453 lines
15 KiB
Text
453 lines
15 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.StandardModel.Basic
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import Mathlib.Data.Complex.Exponential
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import Mathlib.Tactic.Polyrith
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import Mathlib.Geometry.Manifold.VectorBundle.Basic
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import Mathlib.Geometry.Manifold.VectorBundle.SmoothSection
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import Mathlib.Geometry.Manifold.Instances.Real
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import Mathlib.RepresentationTheory.Basic
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import Mathlib.Analysis.InnerProductSpace.Basic
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import Mathlib.Geometry.Manifold.ContMDiff.Product
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import Mathlib.Analysis.Complex.RealDeriv
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import Mathlib.Analysis.Calculus.Deriv.Add
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import Mathlib.Analysis.Calculus.Deriv.Pow
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import Mathlib.Algebra.QuadraticDiscriminant
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/-!
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# The Higgs field
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This file defines the basic properties for the higgs field in the standard model.
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-/
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universe v u
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namespace StandardModel
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noncomputable section
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open Manifold
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open Matrix
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open Complex
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open ComplexConjugate
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/-- The complex vector space in which the Higgs field takes values. -/
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abbrev higgsVec := EuclideanSpace ℂ (Fin 2)
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/-- The trivial vector bundle 𝓡² × ℂ². (TODO: Make associated bundle.) -/
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abbrev higgsBundle := Bundle.Trivial spaceTime higgsVec
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instance : SmoothVectorBundle higgsVec higgsBundle (𝓡 4) :=
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Bundle.Trivial.smoothVectorBundle higgsVec 𝓘(ℝ, spaceTime)
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/-- A higgs field is a smooth section of the higgs bundle. -/
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abbrev higgsField : Type := SmoothSection (𝓡 4) higgsVec higgsBundle
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instance : NormedAddCommGroup (Fin 2 → ℂ) := by
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exact Pi.normedAddCommGroup
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section higgsVec
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/-- The continous linear map from the vector space `higgsVec` to `(Fin 2 → ℂ)` acheived by
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casting vectors. -/
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def higgsVecToFin2ℂ : higgsVec →L[ℝ] (Fin 2 → ℂ) where
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toFun x := x
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map_add' x y := by
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simp
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map_smul' a x := by
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simp
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lemma smooth_higgsVecToFin2ℂ : Smooth 𝓘(ℝ, higgsVec) 𝓘(ℝ, Fin 2 → ℂ) higgsVecToFin2ℂ :=
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ContinuousLinearMap.smooth higgsVecToFin2ℂ
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/-- Given an element of `gaugeGroup` the linear automorphism of `higgsVec` it gets taken to. -/
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@[simps!]
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noncomputable def higgsRepMap (g : guageGroup) : higgsVec →ₗ[ℂ] higgsVec where
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toFun S := (g.2.2 ^ 3) • (g.2.1.1 *ᵥ S)
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map_add' S T := by
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simp [Matrix.mulVec_add, smul_add]
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rw [Matrix.mulVec_add, smul_add]
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map_smul' a S := by
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simp [Matrix.mulVec_smul]
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rw [Matrix.mulVec_smul]
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exact smul_comm _ _ _
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/-- The representation of the SM guage group acting on `ℂ²`. -/
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noncomputable def higgsRep : Representation ℂ guageGroup higgsVec where
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toFun := higgsRepMap
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map_mul' U V := by
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apply LinearMap.ext
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intro S
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simp only [higgsRepMap, Prod.snd_mul, Submonoid.coe_inf, Prod.fst_mul, Submonoid.coe_mul,
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LinearMap.coe_mk, AddHom.coe_mk, LinearMap.mul_apply, LinearMap.map_smul_of_tower,
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mulVec_mulVec]
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simp [mul_pow, smul_smul, mul_comm]
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map_one' := by
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apply LinearMap.ext
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intro S
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simp only [higgsRepMap, LinearMap.mul_apply, AddHom.coe_mk, LinearMap.coe_mk]
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change 1 ^ 3 • (1 *ᵥ _) = _
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rw [one_pow, Matrix.one_mulVec]
