The two-loop electroweak bosonic corrections to $sin^2theta_{rm eff}^{rm b}$

The prediction of the effective electroweak mixing angle $sin^2theta_{rm eff}^{rm b}$ in the Standard Model at two-loop accuracy has now been completed by the first calculation of the bosonic two-loop corrections to the $Z{bar b}b$ vertex. Numerical predictions are presented in the form of a fitting formula as function of $M_Z, M_W, M_H, m_t$ and $Delta{alpha}$, ${alpha_{rm s}}$. For central input values, we obtain a relative correction of $Deltakappa_{rm b}^{(alpha^2,rm bos)} = -0.9855 times 10^{-4}$, amounting to about a quarter of the fermionic corrections, and corresponding to $sin^2theta_{rm eff}^{rm b} = 0.232704$. The integration of the corresponding two-loop vertex Feynman integrals with up to three dimensionless parameters in Minkowskian kinematics has been performed with two approaches: (i) Sector decomposition, implemented in the packages FIESTA 3 and SecDec 3, and (ii) Mellin-Barnes representations, implemented in AMBRE 3/MB and the new package MBnumerics.