84 lines
2.4 KiB
Text
84 lines
2.4 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 HepLean.AnomalyCancellation.MSSMNu.Basic
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/-!
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# The definition of the solution B₃ and properties thereof
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We define `B₃` 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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[Allanach, Madigan and Tooby-Smith][Allanach:2021yjy]
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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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/-- `B₃` is the charge which is $B-L$ in all families, but with the third
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family of the opposite sign. -/
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def B₃AsCharge : 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 => - 1
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| 1, 1 => - 1
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| 1, 2 => 1
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| 2, 0 => - 1
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| 2, 1 => - 1
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| 2, 2 => 1
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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 => 3
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| 4, 1 => 3
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| 4, 2 => - 3
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| 5, 0 => 3
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| 5, 1 => 3
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| 5, 2 => - 3,
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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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/-- `B₃` as a solution. -/
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def B₃ : MSSMACC.Sols :=
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MSSMACC.AnomalyFreeMk B₃AsCharge (by with_unfolding_all rfl)
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(by with_unfolding_all rfl) (by with_unfolding_all rfl) (by with_unfolding_all rfl)
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(by with_unfolding_all rfl) (by with_unfolding_all rfl)
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lemma B₃_val : B₃.val = B₃AsCharge := by
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rfl
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lemma doublePoint_B₃_B₃ (R : MSSMACC.LinSols) : cubeTriLin B₃.val B₃.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 [B₃_val]
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rw [B₃AsCharge]
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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_neg, neg_mul, Hd_apply,
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Fin.reduceFinMk, Hu_apply]
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have hLin := R.linearSol
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simp only [MSSMACC_numberLinear, MSSMACC_linearACCs, Nat.reduceMul, Fin.isValue,
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Fin.reduceFinMk] at hLin
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have h0 := hLin 0
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have h2 := hLin 2
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simp only [Fin.isValue, Fin.sum_univ_three, Prod.mk_zero_zero, Prod.mk_one_one, LinearMap.coe_mk,
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AddHom.coe_mk] at h0 h2
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linear_combination (norm := ring_nf) 9 * (h0) - 24 * (h2)
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simp only [Fin.isValue, Prod.mk_zero_zero, Prod.mk_one_one, add_add_sub_cancel, add_neg_cancel]
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end MSSMACC
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