Measurement of $CP$ asymmetries and branching fractions in a time-dependent Dalitz analysis of $B^0\to (\rho \pi {)}^0$ and a constraint on the quark mixing angle ${\varphi }_2$

We present the results of a time-dependent Dalitz plot analysis of $B^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ decays based on a $414{\mathrm{fb}}^{-1}$ data sample that contains $449\times {10}^{6}B\overline{B}$ pairs. The data were collected on the ${\rm Y}(4S)$ resonance with the Belle detector at the KEKB asymmetric energy $e^{+}{e}^{-}$ collider. Combining our analysis with information on charged $B$ decay modes, we perform a full Dalitz and isospin analysis and obtain a constraint on the quark mixing angle ${\varphi }_2$, $68°<{\varphi }_2<95°$ at the 68.3% confidence level for the ${\varphi }_2$ solution consistent with the standard model. A large standard-model–disfavored region also remains. The branching fractions for the decay processes $B^0\to {\rho }^{\pm }(770){\pi }^{\mp }$ and $B^0\to {\rho }^0(770){\pi }^0$ are measured to be $(22.6\pm 1.1[\mathrm{stat}]\pm 4.4[\mathrm{syst}])\times {10}^{-6}$ and $(3.0\pm 0.5[\mathrm{stat}]\pm 0.7[\mathrm{syst}])\times {10}^{-6}$, respectively. These are the first branching fraction measurements of the process $B^0\to \rho (770)\pi$ with the lowest resonance $\rho (770)$ explicitly separated from the radial excitations.

Figure 1

The relation between three pions in the rest frame of ${\rho }^{\kappa }$.

Figure 2

Distribution of signal Monte Carlo simulation (without detector efficiency and smearing) in the usual Dalitz plot. The dashed line is the kinematic boundary, while the hatched region corresponds to the region rejected by the mass cut described in Sec. 2.

Figure 3

Distribution of $q\overline{q}$ background (from the data $M_{\mathrm{bc}}$ sideband) in the usual Dalitz plot (left panel) and square Dalitz plot (right panel).

Figure 4

Distribution of signal Monte Carlo simulation (without detector efficiency and smearing) in the square Dalitz plot. The solid, dashed, and dotted lines correspond to the isocontours of $\sqrt{s_{\kappa }}=0.5\mathrm{GeV}$, 1.0 GeV, and 1.5 GeV, respectively, for each ${\rho }^{\kappa }$ resonance.

Figure 5

The $M_{\mathrm{bc}}$ (a) and $\Delta E$ (b) distributions within the $\Delta E$ and $M_{\mathrm{bc}}$ signal regions. The histograms are cumulative. Solid, dot-dashed, dotted, and dashed hatched histograms correspond to correctly reconstructed signal, SCF, $B\overline{B}$, and continuum PDFs, respectively.

Figure 6

Mass distributions and fitted line shapes in ${\rho }^{+}{\pi }^{-}$ (left panel), ${\rho }^{-}{\pi }^{+}$ (middle panel), and ${\rho }^0{\pi }^0$ (right panel) enhanced regions. The histograms are cumulative. Solid, dot-dashed, dotted, and dashed hatched histograms correspond to correctly reconstructed signal, SCF, $B\overline{B}$, and continuum PDFs, respectively. Note that there are feed-downs from other quasi-two-body components than those of interest, especially in the high mass regions. For example, the high mass region ($m_0\gtrsim 1.0\mathrm{GeV}/c^2$) of the ${\rho }^0{\pi }^0$ enhanced region (right panel) includes large contributions from ${\rho }^{\pm }{\pi }^{\mp }$.

Figure 7

A schematic figure of the fit result of the line shape and the contributions from radial excitations. Note that our definition of $F_{\pi }(s)$ does not include the factor $1/(1+\beta +\gamma )$ as in Eq. (17). The inset shows the high mass region, $m_{\pi \pi }>1.15\mathrm{GeV}/c^2$, on a semilog scale where the interference between the $\rho (770)$ and radial excitations is visible. One can see that the $\rho (770)$ and $\rho (1450)$ destructively interfere with each other near $\sqrt{s}\equiv m_{\pi \pi }=1.4\mathrm{GeV}/c^2$, which means that the $\rho (1450)$ has a large impact on the phase of $F_{\pi }(s)$, although the absolute value of $|F_{\pi }(s)|$ is not much affected.

Figure 8

Distributions of ${\theta }^{\prime }$ (a) and $m^{\prime }$ (b) with fit results. The histograms are cumulative. Solid, dot-dashed, dotted, and dashed hatched histograms correspond to correctly reconstructed signal, SCF, $B\overline{B}$, and continuum PDFs, respectively.

Figure 9

Mass (upper panels) and helicity (lower panels) distribution of ${\rho }^{+}{\pi }^{-}$ (left panels), ${\rho }^{-}{\pi }^{+}$ (middle panels), and ${\rho }^0{\pi }^0$ (right panels) enhanced regions. The histograms are cumulative. Solid, dot-dashed, dotted, and dashed hatched histograms correspond to correctly reconstructed signal, SCF, $B\overline{B}$, and continuum PDFs, respectively.

Figure 10

Proper time distributions of good-tag ($r>0.5$) regions for $f_{\mathrm{tag}}=B^0$ (upper plots) and $f_{\mathrm{tag}}={\overline{B}}^0$ (middle upper plots), in ${\rho }^{+}{\pi }^{-}$ (left plots), ${\rho }^{-}{\pi }^{+}$ (middle plots), ${\rho }^0{\pi }^0$ (right plots) enhanced regions, where solid (red), dotted, and dashed curves correspond to signal, continuum, and $B\overline{B}$ PDFs. The middle lower and lower plots show the background-subtracted asymmetries in the good-tag ($r>0.5$) and poor-tag ($r<0.5$) regions, respectively. The significant asymmetry in the ${\rho }^{-}{\pi }^{+}$ enhanced region (middle plots) corresponds to a nonzero value of $U_{-}^{-}$.

Figure 11

Contour plot of the confidence level for the direct $CP$ violation parameters ${\mathcal{A}}_{\rho \pi }^{+-}$ vs ${\mathcal{A}}_{\rho \pi }^{-+}$.

Figure 12

$1-\mathrm{C}.\mathrm{L}.$ vs ${\varphi }_2$. Dotted and solid curves correspond to the result from the time-dependent Dalitz plot analysis only and that from the Dalitz and an isospin (pentagon) combined analysis, respectively.

Figure 13

Complex pentagons formed from the $B\to \rho \pi$ decay amplitudes.