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feat: Every CKM can be written with standard parameters

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jstoobysmith 2024-04-26 10:32:03 -04:00
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/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license.
Authors: Joseph Tooby-Smith
-/
import HepLean.FlavorPhysics.CKMMatrix.Basic
import HepLean.FlavorPhysics.CKMMatrix.Rows
import HepLean.FlavorPhysics.CKMMatrix.Relations
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import HepLean.FlavorPhysics.CKMMatrix.PhaseFreedom
open Matrix Complex
noncomputable section
namespace CKMMatrix
open ComplexConjugate
-- A
def Rubud (V : CKMMatrix) : ℂ := [V]ub / [V]ud
scoped[CKMMatrix] notation (name := ub_ud_ratio) "[" V "]ub|ud" => Rubud V
-- B
def Rusud (V : CKMMatrix) : ℂ := [V]us / [V]ud
scoped[CKMMatrix] notation (name := us_ud_ratio) "[" V "]us|ud" => Rusud V
def Rudus (V : CKMMatrix) : ℂ := [V]ud / [V]us
scoped[CKMMatrix] notation (name := ud_us_ratio) "[" V "]ud|us" => Rudus V
def Rubus (V : CKMMatrix) : ℂ := [V]ub / [V]us
scoped[CKMMatrix] notation (name := ub_us_ratio) "[" V "]ub|us" => Rubus V
-- D
def Rcdcb (V : CKMMatrix) : ℂ := [V]cd / [V]cb
scoped[CKMMatrix] notation (name := cd_cb_ratio) "[" V "]cd|cb" => Rcdcb V
lemma Rcdcb_mul_cb {V : CKMMatrix} (h : [V]cb ≠ 0): [V]cd = Rcdcb V * [V]cb := by
rw [Rcdcb]
exact (div_mul_cancel₀ (V.1 1 0) h).symm
-- C'
def Rcscb (V : CKMMatrix) : ℂ := [V]cs / [V]cb
scoped[CKMMatrix] notation (name := cs_cb_ratio) "[" V "]cs|cb" => Rcscb V
lemma Rcscb_mul_cb {V : CKMMatrix} (h : [V]cb ≠ 0): [V]cs = Rcscb V * [V]cb := by
rw [Rcscb]
exact (div_mul_cancel₀ [V]cs h).symm
def Rtb!cbud (V : CKMMatrix) : ℂ := conj [V]tb / ([V]cb * [V]ud)
scoped[CKMMatrix] notation (name := tb_cb_ud_ratio) "[" V "]tb*|cb|ud" => Rtb!cbud V
def Rtb!cbus (V : CKMMatrix) : ℂ := conj [V]tb / ([V]cb * [V]us)
scoped[CKMMatrix] notation (name := tb_cb_us_ratio) "[" V "]tb*|cb|us" => Rtb!cbus V
lemma orthog_fst_snd_row_ratios {V : CKMMatrix} (hb : [V]ud ≠ 0) (ha : [V]cb ≠ 0) :
[V]cd|cb + [V]cs|cb * conj ([V]us|ud) + conj ([V]ub|ud) = 0 := by
have h1 : ([V]cd * conj ([V]ud) + [V]cs * conj ([V]us)
+ [V]cb * conj ([V]ub)) / ([V]cb * conj ([V]ud)) = 0 := by
rw [fst_row_snd_row V ]
simp only [Fin.isValue, zero_div]
rw [← div_add_div_same, ← div_add_div_same] at h1
rw [mul_div_mul_comm, mul_div_mul_comm, mul_div_mul_comm] at h1
rw [div_self, div_self] at h1
change Rcdcb V * 1 + _ + _ = _ at h1
have h2 : (starRingEnd ℂ) (V.1 0 2) / (starRingEnd ℂ) (V.1 0 0) = conj (Rubud V) := by
simp [Rubud]
