feat: Add results related to MSSM

HepLean/AnomalyCancellation/MSSMNu/LineY3B3.lean

@@ -0,0 +1,92 @@
+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.AnomalyCancellation.MSSMNu.Basic
+import HepLean.AnomalyCancellation.MSSMNu.Y3
+import HepLean.AnomalyCancellation.MSSMNu.B3
+import Mathlib.Tactic.Polyrith
+/-!
+# The line through B₃ and Y₃
+
+We give properties of lines through `B₃` and `Y₃`. We show that every point on this line
+is a solution to the quadratic  `lineY₃B₃Charges_quad` and a double point of the cubic
+`lineY₃B₃_doublePoint`.
+
+# References
+
+The main reference for the material in this file is:
+
+- https://arxiv.org/pdf/2107.07926.pdf
+
+-/
+
+universe v u
+
+namespace MSSMACC
+open MSSMCharges
+open MSSMACCs
+open BigOperators
+
+/-- The line through $Y_3$ and $B_3$ as `LinSols`. -/
+def lineY₃B₃Charges (a b : ℚ) : MSSMACC.LinSols := a • Y₃.1.1 + b • B₃.1.1
+
+lemma lineY₃B₃Charges_quad (a b : ℚ) : accQuad (lineY₃B₃Charges a b).val = 0 := by
+  change accQuad (a • Y₃.val + b • B₃.val) = 0
+  rw [accQuad]
+  rw [quadBiLin.toHomogeneousQuad_add]
+  rw [quadBiLin.toHomogeneousQuad.map_smul]
+  rw [quadBiLin.toHomogeneousQuad.map_smul]
+  rw [quadBiLin.map_smul₁, quadBiLin.map_smul₂]
+  erw [quadSol Y₃.1, quadSol B₃.1]
+  simp
+  apply Or.inr ∘ Or.inr
+  rfl
+
+lemma lineY₃B₃Charges_cubic (a b : ℚ) : accCube (lineY₃B₃Charges a b).val = 0 := by
+  change accCube (a • Y₃.val + b • B₃.val) = 0
+  rw [accCube]
+  rw [cubeTriLin.toCubic_add]
+  rw [cubeTriLin.toCubic.map_smul]
+  rw [cubeTriLin.toCubic.map_smul]
+  rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₂, cubeTriLin.map_smul₃]
+  rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₂, cubeTriLin.map_smul₃]
+  erw [Y₃.cubicSol,  B₃.cubicSol]
+  rw [show cubeTriLin (Y₃.val, Y₃.val, B₃.val) = 0 by rfl]
+  rw [show cubeTriLin (B₃.val, B₃.val, Y₃.val) = 0 by rfl]
+  simp
+
+/-- The line through $Y_3$ and $B_3$ as `Sols`. -/
+def lineY₃B₃ (a b : ℚ) : MSSMACC.Sols :=
+  AnomalyFreeMk' (lineY₃B₃Charges a b) (lineY₃B₃Charges_quad a b) (lineY₃B₃Charges_cubic a b)
+
+lemma doublePoint_Y₃_B₃ (R : MSSMACC.LinSols) :
+    cubeTriLin (Y₃.val, B₃.val, R.val) = 0 := by
+  rw [← TriLinearSymm.toFun_eq_coe]
+  simp only [cubeTriLin, cubeTriLinToFun, MSSMSpecies_numberCharges]
+  rw [Fin.sum_univ_three]
+  rw [B₃_val, Y₃_val]
+  rw [B₃AsCharge, Y₃AsCharge]
+  repeat rw [toSMSpecies_toSpecies_inv]
+  rw [Hd_toSpecies_inv, Hu_toSpecies_inv]
+  rw [Hd_toSpecies_inv, Hu_toSpecies_inv]
+  simp
+  have hLin := R.linearSol
+  simp at hLin
+  have h1 := hLin 1
+  have h2 := hLin 2
+  have h3 := hLin 3
+  simp [Fin.sum_univ_three] at h1 h2 h3
+  linear_combination -(12 * h2) + 9 * h1 + 3 * h3
+
+lemma lineY₃B₃_doublePoint (R : MSSMACC.LinSols) (a b : ℚ) :
+    cubeTriLin ((lineY₃B₃ a b).val, (lineY₃B₃ a b).val, R.val) = 0 := by
+  change cubeTriLin (a • Y₃.val + b • B₃.val , a • Y₃.val + b • B₃.val, R.val ) = 0
+  rw [cubeTriLin.map_add₂, cubeTriLin.map_add₁, cubeTriLin.map_add₁]
+  repeat rw [cubeTriLin.map_smul₂, cubeTriLin.map_smul₁]
+  rw [doublePoint_B₃_B₃, doublePoint_Y₃_Y₃, doublePoint_Y₃_B₃]
+  rw [cubeTriLin.swap₁, doublePoint_Y₃_B₃]
+  simp
+
+end MSSMACC