Theory of Pre-Asymptotic Effects in Weak Inclusive Decays

TL;DR

This paper discusses the theoretical framework of pre-asymptotic effects in weak inclusive decays, emphasizing the role of the operator product expansion and quark-hadron duality, with insights into the asymptotic nature of the expansion.

Contribution

It provides a systematic analysis of pre-asymptotic effects using OPE/HQET expansion and proves the asymptotic nature of the operator product expansion in this context.

Findings

Operator product expansion is asymptotic.

High-order terms influence duality accuracy.

Divergence of high-order terms affects theoretical predictions.

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 $\alpha_s$ plane, with terms of the type $\exp [-\exp (1/\alpha_s )]$.