Proof of factorization for B → Dπ

An all orders proof of factorization is given for B̄⁰→D⁺π^- and B^-→D⁰π^-, in the limit where the heavy quark masses approach infinity. In particular, three types of factorization are distinguished. First, the generalized factorization of matrix elements of four quark operators into a form factor describing the B → D transition and a convolution of a short distance coefficient with the pion wave function is proved. A second type of factorization, between hard and infrared scales, is related to defining the correct effective theory. Finally, a third type is factorization theorems between soft and collinear degrees of freedom.