refactor: lint
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jstoobysmith
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8 changed files with 343 additions and 163 deletions
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@ -47,3 +47,10 @@ import HepLean.AnomalyCancellation.SMNu.PlusU1.HyperCharge
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import HepLean.AnomalyCancellation.SMNu.PlusU1.PlaneNonSols
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import HepLean.AnomalyCancellation.SMNu.PlusU1.QuadSol
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import HepLean.AnomalyCancellation.SMNu.PlusU1.QuadSolToSol
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import HepLean.FlavorPhysics.CKMMatrix.Basic
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import HepLean.FlavorPhysics.CKMMatrix.Invariants
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import HepLean.FlavorPhysics.CKMMatrix.PhaseFreedom
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import HepLean.FlavorPhysics.CKMMatrix.Relations
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import HepLean.FlavorPhysics.CKMMatrix.Rows
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import HepLean.FlavorPhysics.CKMMatrix.StandardParameterization.Basic
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import HepLean.FlavorPhysics.CKMMatrix.StandardParameterization.StandardParameters
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@ -6,12 +6,27 @@ Authors: Joseph Tooby-Smith
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import Mathlib.LinearAlgebra.UnitaryGroup
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import Mathlib.Data.Complex.Exponential
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import Mathlib.Tactic.Polyrith
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/-!
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# The CKM Matrix
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The definition of the type of CKM matries as unitary $3×3$-matries.
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An equivalence relation on CKM matrices is defined, where two matrices are equivalent if they are
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related by phase shifts.
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The notation `[V]ud` etc can be used for the elements of a CKM matrix, and
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`[V]ud|us` etc for the ratios of elements.
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-/
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open Matrix Complex
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noncomputable section
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/-- Given three real numbers `a b c` the complex matrix with `exp (I * a)` etc on the
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leading diagonal. -/
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@[simp]
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def phaseShiftMatrix (a b c : ℝ) : Matrix (Fin 3) (Fin 3) ℂ :=
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![![cexp (I * a), 0, 0], ![0, cexp (I * b), 0], ![0, 0, cexp (I * c)]]
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@ -39,6 +54,8 @@ lemma phaseShiftMatrix_mul (a b c d e f : ℝ) :
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change cexp (I * ↑c) * 0 = 0
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simp
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/-- Given three real numbers `a b c` the unitary matrix with `exp (I * a)` etc on the
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leading diagonal. -/
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@[simps!]
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def phaseShift (a b c : ℝ) : unitaryGroup (Fin 3) ℂ :=
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⟨phaseShiftMatrix a b c,
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@ -50,6 +67,7 @@ def phaseShift (a b c : ℝ) : unitaryGroup (Fin 3) ℂ :=
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lemma phaseShift_coe_matrix (a b c : ℝ) : ↑(phaseShift a b c) = phaseShiftMatrix a b c := rfl
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/-- The equivalence relation between CKM matrices. -/
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def phaseShiftRelation (U V : unitaryGroup (Fin 3) ℂ) : Prop :=
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∃ a b c e f g , U = phaseShift a b c * V * phaseShift e f g
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@ -90,11 +108,12 @@ lemma phaseShiftRelation_trans {U V W : unitaryGroup (Fin 3) ℂ} :
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rw [add_comm k e, add_comm l f, add_comm m g]
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def phaseShiftEquivRelation : Equivalence phaseShiftRelation where
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lemma phaseShiftEquivRelation : Equivalence phaseShiftRelation where
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refl := phaseShiftRelation_refl
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symm := phaseShiftRelation_symm
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trans := phaseShiftRelation_trans
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/-- The type of CKM matrices. -/
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def CKMMatrix : Type := unitaryGroup (Fin 3) ℂ
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lemma CKMMatrix_ext {U V : CKMMatrix} (h : U.val = V.val) : U = V := by
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@ -102,18 +121,37 @@ lemma CKMMatrix_ext {U V : CKMMatrix} (h : U.val = V.val) : U = V := by
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simp at h
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subst h
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rfl
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/-- The `ud`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := ud_element) "[" V "]ud" => V.1 0 0
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/-- The `us`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := us_element) "[" V "]us" => V.1 0 1
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/-- The `ub`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := ub_element) "[" V "]ub" => V.1 0 2
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/-- The `cd`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := cd_element) "[" V "]cd" => V.1 1 0
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/-- The `cs`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := cs_element) "[" V "]cs" => V.1 1 1
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/-- The `cb`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := cb_element) "[" V "]cb" => V.1 1 2
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/-- The `td`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := td_element) "[" V "]td" => V.1 2 0
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/-- The `ts`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := ts_element) "[" V "]ts" => V.1 2 1
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/-- The `tb`th element of the CKM matrix. -/
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scoped[CKMMatrix] notation (name := tb_element) "[" V "]tb" => V.1 2 2
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instance CKMMatrixSetoid : Setoid CKMMatrix := ⟨phaseShiftRelation, phaseShiftEquivRelation⟩
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/-- The matrix obtained from `V` by shifting the phases of the fermions. -/
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@[simps!]
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def phaseShiftApply (V : CKMMatrix) (a b c d e f : ℝ) : CKMMatrix :=
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phaseShift a b c * ↑V * phaseShift d e f
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@ -127,87 +165,105 @@ lemma equiv (V : CKMMatrix) (a b c d e f : ℝ) :
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lemma ud (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 0 0 = cexp (a * I + d * I) * V.1 0 0 := by
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simp
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_one, head_cons, zero_mul, add_zero, cons_val_two, tail_cons, head_fin_const, mul_zero,
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tail_val', head_val']
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change _ + _ * _ * 0 = _
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rw [exp_add]
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ring_nf
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lemma us (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 0 1 = cexp (a * I + e * I) * V.1 0 1 := by
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simp
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_one, head_cons, zero_mul, add_zero, cons_val_two, tail_cons, mul_zero, zero_add,
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head_fin_const]
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rw [exp_add]
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ring_nf
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lemma ub (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 0 2 = cexp (a * I + f * I) * V.1 0 2 := by
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simp
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_one, head_cons, zero_mul, add_zero, cons_val_two, tail_cons, mul_zero, head_fin_const,
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zero_add]
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rw [exp_add]
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ring_nf
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lemma cd (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 1 0= cexp (b * I + d * I) * V.1 1 0 := by
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simp
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_one, head_fin_const, zero_mul, head_cons, zero_add, cons_val_two, tail_cons, add_zero,
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mul_zero, tail_val', head_val']
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change _ + _ * _ * 0 = _
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rw [exp_add]
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ring_nf
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lemma cs (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 1 1 = cexp (b * I + e * I) * V.1 1 1 := by
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simp
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_one, head_fin_const, zero_mul, head_cons, zero_add, cons_val_two, tail_cons, add_zero,
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mul_zero]
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rw [exp_add]
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ring_nf
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lemma cb (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 1 2 = cexp (b * I + f * I) * V.1 1 2 := by
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simp
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_one, head_fin_const, zero_mul, head_cons, zero_add, cons_val_two, tail_cons, add_zero,
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mul_zero]
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rw [exp_add]
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ring_nf
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lemma td (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 2 0= cexp (c * I + d * I) * V.1 2 0 := by
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simp
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_two, tail_val', head_val', cons_val_one, head_cons, tail_cons, head_fin_const,
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zero_mul, add_zero, mul_zero]
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change (0 * _ + _ ) * _ + (0 * _ + _ ) * 0 = _
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simp
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simp only [Fin.isValue, zero_mul, zero_add, mul_zero, add_zero]
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rw [exp_add]
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ring_nf
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lemma ts (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 2 1 = cexp (c * I + e * I) * V.1 2 1 := by
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simp
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_two, tail_val', head_val', cons_val_one, head_cons, tail_cons, head_fin_const,
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zero_mul, add_zero, mul_zero, zero_add]
