Evidence against naive truncations of the OPE from $e^{+}{e}^{-}\to$ hadrons below charm

Abstract

The operator product expansion (OPE), truncated in dimension, is employed in many contexts. An example is the extraction of the strong coupling, ${\alpha }_s$, from hadronic $\tau$-decay data, using a variety of analysis methods based on finite-energy sum rules. Here, we reconsider a long-used method, which parametrizes non-perturbative contributions to the $I=1$ vector and axial vacuum polarizations with the OPE, setting several higher-dimension coefficients to zero in order to implement the method in practice. The assumption that doing this has a negligible effect on the value of ${\alpha }_s$ is tantamount to the assumption that the low-dimension part of the OPE converges rapidly with increasing dimension near the $\tau$ mass. Were this assumption valid, it would certainly have to be valid at energies above the $\tau$ mass as well. It follows that the method can be tested using data obtained from $e^+e^-\to\mbox{hadrons}$, as they are not limited by the kinematic constraints of $\tau$ decays. We carry out such an investigation using a recent high-precision compilation for the $R$-ratio, arguing that it provides insights into the validity of the strategy, even if it probes a different, though related channel. We find that $e^{+}{e}^{-}$-based tests call into question the implied assumption of rapid convergence of the low-dimension part of the OPE around the $\tau$ mass, and thus underscore the need to restrict finite-energy sum-rule analyses to observables which receive only contributions from lower-order terms in the OPE.

23 pages, 10 figures

References in corpus (10)

• The muon $g-2$ and $\alpha (M_Z^2)$: a new data-based analysis
• alpha_s and the tau hadronic width: fixed-order, contour-improved and higher-order perturbation theory
• The strong coupling from the revised ALEPH data for hadronic $\tau$ decays
• Possible duality violations in tau decay and their impact on the determination of alpha_s
• Hyperasymptotics and quark-hadron duality violations in QCD
• The strong coupling from hadronic $\tau$ decays: a critical appraisal
• The strong coupling from $e^{+}{e}^{-}\to$ hadrons below charm
• Status of the Hadronic tau Decay Determination of |V_{us}|
• Determining ${\alpha }_s$ from hadronic $\tau$ decay: the pitfalls of truncating the OPE
• The strong coupling: a theoretical perspective

Cited by in corpus (7)

• The strong coupling from an improved $\tau$ vector isovector spectral function
• Reconciling the Contour-Improved and Fixed-Order Approaches for $\tau$ Hadronic Spectral Moments II: Renormalon Norm and Application in ${\alpha }_s$ Determinations
• Determination of perturbative QCD coupling from ALEPH $\tau$ decay data using pinched Borel-Laplace and Finite Energy Sum Rules
• Violations of Quark-Hadron Duality in Low-Energy Determinations of ${\alpha }_s$
• Borel-Laplace Sum Rules with $\tau$ decay data, using OPE with improved anomalous dimensions
• Conformal mappings in perturbative QCD
• Spectral sum of current correlators from lattice QCD