Measurement of $\eta \eta$ production in two-photon collisions

We report the first measurement of the differential cross section for the process $\gamma \gamma \to \eta \eta$ in the kinematic range above the $\eta \eta$ threshold, $1.096\mathrm{GeV}<W<3.8\mathrm{GeV}$ over nearly the entire solid angle range, $|\cos {\theta }^{*}|\le 0.9$ or $\le 1.0$ depending on $W$, where $W$ and ${\theta }^{*}$ are the energy and $\eta$ scattering angle, respectively, in the $\gamma \gamma$ center-of-mass system. The results are based on a $393{\mathrm{fb}}^{-1}$ data sample collected with the Belle detector at the KEKB $e^{+}{e}^{-}$ collider. In the $W$ range $1.1–2.0\mathrm{GeV}/c^2$ we perform an analysis of resonance amplitudes for various partial waves, and at higher energy we compare the energy and the angular dependences of the cross section with predictions of theoretical models and extract contributions of the ${\chi }_{cJ}$ charmonia.

Figure 1

Two-dimensional $W$ and $|\cos {\theta }^{*}|$ distribution for the $\eta \eta$ candidates in data. The same distribution is viewed from two different directions. The backgrounds are not subtracted.

Figure 2

(a) Two-dimensional $M_{\gamma \gamma 1}$ vs $M_{\gamma \gamma 2}$ distribution of the four-photon candidates in data. We take $M_{\gamma \gamma 1}<M_{\gamma \gamma 2}$. The $p_t$-balance cut with $p_t<0.1\mathrm{GeV}/c$ is applied using the photon momenta before the $m_{\eta }$ mass correction, which reduces backgrounds. Red, blue, and green boxes show the signal, sideband A, and sideband B regions, respectively. (b) A one-dimensional projection of the same distribution where the $\gamma \gamma$ pair on the opposite side is required to be in the signal mass region $0.52–0.57\mathrm{GeV}/c^2$. The vertical red and blue lines show the signal and two sideband A regions, respectively. Note that there are two entries per event in the signal region.

Figure 3

$W$ distributions of the yields in the signal region (closed circles with error bars, $Y_s$) and estimates of backgrounds of the $\eta$ non-$\eta$ component [solid histogram, $0.5(Y_{bA}-Y_{bB})$] and the non-$\eta$ non-$\eta$ component (dashed histogram, $0.25Y_{bB}$). Points with error bars show the estimated total background $0.5Y_{bA}–0.25Y_{bB}$.

Figure 4

Energy dependence of $R$ defined in Eq. (1). It indicates the level of $\eta \eta X$ background contamination, for the experimental data (after the sideband subtraction, points with error bars) and signal MC (short horizontal bars).

Figure 5

Two-dimensional dependence of the efficiency on $(W,|\cos {\theta }^{*}|)$.

Figure 6

Angular dependence of the differential cross sections for nine selected $W$ bins indicated. The bin sizes are summarized in Table 2.

Figure 7

The cross section integrated (a) over $|\cos {\theta }^{*}|<0.9$ and (b) over $|\cos {\theta }^{*}|<1.0$ for $W<2.0\mathrm{GeV}$. Errors are statistical only. The dotted curve shows the size of the systematic uncertainty.

Figure 8

Spectra of ${\widehat{S}}^2$, ${\widehat{D}}_0^2$, and ${\widehat{D}}_2^2$ for the SD fit. Those for the SDG fit are nearly identical with larger statistical errors. The error bars shown are statistical errors that do not include correlations.

Figure 9

Spectra of ${\widehat{G}}_0^2$, ${\widehat{G}}_2^2$, and ${\widehat{G}}_0^2\pm {\widehat{G}}_2^2$. The error bars shown are statistical errors that do not include correlations.

Figure 10

Differential cross sections (points with error bars) and fitted curves for the nominal fit in the low mass region (solid curve). Dotted (dot-dashed) curves are $|S{|}^2$ ($|D_2{|}^2$) from the fit. The vertical error bars are statistical only.

Figure 11

Total cross section (points with error bars) ($|\cos {\theta }^{*}|<1.0$) and fitted curves for the nominal fit in the low mass region (solid curve). Dotted (dot-dashed) curves are $|S{|}^2$ ($|D_2{|}^2$) from the fit. The vertical errors are statistical only.

Figure 12

${\widehat{S}}^2$, ${\widehat{D}}_0^2$, and ${\widehat{D}}_2^2$ and fitted curves for the nominal fit in the low mass region (solid curve). The points with error bars are the results of the $W$-independent fits (the same as those in Fig. 8). The vertical error bars are statistical only.

Figure 13

Differential cross sections (points with error bars) in the energy bins indicated and fitted curves for the nominal fit in the high mass region (solid curve). Dotted (dot-dashed) curves are $|S{|}^2$ ($|D_2{|}^2$) from the fit. The vertical error bars are statistical only.

Figure 14

Total cross sections (points with error bars) ($|\cos {\theta }^{*}|<1.0$) and fitted curves for the nominal fit in the high mass region (solid curve). Dotted (dot-dashed) curves are $|S{|}^2$ ($|D_2{|}^2$) from the fit. The vertical error bars are statistical only.

Figure 15

${\widehat{S}}^2$, ${\widehat{D}}_0^2$, and ${\widehat{D}}_2^2$ and fitted curves for the nominal fit in the high mass region (solid curve). The points with error bars are the results of the $W$-independent fits (the same as those in Fig. 8). The vertical error bars are statistical only.

Figure 16

The angular dependence of the differential cross sections in different $W$ regions, with the normalization to the cross section integrated over $|\cos {\theta }^{*}|<0.9$. The solid and dashed curves are proportional to $1/{sin}^4{\theta }^{*}$ and $1/{sin}^6{\theta }^{*}$, respectively, normalized similarly.

Figure 17

(a) The $W$ dependence of the cross sections ($|\cos {\theta }^{*}|<0.8$) for the ${\pi }^0{\pi }^0$ (open squares) [11] and $\eta \eta$ (closed circles) processes. The curve is the power-law fit for the latter process. (b) The $W$ dependence of the cross section ratio of $\eta \eta$ to ${\pi }^0{\pi }^0$ ($|\cos {\theta }^{*}|<0.8$). The line is the average in the 2.4–3.3 GeV range. The error bars are only statistical in the above figures.

Figure 18

The $W$ distribution in the charmonium region ($|\cos {\theta }^{*}|<0.4$). The fit does not take interference into account.