/- Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Joseph Tooby-Smith -/ import HepLean.FlavorPhysics.CKMMatrix.Basic import HepLean.FlavorPhysics.CKMMatrix.Rows import HepLean.FlavorPhysics.CKMMatrix.Invariants import Mathlib.Analysis.SpecialFunctions.Complex.Arg /-! # Standard parameterization for the CKM Matrix This file defines the standard parameterization of CKM matrices in terms of four real numbers `θ₁₂`, `θ₁₃`, `θ₂₃` and `δ₁₃`. We will show that every CKM matrix can be written within this standard parameterization in the file `FlavorPhysics.CKMMatrix.StandardParameters`. -/ open Matrix Complex open ComplexConjugate open CKMMatrix noncomputable section /-- Given four reals `θ₁₂ θ₁₃ θ₂₃ δ₁₃` the standard parameterization of the CKM matrix as a `3×3` complex matrix. -/ def standParamAsMatrix (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : Matrix (Fin 3) (Fin 3) ℂ := ![![Real.cos θ₁₂ * Real.cos θ₁₃, Real.sin θ₁₂ * Real.cos θ₁₃, Real.sin θ₁₃ * exp (-I * δ₁₃)], ![(-Real.sin θ₁₂ * Real.cos θ₂₃) - (Real.cos θ₁₂ * Real.sin θ₁₃ * Real.sin θ₂₃ * exp (I * δ₁₃)), Real.cos θ₁₂ * Real.cos θ₂₃ - Real.sin θ₁₂ * Real.sin θ₁₃ * Real.sin θ₂₃ * exp (I * δ₁₃), Real.sin θ₂₃ * Real.cos θ₁₃], ![Real.sin θ₁₂ * Real.sin θ₂₃ - Real.cos θ₁₂ * Real.sin θ₁₃ * Real.cos θ₂₃ * exp (I * δ₁₃), (-Real.cos θ₁₂ * Real.sin θ₂₃) - (Real.sin θ₁₂ * Real.sin θ₁₃ * Real.cos θ₂₃ * exp (I * δ₁₃)), Real.cos θ₂₃ * Real.cos θ₁₃]] open CKMMatrix lemma standParamAsMatrix_unitary (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : ((standParamAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃)ᴴ * standParamAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃) = 1 := by funext j i simp only [standParamAsMatrix, neg_mul, Fin.isValue] rw [mul_apply] have h1 := exp_ne_zero (I * ↑δ₁₃) fin_cases j <;> rw [Fin.sum_univ_three] simp only [Fin.zero_eta, Fin.isValue, conjTranspose_apply, cons_val', cons_val_zero, empty_val', cons_val_fin_one, star_mul', RCLike.star_def, conj_ofReal, cons_val_one, head_cons, star_sub, star_neg, ← exp_conj, _root_.map_mul, conj_I, neg_mul, cons_val_two, tail_cons, head_fin_const] simp [conj_ofReal] rw [exp_neg ] fin_cases i <;> simp · ring_nf field_simp rw [sin_sq, sin_sq, sin_sq] ring · ring_nf field_simp rw [sin_sq, sin_sq] ring · ring_nf field_simp rw [sin_sq] ring simp only [Fin.mk_one, Fin.isValue, conjTranspose_apply, cons_val', cons_val_one, head_cons, empty_val', cons_val_fin_one, cons_val_zero, star_mul', RCLike.star_def, conj_ofReal, star_sub, ← exp_conj, _root_.map_mul, conj_I, neg_mul, cons_val_two, tail_cons, head_fin_const, star_neg] simp [conj_ofReal] rw [exp_neg] fin_cases i <;> simp · ring_nf field_simp rw [sin_sq, sin_sq] ring · ring_nf field_simp rw [sin_sq, sin_sq, sin_sq] ring · ring_nf field_simp rw [sin_sq] ring simp only [Fin.reduceFinMk, Fin.isValue, conjTranspose_apply, cons_val', cons_val_two, tail_cons, head_cons, empty_val', cons_val_fin_one, cons_val_zero, star_mul', RCLike.star_def, conj_ofReal, ← exp_conj, map_neg, _root_.map_mul, conj_I, neg_mul, neg_neg, cons_val_one, head_fin_const] simp [conj_ofReal] rw [exp_neg] fin_cases i <;> simp · ring_nf rw [sin_sq] ring · ring_nf rw [sin_sq] ring · ring_nf field_simp rw [sin_sq, sin_sq] ring /-- A CKM Matrix from four reals `θ₁₂`, `θ₁₃`, `θ₂₃`, and `δ₁₃`. This is the standard parameterization of CKM matrices. -/ def standParam (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : CKMMatrix := ⟨standParamAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃, by rw [mem_unitaryGroup_iff'] exact standParamAsMatrix_unitary θ₁₂ θ₁₃ θ₂₃ δ₁₃⟩ namespace standParam lemma cross_product_t (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : [standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃]t = (conj [standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃]u ×₃ conj [standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃]c) := by