refactor: Major refactor of CKMMatrix
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jstoobysmith
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9 changed files with 1096 additions and 1512 deletions
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/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.FlavorPhysics.CKMMatrix.Basic
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import HepLean.FlavorPhysics.CKMMatrix.Rows
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import HepLean.FlavorPhysics.CKMMatrix.PhaseFreedom
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import HepLean.FlavorPhysics.CKMMatrix.Invariants
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import Mathlib.Analysis.SpecialFunctions.Complex.Arg
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open Matrix Complex
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open ComplexConjugate
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open CKMMatrix
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noncomputable section
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-- to be renamed stanParamAsMatrix ...
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def standardParameterizationAsMatrix (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : Matrix (Fin 3) (Fin 3) ℂ :=
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![![Real.cos θ₁₂ * Real.cos θ₁₃, Real.sin θ₁₂ * Real.cos θ₁₃, Real.sin θ₁₃ * exp (-I * δ₁₃)],
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![(-Real.sin θ₁₂ * Real.cos θ₂₃) - (Real.cos θ₁₂ * Real.sin θ₁₃ * Real.sin θ₂₃ * exp (I * δ₁₃)),
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Real.cos θ₁₂ * Real.cos θ₂₃ - Real.sin θ₁₂ * Real.sin θ₁₃ * Real.sin θ₂₃ * exp (I * δ₁₃),
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Real.sin θ₂₃ * Real.cos θ₁₃],
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![Real.sin θ₁₂ * Real.sin θ₂₃ - Real.cos θ₁₂ * Real.sin θ₁₃ * Real.cos θ₂₃ * exp (I * δ₁₃),
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(-Real.cos θ₁₂ * Real.sin θ₂₃) - (Real.sin θ₁₂ * Real.sin θ₁₃ * Real.cos θ₂₃ * exp (I * δ₁₃)),
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Real.cos θ₂₃ * Real.cos θ₁₃]]
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open CKMMatrix
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lemma standardParameterizationAsMatrix_unitary (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
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((standardParameterizationAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃)ᴴ * standardParameterizationAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃) = 1 := by
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funext j i
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simp only [standardParameterizationAsMatrix, neg_mul, Fin.isValue]
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rw [mul_apply]
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have h1 := exp_ne_zero (I * ↑δ₁₃)
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fin_cases j <;> rw [Fin.sum_univ_three]
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simp only [Fin.zero_eta, Fin.isValue, conjTranspose_apply, cons_val', cons_val_zero, empty_val',
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cons_val_fin_one, star_mul', RCLike.star_def, conj_ofReal, cons_val_one, head_cons, star_sub,
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star_neg, ← exp_conj, _root_.map_mul, conj_I, neg_mul, cons_val_two, tail_cons, head_fin_const]
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simp [conj_ofReal]
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rw [exp_neg ]
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fin_cases i <;> simp
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· ring_nf
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field_simp
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rw [sin_sq, sin_sq, sin_sq]
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ring
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· ring_nf
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field_simp
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rw [sin_sq, sin_sq]
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ring
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· ring_nf
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field_simp
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rw [sin_sq]
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ring
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simp only [Fin.mk_one, Fin.isValue, conjTranspose_apply, cons_val', cons_val_one, head_cons,
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empty_val', cons_val_fin_one, cons_val_zero, star_mul', RCLike.star_def, conj_ofReal, star_sub,
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← exp_conj, _root_.map_mul, conj_I, neg_mul, cons_val_two, tail_cons, head_fin_const, star_neg]
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simp [conj_ofReal]
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rw [exp_neg]
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fin_cases i <;> simp
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· ring_nf
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field_simp
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rw [sin_sq, sin_sq]
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ring
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· ring_nf
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field_simp
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rw [sin_sq, sin_sq, sin_sq]
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ring
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· ring_nf
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field_simp
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rw [sin_sq]
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ring
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simp only [Fin.reduceFinMk, Fin.isValue, conjTranspose_apply, cons_val', cons_val_two, tail_cons,
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head_cons, empty_val', cons_val_fin_one, cons_val_zero, star_mul', RCLike.star_def, conj_ofReal,
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← exp_conj, map_neg, _root_.map_mul, conj_I, neg_mul, neg_neg, cons_val_one, head_fin_const]
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simp [conj_ofReal]
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rw [exp_neg]
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fin_cases i <;> simp
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· ring_nf
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rw [sin_sq]
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ring
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· ring_nf
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rw [sin_sq]
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ring
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· ring_nf
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field_simp
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rw [sin_sq, sin_sq]
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ring
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def sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : CKMMatrix :=
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⟨standardParameterizationAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃, by
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rw [mem_unitaryGroup_iff']
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exact standardParameterizationAsMatrix_unitary θ₁₂ θ₁₃ θ₂₃ δ₁₃⟩
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lemma sP_cross (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
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[sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]t = (conj [sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]u ×₃ conj [sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]c) := by
