Soft-collinear factorization in effective field theory

The factorization of soft and ultrasoft gluons from collinear particles is shown at the level of operators in an effective field theory. Exclusive hadronic factorization and inclusive partonic factorization follow as special cases. The leading-order Lagrangian is derived using power counting and gauge invariance in the effective theory. Several species of gluons are required, and softer gluons appear as background fields to gluons with harder momenta. Two examples are given: the factorization of soft gluons in $\overrightarrow{B}D\pi$ and the soft-collinear convolution for the $\overrightarrow{B}{X}_s\gamma$ spectrum.