have h3 : ((starRingEnd ℂ) (V.1 0 1) / (starRingEnd ℂ) (V.1 0 0)) = conj (Rusud V) := by
simp [Rusud]
rw [h2, h3] at h1
simp at h1
exact h1
exact ha
simpa using hb
lemma orthog_fst_snd_row_ratios_cb_us {V : CKMMatrix} (hb : [V]us ≠ 0) (ha : [V]cb ≠ 0) :
[V]cd|cb * conj ([V]ud|us) + [V]cs|cb + conj ([V]ub|us) = 0 := by
have h1 : ([V]cd * conj ([V]ud) + [V]cs * conj ([V]us)
+ [V]cb * conj ([V]ub)) / ([V]cb * conj ([V]us)) = 0 := by
rw [fst_row_snd_row V ]
simp only [Fin.isValue, zero_div]
rw [← div_add_div_same, ← div_add_div_same] at h1
rw [mul_div_mul_comm, mul_div_mul_comm, mul_div_mul_comm] at h1
rw [div_self, div_self] at h1
change _ + _ * 1 + _ = _ at h1
have h2 : (starRingEnd ℂ) (V.1 0 2) / (starRingEnd ℂ) (V.1 0 1) = conj (Rubus V) := by
simp [Rubus]
have h3 : ((starRingEnd ℂ) (V.1 0 0) / (starRingEnd ℂ) (V.1 0 1)) = conj (Rudus V) := by
simp [Rudus]
rw [h2, h3] at h1
simp at h1
exact h1
exact ha
simpa using hb
def R' (V : CKMMatrix) : ℂ := [V]cs|cb * (1 + normSq [V]us|ud) + conj ([V]ub|ud) * [V]us|ud
def R's (V : CKMMatrix) : ℂ := [V]cd|cb * (normSq [V]ud|us + 1) + conj ([V]ub|us) * [V]ud|us
lemma R'_eq (V : CKMMatrix) :
R' V = [V]cs|cb * (1 + normSq ([V]us|ud)) + conj ([V]ub|ud) * [V]us|ud := by rfl
lemma R's_eq (V : CKMMatrix) :
R's V = [V]cd|cb * (normSq ([V]ud|us) + 1) + conj ([V]ub|us) * [V]ud|us := by rfl
lemma one_plus_normSq_Rusud_neq_zero_ℝ (V : CKMMatrix):
1 + normSq ([V]us|ud) ≠ 0 := by
have h1 : 0 ≤ (normSq ([V]us|ud)) := normSq_nonneg ([V]us|ud)
have h2 : 0 < 1 + normSq ([V]us|ud) := by linarith
by_contra hn
have h3 := lt_of_lt_of_eq h2 hn
simp at h3
lemma normSq_Rudus_plus_one_neq_zero_ℝ (V : CKMMatrix):
normSq ([V]ud|us) + 1 ≠ 0 := by
have h1 : 0 ≤ (normSq ([V]ud|us)) := normSq_nonneg ([V]ud|us)
have h2 : 0 < normSq ([V]ud|us) + 1 := by linarith
by_contra hn
have h3 := lt_of_lt_of_eq h2 hn
simp at h3
lemma one_plus_normSq_Rusud_neq_zero_ℂ (V : CKMMatrix):
1 + (normSq ([V]us|ud) : ℂ) ≠ 0 := by
by_contra hn
have h1 := one_plus_normSq_Rusud_neq_zero_ℝ V
simp at h1
rw [← ofReal_inj] at h1
simp_all only [ofReal_add, ofReal_one, ofReal_zero, not_true_eq_false]
lemma normSq_Rudus_plus_one_neq_zero_ℂ (V : CKMMatrix):
(normSq ([V]ud|us) : ℂ) + 1 ≠ 0 := by
by_contra hn
have h1 := normSq_Rudus_plus_one_neq_zero_ℝ V
simp at h1
rw [← ofReal_inj] at h1
simp_all only [ofReal_add, ofReal_one, ofReal_zero, not_true_eq_false]
lemma Rcscb_of_R' (V : CKMMatrix) :
[V]cs|cb = (R' V - conj [V]ub|ud * [V]us|ud) / (1 + normSq [V]us|ud) := by
have h2 : R' V - conj ([V]ub|ud) * [V]us|ud = [V]cs|cb * (1 + normSq [V]us|ud) := by
linear_combination R'_eq V
rw [h2]
rw [mul_div_cancel_right₀]
exact one_plus_normSq_Rusud_neq_zero_ℂ V
