Measurement of the $\omega \to {\pi }^0{e}^{+}{e}^{-}$ and $\eta \to e^{+}{e}^{-}\gamma$ Dalitz decays with the A2 setup at the Mainz Microtron

The Dalitz decays $\eta \to e^{+}{e}^{-}\gamma$ and $\omega \to {\pi }^0{e}^{+}{e}^{-}$ have been measured in the $\gamma p\to \eta p$ and $\gamma p\to \omega p$ reactions, respectively, with the A2 tagged-photon facility at the Mainz Microtron. The value obtained for the slope parameter of the electromagnetic transition form factor of $\eta , {\mathrm{\Lambda }}_{\eta }^{-2}=(1.97\pm 0.{11}_{\mathrm{tot}}){\mathrm{GeV}}^{-2}$, is in good agreement with previous measurements of the $\eta \to e^{+}{e}^{-}\gamma$ and $\eta \to {\mu }^{+}{\mu }^{-}\gamma$ decays. The uncertainty obtained in the value of ${\mathrm{\Lambda }}_{\eta }^{-2}$ is lower than in previous results based on the $\eta \to e^{+}{e}^{-}\gamma$ decay. The value obtained for the $\omega$ slope parameter, ${\mathrm{\Lambda }}_{\omega {\pi }^0}^{-2}=(1.99\pm 0.{21}_{\mathrm{tot}}){\mathrm{GeV}}^{-2}$, is somewhat lower than previous measurements based on $\omega \to {\pi }^0{\mu }^{+}{\mu }^{-}$, but the results for the $\omega$ transition form factor are in better agreement with theoretical calculations, compared to earlier experiments.

Figure 1

A general sketch of the Crystal Ball, TAPS, and particle identification (PID) detectors.

Figure 2

Comparison of the $e^{+/-}dE/dx$ of the PID for experimental $\eta \to e^{+}{e}^{-}\gamma$ decays and their MC simulation. The two-dimensional density distribution (with logarithmic scale along plot axis $z$) for the $e^{+/-}dE/dx$ of the PID versus the energy of the corresponding clusters in the CB is shown in panel (a) for the experimental data and in panel (b) for the MC simulation. The $e^{+/-}dE/dx$ distributions for the experimental data (crosses) and the MC simulation (blue solid line) are compared in panel (c).

Figure 3

(a), (c) Two-dimensional density distributions for events selected as the $\gamma p\to {\pi }^0{e}^{+}{e}^{-}p$ candidates obtained from the MC simulations of $\omega \to {\pi }^0{e}^{+}{e}^{-}$ decays and (b), (d) $\omega \to {\pi }^0{\pi }^{+}{\pi }^{-}$ decays causing background in the vicinity of the $\omega$ mass. Distributions (a) and (b) plot the number $N$ of crystals forming the two clusters ascribed to $e^{+/-}$ versus the cluster energies. Distributions (c) and (d) plot the effective radius $R$ of the two clusters ascribed to $e^{+/-}$ versus the cluster energies. The cuts tested for suppressing the $\omega \to {\pi }^0{\pi }^{+}{\pi }^{-}$ background are shown by red dashed lines for looser cuts and by blue solid lines for tighter cuts. The cuts on $N$ discard all events for which at least one of the $e^{+/-}$ clusters has an energy larger than the values shown by the cut lines. The cuts on effective radius discard all events for which at least one of the $e^{+/-}$ clusters has $R$ larger than the values shown by the cut lines.

Figure 4

$m\left(e^{+}{e}^{-}\gamma \right)$ invariant-mass distributions obtained for the $m\left(e^{+}{e}^{-}\right)$ range from 30 to 100 MeV/$c^2$ with $\gamma p\to e^{+}{e}^{-}\gamma p$ candidates initially selected with the kinematic fit and allowing both $e^{+}$ and $e^{-}$ to be identified with the same PID element: (a) MC simulation of $\gamma p\to \eta p\to e^{+}{e}^{-}\gamma p$ fitted with a Gaussian, (b) experimental events (Run I) after subtracting the random background and the remaining background from $\gamma p\to \eta p\to \gamma \gamma p$. The distribution for the $\eta \to \gamma \gamma$ background, shown by a red solid line, is normalized to the number of subtracted events. The experimental distribution is fitted with the sum of a Gaussian for the $\eta \to e^{+}{e}^{-}\gamma$ peak and a polynomial of order four for the background. The total fit is depicted by a solid line, and the dashed line shows the background under the peak.

Figure 5

Same as Fig. 4, but requiring both $e^{+}$ and $e^{-}$ to be identified by different PID elements.