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change (0 * _ + _) * _ = _
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rw [exp_add]
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ring_nf
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lemma tb (V : CKMMatrix) (a b c d e f : ℝ) :
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(phaseShiftApply V a b c d e f).1 2 2 = cexp (c * I + f * I) * V.1 2 2 := by
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simp
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simp only [Fin.isValue, phaseShiftApply_coe]
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rw [mul_apply, Fin.sum_univ_three]
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rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
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simp
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simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
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cons_val_two, tail_val', head_val', cons_val_one, head_cons, tail_cons, head_fin_const,
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zero_mul, add_zero, mul_zero, zero_add]
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change (0 * _ + _) * _ = _
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rw [exp_add]
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ring_nf
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@ -215,13 +271,13 @@ lemma tb (V : CKMMatrix) (a b c d e f : ℝ) :
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end phaseShiftApply
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/-- The aboslute value of the `(i,j)`th element of `V`. -/
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@[simp]
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def VAbs' (V : unitaryGroup (Fin 3) ℂ) (i j : Fin 3) : ℝ := Complex.abs (V i j)
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lemma VAbs'_equiv (i j : Fin 3) (V U : CKMMatrix) (h : V ≈ U) :
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VAbs' V i j = VAbs' U i j := by
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simp
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simp only [VAbs']
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obtain ⟨a, b, c, e, f, g, h⟩ := h
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rw [h]
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simp only [Submonoid.coe_mul, phaseShift_coe_matrix]
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@ -236,33 +292,52 @@ lemma VAbs'_equiv (i j : Fin 3) (V U : CKMMatrix) (h : V ≈ U) :
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all_goals change Complex.abs (0 * _ + _) = _
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all_goals simp [Complex.abs_exp]
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/-- The absolute value of the `(i,j)`th any representative of `⟦V⟧`. -/
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def VAbs (i j : Fin 3) : Quotient CKMMatrixSetoid → ℝ :=
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Quotient.lift (fun V => VAbs' V i j) (VAbs'_equiv i j)
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/-- The absolute value of the `ud`th element of a representative of an equivalence class of
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CKM matrices. -/
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@[simp]
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abbrev VudAbs := VAbs 0 0
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/-- The absolute value of the `us`th element of a representative of an equivalence class of
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CKM matrices. -/
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@[simp]
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abbrev VusAbs := VAbs 0 1
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/-- The absolute value of the `ub`th element of a representative of an equivalence class of
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CKM matrices. -/
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@[simp]
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abbrev VubAbs := VAbs 0 2
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/-- The absolute value of the `cd`th element of a representative of an equivalence class of
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CKM matrices. -/
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@[simp]
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abbrev VcdAbs := VAbs 1 0
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/-- The absolute value of the `cs`th element of a representative of an equivalence class of
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CKM matrices. -/
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@[simp]
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abbrev VcsAbs := VAbs 1 1
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/-- The absolute value of the `cb`th element of a representative of an equivalence class of
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CKM matrices. -/
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@[simp]
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abbrev VcbAbs := VAbs 1 2
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/-- The absolute value of the `td`th element of a representative of an equivalence class of
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CKM matrices. -/
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@[simp]
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abbrev VtdAbs := VAbs 2 0
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/-- The absolute value of the `ts`th element of a representative of an equivalence class of
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CKM matrices. -/
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@[simp]
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abbrev VtsAbs := VAbs 2 1
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/-- The absolute value of the `tb`th element of a representative of an equivalence class of
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CKM matrices. -/
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@[simp]
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abbrev VtbAbs := VAbs 2 2
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@ -273,48 +348,50 @@ open ComplexConjugate
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section ratios
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/-- The ratio of the `ub` and `ud` elements of a CKM matrix. -/
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def Rubud (V : CKMMatrix) : ℂ := [V]ub / [V]ud
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/-- The ratio of the `ub` and `ud` elements of a CKM matrix. -/
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scoped[CKMMatrix] notation (name := ub_ud_ratio) "[" V "]ub|ud" => Rubud V
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/-- The ratio of the `us` and `ud` elements of a CKM matrix. -/
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def Rusud (V : CKMMatrix) : ℂ := [V]us / [V]ud
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/-- The ratio of the `us` and `ud` elements of a CKM matrix. -/
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scoped[CKMMatrix] notation (name := us_ud_ratio) "[" V "]us|ud" => Rusud V
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/-- The ratio of the `ud` and `us` elements of a CKM matrix. -/
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def Rudus (V : CKMMatrix) : ℂ := [V]ud / [V]us
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/-- The ratio of the `ud` and `us` elements of a CKM matrix. -/
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scoped[CKMMatrix] notation (name := ud_us_ratio) "[" V "]ud|us" => Rudus V
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/-- The ratio of the `ub` and `us` elements of a CKM matrix. -/
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def Rubus (V : CKMMatrix) : ℂ := [V]ub / [V]us
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/-- The ratio of the `ub` and `us` elements of a CKM matrix. -/
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scoped[CKMMatrix] notation (name := ub_us_ratio) "[" V "]ub|us" => Rubus V
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/-- The ratio of the `cd` and `cb` elements of a CKM matrix. -/
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def Rcdcb (V : CKMMatrix) : ℂ := [V]cd / [V]cb
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/-- The ratio of the `cd` and `cb` elements of a CKM matrix. -/
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scoped[CKMMatrix] notation (name := cd_cb_ratio) "[" V "]cd|cb" => Rcdcb V
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lemma Rcdcb_mul_cb {V : CKMMatrix} (h : [V]cb ≠ 0): [V]cd = Rcdcb V * [V]cb := by
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rw [Rcdcb]
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exact (div_mul_cancel₀ (V.1 1 0) h).symm
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/-- The ratio of the `cs` and `cb` elements of a CKM matrix. -/
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def Rcscb (V : CKMMatrix) : ℂ := [V]cs / [V]cb
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/-- The ratio of the `cs` and `cb` elements of a CKM matrix. -/
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scoped[CKMMatrix] notation (name := cs_cb_ratio) "[" V "]cs|cb" => Rcscb V
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lemma Rcscb_mul_cb {V : CKMMatrix} (h : [V]cb ≠ 0): [V]cs = Rcscb V * [V]cb := by
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rw [Rcscb]
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exact (div_mul_cancel₀ [V]cs h).symm
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def Rtb!cbud (V : CKMMatrix) : ℂ := conj [V]tb / ([V]cb * [V]ud)
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||||
|
||||
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
|
||||
|
||||
|
||||
end ratios
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -7,7 +7,18 @@ import HepLean.FlavorPhysics.CKMMatrix.Basic
|
|||
import HepLean.FlavorPhysics.CKMMatrix.Rows
|
||||
import HepLean.FlavorPhysics.CKMMatrix.PhaseFreedom
|
||||
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
|
||||
/-!
|
||||
# Invariants of the CKM Matrix
|
||||
|
||||
The CKM matrix is only defined up to an equivalence.
|
||||
|
||||
This file defines some invariants of the CKM matrix, which are well-defined with respect to
|
||||
this equivalence.
|
||||
|
||||
Of note, this file defines the complex jarlskog invariant.
|
||||
|
||||
|
||||
-/
|
||||
open Matrix Complex
|
||||
open ComplexConjugate
|
||||
open CKMMatrix
|
||||
|
|
@ -17,6 +28,7 @@ noncomputable section
|
|||
namespace Invariant
|
||||
|
||||
|
||||
/-- The complex jarlskog invariant for a CKM matrix. -/
|
||||
@[simps!]
|
||||
def jarlskogℂCKM (V : CKMMatrix) : ℂ :=
|
||||
[V]us * [V]cb * conj [V]ub * conj [V]cs
|
||||
|
|
@ -36,13 +48,19 @@ lemma jarlskogℂCKM_equiv (V U : CKMMatrix) (h : V ≈ U) :
|
|||
field_simp
|
||||
ring
|
||||
|
||||
/-- The complex jarlskog invariant for an equivalence class of CKM matrices. -/
|
||||
@[simp]
|
||||
def jarlskogℂ : Quotient CKMMatrixSetoid → ℂ :=
|
||||
Quotient.lift jarlskogℂCKM jarlskogℂCKM_equiv
|
||||
|
||||
/-- An invariant for CKM mtrices corresponding to the square of the absolute values
|
||||
of the `us`, `ub` and `cb` elements multipled together divied by `(VudAbs V ^ 2 + VusAbs V ^2)` .
|
||||
-/
|
||||
def VusVubVcdSq (V : Quotient CKMMatrixSetoid) : ℝ :=
|
||||
VusAbs V ^ 2 * VubAbs V ^ 2 * VcbAbs V ^2 / (VudAbs V ^ 2 + VusAbs V ^2)
|
||||
|
||||
/-- An invariant for CKM matrices. The argument of this invariant is `δ₁₃` in the
|
||||
standard parameterization. -/
|
||||
def mulExpδ₁₃ (V : Quotient CKMMatrixSetoid) : ℂ :=
|
||||
jarlskogℂ V + VusVubVcdSq V
|
||||
|
||||
|
|
|
|||
|
|
@ -8,7 +8,19 @@ import HepLean.FlavorPhysics.CKMMatrix.Rows
|
|||
import HepLean.FlavorPhysics.CKMMatrix.Relations
|
||||
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
|
||||
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
|
||||
/-!
|
||||
# Phase freedom of the CKM Matrix
|
||||
|
||||
The CKM matrix is only defined up to an equivalence. This leads to a freedom
|
||||
to shift the phases of the matrices elements of the CKM matrix.
|
||||
|
||||
In this file we define two sets of conditions on the CKM matrices
|
||||
`fstRowThdColRealCond` which we show can be satisfied by any CKM matrix up to equivalence
|
||||
and `ubOnePhaseCond` which we show can be satisfied by any CKM matrix up to equivalence as long as
|
||||
the ub element as absolute value 1.
