Study of the $e^{+}{e}^{-}\to K^{+}{K}^{-}$ reaction in the energy range from 2.6 to 8.0 GeV

The $e^{+}{e}^{-}\to K^{+}{K}^{-}$ cross section and charged-kaon electromagnetic form factor are measured in the $e^{+}{e}^{-}$ center-of-mass energy range ($E$) from 2.6 to 8.0 GeV using the initial-state radiation technique with an undetected photon. The study is performed using $469{\mathrm{fb}}^{-1}$ of data collected with the BABAR detector at the PEP-II2 $e^{+}{e}^{-}$ collider at center-of-mass energies near 10.6 GeV. The form factor is found to decrease with energy faster than $1/E^2$ and approaches the asymptotic QCD prediction. Production of the $K^{+}{K}^{-}$ final state through the $J/\psi$ and $\psi (2S)$ intermediate states is observed. The results for the kaon form factor are used together with data from other experiments to perform a model-independent determination of the relative phases between electromagnetic (single-photon) and strong amplitudes in $J/\psi$ and $\psi (2S)\to K^{+}{K}^{-}$ decays. The values of the branching fractions measured in the reaction $e^{+}{e}^{-}\to K^{+}{K}^{-}$ are shifted relative to their true values due to interference between resonant and nonresonant amplitudes. The values of these shifts are determined to be about $\pm 5\%$ for the $J/\psi$ meson and $\pm 15\%$ for the $\psi (2S)$ meson.

Figure 1

The $p_{\mathrm{T},K^{+}{K}^{-}}$ distribution for simulated $e^{+}{e}^{-}\to K^{+}{K}^{-}\gamma$ events. The arrow indicates $p_{\mathrm{T},K^{+}{K}^{-}}=0.15\mathrm{GeV}/c$.

Figure 2

The $M_{\mathrm{miss}}^2$ distribution for simulated $e^{+}{e}^{-}\to K^{+}{K}^{-}\gamma$ events. The arrows indicate $|M_{\mathrm{miss}}^2|=1{\mathrm{GeV}}^2/c^4$.

Figure 3

The $K^{+}{K}^{-}$ mass spectrum for selected $K^{+}{K}^{-}\gamma$ candidates. The peaks near 3.1, 3.4, and $3.7\mathrm{GeV}/c^2$ are from $J/\psi$, ${\chi }_{c0}$, and $\psi (2S)$ decays to $K^{+}{K}^{-}$, respectively. The shaded histogram shows events with at least one identified muon candidate.

Figure 4

The $M_{\mathrm{miss}}^2$ distributions for data (points with error bars) in three $M_{K^{+}{K}^{-}}$ intervals: (a) $5.5–6.0\mathrm{GeV}/c^2$, (b) $6.5–7.0\mathrm{GeV}/c^2$, and (c) $7.5–8.0\mathrm{GeV}/c^2$. The solid histogram is the result of the fit described in the text. The dashed blue, dotted red, and shaded histograms show the contributions of signal, muon background, and $\mathrm{ISR}+\mathrm{two}\text{−}\mathrm{photon}$ background, respectively.

Figure 5

The distributions of $M_{\mathrm{miss}}^2$ versus $p_{\mathrm{T},K^{+}{K}^{-}}$ for (a) data events, (b) simulated signal events, and (c) simulated ISR background events. Events in the $J/\psi$ and ${\chi }_{c0}$ mass regions, $3.05<M_{K^{+}{K}^{-}}<3.15\mathrm{GeV}/c^2$ and $3.38<M_{K^{+}{K}^{-}}<3.46\mathrm{GeV}/c^2$, are excluded from the distributions; regions near the ${\chi }_{c2}$ and $\psi (2S)$ are not excluded, since their signal content is quite small. The lines indicate the boundaries of the signal region (bottom left rectangle) and the sideband region (top right rectangle).

Figure 6

The $M_{K^{+}{K}^{-}}$ spectra for data (points with error bars) in the vicinity of the ${\chi }_{c0}$ and ${\chi }_{c2}$ resonances for the $M_{\mathrm{miss}}^2–p_{\mathrm{T},K^{+}{K}^{-}}$ (a) signal region and (b) sideband region. The solid histograms result from the fits described in the text. The dashed histograms represent the nonresonant contributions.

Figure 7

The $M_{\mathrm{miss}}^2$ distribution for data (points with error bars) with $2.8<M_{K^{+}{K}^{-}}<3.0\mathrm{GeV}/c^2$ selected with all the criteria described in Sec. 3 except for the requirement $|M_{\mathrm{miss}}^2|<1{\mathrm{GeV}}^2/c^4$. The solid histogram is a sum of signal and background distributions obtained from MC simulation. The dashed histogram shows the distribution for two-photon events, and the shaded (almost invisible) histogram shows the small contribution of all other background processes. The inset shows an enlarged view of the region $-2<M_{\mathrm{miss}}^2<10{\mathrm{GeV}}^2/c^4$. Above $10{\mathrm{GeV}}^2/c^4$ the solid and dashed histograms are indistinguishable.

Figure 8

The $\cos {\theta }_K$ distribution for data (points with error bars) and reweighted simulated events (histogram) with $M_{\mathrm{miss}}^2>20{\mathrm{GeV}}^2/c^4$ from the $K^{+}{K}^{-}$ mass range $2.8–3.0\mathrm{GeV}/c^2$.

Figure 9

The $K^{+}{K}^{-}$ mass dependence of the detection efficiency for simulated $e^{+}{e}^{-}\to K^{+}{K}^{-}\gamma$ events.

Figure 10

The $e^{+}{e}^{-}\to K^{+}{K}^{-}$ cross section measured in this work.

Figure 11

The charged-kaon electromagnetic form factor measured in this analysis [BABAR (SA ISR)] in comparison with previous measurements: CLEO [8], Seth et al. [9], and BABAR (LA ISR) [3] in the mass region $2.6–5.0\mathrm{GeV}/c^2$. Only statistical uncertainties are shown.

Figure 12

The scaled charged-kaon electromagnetic form factor measured in this analysis [BABAR (SA ISR)] and in other experiments: CLEO [8] and Seth et al. [9]. Our data are approximated by a smooth curve. The dotted blue curve shows the LO pQCD prediction for the form factor obtained with the asymptotic kaon distribution amplitude. The dot-dashed blue and dashed red curves are the NLO pQCD predictions obtained with the asymptotic and Chernyak-Zhitnitsky distribution amplitudes, respectively. The inset shows an enlarged version of the mass region $3.4–4.2\mathrm{GeV}/c^2$.

Figure 13

The $K^{+}{K}^{-}$ mass spectra in the regions near the $J/\psi$ (a) and $\psi (2S)$ (b) resonances. The curves exhibit the results of the fits described in the text.