The QCD improved electroweak parameter $\rho$

Abstract

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 $\Delta \rho {|}_{{\mathrm{N}}^3\mathrm{LO}}=\left(8.257_{-0.012}^{+0.045}\right)\times 10^{-3}$ 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 $\sim \pm 9\%$ at the two-loop level, which changes to $\sim \pm 4\%$ 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 $\Delta \rho {|}_{{\mathrm{N}}^3\mathrm{LO}}$ shall be almost fixed to $8.228\times 10^{-3}$, 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 $\delta M_W{|}_{{\mathrm{N}}^3\mathrm{LO}}=0.7$ MeV and $\delta {\sin }^2{\theta }_{\mathrm{eff}}{|}_{{\mathrm{N}}^3\mathrm{LO}}=-0.4\times 10^{-5}$.

8 pages, 2 figures. Discussions improved. To be published in Phys.Rev.D

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• QCD corrections to the $B_c$ to charmonia semi-leptonic decays
• Setting the Renormalization Scale in pQCD: Comparisons of the Principle of Maximum Conformality with the Sequential Extended Brodsky-Lepage-Mackenzie Approach
• Renormalization group improved pQCD prediction for ${\rm Y}(1S)$ leptonic decay
• General Properties on Applying the Principle of Minimum Sensitivity to High-order Perturbative QCD Predictions
• Elimination of QCD Renormalization Scale and Scheme Ambiguities
• A new analysis of the pQCD contributions to the electroweak parameter $\rho$ using the single-scale approach of principle of maximum conformality
• The Higgs-Boson Decay $H\to gg$ to Order ${\alpha }_s^5$ under the mMOM-Scheme