Measurement of ${\eta }_c(1S)$, ${\eta }_c(2S)$, and nonresonant ${\eta }^{\prime }{\pi }^{+}{\pi }^{-}$ production via two-photon collisions

We report the measurement of $\gamma \gamma \to {\eta }_c(1S),{\eta }_c(2S)\to {\eta }^{\prime }{\pi }^{+}{\pi }^{-}$ with ${\eta }^{\prime }$ decays to $\gamma \rho$ and $\eta {\pi }^{+}{\pi }^{-}$ using $941{\mathrm{fb}}^{-1}$ of data collected with the Belle detector at the KEKB asymmetric-energy $e^{+}{e}^{-}$ collider. The ${\eta }_c(1S)$ mass and width are measured to be $M=[2984.6\pm 0.7(\mathrm{stat})\pm 2.2(\mathrm{syst})\pm 0.3(\mathrm{model})]\mathrm{MeV}/c^2$ and $\mathrm{\Gamma }=[30.8_{-2.2}^{+2.3}(\mathrm{stat})\pm 2.5(\mathrm{syst})\pm 1.4(\mathrm{model})]\mathrm{MeV}$, respectively. First observation of ${\eta }_c(2S)\to {\eta }^{\prime }{\pi }^{+}{\pi }^{-}$ with a significance of $5.5\sigma$ including systematic error is obtained, and the ${\eta }_c(2S)$ mass is measured to be $M=[3635.1\pm 3.7(\mathrm{stat})\pm 2.9(\mathrm{syst})\pm 0.4(\mathrm{model})]\mathrm{MeV}/c^2$. The products of the two-photon decay width and branching fraction ($\mathcal{B}$) of decays to ${\eta }^{\prime }{\pi }^{+}{\pi }^{-}$ are determined to be ${\mathrm{\Gamma }}_{\gamma \gamma }$${\mathrm{\Gamma }}_{\gamma \gamma }\mathcal{B}=[65.4\pm 2.6(\mathrm{stat})\pm 7.8(\mathrm{syst})]\mathrm{eV}$ for ${\eta }_c(1S)$ and $[{5.6}_{-1.1}^{+1.2}(\mathrm{stat})\pm 1.1(\mathrm{syst})]\mathrm{eV}$ for ${\eta }_c(2S)$. The cross sections for $\gamma \gamma \to {\eta }^{\prime }{\pi }^{+}{\pi }^{-}$ and ${\eta }^{\prime }{f}_2(1270)$ are measured for the first time.

Figure 1

The $|\mathrm{\Sigma }{p}_t^{*}|$ distributions in the ${\eta }_c(1S)$ $[{\eta }_c(2S)]$ signal region for (a) [(b)] the $\eta \pi \pi$ mode and (c) [(d)] the $\gamma \rho$ mode. The solid points with error bars are data. The solid red line is the fit; the blue dashed-dot and green dashed lines, respectively, show the signal in MC and the background in data.

Figure 2

The invariant mass distributions of (a) $\eta {\pi }^{+}{\pi }^{-}$ and $\gamma {\rho }^0$ (b) for the ${\eta }^{\prime }{\pi }^{+}{\pi }^{-}$ candidate events. Solid red line is the fit. The blue dashed-dot and green dashed lines are the signal and background, respectively.

Figure 3

The ${\eta }^{\prime }{\pi }^{+}{\pi }^{-}$ invariant mass distribution for the candidate events with ${\eta }^{\prime }$ decays to (a) $\eta {\pi }^{+}{\pi }^{-}$ and (b) $\gamma \rho$. Large ${\eta }_c(1S)$ signal and evident excess in the ${\eta }_c(2S)$ region (as arrow pointed) are seen.

Figure 4

The invariant mass distribution for the ${\eta }^{\prime }{\pi }^{+}{\pi }^{-}$ candidates for (a) the $\eta \pi \pi$ mode and (b) the $\gamma \rho$ mode, in the ${\eta }_c(1S)$ region. The dots with error bars are data. The red solid line is the fit; the blue dashed line is fitted signal for ${\eta }_c(1S)$. The green dot, cyan long-dashed, and magenta dashed-dot lines are the $NR$, ${\eta }^{\prime }\text{−}sdb$ and $b_{\mathrm{any}}$ ($b_{\mathrm{any}}+{\eta }^{\prime }\text{−}sdb$ merged into the magenta dashed-dot line for the $\gamma \rho$ mode) background components, respectively.

