First Evidence for $\cos 2\beta >0$ and Resolution of the Cabibbo-Kobayashi-Maskawa Quark-Mixing Unitarity Triangle Ambiguity

We present first evidence that the cosine of the $CP$-violating weak phase $2\beta$ is positive, and hence exclude trigonometric multifold solutions of the Cabibbo-Kobayashi-Maskawa (CKM) Unitarity Triangle using a time-dependent Dalitz plot analysis of $B^0\to D^{(*)}{h}^0$ with $D\to K_S^0{\pi }^{+}{\pi }^{-}$ decays, where $h^0\in \{{\pi }^0,\eta ,\omega \}$ denotes a light unflavored and neutral hadron. The measurement is performed combining the final data sets of the BABAR and Belle experiments collected at the $\mathrm{\Upsilon }(4S)$ resonance at the asymmetric-energy $B$ factories PEP-II at SLAC and KEKB at KEK, respectively. The data samples contain $(471\pm 3)\times {10}^{6}B\overline{B}$ pairs recorded by the BABAR detector and $(772\pm 11)\times {10}^{6}B\overline{B}$ pairs recorded by the Belle detector. The results of the measurement are $\sin 2\beta =0.80\pm 0.14(\mathrm{stat})\pm 0.06(\mathrm{syst})\pm 0.03(\text{model})$ and $\cos 2\beta =0.91\pm 0.22(\mathrm{stat})\pm 0.09(\mathrm{syst})\pm 0.07(\text{model})$. The result for the direct measurement of the angle $\beta$ of the CKM Unitarity Triangle is $\beta =[22.5\pm 4.4(\mathrm{stat})\pm 1.2(\mathrm{syst})\pm 0.6(\text{model})]°$. The measurement assumes no direct $CP$ violation in $B^0\to D^{(*)}{h}^0$ decays. The quoted model uncertainties are due to the composition of the $D^0\to K_S^0{\pi }^{+}{\pi }^{-}$ decay amplitude model, which is newly established by performing a Dalitz plot amplitude analysis using a high-statistics $e^{+}{e}^{-}\to c\overline{c}$ data sample. $CP$ violation is observed in $B^0\to D^{(*)}{h}^0$ decays at the level of 5.1 standard deviations. The significance for $\cos 2\beta >0$ is 3.7 standard deviations. The trigonometric multifold solution $\pi /2-\beta =(68.1\pm 0.7)°$ is excluded at the level of 7.3 standard deviations. The measurement resolves an ambiguity in the determination of the apex of the CKM Unitarity Triangle.

Figure 1

The Dalitz plot data distributions (points with error bars) for $D^0\to K_S^0{\pi }^{+}{\pi }^{-}$ from $D^{*+}\to D^0{\pi }_s^{+}$ decays reconstructed from Belle $e^{+}{e}^{-}\to c\overline{c}$ data, and projections of the Dalitz fit. The red solid lines show the projections of the total fit function including background, and the grey regions show projections of the background.

Figure 2

Data distributions for (a) $M_{\mathrm{bc}}^{\prime }$, (b) $\mathrm{\Delta }E$, and (c) ${\mathcal{C}}_{N{N}_{\mathrm{out}}}^{\prime }$ (points with error bars) for the BABAR and Belle data samples combined. The solid black lines represent projections of the total fit function, and the colored dotted lines show the signal and background components of the fit as indicated in the legend. In plotting the $M_{\mathrm{bc}}^{\prime }$, $\mathrm{\Delta }E$, and ${\mathcal{C}}_{N{N}_{\mathrm{out}}}^{\prime }$ distributions, each of the other two observables are required to satisfy $M_{\mathrm{bc}}^{\prime }>5.272\mathrm{GeV}{c}^2$, $|\mathrm{\Delta }E|<100\mathrm{MeV}$, or $0<{\mathcal{C}}_{N{N}_{\mathrm{out}}}^{\prime }<8$ to select signal-enhanced regions.

Figure 3

Distributions of the proper-time interval (data points with error bars) and the corresponding asymmetries for $B^0\to D^{(*)}{h}^0$ candidates associated with high-quality flavor tags for two different regions of the $D\to K_S^0{\pi }^{+}{\pi }^{-}$ phase space and for the BABAR and Belle data samples combined. The background has been subtracted using the ${}_s\mathcal{P}lot$ technique [50], with weights obtained from the fit presented in Fig. 2.