Rare radiative exclusive<i>B</i>decays in soft-collinear effective theory

Abstract

We consider rare radiative B decays such as $\overline{B}\ensuremath{\rightarrow}{K}^{*}\ensuremath{\gamma}$ or $\overline{B}\ensuremath{\rightarrow}\ensuremath{\rho}\ensuremath{\gamma}$ in soft-collinear effective theory (SCET) and show that the decay amplitudes are factorized to all orders in ${\ensuremath{\alpha}}_{s}$ and at leading order in ${\ensuremath{\Lambda}}_{\mathrm{QCD}}{/m}_{b}.$ By employing two-step matching, we classify the operators for radiative B decays in powers of a small parameter $\ensuremath{\lambda}(\ensuremath{\sim}\sqrt{{\ensuremath{\Lambda}}_{\mathrm{QCD}}{/m}_{b}})$ and obtain the relevant operators to order $\ensuremath{\lambda}$ in ${\mathrm{SCET}}_{\mathrm{I}}.$ These operators are constructed with or without spectator quarks including the four-quark operators contributing to annihilation and W-exchange channels. And we employ ${\mathrm{SCET}}_{\mathrm{II}}$ where the small parameter becomes of order ${\ensuremath{\Lambda}}_{\mathrm{QCD}}{/m}_{b},$ and evolve the operators in order to compute the decay amplitudes for rare radiative decays in soft-collinear effective theory. We show explicitly that the contributions from the annihilation channels and the W-exchange channels vanish at leading order in SCET. We present the factorized result for the decay amplitudes in rare radiative B decays at leading order in SCET and at next-to-leading order in ${\ensuremath{\alpha}}_{s}.$