  Soft-collinear effective theory is used to prove factorization of the
B->gamma+l+nu decay amplitude at leading power in Lambda/m_b, including a
demonstration of the absence of non-valence Fock states and of the finiteness
of the convolution integral in the factorization formula. Large logarithms
entering the hard-scattering kernel are resummed by performing a two-step
perturbative matching onto the low-energy effective theory, and by solving
evolution equations derived from the renormalization properties of the
leading-order B-meson light-cone distribution amplitude. As a byproduct, the
evolution equation for heavy-collinear current operators in soft-collinear
effective theory is derived.
