Reanalysis of the higher order perturbative QCD corrections to hadronic $Z$ decays using the principle of maximum conformality

A complete calculation of the $\mathcal{O}({\alpha }_s^4)$ perturbative quantum chromodynamics (pQCD) corrections to the hadronic decay width of the $Z$ boson has recently been performed by Baikov et al. [Phys. Rev. Lett. 108, 222003 (2012)]. 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 Brodsky-Lepage-Mackenzie 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.

Figure 1

The nonsinglet contribution $r_{\mathrm{NS}}$ up to four-loop QCD corrections versus ${\mu }_r^{\text{init}}$ before and after PMC scale setting. After PMC scale setting, the curves for NLO, ${\mathrm{N}}^2\mathrm{LO}$, and ${\mathrm{N}}^3\mathrm{LO}$ almost coincide with each other.

Figure 2

The vector-singlet contribution $r_S^V$ up to two-loop QCD corrections versus the initial renormalization scale ${\mu }_r^{\text{init}}$ before and after PMC scale setting.

Figure 3

The axial-singlet contribution $r_S^A$ up to two-loop QCD corrections versus the initial renormalization scale ${\mu }_r^{\text{init}}$ before and after PMC scale setting.

Figure 4

The values of $r_{\mathrm{NS}}^{(n)}=1+\sum _{i=1}^n{C}_i^{\mathrm{NS}}{a}_s^i$ and their errors $\pm |C_n^{\mathrm{NS}}{a}_s^n{|}_{\text{MAX}}$. The diamonds and the crosses are for conventional (Conv.) and PMC scale settings, respectively. The central values assume the initial scale choice ${\mu }_r^{\text{init}}=M_Z$.