lemma Rcdcb_of_R's (V : CKMMatrix) :
[V]cd|cb = (R's V - conj [V]ub|us * [V]ud|us) / (normSq [V]ud|us + 1) := by
have h2 : R's V - conj ([V]ub|us) * [V]ud|us = [V]cd|cb * (normSq [V]ud|us + 1) := by
linear_combination R's_eq V
rw [h2]
rw [mul_div_cancel_right₀]
exact normSq_Rudus_plus_one_neq_zero_ℂ V
lemma Rcdcb_R'_orthog {V : CKMMatrix} (hb : [V]ud ≠ 0) (ha : [V]cb ≠ 0) :
[V]cd|cb + (conj [V]ub|ud + R' V * conj [V]us|ud ) / (1 + normSq [V]us|ud) = 0 := by
have h1 := orthog_fst_snd_row_ratios hb ha
rw [Rcscb_of_R'] at h1
rw [div_mul_eq_mul_div] at h1
rw [add_assoc] at h1
rw [div_add'] at h1
have h2 : (R' V - conj [V]ub|ud * [V]us|ud) * conj [V]us|ud +
conj [V]ub|ud * (1 + normSq [V]us|ud) = conj [V]ub|ud + R' V * conj [V]us|ud := by
rw [← mul_conj]
ring
rw [h2] at h1
exact h1
exact one_plus_normSq_Rusud_neq_zero_ℂ V
lemma Rcdcb_of_R' {V : CKMMatrix} (hb : [V]ud ≠ 0) (ha : [V]cb ≠ 0) :
[V]cd|cb = - (conj [V]ub|ud + R' V * conj [V]us|ud ) / (1 + normSq [V]us|ud) := by
linear_combination (Rcdcb_R'_orthog hb ha)
lemma Rcscb_R's_orthog {V : CKMMatrix} (hb : [V]us ≠ 0) (ha : [V]cb ≠ 0) :
[V]cs|cb + (conj [V]ub|us + R's V * conj [V]ud|us ) / (normSq [V]ud|us + 1) = 0 := by
have h1 := orthog_fst_snd_row_ratios_cb_us hb ha
rw [Rcdcb_of_R's] at h1
rw [div_mul_eq_mul_div] at h1
rw [add_assoc, add_comm [V]cs|cb, ← add_assoc] at h1
rw [div_add', add_comm] at h1
have h2 : (R's V - conj [V]ub|us * [V]ud|us) * conj [V]ud|us +
(starRingEnd ℂ) [V]ub|us * ((normSq [V]ud|us) + 1)= conj [V]ub|us + R's V * conj [V]ud|us := by
rw [← mul_conj]
ring
rw [h2] at h1
exact h1
exact normSq_Rudus_plus_one_neq_zero_ℂ V
lemma Rcscb_of_R's {V : CKMMatrix} (hb : [V]us ≠ 0) (ha : [V]cb ≠ 0) :
[V]cs|cb = - (conj [V]ub|us + R's V * conj [V]ud|us ) / (normSq [V]ud|us + 1) := by
linear_combination (Rcscb_R's_orthog hb ha)
lemma R'_of_Rcscb_Rcdcb {V : CKMMatrix} (hb : [V]ud ≠ 0) (ha : [V]cb ≠ 0) :
R' V = [V]cs|cb - [V]us|ud * [V]cd|cb := by
rw [Rcdcb_of_R' hb ha, Rcscb_of_R']
have h1 := one_plus_normSq_Rusud_neq_zero_ℂ V
have h2 : conj [V]ud ≠ 0 := (AddEquivClass.map_ne_zero_iff starRingAut).mpr hb
field_simp
rw [Rubud, map_div₀, Rusud, map_div₀, map_div₀]
simp
rw [normSq_eq_conj_mul_self, normSq_eq_conj_mul_self]
field_simp
ring
lemma R's_of_Rcscb_Rcdcb {V : CKMMatrix} (hb : [V]us ≠ 0) (ha : [V]cb ≠ 0) :
R's V = [V]cd|cb - [V]ud|us * [V]cs|cb := by
rw [Rcscb_of_R's hb ha, Rcdcb_of_R's]
have h1 := normSq_Rudus_plus_one_neq_zero_ℂ V
have h2 : conj [V]us ≠ 0 := (AddEquivClass.map_ne_zero_iff starRingAut).mpr hb
field_simp
rw [Rubus, map_div₀, Rudus, map_div₀, map_div₀]
simp
