Operator product expansion and analyticity

TL;DR

This paper analyzes the use of operator product expansion in QCD, focusing on the bounds of the remainder term and its dependence on complex plane angles, with implications for alculation of lpha_s from u decay.

Contribution

It provides explicit bounds on the OPE remainder term, highlighting their sensitivity to angle and function class assumptions, challenging previous constant remainder assumptions.

Findings

Bounds on the OPE remainder depend on deflection angle.

Assumption of constant remainder across the cut is unjustified.

Results impact alculation of lpha_s from u decay.

Abstract

We discuss the current use of the operator product expansion in QCD calculations. Treating the OPE as an expansion in inverse powers of an energy-squared variable (with possible exponential terms added), approximating the vacuum expectation value of the operator product by several terms and assuming a bound on the remainder along the euclidean region, we observe how the bound varies with increasing deflection from the euclidean ray down to the cut (Minkowski region). We argue that the assumption that the remainder is constant for all angles in the cut complex plane is not justified. Making specific assumptions on the properties of the expanded function, we obtain bounds on the remainder in explicit form and show that they are very sensitive both to the deflection angle and to the class of functions chosen. The results obtained are discussed in connetcion with calculations of the coupling constant \alpha_{s} from the \tau decay.