Corrections of order ${\cal O}(G_F M_t^2 \as^2)$ to the $ρ$ parameter

TL;DR

This work addresses the precision of the electroweak $\rho$ parameter by computing the three-loop QCD corrections of order $O(G_F M_t^2 \alpha_s^2)$ from top-bottom quark loops. The authors perform a three-loop calculation of the $W$ and $Z$ self-energies in dimensional regularization with $m_b=0$, reduce the arising integrals to three master diagrams via integration-by-parts, and derive $\delta\rho$ in both MS-bar and on-shell mass schemes, finding results that differ significantly from a previous calculation by Avdeev et al. The final expressions show substantial corrections: in the MS-bar scheme $\delta\rho_{\overline{MS}} = 3 x_t \left( 1 - 0.19325 \frac{\alpha_s}{\pi} - 3.9696 \left(\frac{\alpha_s}{\pi}\right)^2 \right)$ and in the on-shell scheme $\delta\rho_{OS} = 3 X_t \left( 1 - 2.8599 \frac{\alpha_s}{\pi} - 14.594 \left(\frac{\alpha_s}{\pi}\right)^2 \right)$. These sizable corrections resemble or exceed the two-loop electroweak contributions for typical mass ratios and have important implications for precision predictions of $M_W$ and $\sin^2\Theta_{\text{eff}}$, while also illustrating reduced scheme dependence when higher orders are included.

Abstract

The three-loop QCD corrections to the $ρ$ parameter from top and bottom quark loops are calculated. The result differs from the one recently calculated by Avdeev et al. As function of the pole mass the numerical value is given by $\drho=\frac{3G_F M_t^2}{8\sqrt{2}π^2}(1- 2.8599 \api - 14.594 (\api)^2 )$.

Figures (1)

• Figure 1: Renormalization scale dependence of $\delta\rho_{OS}$ (a) and $\delta\rho_{\scriptsize\overline{\hbox{MS}}}$ (b).