A Study of Ultraviolet Renormalon Ambiguities in the Determination of $\as$ from $τ$ Decay

The divergent large-order behaviour of the perturbative series relevant for the determination of $\as$ from $\tau$ decay is controlled by the leading ultraviolet (UV) renormalon. Even in the absence of the first infrared (IR) renormalon, an ambiguity of order $\Lambda^2/m_\tau^2$ is introduced. We make a quantitative study of the practical implications of this ambiguity. We discuss the magnitude of UV renormalon corrections obtained in the large-$N_f$ limit, which, although unrealistic, is nevertheless interesting to some extent. We then study a number of improved approximants for the perturbative series, based on a change of variable in the Borel representation, such as to displace the leading UV renormalon singularity at a larger distance from the origin than the first IR renormalon. The spread of the resulting values of $\as(m^2_\tau)$ obtained by different approximants, at different renormalization scales, is exhibited as a measure of the underlying ambiguities. Finally, on the basis of mathematical models, we discuss the prospects of an actual improvement, given the signs and magnitudes of the computed coefficients, the size of $\as(m^2_\tau)$ and what is known of the asymptotic properties of the series. Our conclusion is that a realistic estimate of the theoretical error cannot go below $\delta\as(m^2_\tau) \sim \pm 0.060$, or $\delta\as(m^2_{\sss Z}) \sim \pm 0.006$.