|
||||
|
||||
|
||||
-/
|
||||
open Matrix Complex
|
||||
|
||||
|
||||
|
|
@ -104,7 +116,7 @@ lemma shift_cd_phase_pi {V : CKMMatrix} (h1 : c + d = Real.pi - arg [V]cd) :
|
|||
rfl
|
||||
|
||||
lemma shift_cross_product_phase_zero {V : CKMMatrix} {τ : ℝ}
|
||||
(hτ : cexp (τ * I) • (conj [V]u ×₃ conj [V]c) = [V]t) (h1 : τ = - u - c - t - d - s - b) :
|
||||
(hτ : cexp (τ * I) • (conj [V]u ×₃ conj [V]c) = [V]t) (h1 : τ = - u - c - t - d - s - b) :
|
||||
[phaseShiftApply V u c t d s b]t =
|
||||
conj [phaseShiftApply V u c t d s b]u ×₃ conj [phaseShiftApply V u c t d s b]c := by
|
||||
change _ = phaseShiftApply.ucCross _ _ _ _ _ _ _
|
||||
|
|
@ -118,7 +130,7 @@ lemma shift_cross_product_phase_zero {V : CKMMatrix} {τ : ℝ}
|
|||
rw [← hτ0]
|
||||
rw [← mul_assoc, ← exp_add, h1]
|
||||
congr 2
|
||||
simp
|
||||
simp only [ofReal_sub, ofReal_neg]
|
||||
ring
|
||||
· simp
|
||||
rw [phaseShiftApply.ucCross_snd]
|
||||
|
|
@ -128,7 +140,7 @@ lemma shift_cross_product_phase_zero {V : CKMMatrix} {τ : ℝ}
|
|||
rw [← hτ0]
|
||||
rw [← mul_assoc, ← exp_add, h1]
|
||||
congr 2
|
||||
simp
|
||||
simp only [ofReal_sub, ofReal_neg]
|
||||
ring
|
||||
· simp
|
||||
rw [phaseShiftApply.ucCross_thd]
|
||||
|
|
@ -138,23 +150,28 @@ lemma shift_cross_product_phase_zero {V : CKMMatrix} {τ : ℝ}
|
|||
rw [← hτ0]
|
||||
rw [← mul_assoc, ← exp_add, h1]
|
||||
congr 2
|
||||
simp
|
||||
simp only [ofReal_sub, ofReal_neg]
|
||||
ring
|
||||
|
||||
end phaseShiftApply
|
||||
|
||||
variable (a b c d e f : ℝ)
|
||||
|
||||
-- rename
|
||||
/-- A proposition which is satisfied by a CKM matrix if its `ud`, `us`, `cb` and `tb` elements
|
||||
are positive and real, and there is no phase difference between the `t`th-row and
|
||||
the cross product of the conjugates of the `u`th and `c`th rows. -/
|
||||
def fstRowThdColRealCond (U : CKMMatrix) : Prop := [U]ud = VudAbs ⟦U⟧ ∧ [U]us = VusAbs ⟦U⟧
|
||||
∧ [U]cb = VcbAbs ⟦U⟧ ∧ [U]tb = VtbAbs ⟦U⟧ ∧ [U]t = conj [U]u ×₃ conj [U]c
|
||||
|
||||
-- rename
|
||||
/-- A proposition which is satisfied by a CKM matrix `ub` is one, `ud`, `us` and `cb` are zero,
|
||||
there is no phase difference between the `t`th-row and
|
||||
the cross product of the conjugates of the `u`th and `c`th rows, and the `cd`th and `cs`th
|
||||
elements are real and related in a set way.
|
||||
-/
|
||||
def ubOnePhaseCond (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)
|
||||
|
||||
-- bad name for this lemma
|
||||
lemma fstRowThdColRealCond_shift_solution {V : CKMMatrix} (h1 : a + d = - arg [V]ud)
|
||||
(h2 : a + e = - arg [V]us) (h3 : b + f = - arg [V]cb)
|
||||
(h4 : c + f = - arg [V]tb) (h5 : τ = - a - b - c - d - e - f) :
|
||||
|
|
@ -219,7 +236,7 @@ lemma fstRowThdColRealCond_holds_up_to_equiv (V : CKMMatrix) :
|
|||
(- arg [V]us)
|
||||
(τ - arg [V]ud - arg [V]us - arg [V]cb - arg [V]tb)
|
||||
have hUV : ⟦U⟧ = ⟦V⟧ := by
|
||||
simp
|
||||
simp only [Quotient.eq]
|
||||
symm
|
||||
exact phaseShiftApply.equiv _ _ _ _ _ _ _
|
||||
use U
|
||||
|
|
@ -252,7 +269,7 @@ lemma ubOnePhaseCond_hold_up_to_equiv_of_ub_one {V : CKMMatrix} (hb : ¬ ([V]ud
|
|||
(Real.pi - arg [V]cd ) (- arg [V]cs) (- arg [V]ub )
|
||||
use U
|
||||
have hUV : ⟦U⟧ = ⟦V⟧ := by
|
||||
simp
|
||||
simp only [Quotient.eq]
|
||||
symm
|
||||
exact phaseShiftApply.equiv _ _ _ _ _ _ _
|
||||
apply And.intro
|
||||
|
|
@ -288,7 +305,7 @@ lemma ubOnePhaseCond_hold_up_to_equiv_of_ub_one {V : CKMMatrix} (hb : ¬ ([V]ud
|
|||
simpa using h1
|
||||
apply And.intro
|
||||
· have hτ : [V]t = cexp ((0 : ℝ) * I) • (conj ([V]u) ×₃ conj ([V]c)) := by
|
||||
simp
|
||||
simp only [ofReal_zero, zero_mul, exp_zero, one_smul]
|
||||
exact hV.2.2.2.2
|
||||
apply shift_cross_product_phase_zero _ _ _ _ _ _ hτ.symm
|
||||
ring
|
||||
|
|
@ -304,11 +321,12 @@ lemma ubOnePhaseCond_hold_up_to_equiv_of_ub_one {V : CKMMatrix} (hb : ¬ ([V]ud
|
|||
|
||||
|
||||
lemma cd_of_fstRowThdColRealCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
|
||||
(hV : fstRowThdColRealCond 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
|
||||
(hV : fstRowThdColRealCond 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
|
||||
simp only [ofReal_zero, zero_mul, exp_zero, one_smul]
|
||||
exact hV.2.2.2.2
|
||||
rw [cd_of_ud_us_ub_cb_tb hb hτ]
|
||||
rw [hV.1, hV.2.1, hV.2.2.1, hV.2.2.2.1]
|
||||
|
|
@ -323,11 +341,12 @@ lemma cd_of_fstRowThdColRealCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠
|
|||
ring_nf
|
||||
|
||||
lemma cs_of_fstRowThdColRealCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
|
||||
(hV : fstRowThdColRealCond 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
|
||||
(hV : fstRowThdColRealCond 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
|
||||
simp only [ofReal_zero, zero_mul, exp_zero, one_smul]
|
||||
exact hV.2.2.2.2
|
||||
rw [cs_of_ud_us_ub_cb_tb hb hτ]
|
||||
rw [hV.1, hV.2.1, hV.2.2.1, hV.2.2.2.1]
|
||||
|
|
|
|||
|
|
@ -6,7 +6,13 @@ Authors: Joseph Tooby-Smith
|
|||
import HepLean.FlavorPhysics.CKMMatrix.Basic
|
||||
import HepLean.FlavorPhysics.CKMMatrix.Rows
|
||||
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
|
||||
/-!
|
||||
# Relations for the CKM Matrix
|
||||
|
||||
This file contains a collection of relations and properties between the elements of the CKM
|
||||
matrix.
|
||||
|
||||
-/
|
||||
|
||||
open Matrix Complex
|
||||
|
||||
|
|
@ -55,7 +61,6 @@ lemma thd_row_normalized_normSq (V : CKMMatrix) :
|
|||
repeat rw [← Complex.sq_abs]
|
||||
exact V.thd_row_normalized_abs
|
||||
|
||||
-- rename
|
||||
lemma normSq_Vud_plus_normSq_Vus (V : CKMMatrix) :
|
||||
normSq [V]ud + normSq [V]us = 1 - normSq [V]ub := by
|
||||
linear_combination (fst_row_normalized_normSq V)
|
||||
|
|
@ -99,6 +104,26 @@ lemma normSq_Vud_plus_normSq_Vus_neq_zero_ℝ {V : CKMMatrix} (hb : [V]ud ≠ 0
|
|||
have h2 : ¬ 0 ≤ ( -1 : ℝ) := by simp
|
||||
exact h2 h3
|
||||
|
||||
lemma VAbsub_neq_zero_Vud_Vus_neq_zero {V : Quotient CKMMatrixSetoid}
|
||||
(hV : VAbs 0 2 V ≠ 1) :(VudAbs V ^ 2 + VusAbs V ^ 2) ≠ 0 := by
|
||||
obtain ⟨V⟩ := V
|
||||
change VubAbs ⟦V⟧ ≠ 1 at hV
|
||||
simp only [VubAbs, VAbs, VAbs', Fin.isValue, Quotient.lift_mk] at hV
|
||||
rw [← ud_us_neq_zero_iff_ub_neq_one V] at hV
|
||||
simpa [← Complex.sq_abs] using (normSq_Vud_plus_normSq_Vus_neq_zero_ℝ hV)
|
||||
|
||||
lemma VAbsub_neq_zero_sqrt_Vud_Vus_neq_zero {V : Quotient CKMMatrixSetoid}
|
||||
(hV : VAbs 0 2 V ≠ 1) : √(VudAbs V ^ 2 + VusAbs V ^ 2) ≠ 0 := by
|
||||
obtain ⟨V⟩ := V
|
||||
rw [Real.sqrt_ne_zero (Left.add_nonneg (sq_nonneg _) (sq_nonneg _))]
|
||||
change VubAbs ⟦V⟧ ≠ 1 at hV
|
||||
simp only [VubAbs, VAbs, VAbs', Fin.isValue, Quotient.lift_mk] at hV
|
||||
rw [← ud_us_neq_zero_iff_ub_neq_one V] at hV
|
||||
simpa [← Complex.sq_abs] using (normSq_Vud_plus_normSq_Vus_neq_zero_ℝ hV)
|
||||
|
||||
|
||||
|
||||
|
||||
lemma normSq_Vud_plus_normSq_Vus_neq_zero_ℂ {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) :