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simp only [one_smul, LinearMap.one_apply]
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end higgsVec
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namespace higgsField
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open Complex Real
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/-- Given a `higgsField`, the corresponding map from `spaceTime` to `higgsVec`. -/
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def toHiggsVec (φ : higgsField) : spaceTime → higgsVec := φ
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lemma toHiggsVec_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, higgsVec) φ.toHiggsVec := by
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intro x0
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have h1 := φ.contMDiff x0
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rw [Bundle.contMDiffAt_section] at h1
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have h2 :
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(fun x => ((trivializationAt higgsVec (Bundle.Trivial spaceTime higgsVec) x0)
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{ proj := x, snd := φ x }).2) = φ := by
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rfl
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simp only [h2] at h1
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exact h1
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lemma higgsVecToFin2ℂ_toHiggsVec (φ : higgsField) : higgsVecToFin2ℂ ∘ φ.toHiggsVec = φ := by
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ext x
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rfl
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lemma toVec_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, Fin 2 → ℂ) φ := by
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rw [← φ.higgsVecToFin2ℂ_toHiggsVec]
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exact Smooth.comp smooth_higgsVecToFin2ℂ (φ.toHiggsVec_smooth)
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lemma comp_smooth (φ : higgsField) :
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∀ i, Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℂ) (fun (x : spaceTime) => (φ x i)) := by
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rw [← smooth_pi_space]
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exact φ.toVec_smooth
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lemma comp_re_smooth (φ : higgsField) (i : Fin 2):
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Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) (reCLM ∘ (fun (x : spaceTime) => (φ x i))) :=
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Smooth.comp (ContinuousLinearMap.smooth reCLM) (φ.comp_smooth i)
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lemma comp_im_smooth (φ : higgsField) (i : Fin 2):
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Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) (imCLM ∘ (fun (x : spaceTime) => (φ x i))) :=
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Smooth.comp (ContinuousLinearMap.smooth imCLM) (φ.comp_smooth i)
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/-- Given a `higgsField`, the map `spaceTime → ℝ` obtained by taking the square norm of the
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higgs vector. -/
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@[simp]
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def normSq (φ : higgsField) : spaceTime → ℝ := fun x => ( ‖φ x‖ ^ 2)
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lemma normSq_expand (φ : higgsField) :
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φ.normSq = fun x => (conj (φ x 0) * (φ x 0) + conj (φ x 1) * (φ x 1) ).re := by
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funext x
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simp only [normSq, add_re, mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
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rw [@norm_sq_eq_inner ℂ]
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simp
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lemma normSq_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) φ.normSq := by
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rw [normSq_expand]
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refine Smooth.add ?_ ?_
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simp only [mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
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refine Smooth.add ?_ ?_
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refine Smooth.smul ?_ ?_
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exact φ.comp_re_smooth 0
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exact φ.comp_re_smooth 0
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refine Smooth.smul ?_ ?_
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exact φ.comp_im_smooth 0
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exact φ.comp_im_smooth 0
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simp only [mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add]
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refine Smooth.add ?_ ?_
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refine Smooth.smul ?_ ?_
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exact φ.comp_re_smooth 1
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exact φ.comp_re_smooth 1
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refine Smooth.smul ?_ ?_
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exact φ.comp_im_smooth 1
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exact φ.comp_im_smooth 1
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lemma normSq_nonneg (φ : higgsField) (x : spaceTime) : 0 ≤ φ.normSq x := by
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simp only [normSq, ge_iff_le, norm_nonneg, pow_nonneg]
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lemma normSq_zero (φ : higgsField) (x : spaceTime) : φ.normSq x = 0 ↔ φ x = 0 := by
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simp only [normSq, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff, norm_eq_zero]
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/-- The Higgs potential of the form `- μ² * |φ|² + λ * |φ|⁴`. -/
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@[simp]
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def potential (φ : higgsField) (μSq lambda : ℝ ) (x : spaceTime) : ℝ :=
