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

Measurement of the $ω おめが \to {π ぱい}^{0} e^{+} e^{-}$ and $η いーた \to e^{+} e^{-} γ がんま$ Dalitz decays with the A2 setup at the Mainz Microtron

P. Adlarson et al. (A2 Collaboration at MAMI)

Phys. Rev. C 95, 035208 – Published 22 March 2017

Abstract

The Dalitz decays $η いーた \to e^{+} e^{-} γ がんま$ and $ω おめが \to {π ぱい}^{0} e^{+} e^{-}$ have been measured in the $γ がんま p \to η いーた p$ and $γ がんま p \to ω おめが p$ 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 $η いーた, {Λ らむだ}_{η いーた}^{- 2} = (1.97 \pm 0 . 11_{tot}) {GeV}^{- 2}$ , is in good agreement with previous measurements of the $η いーた \to e^{+} e^{-} γ がんま$ and $η いーた \to {μ みゅー}^{+} {μ みゅー}^{-} γ がんま$ decays. The uncertainty obtained in the value of ${Λ らむだ}_{η いーた}^{- 2}$ is lower than in previous results based on the $η いーた \to e^{+} e^{-} γ がんま$ decay. The value obtained for the $ω おめが$ slope parameter, ${Λ らむだ}_{ω おめが {π ぱい}^{0}}^{- 2} = (1.99 \pm 0 . 21_{tot}) {GeV}^{- 2}$ , is somewhat lower than previous measurements based on $ω おめが \to {π ぱい}^{0} {μ みゅー}^{+} {μ みゅー}^{-}$ , but the results for the $ω おめが$ transition form factor are in better agreement with theoretical calculations, compared to earlier experiments.

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Received 15 September 2016

DOI:https://doi.org/10.1103/PhysRevC.95.035208

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Form factors

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Vol. 95, Iss. 3 — March 2017

