  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
$\delta \sin^2{\theta}_{\rm eff}|_{\rm N^3LO}=-0.4\times10^{-5}$.

High Energy Physics - Phenomenology