CPViolation inB±→γπ±π+π−

We consider $\mathrm{CP}$-violating effects in decays of the type $B^{\pm }\to \gamma a_{1,2}^{\pm }$, where $a_{1,2}$ are the $J^P=1^{+} \mathrm{and} 2^{+}$ resonances, each decaying to the common final state via $a_{1,2}^{\pm }\to {\pi }^{\pm }{\rho }^0$. The resonances enhance the $\mathrm{CP}$ asymmetries and also knowledge of their masses and widths facilitates calculations of the effects. Several types of $\mathrm{CP}$ asymmetries are sizable (∼10%-30%); these large asymmetries together with a model calculation of the ratio between the penguin and the annihilation amplitudes, can therefore provide a possible method for measuring the angle $\alpha$ in the unitarity triangle.