|
||||
(normSq [V]ud : ℂ) + normSq [V]us ≠ 0 := by
|
||||
have h1 := normSq_Vud_plus_normSq_Vus_neq_zero_ℝ hb
|
||||
|
|
@ -186,7 +211,7 @@ lemma conj_Vtb_mul_Vud {V : CKMMatrix} {τ : ℝ}
|
|||
ring
|
||||
rw [h2, V.Vcd_mul_conj_Vud]
|
||||
rw [normSq_eq_conj_mul_self, normSq_eq_conj_mul_self]
|
||||
simp
|
||||
simp only [Fin.isValue, neg_mul]
|
||||
ring
|
||||
|
||||
lemma conj_Vtb_mul_Vus {V : CKMMatrix} {τ : ℝ}
|
||||
|
|
@ -202,7 +227,7 @@ lemma conj_Vtb_mul_Vus {V : CKMMatrix} {τ : ℝ}
|
|||
ring
|
||||
rw [h2, V.Vcs_mul_conj_Vus]
|
||||
rw [normSq_eq_conj_mul_self, normSq_eq_conj_mul_self]
|
||||
simp
|
||||
simp only [Fin.isValue, neg_mul]
|
||||
ring
|
||||
|
||||
|
||||
|
|
@ -224,6 +249,7 @@ lemma cd_of_ud_us_ub_cb_tb {V : CKMMatrix} (h : [V]ud ≠ 0 ∨ [V]us ≠ 0)
|
|||
field_simp
|
||||
ring
|
||||
|
||||
|
||||
end rows
|
||||
|
||||
|
||||
|
|
@ -247,25 +273,6 @@ lemma VAbs_leq_one (i j : Fin 3) (V : Quotient CKMMatrixSetoid) : VAbs i j V ≤
|
|||
|
||||
end individual
|
||||
|
||||
lemma VAbs_thd_neq_one_fst_snd_sq_neq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3}
|
||||
(hV : VAbs i 2 V ≠ 1) : (VAbs i 0 V ^ 2 + VAbs i 1 V ^ 2) ≠ 0 := by
|
||||
have h1 : 1 - VAbs i 2 V ^ 2 = VAbs i 0 V ^ 2 + VAbs i 1 V ^ 2 := by
|
||||
linear_combination - (VAbs_sum_sq_row_eq_one V i)
|
||||
rw [← h1]
|
||||
by_contra h
|
||||
have h2 : VAbs i 2 V ^2 = 1 := by
|
||||
nlinarith
|
||||
simp only [Fin.isValue, sq_eq_one_iff] at h2
|
||||
have h3 : 0 ≤ VAbs i 2 V := VAbs_ge_zero i 2 V
|
||||
have h4 : VAbs i 2 V = 1 := by
|
||||
nlinarith
|
||||
exact hV h4
|
||||
|
||||
lemma VAbs_thd_neq_one_sqrt_fst_snd_sq_neq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3}
|
||||
(hV : VAbs i 2 V ≠ 1) : √(VAbs i 0 V ^ 2 + VAbs i 1 V ^ 2) ≠ 0 := by
|
||||
rw [Real.sqrt_ne_zero (Left.add_nonneg (sq_nonneg (VAbs i 0 V)) (sq_nonneg (VAbs i 1 V)))]
|
||||
exact VAbs_thd_neq_one_fst_snd_sq_neq_zero hV
|
||||
|
||||
section columns
|
||||
|
||||
|
||||
|
|
@ -316,7 +323,7 @@ lemma cs_of_ud_us_zero {V : CKMMatrix} (ha : ¬ ([V]ud ≠ 0 ∨ [V]us ≠ 0)) :
|
|||
simp at h1
|
||||
simp [VAbs]
|
||||
linear_combination h1
|
||||
simp
|
||||
simp only [VcdAbs, Fin.isValue, sub_nonneg, sq_le_one_iff_abs_le_one]
|
||||
rw [@abs_le]
|
||||
have h1 := VAbs_leq_one 1 0 ⟦V⟧
|
||||
have h0 := VAbs_ge_zero 1 0 ⟦V⟧
|
||||
|
|
|
|||
|
|
@ -6,7 +6,16 @@ Authors: Joseph Tooby-Smith
|
|||
import HepLean.FlavorPhysics.CKMMatrix.Basic
|
||||
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
|
||||
import Mathlib.LinearAlgebra.CrossProduct
|
||||
/-!
|
||||
# Rows for the CKM Matrix
|
||||
|
||||
This file contains the definition extracting the rows of the CKM matrix and
|
||||
proves some properties between them.
|
||||
|
||||
The first row can be extracted as `[V]u` for a CKM matrix `V`.
|
||||
|
||||
|
||||
-/
|
||||
|
||||
open Matrix Complex
|
||||
|
||||
|
|
@ -16,17 +25,26 @@ noncomputable section
|
|||
namespace CKMMatrix
|
||||
|
||||
|
||||
/-- The `u`th row of the CKM matrix. -/
|
||||
def uRow (V : CKMMatrix) : Fin 3 → ℂ := ![[V]ud, [V]us, [V]ub]
|
||||
|
||||
/-- The `u`th row of the CKM matrix. -/
|
||||
scoped[CKMMatrix] notation (name := u_row) "[" V "]u" => uRow V
|
||||
|
||||
/-- The `c`th row of the CKM matrix. -/
|
||||
def cRow (V : CKMMatrix) : Fin 3 → ℂ := ![[V]cd, [V]cs, [V]cb]
|
||||
|
||||
/-- The `c`th row of the CKM matrix. -/
|
||||
scoped[CKMMatrix] notation (name := c_row) "[" V "]c" => cRow V
|
||||
|
||||
/-- The `t`th row of the CKM matrix. -/
|
||||
def tRow (V : CKMMatrix) : Fin 3 → ℂ := ![[V]td, [V]ts, [V]tb]
|
||||
|
||||
/-- The `t`th row of the CKM matrix. -/
|
||||
scoped[CKMMatrix] notation (name := t_row) "[" V "]t" => tRow V
|
||||
|
||||
lemma uRow_normalized (V : CKMMatrix) : conj [V]u ⬝ᵥ [V]u = 1 := by
|
||||
simp
|
||||
simp only [vec3_dotProduct, Fin.isValue, Pi.conj_apply]
|
||||
have hV := V.prop
|
||||
rw [mem_unitaryGroup_iff] at hV
|
||||
have ht := congrFun (congrFun hV 0) 0
|
||||
|
|
@ -35,7 +53,7 @@ lemma uRow_normalized (V : CKMMatrix) : conj [V]u ⬝ᵥ [V]u = 1 := by
|
|||
exact ht
|
||||
|
||||
lemma cRow_normalized (V : CKMMatrix) : conj [V]c ⬝ᵥ [V]c = 1 := by
|
||||
simp
|
||||
simp only [vec3_dotProduct, Fin.isValue, Pi.conj_apply]
|
||||
have hV := V.prop
|
||||
rw [mem_unitaryGroup_iff] at hV
|
||||
have ht := congrFun (congrFun hV 1) 1
|
||||
|
|
@ -44,7 +62,7 @@ lemma cRow_normalized (V : CKMMatrix) : conj [V]c ⬝ᵥ [V]c = 1 := by
|
|||
exact ht
|
||||
|
||||
lemma uRow_cRow_orthog (V : CKMMatrix) : conj [V]u ⬝ᵥ [V]c = 0 := by
|
||||
simp
|
||||
simp only [vec3_dotProduct, Fin.isValue, Pi.conj_apply]
|
||||
have hV := V.prop
|
||||
rw [mem_unitaryGroup_iff] at hV
|
||||
have ht := congrFun (congrFun hV 1) 0
|
||||
|
|
@ -53,7 +71,7 @@ lemma uRow_cRow_orthog (V : CKMMatrix) : conj [V]u ⬝ᵥ [V]c = 0 := by
|
|||
exact ht
|
||||
|
||||
lemma uRow_tRow_orthog (V : CKMMatrix) : conj [V]u ⬝ᵥ [V]t = 0 := by
|
||||
simp
|
||||
simp only [vec3_dotProduct, Fin.isValue, Pi.conj_apply]
|
||||
have hV := V.prop
|
||||
rw [mem_unitaryGroup_iff] at hV
|
||||
have ht := congrFun (congrFun hV 2) 0
|
||||
|
|
@ -62,7 +80,7 @@ lemma uRow_tRow_orthog (V : CKMMatrix) : conj [V]u ⬝ᵥ [V]t = 0 := by
|
|||
exact ht
|
||||
|
||||
lemma cRow_uRow_orthog (V : CKMMatrix) : conj [V]c ⬝ᵥ [V]u = 0 := by
|
||||
simp
|
||||
simp only [vec3_dotProduct, Fin.isValue, Pi.conj_apply]
|
||||
have hV := V.prop
|
||||
rw [mem_unitaryGroup_iff] at hV
|
||||
have ht := congrFun (congrFun hV 0) 1
|
||||
|
|
@ -71,7 +89,7 @@ lemma cRow_uRow_orthog (V : CKMMatrix) : conj [V]c ⬝ᵥ [V]u = 0 := by
|
|||
exact ht
|
||||
|
||||
lemma cRow_tRow_orthog (V : CKMMatrix) : conj [V]c ⬝ᵥ [V]t = 0 := by
|
||||
simp
|
||||
simp only [vec3_dotProduct, Fin.isValue, Pi.conj_apply]
|
||||
have hV := V.prop
|
||||
rw [mem_unitaryGroup_iff] at hV
|
||||
have ht := congrFun (congrFun hV 2) 1
|
||||
|
|
@ -80,7 +98,7 @@ lemma cRow_tRow_orthog (V : CKMMatrix) : conj [V]c ⬝ᵥ [V]t = 0 := by
|
|||
exact ht
|
||||
|
||||
lemma tRow_normalized (V : CKMMatrix) : conj [V]t ⬝ᵥ [V]t = 1 := by
|
||||
simp
|
||||
simp only [vec3_dotProduct, Fin.isValue, Pi.conj_apply]
|
||||
have hV := V.prop
|
||||
rw [mem_unitaryGroup_iff] at hV
|
||||
have ht := congrFun (congrFun hV 2) 2
|
||||
|
|
@ -89,7 +107,7 @@ lemma tRow_normalized (V : CKMMatrix) : conj [V]t ⬝ᵥ [V]t = 1 := by
|
|||
exact ht
|
||||
|
||||
lemma tRow_uRow_orthog (V : CKMMatrix) : conj [V]t ⬝ᵥ [V]u = 0 := by
|
||||
simp
|
||||
simp only [vec3_dotProduct, Fin.isValue, Pi.conj_apply]
|
||||
have hV := V.prop
|
||||
rw [mem_unitaryGroup_iff] at hV
|
||||
have ht := congrFun (congrFun hV 0) 2
|
||||
|
|
@ -98,7 +116,7 @@ lemma tRow_uRow_orthog (V : CKMMatrix) : conj [V]t ⬝ᵥ [V]u = 0 := by
|
|||
exact ht
|
||||
|
||||
lemma tRow_cRow_orthog (V : CKMMatrix) : conj [V]t ⬝ᵥ [V]c = 0 := by
|
||||
simp
|
||||
