The two-loop electroweak bosonic corrections to sin2⁡θeffb

Abstract

The prediction of the effective electroweak mixing angle sin 2 θ b eff in the Standard Model at two-loop accuracy has now been completed by the first calculation of the bosonic two-loop corrections to the Zbb vertex. Numerical predictions are presented in the form of a fitting formula as function of M Z , M W , M H , m t and ∆α, α s . For central input values, we obtain a relative correction of ∆κ (α 2 ,bos) b = −0.9855 × 10 −4 , amounting to about a quarter of the fermionic corrections, and corresponding to sin 2 θ b eff = 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. arXiv:1607.08375v2 [hep-ph] 30 Sep 2016 * In fitting programs like Gfitter [ 14] it is assumed that the validity of (3) was established by data preparation. In this respect we would like to mention that a natural language for the unfolding of measured cross-sections into pseudoobservables (not discussed here) and the relation of pseudo-observables to theory predictions has been worked out in the S-matrix approach [15-17]. We will discuss this topic elsewhere.

Numerical integration of massive two-loop Mellin-Barnes integrals in Minkowskian regions

“…Difficult cases like thresholds are also treatable now. For Z → bb, MBnumerics.m turned out to be a very strong and effective tool [58,59].…”

“…In these proceedings a new approach to the numerical calculation of MB integrals is presented which has been developed during the work on the Z → bb vertex, aiming in evaluation of complete two-loop electroweak corrections to this process, see [58,59]. Historically, the MB.m package has been developed and first applied in the Bhabha massive QED 2-loop calculations, as a cross-check for analytical Master Integrals and their asymptotic expansions [60,61,62,63].…”

“…1 Initial tensors of rank 5 and 4 are reduced easily to objects of maximally rank 3 tensors [58,59].…”

“…It is interesting to note that at the the kinematic point s/M 2 Z = 1 + iε (ε is an arbitrarily small parameter chosing the correct sheet), which is a point explored in the Z → bb studies [58,59], shifts works well. To see this, we shift z 2 variable, z 2 = z 20 + n. The integral is now a discrete function of the number of shifts n:…”

The Global Electroweak and Higgs Fits in the LHC era

“…asymmetries has been updated with the 2-loop bosonic contributions to sin θ b eff from [7]. Using the preliminary combinations detailed in the previous lines we present in this section an (equally preliminary) update of the electroweak fit.…”