Helicity fractions of $W$ bosons from top quark decays at next-to-next-to-leading order in QCD

Decay rates of unpolarized top quarks into longitudinally and transversally polarized $W$ bosons are calculated to second order in the strong coupling constant ${\alpha }_s$. Including the finite bottom quark mass and electroweak effects, the standard model predictions for the $W$-boson helicity fractions are ${\mathcal{F}}_L=0.687(5)$, ${\mathcal{F}}_{+}=0.0017(1)$, and ${\mathcal{F}}_{-}=0.311(5)$.

Figure 1

Sample three-loop diagrams. Thick and thin lines denote top and bottom quarks, respectively. Wavy lines denote $W$ bosons and curly lines denote gluons. In the closed fermion loop all quark flavors have to be considered.