106 lines
3.9 KiB
Text
106 lines
3.9 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.HiggsBoson.Basic
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/-!
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# The Two Higgs Doublet Model
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The two Higgs doublet model is the standard model plus an additional Higgs doublet.
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Currently this file contains the definition of the 2HDM potential.
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-/
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namespace TwoHDM
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open StandardModel
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open ComplexConjugate
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open HiggsField
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noncomputable section
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/-- The potential of the two Higgs doublet model. -/
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def potential (Φ1 Φ2 : HiggsField) (m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵₄ : ℝ)
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(m₁₂2 𝓵₅ 𝓵₆ 𝓵₇ : ℂ) (x : SpaceTime) : ℝ :=
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m₁₁2 * ‖Φ1‖_H ^ 2 x + m₂₂2 * ‖Φ2‖_H ^ 2 x - (m₁₂2 * ⟪Φ1, Φ2⟫_H x + conj m₁₂2 * ⟪Φ2, Φ1⟫_H x).re
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+ 1/2 * 𝓵₁ * ‖Φ1‖_H ^ 2 x * ‖Φ1‖_H ^ 2 x + 1/2 * 𝓵₂ * ‖Φ2‖_H ^ 2 x * ‖Φ2‖_H ^ 2 x
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+ 𝓵₃ * ‖Φ1‖_H ^ 2 x * ‖Φ2‖_H ^ 2 x
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+ 𝓵₄ * ‖⟪Φ1, Φ2⟫_H x‖ ^ 2 + (1/2 * 𝓵₅ * ⟪Φ1, Φ2⟫_H x ^ 2 + 1/2 * conj 𝓵₅ * ⟪Φ2, Φ1⟫_H x ^ 2).re
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+ (𝓵₆ * ‖Φ1‖_H ^ 2 x * ⟪Φ1, Φ2⟫_H x + conj 𝓵₆ * ‖Φ1‖_H ^ 2 x * ⟪Φ2, Φ1⟫_H x).re
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+ (𝓵₇ * ‖Φ2‖_H ^ 2 x * ⟪Φ1, Φ2⟫_H x + conj 𝓵₇ * ‖Φ2‖_H ^ 2 x * ⟪Φ2, Φ1⟫_H x).re
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namespace potential
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variable (Φ1 Φ2 : HiggsField)
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variable (m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵₄ : ℝ)
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variable (m₁₂2 𝓵₅ 𝓵₆ 𝓵₇ : ℂ)
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/-!
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## Simple properties of the potential
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-/
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/-- Swapping `Φ1` with `Φ2`, and a number of the parameters (with possible conjugation) leads
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to an identical potential. -/
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lemma swap_fields :
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potential Φ1 Φ2 m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵₄ m₁₂2 𝓵₅ 𝓵₆ 𝓵₇
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= potential Φ2 Φ1 m₂₂2 m₁₁2 𝓵₂ 𝓵₁ 𝓵₃ 𝓵₄ (conj m₁₂2) (conj 𝓵₅) (conj 𝓵₇) (conj 𝓵₆) := by
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funext x
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simp only [potential, HiggsField.normSq, Complex.add_re, Complex.mul_re, Complex.conj_re,
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Complex.conj_im, neg_mul, sub_neg_eq_add, one_div, Complex.norm_eq_abs, Complex.inv_re,
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Complex.re_ofNat, Complex.normSq_ofNat, div_self_mul_self', Complex.inv_im, Complex.im_ofNat,
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neg_zero, zero_div, zero_mul, sub_zero, Complex.mul_im, add_zero, mul_neg, Complex.ofReal_pow,
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RingHomCompTriple.comp_apply, RingHom.id_apply]
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ring_nf
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simp only [one_div, add_left_inj, add_right_inj, mul_eq_mul_left_iff]
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apply Or.inl
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rw [HiggsField.innerProd, HiggsField.innerProd, ← InnerProductSpace.conj_symm, Complex.abs_conj]
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/-- If `Φ₂` is zero the potential reduces to the Higgs potential on `Φ₁`. -/
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lemma right_zero : potential Φ1 0 m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵₄ m₁₂2 𝓵₅ 𝓵₆ 𝓵₇ =
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StandardModel.HiggsField.potential Φ1 (- m₁₁2) (𝓵₁/2) := by
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funext x
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simp only [potential, normSq, ContMDiffSection.coe_zero, Pi.zero_apply, norm_zero, ne_eq,
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OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, mul_zero, add_zero, innerProd_right_zero,
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innerProd_left_zero, Complex.zero_re, sub_zero, one_div, Complex.ofReal_pow,
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Complex.ofReal_zero, HiggsField.potential, neg_neg, add_right_inj, mul_eq_mul_right_iff,
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pow_eq_zero_iff, norm_eq_zero, or_self_right]
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ring_nf
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simp only [true_or]
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/-- If `Φ₁` is zero the potential reduces to the Higgs potential on `Φ₂`. -/
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lemma left_zero : potential 0 Φ2 m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵₄ m₁₂2 𝓵₅ 𝓵₆ 𝓵₇ =
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StandardModel.HiggsField.potential Φ2 (- m₂₂2) (𝓵₂/2) := by
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rw [swap_fields, right_zero]
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/-!
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## Potential bounded from below
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-/
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/-! TODO: Prove bounded properties of the 2HDM potential. -/
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/-!
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## Smoothness of the potential
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-/
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/-! TODO: Prove smoothness properties of the 2HDM potential. -/
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/-!
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## Invariance of the potential under gauge transformations
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-/
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/-! TODO: Prove invariance properties of the 2HDM potential. -/
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end potential
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end
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end TwoHDM
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