross sections where possible. In particular, we isolate the contribution from $e^{+}{e}^{-}\to \varphi (1020)f_0(980)$ and study its structure near threshold. In the charmonium region, we observe the $J/\psi$ in all three final states and several intermediate states, as well as the $\psi (2S)$ in some modes, and measure the corresponding branching fractions. We see no signal for the $Y(4260)$ and obtain an upper limit of ${\mathcal{B}}_{Y(4260)\to \varphi {\pi }^{+}{\pi }^{-}}·{\Gamma }_{ee}^Y<0.4\mathrm{eV}$ at 90% C.L.

Figure 1

Distribution of ${\chi }^2$ from the three-constraint fit for $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ candidates in the data (points). The open histogram is the distribution for simulated signal events, normalized as described in the text. The cross-hatched (hatched) histogram represents the background from non-ISR events (plus that from ISR $4\pi$ events), estimated as described in the text.

Figure 2

The invariant-mass distribution for $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ candidates in the data (points): the cross-hatched, hatched and open histograms represent, cumulatively, the non-ISR background, the contribution from ISR ${\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}$ events, and the ISR background from the control region of Fig. 1.

Figure 3

(a) The invariant-mass distributions for simulated $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ events in the phase space model, reconstructed in the signal (open) and control (hatched) regions of Fig. 1; (b) net reconstruction and selection efficiency as a function of mass obtained from this simulation (the curve represents a 3rd order polynomial fit).

Figure 4

The $e^{+}{e}^{-}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ cross section as a function of the effective $e^{+}{e}^{-}$ c.m. energy measured with ISR data at BABAR (dots). The direct measurements from DM1 [13] are shown as the open circles. Only statistical errors are shown.

Figure 5

(a) Invariant mass of the $K^{-}{\pi }^{+}$ pair versus that of the $K^{+}{\pi }^{-}$ pair; (b) sum of projections of (a); (c) sum of projections of the $K^{*0}$ bands of (a), with events in the overlap region taken only once. The $\varphi {\pi }^{+}{\pi }^{-}$ and $KK\rho$ are vetoed.

Figure 6

(a) The $K^{*0}\pi$ invariant-mass distribution; (b) the $K^{*0}\pi$ mass versus $K^{*0}K$ mass.

Figure 7

(a) Invariant mass of the $K^{\pm }{\pi }^{+}{\pi }^{-}$ combinations versus that of the ${\pi }^{+}{\pi }^{-}$ pair; (b) the $K^{\pm }{\pi }^{+}{\pi }^{-}$ and (c) ${\pi }^{+}{\pi }^{-}$ mass projections of (a).

Figure 8

(a) The $K^{\pm }{\pi }^{\mp }$ mass distribution (two entries per event) for all selected $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ events: the solid line represents a fit including two resonances and a polynomial function (see text), shown separately as the dashed line; (b) the $e^{+}{e}^{-}\to K^{*0}K\pi$ cross section obtained from the $K^{*0}(892)$ signal by a similar fit in each $25\mathrm{MeV}/c^2$ mass bin.

Figure 9

(a) The ${\pi }^{+}{\pi }^{-}$ vs the $K^{+}{K}^{-}$ invariant masses for all selected $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ events; (b) the ${\pi }^{+}{\pi }^{-}$ invariant-mass projections for events in the $\varphi$ peak (open histogram), sidebands (hatched) and background control region (cross-hatched); (c) the $K^{+}{K}^{-}$ mass projections for all events (open) and control region (cross-hatched); (d) the difference between the open and the sum of the other histograms in (b).

Figure 10

(a) The $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ invariant-mass distributions from the $\varphi {\pi }^{+}{\pi }^{-}$ simulation described in the text, reconstructed in the signal (open) and control (hatched) regions; (b) net reconstruction and selection efficiency as a function of mass: the solid line represents a cubic fit, and the dashed line the corresponding fit for the space phase model shown in Fig. 3.

Figure 11

The $e^{+}{e}^{-}\to \varphi {\pi }^{+}{\pi }^{-}$ cross section as a function of the effective $e^{+}{e}^{-}$ c.m. energy.

Figure 12

Distributions of the cosines of (a) the $\varphi$ production angle, (b) the pion helicity angle, and (c) the kaon helicity angle (see text) for $e^{+}{e}^{-}\to \varphi {\pi }^{+}{\pi }^{-}$ events: the lines represent the distributions expected if the ${\pi }^{+}{\pi }^{-}$ system recoiling against a vector $\varphi$ meson is produced in an $S$-wave, normalized to the number of events in the data.

Figure 13

The $e^{+}{e}^{-}\to \varphi (1020)f_0(980)$ cross section as a function of the effective $e^{+}{e}^{-}$ c.m. energy obtained from the $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ state.

