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

Abstract

We derive the triply differential spectrum for the inclusive rare decay B → X s ℓ + ℓ − in the shape function region, in which X s is jet-like with m 2 X < ∼ m b Λ QCD . Experimental cuts make this a relevant region. The perturbative and non-perturbative parts of the matrix elements can be defined with the Soft-Collinear Effective Theory, which is used to incorporate α 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 → X s γ and B → X u ℓν decays. Parts of the usual α s (m b ) corrections go into the jet function at a lower scale, and parts go into the non-perturbative shape function. For B → X s ℓ + ℓ − , the perturbative series in α s are of a different character above and below µ = m b . We introduce a "split matching" method that allows the series in these regions to be treated independently.

NonperturbativemXcut effects inB→Xsl+l−observables

“…If subleading contributions from other operators are considered, it can be necessary to count q 2 as parametrically small and to treat the photon as collinear particle. In this case there will be additional four-quark operators with collinear quarks coupling to the collinear photon, giving rise to additional subleading four-quark shape functions [14,52]. We shall restrict our discussion to tree level and the primary subleading shape functions.…”

“…This is achieved by the "split matching" procedure introduced in Ref. [14] in the context of matching on to SCET.…”

“…It was first applied to B → X s ℓ + ℓ − in Refs. [14,15] to study systematically the effect of the m cut X on the q 2 spectrum and forwardbackward asymmetry. In Ref.…”

Heavy-to-light currents at NNLO in SCET and semi-inclusive decay

“…The semi-inclusive decay rates can be calculated by matching the effective weak interaction Hamiltonian to soft-collinear effective theory. At the leading order in the Λ QCD /m b expansion, the decay rates can be factorized into process-dependent hard functions h [0] , related to physics at the hard scale µ ∼ m b and above, a universal jet function J, related to physics at the intermediate hard-collinear scale µ hc ∼ m b Λ QCD , as well as a universal non-perturbative shape function S, describing the internal soft dynamics of the B meson, with the following schematic form [36,37] …”

Rare K- (vs.) B-decays

“…In the SM one finds [30] Like in the case ofB → X s γ, experimental cuts complicate the theoretical description of B → X s + − as they make the measured decay distributions sensitive to the non-perturbative shape function [34]. In particular, putting an upper cut M cut…”