30 years, some 700 integrals, and 1 dessert, or: Electroweak two-loop corrections to the Zbb vertex

The one-loop corrections to the weak mixing angle ${\sin }^2{\theta }_{eff}^b$ derived from the $Z\bar{b}b$ vertex, are known since 1985. It took another 30 years to calculate the complete electroweak two-loop corrections to ${\sin }^2{\theta }_{eff}^b$. The main obstacle was the calculation of the O(700) bosonic two-loop vertex integrals with up to three mass scales, at $s=M_Z^2$. We did not perform the usual integral reduction and master evaluation, but chose a completely numerical approach, using two different calculational chains. One method relies on publicly available sector decomposition implementations. Further, we derived Mellin-Barnes (MB) representations, exploring the publicly available MB suite. We had to supplement the MB suite by two new packages: AMBRE~3, a Mathematica program, for the efficient treatment of non-planar integrals and MBnumerics for advanced numerics in the Minkowskian space-time. Our preliminary result for LL2016, the “dessert”, for the electroweak bosonic two-loop contributions to ${\sin }^2{\theta }_{eff}^b$ is: $\Delta \sin^2\theta_{eff}^{b(\alpha^2,\rm bos)} = \sin^2\theta_W ~ \Delta\kappa_b^{(\alpha^2,bos)}$,with$\Delta\kappa_b^{(\alpha^2,bos)} = -1.0276 x 10^{-4}$. This contribution is about a quarter of the corresponding fermionic corrections and of about the same magnitude as several of the known higher-order QCD corrections. The ${\sin }^2{\theta }_{eff}^b$ is now predicited in the Standard Model with a relative error of ${10}^{-4}$ [1].