rw [normSq_eq_conj_mul_self, normSq_eq_conj_mul_self]
field_simp
ring
lemma R'_of_Rtb!cbud {V : CKMMatrix} (hb : [V]ud ≠ 0) (ha : [V]cb ≠ 0)
{τ : ℝ} (hτ : [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c))) :
R' V = cexp (τ * I) * [V]tb*|cb|ud := by
have h1 : cexp (- τ * I) = conj (cexp (τ * I)) := by
rw [← exp_conj]
simp only [neg_mul, _root_.map_mul, conj_I, mul_neg]
rw [conj_ofReal]
have h2 : [V]tb*|cb|ud = cexp (- τ * I) * R' V := by
rw [h1, R'_of_Rcscb_Rcdcb hb ha]
have h1 := congrFun hτ 2
simp [crossProduct, tRow, uRow, cRow] at h1
apply congrArg conj at h1
simp at h1
rw [Rtb!cbud, Rcscb, Rusud, Rcdcb, h1]
field_simp
ring
rw [h2, ← mul_assoc, ← exp_add]
simp
lemma R's_of_Rtb!cbus {V : CKMMatrix} (hb : [V]us ≠ 0) (ha : [V]cb ≠ 0)
{τ : ℝ} (hτ : [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c))) :
R's V = - cexp (τ * I) * [V]tb*|cb|us := by
have h1 : cexp (- τ * I) = conj (cexp (τ * I)) := by
rw [← exp_conj]
simp only [neg_mul, _root_.map_mul, conj_I, mul_neg]
rw [conj_ofReal]
have h2 : [V]tb*|cb|us = - cexp (- τ * I) * R's V := by
rw [h1, R's_of_Rcscb_Rcdcb hb ha]
have h1 := congrFun hτ 2
simp [crossProduct, tRow, uRow, cRow] at h1
apply congrArg conj at h1
simp at h1
rw [Rtb!cbus, Rcscb, Rudus, Rcdcb, h1]
field_simp
ring
rw [h2]
simp
rw [← mul_assoc, ← exp_add]
simp
lemma over_normSq_Rusud (V : CKMMatrix) (hb : [V]ud ≠ 0) (a : ℂ) : a / (1 + normSq [V]us|ud) =
(a * normSq [V]ud) / (normSq [V]ud + normSq [V]us) := by
rw [Rusud]
field_simp
rw [one_add_div]
field_simp
simp
exact hb
lemma over_normSq_Rudus (V : CKMMatrix) (hb : [V]us ≠ 0) (a : ℂ) : a / (normSq [V]ud|us + 1) =
(a * normSq [V]us) / (normSq [V]ud + normSq [V]us) := by
rw [Rudus]
field_simp
rw [div_add_one]
field_simp
simp
exact hb
lemma cd_of_ud_neq_zero {V : CKMMatrix} (hb : [V]ud ≠ 0) (ha : [V]cb ≠ 0)
{τ : ℝ} (hτ : [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c))) :
[V]cd = - (conj [V]ub * [V]ud * [V]cb + cexp (τ * I) * conj [V]tb * conj [V]us) /
(normSq [V]ud + normSq [V]us) := by
obtain hτ2 := R'_of_Rtb!cbud hb ha hτ
rw [Rcdcb_mul_cb ha]
rw [Rcdcb_of_R' hb ha, hτ2]
rw [over_normSq_Rusud V hb, div_mul_eq_mul_div]
congr 1
rw [normSq_eq_conj_mul_self, Rubud, map_div₀, Rusud, map_div₀, Rtb!cbud]
have h1 : conj [V]ud ≠ 0 := (AddEquivClass.map_ne_zero_iff starRingAut).mpr hb
field_simp
ring
lemma cs_of_ud_neq_zero {V : CKMMatrix} (hb : [V]ud ≠ 0) (ha : [V]cb ≠ 0)
{τ : ℝ} (hτ : [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c))) :
[V]cs = (- conj [V]ub * [V]us * [V]cb +
cexp (τ * I) * conj [V]tb * conj [V]ud ) /(normSq [V]ud + normSq [V]us) := by
have hτ2 := R'_of_Rtb!cbud hb ha hτ
rw [Rcscb_mul_cb ha]
