Resumming phase space logarithms in inclusive semileptonic B decays

We study logarithms of the form ln(m(q)/m(b)) which arise in the inclusive semileptonic decay of a bottom quark to a quark of mass m(q). We use the renormalization group to resum the leading radiative corrections to these terms, of the form m(q)(2) alpha(s)(n)ln(n)(m(q)/m(b)), m(q)(3) alpha(s)(n+1)ln(n)(m(q)/m(b)), and m(q)(4) alpha(s)(n)ln(n+1)(m(q)/m(b)). The first two resummations are trivial, while the last involves a nontrivial mixing of four-Fermi operators in the 1/m(b) expansion. We illustrate this technique in a toy model in which the semileptonic decay is mediated by a vector interaction, before treating the more complicated case of left-handed decay.