Heavy quark expansion in beauty and its decays

TL;DR

This work outlines a QCD-grounded heavy-quark expansion (HQE) framework that combines the Wilson operator product expansion with a nonrelativistic treatment of the heavy quark. It clarifies the proper definition of heavy-quark masses, introduces the kinetic and chromomagnetic operators $\mu_\pi^2$ and $\mu_G^2$, and derives controlled $1/m_Q$ and perturbative corrections to inclusive and exclusive $B$-decays, enabling precise extractions of CKM elements such as $|V_{cb}|$ and $|V_{ub}|$. The analysis emphasizes that pole masses are ill-suited for HQE, advocating short-distance masses $m_Q(\mu)$ and careful treatment of duality and renormalization. The work demonstrates that inclusive widths are largely free of $1/m_Q$ nonperturbative corrections and that the zero-recoil exclusive determination of $|V_{cb}|$ is robust, while acknowledging ongoing challenges in $1/m_Q$-suppressed effects, duality violations, and the precise determination of nonperturbative parameters. Overall, the HQE provides a rigorous, quantitatively reliable approach to heavy-flavor phenomenology with significant implications for CKM physics and tests of QCD in the heavy-quark regime.

Abstract

I give an introduction to the QCD-based theory of the heavy flavor hadrons and their weak decays. Trying to remain at the next-to-elementary level and skip technicalities, I concentrate on the qualitative description of the most important applications and physical meaning of the theoretical statements. The numerical results of the dedicated theoretical analyses of extracting V_cb are given and the possibilities to determine V_ub in future are discussed. At the same time I describe in simple language subtle peculiarities distinguishing actual QCD of heavy quarks from naive quantum mechanical treatment often applied to heavy flavor hadrons. These subtleties are often mistreated. Particular attention is paid to the concept of the heavy quark mass and its evaluation, to the kinetic operator and the question of the 1/m_Q corrections to inclusive widths of heavy flavor hadrons. I argue that the properly defined b quark mass is known with a good accuracy from the \bar{b} b threshold cross section.

Figures (5)

• Figure 1: Quark diagrams for the exclusive $B\rightarrow D^{(*)}\:$ ( a) and generic ( b) semileptonic decays.
• Figure 2: Perturbative diagrams leading to renormalization of the heavy quark mass. The contribution of the gluon momenta below $m_Q$ expresses the classical Coulomb self-energy of the colored particle. The number of bubble insertions into the gluon propagator can be arbitrary generating corrections in all orders in $\alpha_s$. The factorial growth of the coefficients produces the IR renormalon uncertainty in $m_Q^{\rm pole}$ of order $\Lambda_{\rm QCD}$.
• Figure 3: Perturbative diagrams determining the high-energy asymptotics of the heavy quark transition amplitudes and renormalization of the local operators.
• Figure 4: Feynman diagrams contributing to the one-loop renormalization of the kinetic operator $\bar{Q} (i{\vec{D}})^2 Q$. Dashed line denotes gluon, and dark box represents the operator.
• Figure 5: The integrated fraction of the $b\rightarrow u \,\ell \nu$ events $\Phi(M_x)$. ( a): Dependence on $m_b$. Long-dashed, solid, and short-dashed lines are $m_b(1\,\hbox{GeV})=4.58\,\hbox{GeV}$, $4.63\,\hbox{GeV}$ and $4.68\,\hbox{GeV}$. ( b): Dependence on $\mu_\pi^2$. Long-dashed, solid, and short-dashed lines correspond to $\mu_\pi^2 =0.3\,\hbox{GeV}^2$, $0.5\,\hbox{GeV}^2$ and $0.7\,\hbox{GeV}^2$.