The QCD improved electroweak parameter $\rho$

In the present paper, we make a detailed analysis for the QCD corrections to the electroweak $\rho$ parameter by applying the principle of maximum conformality (PMC). As a comparison, we show that under the conventional scale setting, we have $\Delta\rho|_{\rm N^3LO} = \left(8.257^{+0.045}_{-0.012}\right) \times10^{-3}$ by varying the scale $\mu_{r}\in[M_{t}/2$, $2M_{t}]$. By defining a ratio, $\Delta R=\Delta\rho/3X_t-1$, 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 $\rho$ parameter. On the other hand, by applying the PMC scale setting, the four-loop estimation $\Delta\rho|_{\rm N^3LO}$ 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 $\rho$ parameter has also been greatly improved due to the elimination of the divergent renormalon terms. As applications of the present QCD improved $\rho$ parameter, we show the shifts of the $W$-boson mass and the effective leptonic weak-mixing angle due to $\Delta\rho$ can be reduced to $\delta M_{W}|_{\rm N^3LO} =0.7$ MeV and $\delta \sin^2{\theta}_{\rm eff}|_{\rm N^3LO}=-0.4\times10^{-5}$.