Figure 6

Same as Fig. 5, but after applying the $dE/dx$ PID cut selecting only events with the elements having an energy deposit corresponding to a single $e^{+/-}$, and requiring the final-state proton to be detected.

Figure 7

The $\eta \to \gamma {\gamma }^{*}\to \gamma e^{+}{e}^{-}$ angular dependence (in the $\eta$ rest frame) of the virtual photon decaying into a lepton pair, with ${\theta }^{*}$ being the angle between the direction of one of the leptons in the virtual-photon (or the dilepton) rest frame and the direction of the dilepton system (which is opposite to the $\gamma$ direction): (a) experimental events from the $\eta \to \gamma e^{+}{e}^{-}$ peak, (b) angular acceptance based on the MC simulation, (c) the experimental spectrum corrected for the acceptance and normalized for comparing to the $1+{cos}^2{\theta }^{*}$ dependence (shown by a red dashed line). Because $e^{+}$ and $e^{-}$ cannot be separated in the present experiment, the angles of both leptons were used, resulting in a symmetric shape with respect to $cos{\theta }^{*}=0$.

Figure 8

Same as Fig. 4, but for $m\left(e^{+}{e}^{-}\right)=(360\pm 20)$ MeV/$c^2$. The $\eta \to \gamma \gamma$ background is not shown in panel (b) because it is negligibly small.

Figure 9

Same as Fig. 8, but after applying the softer cuts on $N$ and $R$ (depicted by red dashed lines in Fig. 3).

Figure 10

Same as Fig. 8, but after applying the tighter cuts on $N$ and $R$ (depicted by blue solid lines in Fig. 3).

Figure 11

$m\left({\pi }^0{e}^{+}{e}^{-}\right)$ invariant-mass distributions obtained for the $m\left(e^{+}{e}^{-}\right)$ range from 30 to 100 MeV/$c^2$ with $\gamma p\to {\pi }^0{e}^{+}{e}^{-}p$ candidates initially selected with the kinematic fit (a) MC simulation of $\gamma p\to \omega p\to {\pi }^0{e}^{+}{e}^{-}p$ fitted with a Gaussian; (b) experimental events (Run I) after subtracting the random background and the remaining background from $\omega \to {\pi }^0\gamma$ decays (with both the ${\pi }^0\to \gamma \gamma$ and ${\pi }^0\to e^{+}{e}^{-}\gamma$ decay modes). The distribution for the $\omega \to {\pi }^0\gamma$ background, shown by a red solid line, is normalized to the number of subtracted events. The experimental distribution is fitted with the sum of a Gaussian for the $\omega \to {\pi }^0{e}^{+}{e}^{-}$ peak and a polynomial of order five for the background. The total fit is depicted by a solid line, and the dashed line shows the background under the peak.

Figure 12

Same as Fig. 11, but after applying the softer cuts on $N$ and $R$ (depicted by red dashed lines in Fig. 3) and the $dE/dx$ cut rejecting events with twice the magnitude of a signal from a single $e^{+/-}$ in one PID element.

Figure 13

The $\omega \to {\pi }^0{\gamma }^{*}\to {\pi }^0{e}^{+}{e}^{-}$ angular dependence (in the $\omega$ rest frame) of the virtual photon decaying into a lepton pair, with ${\theta }^{*}$ being the angle between the direction of one of the leptons in the virtual-photon (or the dilepton) rest frame and the direction of the dilepton system (which is opposite to the ${\pi }^0$ direction): (a) experimental events from the $\omega \to {\pi }^0{e}^{+}{e}^{-}$ peak; (b) angular acceptance based on the MC simulation; (c) experimental spectrum corrected for the acceptance and normalized for comparing to the $1+{cos}^2{\theta }^{*}$ dependence (shown by a red dashed line). Because $e^{+}$ and $e^{-}$ cannot be separated in the present experiment, the angles of both leptons were used, resulting in a symmetric shape with respect to $cos{\theta }^{*}=0$.

Figure 14

Same as Fig. 11, but for the $m\left(e^{+}{e}^{-}\right)$ range from 150 to 400 MeV/$c^2$ and the tighter cuts on $N$ and $R$ (depicted by blue solid lines in Fig. 3). The $\omega \to {\pi }^0\gamma$ background is not shown in panel (b) because it is negligibly small.

Figure 15

Same as Fig. 14, but after applying the $dE/dx$ cut rejecting events with twice the magnitude of a signal from a single $e^{+/-}$ in one PID element.