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have h1 := exp_ne_zero (I * ↑δ₁₃)
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funext i
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fin_cases i
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· simp only [tRow, sP, standardParameterizationAsMatrix, neg_mul, exp_neg,
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Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, tail_cons,
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head_fin_const, cons_val_one, head_cons, Fin.zero_eta, crossProduct, uRow, cRow,
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LinearMap.mk₂_apply, Pi.conj_apply, _root_.map_mul, map_inv₀, ← exp_conj, conj_I, conj_ofReal,
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inv_inv, map_sub, map_neg]
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field_simp
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ring_nf
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rw [sin_sq]
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ring
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· simp only [tRow, sP, standardParameterizationAsMatrix, neg_mul, exp_neg, Fin.isValue, cons_val',
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cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, tail_cons, head_fin_const,
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cons_val_one, head_cons, Fin.mk_one, crossProduct, uRow, cRow, LinearMap.mk₂_apply,
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Pi.conj_apply, _root_.map_mul, conj_ofReal, map_inv₀, ← exp_conj, conj_I, inv_inv, map_sub,
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map_neg]
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field_simp
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ring_nf
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rw [sin_sq]
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ring
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· simp only [tRow, sP, standardParameterizationAsMatrix, neg_mul, exp_neg, Fin.isValue,
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cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, tail_cons, head_fin_const,
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cons_val_one, head_cons, Fin.reduceFinMk, crossProduct, uRow, cRow, LinearMap.mk₂_apply,
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Pi.conj_apply, _root_.map_mul, conj_ofReal, map_inv₀, ← exp_conj, conj_I, inv_inv, map_sub,
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map_neg]
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field_simp
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ring_nf
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rw [sin_sq]
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ring
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lemma eq_sP (U : CKMMatrix) {θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ} (hu : [U]u = [sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]u)
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(hc : [U]c = [sP θ₁₂ θ₁₃ θ₂₃ δ₁₃]c) (hU : [U]t = conj [U]u ×₃ conj [U]c) :
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U = sP θ₁₂ θ₁₃ θ₂₃ δ₁₃ := by
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apply ext_Rows hu hc
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rw [hU, sP_cross, hu, hc]
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lemma eq_phases_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ δ₁₃' : ℝ) (h : cexp (δ₁₃ * I) = cexp (δ₁₃' * I)) :
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sP θ₁₂ θ₁₃ θ₂₃ δ₁₃ = sP θ₁₂ θ₁₃ θ₂₃ δ₁₃' := by
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simp [sP, standardParameterizationAsMatrix]
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apply CKMMatrix_ext
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simp
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rw [show exp (I * δ₁₃) = exp (I * δ₁₃') by rw [mul_comm, h, mul_comm]]
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rw [show cexp (-(I * ↑δ₁₃)) = cexp (-(I * ↑δ₁₃')) by rw [exp_neg, exp_neg, mul_comm, h, mul_comm]]
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namespace Invariant
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lemma VusVubVcdSq_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂)
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(h2 : 0 ≤ Real.cos θ₁₃) (h3 : 0 ≤ Real.sin θ₂₃) (h4 : 0 ≤ Real.cos θ₁₂) :
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VusVubVcdSq ⟦sP θ₁₂ θ₁₃ θ₂₃ δ₁₃⟧ =
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Real.sin θ₁₂ ^ 2 * Real.cos θ₁₃ ^ 2 * Real.sin θ₁₃ ^ 2 * Real.sin θ₂₃ ^ 2 := by
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simp only [VusVubVcdSq, VusAbs, VAbs, VAbs', Fin.isValue, sP, standardParameterizationAsMatrix,
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neg_mul, Quotient.lift_mk, cons_val', cons_val_one, head_cons,
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empty_val', cons_val_fin_one, cons_val_zero, _root_.map_mul, VubAbs, cons_val_two, tail_cons,
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VcbAbs, VudAbs, Complex.abs_ofReal]
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by_cases hx : Real.cos θ₁₃ ≠ 0
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· rw [Complex.abs_exp]
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simp
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rw [_root_.abs_of_nonneg h1, _root_.abs_of_nonneg h3, _root_.abs_of_nonneg h2,
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_root_.abs_of_nonneg h4]
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simp [sq]
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ring_nf
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nth_rewrite 2 [Real.sin_sq θ₁₂]
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ring_nf
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field_simp
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ring
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· simp at hx
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rw [hx]
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simp
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lemma mulExpδ₃_sP (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂)
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(h2 : 0 ≤ Real.cos θ₁₃) (h3 : 0 ≤ Real.sin θ₂₃) (h4 : 0 ≤ Real.cos θ₁₂) :
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mulExpδ₃ ⟦sP θ₁₂ θ₁₃ θ₂₃ δ₁₃⟧ =
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sin θ₁₂ * cos θ₁₃ ^ 2 * sin θ₂₃ * sin θ₁₃ * cos θ₁₂ * cos θ₂₃ * cexp (I * δ₁₃) := by
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rw [mulExpδ₃, VusVubVcdSq_sP _ _ _ _ h1 h2 h3 h4 ]
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simp only [jarlskogℂ, sP, standardParameterizationAsMatrix, neg_mul,
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Quotient.lift_mk, jarlskogℂCKM, Fin.isValue, cons_val', cons_val_one, head_cons,
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empty_val', cons_val_fin_one, cons_val_zero, cons_val_two, tail_cons, _root_.map_mul, ←
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exp_conj, map_neg, conj_I, conj_ofReal, neg_neg, map_sub]
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simp
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ring_nf
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rw [exp_neg]
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have h1 : cexp (I * δ₁₃) ≠ 0 := exp_ne_zero _
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field_simp
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end Invariant
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end
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