Study of $B^0\to {\rho }^0{\rho }^0$ decays, implications for the CKM angle ${\varphi }_2$ and search for other $B^0$ decay modes with a four-pion final state

We present a study of the branching fraction of the decay $B^0\to {\rho }^0{\rho }^0$ and the fraction of longitudinally polarized ${\rho }^0$ mesons in this decay. The results are obtained from the final data sample containing $772\times 10^6$ $B\overline{B}$ pairs collected at the $\mathrm{\Upsilon }(4S)$ resonance with the Belle detector at the KEKB asymmetric-energy $e^{+}{e}^{-}$ collider. We find $166\pm 59$ $B^0\to {\rho }^0{\rho }^0$ events (including systematic uncertainties), corresponding to a branching fraction of $\mathcal{B}(B^0\to {\rho }^0{\rho }^0)=(1.02\pm 0.30(\text{stat})\pm 0.15(\text{syst}))\times 10^{-6}$ with a significance of 3.4 standard deviations and a longitudinal polarization fraction $f_L=0.21_{-0.22}^{+0.18}(\text{stat})\pm 0.15(\text{syst})$. We use the longitudinal polarization fraction to determine the Cabibbo-Kobayashi-Maskawa matrix angle ${\varphi }_2=(84.9\pm 13.5)°$ through an isospin analysis in the $B\to \rho \rho$ system. We furthermore find $125\pm 41$ $B^0\to f_0{\rho }^0$ events, corresponding to $\mathcal{B}(B^0\to f_0{\rho }^0)\times \mathcal{B}(f_0\to {\pi }^{+}{\pi }^{-})=(0.78\pm 0.22(\text{stat})\pm 0.11(\text{syst}))\times 10^{-6}$, with a significance of 3.1 standard deviations. We find no other significant contribution with the same final state, and set upper limits at 90% confidence level on the (product) branching fractions, $\mathcal{B}(B^0\to {\pi }^{+}{\pi }^{-}{\pi }^{+}{\pi }^{-})<11.2\times 10^{-6}$, $\mathcal{B}(B^0\to {\rho }^0{\pi }^{+}{\pi }^{-})<12.0\times 10^{-6}$, $\mathcal{B}(B^0\to f_0{\pi }^{+}{\pi }^{-})\times \mathcal{B}(f_0\to {\pi }^{+}{\pi }^{-})<3.0\times 10^{-6}$ and $\mathcal{B}(B^0\to f_0{f}_0)\times \mathcal{B}(f_0\to {\pi }^{+}{\pi }^{-}{)}^2<0.2\times 10^{-6}$.

Figure 1

Leading-order tree (left) and penguin (right) diagrams for the decay $B^0\to {\rho }^0{\rho }^0$.

Figure 2

Definition of the helicity angles ${\theta }_{Hk}$ for each $\rho$, identified by index $k=1,2$.

Figure 3

Simulated MC and off-resonance distributions for the quantities used to construct the Fisher discriminant ${\mathcal{F}}_{S/B}$, which is shown in the lower right plot. The solid (blue) histograms show the distribution for $B\overline{B}$ MC events, while the dashed (red) histograms show the distributions for events from off-resonance data, both normalized to the same area. The vertical line indicates the requirement of $|\cos {\theta }_{\mathrm{TB},\mathrm{TO}}|<0.9$ being applied before the training of the Fisher.

Figure 4

Projections of the fit to the data. The points represent the data and the solid curves represent the fit result. The shaded (red) areas show the $B^0\to {\rho }^0{\rho }^0$ and the (orange) long dashed curves the $B^0\to f_0{\rho }^0$ contribution. The bright-shaded (cyan) areas show all four-pion final states, the (dark green) dashed curves show the nonpeaking $B\overline{B}$ contribution and the (bright green) dash-dotted curves show the total nonpeaking background. Panel (a) shows the $\mathrm{\Delta }E$ projection in the ${\mathcal{F}}_{S/B}>0.5$ region; (d) shows the ${\mathcal{F}}_{S/B}$ projection in the $|\mathrm{\Delta }E|<0.02\mathrm{GeV}$ region. Here, the (dark red) dash double-dotted curve shows only the continuum background contribution. Panels (b) and (c) show the $m_{{\pi }^{+}{\pi }^{-}}$ projections and (e) and (f) the $\cos {\theta }_H$ projections in the ${\mathcal{F}}_{S/B}>0.5$ and $|\mathrm{\Delta }E|<0.02\mathrm{GeV}$ region. The residuals are plotted below each distribution.

Figure 5

Top: Likelihood scans of $\mathcal{B}(B^0\to {\rho }^0{\rho }^0)$, $f_L$ and $\mathcal{B}(B^0\to f_0{\rho }^0)$ convolved with the corresponding systematic uncertainty. Bottom: 2 D likelihood scans of $\mathcal{B}(B^0\to {\rho }^0{\rho }^0)$ versus $f_L$ (left and middle) and versus $B^0\to f_0{\rho }^0$ (right). The statistical likelihood scan of $\mathcal{B}(B^0\to {\rho }^0{\rho }^0)$ versus $f_L$ (left) demonstrates the expected insensitivity of the likelihood to $f_L$ for $\mathcal{B}(B^0\to {\rho }^0{\rho }^0)=0$. The middle and right scans are convoluted with the corresponding systematic uncertainties. The minimum (white spot) presents the fit result obtained from this measurement.

Figure 6

Projections of the fit to the data into a ${\rho }^0{\rho }^0$ window $0.60\mathrm{GeV}/{\mathrm{c}}^2<m_{{\pi }^{+}{\pi }^{-}}<0.88\mathrm{GeV}/{\mathrm{c}}^2$ (upper row) and a $f_0{\rho }^0$ window $0.6\mathrm{GeV}/{\mathrm{c}}^2<M_k<0.88\mathrm{GeV}/{\mathrm{c}}^2$ and $0.88\mathrm{GeV}/{\mathrm{c}}^2<M_l<1.08\mathrm{GeV}/{\mathrm{c}}^2$ (lower row). The points represent the data and the solid curves represent the fit result. The shaded (red) areas show the $B^0\to {\rho }^0{\rho }^0$ and the (orange) long dashed curves the $B^0\to f_0{\rho }^0$ contribution. The bright shaded (cyan) areas show all four-pion final states, the (dark green) dashed curves show the nonpeaking $B\overline{B}$ contribution and the (bright green) dash-dotted curves show the total nonpeaking background. Panels (a) and (d) show the projection into $\mathrm{\Delta }E$ for ${\mathcal{F}}_{S/B}>0.5$ and (b), (c) and (e), (f) show into $\cos {\theta }_H$ for ${\mathcal{F}}_{S/B}>0.5$ and $|\mathrm{\Delta }E|<0.02\mathrm{GeV}$. The residuals are plotted below each distribution.

Figure 7

$1-\mathrm{C}\mathrm{.}\mathrm{L}\mathrm{.}$ versus ${\varphi }_2$ obtained from an isospin analysis from $B\to \rho \rho$ decays. The horizontal line shows the 68% confidence level (C.L.).