Heavy Quark Expansion (HQE)

• Heavy Quark Expansion (HQE) is an operator product expansion in inverse heavy quark mass that systematically separates short- and long-distance QCD dynamics.
• It employs the optical theorem and local operator expansions to compute inclusive decay rates and heavy-light meson mass spectra with controlled uncertainties.
• Its precise predictions in flavor physics, particularly for bottom and charm quark systems, have advanced our understanding while highlighting non-perturbative challenges.

The Heavy Quark Expansion (HQE) is an operator product expansion in inverse powers of the heavy quark mass, central to modern theoretical treatments of heavy hadron properties such as inclusive decay rates, lifetimes, and mass spectra. In quantum chromodynamics (QCD), the HQE systematically separates short-distance (perturbative) and long-distance (non-perturbative) dynamics, allowing for precise computations with controlled theoretical uncertainties for systems containing bottom or charm quarks. This article reviews the formalism, technical structure, lattice and phenomenological determinations, and contemporary precision developments of the HQE, emphasizing its foundational role and limitations in current flavor physics.

1. Operator Product Expansion and Formal Structure

The HQE expresses observables involving a heavy quark $Q$ (typically $b$ or $c$) as an asymptotic series in $1/m_Q$, the inverse heavy-quark mass. For inclusive decay rates or heavy-light meson masses, the HQE arises from the optical theorem applied to the time-ordered product of weak Hamiltonian insertions, leading to an expansion in local operators: [ \Gamma(H_Q \to X) = \frac{G_F^2\, m_Q^5}{192\pi^3} \left[ c_3 \frac{\langle H