New determination of the $\eta$ transition form factor in the Dalitz decay $\eta \to e^{+}{e}^{-}\gamma$ with the Crystal Ball/TAPS detectors at the Mainz Microtron

The Dalitz decay $\eta \to e^{+}{e}^{-}\gamma$ has been measured in the $\gamma p\to \eta p$ reaction with the Crystal Ball and TAPS multiphoton spectrometers, together with the photon-tagging facility at the Mainz Microtron MAMI. The experimental statistic used in this work is one order of magnitude greater than in any previous measurement of $\eta \to e^{+}{e}^{-}\gamma$. The value obtained for the slope parameter ${\Lambda }^{-2}$ of the $\eta$ transition form factor, ${\Lambda }^{-2}=(1.95\pm 0.{15}_{\mathrm{stat}}\pm 0.{10}_{\mathrm{syst}})$ GeV${}^{-2}$, is in good agreement with recent measurements conducted in $\eta \to e^{+}{e}^{-}\gamma$ and $\eta \to {\mu }^{+}{\mu }^{-}\gamma$ decays, as well as with recent form-factor calculations. The uncertainty obtained in the value of ${\Lambda }^{-2}$ is lower compared to results from previous measurements of the $\eta \to e^{+}{e}^{-}\gamma$ decay.

Figure 1

Feynman diagrams illustrating (a) the $P\to {\gamma }^{*}\gamma$ transition in VMD and (b) one of the leading contributions to the correction for ${(g-2)}_{\mu }$ owing to the hadronic light-by-light scattering.

Figure 2

Feynman diagrams showing the processes with the $P\to {\gamma }^{*}\gamma$ transition that can be used for investigating TFFs of light pseudoscalar mesons $P$ in three different $q^2$ regions: (a) spacelike ($q^2<0$ GeV${}^2$), (b) timelike [${\left(2m_l\right)}^2<q^2<m_P^2$], and (c) timelike ($q^2>m_P^2$).

Figure 3

A general sketch of the CB, TAPS, and PID detectors.

Figure 4

(a) Difference between the generated and reconstructed invariant mass $m\left(e^{+}{e}^{-}\right)$ for the MC simulation of $\gamma p\to \eta p\to e^{+}{e}^{-}\gamma p$ shown for three-cluster (dashed line) and four-cluster (solid line) events, where both the distributions are normalized to each other. (b) The $\gamma p\to \eta p\to e^{+}{e}^{-}\gamma p$ acceptance as a function of $m\left(e^{+}{e}^{-}\right)$ shown for all (solid squares) and for only four-cluster (open circles) events. (c),(d) Distributions of the invariant mass $m\left(e^{+}{e}^{-}\right)$ as a function of the invariant mass $m\left(e^{+}{e}^{-}\gamma \right)$ for the experimental $\gamma p\to e^{+}{e}^{-}\gamma p$ candidates shown for all and only four-cluster events, respectively.

Figure 5

$m\left(e^{+}{e}^{-}\gamma \right)$ invariant-mass distributions obtained for $m\left(e^{+}{e}^{-}\right)=(45\pm 5)$ MeV/$c^2$ by using both the three- and the four-cluster events. (a) MC simulation of the background reaction $\gamma p\to {\pi }^0{\pi }^{0}p$ with a polynomial fit. (b) MC simulation of $\gamma p\to \eta p\to e^{+}{e}^{-}\gamma p$ with a Gaussian fit. (c) Experimental events (crosses) after subtracting the random and empty-target background; background expected from $\gamma p\to \eta p\to \gamma \gamma p$ (solid line). (d) Experimental events from (c) after subtracting the $\gamma p\to \eta p\to \gamma \gamma p$ background with fitting the sum of a Gaussian and a polynomial.

Figure 6

Same as Fig. 5, but for the four-cluster events only.

Figure 7

Same as Fig. 5, but for $m\left(e^{+}{e}^{-}\right)=(370\pm 10)$ MeV/$c^2$.

Figure 8

Same as Fig. 7, but for the four-cluster events only.

Figure 9

(a) Results for $d\Gamma (\eta \to e^{+}{e}^{-}\gamma )/dm\left(e^{+}{e}^{-}\right)$ obtained by using the four-cluster events only (solid squares) and both the three- and four-cluster events (open circles). The QED prediction with $|F_{\eta }{|}^2=1$ and the QED prediction including the $\eta$ TFF with ${\Lambda }^{-2}=1.95$ GeV${}^{-2}$ are shown by a dashed and a solid line, respectively. (b) Main results (based on the four-cluster events only) of this work (solid squares) for $|F_{\eta }\left(m_{e^{+}{e}^{-}}\right){|}^2$ compared to the results based on all events (open circles); the solid line shows a fit of the $\eta$ TFF parametrized as Eq. (2) to the main results. (c) Fit (solid line) to results for $|F_{\eta }\left(m_{e^{+}{e}^{-}}\right){|}^2$ obtained from combining the three- and four-cluster events and used then to estimate the systematic uncertainty in the ${\Lambda }^{-2}$ value.

Figure 10

Results of this work (solid squares) for the $\eta$ TFF, $|F_{\eta }\left(m_{ll}\right){|}^2$, compared to other recent measurements and theoretical predictions: former data of the A2 Collaboration [7] [open circles in (a)] and the NA60 in peripheral In-In data [8] [open squares in (b)], calculations of Ref. [23] [dash-dotted line in (a)], Ref. [24] [red dashed line with an error band in (a)], and Ref. [25] [dotted line with an error band in (b)]. The solid line is the fit from Fig. 9 rescaled so that $p0=1$.