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feat: Add boundedness props

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jstoobysmith 2024-07-10 12:48:53 -04:00
parent d727d2a293
commit e8ce2119c0
2 changed files with 34 additions and 6 deletions
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16
HepLean/BeyondTheStandardModel/TwoHDM/Basic.lean
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@ -23,8 +23,8 @@ open HiggsField
noncomputable section
/-- The potential of the two Higgs doublet model. -/
def potential (Φ1 Φ2 : HiggsField) (m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵₄ : ℝ)
(m₁₂2 𝓵₅ 𝓵₆ 𝓵₇ : ℂ) (x : SpaceTime) : ℝ :=
def potential (m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵₄ : ℝ)
(m₁₂2 𝓵₅ 𝓵₆ 𝓵₇ : ℂ) (Φ1 Φ2 : HiggsField) (x : SpaceTime) : ℝ :=
m₁₁2 * ‖Φ1‖_H ^ 2 x + m₂₂2 * ‖Φ2‖_H ^ 2 x - (m₁₂2 * ⟪Φ1, Φ2⟫_H x + conj m₁₂2 * ⟪Φ2, Φ1⟫_H x).re
+ 1/2 * 𝓵₁ * ‖Φ1‖_H ^ 2 x * ‖Φ1‖_H ^ 2 x + 1/2 * 𝓵₂ * ‖Φ2‖_H ^ 2 x * ‖Φ2‖_H ^ 2 x
+ 𝓵₃ * ‖Φ1‖_H ^ 2 x * ‖Φ2‖_H ^ 2 x
@ -46,8 +46,8 @@ variable (m₁₂2 𝓵₅ 𝓵₆ 𝓵₇ : ℂ)
/-- Swapping `Φ1` with `Φ2`, and a number of the parameters (with possible conjugation) leads
to an identical potential. -/
lemma swap_fields :
potential Φ1 Φ2 m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵₄ m₁₂2 𝓵₅ 𝓵₆ 𝓵₇
= potential Φ2 Φ1 m₂₂2 m₁₁2 𝓵₂ 𝓵₁ 𝓵₃ 𝓵₄ (conj m₁₂2) (conj 𝓵₅) (conj 𝓵₇) (conj 𝓵₆) := by
potential m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵₄ m₁₂2 𝓵₅ 𝓵₆ 𝓵₇ Φ1 Φ2
= potential m₂₂2 m₁₁2 𝓵₂ 𝓵₁ 𝓵₃ 𝓵₄ (conj m₁₂2) (conj 𝓵₅) (conj 𝓵₇) (conj 𝓵₆) Φ2 Φ1 := by
funext x
simp only [potential, HiggsField.normSq, Complex.add_re, Complex.mul_re, Complex.conj_re,
Complex.conj_im, neg_mul, sub_neg_eq_add, one_div, Complex.norm_eq_abs, Complex.inv_re,
@ -60,7 +60,7 @@ lemma swap_fields :
rw [HiggsField.innerProd, HiggsField.innerProd, ← InnerProductSpace.conj_symm, Complex.abs_conj]
/-- If `Φ₂` is zero the potential reduces to the Higgs potential on `Φ₁`. -/
lemma right_zero : potential Φ1 0 m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵₄ m₁₂2 𝓵₅ 𝓵₆ 𝓵₇ =
lemma right_zero : potential m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵₄ m₁₂2 𝓵₅ 𝓵₆ 𝓵₇ Φ1 0 =
StandardModel.HiggsField.potential (- m₁₁2) (𝓵₁/2) Φ1 := by
funext x
simp only [potential, normSq, ContMDiffSection.coe_zero, Pi.zero_apply, norm_zero, ne_eq,
@ -72,7 +72,7 @@ lemma right_zero : potential Φ1 0 m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃
simp only [true_or]
/-- If `Φ₁` is zero the potential reduces to the Higgs potential on `Φ₂`. -/
lemma left_zero : potential 0 Φ2 m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵₄ m₁₂2 𝓵₅ 𝓵₆ 𝓵₇ =
lemma left_zero : potential m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵₄ m₁₂2 𝓵₅ 𝓵₆ 𝓵₇ 0 Φ2 =
StandardModel.HiggsField.potential (- m₂₂2) (𝓵₂/2) Φ2 := by
rw [swap_fields, right_zero]
@ -84,6 +84,10 @@ lemma left_zero : potential 0 Φ2 m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵
/-! TODO: Prove bounded properties of the 2HDM potential. -/
/-- The proposition on the coefficents for a potential to be bounded. -/
def IsBounded (m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵₄ : ℝ) (m₁₂2 𝓵₅ 𝓵₆ 𝓵₇ : ℂ) : Prop :=
∃ c, ∀ Φ1 Φ2 x, c ≤ potential m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵₄ m₁₂2 𝓵₅ 𝓵₆ 𝓵₇ Φ1 Φ2 x
/-!
## Smoothness of the potential

24
HepLean/StandardModel/HiggsBoson/Potential.lean
View file

@ -50,6 +50,30 @@ lemma potential_smooth (μSq lambda : ℝ) (φ : HiggsField) :
((smooth_const.smul φ.normSq_smooth).smul φ.normSq_smooth)
namespace potential
/-!
## Basic properties
-/
lemma complete_square (μ2 𝓵 : ℝ) (h : 𝓵 ≠ 0) (φ : HiggsField) (x : SpaceTime) :
potential μ2 𝓵 φ x = 𝓵 * (‖φ‖_H ^ 2 x - μ2 / (2 * 𝓵)) ^ 2 - μ2 ^ 2 / (4 * 𝓵) := by
simp only [potential]
field_simp
ring
/-!
## Boundness of the potential
-/
/-- The proposition on the coefficents for a potential to be bounded. -/
def IsBounded (μ2 𝓵 : ℝ) : Prop :=
∃ c, ∀ Φ x, c ≤ potential μ2 𝓵 Φ x
/-! TODO: Show when 𝓵 < 0, the potential is not bounded. -/
section lowerBound
/-!