higgsMassRatio

formal source

 118noncomputable def wMassSq_over_vev (g : ℝ) (v : ℝ) : ℝ := g^2 * v^2 / 4
 119
 120/-- The Higgs-to-W mass ratio: m_H / m_W = 2/g = 2·√(m_Z²/v²)/sin(θ_W). -/
 121noncomputable def higgsMassRatio (g : ℝ) (hg : g > 0) : ℝ := 2 / g
 122
 123/-- With g = 2·m_W/v ≈ 2·80.4/246 ≈ 0.654:
 124    m_H / m_W = 2/g ≈ 2/0.654 ≈ 3.06 ... but observed is 125.2/80.4 ≈ 1.557.
 125
 126    The discrepancy: J″(1) = 1 gives the CURVATURE at the minimum, but the
 127    actual quartic coupling λ is renormalized by EW loop corrections.
 128    At the EW scale, λ_physical ≈ λ_RS · (1 - corrections).
 129    The loop correction: λ_ren ≈ λ_RS · sin²θ_W / (1 - sin²θ_W) × (factor)
 130
 131    More precisely: the RS mass formula for the Higgs uses:
 132    m_H² = 2λv² where λ = (3 - φ)/3 · sin²θ_W (from the Q₃ reduction)
 133    This gives m_H ≈ v · √(2(3-φ)/3 · sin²θ_W) -/
 134noncomputable def sin2ThetaW_RS : ℝ := (3 - phi) / 6
 135
 136theorem sin2ThetaW_RS_val : sin2ThetaW_RS = (3 - phi) / 6 := rfl
 137
 138/-- sin²θ_W^RS is positive. -/
 139theorem sin2ThetaW_RS_pos : 0 < sin2ThetaW_RS := by
 140  unfold sin2ThetaW_RS
 141  apply div_pos
 142  · linarith [phi_lt_onePointSixTwo]
 143  · norm_num
 144
 145/-- sin²θ_W^RS is less than 0.5. -/
 146theorem sin2ThetaW_RS_lt_half : sin2ThetaW_RS < 1/2 := by
 147  unfold sin2ThetaW_RS
 148  rw [div_lt_iff₀ (by norm_num : (0:ℝ) < 6)]
 149  linarith [phi_gt_onePointSixOne]
 150
 151/-- The RS prediction for sin²θ_W: ≈ (3-1.618)/6 ≈ 0.230. -/