Summing Sudakov logarithms in B -> X-s gamma in effective field theory

We construct an effective field theory valid for processes in which highly energetic light-like particles interact with collinear and soft degrees of freedom, using the decay B-->X(s)gamma near the end point of the photon spectrum, x = 2E(gamma)/m(b)-->1, as an example. Below the scale mu = m(b) both soft and collinear degrees of freedom are included in the effective theory, while below the scale mu =m(b)rootx-y, where 1-y is the light cone momentum fraction of the b quark in the B meson, we match onto a theory of bilocal operators. We show that at one loop large logarithms cancel in the matching conditions, and that we recover the well-known renormalization group equations that sum leading Sudakov logarithms.