432 lines
16 KiB
Text
432 lines
16 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 as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import Mathlib.Algebra.QuadraticDiscriminant
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import HepLean.StandardModel.HiggsBoson.PointwiseInnerProd
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import HepLean.Meta.Informal.Basic
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/-!
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# The potential of the Higgs field
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We define the potential of the Higgs field.
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We show that the potential is a smooth function on spacetime.
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-/
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noncomputable section
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namespace StandardModel
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namespace HiggsField
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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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open SpaceTime
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/-!
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## The Higgs potential
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-/
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/-- The parameters of the Higgs potential. -/
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structure Potential where
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/-- The mass-squared of the Higgs boson. -/
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μ2 : ℝ
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/-- The quartic coupling of the Higgs boson. Usually denoted λ. -/
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𝓵 : ℝ
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namespace Potential
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variable (P : Potential)
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/-- The function corresponding to the Higgs potential. -/
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def toFun (φ : HiggsField) (x : SpaceTime) : ℝ :=
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- P.μ2 * ‖φ‖_H ^ 2 x + P.𝓵 * ‖φ‖_H ^ 2 x * ‖φ‖_H ^ 2 x
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/-- The potential is smooth. -/
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lemma toFun_smooth (φ : HiggsField) :
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Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℝ) (fun x => P.toFun φ x) := by
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simp only [toFun, normSq, neg_mul]
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exact (smooth_const.smul φ.normSq_smooth).neg.add
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((smooth_const.smul φ.normSq_smooth).smul φ.normSq_smooth)
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/-- The Higgs potential formed by negating the mass squared and the quartic coupling. -/
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def neg : Potential where
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μ2 := - P.μ2
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𝓵 := - P.𝓵
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@[simp]
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lemma toFun_neg (φ : HiggsField) (x : SpaceTime) : P.neg.toFun φ x = - P.toFun φ x := by
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simp only [toFun, neg, neg_neg, normSq, neg_mul, neg_add_rev]
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ring
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@[simp]
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lemma μ2_neg : P.neg.μ2 = - P.μ2 := by rfl
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@[simp]
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lemma 𝓵_neg : P.neg.𝓵 = - P.𝓵 := by rfl
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/-!
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## Basic properties
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-/
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@[simp]
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lemma toFun_zero (x : SpaceTime) : P.toFun HiggsField.zero x = 0 := by
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simp [toFun, zero, ofReal]
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lemma complete_square (h : P.𝓵 ≠ 0) (φ : HiggsField) (x : SpaceTime) :
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P.toFun φ x = P.𝓵 * (‖φ‖_H ^ 2 x - P.μ2 / (2 * P.𝓵)) ^ 2 - P.μ2 ^ 2 / (4 * P.𝓵) := by
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simp only [toFun]
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field_simp
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ring
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/-- The quadratic equation satisfied by the Higgs potential at a spacetime point `x`. -/
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lemma as_quad (φ : HiggsField) (x : SpaceTime) :
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P.𝓵 * ‖φ‖_H ^ 2 x * ‖φ‖_H ^ 2 x + (- P.μ2) * ‖φ‖_H ^ 2 x + (- P.toFun φ x) = 0 := by
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simp only [normSq, neg_mul, toFun, neg_add_rev, neg_neg]
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ring
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/-- The Higgs potential is zero iff and only if the higgs field is zero, or the
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higgs field has norm-squared `P.μ2 / P.𝓵`, assuming `P.𝓁 = 0`. -/
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lemma toFun_eq_zero_iff (h : P.𝓵 ≠ 0) (φ : HiggsField) (x : SpaceTime) :
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P.toFun φ x = 0 ↔ φ x = 0 ∨ ‖φ‖_H ^ 2 x = P.μ2 / P.𝓵 := by
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refine Iff.intro (fun hV => ?_) (fun hD => ?_)
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· have h1 := P.as_quad φ x
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rw [hV] at h1
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have h2 : ‖φ‖_H ^ 2 x * (P.𝓵 * ‖φ‖_H ^ 2 x + - P.μ2) = 0 := by
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linear_combination h1
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simp only [normSq, mul_eq_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, pow_eq_zero_iff,
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norm_eq_zero] at h2
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cases' h2 with h2 h2
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· simp_all
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· apply Or.inr
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field_simp at h2 ⊢
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ring_nf
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linear_combination h2
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· cases' hD with hD hD
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· simp [toFun, hD]
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· simp only [toFun, neg_mul]
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rw [hD]
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field_simp
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/-!