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- μSq * φ.normSq x + lambda * φ.normSq x * φ.normSq x
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lemma potential_smooth (φ : higgsField) (μSq lambda : ℝ ) :
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Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) (fun x => φ.potential μSq lambda x) := by
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simp only [potential, normSq, neg_mul]
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exact Smooth.add
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(Smooth.neg (Smooth.smul smooth_const φ.normSq_smooth))
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(Smooth.smul (Smooth.smul smooth_const φ.normSq_smooth) φ.normSq_smooth)
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/-- A higgs field is constant if it is equal for all `x` `y` in `spaceTime`. -/
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def isConst (Φ : higgsField) : Prop := ∀ x y, Φ x = Φ y
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/-- Given a vector `ℂ²` the constant higgs field with value equal to that
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section. -/
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noncomputable def const (φ : higgsVec) : higgsField where
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toFun := fun _ => φ
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contMDiff_toFun := by
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intro x
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rw [Bundle.contMDiffAt_section]
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exact smoothAt_const
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lemma normSq_const (φ : higgsVec) : (const φ).normSq = fun x => (norm φ) ^ 2 := by
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simp only [normSq, const]
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funext x
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simp
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lemma potential_const (φ : higgsVec) (μSq lambda : ℝ ) :
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(const φ).potential μSq lambda = fun x => - μSq * (norm φ) ^ 2 + lambda * (norm φ) ^ 4 := by
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unfold potential
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rw [normSq_const]
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ring_nf
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/-- Given a element `v : ℝ` the `higgsField` `(0, v/√2)`. -/
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def constStd (v : ℝ) : higgsField := const ![0, v/√2]
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lemma normSq_constStd (v : ℝ) : (constStd v).normSq = fun x => v ^ 2 / 2 := by
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simp only [normSq_const, constStd]
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funext x
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rw [@PiLp.norm_sq_eq_of_L2]
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rw [Fin.sum_univ_two]
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simp
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/-- The higgs potential as a function of `v : ℝ` when evaluated on a `constStd` field. -/
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def potentialConstStd (μSq lambda v : ℝ) : ℝ := - μSq /2 * v ^ 2 + lambda /4 * v ^ 4
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lemma potential_constStd (v μSq lambda : ℝ) :
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(constStd v).potential μSq lambda = fun _ => potentialConstStd μSq lambda v := by
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unfold potential potentialConstStd
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rw [normSq_constStd]
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simp only [neg_mul]
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ring_nf
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lemma smooth_potentialConstStd (μSq lambda : ℝ) :
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Smooth 𝓘(ℝ, ℝ) 𝓘(ℝ, ℝ) (fun v => potentialConstStd μSq lambda v) := by
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simp only [potentialConstStd]
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have h1 (v : ℝ) : v ^ 4 = v * v * v * v := by
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ring
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simp [sq, h1]
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refine Smooth.add ?_ ?_
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exact Smooth.smul smooth_const (Smooth.smul smooth_id smooth_id)
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exact Smooth.smul smooth_const
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(Smooth.smul (Smooth.smul (Smooth.smul smooth_id smooth_id) smooth_id) smooth_id)
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lemma deriv_potentialConstStd (μSq lambda v : ℝ) :
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deriv (fun v => potentialConstStd μSq lambda v) v = - μSq * v + lambda * v ^ 3 := by
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simp only [potentialConstStd]
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rw [deriv_add, deriv_mul, deriv_mul, deriv_const, deriv_const, deriv_pow, deriv_pow]
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simp only [zero_mul, Nat.cast_ofNat, Nat.succ_sub_succ_eq_sub, tsub_zero, pow_one, zero_add,
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neg_mul]
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ring
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exact differentiableAt_const _
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exact differentiableAt_pow _
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exact differentiableAt_const _
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exact differentiableAt_pow _
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exact DifferentiableAt.const_mul (differentiableAt_pow _) _
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exact DifferentiableAt.const_mul (differentiableAt_pow _) _
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lemma deriv_potentialConstStd_zero (μSq lambda v : ℝ) (hLam : 0 < lambda)
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(hv : deriv (fun v => potentialConstStd μSq lambda v) v = 0) : v = 0 ∨ v ^ 2 = μSq/lambda:= by