Figure 14

Distribution of ${\chi }^2$ from the six-constraint fit for $K^{+}{K}^{-}{\pi }^0{\pi }^0$ candidates in the data (points). The open histogram is the distribution for simulated signal events, normalized as described in the text. The cross-hatched, hatched and dashed histograms represent, cumulatively, the backgrounds from non-ISR events, ISR ${\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$ events, and ISR $K^{+}{K}^{-}{\pi }^0$, $K^{+}{K}^{-}\eta$ and $K^{+}{K}^{-}{\pi }^0{\pi }^0{\pi }^0$ events.

Figure 15

Invariant-mass distribution for $K^{+}{K}^{-}{\pi }^0{\pi }^0$ candidates in the data (points). The cross-hatched, hatched and open histograms represent, cumulatively, the non-ISR background, the contribution from ISR ${\pi }^{+}{\pi }^{-}{\pi }^0{\pi }^0$ events, and the ISR background from the control region of Fig. 14.

Figure 16

(a) Invariant-mass distribution for simulated $K^{+}{K}^{-}{\pi }^0{\pi }^0$ events in the signal (open) and control (hatched) regions (see Fig. 14); (b) net reconstruction and selection efficiency as a function of mass obtained from this simulation (the curve represents a third-order polynomial fit).

Figure 17

The $e^{+}{e}^{-}\to K^{+}{K}^{-}{\pi }^0{\pi }^0$ cross section as a function of the effective $e^{+}{e}^{-}$ c.m. energy measured with ISR data at BABAR. The errors are statistical only.

Figure 18

(a) Invariant mass of the $K^{-}{\pi }^0$ pair versus that of the $K^{+}{\pi }^0$ pair in selected $K^{+}{K}^{-}{\pi }^0{\pi }^0$ events (two entries per event); (b) sum of projections of (a) (dots, four entries per event). The curve represents the result of the fit described in the text. The hatched histogram is the $K^{\pm }{\pi }^0$ distribution for events in which the other $K^{\mp }{\pi }^0$ combination is within the $K^{*\pm }(892)$ bands indicated in (a), with events in the overlap region taken only once.

Figure 19

(a) Scatter plot of the ${\pi }^0{\pi }^0$ invariant mass vs the $K^{+}{K}^{-}$ invariant mass for all selected $K^{+}{K}^{-}{\pi }^0{\pi }^0$ events; (b) the ${\pi }^0{\pi }^0$ invariant-mass projections for events in the $\varphi$ peak (open histogram), sidebands (cross-hatched) and control region (hatched); (c) the $K^{+}{K}^{-}$ mass projection for events in the signal (open) and control (hatched) regions; (d) difference between the open and other histograms in (b).

Figure 20

Cross section for the reaction $e^{+}{e}^{-}\to \varphi (1020)f_0(980)$ as a function of effective $e^{+}{e}^{-}$ c.m. energy obtained from the $K^{+}{K}^{-}{\pi }^0{\pi }^0$ final state.

Figure 21

Distribution of ${\chi }^2$ from the three-constraint fit for $K^{+}{K}^{-}{K}^{+}{K}^{-}$ candidates in the data (points). The open histogram is the distribution for simulated signal events, normalized as described in the text. The hatched histogram represents the background from non-ISR events, estimated as described in the text. The cross-hatched histogram is for simulated ISR $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ events.

Figure 22

Invariant-mass distribution for $K^{+}{K}^{-}{K}^{+}{K}^{-}$ candidates in the data (points). The hatched and open histograms represent, cumulatively, the non-ISR background and the ISR background from the control region, which is dominated by the contribution from ISR $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ events.

Figure 23

(a) Invariant-mass distributions for simulated $K^{+}{K}^{-}{K}^{+}{K}^{-}$ events in the signal (open) and control (hatched) regions (see Fig. 21); (b) net reconstruction and selection efficiency as a function of mass obtained from this simulation (the curve represents a 3rd order polynomial fit).

Figure 24

The $e^{+}{e}^{-}\to K^{+}{K}^{-}{K}^{+}{K}^{-}$ cross section as a function of the effective $e^{+}{e}^{-}$ c.m. energy measured with ISR data at BABAR. The errors are statistical only.

Figure 25

Invariant-mass distributions for all $K^{+}{K}^{-}$ pairs in selected $e^{+}{e}^{-}\to K^{+}{K}^{-}{K}^{+}{K}^{-}$ events (open histogram) and for the combination in each event closest to the $\varphi$-meson mass (hatched).

Figure 26

Invariant-mass distribution for $K^{+}{K}^{-}$ pairs in events in which the other $K^{+}{K}^{-}$ pair has mass closest to and within $10\mathrm{MeV}/c^2$ of the nominal $\varphi$ mass (open histogram). The hatched histogram is for the subset with a $K^{+}{K}^{-}{K}^{+}{K}^{-}$ mass in the $J/\psi$ peak.

