Cw. Bauer et al., An effective field theory for collinear and soft gluons: Heavy to light decays - art. no. 114020, PHYS REV D, 6311(11), 2001, pp. 4020
We construct the Lagrangian for an effective theory of highly energetic qua
rks with energy Q, interacting with collinear and soft gluons. This theory
has two low energy scales, the transverse momentum of the collinear particl
es, p(perpendicular to), and the scale p(perpendicular to)(2)/Q. The heavy
to light currents are matched onto operators in the effective theory at one
loop and the renormalization group equations for the corresponding Wilson
coefficients are solved. This running is used to sum Sudakov logarithms in
inclusive B-->X(s)gamma and B-->X(u)l<(<nu>)over bar> decays. We also show
that the interactions with collinear gluons preserve the relations for the
soft part of the form factors for heavy-to-light decays found by Charles et
nl. [Phys. Rev. D 60, 014001 (1999)]. establishing these relations in the
large energy limit of QCD.