beta0_pos

formal source

  29def beta0 (N_f : ℕ) : ℝ := (11 * (N_c : ℝ) - 2 * (N_f : ℝ)) / (12 * Real.pi)
  30
  31/-- b₀ > 0 for N_f ≤ 16 (asymptotic freedom). -/
  32theorem beta0_pos (N_f : ℕ) (hNf : N_f ≤ 16) : 0 < beta0 N_f := by
  33  unfold beta0 N_c
  34  apply div_pos _ (mul_pos (by norm_num : (0:ℝ) < 12) Real.pi_pos)
  35  have h : (N_f : ℝ) ≤ 16 := by exact_mod_cast hNf
  36  simp only [N_c, Nat.cast_ofNat]
  37  nlinarith
  38
  39/-- At N_f = 5: b₀ = 23/(12π). -/
  40theorem beta0_at_Z : beta0 N_f_Z = 23 / (12 * Real.pi) := by
  41  unfold beta0 N_c N_f_Z; ring
  42
  43/-- α_s at M_Z from RS. -/
  44def alpha_s_MZ : ℝ := alpha_s_geom
  45
  46theorem alpha_s_MZ_pos : 0 < alpha_s_MZ := by unfold alpha_s_MZ alpha_s_geom; norm_num
  47
  48/-- α_s(M_Z) is between 0.11 and 0.12. -/
  49theorem alpha_s_MZ_bounds : 0.11 < alpha_s_MZ ∧ alpha_s_MZ < 0.12 := by
  50  unfold alpha_s_MZ alpha_s_geom; constructor <;> norm_num
  51
  52/-- Z boson mass in GeV. -/
  53def M_Z_GeV : ℝ := 91.1876
  54
  55/-- Running α_s at scale Q (one-loop). -/
  56def alpha_s_running (Q_GeV : ℝ) : ℝ :=
  57  alpha_s_MZ / (1 + beta0 N_f_Z * alpha_s_MZ * Real.log (Q_GeV ^ 2 / M_Z_GeV ^ 2) / (2 * Real.pi))
  58
  59/-- For Q < M_Z, ln(Q²/M_Z²) < 0, so α_s increases (asymptotic freedom in reverse). -/
  60theorem low_Q_log_negative (Q : ℝ) (hQ : 0 < Q) (h : Q < M_Z_GeV) :
  61    Real.log (Q ^ 2 / M_Z_GeV ^ 2) < 0 := by
  62  apply Real.log_neg