rw [Rcscb_of_R', hτ2]
rw [over_normSq_Rusud V hb, div_mul_eq_mul_div]
congr 1
rw [normSq_eq_conj_mul_self, Rusud, Rtb!cbud, Rubud, map_div₀]
have h1 : conj [V]ud ≠ 0 := (AddEquivClass.map_ne_zero_iff starRingAut).mpr hb
field_simp
ring
lemma cd_of_us_neq_zero {V : CKMMatrix} (hb : [V]us ≠ 0) (ha : [V]cb ≠ 0)
{τ : ℝ} (hτ : [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c))) :
[V]cd =
- (conj [V]ub * [V]ud * [V]cb + cexp (τ * I) * conj [V]tb * conj [V]us) /
(normSq [V]ud + normSq [V]us) := by
have hτ2 := R's_of_Rtb!cbus hb ha hτ
rw [Rcdcb_mul_cb ha]
rw [Rcdcb_of_R's, hτ2]
rw [over_normSq_Rudus V hb, div_mul_eq_mul_div]
congr 1
rw [normSq_eq_conj_mul_self, Rubus, map_div₀, Rudus, Rtb!cbus]
have h1 : conj [V]us ≠ 0 := (AddEquivClass.map_ne_zero_iff starRingAut).mpr hb
field_simp
ring
lemma cs_of_us_neq_zero {V : CKMMatrix} (hb : [V]us ≠ 0) (ha : [V]cb ≠ 0)
{τ : ℝ} (hτ : [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c))) :
[V]cs = ( - conj [V]ub * [V]us * [V]cb +
cexp (τ * I) * conj [V]tb * conj [V]ud ) / (normSq [V]ud + normSq [V]us) := by
have hτ2 := R's_of_Rtb!cbus hb ha hτ
rw [Rcscb_mul_cb ha]
rw [Rcscb_of_R's hb ha, hτ2]
rw [over_normSq_Rudus V hb, div_mul_eq_mul_div]
congr 1
rw [normSq_eq_conj_mul_self, Rudus, map_div₀, Rtb!cbus, Rubus, map_div₀]
have h1 : conj [V]us ≠ 0 := (AddEquivClass.map_ne_zero_iff starRingAut).mpr hb
field_simp
ring
lemma cd_of_us_or_ud_neq_zero {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb ≠ 0)
{τ : ℝ} (hτ : [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c))) :
[V]cd = - (conj [V]ub * [V]ud * [V]cb + cexp (τ * I) * conj [V]tb * conj [V]us) /
(normSq [V]ud + normSq [V]us) := by
cases' hb with hb hb
exact cd_of_ud_neq_zero hb ha hτ
exact cd_of_us_neq_zero hb ha hτ
lemma cs_of_us_or_ud_neq_zero {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb ≠ 0)
{τ : ℝ} (hτ : [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c))) :
[V]cs = (- conj [V]ub * [V]us * [V]cb +
cexp (τ * I) * conj [V]tb * conj [V]ud ) / (normSq [V]ud + normSq [V]us) := by
cases' hb with hb hb
exact cs_of_ud_neq_zero hb ha hτ
exact cs_of_us_neq_zero hb ha hτ
lemma cd_of_us_or_ud_neq_zero_UCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb ≠ 0)
(hV : UCond₁ V) : [V]cd = (- VtbAbs ⟦V⟧ * VusAbs ⟦V⟧ / (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2)) +
(- VubAbs ⟦V⟧ * VudAbs ⟦V⟧ * VcbAbs ⟦V⟧ / (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2 )) * cexp (- arg [V]ub * I)
:= by
have hτ : [V]t = cexp ((0 : ℝ) * I) • (conj ([V]u) ×₃ conj ([V]c)) := by
simp
exact hV.2.2.2.2
rw [cd_of_us_or_ud_neq_zero hb ha hτ]
rw [hV.1, hV.2.1, hV.2.2.1, hV.2.2.2.1]