simp only [vec3_dotProduct, Fin.isValue, Pi.conj_apply]
|
||||
have hV := V.prop
|
||||
rw [mem_unitaryGroup_iff] at hV
|
||||
have ht := congrFun (congrFun hV 1) 2
|
||||
|
|
@ -117,19 +135,20 @@ lemma cRow_cross_tRow_conj (V : CKMMatrix) : conj (conj [V]c ×₃ conj [V]t) =
|
|||
fin_cases i <;> simp
|
||||
|
||||
lemma uRow_cross_cRow_normalized (V : CKMMatrix) :
|
||||
conj (conj [V]u ×₃ conj [V]c) ⬝ᵥ (conj [V]u ×₃ conj [V]c) = 1 := by
|
||||
conj (conj [V]u ×₃ conj [V]c) ⬝ᵥ (conj [V]u ×₃ conj [V]c) = 1 := by
|
||||
rw [uRow_cross_cRow_conj, cross_dot_cross]
|
||||
rw [dotProduct_comm, uRow_normalized, dotProduct_comm, cRow_normalized]
|
||||
rw [dotProduct_comm, cRow_uRow_orthog, dotProduct_comm, uRow_cRow_orthog]
|
||||
simp
|
||||
|
||||
lemma cRow_cross_tRow_normalized (V : CKMMatrix) :
|
||||
conj (conj [V]c ×₃ conj [V]t) ⬝ᵥ (conj [V]c ×₃ conj [V]t) = 1 := by
|
||||
conj (conj [V]c ×₃ conj [V]t) ⬝ᵥ (conj [V]c ×₃ conj [V]t) = 1 := by
|
||||
rw [cRow_cross_tRow_conj, cross_dot_cross]
|
||||
rw [dotProduct_comm, cRow_normalized, dotProduct_comm, tRow_normalized]
|
||||
rw [dotProduct_comm, tRow_cRow_orthog, dotProduct_comm, cRow_tRow_orthog]
|
||||
simp
|
||||
|
||||
/-- A map from `Fin 3` to each row of a CKM matrix. -/
|
||||
@[simp]
|
||||
def rows (V : CKMMatrix) : Fin 3 → Fin 3 → ℂ := fun i =>
|
||||
match i with
|
||||
|
|
@ -137,7 +156,7 @@ def rows (V : CKMMatrix) : Fin 3 → Fin 3 → ℂ := fun i =>
|
|||
| 1 => cRow V
|
||||
| 2 => tRow V
|
||||
|
||||
def rowsLinearlyIndependent (V : CKMMatrix) : LinearIndependent ℂ (rows V) := by
|
||||
lemma rows_linearly_independent (V : CKMMatrix) : LinearIndependent ℂ (rows V) := by
|
||||
apply Fintype.linearIndependent_iff.mpr
|
||||
intro g hg
|
||||
rw [Fin.sum_univ_three] at hg
|
||||
|
|
@ -160,9 +179,10 @@ def rowsLinearlyIndependent (V : CKMMatrix) : LinearIndependent ℂ (rows V) :=
|
|||
lemma rows_card : Fintype.card (Fin 3) = FiniteDimensional.finrank ℂ (Fin 3 → ℂ) := by
|
||||
simp only [Fintype.card_fin, FiniteDimensional.finrank_fintype_fun_eq_card]
|
||||
|
||||
/-- The rows of a CKM matrix as a basis of `ℂ³`. -/
|
||||
@[simps!]
|
||||
noncomputable def rowBasis (V : CKMMatrix) : Basis (Fin 3) ℂ (Fin 3 → ℂ) :=
|
||||
basisOfLinearIndependentOfCardEqFinrank (rowsLinearlyIndependent V) rows_card
|
||||
basisOfLinearIndependentOfCardEqFinrank (rows_linearly_independent V) rows_card
|
||||
|
||||
lemma cRow_cross_tRow_eq_uRow (V : CKMMatrix) :
|
||||
∃ (κ : ℝ), [V]u = cexp (κ * I) • (conj [V]c ×₃ conj [V]t) := by
|
||||
|
|
@ -197,7 +217,7 @@ lemma cRow_cross_tRow_eq_uRow (V : CKMMatrix) :
|
|||
swap
|
||||
have hx : Complex.abs (g 0) = -1 := by
|
||||
rw [← ofReal_inj]
|
||||
simp
|
||||
simp only [Fin.isValue, ofReal_neg, ofReal_one]
|
||||
exact h2
|
||||
have h3 := Complex.abs.nonneg (g 0)
|
||||
simp_all
|
||||
|
|
@ -207,7 +227,7 @@ lemma cRow_cross_tRow_eq_uRow (V : CKMMatrix) :
|
|||
rw [← hg]
|
||||
rw [@smul_smul]
|
||||
rw [inv_mul_cancel]
|
||||
simp
|
||||
simp only [one_smul]
|
||||
by_contra hn
|
||||
rw [hn] at h2
|
||||
simp at h2
|
||||
|
|
@ -256,7 +276,7 @@ lemma uRow_cross_cRow_eq_tRow (V : CKMMatrix) :
|
|||
swap
|
||||
have hx : Complex.abs (g 2) = -1 := by
|
||||
rw [← ofReal_inj]
|
||||
simp
|
||||
simp only [Fin.isValue, ofReal_neg, ofReal_one]
|
||||
exact h2
|
||||
have h3 := Complex.abs.nonneg (g 2)
|
||||
simp_all
|
||||
|
|
@ -266,7 +286,7 @@ lemma uRow_cross_cRow_eq_tRow (V : CKMMatrix) :
|
|||
rw [← hg]
|
||||
rw [@smul_smul]
|
||||
rw [inv_mul_cancel]
|
||||
simp
|
||||
simp only [one_smul]
|
||||
by_contra hn
|
||||
rw [hn] at h2
|
||||
simp at h2
|
||||
|
|
@ -283,23 +303,6 @@ lemma uRow_cross_cRow_eq_tRow (V : CKMMatrix) :
|
|||
rw [hx, hτ]
|
||||
|
||||
|
||||
def uRow₁₂ (V : CKMMatrix) : Fin 2 → ℂ := ![[V]ud, [V]us]
|
||||
|
||||
def cRow₁₂ (V : CKMMatrix) : Fin 2 → ℂ := ![[V]cd, [V]cs]
|
||||
scoped[CKMMatrix] notation (name := c₁₂_row) "[" V "]c₁₂" => cRow₁₂ V
|
||||
|
||||
lemma cRow₁₂_normalized_of_cb_zero {V : CKMMatrix} (hcb : [V]cb = 0) :
|
||||
conj [V]c₁₂ ⬝ᵥ [V]c₁₂ = 1 := by
|
||||
simp
|
||||
have hV := V.prop
|
||||
rw [mem_unitaryGroup_iff] at hV
|
||||
have ht := congrFun (congrFun hV 1) 1
|
||||
simp [Matrix.mul_apply, Fin.sum_univ_three] at ht
|
||||
rw [hcb] at ht
|
||||
rw [mul_comm (V.1 1 0) _, mul_comm (V.1 1 1) _] at ht
|
||||
simp at ht
|
||||
exact ht
|
||||
|
||||
lemma ext_Rows {U V : CKMMatrix} (hu : [U]u = [V]u) (hc : [U]c = [V]c) (ht : [U]t = [V]t) :
|
||||
U = V := by
|
||||
apply CKMMatrix_ext
|
||||
|
|
@ -319,7 +322,7 @@ open CKMMatrix
|
|||
|
||||
variable (V : CKMMatrix) (a b c d e f : ℝ)
|
||||
|
||||
|
||||
/-- The cross product of the conjugate of the `u` and `c` rows of a CKM matrix. -/
|
||||
def ucCross : Fin 3 → ℂ :=
|
||||
conj [phaseShiftApply V a b c d e f]u ×₃ conj [phaseShiftApply V a b c d e f]c
|
||||
|
||||
|
|
@ -350,39 +353,39 @@ lemma ucCross_thd (V : CKMMatrix) : (ucCross V a b c d e f) 2 =
|
|||
lemma uRow_mul (V : CKMMatrix) (a b c : ℝ) :
|
||||
[phaseShiftApply V a b c 0 0 0]u = cexp (a * I) • [V]u := by
|
||||
funext i
|
||||
simp
|
||||
simp only [Pi.smul_apply, smul_eq_mul]
|
||||
fin_cases i <;>
|
||||
change (phaseShiftApply V a b c 0 0 0).1 0 _ = _
|
||||
rw [ud, uRow]
|
||||
simp
|
||||
simp only [ofReal_zero, zero_mul, add_zero, Fin.isValue, Fin.zero_eta, cons_val_zero]
|
||||
rw [us, uRow]
|
||||
simp
|
||||
simp only [ofReal_zero, zero_mul, add_zero, Fin.isValue, Fin.mk_one, cons_val_one, head_cons]
|
||||
rw [ub, uRow]
|
||||
simp
|
||||
|
||||
lemma cRow_mul (V : CKMMatrix) (a b c : ℝ) :
|
||||
[phaseShiftApply V a b c 0 0 0]c = cexp (b * I) • [V]c := by
|
||||
funext i
|
||||
simp
|
||||
simp only [Pi.smul_apply, smul_eq_mul]
|
||||
fin_cases i <;>
|
||||
change (phaseShiftApply V a b c 0 0 0).1 1 _ = _
|
||||
rw [cd, cRow]
|
||||
simp
|
||||
simp only [ofReal_zero, zero_mul, add_zero, Fin.isValue, Fin.zero_eta, cons_val_zero]
|
||||
rw [cs, cRow]
|
||||
simp
|
||||
simp only [ofReal_zero, zero_mul, add_zero, Fin.isValue, Fin.mk_one, cons_val_one, head_cons]
|
||||
rw [cb, cRow]
|
||||
simp
|
||||
|
||||
lemma tRow_mul (V : CKMMatrix) (a b c : ℝ) :
|
||||
[phaseShiftApply V a b c 0 0 0]t = cexp (c * I) • [V]t := by
|
||||
funext i
|
||||
simp
|
||||
simp only [Pi.smul_apply, smul_eq_mul]
|
||||
fin_cases i <;>
|
||||
change (phaseShiftApply V a b c 0 0 0).1 2 _ = _
|
||||
rw [td, tRow]
|
||||
simp
|
||||
simp only [ofReal_zero, zero_mul, add_zero, Fin.isValue, Fin.zero_eta, cons_val_zero]
|
||||
rw [ts, tRow]
|
||||
simp
|
||||
simp only [ofReal_zero, zero_mul, add_zero, Fin.isValue, Fin.mk_one, cons_val_one, head_cons]
|
||||
rw [tb, tRow]
|
||||
simp
|
||||
|
||||
|
|
|
|||
|
|
@ -8,14 +8,25 @@ import HepLean.FlavorPhysics.CKMMatrix.Rows
|
|||
import HepLean.FlavorPhysics.CKMMatrix.PhaseFreedom
|
||||
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
|
||||
|
||||
-- to be renamed stanParamAsMatrix ...
