On the use of the Operator Product Expansion in finite-energy sum rules for light-quark correlators

TL;DR

This paper critically examines the assumptions behind tau-based finite-energy sum-rule analyses, revealing that neglecting duality violations and higher-dimensional OPE contributions leads to unreliable determinations of the strong coupling constant.

Contribution

The paper introduces new tools to evaluate the impact of omitted higher-dimensional OPE terms and demonstrates the failure of the truncated OPE strategy in tau decay analyses.

Findings

Truncated OPE strategy is unreliable for strong coupling extraction.

Duality violations are significant at energies around 2-3 GeV$^2$.

New assessment tools reveal limitations of previous assumptions.

Abstract

Tau-based finite-energy sum-rule (FESR) analyses often assume that scales $s_0\sim m_\tau^2$ are large enough that (i) integrated duality violations (DVs) can be neglected, and (ii) contributions from non-perturbative OPE condensates of dimension $D$ scale as $(\Lambda_{\rm QCD}/m_\tau )^D$, allowing the OPE series to be truncated at low dimension. The latter assumption is not necessarily valid since the OPE series is not convergent, while the former is open to question given experimental results for the electromagnetic, $I=1$ vector ($V$), $I=1$ axial vector ($A$) and $I=1$ $V+A$ current spectral functions, which show DV oscillations with amplitudes comparable in size to the corresponding $\alpha_s$-dependent perturbative contributions at $s\sim2-3$ GeV$^2$. Here, we discuss recently introduced new tools for assessing the numerical relevance of omitted higher-$D$ OPE contributions. Applying these to the ``truncated OPE'' strategy used in Refs.[1,2] and earlier work by the same authors, we find that this strategy fails to yield reliable results for the strong coupling from hadronic $\tau$ decays.