Shape-function effects and split matching inB→Xsℓ+ℓ−

Abstract

We derive the triply differential spectrum for the inclusive rare decay $B\ensuremath{\rightarrow}{X}_{s}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$ in the shape-function region, in which ${X}_{s}$ is jetlike with ${m}_{X}^{2}\ensuremath{\lesssim}{m}_{b}{\ensuremath{\Lambda}}_{\mathrm{QCD}}$. Experimental cuts make this a relevant region. The perturbative and nonperturbative parts of the matrix elements can be defined with the soft-collinear effective theory, which is used to incorporate ${\ensuremath{\alpha}}_{s}$ corrections consistently. We prove that, with a suitable power counting for the dilepton invariant mass, the same universal jet and shape functions appear as in $B\ensuremath{\rightarrow}{X}_{s}\ensuremath{\gamma}$ and $B\ensuremath{\rightarrow}{X}_{u}\ensuremath{\ell}\overline{\ensuremath{\nu}}$ decays. Parts of the usual ${\ensuremath{\alpha}}_{s}({m}_{b})$ corrections go into the jet function at a lower scale, and parts go into the nonperturbative shape function. For $B\ensuremath{\rightarrow}{X}_{s}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$, the perturbative series in ${\ensuremath{\alpha}}_{s}$ are of a different character above and below $\ensuremath{\mu}={m}_{b}$. We introduce a ``split matching'' method that allows the series in these regions to be treated independently.