def
definition
alpha_s_running
show as:
view math explainer →
open explainer
Read the cached plain-language explainer.
open lean source
IndisputableMonolith.Physics.QCDRGE.AlphaRunning on GitHub at line 56.
browse module
All declarations in this module, on Recognition.
explainer page
depends on
used by
formal source
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
63 · apply div_pos (sq_pos_of_pos hQ)
64 exact sq_pos_of_pos (by unfold M_Z_GeV; norm_num)
65 · rw [div_lt_one (by unfold M_Z_GeV; positivity)]
66 nlinarith [sq_nonneg Q, sq_nonneg M_Z_GeV, sq_abs (Q - M_Z_GeV)]
67
68/-- At Q = M_Z the log vanishes so α_s = α_s(M_Z). -/
69theorem log_at_MZ : Real.log (M_Z_GeV ^ 2 / M_Z_GeV ^ 2) = 0 := by
70 rw [div_self (by unfold M_Z_GeV; positivity)]; exact Real.log_one
71
72/-- Asymptotic freedom: α_s gets STRONGER at lower energies. -/
73theorem asymptotic_freedom :
74 0 < beta0 N_f_Z := beta0_pos N_f_Z (by unfold N_f_Z; norm_num)
75
76end
77
78end AlphaRunning
79end QCDRGE
80end Physics
81end IndisputableMonolith