have h1 := exp_ne_zero (I * ↑δ₁₃) funext i fin_cases i · simp only [tRow, standParam, standParamAsMatrix, neg_mul, exp_neg, Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, tail_cons, head_fin_const, cons_val_one, head_cons, Fin.zero_eta, crossProduct, uRow, cRow, LinearMap.mk₂_apply, Pi.conj_apply, _root_.map_mul, map_inv₀, ← exp_conj, conj_I, conj_ofReal, inv_inv, map_sub, map_neg] field_simp ring_nf rw [sin_sq] ring · simp only [tRow, standParam, standParamAsMatrix, neg_mul, exp_neg, Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, tail_cons, head_fin_const, cons_val_one, head_cons, Fin.mk_one, crossProduct, uRow, cRow, LinearMap.mk₂_apply, Pi.conj_apply, _root_.map_mul, conj_ofReal, map_inv₀, ← exp_conj, conj_I, inv_inv, map_sub, map_neg] field_simp ring_nf rw [sin_sq] ring · simp only [tRow, standParam, standParamAsMatrix, neg_mul, exp_neg, Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, cons_val_two, tail_cons, head_fin_const, cons_val_one, head_cons, Fin.reduceFinMk, crossProduct, uRow, cRow, LinearMap.mk₂_apply, Pi.conj_apply, _root_.map_mul, conj_ofReal, map_inv₀, ← exp_conj, conj_I, inv_inv, map_sub, map_neg] field_simp ring_nf rw [sin_sq] ring lemma eq_rows (U : CKMMatrix) {θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ} (hu : [U]u = [standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃]u) (hc : [U]c = [standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃]c) (hU : [U]t = conj [U]u ×₃ conj [U]c) : U = standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃ := by apply ext_Rows hu hc rw [hU, cross_product_t, hu, hc] lemma eq_exp_of_phases (θ₁₂ θ₁₃ θ₂₃ δ₁₃ δ₁₃' : ℝ) (h : cexp (δ₁₃ * I) = cexp (δ₁₃' * I)) : standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃ = standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃' := by simp [standParam, standParamAsMatrix] apply CKMMatrix_ext simp only rw [show exp (I * δ₁₃) = exp (I * δ₁₃') by rw [mul_comm, h, mul_comm]] rw [show cexp (-(I * ↑δ₁₃)) = cexp (-(I * ↑δ₁₃')) by rw [exp_neg, exp_neg, mul_comm, h, mul_comm]] open Invariant in lemma VusVubVcdSq_eq (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂) (h2 : 0 ≤ Real.cos θ₁₃) (h3 : 0 ≤ Real.sin θ₂₃) (h4 : 0 ≤ Real.cos θ₁₂) : VusVubVcdSq ⟦standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃⟧ = Real.sin θ₁₂ ^ 2 * Real.cos θ₁₃ ^ 2 * Real.sin θ₁₃ ^ 2 * Real.sin θ₂₃ ^ 2 := by simp only [VusVubVcdSq, VusAbs, VAbs, VAbs', Fin.isValue, standParam, standParamAsMatrix, neg_mul, Quotient.lift_mk, cons_val', cons_val_one, head_cons, empty_val', cons_val_fin_one, cons_val_zero, _root_.map_mul, VubAbs, cons_val_two, tail_cons, VcbAbs, VudAbs, Complex.abs_ofReal] by_cases hx : Real.cos θ₁₃ ≠ 0 · rw [Complex.abs_exp] simp only [neg_re, mul_re, I_re, ofReal_re, zero_mul, I_im, ofReal_im, mul_zero, sub_self, neg_zero, Real.exp_zero, mul_one, _root_.sq_abs] rw [_root_.abs_of_nonneg h1, _root_.abs_of_nonneg h3, _root_.abs_of_nonneg h2, _root_.abs_of_nonneg h4] simp [sq] ring_nf nth_rewrite 2 [Real.sin_sq θ₁₂] ring_nf field_simp ring · simp at hx rw [hx] simp open Invariant in lemma mulExpδ₁₃_eq (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂) (h2 : 0 ≤ Real.cos θ₁₃) (h3 : 0 ≤ Real.sin θ₂₃) (h4 : 0 ≤ Real.cos θ₁₂) : mulExpδ₁₃ ⟦standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃⟧ = sin θ₁₂ * cos θ₁₃ ^ 2 * sin θ₂₃ * sin θ₁₃ * cos θ₁₂ * cos θ₂₃ * cexp (I * δ₁₃) := by rw [mulExpδ₁₃, VusVubVcdSq_eq _ _ _ _ h1 h2 h3 h4 ] simp only [jarlskogℂ, standParam, standParamAsMatrix, neg_mul, Quotient.lift_mk, jarlskogℂCKM, Fin.isValue, cons_val', cons_val_one, head_cons, empty_val', cons_val_fin_one, cons_val_zero, cons_val_two, tail_cons, _root_.map_mul, ← exp_conj, map_neg, conj_I, conj_ofReal, neg_neg, map_sub] simp only [ofReal_sin, ofReal_cos, ofReal_mul, ofReal_pow] ring_nf rw [exp_neg] have h1 : cexp (I * δ₁₃) ≠ 0 := exp_ne_zero _ field_simp end standParam end