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.
In plain language
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(ιτ, 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) = (1 − 5αι_τ²). The NNLO correction is governed by the same universal modulus at the next perturbative order: per the Lean source TauHiggs2.lean (lines 597–600), the k-th perturbative order receives a W₃(4)^k factor — one lemniscate traversal at NLO, double traversal at NNLO — so the NNLO correction is (1 + α²W₃(4)²ι_τ⁴) = (1 + 25α²ι_τ⁴). The full NNLO formula (IV.D337) gives −0.7 ppm from the PDG value sin²θ_W = 0.23122. There is no W₃(3) factor in the NNLO term; the universal modulus across orders is W₃(4) = 5 alone. (Earlier drafts of this page erroneously introduced a “W₃(3) = 4” symbol in the NNLO formula — that was a bug; the canonical W₃(3) = 17 per Lean TauLib/BookI/CF/WindowAlgebra.lean:w3_at_3, and W₃(3) does not appear in the Weinberg derivation at any order.) This is part of the electroweak precision programme in Book IV that also produces M_W at −0.5 ppm (IV.T177); sin²θ_W is the most precisely derived EW parameter.
Result Statement
IV.D337: sin²θ_W at NNLO, governed by Window Universality W₃(4) = 5 (single universal modulus, with W₃(4)^k at the k-th perturbative order), at −0.7 ppm from PDG value. Lean-certified: TauLib/BookIV/Electroweak/WeinbergNLO.lean:consecutive_window_integers proves W₃(3) = 17, W₄(3) = 18, W₅(3) = 19; the W₃(4) = 5 modulus is at TauLib/BookI/CF/WindowAlgebra.lean:w3_at_4.
Cross-references
Glossary terms
Cross-domain. This result references glossary terms across 2 domains — a bridge between the framework's physics and metaphysics layers.
Physics: Weak mixing angle sin²θ_W
Metaphysics: Universal (structural position)