Measurement of the Branching Fraction, Polarization, and $CP$ Asymmetry for $B^0\to {\rho }^{+}{\rho }^{-}$ Decays, and Determination of the Cabibbo-Kobayashi-Maskawa Phase ${\varphi }_2$

We have measured the branching fraction $\mathcal{B}$, longitudinal polarization fraction $f_L$, and $CP$ asymmetry coefficients $\mathcal{A}$ and $\mathcal{S}$ for $B^0\to {\rho }^{+}{\rho }^{-}$ decays with the Belle detector at the KEKB $e^{+}{e}^{-}$ collider using $253{\mathrm{fb}}^{-1}$ of data. We obtain $\mathcal{B}=[22.8\pm 3.8(\mathrm{stat}{)}_{-2.6}^{+2.3}(\mathrm{syst})]\times {10}^{-6}$, $f_L={0.941}_{-0.040}^{+0.034}(\mathrm{stat})\pm 0.030(\mathrm{syst})$, $\mathcal{A}=0.00\pm 0.30(\mathrm{stat})\pm 0.09(\mathrm{syst})$, and $\mathcal{S}=0.08\pm 0.41(\mathrm{stat})\pm 0.09(\mathrm{syst})$. These values are used to constrain the Cabibbo-Kobayashi-Maskawa phase ${\varphi }_2$; the solution consistent with the standard model is ${\varphi }_2=(88\pm 17)°$ or $59°<{\varphi }_2<115°$ at 90% C.L.

Figure 1

(a) $M_{\mathrm{bc}}$ for events with $\Delta E\in (-0.10,0.06)\mathrm{GeV}$. (b) $\Delta E$ for $M_{\mathrm{bc}}\in (5.27,5.29)\mathrm{GeV}/c^2$. The curves show fit projections: the dashed curve is ${\rho }^{+}{\rho }^{-}+\rho \pi \pi$, the dotted curve is $q\overline{q}$, the dot-dashed curve is $b\to c$, the small solid curve is $b\to u$, and the large solid curve is the total.

Figure 2

(a) $M_{{\pi }^{\pm }{\pi }^0}$ for events in the $M_{\mathrm{bc}}-\Delta E$ signal region that satisfy $M_{\pi {\pi }^0}(\mathrm{\text{not fit}})\in (0.62,0.92)\mathrm{GeV}/c^2$. (b) Sum of $\cos {\theta }_{\pm }$ distributions for events in the signal region that satisfy $M_{{\pi }^{\pm }{\pi }^0}(\mathrm{\text{both}})\in (0.62,0.92)\mathrm{GeV}/c^2$. The curves show fit projections: the dashed curve is ${\rho }^{+}{\rho }^{-}$, the dot-dashed curve is $\rho \pi \pi$, the dotted curve is $q\overline{q}+(b\to c)+(b\to u)$, and the solid curve is the total.

Figure 3

The $\Delta t$ distribution of events in the $M_{\mathrm{bc}}-\Delta E$ signal region that satisfy $M_{{\pi }^{\pm }{\pi }^0}\in (-0.62,0.92)\mathrm{GeV}/c^2$, and projections of the fit. (a) $q=+1$ tags; (b) $q=-1$ tags; (c) raw $CP$ asymmetry for good tags ($0.5<r<1.0$); (d) $1-\mathrm{C}.\mathrm{L}.$ vs ${\varphi }_2$. In (a), the hatched region (dashed line) shows signal (background) events. In (d), the solid (dashed) horizontal line denotes $\mathrm{C}.\mathrm{L}.=68.3\%$ (90%).