Measurement of the $\gamma {\gamma }^{*}\to \eta$ and $\gamma {\gamma }^{*}\to {\eta }^{\prime }$ transition form factors

We study the reactions $e^{+}{e}^{-}\to e^{+}{e}^{-}{\eta }^{(\prime )}$ in the single-tag mode and measure the $\gamma {\gamma }^{*}\to {\eta }^{(\prime )}$ transition form factors in the momentum-transfer range from 4 to $40{\mathrm{GeV}}^2$. The analysis is based on $469{\mathrm{fb}}^{-1}$ of integrated luminosity collected at PEP-II with the BABAR detector at $e^{+}{e}^{-}$ center-of-mass energies near 10.6 GeV.

Figure 1

The diagram for the $e^{+}{e}^{-}\to e^{+}{e}^{-}P$ two-photon production process, where $P$ is a pseudoscalar meson.

Figure 2

The $|\cos {\theta }_{e\eta }^{*}|$ distribution for data events (solid histogram). The shaded histogram shows the same distributions for the $e^{+}{e}^{-}\to e^{+}{e}^{-}\eta$ simulation. Events with $|\cos {\theta }_{e\eta }^{*}|>0.99$ (indicated by the arrow) are retained.

Figure 3

The $r$ distributions for $e^{+}{e}^{-}\to e^{+}{e}^{-}\eta$ data (solid-line histogram) and signal simulation (shaded histogram). The arrows indicate the region used to select event candidates ($-0.025<r<0.05$).

Figure 4

The (a) ${\pi }^{+}{\pi }^{-}{\pi }^0$ and (b) ${\pi }^{+}{\pi }^{-}\eta$ mass spectra for data events with $4<Q^2<40{\mathrm{GeV}}^2$. The solid curves are the results of the fits described in Sec. 4. The dashed curves represent non-peaking background.

Figure 5

The ${\pi }^{+}{\pi }^{-}{\pi }^0$ and ${\pi }^{+}{\pi }^{-}\eta$ mass spectra for simulated (a) $e^{+}{e}^{-}\to e^{+}{e}^{-}\eta$ and (b) $e^{+}{e}^{-}\to e^{+}{e}^{-}{\eta }^{\prime }$ events, respectively. The curves represent the resolution functions described in the text.

Figure 6

The ${\pi }^{+}{\pi }^{-}{\pi }^0$ mass spectra for data events with $-0.025<r<0.025$ for three representative $Q^2$ intervals. The solid curves are the fit results. The dashed curves represent non-peaking background.

Figure 7

The ${\pi }^{+}{\pi }^{-}\eta$ mass spectra for data events for three representative $Q^2$ intervals. The solid curves are the fit results. The dashed lines represent non-peaking background.

Figure 8

The two-photon invariant mass spectra for (a) $\eta$ and (b) ${\eta }^{\prime }$ events with two extra photons. The solid histograms represent the fit results. The dashed curves are the fitted distributions for events without an extra ${\pi }^0$.

Figure 9

The distribution of the $\eta {\pi }^0$ invariant mass for $\eta$ events with an extra ${\pi }^0$. The background from non-$\eta {\pi }^0$ events is subtracted.

Figure 10

(a) The $r$ distribution for data events containing an $\eta$ (points with error bars). The dashed histogram shows the fit results. The shaded histogram is the fitted background contribution from the processes $e^{+}{e}^{-}\to e^{+}{e}^{-}\eta {\pi }^0$ and $e^{+}{e}^{-}\to e^{+}{e}^{-}{\eta }^{\prime }\to e^{+}{e}^{-}{\pi }^0{\pi }^0\eta$. (b) The $r$ distribution for data events containing an ${\eta }^{\prime }$ (points with error bars). The solid histogram is the simulated distribution for events from the signal $e^{+}{e}^{-}\to e^{+}{e}^{-}{\eta }^{\prime }$ process normalized to the number of data events.

Figure 11

The detection efficiencies for (a) $e^{+}{e}^{-}\to e^{+}{e}^{-}\eta$ with $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$ and ${\pi }^0\to 2\gamma$ and (b) $e^{+}{e}^{-}\to e^{+}{e}^{-}{\eta }^{\prime }$ with $\eta \to {\pi }^{+}{\pi }^{-}\eta$ and $\eta \to 2\gamma$ as functions of the momentum transfer squared for events with a tagged electron (squares), a tagged positron (triangles), and their sum (circles). In the region $Q^2<6{\mathrm{GeV}}^2$, where the electron-tag efficiency is close to zero, the sum and the positron-tag efficiencies coincide.

Figure 12

The correction to the MC-estimated ${\pi }^0$ reconstruction efficiency ${\delta }_{{\pi }^0}$ as a function of $Q^2$ for the $e^{+}{e}^{-}\to e^{+}{e}^{-}\eta$ process.

Figure 13

The ratio of the cross sections for the $e^{+}{e}^{-}\to e^{+}{e}^{-}\eta$ process measured with electron and positron tags as a function of $Q^2$.

Figure 14

The differential cross sections for (a) $e^{+}{e}^{-}\to e^{+}{e}^{-}\eta$ and (b) $e^{+}{e}^{-}\to e^{+}{e}^{-}{\eta }^{\prime }$ from the present analysis compared to those from the CLEO experiment [9]. The asterisk near the label “CLEO” in this and the next figures indicates that the original CLEO results obtained in different ${\eta }^{(\prime )}$ decay modes were averaged assuming that systematic errors for different modes are not correlated. In the present analysis the cross sections are measured with the restriction $|q_2^2|<0.6{\mathrm{GeV}}^2$. In the CLEO analysis, the cross sections have been obtained using the vector dominance model for the $q_2^2$ dependence in simulation.

Figure 15

The transition form factors multiplied by $Q^2$ for (a) $\gamma {\gamma }^{*}\to \eta$ and (b) $\gamma {\gamma }^{*}\to {\eta }^{\prime }$.

Figure 16

The transition form factors multiplied by $Q^2$ for (a) $\gamma {\gamma }^{*}\to \eta$ and (b) $\gamma {\gamma }^{*}\to {\eta }^{\prime }$. The solid line shows the result of the fit to BABAR data by the function given by Eq. (16). The dashed lines indicate the average form factor values over the data points with $Q^2>14{\mathrm{GeV}}^2$.

Figure 17

The $\gamma {\gamma }^{*}\to |n⟩$ transition form factor multiplied by $3Q^2/5$ in comparison with the $\gamma {\gamma }^{*}\to {\pi }^0$ transition form factor [1]. The dashed line indicates the asymptotic limit for the ${\pi }^0$ form factor. The dotted, dash-dotted, and solid curves show predictions of Ref. 29 for the asymptotic DA [30], the Chernyak-Zhitnitsky ${\pi }^0$ DA [31], and the ${\pi }^0$ DA from Ref. 32, respectively.

Figure 18

The $\gamma {\gamma }^{*}\to |s⟩$ transition form factor multiplied by $Q^2$. The dashed line indicates the asymptotic limit for the form factor. The dotted curve shows the prediction [29] for the asymptotic DA [30].