Three-Loop Leading Top Mass Contributions to the rho Parameter
J. J. van der Bij, K. G. Chetyrkin, M. Faisst, G. Jikia, T. Seidensticker
TL;DR
The paper addresses the leading three-loop corrections from a heavy top quark to the electroweak rho parameter in the limit $M_t \gg M_W, M_Z$ with a massless Higgs ($M_H=0$). It employs an external-momentum zero-momentum expansion to reduce three-loop diagrams to vacuum integrals and computes the self-energy (renormalization) pieces, delivering analytic results for the terms of order $X_t^3$ and $X_t^2 \alpha_s$, where $X_t = G_F M_t^2/(8\sqrt{2}\pi^2)$. The main findings show the three-loop electroweak correction is sizeable relative to the leading two-loop term (about 36%), but still smaller than the subleading two-loop $M_t^2 M_Z^2$ contributions (about 2%), with the mixed $O(X_t^2 \alpha_s)$ piece being an order of magnitude smaller. This work reduces theoretical uncertainty in precision electroweak tests and informs the perturbative behavior of heavy-top effects in the rho parameter, contributing to a clearer understanding of high-order Standard Model radiative corrections.
Abstract
We present analytical results for the leading contributions of the top quark to the electroweak rho parameter at order GF^3 Mt^6 and GF^2 Mt^4 alpha_s. The Higgs boson and the gauge bosons are taken to be massless in this limit. The correction of order GF^3 Mt^6 is found to be sizeable in comparison to the the leading two-loop GF^2 Mt^4 correction, however it is much smaller than the subleading GF^2 Mt^2 MZ^2 correction.