Theory of Pre-Asymptotic Effects in Weak Inclusive Decays

TL;DR

The work develops a coherent, QCD‑based framework (OPE/HQET) for preasymptotic effects in weak inclusive decays of heavy flavors. It establishes a robust separation of short‑distance coefficients and soft matrix elements, yielding a CGG/BUV theorem that eliminates $1/m_Q$ corrections in total semileptonic widths and ties nonperturbative physics to a small set of parameters like $\mu_G^2$ and $\mu_\pi^2$. It introduces the heavy‑quark light‑cone distribution function $F(x)$ to describe primordial motion, shows its evolution toward a temporal function $G(x)$ in the SV limit, and connects this to end‑point spectra and duality. The discussion also covers hard gluon effects with Sudakov resummation, the ambiguity of the pole mass due to renormalons, and the theoretical boundaries of duality, highlighting both practical predictive power and fundamental questions for future work.

Abstract

I give an introduction to the theory of preasymptotic effects based on the systematic OPE/HQET expansion in $1/m_Q$ where $m_Q$ is the heavy quark mass. The general idea is explained in two most instructive examples, with an emphasis on pedagogical aspects. Some important results of the last year are reviewed. In discussing the issue of the quark-hadron duality, one of the basic ingredients of the theory, I prove that the operator product expansion {\em per se} is an asymptotic expansion. The behavior of the high order terms in this expansion determines the onset of duality and the accuracy of the duality relations. The factorial divergence of the high-order terms in OPE implies a sophisticated analytical structure in the $α_s$ plane, with terms of the type $\exp [-\exp (1/α_s )]$.