W and Z Masses in Pion Units: Withdrawal of a Statistical Claim, and the One-Loop Question That Remains

The W and Z boson masses, expressed in units of the neutral pion mass (mπ⁰ = 134. 9768 ± 0. 0005 MeV, PDG 2025), are 595. 430 and 675. 580. Replacing these by the integers 595 and 675 gives sin²θW = 1 − (595/675) ² = 0. 22299, against a measured on-shell value of 0. 22290 ± 0. 00030. v1 of this record presented that agreement as statistically significant. This version withdraws that presentation for the reasons given in the correction notice above, and states three facts v1 did not. First, the pion does not appear in the result. In the on-shell scheme sin²θW is *defined* as 1 − (mW/mZ) ², so the base mass cancels identically; the content of the claim is that mW/mZ ≈ 119/135, and mπ enters only as the quantity that made the integers visible. Second, the integer 675 is not the nearest integer to 675. 580 — 676 is. v1 acknowledged this selection in a footnote and called it mildly post-hoc. It is promoted here because it is load-bearing: nZ = 676 gives sin²θW = 0. 22529, which is 7. 96σ from measurement, while nZ = 675 gives 0. 30σ. Third, the rounding improves the agreement. The exact tree-level value from PDG masses is 1 − (mW/mZ) ² = 0. 22320, which sits 1. 01σ from the measured on-shell value. The rounded integers give 0. 30σ. **The approximation outperforms the exact calculation**, because the rounding error happens to point in the same direction as the radiative correction that a tree-level expression omits. The ρ (770), with integers 104/118, returns 0. 22321 — essentially the exact tree-level value — and is scored *worse* by v1's criterion for that reason. What remains is a well-posed open calculation, stated in v1 §5 and preserved here: whether standard one-loop electroweak corrections applied to 0. 22299 as a tree-level input converge to the MS-bar value 0. 23122 ± 0. 00003. That calculation is routine for a precision electroweak theorist, has not been performed, and would settle the question either way. **Keywords: ** weak mixing angle · W boson · Z boson · pion mass · look-elsewhere effect · null distribution · post-hoc selection · correction · negative result