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Figure 1
A general sketch of the Crystal Ball, TAPS, and particle identification (PID) detectors.
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Figure 2
Comparison of the $e^{+ / -} d E / d x$ of the PID for experimental $η いーた \to e^{+} e^{-} γ がんま$ decays and their MC simulation. The two-dimensional density distribution (with logarithmic scale along plot axis $z$ ) for the $e^{+ / -} d E / d x$ 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 $e^{+ / -} d E / d x$ distributions for the experimental data (crosses) and the MC simulation (blue solid line) are compared in panel (c).
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Figure 3
(a), (c) Two-dimensional density distributions for events selected as the $γ がんま p \to {π ぱい}^{0} e^{+} e^{-} p$ candidates obtained from the MC simulations of $ω おめが \to {π ぱい}^{0} e^{+} e^{-}$ decays and (b), (d) $ω おめが \to {π ぱい}^{0} {π ぱい}^{+} {π ぱい}^{-}$ decays causing background in the vicinity of the $ω おめが$ 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 $ω おめが \to {π ぱい}^{0} {π ぱい}^{+} {π ぱい}^{-}$ 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.
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Figure 4
$m (e^{+} e^{-} γ がんま)$ invariant-mass distributions obtained for the $m (e^{+} e^{-})$ range from 30 to 100 MeV/ $c^{2}$ with $γ がんま p \to e^{+} e^{-} γ がんま p$ 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 $γ がんま p \to η いーた p \to e^{+} e^{-} γ がんま p$ fitted with a Gaussian, (b) experimental events (Run I) after subtracting the random background and the remaining background from $γ がんま p \to η いーた p \to γ がんま γ がんま p$ . The distribution for the $η いーた \to γ がんま γ がんま$ 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 $η いーた \to e^{+} e^{-} γ がんま$ 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.
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Figure 5
Same as Fig. 4, but requiring both $e^{+}$ and $e^{-}$ to be identified by different PID elements.
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Figure 6
Same as Fig. 5, but after applying the $d E / d x$ 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.
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Figure 7
The $η いーた \to γ がんま {γ がんま}^{*} \to γ がんま e^{+} e^{-}$ angular dependence (in the $η いーた$ rest frame) of the virtual photon decaying into a lepton pair, with ${θ しーた}^{*}$ 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 $γ がんま$ direction): (a) experimental events from the $η いーた \to γ がんま e^{+} e^{-}$ peak, (b) angular acceptance based on the MC simulation, (c) the experimental spectrum corrected for the acceptance and normalized for comparing to the $1 + {cos}^{2} {θ しーた}^{*}$ 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 $cos {θ しーた}^{*} = 0$ .
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Figure 8
Same as Fig. 4, but for $m (e^{+} e^{-}) = (360 \pm 20)$ MeV/ $c^{2}$ . The $η いーた \to γ がんま γ がんま$ background is not shown in panel (b) because it is negligibly small.
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Figure 9
Same as Fig. 8, but after applying the softer cuts on $N$ and $R$ (depicted by red dashed lines in Fig. 3).
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Figure 10
Same as Fig. 8, but after applying the tighter cuts on $N$ and $R$ (depicted by blue solid lines in Fig. 3).
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Figure 11
$m ({π ぱい}^{0} e^{+} e^{-})$ invariant-mass distributions obtained for the $m (e^{+} e^{-})$ range from 30 to 100 MeV/ $c^{2}$ with $γ がんま p \to {π ぱい}^{0} e^{+} e^{-} p$ candidates initially selected with the kinematic fit (a) MC simulation of $γ がんま p \to ω おめが p \to {π ぱい}^{0} e^{+} e^{-} p$ fitted with a Gaussian; (b) experimental events (Run I) after subtracting the random background and the remaining background from $ω おめが \to {π ぱい}^{0} γ がんま$ decays (with both the ${π ぱい}^{0} \to γ がんま γ がんま$ and ${π ぱい}^{0} \to e^{+} e^{-} γ がんま$ decay modes). The distribution for the $ω おめが \to {π ぱい}^{0} γ がんま$ 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 $ω おめが \to {π ぱい}^{0} e^{+} e^{-}$ 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.
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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 $d E / d x$ cut rejecting events with twice the magnitude of a signal from a single $e^{+ / -}$ in one PID element.
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Figure 13
The $ω おめが \to {π ぱい}^{0} {γ がんま}^{*} \to {π ぱい}^{0} e^{+} e^{-}$ angular dependence (in the $ω おめが$ rest frame) of the virtual photon decaying into a lepton pair, with ${θ しーた}^{*}$ 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 ${π ぱい}^{0}$ direction): (a) experimental events from the $ω おめが \to {π ぱい}^{0} e^{+} e^{-}$ peak; (b) angular acceptance based on the MC simulation; (c) experimental spectrum corrected for the acceptance and normalized for comparing to the $1 + {cos}^{2} {θ しーた}^{*}$ 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 $cos {θ しーた}^{*} = 0$ .
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Figure 14
Same as Fig. 11, but for the $m (e^{+} e^{-})$ 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 $ω おめが \to {π ぱい}^{0} γ がんま$ background is not shown in panel (b) because it is negligibly small.
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Figure 15
Same as Fig. 14, but after applying the $d E / d x$ cut rejecting events with twice the magnitude of a signal from a single $e^{+ / -}$ in one PID element.
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Figure 16
Comparison of the $| F_{η いーた} (m_{ℓ^{+} ℓ^{-}}) |^{2}$ 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 $p 0$ reflects the general normalization of the data points, and $p 1$ is the slope parameter ${Λ らむだ}^{- 2} [{GeV}^{- 2}]$ . 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).
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Figure 17
$| F_{η いーた} (m_{ℓ^{+} ℓ^{-}}) |^{2}$ results (black filled triangles) combined from Run I and Run II and their pole-approximation fit (black solid line, with $p 0$ and $p 1$ being the normalization and the slope parameter ${Λ らむだ}^{- 2}$ , 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).
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Figure 18
Comparison of the $| F_{ω おめが {π ぱい}^{0}} (m_{ℓ^{+} ℓ^{-}}) |^{2}$ 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 $ω おめが \to 3 π ぱい$ partial wave $f_{1} (s) = α あるふぁ Ω おめが (s)$ inside the dispersion integral is shown by cyan dashed lines, and the solution using the full rescattering of $3 π ぱい$ 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 $p 0$ reflects the general normalization of the data points, and $p 1$ is the slope parameter ${Λ らむだ}^{- 2}$ . 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).
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Figure 19
$| F_{ω おめが {π ぱい}^{0}} (m_{ℓ^{+} ℓ^{-}}) |^{2}$ results (black filled triangles) combined from Run I and Run II and their pole-approximation fit (black solid line, with $p 0$ and $p 1$ being the normalization and the slope parameter ${Λ らむだ}^{- 2}$ , respectively) are compared to previous measurements and various theoretical calculations. The results by Lepton- $G$ [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 $3 π ぱい$ 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 $f_{1} (t)$ 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 $ω おめが {π ぱい}^{0}$ 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 $| F_{ω おめが {π ぱい}^{0}} (m_{ℓ^{+} ℓ^{-}}) |^{2}$ results is shown in panel (c) for the solutions with one (magenta long-dashed line) and with two (cyan dash-double-dotted line) terms.
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Physical Review C

covering nuclear physics

Measurement of the $ω おめが \to {π ぱい}^{0} e^{+} e^{-}$ and $η いーた \to e^{+} e^{-} γ がんま$ Dalitz decays with the A2 setup at the Mainz Microtron

P. Adlarson et al. (A2 Collaboration at MAMI)

Phys. Rev. C 95, 035208 – Published 22 March 2017

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Physical Review C

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Measurement of the ωおめが→πぱい0e+e− and ηいーた→e+e−γがんま Dalitz decays with the A2 setup at the Mainz Microtron

P. Adlarson et al. (A2 Collaboration at MAMI)

Phys. Rev. C 95, 035208 – Published 22 March 2017

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Measurement of the $ω おめが \to {π ぱい}^{0} e^{+} e^{-}$ and $η いーた \to e^{+} e^{-} γ がんま$ Dalitz decays with the A2 setup at the Mainz Microtron