Measurement of the $\eta$ and ${\eta }^{\prime }$ transition form factors at $q^2=112{\mathrm{GeV}}^2$

We report a study of the processes $e^{+}{e}^{-}\to \eta \gamma$ and $e^{+}{e}^{-}\to {\eta }^{\prime }\gamma$ at a center-of-mass energy of 10.58 GeV, using a $232{\mathrm{fb}}^{-1}$ data sample collected with the BABAR detector at the PEP-II collider at SLAC. We observe ${20}_{-5}^{+6}\eta \gamma$ and ${50}_{-7}^{+8}{\eta }^{\prime }\gamma$ events over small backgrounds, and measure the cross sections $\sigma (e^{+}{e}^{-}\to \eta \gamma )={4.5}_{-1.1}^{+1.2}\pm 0.3\mathrm{fb}$ and $\sigma (e^{+}{e}^{-}\to {\eta }^{\prime }\gamma )=5.4\pm 0.8\pm 0.3\mathrm{fb}$. The corresponding transition form factors at $q^2=112{\mathrm{GeV}}^2$ are $q^2|F_{\eta }(q^2)|=0.229\pm 0.030\pm 0.008\mathrm{GeV}$, and $q^2|F_{{\eta }^{\prime }}(q^2)|=0.251\pm 0.019\pm 0.008\mathrm{GeV}$, respectively.

Figure 1

Distribution (left) of ${\chi }_{2\pi {\pi }^0\gamma }^2$ for simulated $e^{+}{e}^{-}\to \eta \gamma \to {\pi }^{+}{\pi }^{-}{\pi }^0\gamma$ signal events. Scatter plots of ${\chi }_{2\pi {\pi }^0\gamma }^2$ versus the ${\pi }^{+}{\pi }^{-}{\pi }^0$ invariant mass for the selected events in data (center) and signal simulation (right).

Figure 2

The $3\pi$ invariant mass spectrum for the $e^{+}{e}^{-}\to \eta \gamma$ candidates in the data. The dashed curve represents the result of the fit described in the text.

Figure 3

Scatter plots of ${\chi }_{2\pi \eta \gamma }^2$ vs $M_{2\pi \eta }$ for the selected events in the data (top) and $e^{+}{e}^{-}\to {\eta }^{\prime }\gamma \to {\pi }^{+}{\pi }^{-}\eta \gamma \to {\pi }^{+}{\pi }^{-}\gamma \gamma \gamma$ signal simulation (bottom).

Figure 4

The ${\pi }^{+}{\pi }^{-}\eta$ invariant mass distribution for the $e^{+}{e}^{-}\to {\eta }^{\prime }\gamma$ candidates in the data with ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}\eta$, $\eta \to \gamma \gamma$. The curve represents the result of the fit described in the text.

Figure 5

Distributions of ${\pi }^{+}{\pi }^{-}{\pi }^0$ invariant mass (4 combinations per event) for selected events in the data (top) and $e^{+}{e}^{-}\to {\eta }^{\prime }\gamma \to {\pi }^{+}{\pi }^{-}\eta \gamma \to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-}{\pi }^0\gamma$ simulation (bottom).

Figure 6

The ${\pi }^{+}{\pi }^{-}\eta$ invariant mass distribution for the $e^{+}{e}^{-}\to {\eta }^{\prime }\gamma$ candidates in the data with ${\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}\eta$, $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$. The curve represents the result of the fit described in the text.

Figure 7

Scatter plot of ${\chi }_{3\pi \gamma \gamma }^2$ versus $M_{3\pi }$ for the selected data events containing an additional photon.

Figure 8

The $3\pi$ invariant mass spectrum for events in the data with ${\chi }_{3\pi \gamma \gamma }^2<25$. The solid vertical lines bound the $\eta$ signal region; the sideband regions are between these and the dashed lines.

Figure 9

Distributions of the ${\pi }^{+}{\pi }^{-}\eta$ invariant mass for selected $e^{+}{e}^{-}\to {\gamma }_{\mathrm{ISR}}J/\psi \to {\gamma }_{\mathrm{ISR}}{\eta }^{\prime }\gamma$ events in the data (points with error bars) and simulation (histogram).

Figure 10

The ${\pi }^{+}{\pi }^{-}{\pi }^0$ invariant mass spectrum for data events in the $\omega$ mass region. The curve is the result of the fit described in the text.

Figure 11

Distributions of the photon polar angle (left), the invariant mass of the two charged pions (middle), and the minimum angle between a charged pion and a photon from the ${\pi }^0$ decay (right) for data (points with error bars) and simulated (solid lines) $e^{+}{e}^{-}\to {\gamma }_{\mathrm{ISR}}\omega$ events. The simulated background is shown as the (very small) shaded histograms, and the dashed lines show the distributions for simulated $e^{+}{e}^{-}\to \eta \gamma$ events.

Figure 12

Distributions of the angle between the two photons from the ${\pi }^0\to \gamma \gamma$ or $\eta \to \gamma \gamma$ decay (left), and the minimum (middle) and maximum (right) energy of the decay photons, for data (points with error bars) and simulated (solid lines) $e^{+}{e}^{-}\to {\gamma }_{\mathrm{ISR}}\omega$ events, simulated background events (small shaded histograms), and simulated $e^{+}{e}^{-}\to {\eta }^{\prime }\gamma \to {\pi }^{+}{\pi }^{-}\eta$ events (dashed lines).

Figure 13

The magnitudes of the $\eta \gamma$ (top) and ${\eta }^{\prime }\gamma$ (bottom) transition form factors measured in this work (filled circle) and by CLEO [13] (triangles). The shaded boxes indicate the ranges of form-factor values calculated according to Eq. (2) with the decay constants from Refs. [3, 4, 5, 6, 7, 8].