Measurement of $CP$ observables in $B^{\pm }\to D_{CP}{K}^{\pm }$ decays and constraints on the CKM angle $\gamma$

Using the entire sample of $467\times {10}^6$ ${\rm Y}(4S)\to B\overline{B}$ decays collected with the BABAR detector at the PEP-II asymmetric-energy $B$ factory at the SLAC National Accelerator Laboratory, we perform an analysis of $B^{\pm }\to D{K}^{\pm }$ decays, using decay modes in which the neutral $D$ meson decays to either $CP$-eigenstates or non-$CP$-eigenstates. We measure the partial decay rate charge asymmetries for $CP$-even and $CP$-odd $D$ final states to be $A_{CP+}=0.25\pm 0.06\pm 0.02$ and $A_{CP-}=-0.09\pm 0.07\pm 0.02$, respectively, where the first error is the statistical and the second is the systematic uncertainty. The parameter $A_{CP+}$ is different from zero with a significance of 3.6 standard deviations, constituting evidence for direct $CP$ violation. We also measure the ratios of the charged-averaged $B$ partial decay rates in $CP$ and non-$CP$ decays, $R_{CP+}=1.18\pm 0.09\pm 0.05$ and $R_{CP-}=1.07\pm 0.08\pm 0.04$. We infer frequentist confidence intervals for the angle $\gamma$ of the unitarity triangle, for the strong phase difference ${\delta }_B$, and for the amplitude ratio $r_B$, which are related to the $B^{-}\to D{K}^{-}$ decay amplitude by $r_B{e}^{i({\delta }_B-\gamma )}=A(B^{-}\to {\overline{D}}^0{K}^{-})/A(B^{-}\to D^0{K}^{-})$. Including statistical and systematic uncertainties, we obtain $0.24<r_B<0.45$ ($0.06<r_B<0.51$) and, modulo 180°, $11.3°<\gamma <22.7°$ or $80.8°<\gamma <99.2°$ or $157.3°<\gamma <168.7°$ ($7.0°<\gamma <173.0°$) at the 68% (95%) confidence level.

Figure 1

Distributions of the difference between the $D$ candidate’s invariant mass and the nominal $D^0$ mass [25], as measured in the $B^{\pm }\to D{h}^{\pm }$ samples. All selection criteria described in Sec. 4, except that on the $D$ invariant mass $M_D$, have been applied, including the ${\chi }^2$-based candidate selection. In addition we reduce the background by requiring the fit variables to satisfy $m_{\mathrm{ES}}>5.27\mathrm{GeV}/c^2$, $\Delta E>-0.05\mathrm{GeV}$, and $\mathcal{F}>-0.25$. The $\Delta M_D$ selection requirements are depicted by the vertical lines.

Figure 2

$\Delta E$ projections of the fits to the data, split into subsets of definite $CP$ of the $D$ candidate and charge of the $B$ candidate: (a) $B^{-}\to D_{CP+}{K}^{-}$, (b) $B^{+}\to D_{CP+}{K}^{+}$, (c) $B^{-}\to D_{CP-}{K}^{-}$, (d) $B^{+}\to D_{CP-}{K}^{+}$. The curves are the full PDF (solid, blue), and $B\to D\pi$ (dash-dotted, green) stacked on the remaining backgrounds (dotted, purple). The region between the solid and the dash-dotted lines represents the $B\to DK$ contribution. We show the subsets of the data sample in which the track $h$ from the $B$ decay is identified as a kaon. We require candidates to lie inside the signal-enriched region defined in Sec. 4, except for the plotted variable.

Figure 3

$m_{\mathrm{ES}}$ projections of the fits to the data, split into subsets of definite $CP$ of the $D$ candidate and charge of the $B$ candidate: (a) $B^{-}\to D_{CP+}{K}^{-}$, (b) $B^{+}\to D_{CP+}{K}^{+}$, (c) $B^{-}\to D_{CP-}{K}^{-}$, (d) $B^{+}\to D_{CP-}{K}^{+}$. We show the subsets of the data sample in which the track $h$ from the $B$ decay is identified as a kaon. See caption of Fig. 2 for line definitions. Only a subrange of the whole fit range is shown in order to provide a closer view of the signal peak.

Figure 4

$\mathcal{F}$ projections of the fits to the data, split into subsets of definite $CP$ of the $D$ candidate and charge of the $B$ candidate: (a) $B^{-}\to D_{CP+}{K}^{-}$, (b) $B^{+}\to D_{CP+}{K}^{+}$, (c) $B^{-}\to D_{CP-}{K}^{-}$, (d) $B^{+}\to D_{CP-}{K}^{+}$. We show the subsets of the data sample in which the track $h$ from the $B$ decay is identified as a kaon. See caption of Fig. 2 for line definitions.

Figure 5

Projections of (a) $\Delta E$, (b) $m_{\mathrm{ES}}$, and (c) $\mathcal{F}$ variables of the fit to the $B^{\pm }\to D{h}^{\pm }$, $D^0\to K^{-}{\pi }^{+}$ flavor mode. No requirements are put on the PID of the track $h$ from the $B$ decay and on the fit variables not plotted. See caption of Fig. 2 for line definitions.

Figure 6

1-CL as a function of $\gamma$ (top) and $r_B$ (bottom). Both statistical and systematic uncertainties are taken into account. For the angle $\gamma$, the plot is identical in the range [180°, 360°]. The horizontal lines show the 68% CL (dashed) and the 95% CL (dotted). Because of the symmetry of Eqs. (3, 4) the plot for the strong phase ${\delta }_B$ is identical to the one for $\gamma$.

Figure 7

Contours at 68% (dotted, red) and 95% (solid, green) 2-dimensional CL in the $(\gamma ,r_B)$ and $(\gamma ,{\delta }_B)$ planes. See also the caption of Fig. 6 regarding symmetries.