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## The descriminant
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-/
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/-- The discrimiant of the quadratic equation formed by the Higgs potential. -/
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def quadDiscrim (φ : HiggsField) (x : SpaceTime) : ℝ := discrim P.𝓵 (- P.μ2) (- P.toFun φ x)
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/-- The discriminant of the quadratic formed by the potential is non-negative. -/
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lemma quadDiscrim_nonneg (h : P.𝓵 ≠ 0) (φ : HiggsField) (x : SpaceTime) :
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0 ≤ P.quadDiscrim φ x := by
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have h1 := P.as_quad φ x
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rw [mul_assoc, quadratic_eq_zero_iff_discrim_eq_sq] at h1
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· simp only [h1, ne_eq, quadDiscrim, div_eq_zero_iff, OfNat.ofNat_ne_zero, or_false]
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exact sq_nonneg (2 * P.𝓵 * ‖φ‖_H ^ 2 x + - P.μ2)
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· exact h
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lemma quadDiscrim_eq_sqrt_mul_sqrt (h : P.𝓵 ≠ 0) (φ : HiggsField) (x : SpaceTime) :
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P.quadDiscrim φ x = Real.sqrt (P.quadDiscrim φ x) * Real.sqrt (P.quadDiscrim φ x) :=
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(Real.mul_self_sqrt (P.quadDiscrim_nonneg h φ x)).symm
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lemma quadDiscrim_eq_zero_iff (h : P.𝓵 ≠ 0) (φ : HiggsField) (x : SpaceTime) :
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P.quadDiscrim φ x = 0 ↔ P.toFun φ x = - P.μ2 ^ 2 / (4 * P.𝓵) := by
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rw [quadDiscrim, discrim]
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refine Iff.intro (fun hD => ?_) (fun hV => ?_)
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· field_simp
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linear_combination hD
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· field_simp [hV]
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lemma quadDiscrim_eq_zero_iff_normSq (h : P.𝓵 ≠ 0) (φ : HiggsField) (x : SpaceTime) :
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P.quadDiscrim φ x = 0 ↔ ‖φ‖_H ^ 2 x = P.μ2 / (2 * P.𝓵) := by
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rw [P.quadDiscrim_eq_zero_iff h]
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refine Iff.intro (fun hV => ?_) (fun hF => ?_)
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· have h1 := P.as_quad φ x
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rw [mul_assoc, quadratic_eq_zero_iff_of_discrim_eq_zero h
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((P.quadDiscrim_eq_zero_iff h φ x).mpr hV)] at h1
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simp_rw [h1, neg_neg]
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· rw [toFun, hF]
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field_simp
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ring
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lemma neg_𝓵_quadDiscrim_zero_bound (h : P.𝓵 < 0) (φ : HiggsField) (x : SpaceTime) :
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P.toFun φ x ≤ - P.μ2 ^ 2 / (4 * P.𝓵) := by
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have h1 := P.quadDiscrim_nonneg (ne_of_lt h) φ x
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simp only [quadDiscrim, discrim, even_two, Even.neg_pow] at h1
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ring_nf at h1
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rw [← neg_le_iff_add_nonneg',
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show P.𝓵 * P.toFun φ x * 4 = (- 4 * P.𝓵) * (- P.toFun φ x) by ring] at h1
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have h2 := le_neg_of_le_neg <| (div_le_iff₀' (by linarith : 0 < - 4 * P.𝓵)).mpr h1
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ring_nf at h2 ⊢
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exact h2
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lemma pos_𝓵_quadDiscrim_zero_bound (h : 0 < P.𝓵) (φ : HiggsField) (x : SpaceTime) :
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- P.μ2 ^ 2 / (4 * P.𝓵) ≤ P.toFun φ x := by
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have h1 := P.neg.neg_𝓵_quadDiscrim_zero_bound (by simpa [neg] using h) φ x
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simp only [toFun_neg, μ2_neg, even_two, Even.neg_pow, 𝓵_neg, mul_neg, neg_div_neg_eq] at h1
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rw [neg_le, neg_div'] at h1
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exact h1
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/-- If `P.𝓵` is negative, then if `P.μ2` is greater then zero, for all space-time points,
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the potential is negative `P.toFun φ x ≤ 0`. -/
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lemma neg_𝓵_toFun_neg (h : P.𝓵 < 0) (φ : HiggsField) (x : SpaceTime) :
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(0 < P.μ2 ∧ P.toFun φ x ≤ 0) ∨ P.μ2 ≤ 0 := by
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by_cases hμ2 : P.μ2 ≤ 0
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· simp [hμ2]
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simp only [toFun, normSq, neg_mul, neg_add_le_iff_le_add, add_zero, hμ2, or_false]
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apply And.intro (lt_of_not_ge hμ2)
