Summing Sudakov logarithms in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo>→</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mi>γ</mml:mi></mml:math>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 $\stackrel{\ensuremath{\rightarrow}}{B}{X}_{s}\ensuremath{\gamma}$ near the end point of the photon spectrum, ${x=2E}_{\ensuremath{\gamma}}{/m}_{b}\ensuremath{\rightarrow}1,$ as an example. Below the scale $\ensuremath{\mu}{=m}_{b}$ both soft and collinear degrees of freedom are included in the effective theory, while below the scale $\ensuremath{\mu}{=m}_{b}\sqrt{x\ensuremath{-}y},$ where $1\ensuremath{-}y$ 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.