|
||||
/-- Given four reals `θ₁₂ θ₁₃ θ₂₃ δ₁₃` the standard paramaterization 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)),
|
||||
|
|
@ -87,6 +98,8 @@ lemma standParamAsMatrix_unitary (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
|
|||
rw [sin_sq, sin_sq]
|
||||
ring
|
||||
|
||||
/-- Given four reals `θ₁₂ θ₁₃ θ₂₃ δ₁₃` the standard paramaterization of the CKM matrix
|
||||
as a CKM matrix. -/
|
||||
def standParam (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : CKMMatrix :=
|
||||
⟨standParamAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃, by
|
||||
rw [mem_unitaryGroup_iff']
|
||||
|
|
@ -137,13 +150,13 @@ lemma eq_exp_of_phases (θ₁₂ θ₁₃ θ₂₃ δ₁₃ δ₁₃' : ℝ) (h
|
|||
standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃ = standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃' := by
|
||||
simp [standParam, standParamAsMatrix]
|
||||
apply CKMMatrix_ext
|
||||
simp
|
||||
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) :
|
||||
(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,
|
||||
|
|
@ -152,7 +165,8 @@ lemma VusVubVcdSq_eq (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Rea
|
|||
VcbAbs, VudAbs, Complex.abs_ofReal]
|
||||
by_cases hx : Real.cos θ₁₃ ≠ 0
|
||||
· rw [Complex.abs_exp]
|
||||
simp
|
||||
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]
|
||||
|
|
@ -175,7 +189,7 @@ lemma mulExpδ₁₃_eq (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤
|
|||
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
|
||||
simp only [ofReal_sin, ofReal_cos, ofReal_mul, ofReal_pow]
|
||||
ring_nf
|
||||
rw [exp_neg]
|
||||
have h1 : cexp (I * δ₁₃) ≠ 0 := exp_ne_zero _
|
||||
|
|
|
|||
|
|
@ -9,33 +9,63 @@ import HepLean.FlavorPhysics.CKMMatrix.PhaseFreedom
|
|||
import HepLean.FlavorPhysics.CKMMatrix.Invariants
|
||||
import HepLean.FlavorPhysics.CKMMatrix.StandardParameterization.Basic
|
||||
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
|
||||
/-!
|
||||
# Standard parameters for the CKM Matrix
|
||||
|
||||
Given a CKM matrix `V` we can extract four real numbers `θ₁₂`, `θ₁₃`, `θ₂₃` and `δ₁₃`.
|
||||
These, when used in the standard parameterization return `V` up to equivalence.
|
||||
|
||||
This leads to the theorem `standParam.exists_for_CKMatrix` which says that up to equivalence every
|
||||
CKM matrix can be written using the standard parameterization.
|
||||
|
||||
|
||||
-/
|
||||
open Matrix Complex
|
||||
open ComplexConjugate
|
||||
open CKMMatrix
|
||||
|
||||
noncomputable section
|
||||
|
||||
/-- Given a CKM matrix `V` the real number corresponding to `sin θ₁₂` in the
|
||||
standard parameterization. --/
|
||||
def S₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := VusAbs V / (√ (VudAbs V ^ 2 + VusAbs V ^ 2))
|
||||
|
||||
/-- Given a CKM matrix `V` the real number corresponding to `sin θ₁₃` in the
|
||||
standard parameterization. --/
|
||||
def S₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := VubAbs V
|
||||
|
||||
/-- Given a CKM matrix `V` the real number corresponding to `sin θ₂₃` in the
|
||||
standard parameterization. --/
|
||||
def S₂₃ (V : Quotient CKMMatrixSetoid) : ℝ :=
|
||||
if VubAbs V = 1 then VcdAbs V
|
||||
else VcbAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2)
|
||||
|
||||
/-- Given a CKM matrix `V` the real number corresponding to `θ₁₂` in the
|
||||
standard parameterization. --/
|
||||
def θ₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := Real.arcsin (S₁₂ V)
|
||||
|
||||
/-- Given a CKM matrix `V` the real number corresponding to `θ₁₃` in the
|
||||
standard parameterization. --/
|
||||
def θ₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.arcsin (S₁₃ V)
|
||||
|
||||
/-- Given a CKM matrix `V` the real number corresponding to `θ₂₃` in the
|
||||
standard parameterization. --/
|
||||
def θ₂₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.arcsin (S₂₃ V)
|
||||
|
||||
/-- Given a CKM matrix `V` the real number corresponding to `cos θ₁₂` in the
|
||||
standard parameterization. --/
|
||||
def C₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₁₂ V)
|
||||
|
||||
/-- Given a CKM matrix `V` the real number corresponding to `cos θ₁₃` in the
|
||||
standard parameterization. --/
|
||||
def C₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₁₃ V)
|
||||
|
||||
/-- Given a CKM matrix `V` the real number corresponding to `sin θ₂₃` in the
|
||||
standard parameterization. --/
|
||||
def C₂₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₂₃ V)
|
||||
|
||||
/-- Given a CKM matrix `V` the real number corresponding to the phase `δ₁₃` in the
|
||||
standard parameterization. --/
|
||||
def δ₁₃ (V : Quotient CKMMatrixSetoid) : ℝ :=
|
||||
arg (Invariant.mulExpδ₁₃ V)
|
||||
|
||||
|
|
@ -69,7 +99,7 @@ lemma S₁₂_leq_one (V : Quotient CKMMatrixSetoid) : S₁₂ V ≤ 1 := by
|
|||
apply Or.inl
|
||||
simp_all
|
||||
rw [Real.le_sqrt (VAbs_ge_zero 0 1 V) (le_of_lt h3)]
|
||||
simp
|
||||
simp only [Fin.isValue, le_add_iff_nonneg_left]
|
||||
exact sq_nonneg (VAbs 0 0 V)
|
||||
|
||||
lemma S₁₃_leq_one (V : Quotient CKMMatrixSetoid) : S₁₃ V ≤ 1 :=
|
||||
|
|
@ -91,7 +121,7 @@ lemma S₂₃_leq_one (V : Quotient CKMMatrixSetoid) : S₂₃ V ≤ 1 := by
|
|||
simp_all
|
||||
rw [Real.le_sqrt (VAbs_ge_zero 1 2 V) (le_of_lt h3)]
|
||||
rw [VudAbs_sq_add_VusAbs_sq, ← VcbAbs_sq_add_VtbAbs_sq]
|
||||
simp
|
||||
simp only [Fin.isValue, VcbAbs, VtbAbs, le_add_iff_nonneg_right]
|
||||
exact sq_nonneg (VAbs 2 2 V)
|
||||
|
||||
lemma S₁₂_eq_sin_θ₁₂ (V : Quotient CKMMatrixSetoid) : Real.sin (θ₁₂ V) = S₁₂ V :=
|
||||
|
|
@ -158,19 +188,19 @@ lemma C₂₃_eq_ℂcos_θ₂₃ (V : Quotient CKMMatrixSetoid) : Complex.cos (
|
|||
lemma complexAbs_cos_θ₁₂ (V : Quotient CKMMatrixSetoid) : Complex.abs (Complex.cos (θ₁₂ V)) =
|
||||
cos (θ₁₂ V):= by
|
||||
rw [C₁₂_eq_ℂcos_θ₁₂, Complex.abs_ofReal]
|
||||
simp
|
||||
simp only [ofReal_inj, abs_eq_self]
|
||||
exact Real.cos_arcsin_nonneg _
|
||||
|
||||
lemma complexAbs_cos_θ₁₃ (V : Quotient CKMMatrixSetoid) : Complex.abs (Complex.cos (θ₁₃ V)) =
|
||||