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rw [deriv_potentialConstStd] at hv
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ring_nf at hv
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have h1 : v * (- μSq + lambda * v ^ 2) = 0 := by
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ring_nf
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linear_combination hv
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simp at h1
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cases' h1 with h1 h1
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simp_all
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apply Or.inr
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field_simp
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linear_combination h1
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lemma potentialConstStd_bounded' (μSq lambda v x : ℝ) (hLam : 0 < lambda) :
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potentialConstStd μSq lambda v = x → - μSq ^ 2 / (4 * lambda) ≤ x := by
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simp only [potentialConstStd]
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intro h
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let y := v ^2
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have h1 : lambda / 4 * y * y + (- μSq / 2) * y + (-x) = 0 := by
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simp [y]
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linear_combination h
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rw [quadratic_eq_zero_iff_discrim_eq_sq] at h1
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simp [discrim] at h1
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have h2 : 0 ≤ μSq ^ 2 / 2 ^ 2 + 4 * (lambda / 4) * x := by
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rw [h1]
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exact sq_nonneg (2 * (lambda / 4) * y + -μSq / 2)
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ring_nf at h2
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rw [← neg_le_iff_add_nonneg'] at h2
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have h4 := (div_le_iff' hLam).mpr h2
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ring_nf at h4
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ring_nf
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exact h4
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simp only [ne_eq, div_eq_zero_iff, OfNat.ofNat_ne_zero, or_false]
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exact OrderIso.mulLeft₀.proof_1 lambda hLam
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lemma potentialConstStd_bounded (μSq lambda v : ℝ) (hLam : 0 < lambda) :
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- μSq ^ 2 / (4 * lambda) ≤ potentialConstStd μSq lambda v := by
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apply potentialConstStd_bounded' μSq lambda v (potentialConstStd μSq lambda v) hLam
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rfl
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lemma potentialConstStd_IsMinOn_of_eq_bound (μSq lambda v : ℝ) (hLam : 0 < lambda)
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(hv : potentialConstStd μSq lambda v = - μSq ^ 2 / (4 * lambda)) :
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IsMinOn (potentialConstStd μSq lambda) Set.univ v := by
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rw [isMinOn_univ_iff]
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intro x
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rw [hv]
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exact potentialConstStd_bounded μSq lambda x hLam
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lemma potentialConstStd_vsq_of_eq_bound (μSq lambda v : ℝ) (hLam : 0 < lambda) :
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potentialConstStd μSq lambda v = - μSq ^ 2 / (4 * lambda) ↔ v ^ 2 = μSq / lambda := by
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apply Iff.intro
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intro h
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simp [potentialConstStd] at h
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field_simp at h
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have h1 : (8 * lambda ^ 2) * v ^ 2 * v ^ 2 + (- 16 * μSq * lambda ) * v ^ 2
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+ (8 * μSq ^ 2) = 0 := by
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linear_combination h
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have hd : discrim (8 * lambda ^ 2) (- 16 * μSq * lambda) (8 * μSq ^ 2) = 0 := by
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simp [discrim]
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ring_nf
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rw [quadratic_eq_zero_iff_of_discrim_eq_zero _ hd] at h1
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field_simp at h1 ⊢
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ring_nf at h1
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have hx : 16 * lambda ≠ 0 := by
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simp [hLam]
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exact OrderIso.mulLeft₀.proof_1 lambda hLam
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apply mul_left_cancel₀ hx
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ring_nf
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rw [← h1]
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ring
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simp only [ne_eq, mul_eq_zero, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff, false_or]
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exact OrderIso.mulLeft₀.proof_1 lambda hLam
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intro h
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simp [potentialConstStd, h]
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have hv : v ^ 4 = v^2 * v^2 := by
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ring
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rw [hv, h]
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field_simp
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ring
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lemma potentialConstStd_IsMinOn (μSq lambda v : ℝ) (hLam : 0 < lambda) (hμSq : 0 ≤ μSq) :
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IsMinOn (potentialConstStd μSq lambda) Set.univ v ↔ v ^ 2 = μSq / lambda := by
