The two-loop electroweak bosonic corrections to ${\sin }^2{\theta }_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\mathrm{b}}$

The prediction of the effective electroweak mixing angle $\sin^2\theta_{\rm eff}^{\rm b}$ 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 $\mathrm{\Delta }\alpha$, ${\alpha_{\rm s}}$\mathrm{.}Forcentralinputvalues,weobtainarelativecorrectionof$\Delta\kappa_{\rm b}^{(\alpha^2,\rm bos)} = -0.9855 \times 10^{-4}$, amounting to about a quarter of the fermionic corrections, and corresponding to ${\sin }^2{\theta }_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\mathrm{b}}=0.232704$. 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.