Bosonic Corrections to The Effective Weak Mixing Angle at O(alpha^2)
M. Awramik, M. Czakon, A. Freitas
TL;DR
This work completes the bosonic two-loop electroweak corrections to the effective leptonic weak mixing angle, $\sin^2\theta^{\text{lept}}_{\text{eff}}$, by calculating the full ${\mathcal{O}}(\alpha^2)$ contributions to $\Delta\kappa$. Using expansions in $s_W^2$ and $s_H^2$, expansion-by-regions techniques, and IBP/Laporta reduction to master integrals (evaluated via multiple numerical methods), the authors obtain the complete bosonic piece and confirm gauge-parameter cancellations. The bosonic corrections are found to be tiny, of order $10^{-6}$, with Higgs-mass dependence yielding a slight reduction with increasing $M_H$ and significant cancellations with the $W$-mass contribution; the net effect on $\sin^2\theta^{\text{lept}}_{\text{eff}}$ is at the $10^{-5}$ level. The result supports the reliability of existing fermionic fits for current precision and sets a conservative theoretical error of $4.7\times 10^{-5}$, while highlighting the need for future higher-order calculations for ultimate collider precision.
Abstract
We present the complete bosonic contributions to the effective weak mixing angle, $\seff$, at the two-loop level in the electroweak interactions. We find their size to be about three times smaller than inferred from simple estimates from lower orders. In particular, for a Higgs boson mass, $M_H$, of 100 GeV they amount to $4 \times 10^{-6}$, and drop down by about an order of magnitude for $M_H = 200$ GeV. We estimate the intrinsic error of the theory prediction of $\seff$ to be $4.7 \times 10^{-5}$.