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have h1 : 0 ≤ P.μ2 * ‖φ x‖ ^ 2 := by
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refine Left.mul_nonneg ?ha ?hb
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· exact le_of_not_ge hμ2
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· exact sq_nonneg ‖φ x‖
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refine le_trans ?_ h1
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exact mul_nonpos_of_nonpos_of_nonneg (mul_nonpos_of_nonpos_of_nonneg (le_of_lt h)
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(sq_nonneg ‖φ x‖)) (sq_nonneg ‖φ x‖)
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/-- If `P.𝓵` is bigger then zero, then if `P.μ2` is less then zero, for all space-time points,
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the potential is positive `0 ≤ P.toFun φ x`. -/
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lemma pos_𝓵_toFun_pos (h : 0 < P.𝓵) (φ : HiggsField) (x : SpaceTime) :
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(P.μ2 < 0 ∧ 0 ≤ P.toFun φ x) ∨ 0 ≤ P.μ2 := by
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simpa using P.neg.neg_𝓵_toFun_neg (by simpa using h) φ x
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lemma neg_𝓵_sol_exists_iff (h𝓵 : P.𝓵 < 0) (c : ℝ) : (∃ φ x, P.toFun φ x = c) ↔ (0 < P.μ2 ∧ c ≤ 0) ∨
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(P.μ2 ≤ 0 ∧ c ≤ - P.μ2 ^ 2 / (4 * P.𝓵)) := by
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refine Iff.intro (fun ⟨φ, x, hV⟩ => ?_) (fun h => ?_)
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· rcases P.neg_𝓵_toFun_neg h𝓵 φ x with hr | hr
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· rw [← hV]
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exact Or.inl hr
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· rw [← hV]
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exact Or.inr (And.intro hr (P.neg_𝓵_quadDiscrim_zero_bound h𝓵 φ x))
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· simp only [toFun, neg_mul]
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simp only [← sub_eq_zero, sub_zero]
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ring_nf
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let a := (P.μ2 - Real.sqrt (discrim P.𝓵 (- P.μ2) (- c))) / (2 * P.𝓵)
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have ha : 0 ≤ a := by
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simp only [discrim, even_two, Even.neg_pow, mul_neg, sub_neg_eq_add, a]
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rw [div_nonneg_iff]
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refine Or.inr (And.intro ?_ ?_)
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· rw [sub_nonpos]
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by_cases hμ : P.μ2 < 0
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· have h1 : 0 ≤ √(P.μ2 ^ 2 + 4 * P.𝓵 * c) := Real.sqrt_nonneg (P.μ2 ^ 2 + 4 * P.𝓵 * c)
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linarith
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· refine Real.le_sqrt_of_sq_le ?_
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rw [le_add_iff_nonneg_right]
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refine mul_nonneg_of_nonpos_of_nonpos ?_ ?_
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· refine mul_nonpos_of_nonneg_of_nonpos ?_ ?_
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· linarith
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· linarith
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· rcases h with h | h
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· linarith
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· have h1 : P.μ2 = 0 := by linarith
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rw [h1] at h
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simpa using h.2
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· linarith
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use (ofReal a)
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use 0
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rw [ofReal_normSq ha]
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trans P.𝓵 * a * a + (- P.μ2) * a + (- c)
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· ring
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have hd : 0 ≤ (discrim P.𝓵 (- P.μ2) (-c)) := by
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simp only [discrim, even_two, Even.neg_pow, mul_neg, sub_neg_eq_add]
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rcases h with h | h
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· refine Left.add_nonneg (sq_nonneg P.μ2) ?_
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refine mul_nonneg_of_nonpos_of_nonpos ?_ h.2
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linarith
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· rw [← @neg_le_iff_add_nonneg']
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rw [← le_div_iff_of_neg']
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· exact h.2
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· linarith
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have hdd : discrim P.𝓵 (- P.μ2) (-c) = Real.sqrt (discrim P.𝓵 (- P.μ2) (-c))
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* Real.sqrt (discrim P.𝓵 (- P.μ2) (-c)) := by
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exact (Real.mul_self_sqrt hd).symm
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rw [mul_assoc]
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refine (quadratic_eq_zero_iff (ne_of_gt h𝓵).symm hdd _).mpr ?_
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simp only [neg_neg, or_true, a]
