The QCD improved electroweak parameter $ρ$

In the present paper, we make a detailed analysis for the QCD corrections to the electroweak $ρ$ parameter by applying the principle of maximum conformality (PMC). As a comparison, we show that under the conventional scale setting, we have $Δρ|_{\rm N^3LO} = \left(8.257^{+0.045}_{-0.012}\right) \times10^{-3}$ by varying the scale $μ_{r}\in[M_{t}/2$, $2M_{t}]$. By defining a ratio, $ΔR=Δρ/3X_t-1$, which shows the relative importance of the QCD corrections, it is found that its scale error is $\sim \pm9 \%$ at the two-loop level, which changes to $\sim\pm4\%$ at the three-loop level and $\sim \pm 2.5\%$ at the four-loop level, respectively. These facts well explain why the conventional scale uncertainty constitutes an important error for estimating the $ρ$ parameter. On the other hand, by applying the PMC scale setting, the four-loop estimation $Δρ|_{\rm N^3LO}$ shall be almost fixed to $8.228\times10^{-3}$, which indicates that the conventional scale error has been eliminated. We observe the pQCD convergence for the $ρ$ parameter has also been greatly improved due to the elimination of the divergent renormalon terms. As applications of the present QCD improved $ρ$ parameter, we show the shifts of the $W$-boson mass and the effective leptonic weak-mixing angle due to $Δρ$ can be reduced to $δM_{W}|_{\rm N^3LO} =0.7$ MeV and $δ\sin^2θ_{\rm eff}|_{\rm N^3LO}=-0.4\times10^{-5}$.

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