cos (θ₁₃ V):= by
|
||||
rw [C₁₃_eq_ℂcos_θ₁₃, Complex.abs_ofReal]
|
||||
simp
|
||||
simp only [ofReal_inj, abs_eq_self]
|
||||
exact Real.cos_arcsin_nonneg _
|
||||
|
||||
lemma complexAbs_cos_θ₂₃ (V : Quotient CKMMatrixSetoid) : Complex.abs (Complex.cos (θ₂₃ V)) =
|
||||
cos (θ₂₃ V):= by
|
||||
rw [C₂₃_eq_ℂcos_θ₂₃, Complex.abs_ofReal]
|
||||
simp
|
||||
simp only [ofReal_inj, abs_eq_self]
|
||||
exact Real.cos_arcsin_nonneg _
|
||||
|
||||
lemma S₁₂_sq_add_C₁₂_sq (V : Quotient CKMMatrixSetoid) : S₁₂ V ^ 2 + C₁₂ V ^ 2 = 1 := by
|
||||
|
|
@ -198,12 +228,12 @@ lemma C₁₂_eq_Vud_div_sqrt {V : Quotient CKMMatrixSetoid} (ha : VubAbs V ≠
|
|||
C₁₂ V = VudAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2) := by
|
||||
rw [C₁₂, θ₁₂, Real.cos_arcsin, S₁₂, div_pow, Real.sq_sqrt]
|
||||
rw [one_sub_div]
|
||||
simp
|
||||
simp only [VudAbs, Fin.isValue, VusAbs, add_sub_cancel_right]
|
||||
rw [Real.sqrt_div]
|
||||
rw [Real.sqrt_sq]
|
||||
exact VAbs_ge_zero 0 0 V
|
||||
exact sq_nonneg (VAbs 0 0 V)
|
||||
exact VAbs_thd_neq_one_fst_snd_sq_neq_zero ha
|
||||
exact VAbsub_neq_zero_Vud_Vus_neq_zero ha
|
||||
exact (Left.add_nonneg (sq_nonneg (VAbs 0 0 V)) (sq_nonneg (VAbs 0 1 V)))
|
||||
|
||||
--rename
|
||||
|
|
@ -222,7 +252,7 @@ lemma C₂₃_of_Vub_neq_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V ≠ 1
|
|||
rw [Real.sqrt_div (sq_nonneg (VAbs 2 2 V))]
|
||||
rw [Real.sqrt_sq (VAbs_ge_zero 2 2 V)]
|
||||
rw [VcbAbs_sq_add_VtbAbs_sq, ← VudAbs_sq_add_VusAbs_sq ]
|
||||
exact VAbs_thd_neq_one_fst_snd_sq_neq_zero ha
|
||||
exact VAbsub_neq_zero_Vud_Vus_neq_zero ha
|
||||
exact (Left.add_nonneg (sq_nonneg (VAbs 0 0 V)) (sq_nonneg (VAbs 0 1 V)))
|
||||
|
||||
end cosines
|
||||
|
|
@ -235,12 +265,12 @@ lemma VudAbs_eq_C₁₂_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VudAbs V =
|
|||
change VAbs 0 0 V = C₁₂ V * C₁₃ V
|
||||
rw [VAbs_thd_eq_one_fst_eq_zero ha]
|
||||
rw [C₁₃, θ₁₃, Real.cos_arcsin, S₁₃, ha]
|
||||
simp
|
||||
simp only [one_pow, sub_self, Real.sqrt_zero, mul_zero]
|
||||
rw [C₁₂_eq_Vud_div_sqrt ha, C₁₃, θ₁₃, Real.cos_arcsin, S₁₃]
|
||||
have h1 : 1 - VubAbs V ^ 2 = VudAbs V ^ 2 + VusAbs V ^ 2 := by
|
||||
linear_combination - (VAbs_sum_sq_row_eq_one V 0)
|
||||
rw [h1, mul_comm]
|
||||
exact (mul_div_cancel₀ (VudAbs V) (VAbs_thd_neq_one_sqrt_fst_snd_sq_neq_zero ha)).symm
|
||||
exact (mul_div_cancel₀ (VudAbs V) (VAbsub_neq_zero_sqrt_Vud_Vus_neq_zero ha)).symm
|
||||
|
||||
lemma VusAbs_eq_S₁₂_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VusAbs V = S₁₂ V * C₁₃ V := by
|
||||
rw [C₁₃, θ₁₃, Real.cos_arcsin, S₁₂, S₁₃]
|
||||
|
|
@ -254,7 +284,7 @@ lemma VusAbs_eq_S₁₂_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VusAbs V =
|
|||
rw [← h1]
|
||||
simp only [Real.sqrt_zero, div_zero, mul_zero]
|
||||
exact VAbs_thd_eq_one_snd_eq_zero ha
|
||||
have h2 := VAbs_thd_neq_one_sqrt_fst_snd_sq_neq_zero ha
|
||||
have h2 := VAbsub_neq_zero_sqrt_Vud_Vus_neq_zero ha
|
||||
exact (mul_div_cancel₀ (VusAbs V) h2).symm
|
||||
|
||||
lemma VubAbs_eq_S₁₃ (V : Quotient CKMMatrixSetoid) : VubAbs V = S₁₃ V := rfl
|
||||
|
|
@ -262,20 +292,20 @@ lemma VubAbs_eq_S₁₃ (V : Quotient CKMMatrixSetoid) : VubAbs V = S₁₃ V :=
|
|||
lemma VcbAbs_eq_S₂₃_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VcbAbs V = S₂₃ V * C₁₃ V := by
|
||||
by_cases ha : VubAbs V = 1
|
||||
rw [C₁₃_of_Vub_eq_one ha]
|
||||
simp
|
||||
simp only [VcbAbs, Fin.isValue, mul_zero]
|
||||
exact VAbs_fst_col_eq_one_snd_eq_zero ha
|
||||
rw [S₂₃_of_Vub_neq_one ha, C₁₃_eq_add_sq]
|
||||
rw [mul_comm]
|
||||
exact (mul_div_cancel₀ (VcbAbs V) (VAbs_thd_neq_one_sqrt_fst_snd_sq_neq_zero ha)).symm
|
||||
exact (mul_div_cancel₀ (VcbAbs V) (VAbsub_neq_zero_sqrt_Vud_Vus_neq_zero ha)).symm
|
||||
|
||||
lemma VtbAbs_eq_C₂₃_mul_C₁₃ (V : Quotient CKMMatrixSetoid) : VtbAbs V = C₂₃ V * C₁₃ V := by
|
||||
by_cases ha : VubAbs V = 1
|
||||
rw [C₁₃_of_Vub_eq_one ha]
|
||||
simp
|
||||
simp only [VtbAbs, Fin.isValue, mul_zero]
|
||||
exact VAbs_fst_col_eq_one_thd_eq_zero ha
|
||||
rw [C₂₃_of_Vub_neq_one ha, C₁₃_eq_add_sq]
|
||||
rw [mul_comm]
|
||||
exact (mul_div_cancel₀ (VtbAbs V) (VAbs_thd_neq_one_sqrt_fst_snd_sq_neq_zero ha)).symm
|
||||
exact (mul_div_cancel₀ (VtbAbs V) (VAbsub_neq_zero_sqrt_Vud_Vus_neq_zero ha)).symm
|
||||
|
||||
lemma VubAbs_of_cos_θ₁₃_zero {V : Quotient CKMMatrixSetoid} (h1 : Real.cos (θ₁₃ V) = 0) :
|
||||
VubAbs V = 1 := by
|
||||
|
|
@ -288,7 +318,7 @@ lemma VubAbs_of_cos_θ₁₃_zero {V : Quotient CKMMatrixSetoid} (h1 : Real.cos
|
|||
rw [h2] at h3
|
||||
simp at h3
|
||||
linarith
|
||||
simp
|
||||
simp only [VubAbs, Fin.isValue, sub_nonneg, sq_le_one_iff_abs_le_one]
|
||||
rw [_root_.abs_of_nonneg (VAbs_ge_zero 0 2 V)]
|
||||
exact VAbs_leq_one 0 2 V
|
||||
|
||||
|
|
@ -309,7 +339,8 @@ open Invariant
|
|||
|
||||
lemma mulExpδ₁₃_on_param_δ₁₃ (V : CKMMatrix) (δ₁₃ : ℝ) :
|
||||
mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
|
||||
sin (θ₁₂ ⟦V⟧) * cos (θ₁₃ ⟦V⟧) ^ 2 * sin (θ₂₃ ⟦V⟧) * sin (θ₁₃ ⟦V⟧) * cos (θ₁₂ ⟦V⟧) * cos (θ₂₃ ⟦V⟧) * cexp (I * δ₁₃) := by
|
||||
sin (θ₁₂ ⟦V⟧) * cos (θ₁₃ ⟦V⟧) ^ 2 * sin (θ₂₃ ⟦V⟧) * sin (θ₁₃ ⟦V⟧)
|
||||
* cos (θ₁₂ ⟦V⟧) * cos (θ₂₃ ⟦V⟧) * cexp (I * δ₁₃) := by
|
||||
refine mulExpδ₁₃_eq _ _ _ _ ?_ ?_ ?_ ?_
|
||||
rw [S₁₂_eq_sin_θ₁₂]
|
||||
exact S₁₂_nonneg _
|
||||
|
|
@ -321,14 +352,15 @@ lemma mulExpδ₁₃_on_param_δ₁₃ (V : CKMMatrix) (δ₁₃ : ℝ) :
|
|||
lemma mulExpδ₁₃_on_param_eq_zero_iff (V : CKMMatrix) (δ₁₃ : ℝ) :
|
||||
mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
|
||||
VudAbs ⟦V⟧ = 0 ∨ VubAbs ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0 := by
|
||||
rw [VudAbs_eq_C₁₂_mul_C₁₃, VubAbs_eq_S₁₃, VusAbs_eq_S₁₂_mul_C₁₃, VcbAbs_eq_S₂₃_mul_C₁₃, VtbAbs_eq_C₂₃_mul_C₁₃,
|
||||
← ofReal_inj,
|
||||
rw [VudAbs_eq_C₁₂_mul_C₁₃, VubAbs_eq_S₁₃, VusAbs_eq_S₁₂_mul_C₁₃, VcbAbs_eq_S₂₃_mul_C₁₃,
|
||||
VtbAbs_eq_C₂₃_mul_C₁₃, ← ofReal_inj,
|
||||
← ofReal_inj, ← ofReal_inj, ← ofReal_inj, ← ofReal_inj]
|
||||
simp only [ofReal_mul]
|
||||
rw [← S₁₃_eq_ℂsin_θ₁₃, ← S₁₂_eq_ℂsin_θ₁₂, ← S₂₃_eq_ℂsin_θ₂₃,
|
||||