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lemma pos_𝓵_sol_exists_iff (h𝓵 : 0 < P.𝓵) (c : ℝ) : (∃ φ x, P.toFun φ x = c) ↔ (P.μ2 < 0 ∧ 0 ≤ c) ∨
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(0 ≤ P.μ2 ∧ - P.μ2 ^ 2 / (4 * P.𝓵) ≤ c) := by
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have h1 := P.neg.neg_𝓵_sol_exists_iff (by simpa using h𝓵) (- c)
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simp only [toFun_neg, neg_inj, μ2_neg, Left.neg_pos_iff, Left.neg_nonpos_iff, even_two,
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Even.neg_pow, 𝓵_neg, mul_neg, neg_div_neg_eq] at h1
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rw [neg_le, neg_div'] at h1
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exact h1
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/-!
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## Boundness of the potential
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-/
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/-- The proposition on the coefficents for a potential to be bounded. -/
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def IsBounded : Prop :=
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∃ c, ∀ Φ x, c ≤ P.toFun Φ x
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/-- If the potential is bounded, then `P.𝓵` is non-negative. -/
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lemma isBounded_𝓵_nonneg (h : P.IsBounded) : 0 ≤ P.𝓵 := by
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by_contra hl
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rw [not_le] at hl
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obtain ⟨c, hc⟩ := h
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by_cases hμ : P.μ2 ≤ 0
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· by_cases hcz : c ≤ - P.μ2 ^ 2 / (4 * P.𝓵)
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· have hcm1 : ∃ φ x, P.toFun φ x = c - 1 := by
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rw [P.neg_𝓵_sol_exists_iff hl (c - 1)]
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apply Or.inr
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simp_all
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linarith
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obtain ⟨φ, x, hφ⟩ := hcm1
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have hc2 := hc φ x
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rw [hφ] at hc2
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linarith
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· rw [not_le] at hcz
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have hcm1 : ∃ φ x, P.toFun φ x = - P.μ2 ^ 2 / (4 * P.𝓵) - 1 := by
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rw [P.neg_𝓵_sol_exists_iff hl _]
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apply Or.inr
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simp_all
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obtain ⟨φ, x, hφ⟩ := hcm1
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have hc2 := hc φ x
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rw [hφ] at hc2
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linarith
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· rw [not_le] at hμ
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by_cases hcz : c ≤ 0
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· have hcm1 : ∃ φ x, P.toFun φ x = c - 1 := by
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rw [P.neg_𝓵_sol_exists_iff hl (c - 1)]
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apply Or.inl
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simp_all
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linarith
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obtain ⟨φ, x, hφ⟩ := hcm1
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have hc2 := hc φ x
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rw [hφ] at hc2
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linarith
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· have hcm1 : ∃ φ x, P.toFun φ x = 0 := by
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rw [P.neg_𝓵_sol_exists_iff hl 0]
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apply Or.inl
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simp_all
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obtain ⟨φ, x, hφ⟩ := hcm1
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have hc2 := hc φ x
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rw [hφ] at hc2
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linarith
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/-- If `P.𝓵` is positive, then the potential is bounded. -/
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lemma isBounded_of_𝓵_pos (h : 0 < P.𝓵) : P.IsBounded := by
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simp only [IsBounded]
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have h2 := P.pos_𝓵_quadDiscrim_zero_bound h
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by_contra hn
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simp only [not_exists, not_forall, not_le] at hn
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obtain ⟨φ, x, hx⟩ := hn (-P.μ2 ^ 2 / (4 * P.𝓵))
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have h2' := h2 φ x
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linarith
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informal_lemma isBounded_iff_of_𝓵_zero where
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physics :≈ "When there is no quartic coupling, the potential is bounded iff the mass squared is
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non-positive."
