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

## TL;DR

This paper challenges the common assumption that truncating the operator product expansion (OPE) at low dimensions has negligible effects on extracting the strong coupling constant, using $e^+e^-$ data to test the validity of this assumption.

## Contribution

It provides an empirical test of the OPE truncation assumptions using high-precision $e^+e^-$ data, revealing potential issues with the rapid convergence assumption near the $ au$ mass.

## Key findings

- Questions the validity of rapid convergence of low-dimension OPE terms
- Highlights the limitations of current finite-energy sum rule methods
- Recommends restricting analyses to observables dominated by lower-order OPE terms

## 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.