Study of $K_S^0$ pair production in single-tag two-photon collisions

We report a measurement of the cross section for $K_S^0$ pair production in single-tag two-photon collisions, ${\gamma }^{*}\gamma \to K_S^0{K}_S^0$, for $Q^2$ up to $30{\mathrm{GeV}}^2$, where $Q^2$ is the negative of the invariant mass squared of the tagged photon. The measurement covers the kinematic range $1.0\mathrm{GeV}<W<2.6\mathrm{GeV}$ and $|\cos {\theta }^{*}|<1.0$ for the total energy and kaon scattering angle, respectively, in the ${\gamma }^{*}\gamma$ center-of-mass system. These results are based on a data sample of $759{\mathrm{fb}}^{-1}$ collected with the Belle detector at the KEKB asymmetric-energy $e^{+}{e}^{-}$ collider. For the first time, the transition form factor of the $f_2^{\prime }(1525)$ meson is measured separately for the helicity-0, -1, and -2 components and also compared with theoretical calculations. We have derived the cross section for the process for $W<2.6\mathrm{GeV}$ from 121 signal candidate events. Finally, the ${\gamma }^{*}\gamma$ partial decay widths of the ${\chi }_{c0}$ and ${\chi }_{c2}$ mesons are measured as a function of $Q^2$ based on 10 candidate events in total.

Figure 1

Feynman diagram for the process $e^{+}{e}^{-}\to e(e)K_S^0{K}_S^0$ and definition of the eight four-momenta.

Figure 2

Definition of the ${\gamma }^{*}\gamma$ c.m. coordinate system for ${\gamma }^{*}\gamma \to K_S^0{K}_S^0$. The “incident” ${\gamma }^{*}$ has momentum along the $z^{*}$ axis with $p_{z^{*}}>0$, the tagging $e^{\pm }$ is in the $x^{*}{z}^{*}$ plane with $p_{\mathrm{tag}{x}^{*}}>0$, and the forward-going $K_S^0$ (i.e., having $p_{K{z}^{*}}>0$) is produced at angles (${\theta }^{*}$, ${\phi }^{*}$).

Figure 3

The experimental $W$ distributions of the signal candidates at $2{\mathrm{GeV}}^2$ ($3{\mathrm{GeV}}^2$) $<Q^2<30{\mathrm{GeV}}^2$ as indicated by the asterisks (dashed histogram). Backgrounds have not been subtracted.

Figure 4

(a) The $Q^2$ distributions for the data samples at $W\le 4.0\mathrm{GeV}$. The asterisks and the dashed histogram are for the p-tag and e-tag samples, respectively. (b) The corresponding distributions from the signal MC events. Statistics of the MC figures are arbitrary, but the scale is common for the e- and p-tags in each panel, so that their ratio can be compared between MC and data.

Figure 5

(a),(b),(c) Reconstructed ${\pi }^{+}{\pi }^{-}$ invariant mass, as measured at each determined decay vertex for the data, in three different $W$ ranges, as indicated above each panel. (d),(e),(f) The corresponding distributions from the signal MC.

Figure 6

The distributions for experimental signal candidates (top row) and signal MC (bottom row) for (a),(d) the cosine of the laboratory polar angle of the tagging electron, (b),(e) the cosine of the laboratory polar angle of the two $K_S^0$ candidates (two entries per event), and (c),(f) the laboratory energy of the two $K_S^0$ candidates (two entries per event).

Figure 7

Distribution of $p_t$ balance vs. $E_{\text{ratio}}$ for the experimental samples to which the selection criteria other than those related to the illustrated variables have been applied. The $W$ region for the samples is shown in each panel. The half-ellipse and the rectangle in (a) show the signal and control regions, respectively.

