Dalitz plot analysis of the $D^{+}\to {\pi }^{-}{\pi }^{+}{\pi }^{+}$ decay

Using $281{\mathrm{pb}}^{-1}$ of data recorded by the CLEO-c detector in $e^{+}{e}^{-}$ collisions at the $\psi (3770)$, corresponding to $0.78\times {10}^6$ $D^{+}{D}^{-}$ pairs, we investigate the substructure of the decay $D^{+}\to {\pi }^{-}{\pi }^{+}{\pi }^{+}$ using the Dalitz plot technique. We find that our data are consistent with the following intermediate states: $\rho (770){\pi }^{+}$, $f_2(1270){\pi }^{+}$, $f_0(1370){\pi }^{+}$, $f_0(1500){\pi }^{+}$, $f_0(980){\pi }^{+}$, and $\sigma {\pi }^{+}$. We confirm large $S$ wave contributions at low $\pi \pi$ mass. We set upper limits on contributions of other possible intermediate states. We consider three models of the $\pi \pi$ $S$ wave and find that all of them adequately describe our data.

Figure 1

The $m_{\mathrm{BC}}$ vs. $\Delta E$ distribution of events passing all selection requirements described in the text. The center box shows the signal region for the Dalitz plot analysis. The two hatched boxes show the sidebands. The vertical and horizontal lines restrict the regions of events plotted in Figs. 2 and 3, respectively.

Figure 2

The $m_{\mathrm{BC}}$ distribution of events from the $|\Delta E|<2\sigma (\Delta E)$ range. Dashed curve shows a contribution from the background, dotted curve is a Gaussian part of the crystal ball function for the signal shape, and solid curve is total, signal plus background. Events between arrows are selected for the Dalitz plot analysis.

Figure 3

The $\Delta E$ distribution of events from the $|m_{\mathrm{BC}}-m_D|<2\sigma (m_{\mathrm{BC}})$ range. Events between the arrows are selected for the Dalitz plot.

Figure 4

The $m({\pi }^{+}{\pi }^{-}{)}_{\mathrm{Low}}$ distribution of events preselected for the Dalitz plot. A clear signal for the $K_S^0\to {\pi }^{+}{\pi }^{-}$ decay is observed. Events in the range between the arrows, $0.2<m^2({\pi }^{+}{\pi }^{-}{)}_{\mathrm{Low}}<0.3(\mathrm{GeV}/c^2{)}^2$, are discarded from the Dalitz plot analysis.

Figure 5

The Dalitz plot for $D^{+}\to {\pi }^{-}{\pi }^{+}{\pi }^{+}$ candidates.

Figure 6

The adaptive binning scheme.

Figure 7

The signal p.d.f. for the isobar model fit described in the text.

Figure 8

Projection of the Dalitz plot onto the $m^2({\pi }^{+}{\pi }^{-})$ axis (two combinations per $D^{+}$ candidate) for CLEO-c data (points) and isobar model fit (histograms) showing the various components.

Figure 9

Projection of the Dalitz plot onto the $m^2({\pi }^{+}{\pi }^{+})$ axis for CLEO-c data (points) and isobar model fit (histograms) showing the various components.

Figure 10

Projection of the Dalitz plot onto the $m^2({\pi }^{+}{\pi }^{-})$ axis (two combinations per $D^{+}$ candidate) for CLEO-c data (points) and Schechter model fit #S3 (histograms) showing the various components.

Figure 11

Projection of the Dalitz plot onto the $m^2({\pi }^{+}{\pi }^{+})$ axis for CLEO-c data (points) and Schechter model fit #S3 (histograms) showing the various components.

Figure 12

The isolated $S$ wave contribution of Schechter model fit #S3 on the Dalitz plot.

Figure 13

The ${\pi }^{+}{\pi }^{-}$ scattering phase $\delta (m)$, Eq. (a12), calculated for parameters from Schechter model fit #S3 to the $D^{+}\to {\pi }^{-}{\pi }^{+}{\pi }^{+}$ Dalitz plot.

Figure 14

Complex $S$ wave amplitude from Schechter model fit #S3 to the $D^{+}\to {\pi }^{-}{\pi }^{+}{\pi }^{+}$ Dalitz plot. The real and imaginary parts, the magnitude and phase, are shown as a function of ${\pi }^{+}{\pi }^{-}$ mass.

Figure 15

Projection of the Dalitz plot onto the $m^2({\pi }^{+}{\pi }^{-})$ axis for CLEO-c data (points) and Achasov model fit #A2 (histograms) showing the various components.

Figure 16

Projection of the Dalitz plot onto the $m^2({\pi }^{+}{\pi }^{+})$ axis for CLEO-c data (points) and Achasov model fit #A2 (histograms) showing the various components.

Figure 17

Complex $S$ wave amplitude from Achasov model fit #A2 to the $D^{+}\to {\pi }^{-}{\pi }^{+}{\pi }^{+}$ Dalitz plot. The real and imaginary parts, the magnitude and phase, are shown as a function of ${\pi }^{+}{\pi }^{-}$ mass.

Figure 18

Complex $S$ wave amplitude [complex pole for $\sigma$ and Flatté for $f_0(980)$] from isobar model fit to the $D^{+}\to {\pi }^{-}{\pi }^{+}{\pi }^{+}$ Dalitz plot. The real and imaginary parts, the magnitude and phase, are shown as a function of ${\pi }^{+}{\pi }^{-}$ mass.

Figure 19

The ${\pi }^{+}{\pi }^{-}$ $S$ wave absolute amplitude for different models.

Figure 20

The ${\pi }^{+}{\pi }^{-}$ $S$ wave phase for different models.

Figure 21

The background phase in $\pi \pi$ scattering, ${\delta }_B^{\pi \pi }(m)$, from Eq. (a36).

Figure 22

The phase of the resonance $\pi \pi$ scattering, ${\delta }_{\mathrm{res}}(m)$, from Eq. (a34).

Figure 23

The total $\pi \pi$ scattering phase, ${\delta }_0^0(m)$, from Eq. (a31).

Figure 24

The inelasticity, ${\eta }_0^0$, from Eq. (a33) for $2m_{\pi }<m<2\mathrm{GeV}/c^2$.

Figure 25

The background phase in $K\overline{K}$ scattering, ${\delta }_B^{K\overline{K}}$, from Eq. (a44) (solid curve) and its approximation by the phase space factor (dashed curve).

Figure 26

The total background phase in $K\overline{K}\to \pi \pi$ scattering, ${\delta }_B={\delta }_B^{\pi \pi }+{\delta }_B^{K\overline{K}}$, from Eq. (a41).

Figure 27

The loop integral, $L_{K^{+}{K}^{-}}(m,1,0)/16\pi$, from Eq. (a27). The real (solid curve) and imaginary (dashed curve) parts of the complex function are shown.

Figure 28

The loop integral, $L_{{\pi }^{+}{\pi }^{-}}(m,1,0)/16\pi$, from Eq. (a27). The real (solid curve) and imaginary (dashed curve) parts of the complex function are shown.