A new analysis of the pQCD contributions to the electroweak parameter $rho$ using the single-scale approach of principle of maximum conformality

It has been observed that conventional renormalization scheme and scale ambiguities for the pQCD predictions can be eliminated by using the principle of maximum conformality (PMC). However, being the intrinsic nature of any perturbative theory, there are still two types of residual scale dependences due to uncalculated higher-order terms. In the paper, as a step forward of our previous work [Phys.Rev.D {bf 89},116001(2014)], we reanalyze the electroweak $rho$ parameter by using the PMC single-scale approach. Using the PMC conformal series and the Pad$acute{e}$ approximation approach, we observe that the residual scale dependence can be greatly suppressed and then a more precise pQCD prediction up to ${rm N^4LO}$-level can be achieved, e.g. $Deltarho|_{rm PMC}simeq(8.204pm0.012)times10^{-3}$, where the errors are squared averages of those from unknown higher-order terms and $Deltaalpha_s(M_Z)=pm 0.0010$. We then predict the magnitudes of the shifts of the $W$-boson mass and the effective leptonic weak-mixing angle: $delta M_{W}|_{rm N^4LO} =-0.26$ MeV and $delta sin^2{theta}_{rm eff}|_{rm N^4LO}=0.14times10^{-5}$, which are well below the precision anticipated for the future electron-position colliders such as FCC, CEPC and ILC. Thus by measuring those parameters, it is possible to test SM with high precision.