simp [sq, conj_ofReal]
have hx := Vabs_sq_add_neq_zero hb
field_simp
have h1 : conj [V]ub = VubAbs ⟦V⟧ * cexp (- arg [V]ub * I) := by
nth_rewrite 1 [← abs_mul_exp_arg_mul_I [V]ub]
rw [@RingHom.map_mul]
simp [conj_ofReal, ← exp_conj, VAbs]
rw [h1]
ring_nf
lemma cs_of_us_or_ud_neq_zero_UCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb ≠ 0)
(hV : UCond₁ V) : [V]cs = (VtbAbs ⟦V⟧ * VudAbs ⟦V⟧ / (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2))
+ (- VubAbs ⟦V⟧ * VusAbs ⟦V⟧ * VcbAbs ⟦V⟧/ (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2)) * cexp (- arg [V]ub * I)
:= by
have hτ : [V]t = cexp ((0 : ℝ) * I) • (conj ([V]u) ×₃ conj ([V]c)) := by
simp
exact hV.2.2.2.2
rw [cs_of_us_or_ud_neq_zero hb ha hτ]
rw [hV.1, hV.2.1, hV.2.2.1, hV.2.2.2.1]
simp [sq, conj_ofReal]
have hx := Vabs_sq_add_neq_zero hb
field_simp
have h1 : conj [V]ub = VubAbs ⟦V⟧ * cexp (- arg [V]ub * I) := by
nth_rewrite 1 [← abs_mul_exp_arg_mul_I [V]ub]
rw [@RingHom.map_mul]
simp [conj_ofReal, ← exp_conj, VAbs]
rw [h1]
ring_nf
lemma cd_of_cb_zero {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb = 0)
{κ : ℝ} (hκ : [V]u = cexp (κ * I) • (conj [V]c ×₃ conj [V]t)) :
[V]cd = - cexp (κ * I) * conj [V]tb * conj [V]us / (normSq [V]ud + normSq [V]us) := by
have h2 : [V]cd = - cexp (κ * I) * conj [V]us / [V]tb := by
have h1 := congrFun hκ 1
simp [crossProduct, tRow, uRow, cRow] at h1
rw [ha] at h1
apply congrArg conj at h1
simp at h1
rw [h1]
have hx := tb_neq_zero_of_cb_zero_ud_us_neq_zero ha hb
field_simp
have h1 : conj (cexp (↑κ * I)) = cexp (- κ * I) := by
simp [← exp_conj, conj_ofReal]
rw [h1]
rw [← mul_assoc]
rw [← exp_add]
simp
rw [div_td_of_cb_zero_ud_us_neq_zero ha hb] at h2
rw [h2]
congr 1
ring
lemma cs_of_cb_zero {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb = 0)
{κ : ℝ} (hκ1 : [V]u = cexp (κ * I) • (conj [V]c ×₃ conj [V]t)) :
[V]cs = cexp (κ * I) * conj [V]tb * conj [V]ud / (normSq [V]ud + normSq [V]us) := by
have h2 : [V]cs = cexp (κ * I) * conj [V]ud / [V]tb := by
have h1 := congrFun hκ1 0
simp [crossProduct, tRow, uRow, cRow] at h1
rw [ha] at h1
apply congrArg conj at h1
simp at h1
rw [h1]
have hx := tb_neq_zero_of_cb_zero_ud_us_neq_zero ha hb
field_simp
have h1 : conj (cexp (↑κ * I)) = cexp (- κ * I) := by
rw [← exp_conj]
simp only [neg_mul, _root_.map_mul, conj_I, mul_neg]
rw [conj_ofReal]
rw [h1]
rw [← mul_assoc]
rw [← exp_add]
simp
rw [div_td_of_cb_zero_ud_us_neq_zero ha hb] at h2
rw [h2]
congr 1
ring
lemma cd_of_cb_zero_UCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb = 0)
(hV : UCond₁ V) {κ : ℝ} (hκ1 : [V]u = cexp (κ * I) • (conj [V]c ×₃ conj [V]t)) :
