Pathways to a clean $\gamma$ (${\varphi }_3$): From $B$ to super $B$ factories

The implementation of various methods for the determination of $\gamma$ through direct $CP$ violation arising in the interference of $b\to c$ and $b\to u$ processes in charged as well as neutral $B$ meson decays are considered. We show that the inclusion of $D^0$ resulting from $D^{*0}\to D^0+{\pi }^0(\gamma )$, say, via $B\to K{D}^{*0}$ makes a significant difference in the attainable accuracy for $\gamma$. Both exclusive and inclusive decays of the $B^{\pm }$ and $B^0$ to states containing $D^0/{\overline{D}}^0$ followed by both inclusive and exclusive decays of the $D^0$ are discussed. It is shown that with statistics which might be obtained at $B$ factories ($5–10\times {10}^8$ $B$ pairs) a $1\sigma$ determination of $\gamma$ to $\approx \pm 8°$ may be possible, depending on the efficiency of reconstruction, backgrounds, and the details of the decay amplitudes involved. The role of data from a charm factory as well as effects of $D^0$ mixing are discussed. Extraction of $\gamma$ with accuracy that is roughly commensurate with the intrinsic theory error of these methods (i.e., around 0.1%), which is an important goal, will require $>{10}^{10}$ $B$ pairs, namely, a super $B$ factory.

Figure 1

Feynman diagrams for $B^{-}\to K^{-}[{\mathbf{D}}^0\to K^{+}{\pi }^{-}]$. Diagrams (a) and (b) are $b\to c$ transitions where ${\mathbf{D}}^0=D^0$ while (c) is the $b\to u$ transition with ${\mathbf{D}}^0={\overline{D}}^0$. Diagram (a) is color allowed while (b) and (c) are color suppressed; ${\overline{D}}^0\to K^{+}{\pi }^{-}$ is Cabibbo allowed, whereas $D^0\to K^{+}{\pi }^{-}$ is doubly Cabibbo suppressed.

Figure 2

The minimum value of ${\chi }^2$ is shown as a function of $\gamma$ for various combinations of data in the sample calculation. The thin solid line shows the result using just $B^{-}\to K^{-}[{\mathbf{D}}^0\to K^{+}{\pi }^{-}]$ data. The dotted line shows the result using just the $B^{-}\to K^{-}[{\mathbf{D}}^0\to \mathrm{C}\mathrm{P}\mathrm{E}\mathrm{S}-]$ data. The thick solid curve shows the result taking both $B^{-}\to K^{-}[{\mathbf{D}}^0\to K^{+}{\pi }^{-}]$ and $B^{-}\to K^{-}[{\mathbf{D}}^0\to \mathrm{C}\mathrm{P}\mathrm{E}\mathrm{S}-]$ data together. The dashed line shows the result using $B^{-}\to K^{(*)-}[{\mathbf{D}}^{(*)0}\to K^{+}{\pi }^{-}]$. In the dashed-dotted curve, all four of the initial $B^{-}$ decays where the $\mathbf{D}$ decays to the same two final states are considered. Thus, the dashed-dotted curve results from taking together data of the form $B^{-}\to K^{(*)-}[{\mathbf{D}}^{(*)0}\to K^{+}{\pi }^{-}]$ and $B^{-}\to K^{(*)-}[{\mathbf{D}}^{(*)0}\to \mathrm{C}\mathrm{P}\mathrm{E}\mathrm{S}-]$. The long-dashed curve includes only data from two parent $B^{-}$ decays, i.e., $B^{-}\to K^{-}{D}^0$ as well as $B^{-}\to K^{-}[{\mathbf{D}}^{*0}\to {\mathbf{D}}^0+{\pi }^0(\gamma )]$ with either of the two ${\mathbf{D}}^0$ decaying to $K^{+}{\pi }^{-}$ as well as $\mathrm{C}\mathrm{P}\mathrm{E}\mathrm{S}-$. The upper horizontal dashed line indicates the 3-sigma level determination of $\gamma$ with the luminosity required to give the results in Table , which corresponds to current $B$ factories while the lower horizontal dashed line corresponds to a luminosity 50 times greater which may be achieved at future high luminosity $B$ factories.

Figure 3

Some effects on the accuracy of $\gamma$ extraction. Dashed-dotted curve is from Fig. 2 using arbitrarily chosen particular strong phases as in Fig. 2, whereas in the long-dashed curve the ${\chi }^2$ is averaged over randomly chosen strong phases. The thin solid curve is obtained by averaging over randomly chosen phases and also includes the error in DCS modes from Ref. 18 along with a 20% error on CPES modes (see also Table ).

Figure 4

The minimum value of ${\chi }^2$ using the four SCS modes in Eq. (2) is shown with the solid line. The dashed-dotted line shows the results with $D^0\to K^{*+}{K}^{-}$ and $D^0\to K^{*-}{K}^{+}$ together with the four parent $B^{-}\to {\mathbf{D}}^{(*)0}{K}^{(*)-}$ decays.