Figure 5

The invariant mass distribution for the ${\eta }^{\prime }{\pi }^{+}{\pi }^{-}$ candidates for (a) the $\eta \pi \pi$ mode and (b) the $\gamma \rho$ mode, in the ${\eta }_c(2S)$ region. The dots with error bars are data. The red solid line is the fit; the blue dashed line is fitted signal for ${\eta }_c(2S)$. The green dot, cyan long-dashed, and magenta dashed-dot lines are the $NR$, ${\eta }^{\prime }\text{−}sdb$ and $b_{\mathrm{any}}$ ($b_{\mathrm{any}}+{\eta }^{\prime }\text{−}sdb$ merged into the magenta dashed-dot line for the $\gamma \rho$ mode) background components, respectively.

Figure 6

The Dalitz plots for events selected in the ${\eta }_c(1S)$ signal (a) and $sdb$ (b) regions.

Figure 7

(a) [(c)] The invariant mass $M({\eta }^{\prime }{\pi }^{+})$ distributions (two entries per event) [$M({\pi }^{+}{\pi }^{-})$ distributions] in data for the events selected in the ${\eta }_c(1S)$ signal region is drawn as the black solid dots with error bars. The red histogram is for the normalized $sdb$ background events. (b) [(d)] The black solid dots with error bars is the $M({\eta }^{\prime }{\pi }^{+})$ [$M({\pi }^{+}{\pi }^{-})$] distribution in the ${\eta }_c(1S)$ signal region in data after subtraction of the $sdb$ background and the blue histogram normalized to data is for MC events of the ${\eta }_c(1S)$ decays to three-body final state according to PHSP distribution.

Figure 8

Detection efficiency $\epsilon$ as a function of $W$ and $|\cos {\theta }^{*}|$ for $\gamma \gamma \to {\eta }^{\prime }{\pi }^{+}{\pi }^{-}$ with the $\eta {\pi }^{+}{\pi }^{-}$ mode in the regions of (a) $W\in [1.40,2.66)\mathrm{GeV}$ and (b) $W\in [2.66,3.80]\mathrm{GeV}$.

Figure 9

Measured cross section of $\gamma \gamma \to {\eta }^{\prime }{\pi }^{+}{\pi }^{-}$ (including ${\eta }^{\prime }{f}_2(1270)$) for the $\eta \pi \pi$ mode.

Figure 10

Detection efficiency $\epsilon$ as a function of $W$ and $|\cos {\theta }^{*}|$ for $\gamma \gamma \to {\eta }^{\prime }{f}_2(1270)$ in the $\eta \pi \pi$ mode in the $W$ ranges of (a) [2.26, 2.62) GeV and (b) [2.62, 3.80] GeV.

Figure 11

Measured cross sections for $\gamma \gamma \to {\eta }^{\prime }{f}_2(1270)$. The black dots with statistical error bars are the data within (a) $|\cos {\theta }^{*}|<1$ and (b) $|\cos {\theta }^{*}|<0.6$. The red solid lines are fitted curves with the $W$-power index $n=5.1\pm 1.0$ and $n=7.5\pm 2.0$, respectively, assuming a $W$ dependence of $1/W^n$. The green dashed line corresponds to the leading-term QCD prediction for neutral meson pairs ($n=10$).

Figure 12

Cross sections of $\gamma \gamma \to {\eta }^{\prime }{f}_2(1270)$ in $|\cos {\theta }^{*}|$ in three $W$ regions from 2.26 to 3.80 GeV. The normalizer ${\sigma }_0$ is the total cross section in the $|\cos {\theta }^{*}|<0.8$ region. The black solid points are the data with statistical errors. The red solid line, normalized to the data in the same angular range, follows a $1/{\sin }^4{\theta }^{*}$ behavior.

Figure 13

Left panel: cross sections of $\gamma \gamma \to {\eta }^{\prime }{\pi }^{+}{\pi }^{-}$ [including ${\eta }^{\prime }{f}_2(1270)]$ (black solid dots) and $\gamma \gamma \to {\eta }^{\prime }{f}_2(1270)$ (red open dots). Right panel: cross sections of $\gamma \gamma \to {\eta }^{\prime }{\pi }^{+}{\pi }^{-}$ [excluding $\gamma \gamma \to {\eta }^{\prime }{f}_2(1270)$] in the $W$ range above 2.26 GeV. The structure (a) near 1.8 GeV arises from $X(1835)$ and $\eta (1760)$; the structure (b) near 2.1 GeV is perhaps from $\gamma \gamma \to {\eta }^{\prime }{f}_0(980)$ production. In both panels, the error bars are statistical.

Figure 14

Differential cross sections of $\gamma \gamma \to {\eta }^{\prime }{\pi }^{+}{\pi }^{-}$ [excluding ${\eta }^{\prime }{f}_2(1270)$] in $|\cos {\theta }^{*}|$ in three $W$ regions from 2.26 to 3.80 GeV. The red solid line is a uniform distribution normalized to the data. In all panels, the error bars are statistical.