b -> s +γ: A QCD Consistent Analysis of the Photon Energy Distribution

TL;DR

The paper addresses the photonenergy spectrum in inclusive b → sγ decays within a QCD-consistent framework based on the heavy quark expansion and OPE, introducing a two-component description: a soft primordial distribution F(x) and a perturbative hard-gluon component. It formalizes the spectrum as a μ-dependent convolution of the soft and perturbative pieces, incorporating Sudakov resummation and running α_s via a BLM-inspired scheme, and presents explicit expressions for the endpoint region, including running masses m_b(μ) and the kinetic parameter μ_π^2. The study demonstrates that, despite perturbative smearing, the endpoint peak remains visible and can be used to constrain heavy-quark parameters, while current CLEO data yield consistency with theory; it also outlines an Advanced Perturbative Spectrum (APS) approach to go beyond leading-log OPE. The framework clarifies how short- and long-distance effects intertwine in the spectrum and provides a path to incorporate higher-twist effects and more precise data, enabling more accurate determinations of $m_b$ and $ar{Λ}$ in future analyses.

Abstract

The photon energy distribution in the inclusive b -> s+γtransitions is a combination of two components: the first component, soft physics, is determined by the so called primordial distribution function, while the second component, perturbative physics, is governed by the hard gluon emission. A simple ansatz is suggested for the primordial distribution function which obeys the QCD constraints known so far. We then discuss in detail how the hard gluon emission affects the energy distribution. An extension of the Sudakov approximation is worked out incorporating the Brodsky-Lepage-Mackenzie prescription and its generalizations. We explicitly calculate the marriage of nonperturbative with perturbative effects in the way required by OPE, introducing separation scale μ. A few parameters, such as m_b and μ_π^2 affect the shape of the distribution and, thus, can be determined by matching to the experimental data. The data, still scarce, while not giving precise values for these parameters, yield consistency with theory: the current values of the above parameters lie within experimental uncertainty. On the theoretical side we outline a method allowing one to go beyond the practical version of OPE.