Bounding hadronic uncertainties in $c\to u$ decays

Time-dependent $CP$ asymmetry measurements in $D\to h^{+}{h}^{-}$ decays, where $h=\pi$ or $\rho$ can, in principal, be used to constrain the angle ${\beta }_c$ of the $cu$ unitarity triangle up to theoretical uncertainties. Here we discuss the theoretical uncertainty from penguin contributions that can be investigated through the use of isospin analyses. We show that uncertainty from penguin pollution on a measurement of ${\beta }_c$ (or alternatively the mixing phase) in $D^0\to {\pi }^{+}{\pi }^{-}$ (${\rho }^{+}{\rho }^{-}$) decays is 2.7° (4.6°). We also comment on the applicability of this method to $D^0\to \rho \pi$ decays for which measurements of weak phases with a precision below the one degree level may be possible.

Figure 1

Unitarity triangles relating CKM amplitudes for (a) $B_d$, (b) $B_s$ and (c) $D$ decays (not to scale). We label angles with appropriate suffix as $\beta$ (smallest), $\alpha$ (closest to 90°) and $\gamma$. Note that ${\gamma }_c=\gamma \simeq 68°$.

Figure 2

The $I$-spin triangle relations given in Eq. (1).

Figure 3

The constraint on $2\delta {\beta }_c$ obtained from (solid) $D\to \pi \pi$ and (dashed) $D\to \rho \rho$ decays using existing data. Mirror solutions arising from ambiguities in the orientation of the $I$-spin triangles also exist for $-180°\le 2\delta {\beta }_c<0$.

Figure 4

The $I$-spin pentagon relation for the $D\to \rho \pi$ modes given in Eq. (2). Amplitudes for $D^0$ (solid) and ${\overline{D}}^0$ (dotted) are shown. The ${\overline{D}}^0$ pentagon is rotated so that the sums on each side of Eq. (2) coincide with those for $D^0$ decays. These rotated amplitudes are labeled with a tilde. The effect of penguins that can contribute a change, $\delta {\beta }_c$, in the phase of the ${\rho }^0{\pi }^0$ amplitude is illustrated.

Figure 5

(a) $p$-values for minimum ${\chi }^2$ fits to BABAR data for $-3°<{\beta }_c<3°$ about zero, the value expected as outlined in the text. A second, identical peak centered at 180° is also observed in the full ${\beta }_c$ scan, but is not shown here. (b) The corresponding values for $|P|/|T|$, the ratio of magnitudes for the penguin and tree amplitudes for the ${\rho }^0{\pi }^0$ decays. As expected, this ratio is zero at ${\beta }_c=0$ in the absence of $CPV$, with an uncertainty of $\pm 0.01$.