R(s) and hadronic tau-Decays in Order alpha_s^4: technical aspects

TL;DR

The paper addresses the ${O}(\alpha_s^4)$ corrections to $R(s)$ and the hadronic tau width through the vector-current correlator, detailing the perturbative framework and the need for verification. It employs RG evolution with the five-loop anomalous dimension $\gamma^{VV}$ and the four-loop polarization operator, working with $a_s=\alpha_s(\mu^2)/\pi$ and $L=\ln(\mu^2/Q^2)$ to obtain the $L$-dependent pieces and final coefficients. It provides explicit analytic results for $\gamma^{VV}_i$ up to $i=4$ and for the ${\cal O}(\alpha_s^3)$ polarization operator, enabling the construction of the final $R(s)$ pieces $r_0^{V,i}$ and $r_2^{V,i}$ with discussion of transcendental constants. A partial independent cross-check via Padé reconstruction of the polarization function (Hoang et al. 2008) supports the accuracy of the ${O}(\alpha_s^4)$ coefficients, strengthening confidence in these high-order QCD predictions.

Abstract

We report on some technical aspects of our calculation of alpha_s^4 corrections to R(s) and the semi-leptonic tau decay width [1-3]. We discuss the inner structure of the result as well as the issue of its correctness. We demonstrate recently appeared independent evidence positively testing one of two components of the full result.