Prospects for the measurement of the unitarity triangle angle $\gamma$ from $B^0\to D{K}^{+}{\pi }^{-}$ decays

The potential for a precise measurement of the unitarity triangle angle $\gamma$ in future experiments from the decay $B^0\to D{K}^{*0}$ is well known. It has recently been suggested that the sensitivity can be significantly enhanced by analyzing the $B^0\to D{K}^{+}{\pi }^{-}$ Dalitz plot to extract amplitudes relative to those of the flavor-specific decay $B^0\to D_2^{*-}{K}^{+}$. An extension to this method which includes the case where the neutral $D$ meson is reconstructed in suppressed final states is presented. The sensitivity to $\gamma$ is estimated using this method and compared to that obtained using the $B^0\to D{K}^{*0}$ decay alone. Experimental effects, such as background contamination, are also considered. This approach appears to be a highly attractive addition to the family of methods that can be used to determine $\gamma$.

Figure 1

Feynman diagrams for $B^0\to D{K}^{*0}$, via (left) a $\overline{b}\to \overline{c}u\overline{s}$ transition and (right) a $\overline{b}\to \overline{u}c\overline{s}$ transition.

Figure 2

Argand diagrams illustrating the measurements of relative amplitudes and phases from analysis of the Dalitz plots of (left) ${\overline{D}}^0{K}^{+}{\pi }^{-}$ and (right) $D_{CP}{K}^{+}{\pi }^{-}$. In these illustrative examples the following values are used: $\varrho =1.5$, $\Delta =20°$, $\gamma =75°$, ${\delta }_B=45°$, and $r_B=0.4$.

Figure 3

Argand diagrams illustrating the measurements of relative amplitudes and phases from analysis of the Dalitz plots of (left) $D_{\mathrm{fav}}{K}^{+}{\pi }^{-}$ and (right) $D_{\sup }{K}^{+}{\pi }^{-}$. Note that in the latter the amplitudes are rotated by $-{\delta }_D$ to maintain the convention of having the $D_2^{*-}{K}^{+}$ amplitude on the real axis. In these illustrative examples the following values are used: $\varrho =1.5$, $\Delta =20°$, $\gamma =75°$, ${\delta }_B=45°$, $r_B=0.4$, ${\delta }_D=-158°$, and $r_D=0.06$.

Figure 4

High-statistics $B\to D_{\mathrm{fav}}K\pi$ Dalitz-plot distribution generated with our nominal model. Structures due to $D{K}^{*0}(892)$, $D{K}_2^{*0}(1430)$, and $D_2^{*-}(2460)K^{+}$, as well as the nonresonant contribution, are clearly apparent.

Figure 5

Distributions of fitted values of $\gamma$ in the quasi-two-body approach, with different input values of ${\delta }_B$. These results are obtained with $\gamma =60°$ and $r_B=0.4$. The location of each of the generated solutions for $\kappa =1$, i.e. ${\delta }_S={\delta }_B$, is given by a red solid cross. The locations of each of the ambiguous solutions obtained for $\kappa =1$ and $r_D=0$ are given by the intersections of the blue dashed lines.

Figure 6

Distributions of fitted values of $\gamma$ in the quasi-two-body approach with $\kappa$ free (left) and fixed (right). These results are obtained with $\gamma =60°$, ${\delta }_B=90°$, and $r_B=0.4$.

Figure 7

Distributions of fitted values of $\gamma$ in the amplitude (left) and quasi-two-body (right) approaches. These results are obtained with $\gamma =60°$, ${\delta }_B=90°$, and $r_B=0.4$.

Figure 8

Distributions of fitted values of $r_B$ (left), ${\delta }_B$ (middle), and $\gamma$ (right) in the amplitude analysis. These results are obtained with $\gamma =60°$, ${\delta }_B=0°$, and $r_B=0.4$.

Figure 9

Distribution of fitted results for ${\delta }_D$ obtained in the amplitude analysis when fitting with ${\delta }_D$ as a free parameter. The mean and width obtained by fitting the distribution to a Gaussian line shape (solid line) are shown on the plot. The generated value of ${\delta }_D$ is $-158°$.

Figure 10

Distributions of fitted values of $\gamma$ in the Dalitz-plot approach, for $B/S=10$, 50, 100. These results are obtained with $\gamma =60°$, ${\delta }_B=0°$, and $r_B=0.4$.