Weinberg Angle NNLO at −0.7 ppm
The weak mixing angle sin²θ_W is derived at NNLO to −0.7 ppm accuracy using the Window Universality W₃(4) = 5 correction.
Overview
IV.D337 gives the NNLO derivation of sin²θ_W = sin²θ_W^LO [1 + correction(ι_τ, α, W₃(4))] at −0.7 ppm from the PDG value. The NLO and NNLO corrections are both governed by Window Universality: W₃(4) = 5 appears at each order. This is one of three EW precision observables derived at sub-ppm accuracy in the τ-framework.
Detail
The Weinberg angle sin²θ_W ≈ 0.23122 is one of the most precisely measured parameters in particle physics. At LO in the τ-framework, sin²θ_W arises from the A-B sector mixing angle (weak-EM mixing). The LO formula gives a value at a few hundred ppm from experiment. The NLO correction uses the Window Universality: W₃(4) = 5 generates the first correction term (1 − αW₃(4)ι_τ²). The NNLO correction introduces the second Window term W₃(3) = 4: (1 + α²W₃(3)W₃(4)ι_τ⁴). The full NNLO formula (IV.D337) gives −0.7 ppm from the PDG value sin²θ_W = 0.23122. This is part of the electroweak precision programme in Book IV that also produces M_W at −0.5 ppm (IV.T177) and sin²θ_W is the most precisely derived EW parameter.
Result Statement
IV.D337: sin²θ_W at NNLO, governed by Window Universality W₃(4) = 5, at −0.7 ppm from PDG value.