Shape-Function Effects and Split Matching in B-> Xs l+ l-

TL;DR

The paper develops a model-independent, factorized framework for B → Xs ℓℓ in the shape-function region using SCET, showing that a universal shape function governs the nonperturbative dynamics alongside a jet function, with perturbative corrections organized via a novel split matching at two nearby scales. It provides the LO triply differential rate and various doubly differential spectra under q^2 and m_X cuts, incorporating RG evolution and realistic experimental constraints. A key outcome is the demonstrated universality of the shape function across related decays (B→Xsγ, B→Xuℓν), enabling a clean connection between measured spectra and short-distance Wilson coefficients, with quantified perturbative corrections and explicit numerical insights. The results pave the way for precise extractions of C9, C7, and C10 from data with phase-space cuts, and set the stage for including subleading shape-function effects in future work.

Abstract

We derive the triply differential spectrum for the inclusive rare decay B -> Xs l+ l- in the shape function region, in which Xs is jet-like with $mX^2 \lesssim mb Λ_QCD$. 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 alphas corrections consistently. We show that, with a suitable power counting for the dilepton invariant mass, the same universal jet and shape functions appear as in B-> Xs gamma and B-> Xu l nu decays. Parts of the usual alphas(m_b) corrections go into the jet function at a lower scale, and parts go into the non-perturbative shape function. For B -> Xs l+ l-, the perturbative series in alphas are of a different character above and below mu=mb. We introduce a ``split matching'' method that allows the series in these regions to be treated independently.

Figures (6)

• Figure 1: The kinematic range for $p_X^-$ and $p_X^+$ given the experimental cuts of $q^2 < 6\,{\rm GeV}^2$ and $m_X \le 2.0\,{\rm GeV}$ for $B\to X_s\ell^+\ell^-$.
• Figure 2: Graphs from $H_W$ for matching on to SCET.
• Figure 3: Graphs in SCET for the matching computation.
• Figure 4: Time-ordered product for the leading-order factorization theorem.
• Figure 5: Comparison of the real part of Wilson coefficients at $\mu_0=\mu_b=4.8\,{\rm GeV}$ with $m_c/m_b=0.292$, $\overline {m_b}(\mu_0) =4.17\,{\rm GeV}$, and $m_b=4.8\,{\rm GeV}$. For ${\cal C}_9$, ${\cal C}_7$, and ${\cal C}_{10b}$ we take $p_X^+=0$.