Using improved Operator Product Expansion in Borel-Laplace Sum Rules with ALEPH $\tau$ decay data, and determination of pQCD coupling

TL;DR

This paper improves the determination of the strong coupling constant _s by applying an enhanced Operator Product Expansion and various perturbation theories to ALEPH ecay data, leading to more accurate results.

Contribution

It introduces an improved truncated OPE with renormalon-motivated terms and compares different perturbation methods for _s extraction from ecay data.

Findings

_s(m_{ au}^2) = 0.3235^{+0.0138}_{-0.0126} from main methods

The improved OPE better accounts for renormalon effects in sum rules

The results are consistent with previous determinations but with reduced uncertainties.

Abstract

We use improved truncated Operator Product Expansion (OPE) for the Adler function, involving two types of terms with dimension $D=6$, in the double-pinched Borel-Laplace Sum Rules and Finite Energy Sum Rules for the V+A channel strangeless semihadronic $\tau$ decays. The generation of the higher order perturbative QCD terms of the $D=0$ part of the Adler function is carried out using a renormalon-motivated ansatz incorporating the leading UV renormalon and the first two leading IR renormalons. The trunacted $D=0$ part of the Sum Rules is evaluated by two variants of the fixed-order perturbation theory (FO), by Principal Value of the Borel resummation (PV), and by contour-improved perturbation theory (CI). For the experimental V+A channel spectral function we use the ALEPH $\tau$-decay data. We point out that the truncated FO and PV evaluation methods account correctly for the renormalon structure of the Sum Rules, while this is not the case for the truncated CI evaluation. We extract the value of the ${\overline {\rm MS}}$ coupling $\alpha_s(m_{\tau}^2) = 0.3235^{+0.0138}_{-0.0126}$ [$\alpha_s(M_Z^2)=0.1191 \pm 0.0016$] for the average of the two FO methods and the PV method, which we consider as our main result. If we included in the average also CI extraction, the value would be $\alpha_s(m_{\tau}^2) = 0.3299^{+0.0232}_{-0.0225}$ [$\alpha_s(M_Z^2)=0.1199^{+0.0026}_{-0.0028}$]. This work is an extension and improvement of our previous work [Eur.Phys.J.C81 (2021) 10, 930] where we used for the truncated OPE a more naive (and widely used) form and where the extracted values for $\alpha_s(M_Z^2)$ were somewhat lower.