← C₁₃_eq_ℂcos_θ₁₃, ← C₂₃_eq_ℂcos_θ₂₃,← C₁₂_eq_ℂcos_θ₁₂]
|
||||
rw [mulExpδ₁₃_on_param_δ₁₃]
|
||||
simp
|
||||
simp only [mul_eq_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff,
|
||||
ofReal_zero]
|
||||
have h1 := exp_ne_zero (I * δ₁₃)
|
||||
simp_all
|
||||
aesop
|
||||
|
|
@ -348,13 +380,15 @@ lemma mulExpδ₁₃_on_param_neq_zero_arg (V : CKMMatrix) (δ₁₃ : ℝ)
|
|||
cexp (δ₁₃ * I) := by
|
||||
have h1a := mulExpδ₁₃_on_param_δ₁₃ V δ₁₃
|
||||
have habs := mulExpδ₁₃_on_param_abs V δ₁₃
|
||||
have h2 : mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
|
||||
Complex.abs (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) * exp (δ₁₃ * I) := by
|
||||
have h2 : mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = Complex.abs
|
||||
(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) * exp (δ₁₃ * I) := by
|
||||
rw [habs, h1a]
|
||||
ring_nf
|
||||
nth_rewrite 1 [← abs_mul_exp_arg_mul_I (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ )] at h2
|
||||
have habs_neq_zero : (Complex.abs (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) : ℂ) ≠ 0 := by
|
||||
simp
|
||||
nth_rewrite 1 [← abs_mul_exp_arg_mul_I (mulExpδ₁₃
|
||||
⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ )] at h2
|
||||
have habs_neq_zero :
|
||||
(Complex.abs (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) : ℂ) ≠ 0 := by
|
||||
simp only [ne_eq, ofReal_eq_zero, map_eq_zero]
|
||||
exact h1
|
||||
rw [← mul_right_inj' habs_neq_zero]
|
||||
rw [← h2]
|
||||
|
|
@ -387,14 +421,14 @@ lemma on_param_cos_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.c
|
|||
apply Or.inr
|
||||
rfl
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
simp only [mul_zero, add_zero, neg_inj]
|
||||
field_simp
|
||||
ring
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
simp only [mul_zero, add_zero]
|
||||
field_simp
|
||||
ring
|
||||
ring
|
||||
|
|
@ -413,7 +447,7 @@ lemma on_param_cos_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.c
|
|||
rfl
|
||||
ring_nf
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
simp only [mul_zero, add_zero]
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
|
|
@ -441,14 +475,14 @@ lemma on_param_sin_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.s
|
|||
apply Or.inr
|
||||
rfl
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
simp only [mul_zero, add_zero, neg_inj]
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
simp only [mul_zero, add_zero, neg_inj]
|
||||
ring
|
||||
field_simp
|
||||
ring
|
||||
|
|
@ -467,7 +501,7 @@ lemma on_param_sin_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.s
|
|||
apply Or.inr
|
||||
rfl
|
||||
change _ = _ + _ * 0
|
||||
simp
|
||||
simp only [mul_zero, add_zero, neg_inj]
|
||||
ring
|
||||
ring
|
||||
field_simp
|
||||
|
|
@ -480,8 +514,8 @@ lemma eq_standParam_of_fstRowThdColRealCond {V : CKMMatrix} (hb : [V]ud ≠ 0
|
|||
have hb' : VubAbs ⟦V⟧ ≠ 1 := by
|
||||
rw [ud_us_neq_zero_iff_ub_neq_one] at hb
|
||||
simp [VAbs, hb]
|
||||
have h1 : ofReal (√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) * ↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) )
|
||||
= ofReal ((VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) ) := by
|
||||
have h1 : ofReal (√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) *
|
||||
↑√(VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2)) = ofReal ((VAbs 0 0 ⟦V⟧ ^ 2 + VAbs 0 1 ⟦V⟧ ^ 2) ) := by
|
||||
rw [Real.mul_self_sqrt ]
|
||||
apply add_nonneg (sq_nonneg _) (sq_nonneg _)
|
||||
simp at h1
|
||||
|
|
@ -496,14 +530,14 @@ lemma eq_standParam_of_fstRowThdColRealCond {V : CKMMatrix} (hb : [V]ud ≠ 0
|
|||
simp [C₁₂, C₁₃]
|
||||
simp [uRow, standParam, standParamAsMatrix]
|
||||
rw [hV.2.1, VusAbs_eq_S₁₂_mul_C₁₃ ⟦V⟧, ← S₁₂_eq_sin_θ₁₂ ⟦V⟧, C₁₃]
|
||||
simp
|
||||
simp only [ofReal_mul, ofReal_sin, ofReal_cos]
|
||||
simp [uRow, standParam, standParamAsMatrix]
|
||||
nth_rewrite 1 [← abs_mul_exp_arg_mul_I (V.1 0 2)]
|
||||
rw [show Complex.abs (V.1 0 2) = VubAbs ⟦V⟧ from rfl]
|
||||
rw [VubAbs_eq_S₁₃, ← S₁₃_eq_sin_θ₁₃ ⟦V⟧]
|
||||
simp
|
||||
simp only [ofReal_sin, Fin.isValue, mul_eq_mul_left_iff]
|
||||
ring_nf
|
||||
simp
|
||||
simp only [true_or]
|
||||
funext i
|
||||
fin_cases i
|
||||
simp [cRow, standParam, standParamAsMatrix]
|
||||
|
|
@ -543,15 +577,15 @@ lemma eq_standParam_of_ubOnePhaseCond {V : CKMMatrix} (hV : ubOnePhaseCond V) :
|
|||
fin_cases i
|
||||
simp [uRow, standParam, standParamAsMatrix]
|
||||
rw [C₁₃_eq_ℂcos_θ₁₃ ⟦V⟧, C₁₃_of_Vub_eq_one h1, hV.1]
|
||||
simp
|
||||
simp only [ofReal_zero, mul_zero]
|
||||
simp [uRow, standParam, standParamAsMatrix]
|
||||
rw [C₁₃_eq_ℂcos_θ₁₃ ⟦V⟧, C₁₃_of_Vub_eq_one h1, hV.2.1]
|
||||
simp
|
||||
simp only [ofReal_zero, mul_zero]
|
||||
simp [uRow, standParam, standParamAsMatrix]
|
||||
rw [S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧, S₁₃]
|
||||
simp [VAbs]
|
||||
rw [hV.2.2.2.1]
|
||||
simp
|
||||
simp only [_root_.map_one, ofReal_one]
|
||||
funext i
|
||||
fin_cases i
|
||||
simp [cRow, standParam, standParamAsMatrix]
|
||||
|
|
@ -560,13 +594,14 @@ lemma eq_standParam_of_ubOnePhaseCond {V : CKMMatrix} (hV : ubOnePhaseCond V) :
|
|||
rw [C₁₂_eq_ℂcos_θ₁₂ ⟦V⟧, C₁₂_of_Vub_one h1]
|
||||
rw [S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧, S₁₃_of_Vub_one h1]
|
||||
rw [hV.2.2.2.2.2.1]
|
||||
simp
|
||||
simp only [VcdAbs, Fin.isValue, ofReal_zero, zero_mul, neg_zero, ofReal_one, mul_one, one_mul,
|
||||
zero_sub]
|
||||
simp [cRow, standParam, standParamAsMatrix]
|
||||
rw [S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧, S₂₃_of_Vub_eq_one h1]
|
||||
rw [S₁₂_eq_ℂsin_θ₁₂ ⟦V⟧, S₁₂_of_Vub_one h1]
|
||||
rw [C₁₂_eq_ℂcos_θ₁₂ ⟦V⟧, C₁₂_of_Vub_one h1]
|
||||
rw [S₁₃_eq_ℂsin_θ₁₃ ⟦V⟧, S₁₃_of_Vub_one h1]
|
||||
simp
|
||||
simp only [Fin.isValue, ofReal_one, one_mul, ofReal_zero, mul_one, VcdAbs, zero_mul, sub_zero]
|
||||
have h3 : (Real.cos (θ₂₃ ⟦V⟧) : ℂ) = √(1 - S₂₃ ⟦V⟧ ^ 2) := by
|
||||
rw [θ₂₃, Real.cos_arcsin]
|
||||
simp at h3
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue