30 years, 750 integrals, and 1 desert

Abstract

The one-loop corrections to the weak mixing angle sin 2 θ b eff , derived from the Zbb vertex, are known since 1985. It took another 30 years to calculate the complete electroweak two-loop corrections to sin 2 θ b eff . The main obstacle was the calculation of the O(700) bosonic two-loop vertex integrals with up to three mass scales, at s = M 2 Z . 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 θ b eff is: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 θ b eff is now predicited in the Standard Model with a relative error of 10 −4 [1].