/- Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved. Released under Apache 2.0 license. Authors: Joseph Tooby-Smith -/ import HepLean.SpaceTime.Basic import HepLean.StandardModel.Basic import HepLean.StandardModel.HiggsBoson.TargetSpace import Mathlib.Data.Complex.Exponential import Mathlib.Tactic.Polyrith import Mathlib.Geometry.Manifold.VectorBundle.Basic import Mathlib.Geometry.Manifold.VectorBundle.SmoothSection import Mathlib.Geometry.Manifold.Instances.Real import Mathlib.RepresentationTheory.Basic import Mathlib.Analysis.InnerProductSpace.Basic import Mathlib.Analysis.InnerProductSpace.Adjoint import Mathlib.Geometry.Manifold.ContMDiff.Product import Mathlib.Algebra.QuadraticDiscriminant /-! # The Higgs field This file defines the basic properties for the higgs field in the standard model. ## References - We use conventions given in: https://pdg.lbl.gov/2019/reviews/rpp2019-rev-higgs-boson.pdf -/ universe v u namespace StandardModel noncomputable section open Manifold open Matrix open Complex open ComplexConjugate open spaceTime /-- The trivial vector bundle 𝓡² × ℂ². (TODO: Make associated bundle.) -/ abbrev higgsBundle := Bundle.Trivial spaceTime higgsVec instance : SmoothVectorBundle higgsVec higgsBundle spaceTime.asSmoothManifold := Bundle.Trivial.smoothVectorBundle higgsVec 𝓘(ℝ, spaceTime) /-- A higgs field is a smooth section of the higgs bundle. -/ abbrev higgsField : Type := SmoothSection spaceTime.asSmoothManifold higgsVec higgsBundle instance : NormedAddCommGroup (Fin 2 → ℂ) := by exact Pi.normedAddCommGroup /-- Given a vector `ℂ²` the constant higgs field with value equal to that section. -/ noncomputable def higgsVec.toField (φ : higgsVec) : higgsField where toFun := fun _ ↦ φ contMDiff_toFun := by intro x rw [Bundle.contMDiffAt_section] exact smoothAt_const namespace higgsField open Complex Real /-- Given a `higgsField`, the corresponding map from `spaceTime` to `higgsVec`. -/ def toHiggsVec (φ : higgsField) : spaceTime → higgsVec := φ lemma toHiggsVec_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, higgsVec) φ.toHiggsVec := by intro x0 have h1 := φ.contMDiff x0 rw [Bundle.contMDiffAt_section] at h1 have h2 : (fun x => ((trivializationAt higgsVec (Bundle.Trivial spaceTime higgsVec) x0) { proj := x, snd := φ x }).2) = φ := by rfl simp only [h2] at h1 exact h1 lemma toField_toHiggsVec_apply (φ : higgsField) (x : spaceTime) : (φ.toHiggsVec x).toField x = φ x := rfl lemma higgsVecToFin2ℂ_toHiggsVec (φ : higgsField) : higgsVecToFin2ℂ ∘ φ.toHiggsVec = φ := rfl lemma toVec_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, Fin 2 → ℂ) φ := smooth_higgsVecToFin2ℂ.comp φ.toHiggsVec_smooth lemma apply_smooth (φ : higgsField) : ∀ i, Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℂ) (fun (x : spaceTime) => (φ x i)) := (smooth_pi_space).mp (φ.toVec_smooth) lemma apply_re_smooth (φ : higgsField) (i : Fin 2): Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) (reCLM ∘ (fun (x : spaceTime) => (φ x i))) := (ContinuousLinearMap.smooth reCLM).comp (φ.apply_smooth i) lemma apply_im_smooth (φ : higgsField) (i : Fin 2): Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) (imCLM ∘ (fun (x : spaceTime) => (φ x i))) := (ContinuousLinearMap.smooth imCLM).comp (φ.apply_smooth i) /-- Given two `higgsField`, the map `spaceTime → ℂ` obtained by taking their inner product. -/ def innerProd (φ1 φ2 : higgsField) : spaceTime → ℂ := fun x => ⟪φ1 x, φ2 x⟫_ℂ /-- Given a `higgsField`, the map `spaceTime → ℝ` obtained by taking the square norm of the higgs vector. -/ @[simp] def normSq (φ : higgsField) : spaceTime → ℝ := fun x => ( ‖φ x‖ ^ 2) lemma toHiggsVec_norm (φ : higgsField) (x : spaceTime) : ‖φ x‖ = ‖φ.toHiggsVec x‖ := rfl lemma normSq_expand (φ : higgsField) : φ.normSq = fun x => (conj (φ x 0) * (φ x 0) + conj (φ x 1) * (φ x 1) ).re := by funext x simp [normSq, add_re, mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add, @norm_sq_eq_inner ℂ] lemma normSq_smooth (φ : higgsField) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) φ.normSq := by rw [normSq_expand] refine Smooth.add ?_ ?_ simp only [mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add] refine Smooth.add ?_ ?_ refine Smooth.smul ?_ ?_ exact φ.apply_re_smooth 0 exact φ.apply_re_smooth 0 refine Smooth.smul ?_ ?_ exact φ.apply_im_smooth 0 exact φ.apply_im_smooth 0 simp only [mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add] refine Smooth.add ?_ ?_ refine Smooth.smul ?_ ?_ exact φ.apply_re_smooth 1 exact φ.apply_re_smooth 1 refine Smooth.smul ?_ ?_ exact φ.apply_im_smooth 1 exact φ.apply_im_smooth 1 lemma normSq_nonneg (φ : higgsField) (x : spaceTime) : 0 ≤ φ.normSq x := by simp [normSq, ge_iff_le, norm_nonneg, pow_nonneg] lemma normSq_zero (φ : higgsField) (x : spaceTime) : φ.normSq x = 0 ↔ φ x = 0 := by simp [normSq, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff, norm_eq_zero] /-- The Higgs potential of the form `- μ² * |φ|² + λ * |φ|⁴`. -/ @[simp] def potential (φ : higgsField) (μSq lambda : ℝ ) (x : spaceTime) : ℝ := - μSq * φ.normSq x + lambda * φ.normSq x * φ.normSq x lemma potential_smooth (φ : higgsField) (μSq lambda : ℝ) : Smooth 𝓘(ℝ, spaceTime) 𝓘(ℝ, ℝ) (fun x => φ.potential μSq lambda x) := by simp only [potential, normSq, neg_mul] exact (smooth_const.smul φ.normSq_smooth).neg.add ((smooth_const.smul φ.normSq_smooth).smul φ.normSq_smooth) lemma potential_apply (φ : higgsField) (μSq lambda : ℝ) (x : spaceTime) : (φ.potential μSq lambda) x = higgsVec.potential μSq lambda (φ.toHiggsVec x) := by simp [higgsVec.potential, toHiggsVec_norm] ring /-- A higgs field is constant if it is equal for all `x` `y` in `spaceTime`. -/ def isConst (Φ : higgsField) : Prop := ∀ x y, Φ x = Φ y lemma isConst_of_higgsVec (φ : higgsVec) : φ.toField.isConst := by intro x _ simp [higgsVec.toField] lemma isConst_iff_of_higgsVec (Φ : higgsField) : Φ.isConst ↔ ∃ (φ : higgsVec), Φ = φ.toField := Iff.intro (fun h ↦ ⟨Φ 0, by ext x y; rw [← h x 0]; rfl⟩) (fun ⟨φ, hφ⟩ x y ↦ by subst hφ; rfl) lemma normSq_of_higgsVec (φ : higgsVec) : φ.toField.normSq = fun x => (norm φ) ^ 2 := by funext x simp [normSq, higgsVec.toField] lemma potential_of_higgsVec (φ : higgsVec) (μSq lambda : ℝ) : φ.toField.potential μSq lambda = fun _ => higgsVec.potential μSq lambda φ := by simp [higgsVec.potential] unfold potential rw [normSq_of_higgsVec] ring_nf end higgsField end end StandardModel