Nonperturbative Corrections to Inclusive Beauty and Charm Decays: QCD versus Phenomenological Models

TL;DR

The authors develop a QCD-based, self-consistent heavy-quark expansion (HQE) to include nonperturbative corrections in inclusive beauty and charm decays. They show the leading nonperturbative corrections start at $O\left(1/m_Q^2\right)$ via the chromomagnetic operator and, at $O\left(1/m_Q^3\right)$, via four-fermion operators, with matrix elements tied to hadron spectroscopy (e.g., the vector–pseudoscalar mass splitting). These corrections shift observables such as the semileptonic branching ratio: in beauty decays there is a modest reduction (a few percent), while in charm decays the effects can be substantially larger; importantly, simple phenomenological models fail to mimic these leading corrections. The framework provides a systematically improvable, model-independent description and outlines clear directions for extensions to higher orders, baryons, and flavor SU(3) breaking.

Abstract

We present a selfconsistent method for treating nonperturbative effects in inclusive nonleptonic and semileptonic decays of heavy flavour hadrons. These effects give rise to powerlike corrections $\propto 1/m_Q^n\,$, $n \ge 2$ with $m_Q$ denoting the heavy quark mass.The leading correction to the semileptonic branching ratio occurs for n=2. It is expressed in terms of the vector-pseudoscalar mass splitting: $δBR\ind{sl}/BR\ind{sl} \simeq BR\ind{nl}\, \cdot \,6\,(\,(M_V^2-M_P^2)/m_Q^2)\cdot (c_+^2 - c_-^2)/2N_c$ and yields a {\it reduction} of $BR\ind{sl}$. This nonperturbative correction contributes to the nonleptonic width with a sign opposite to that of the perturbative terms that are non-leading in $1/N_c$. In beauty decays the former reduces the latter by 20 \% whereas in charm decays they more or less cancel. This leads to a {\it reduction} of $BR\ind{sl}$ by no more than 10 \% in beauty decays and by a factor of roughly two in charm decays. We confront these results with those obtained from phenomenological models of heavy flavour decays and find that such models are unable to mimic these leading corrections by a specific choice of quark masses or by invoking Fermi motion.