alpha_s_two_loop_b1_zero_eq_one_loop

formal source

  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. -/
  92theorem alpha_s_two_loop_pos_when_denom_pos
  93    (alpha_0 : ℝ) (N_f : ℕ) (mu_GeV mu_0_GeV : ℝ)
  94    (h_alpha_pos : 0 < alpha_0)
  95    (h_denom_pos : 0 < 1 + b0 N_f * alpha_0 * Real.log (mu_GeV ^ 2 / mu_0_GeV ^ 2) +
  96                    (b1 N_f / b0 N_f) * alpha_0 *
  97                    Real.log (1 + b0 N_f * alpha_0 * Real.log (mu_GeV ^ 2 / mu_0_GeV ^ 2))) :
  98    0 < alpha_s_two_loop alpha_0 N_f mu_GeV mu_0_GeV := by
  99  unfold alpha_s_two_loop
 100  exact div_pos h_alpha_pos h_denom_pos
 101
 102/-! ## At-Z bracket sanity check
 103
 104The PDG value `alpha_s(M_Z) = 0.1179 +- 0.0009`. The RS prediction is
 105`alpha_s_geom = 2/17 ≈ 0.11765`, sitting inside the PDG bracket. The two-loop
 106formula, evaluated at mu_0 = M_Z = mu, returns alpha_0 exactly (the log
 107vanishes), so the two-loop framework is self-consistent at the anchor scale.
 108-/
 109