Factorization ofBdecays into two light mesons in soft-collinear effective theory

We show that the amplitudes for B decays into two light mesons at leading order in soft-collinear effective theory are factorized to all orders in ${\alpha }_s.$ We construct gauge-invariant four-quark operators by employing the effective theories ${\mathrm{SCET}}_{\mathrm{I}}$ for $\sqrt{m_b\Lambda }<\mu <m_b,$ and ${\mathrm{SCET}}_{\mathrm{II}}$ for $\mu <\sqrt{m_b\Lambda }$ with $\Lambda \sim 0.5$ GeV. These operators do not allow gluon exchanges between different sectors of the current and the matrix elements are reduced to the products of current matrix elements. The spectator interactions are also factorized at leading order in SCET and to all orders in ${\alpha }_s.$