CP Violation in $B^\pm\to \gamma\pi^\pm\pi^+\pi^-$

Abstract

We consider CP violating effects in decays of the type $B^\pm\to \gamma a^\pm_{1,2}$ where $a_{1,2}$ are the $J^P=1^+$ and $2^+$ resonances each decaying to the common final state via $a^\pm_{1,2} \to\pi^\pm\rho^0$. The resonances enhance the CP asymmetries and also knowledge of their masses and widths facilitates calculations of the effects. Several types of CP asymmetries are sizable ($\sim10$--30\%) requiring about (3--10)${}\times10^8$ $B^\pm$ mesons for detection at the $3\sigma$ level thereby providing a method for measuring the angle $\alpha$ in the unitarity triangle. \bigskip