79 lines
1.9 KiB
Text
79 lines
1.9 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.AnomalyCancellation.MSSMNu.Basic
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/-!
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# The definition of the solution Y₃ and properties thereof
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We define $Y_3$ and show that it is a double point of the cubic.
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# References
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The main reference for the material in this file is:
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- https://arxiv.org/pdf/2107.07926.pdf
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-/
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universe v u
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namespace MSSMACC
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open MSSMCharges
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open MSSMACCs
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open BigOperators
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/-- $Y_3$ is the charge which is hypercharge in all families, but with the third
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family of the opposite sign. -/
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def Y₃AsCharge : MSSMACC.Charges := toSpecies.symm
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⟨fun s => fun i =>
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match s, i with
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| 0, 0 => 1
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| 0, 1 => 1
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| 0, 2 => - 1
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| 1, 0 => -4
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| 1, 1 => -4
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| 1, 2 => 4
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| 2, 0 => 2
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| 2, 1 => 2
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| 2, 2 => - 2
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| 3, 0 => -3
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| 3, 1 => -3
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| 3, 2 => 3
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| 4, 0 => 6
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| 4, 1 => 6
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| 4, 2 => - 6
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| 5, 0 => 0
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| 5, 1 => 0
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| 5, 2 => 0,
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fun s =>
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match s with
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| 0 => -3
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| 1 => 3⟩
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/-- $Y_3$ as a solution. -/
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def Y₃ : MSSMACC.Sols :=
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MSSMACC.AnomalyFreeMk Y₃AsCharge (by rfl) (by rfl) (by rfl) (by rfl) (by rfl) (by rfl)
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lemma Y₃_val : Y₃.val = Y₃AsCharge := by
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rfl
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lemma doublePoint_Y₃_Y₃ (R : MSSMACC.LinSols) :
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cubeTriLin Y₃.val Y₃.val R.val = 0 := by
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simp only [cubeTriLin, TriLinearSymm.mk₃_toFun_apply_apply, cubeTriLinToFun,
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MSSMSpecies_numberCharges, Fin.isValue, Fin.reduceFinMk]
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rw [Fin.sum_univ_three]
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rw [Y₃_val]
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rw [Y₃AsCharge]
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repeat rw [toSMSpecies_toSpecies_inv]
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rw [Hd_toSpecies_inv, Hu_toSpecies_inv]
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simp only [mul_one, Fin.isValue, toSMSpecies_apply, one_mul, mul_neg, neg_mul, neg_neg, mul_zero,
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zero_mul, add_zero, Hd_apply, Fin.reduceFinMk, Hu_apply]
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have hLin := R.linearSol
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simp at hLin
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have h3 := hLin 3
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simp [Fin.sum_univ_three] at h3
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linear_combination 6 * h3
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end MSSMACC
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