Determining $\alpha_s$ from hadronic $\tau$ decay: the pitfalls of truncating the OPE

TL;DR

This paper critically examines the truncated OPE approach used in extracting the strong coupling constant $\,\alpha_s$ from hadronic tau decay data, demonstrating its unreliability and emphasizing the importance of accounting for duality violations.

Contribution

The paper tests the truncated OPE strategy against $e^+e^-$ data and highlights the necessity of including duality-violating effects for accurate $\,\alpha_s$ determination.

Findings

The truncated OPE strategy fails when tested against $e^+e^-$ data.

Duality-violating contributions are significant in spectral functions.

Including duality violations is essential for reliable $\,\alpha_s$ extraction.

Abstract

We discuss sum-rule determinations of $\alpha_s$ from non-strange hadronic $\tau$-decay data. We investigate, in particular, the reliability of the assumptions underlying the "truncated OPE strategy," which specifies a certain treatment of non-perturbative contributions, and which was employed in Refs. [1-3]. Here, we test this strategy by applying the strategy to the $R$-ratio obtained from $e^+e^-$ data, which extend beyond the $\tau$ mass, and, based on the outcome of these tests, we demonstrate the failure of this strategy.We then present a brief overview of new results on the form of duality-violating non-perturbative contributions, which are conspicuously present in the experimentally determined spectral functions. As we show, with the current precision claimed for the extraction of $\alpha_s$, including a representation of duality violations is unavoidable if one wishes to avoid uncontrolled theoretical errors.