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math :≈ "For `P : Potential` then P.IsBounded if and only if P.μ2 ≤ 0.
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That is to say `- P.μ2 * ‖φ‖_H ^ 2 x` is bounded below if and only if `P.μ2 ≤ 0`."
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deps :≈ [`StandardModel.HiggsField.Potential.IsBounded, `StandardModel.HiggsField.Potential]
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/-!
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## Minimum and maximum
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-/
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lemma eq_zero_iff_of_μSq_nonpos_𝓵_pos (h𝓵 : 0 < P.𝓵) (hμ2 : P.μ2 ≤ 0) (φ : HiggsField)
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(x : SpaceTime) : P.toFun φ x = 0 ↔ φ x = 0 := by
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rw [P.toFun_eq_zero_iff (ne_of_lt h𝓵).symm]
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simp only [or_iff_left_iff_imp]
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intro h
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have h1 := div_nonpos_of_nonpos_of_nonneg hμ2 (le_of_lt h𝓵)
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rw [← h] at h1
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have hx := normSq_nonneg φ x
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have hx' : ‖φ‖_H ^ 2 x = 0 := by linarith
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simpa using hx'
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lemma isMinOn_iff_of_μSq_nonpos_𝓵_pos (h𝓵 : 0 < P.𝓵) (hμ2 : P.μ2 ≤ 0) (φ : HiggsField)
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(x : SpaceTime) : IsMinOn (fun (φ, x) => P.toFun φ x) Set.univ (φ, x)
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↔ P.toFun φ x = 0 := by
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have h1 := P.pos_𝓵_sol_exists_iff h𝓵
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rw [isMinOn_univ_iff]
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simp only [Prod.forall]
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refine Iff.intro (fun h => ?_) (fun h => ?_)
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· have h1' : P.toFun φ x ≤ 0 := by
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simpa using h HiggsField.zero 0
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have h1'' : 0 ≤ P.toFun φ x := by
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have hx := (h1 (P.toFun φ x)).mp ⟨φ, x, rfl⟩
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rcases hx with hx | hx
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· exact hx.2
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· have hμ2' : P.μ2 = 0 := by
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linarith
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simpa [hμ2'] using hx.2
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linarith
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· rw [h]
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intro φ' x'
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have h1' := (h1 (P.toFun φ' x')).mp ⟨φ', x', rfl⟩