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apply Iff.intro
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intro h
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have h1 := potentialConstStd_bounded μSq lambda v hLam
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rw [isMinOn_univ_iff] at h
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let vmin := √(μSq / lambda)
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have hvmin : vmin ^ 2 = μSq / lambda := by
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simp [vmin]
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field_simp
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have h2 := h vmin
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have h3 := (potentialConstStd_vsq_of_eq_bound μSq lambda vmin hLam).mpr hvmin
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rw [h3] at h2
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rw [(potentialConstStd_vsq_of_eq_bound μSq lambda v hLam).mp]
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exact (Real.partialOrder.le_antisymm _ _ h1 h2).symm
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intro h
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rw [← potentialConstStd_vsq_of_eq_bound μSq lambda v hLam] at h
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exact potentialConstStd_IsMinOn_of_eq_bound μSq lambda v hLam h
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lemma potentialConstStd_muSq_le_zero_nonneg (μSq lambda v : ℝ) (hLam : 0 < lambda)
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(hμSq : μSq ≤ 0) : 0 ≤ potentialConstStd μSq lambda v := by
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simp [potentialConstStd]
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apply add_nonneg
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field_simp
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refine div_nonneg ?_ (by simp)
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refine neg_nonneg.mpr ?_
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rw [@mul_nonpos_iff]
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simp_all
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apply Or.inr
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exact sq_nonneg v
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rw [mul_nonneg_iff]
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apply Or.inl
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apply And.intro
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refine div_nonneg ?_ (by simp)
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exact le_of_lt hLam
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have hv : v ^ 4 = (v ^ 2) ^ 2 := by ring
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rw [hv]
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exact sq_nonneg (v ^ 2)
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lemma potentialConstStd_zero_muSq_le_zero (μSq lambda v : ℝ) (hLam : 0 < lambda)
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(hμSq : μSq ≤ 0) : potentialConstStd μSq lambda v = 0 ↔ v = 0 := by
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apply Iff.intro
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· intro h
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simp [potentialConstStd] at h
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field_simp at h
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have h1 : v ^ 2 * ((2 * lambda ) * v ^ 2 + (- 4 * μSq )) = 0 := by
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linear_combination h
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simp at h1
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cases' h1 with h1 h1
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exact h1
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have h2 : v ^ 2 = (4 * μSq) / (2 * lambda) := by
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field_simp
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ring_nf
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linear_combination h1
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by_cases hμSqZ : μSq = 0
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rw [hμSqZ] at h2
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simpa using h2
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have h3 : ¬ (0 ≤ 4 * μSq / (2 * lambda)) := by
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rw [div_nonneg_iff]
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simp only [gt_iff_lt, Nat.ofNat_pos, mul_nonneg_iff_of_pos_left]
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rw [not_or]
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apply And.intro
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simp only [not_and, not_le]
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intro hm
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exact (hμSqZ (le_antisymm hμSq hm)).elim
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simp only [not_and, not_le, gt_iff_lt, Nat.ofNat_pos, mul_pos_iff_of_pos_left]
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intro _
|
||
simp_all only [true_or]
|
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rw [← h2] at h3
|
||
refine (h3 ?_).elim
|
||
exact sq_nonneg v
|
||
· intro h
|
||
simp [potentialConstStd, h]
|
||
|
||
|
||
lemma potentialConstStd_IsMinOn_muSq_le_zero (μSq lambda v : ℝ) (hLam : 0 < lambda)
|
||
(hμSq : μSq ≤ 0) :
|
||
IsMinOn (potentialConstStd μSq lambda) Set.univ v ↔ v = 0 := by
|
||
have hx := (potentialConstStd_zero_muSq_le_zero μSq lambda 0 hLam hμSq)
|
||
simp at hx
|
||
apply Iff.intro
|
||
intro h
|
||
rw [isMinOn_univ_iff] at h
|
||
have h1 := potentialConstStd_muSq_le_zero_nonneg μSq lambda v hLam hμSq
|
||
have h2 := h 0
|
||
rw [hx] at h2
|
||
exact (potentialConstStd_zero_muSq_le_zero μSq lambda v hLam hμSq).mp
|
||
(Real.partialOrder.le_antisymm _ _ h1 h2).symm
|
||
intro h
|
||
rw [h, isMinOn_univ_iff, hx]
|
||
intro x
|
||
exact potentialConstStd_muSq_le_zero_nonneg μSq lambda x hLam hμSq
|
||
|
||
|
||
|
||
lemma const_isConst (φ : Fin 2 → ℂ) : (const φ).isConst := by
|
||
intro x _
|
||
simp [const]
|
||
|
||
lemma isConst_iff_exists_const (Φ : higgsField) : Φ.isConst ↔ ∃ φ, Φ = const φ := by
|
||
apply Iff.intro
|
||
intro h
|
||
use Φ 0
|
||
ext x y
|
||
rw [← h x 0]
|
||
rfl
|
||
intro h
|
||
intro x y
|
||
obtain ⟨φ, hφ⟩ := h
|
||
subst hφ
|
||
rfl
|
||
|
||
|
||
end higgsField
|
||
end
|
||
end StandardModel
|