[V]cd = (- VtbAbs ⟦V⟧ * VusAbs ⟦V⟧ / (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2)) * cexp (κ * I) := by
rw [cd_of_cb_zero hb ha hκ1]
rw [hV.1, hV.2.1, hV.2.2.2.1]
simp [sq, conj_ofReal]
have hx := Vabs_sq_add_neq_zero hb
field_simp
ring_nf
simp
lemma cs_of_cb_zero_UCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb = 0)
(hV : UCond₁ V) {κ : ℝ} (hκ1 : [V]u = cexp (κ * I) • (conj [V]c ×₃ conj [V]t)) :
[V]cs = (VtbAbs ⟦V⟧ * VudAbs ⟦V⟧ / (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2)) * cexp (κ * I) := by
rw [cs_of_cb_zero hb ha hκ1]
rw [hV.1, hV.2.1, hV.2.2.2.1]
simp [sq, conj_ofReal]
have hx := Vabs_sq_add_neq_zero hb
field_simp
ring_nf
def UCond₂ (U : CKMMatrix) : Prop := [U]ud = VudAbs ⟦U⟧ ∧ [U]us = VusAbs ⟦U⟧
∧ [U]ub = VubAbs ⟦U⟧ ∧ [U]t = conj [U]u ×₃ conj [U]c
∧ [U]cd = (- VtbAbs ⟦U⟧ * VusAbs ⟦U⟧ / (VudAbs ⟦U⟧ ^2 + VusAbs ⟦U⟧ ^2)) ∧
[U]cs = (VtbAbs ⟦U⟧ * VudAbs ⟦U⟧ / (VudAbs ⟦U⟧ ^2 + VusAbs ⟦U⟧ ^2))
lemma UCond₂_solv {V : CKMMatrix} (h1 : a + d = 0) (h2 : a + e = 0) (h3 : a + f = - arg [V]ub)
(h4 : 0 = - a - b - c - d - e - f) (h5 : b + d = - κ) (h6 : b + e = - κ) :
b = - κ + a ∧
c = κ + arg [V]ub + a ∧
d = - a ∧
e = - a ∧
f = - arg [V]ub - a := by
have hd : d = - a := by
linear_combination h1
subst hd
have he : e = - a := by
linear_combination h2
subst he
simp_all
have hb : b = - κ + a := by
linear_combination h6
subst hb
simp_all
have hf : f = - arg [V]ub - a := by
linear_combination h3
subst hf
simp_all
ring_nf at h4
have hc : c = κ + arg [V]ub + a := by
linear_combination h4
simp_all
lemma UCond₂_exists {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) (ha : [V]cb = 0)
(hV : UCond₁ V) : ∃ (U : CKMMatrix), V ≈ U ∧ UCond₂ U:= by
obtain ⟨κ, hκ⟩ := V.cRow_cross_tRow_eq_uRow
let U : CKMMatrix := phaseShiftApply V 0 (- κ) (κ + arg [V]ub) 0 0 (- arg [V]ub)
use U
have hUV : ⟦U⟧ = ⟦V⟧ := by
simp
symm
exact phaseShiftApply.equiv _ _ _ _ _ _ _
apply And.intro
exact phaseShiftApply.equiv _ _ _ _ _ _ _
apply And.intro
· rw [hUV]
apply ud_eq_abs _ _ _ _ _ _ _
rw [hV.1, arg_ofReal_of_nonneg]
simp
exact Complex.abs.nonneg _
apply And.intro
· rw [hUV]
apply us_eq_abs _ _ _ _ _ _ _
rw [hV.2.1, arg_ofReal_of_nonneg]
simp
exact Complex.abs.nonneg _
apply And.intro
· rw [hUV]
apply ub_eq_abs _ _ _ _ _ _ _
ring
apply And.intro
· have hτ : [V]t = cexp ((0 : ℝ) * I) • (conj ([V]u) ×₃ conj ([V]c)) := by
simp
exact hV.2.2.2.2
apply t_eq_conj _ _ _ _ _ _ hτ.symm
ring
apply And.intro
· rw [phaseShiftApply.cd]
rw [cd_of_cb_zero_UCond hb ha hV hκ]
rw [mul_comm, mul_assoc, ← exp_add, hUV]
simp
· rw [phaseShiftApply.cs]
rw [cs_of_cb_zero_UCond hb ha hV hκ]
rw [mul_comm, mul_assoc, ← exp_add, hUV]
simp
section zero
def UCond₃ (U : CKMMatrix) : Prop :=
[U]ud = 0 ∧ [U]us = 0 ∧ [U]cb = 0 ∧
[U]ub = 1 ∧ [U]t = conj [U]u ×₃ conj [U]c
∧ [U]cd = - VcdAbs ⟦U⟧ ∧ [U]cs = √(1 - VcdAbs ⟦U⟧ ^ 2)
lemma UCond₃_solv {V : CKMMatrix} (h1 : a + f = - arg [V]ub) (h2 : 0 = - a - b - c - d - e - f)
(h3 : b + d = Real.pi - arg [V]cd) (h5 : b + e = - arg [V]cs) :
c = - Real.pi + arg [V]cd + arg [V]cs + arg [V]ub + b ∧
d = Real.pi - arg [V]cd - b ∧
e = - arg [V]cs - b ∧
f = - arg [V]ub - a := by
have hf : f = - arg [V]ub - a := by
linear_combination h1
subst hf
have he : e = - arg [V]cs - b := by
linear_combination h5
have hd : d = Real.pi - arg [V]cd - b := by
linear_combination h3
subst he hd
simp_all
ring_nf at h2
have hc : c = - Real.pi + arg [V]cd + arg [V]cs + arg [V]ub + b := by
linear_combination h2
subst hc
ring
lemma cs_of_ud_us_zero {V : CKMMatrix} (ha : ¬ ([V]ud ≠ 0 ∨ [V]us ≠ 0)) :
VcsAbs ⟦V⟧ = √(1 - VcdAbs ⟦V⟧ ^ 2) := by
have h1 := snd_row_normalized_abs V
symm
rw [Real.sqrt_eq_iff_sq_eq]
simp [not_or] at ha
rw [cb_eq_zero_of_ud_us_zero ha] at h1
simp at h1
simp [VAbs]
linear_combination h1
simp
rw [@abs_le]
have h1 := VAbs_leq_one 1 0 ⟦V⟧
have h0 := VAbs_ge_zero 1 0 ⟦V⟧
simp_all
have hn : -1 ≤ (0 : ℝ) := by simp
exact hn.trans h0
exact VAbs_ge_zero _ _ ⟦V⟧
lemma UCond₃_exists {V : CKMMatrix} (hb :¬ ([V]ud ≠ 0 ∨ [V]us ≠ 0)) (hV : UCond₁ V) :
∃ (U : CKMMatrix), V ≈ U ∧ UCond₃ U:= by
let U : CKMMatrix := phaseShiftApply V 0 0 (- Real.pi + arg [V]cd + arg [V]cs + arg [V]ub)
(Real.pi - arg [V]cd ) (- arg [V]cs) (- arg [V]ub )
use U
have hUV : ⟦U⟧ = ⟦V⟧ := by
simp
symm
exact phaseShiftApply.equiv _ _ _ _ _ _ _
apply And.intro
exact phaseShiftApply.equiv _ _ _ _ _ _ _
apply And.intro
· simp [not_or] at hb
have h1 : VudAbs ⟦U⟧ = 0 := by
rw [hUV]
simp [VAbs, hb]
simp [VAbs] at h1
exact h1
apply And.intro
· simp [not_or] at hb
have h1 : VusAbs ⟦U⟧ = 0 := by
rw [hUV]
simp [VAbs, hb]
simp [VAbs] at h1
exact h1
apply And.intro
· simp [not_or] at hb
have h3 := cb_eq_zero_of_ud_us_zero hb
have h1 : VcbAbs ⟦U⟧ = 0 := by
rw [hUV]
simp [VAbs, h3]
simp [VAbs] at h1
exact h1
apply And.intro
· have hU1 : [U]ub = VubAbs ⟦V⟧ := by
apply ub_eq_abs _ _ _ _ _ _ _
ring
rw [hU1]
have h1:= (ud_us_neq_zero_iff_ub_neq_one V).mpr.mt hb
simpa using h1
apply And.intro
· have hτ : [V]t = cexp ((0 : ℝ) * I) • (conj ([V]u) ×₃ conj ([V]c)) := by
simp
exact hV.2.2.2.2
apply t_eq_conj _ _ _ _ _ _ hτ.symm
ring
apply And.intro
· rw [hUV]
apply cd_eq_neg_abs _ _ _ _ _ _ _
ring
have hcs : [U]cs = VcsAbs ⟦U⟧ := by
rw [hUV]
apply cs_eq_abs _ _ _ _ _ _ _
ring
rw [hcs, hUV, cs_of_ud_us_zero hb]
end zero
end CKMMatrix
end