Figure 27

The $e^{+}{e}^{-}\to \varphi (1020)f_0(980)$ cross section measured in the $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ (circles) and $K^{+}{K}^{-}{\pi }^0{\pi }^0$ (squares) final states. The hatched histogram shows the simulated cross section, assuming no resonant structure. The solid (dashed) line represents the result of the one-resonance (no-resonance) fit described in the text.

Figure 28

Invariant-mass distribution for all selected $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ events lying outside the $K^{*0}(892)$ bands of Fig. 5 (points), and the subset of these events with $0.85<m({\pi }^{+}{\pi }^{-})<1.10\mathrm{GeV}/c^2$ (hatched).

Figure 29

Invariant-mass distribution for all selected $K^{+}{K}^{-}{\pi }^0{\pi }^0$ events lying outside the $K^{*0}(892)$ bands of Fig. 18 (points), and the subset of these events with $0.85<m({\pi }^0{\pi }^0)<1.10\mathrm{GeV}/c^2$ (hatched).

Figure 30

The $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ invariant-mass distribution for all selected events (open histogram), and for those with a ${\pi }^{+}{\pi }^{-}$ mass inside (cross-hatched) or outside (hatched) the $f_0$ band as defined in the text.

Figure 31

The $K^{+}{K}^{-}{\pi }^0{\pi }^0$ invariant-mass distribution for all selected events (open histogram), and for those with a ${\pi }^0{\pi }^0$ mass inside (cross-hatched) or outside (hatched) the $f_0$ band as defined in the text.

Figure 32

The $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ invariant-mass distribution in the $K^{+}{K}^{-}{f}_0(980)$ threshold region for events with a ${\pi }^{+}{\pi }^{-}$ mass inside the $f_0$ band. The lines represent the results of the fits including no (dashed), one (solid) and two (dotted) resonances described in the text.

Figure 33

The $K^{+}{K}^{-}{\pi }^0{\pi }^0$ invariant-mass distribution in the $K^{+}{K}^{-}{f}_0(980)$ threshold region for events with a ${\pi }^0{\pi }^0$ mass inside the $f_0$ band. The lines represent the results of the fits including no (dashed), one (solid) and two (dotted) resonances described in the text.

Figure 34

Raw invariant-mass distribution for all selected $e^{+}{e}^{-}\to K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ events in the charmonium region. The line represents the result of the fit described in the text.

Figure 35

Invariant-mass distribution for $e^{+}{e}^{-}\to K^{+}{K}^{-}{\pi }^0{\pi }^0$ events in the charmonium region, after partial background subtraction. The line represents the result of the fit described in the text.

Figure 36

Raw invariant-mass distribution for all selected $e^{+}{e}^{-}\to K^{+}{K}^{-}{K}^{+}{K}^{-}$ events in the charmonium region. The line represents the result of the fit described in the text.

Figure 37

The $K^{\pm }{\pi }^{\mp }$ invariant-mass versus $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ invariant-mass for events with the other $K^{\mp }{\pi }^{\pm }$ combination in the $K^{*0}(892)$ bands of Fig. 5a. The overlapped region is taken only once.

Figure 38

The $K^{\pm }{\pi }^{\mp }$ mass projection from Fig. 37 for events with a $K^{+}{K}^{-}{\pi }^{+}{\pi }^{-}$ mass within $50\mathrm{MeV}/c^2$ of the $J/\psi$ mass (open histogram) and $50–100\mathrm{MeV}/c^2$ below (hatched).

Figure 39

Raw invariant-mass distributions for candidate $e^{+}{e}^{-}\to \varphi {\pi }^{+}{\pi }^{-}$ events (open histogram) and events in the $\varphi$ sidebands of Fig. 9c (cross-hatched) in the charmonium region. The vertical lines indicate the region used for the $Y(4260)$ search.

Figure 40

Raw invariant-mass distributions for candidate $e^{+}{e}^{-}\to \varphi {\pi }^0{\pi }^0$ events (open histogram) and events in the ${\chi }_{KK{\pi }^0{\pi }^0}^2$ control region (hatched) in the charmonium region.

Figure 41

(a) Raw invariant-mass distribution for candidate $\varphi f_0$, $f_0\to {\pi }^{+}{\pi }^{-}$ events (open histogram) and events in the $\varphi$ sidebands (cross-hatched) in the charmonium region; (b) the ${\pi }^{+}{\pi }^{-}$ invariant-mass distribution for $\varphi {\pi }^{+}{\pi }^{-}$ events from the $J/\psi$ peak of Fig. 39. The line represents the result of the fit described in the text.

Figure 42

Raw invariant-mass distribution for candidate $\varphi f_0$, $f_0\to {\pi }^0{\pi }^0$ events (open histogram) and events in the ${\chi }_{KK{\pi }^0{\pi }^0}^2$ control region (hatched) in the charmonium region.