Figure 5

The thick solid curve shows ${\chi }^2$ as a function of $\gamma$ taking both $B^{-}\to K^{-}[{\mathbf{D}}^0\to K^{+}{\pi }^{-}]$ and $B^{-}\to K^{-}[{\mathbf{D}}^0\to \mathrm{C}\mathrm{P}\mathrm{E}\mathrm{S}-]$ data together as in Fig. 2. The thin solid curve shows the result if ${\zeta }_{K^{+}{\pi }^{-}}$ is determined separately at a $\psi (3770)$ charm factory. The dotted curve includes $B^{-}\to K^{-}[{\mathbf{D}}^0\to K^{+}{\pi }^{-}]$, $B^{-}\to K^{-}[{\mathbf{D}}^0\to K^{*+}{\pi }^{-}]$, $B^{-}\to K^{-}[{\mathbf{D}}^0\to K^{+}{\pi }^{-}+n\pi ]$, $B^{-}\to K^{-}[{\mathbf{D}}^0\to \mathrm{C}\mathrm{P}\mathrm{E}\mathrm{S}-]$, as well as the phase determination from a $\psi (3770)$ charm factory. The dashed-dotted curves consider all these decay modes together with the four parent decays $B^{-}\to K^{*-}{D}^{*0}$ as well as phase determination from a $\psi (3770)$ charm factory.

Figure 6

The results for $B^{-}\to K^{(*)-}{\mathbf{D}}^{(*)0}$ followed by ${\mathbf{D}}^0\to K^{+}{\pi }^{-}$; $\mathrm{C}\mathrm{P}\mathrm{E}\mathrm{S}-$; $K^{+}{\pi }^{-}+n\pi$, $K^{*+}{\pi }^{-}$, where finite $CP$ conserving backgrounds are included. The solid curve is for no background; the dashed curve is for $\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{a}\mathrm{l}/\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}=1$; the dotted curve is for $\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{a}\mathrm{l}/\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}=\frac{1}{4}$; and the long-dashed curve is for $\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{a}\mathrm{l}/\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}=\frac{1}{10}$.

Figure 7

The minimum value of ${\chi }^2$ is shown as a function of $\mathrm{B}\mathrm{r}(B^{-}\to K^{-}{\overline{D}}^0)$ and $\mathrm{B}\mathrm{r}(B^{-}\to K^{*-}{\overline{D}}^0)$ using data for all the combinations of $B^{-}\to K^{(*)-}{\mathbf{D}}^{(*)0}$ followed by ${\mathbf{D}}^0\to K^{+}{\pi }^{-}$; $\mathrm{C}\mathrm{P}\mathrm{E}\mathrm{S}-$; $K^{+}{\pi }^{-}+n\pi$, $K^{*+}{\pi }^{-}$. The dependence on $\mathrm{B}\mathrm{r}(B^{-}\to K^{-}{\overline{D}}^0)$ is indicated by the solid line while the dependence on $\mathrm{B}\mathrm{r}(B^{-}\to K^{*-}{\overline{D}}^0)$ is shown by the dashed curve.

Figure 8

${\chi }^2$ as a function of $\gamma$ for $B^{-}\to K^{-}[D^0\to \mathrm{C}\mathrm{P}\mathrm{E}\mathrm{S}-]$. The dotted curve is the same as in Fig. 2; the long-dashed curve assumes that $b$ has a 10% $1\mathrm{\text{−}}\sigma$ error; the thick line assumes a 100% error; and the thin solid line assumes a 200% error.

Figure 9

The curves are as in Fig. 8 except all four $B^{-}$ decays of the form $B^{-}\to K^{(*)-}{D}^{*(0)}$ are considered.

Figure 10

The solid line shows the fit to the data from $K^{+}{\pi }^{-}$, $K^{+}{\pi }^{-}+n\pi$, $\mathrm{C}\mathrm{P}\mathrm{E}\mathrm{S}-$, and $K^{*+}{\pi }^{-}$ with no $D$ mixing. The dashed line shows the fit, assuming $y=0$ to the data in the case $y=0.01$; the dotted line shows the fit, assuming $y=0$ to the data in the case $y=0.1$, while the dashed-dotted line shows the fit where a nonzero value of $y$ is taken into account where the data is generated with $y=0.1$.

Figure 11

The solid curve is generated using charged $B^{-}$ data and is the same as the solid curve in Fig. 6. The dashed curve uses only neutral data with the two parent decays ${\overline{B}}^0\to [{\overline{K}}^{*0}\to K^{-}{\pi }^0]{\mathbf{D}}^{(*)0}$ and subsequent $\mathbf{D}$ decay to $K^{+}{\pi }^{-}$, $K^{*+}{\pi }^{-}$, $K^{+}{\pi }^{-}+n\pi$, $\mathrm{C}\mathrm{P}\mathrm{E}\mathrm{S}-$. The dotted curve combines the data set used in the solid curve (charged $B$) with the data set used in the dashed curve (neutral $B$).

Figure 12

The solid curve is the same as in Fig. 6 while the dashed one is the ${\chi }^2$ for $\gamma$ determined by the three-body $B^{-}$ decay modes ${\mathbf{D}}^0{K}^{-}{\pi }^0$, ${\mathbf{D}}^{*0}{K}^{-}{\pi }^0$, and ${\mathbf{D}}^0{K}^{*-}{\pi }^0$ using the assumptions discussed in the text.