Abstract
Precise extractions of $\alpha_s$ from $\tau\to {\rm (hadrons)}+\nu_\tau$ and from $e^+e^-\to {\rm (hadrons)}$ below the charm threshold rely on finite energy sum rules (FESRs) where the experimental side is given by integrated spectral function moments. Here we study the renormalons that appear in the Borel transform of polynomial moments in the large-$\beta_0$ limit and in full QCD. In large-$\beta_0$, we establish a direct connection between the renormalons and the perturbative behaviour of moments often employed in the literature. The leading IR singularity is particularly prominent and is behind the fate of moments whose perturbative series are unstable, while those with good perturbative behaviour benefit from partial cancellations of renormalon singularities. The conclusions can be extended to QCD through a convenient scheme transformation to the $C$-scheme together with the use of a modified Borel transform which make the results particularly simple; the leading IR singularity becomes a simple pole, as in large-$\beta_0$. Finally, for the moments that display good perturbative behaviour, we discuss an optimized truncation based on renormalisation scheme (or scale) variation. Our results allow for a deeper understanding of the perturbative behaviour of integrated spectral function moments and provide theoretical support for low-$Q^2$ $\alpha_s$ determinations.
Highlights
Extractions of the strong coupling, αs, at lower energies can be very precise due to increased sensitivity to the higher-order corrections, as long as the nonperturbative contributions are under good control
We are in a position to draw a few conclusions from the study of spectral function moments in large β0 and its truncated form: (i) The Borel transform of δðw0iÞ for polynomial weight functions is less singular than the Borel transform of the Adler function
The Borel transform has a pole at u 1⁄4 2 if and only if the weight function contains a term proportional to x
Summary
Extractions of the strong coupling, αs, at lower energies can be very precise due to increased sensitivity to the higher-order corrections, as long as the nonperturbative contributions are under good control. In order to extract from the data αs and the nonperturbative parameters in a self-consistent way, i.e., without relying on external information, one resorts to the use of several (pseudo) observables Those are built using the fact that any analytic weight function gives rise to a valid FESR, with an experimental side that can be computed from the empirical spectral functions and a theoretical counterpart that can be obtained from the integral along the complex contour. The main purpose of this work is to understand the perturbative behavior of the different integrated spectral function moments at intermediate and high orders by studying the renormalon singularities appearing in their Borel transform. The enhancement and suppression of renormalon singularities identified in large β0 is, present in QCD which explains the similarity between the perturbative behavior of moments in the two cases. In the Appendix A we present our conventions for the QCD β function; Appendix B contains further details about the Borel integral of the moments discussed in this work
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