Recently Ciuchini, Pierini, and Silvestrini proposed a method for constraining CKM parameters in $B\to K\pi \pi$ and $B_s\to K\pi \pi$ through phase measurements of amplitudes involving $I=3/2$ $K^{*}\pi$ final states. We show that complementary information on CKM parameters may be obtained by studying the phases of $\Delta I=1$ $B\to (K^{*}\pi {)}_{I=1/2}$, $B_s\to (K^{*}\overline{K}{)}_{I=1}$ and $B_s\to ({\overline{K}}^{*}K{)}_{I=1}$ amplitudes. Hadronic uncertainties in these constraints from electroweak penguin operators $O_9$ and $O_{10}$, studied using flavor SU(3), are shown to be very small in $B\to K\pi \pi$ and $B_s\to K\pi \pi$ and somewhat larger in $B_s\to K\overline{K}\pi$. The first processes imply a precise linear relation between $\overline{\rho }$ and $\overline{\eta }$, with a measurable slope and an intercept at $\overline{\eta }=0$ involving a theoretical error of 0.03. The decays $B_s\to K\pi \pi$ permit a measurement of $\gamma$ involving a theoretical error below a degree. We note that while time-dependence is required when studying $B^0$ decays at the ${\rm Y}(4S)$, it may not be needed when studying $B_s$ decays at hadronic colliders.

• Received 28 August 2006

DOI:https://doi.org/10.1103/PhysRevD.75.014002