The QCD improved electroweak parameter $\ddot{I}$

Abstract

In the present paper, we make a detailed analysis for the QCD corrections to the electroweak $\ddot{I}$ parameter by applying the principle of maximum conformality (PMC). As a comparison, we show that under the conventional scale setting, we have $\hat{I}”\ddot{I}{|}_{{\mathrm{N}}^3\mathrm{LO}}=\left(8.257_{-0.012}^{+0.045}\right)\times 10^{-3}$ by varying the scale $\hat{I}{¼}_r\in [M_t/2$, $2M_t]$. By defining a ratio, $\hat{I}”R=\hat{I}”\ddot{I}/3X_t-1$, 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 $\ddot{I}$ parameter. On the other hand, by applying the PMC scale setting, the four-loop estimation $\hat{I}”\ddot{I}{|}_{{\mathrm{N}}^3\mathrm{LO}}$ 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 $\ddot{I}$ parameter has also been greatly improved due to the elimination of the divergent renormalon terms. As applications of the present QCD improved $\ddot{I}$ parameter, we show the shifts of the $W$-boson mass and the effective leptonic weak-mixing angle due to $\hat{I}”\ddot{I}$ can be reduced to $\hat{I}´M_W{|}_{{\mathrm{N}}^3\mathrm{LO}}=0.7$ MeV and $\hat{I}´{\sin }^2\hat{I}{¸}_{\mathrm{eff}}{|}_{{\mathrm{N}}^3\mathrm{LO}}=-0.4\times 10^{-5}$.

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