b1_at_Nf5_pos

formal source

  58  linarith
  59
  60/-- At N_f = 5 (canonical between bottom and top thresholds): b1 > 0. -/
  61theorem b1_at_Nf5_pos : 0 < b1 5 := b1_pos_lowNf 5 (by norm_num)
  62
  63/-- One-loop coefficient at N_f = 5 is positive (asymptotic freedom). -/
  64theorem b0_at_Nf5_pos : 0 < b0 5 := by
  65  unfold b0 N_c
  66  apply div_pos _ (by positivity : (0 : ℝ) < 12 * Real.pi)
  67  norm_num
  68
  69/-- The two-loop running closed form. -/
  70def alpha_s_two_loop (alpha_0 : ℝ) (N_f : ℕ) (mu_GeV mu_0_GeV : ℝ) : ℝ :=
  71  let L := Real.log (mu_GeV ^ 2 / mu_0_GeV ^ 2)
  72  let denom := 1 + b0 N_f * alpha_0 * L +
  73               (b1 N_f / b0 N_f) * alpha_0 *
  74               Real.log (1 + b0 N_f * alpha_0 * L)
  75  alpha_0 / denom
  76
  77/-- The two-loop denominator collapses to the one-loop denominator when the
  78    `b1`-induced correction term vanishes. -/
  79theorem alpha_s_two_loop_b1_zero_eq_one_loop
  80    (alpha_0 : ℝ) (mu_GeV mu_0_GeV : ℝ) :
  81    1 + b0 5 * alpha_0 * Real.log (mu_GeV ^ 2 / mu_0_GeV ^ 2) +
  82        0 * alpha_0 * Real.log (1 + b0 5 * alpha_0 *
  83                                Real.log (mu_GeV ^ 2 / mu_0_GeV ^ 2)) =
  84    1 + (11 * (N_c : ℝ) - 2 * 5) / (12 * Real.pi) * alpha_0 *
  85        Real.log (mu_GeV ^ 2 / mu_0_GeV ^ 2) := by
  86  unfold b0 N_c
  87  ring
  88
  89/-- Positivity of the two-loop alpha_s under the MS-bar perturbative window:
  90    when the alpha_0 input is positive and the log scale ratio is bounded
  91    so the denominator stays positive, the running coupling is positive. -/