Summing Sudakov logarithms inB→Xsγin 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 $\overrightarrow{B}{X}_s\gamma$ near the end point of the photon spectrum, ${x=2E}_{\gamma }{/m}_b\to 1,$ 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 $\mu {=m}_b\sqrt{x-y},$ 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.