An effective field theory for collinear and soft gluons: Heavy to light decays

We construct the Lagrangian for an effective theory of highly energetic quarks with energy Q, interacting with collinear and soft gluons. This theory has two low energy scales, the transverse momentum of the collinear particles, ${p}_{\ensuremath{\perp}},$ and the scale ${p}_{\ensuremath{\perp}}^{2}/Q.$ 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 $\stackrel{\ensuremath{\rightarrow}}{B}{X}_{s}\ensuremath{\gamma}$ and $\stackrel{\ensuremath{\rightarrow}}{B}{X}_{u}l\overline{\ensuremath{\nu}}$ 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 al. [Phys. Rev. D 60, 014001 (1999)], establishing these relations in the large energy limit of QCD.