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rcases h1' with h1' | h1'
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· exact h1'.2
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· have hμ2' : P.μ2 = 0 := by
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linarith
|
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simpa [hμ2'] using h1'.2
|
||
|
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lemma isMinOn_iff_field_of_μSq_nonpos_𝓵_pos (h𝓵 : 0 < P.𝓵) (hμ2 : P.μ2 ≤ 0) (φ : HiggsField)
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(x : SpaceTime) : IsMinOn (fun (φ, x) => P.toFun φ x) Set.univ (φ, x)
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↔ φ x = 0 := by
|
||
rw [P.isMinOn_iff_of_μSq_nonpos_𝓵_pos h𝓵 hμ2 φ x]
|
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exact P.eq_zero_iff_of_μSq_nonpos_𝓵_pos h𝓵 hμ2 φ x
|
||
|
||
lemma isMinOn_iff_of_μSq_nonneg_𝓵_pos (h𝓵 : 0 < P.𝓵) (hμ2 : 0 ≤ P.μ2) (φ : HiggsField)
|
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(x : SpaceTime) : IsMinOn (fun (φ, x) => P.toFun φ x) Set.univ (φ, x) ↔
|
||
P.toFun φ x = - P.μ2 ^ 2 / (4 * P.𝓵) := by
|
||
have h1 := P.pos_𝓵_sol_exists_iff h𝓵
|
||
simp only [not_lt.mpr hμ2, false_and, hμ2, true_and, false_or] at h1
|
||
rw [isMinOn_univ_iff]
|
||
simp only [Prod.forall]
|
||
refine Iff.intro (fun h => ?_) (fun h => ?_)
|
||
· obtain ⟨φ', x', hφ'⟩ := (h1 (- P.μ2 ^ 2 / (4 * P.𝓵))).mpr (by rfl)
|
||
have h' := h φ' x'
|
||
rw [hφ'] at h'
|
||
have hφ := (h1 (P.toFun φ x)).mp ⟨φ, x, rfl⟩
|
||
linarith
|
||
· intro φ' x'
|
||
rw [h]
|
||
exact (h1 (P.toFun φ' x')).mp ⟨φ', x', rfl⟩
|
||
|
||
lemma isMinOn_iff_field_of_μSq_nonneg_𝓵_pos (h𝓵 : 0 < P.𝓵) (hμ2 : 0 ≤ P.μ2) (φ : HiggsField)
|
||
(x : SpaceTime) : IsMinOn (fun (φ, x) => P.toFun φ x) Set.univ (φ, x) ↔
|
||
‖φ‖_H ^ 2 x = P.μ2 /(2 * P.𝓵) := by
|
||
rw [P.isMinOn_iff_of_μSq_nonneg_𝓵_pos h𝓵 hμ2 φ x, ← P.quadDiscrim_eq_zero_iff_normSq
|
||
(Ne.symm (ne_of_lt h𝓵)), P.quadDiscrim_eq_zero_iff (Ne.symm (ne_of_lt h𝓵))]
|
||
|
||
theorem isMinOn_iff_field_of_𝓵_pos (h𝓵 : 0 < P.𝓵) (φ : HiggsField) (x : SpaceTime) :
|
||
IsMinOn (fun (φ, x) => P.toFun φ x) Set.univ (φ, x) ↔
|
||
(0 ≤ P.μ2 ∧ ‖φ‖_H ^ 2 x = P.μ2 /(2 * P.𝓵)) ∨ (P.μ2 < 0 ∧ φ x = 0) := by
|
||
by_cases hμ2 : 0 ≤ P.μ2
|
||
· simpa [not_lt.mpr hμ2, hμ2] using P.isMinOn_iff_field_of_μSq_nonneg_𝓵_pos h𝓵 hμ2 φ x
|
||
· simpa [hμ2, lt_of_not_ge hμ2] using P.isMinOn_iff_field_of_μSq_nonpos_𝓵_pos h𝓵 (by linarith) φ x
|
||
|
||
lemma isMaxOn_iff_isMinOn_neg (φ : HiggsField) (x : SpaceTime) :
|
||
IsMaxOn (fun (φ, x) => P.toFun φ x) Set.univ (φ, x) ↔
|
||
IsMinOn (fun (φ, x) => P.neg.toFun φ x) Set.univ (φ, x) := by
|
||
simp only [toFun_neg]
|
||
rw [isMaxOn_univ_iff, isMinOn_univ_iff]
|
||
simp_all only [Prod.forall, neg_le_neg_iff]
|
||
|
||
lemma isMaxOn_iff_field_of_𝓵_neg (h𝓵 : P.𝓵 < 0) (φ : HiggsField) (x : SpaceTime) :
|
||
IsMaxOn (fun (φ, x) => P.toFun φ x) Set.univ (φ, x) ↔
|
||
(P.μ2 ≤ 0 ∧ ‖φ‖_H ^ 2 x = P.μ2 /(2 * P.𝓵)) ∨ (0 < P.μ2 ∧ φ x = 0) := by
|
||
rw [P.isMaxOn_iff_isMinOn_neg,
|
||
P.neg.isMinOn_iff_field_of_𝓵_pos (by simpa using h𝓵)]
|
||
simp
|
||
|
||
end Potential
|
||
|
||
end HiggsField
|
||
|
||
end StandardModel
|
||
end
|