Reanalysis of the Higher Order Perturbative QCD corrections to Hadronic $Z$ Decays using the Principle of Maximum Conformality

TL;DR

This paper demonstrates that applying the principle of maximum conformality (PMC) to perturbative QCD calculations of Z boson decays significantly reduces renormalization scale uncertainties and improves convergence, enabling more precise Standard Model tests.

Contribution

The paper introduces the use of PMC for Z boson decay calculations, showing it reduces scheme dependence and improves perturbative series convergence compared to conventional methods.

Findings

PMC reduces renormalization scale uncertainties.

Improved convergence of perturbative series.

Enhanced precision for Standard Model tests.

Abstract

A complete calculation of the ${\cal O}(\alpha_s^4)$ perturbative QCD corrections to the hadronic decay width of the $Z$-boson has recently been performed by Baikov et al.[1]. In their analysis, Baikov et al. relied on the conventional practice of simply guessing the renormalization scale and taking an arbitrary range to estimate the pQCD uncertainties. This procedure inevitably introduces an arbitrary, scheme-dependent theoretical systematic error in the predictions. In this paper, we show that the renormalization scale uncertainties for hadronic $Z$ decays can be greatly reduced by applying the principle of maximum conformality (PMC), a rigorous extension of the BLM method. The PMC prediction is independent of the choice of renormalization scheme; i.e., it respects renormalization group invariance, and thus it provides an optimal and theoretically rigorous method for setting the renormalization scale. We show that the convergence of the pQCD prediction for the $Z$ hadronic width is greatly improved using the PMC since the divergent renormalon series does not appear. The magnitude of the high-order corrections quickly approach a steady point. The PMC predictions also have the property that any residual dependence on the choice of initial scale is highly suppressed, even for low-order predictions. Thus, one obtains optimal fixed-order predictions for the $Z$-boson hadronic decay rates thus enabling high precision tests of the Standard Model.