Figure 16

Comparison of the $|F_{\eta }\left(m_{{\ell }^{+}{\ell }^{-}}\right){|}^2$ results obtained individually from the analyses of Run I (blue filled triangles) and Run II (red open circles) with each other and with the two solutions for the DA calculations by the Jülich group [20, 21]. The solution without including the $a_2$-meson contribution is shown by a red dotted line with an error band, and the solution involving the $a_2$ contribution is shown by a blue dashed line. The pole-approximation fits (black solid lines) to the results of Run I and Run II are depicted in panels (a) and (b), respectively. The fit parameter $p0$ reflects the general normalization of the data points, and $p1$ is the slope parameter ${\mathrm{\Lambda }}^{-2} \left[{\mathrm{GeV}}^{-2}\right]$. For a better comparison of the magnitudes of total uncertainties from the two data sets, the error bars of Run I are plotted in panel (a) on the top of the error bars of Run II, and the other way around in panel (b).

Figure 17

$|F_{\eta }\left(m_{{\ell }^{+}{\ell }^{-}}\right){|}^2$ results (black filled triangles) combined from Run I and Run II and their pole-approximation fit (black solid line, with $p0$ and $p1$ being the normalization and the slope parameter ${\mathrm{\Lambda }}^{-2}$, respectively) are compared to previous measurements and various theoretical calculations. The former results by the A2 Collaboration from Ref. [17] (open magenta circles) and Ref. [16] (open green diamonds) are shown in panel (a). The results of NA60 obtained in peripheral In-In data [15] and in $p-A$ collisions [12] are shown in panel (b). The calculation from Ref. [47] is shown in panel (a) by a blue dash-dotted line. The most recent DA calculation by the Jülich group [20] is shown in panel (b) by a blue dashed line. The calculations by the Mainz group with Padé approximants are shown in panel (a) for their previous solution [48] (red dashed line with an error band) and in panel (b) for their latest solution [18] (red dotted line with an error band).

Figure 18

Comparison of the $|F_{\omega {\pi }^0}\left(m_{{\ell }^{+}{\ell }^{-}}\right){|}^2$ results obtained individually from the analyses of Run I (blue filled triangles) and Run II (red open circles) with each other and with the two solutions for the DA calculations by the Bonn group [26] shown by error-band borders. The solution using a simplified, VMD-inspired $\omega \to 3\pi$ partial wave $f_1\left(s\right)=\alpha \mathrm{\Omega }\left(s\right)$ inside the dispersion integral is shown by cyan dashed lines, and the solution using the full rescattering of $3\pi$ by magenta dashed lines. The pole-approximation fits (black solid lines) to the results of Run I and Run II are depicted in panels (a) and (b), respectively. The fit parameter $p0$ reflects the general normalization of the data points, and $p1$ is the slope parameter ${\mathrm{\Lambda }}^{-2}$. For a better comparison of the magnitudes of total uncertainties from the two data sets, the error bars of Run I are plotted in panel (a) on the top of the error bars of Run II, and the other way around in panel (b).

Figure 19

$|F_{\omega {\pi }^0}\left(m_{{\ell }^{+}{\ell }^{-}}\right){|}^2$ results (black filled triangles) combined from Run I and Run II and their pole-approximation fit (black solid line, with $p0$ and $p1$ being the normalization and the slope parameter ${\mathrm{\Lambda }}^{-2}$, respectively) are compared to previous measurements and various theoretical calculations. The results by Lepton-$\mathrm{G}$ [23] are shown by open red squares in panel (b). The results of NA60 obtained in peripheral In-In data [15] are shown by open green circles in panels (a) and (c), and from $p-A$ collisions [12] by open green triangles in panel (b). The VMD prediction is shown by a blue dashed line in panel (a). The calculation from Refs. [24, 25] is shown by a red dash-dotted line in panel (a). The DA calculation by the Bonn group [26] for the full $3\pi$ rescattering is shown by error-band borders (magenta dashed lines) in panel (b). Upper and lower bounds by Caprini [30] are shown by cyan dashed lines for two cases of the discontinuity calculated with the partial-wave amplitude $f_1\left(t\right)$ based on (a) the improved $N/D$ model [52], and (b) taken from Ref. [26]. The calculation based on a model-independent method using Canterbury approximants [53] is shown by a magenta long-dashed line with a gray error band. The basic calculation (blue dashed line) from JPAC [27] and the effect from including higher-order terms of the inelastic contributions in the $\omega {\pi }^0$ TFF by fitting them to the NA60 In-In data is shown in panel (c) for the solutions with adding one (black dotted line) and two (red dash-dotted line) terms. A similar effect from including higher-order terms by fitting them to the present $|F_{\omega {\pi }^0}\left(m_{{\ell }^{+}{\ell }^{-}}\right){|}^2$ results is shown in panel (c) for the solutions with one (magenta long-dashed line) and with two (cyan dash-double-dotted line) terms.