c_2_alpha

formal source

  38
  39/-- NLO alpha_s matching coefficient at mu = m_q^MSbar.
  40    Standard result: c_2_alpha = 11/72. -/
  41def c_2_alpha : ℝ := 11 / 72
  42
  43/-- NLO mass matching coefficient at mu = m_q^MSbar.
  44    Standard result: d_2_mass = -89/432 (negative; mass slightly decreases). -/
  45def d_2_mass : ℝ := -89 / 432
  46
  47/-- Matching factor for alpha_s at threshold (NLO accuracy).
  48    Returns alpha_s^(n) given alpha_s^(n+1) at the threshold scale. -/
  49def alpha_match (alpha_above : ℝ) : ℝ :=
  50  alpha_above * (1 + c_2_alpha * (alpha_above / Real.pi) ^ 2)
  51
  52/-- Matching factor for the mass at threshold (NLO accuracy).
  53    Returns m^(n) given m^(n+1) at the threshold scale. -/
  54def mass_match (mass_above alpha_at_threshold : ℝ) : ℝ :=
  55  mass_above * (1 + d_2_mass * (alpha_at_threshold / Real.pi) ^ 2)
  56
  57/-! ## Positivity of matching factors
  58
  59For the small alpha values that occur at the heavy-quark thresholds
  60(alpha_s(m_b) ~ 0.22, alpha_s(m_c) ~ 0.38), the corrections are at the
  61percent level and never threaten positivity. We prove this conditionally
  62on the input being positive and the alpha being below 1 (perturbative). -/
  63
  64theorem alpha_match_pos (alpha_above : ℝ)
  65    (h_pos : 0 < alpha_above) (h_lt : alpha_above < 1) :
  66    0 < alpha_match alpha_above := by
  67  unfold alpha_match
  68  apply mul_pos h_pos
  69  have h_a_over_pi_pos : 0 < alpha_above / Real.pi :=
  70    div_pos h_pos Real.pi_pos
  71  have h_a_over_pi_sq_pos : 0 < (alpha_above / Real.pi) ^ 2 :=