Figure 8

(a),(b) Experimental distribution of $r_{VKi}$ ($r$ coordinate of the ${\pi }^{+}{\pi }^{-}$ vertex point for a $K_S^0$ candidate) for an event in which the other kaon-vertex coordinate satisfies the selection criterion $r_{VKj}>0.3\mathrm{cm}$. The $W$ region for each sample is shown in each panel. The vertical arrows indicate the selection criterion. (c),(d) The corresponding distributions from the signal-MC samples. Statistics of the MC figures are arbitrary.

Figure 9

The $W$ distribution of experimental-data events in (a) the signal region and (b) the control region.

Figure 10

Efficiency (including trigger effects) as estimated from the signal-MC samples. The solid (black) and dashed (red) curves are for e-tag and p-tag events, respectively. Results are shown for five $Q^2$ regions, whose central values are indicated above each panel.

Figure 11

The efficiency-corrected and background-subtracted $W$ dependence of the $e^{+}{e}^{-}$-based cross section in each $Q^2$ bin. The black closed (red open) circles with error bars are for the e-tag (p-tag) measurements. The red p-tag points have been shifted slightly to the right for enhanced visibility.

Figure 12

Total cross sections (integrated over angle) for ${\gamma }^{*}\gamma \to K_S^0{K}_S^0$ in the five $Q^2$ bins indicated in each panel.

Figure 13

$Q^2$ dependence of ${\mathrm{\Gamma }}_{{\gamma }^{*}\gamma }$ for the (a) ${\chi }_{c0}$ and (b) ${\chi }_{c2}$ mesons normalized to ${\mathrm{\Gamma }}_{\gamma \gamma }$ (at $Q^2=0$) [15]. The data point without a dot is based on a zero-event observation, and the upper edge of its error bar corresponds to the value for one event. The overall uncertainties due to the normalization errors of the ${\mathrm{\Gamma }}_{\gamma \gamma }\mathcal{B}(K_S^0{K}_S^0)$ are not shown. The solid and dashed curves, respectively, show the SBG [6] prediction and also one motivated by VDM, assuming $\rho$ dominance.

Figure 14

$|\cos {\theta }^{*}|$ dependence of the normalized differential cross sections and the fitted results in the four $W$ bins indicated in each panel. The lines shown are obtained from the nominal fit (at $⟨Q^2⟩=6.5{\mathrm{GeV}}^2$). Black solid lines show the total, green dotted the $|S{|}^2$ term, blue dashed the $|D_0{|}^2$ term, red long-dashed the $|D_1{|}^2$ term, and magenta dash-dotted the $|D_2{|}^2$ term.

Figure 15

$|{\phi }^{*}|$ dependence of the normalized differential cross sections and the fitted results in the four $W$ bins indicated in each panel. The lines shown result from the nominal fit (at $⟨Q^2⟩=6.5{\mathrm{GeV}}^2$). Black solid line: total, red dotted: $t_0$; blue dashed: $t_1\cos |{\phi }^{*}|$; and magenta dash-dotted: $t_2\cos 2|{\phi }^{*}|$.

Figure 16

Total cross sections (integrated over angle) for ${\gamma }^{*}\gamma \to K_S^0{K}_S^0$ in five $Q^2$ bins as indicated in each panel, together with the fit results described in 6c. Black solid line: total; green dotted: $|S{|}^2$; blue dashed: $|D_0{|}^2$; red long-dashed: $|D_1{|}^2$; and magenta dash-dotted: $|D_2{|}^2$.

Figure 17

The obtained helicity-0, -1, and -2 TFF of the $f_2^{\prime }(1525)$ meson as a function of $Q^2$, assuming Eq. (24). Short (long) vertical bars indicate statistical (combined statistical and systematic) errors. The shaded band corresponds to the overall uncertainty arising from the known errors on ${\mathrm{\Gamma }}_{\gamma \gamma }$. The solid line shows the predicted $Q^2$ dependence in SBG [6].

Figure 18

$Q^2$ dependence of the cross section in the three $W$ regions near the $K_S^0{K}_S^0$ mass threshold, with central values as indicated in the subpanels. Only statistical errors are shown. Solid curves show the predicted $Q^2$ dependence in SBG [6].