B→M1M2: Factorization, charming penguins, strong phases, and polarization

Using the soft-collinear effective theory we derive the factorization theorem for the decays $B\to M_1{M}_2$ with $M_{1,2}=\pi ,K,\rho ,K^{*}$, at leading order in $\Lambda /E_M$ and $\Lambda /m_b$. The results derived here apply even if ${\alpha }_s(E_M\Lambda )$ is not perturbative, and we prove that the physics sensitive to the $E\Lambda$ scale is the same in $B\to M_1{M}_2$ and $B\to M$ form factors. We argue that $c\overline{c}$ penguins could give long-distance effects at leading order. Decays to two transversely polarized vector mesons are discussed. Analyzing $B\to \pi \pi$ we find predictions for $B^0\to {\pi }^0{\pi }^0$ and $|V_{ub}|f_{+}^{B\to \pi }(0)$ as a function of $\gamma$.