Resumming Phase Space Logarithms in Inclusive Semileptonic $B$ Decays

Abstract

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 latter involves a non-trivial 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.

Comments: 18 pages, REVTeX, 8 EPS figures embedded. Final version, to appear in Physical Review D; one reference has been added