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PhysLean/HepLean/AnomalyCancellation/MSSMNu/PlaneY3B3Orthog.lean

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feat: Add results related to MSSM
2024-04-17 16:23:40 -04:00
/-
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.LineY3B3
import HepLean.AnomalyCancellation.MSSMNu.OrthogY3B3
import Mathlib.Tactic.Polyrith
/-!
# Plane Y₃ B₃ and an orthogonal third point
The plane spanned by Y₃, B₃ and third orthogonal point.
# References
- https://arxiv.org/pdf/2107.07926.pdf
-/
universe v u
namespace MSSMACC
open MSSMCharges
open MSSMACCs
open BigOperators
/-- The plane of linear solutions spanned by $Y_3$, $B_3$ and $R$, a point orthogonal
to $Y_3$ and $B_3$. -/
def planeY₃B₃ (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) : MSSMACC.LinSols :=
a • Y₃.1.1 + b • B₃.1.1 + c • R.1
lemma planeY₃B₃_val (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) :
(planeY₃B₃ R a b c).val = a • Y₃.val + b • B₃.val + c • R.val := by
rfl
lemma planeY₃B₃_smul (R : MSSMACC.AnomalyFreePerp) (a b c d : ℚ) :
planeY₃B₃ R (d * a) (d * b) (d * c) = d • planeY₃B₃ R a b c := by
apply ACCSystemLinear.LinSols.ext
change _ = d • (planeY₃B₃ R a b c).val
rw [planeY₃B₃_val, planeY₃B₃_val]
rw [smul_add, smul_add]
rw [smul_smul, smul_smul, smul_smul]
lemma planeY₃B₃_eq (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) (h : a = a' ∧ b = b' ∧ c = c') :
(planeY₃B₃ R a b c) = (planeY₃B₃ R a' b' c') := by
rw [h.1, h.2.1, h.2.2]
lemma planeY₃B₃_val_eq' (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) (hR' : R.val ≠ 0)
(h : (planeY₃B₃ R a b c).val = (planeY₃B₃ R a' b' c').val) :
a = a' ∧ b = b' ∧ c = c' := by
rw [planeY₃B₃_val, planeY₃B₃_val] at h
have h1 := congrArg (fun S => dot (Y₃.val, S)) h
have h2 := congrArg (fun S => dot (B₃.val, S)) h
simp only [ Fin.isValue, ACCSystemCharges.chargesAddCommMonoid_add, Fin.reduceFinMk] at h1 h2
erw [dot.map_add₂, dot.map_add₂] at h1 h2
erw [dot.map_add₂ Y₃.val (a' • Y₃.val + b' • B₃.val) (c' • R.val)] at h1
erw [dot.map_add₂ B₃.val (a' • Y₃.val + b' • B₃.val) (c' • R.val)] at h2
rw [dot.map_add₂] at h1 h2
rw [dot.map_smul₂, dot.map_smul₂, dot.map_smul₂] at h1 h2
rw [dot.map_smul₂, dot.map_smul₂, dot.map_smul₂] at h1 h2
rw [R.perpY₃] at h1
rw [R.perpB₃] at h2
rw [show dot (Y₃.val, Y₃.val) = 216 by rfl] at h1
rw [show dot (B₃.val, B₃.val) = 108 by rfl] at h2
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h1
rw [show dot (B₃.val, Y₃.val) = 108 by rfl] at h2
simp_all
have ha : a = a' := by
linear_combination h1 / 108 + -1 * h2 / 108
have hb : b = b' := by
linear_combination -1 * h1 / 108 + h2 / 54
rw [ha, hb] at h
have h1 := add_left_cancel h
have h1i : c • R.val + (- c') • R.val = 0 := by
rw [h1]
rw [← Module.add_smul]
simp
rw [← Module.add_smul] at h1i
have hR : ∃ i, R.val i ≠ 0 := by
by_contra h
simp at h
have h0 : R.val = 0 := by
funext i
apply h i
exact hR' h0
obtain ⟨i, hi⟩ := hR
have h2 := congrArg (fun S => S i) h1i
change _ = 0 at h2
simp [HSMul.hSMul] at h2
have hc : c + -c' = 0 := by
cases h2 <;> rename_i h2
exact h2
exact (hi h2).elim
have hc : c = c' := by
linear_combination hc
rw [ha, hb, hc]
simp
lemma planeY₃B₃_quad (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) :
accQuad (planeY₃B₃ R a b c).val = c * (2 * a * quadBiLin (Y₃.val, R.val)
+ 2 * b * quadBiLin (B₃.val, R.val) + c * quadBiLin (R.val, R.val)) := by
rw [planeY₃B₃_val]
erw [BiLinearSymm.toHomogeneousQuad_add]
erw [lineY₃B₃Charges_quad]
rw [quadBiLin.toHomogeneousQuad.map_smul]
rw [quadBiLin.map_add₁, quadBiLin.map_smul₁, quadBiLin.map_smul₁]
rw [quadBiLin.map_smul₂, quadBiLin.map_smul₂]
rw [show (BiLinearSymm.toHomogeneousQuad quadBiLin) R.val = quadBiLin (R.val, R.val) by rfl]
ring
lemma planeY₃B₃_cubic (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) :
accCube (planeY₃B₃ R a b c).val = c ^ 2 *
(3 * a * cubeTriLin (R.val, R.val, Y₃.val)
+ 3 * b * cubeTriLin (R.val, R.val, B₃.val) + c * cubeTriLin (R.val, R.val, R.val) ) := by
rw [planeY₃B₃_val]
erw [TriLinearSymm.toCubic_add]
erw [lineY₃B₃Charges_cubic]
erw [lineY₃B₃_doublePoint (c • R.1) a b]
rw [cubeTriLin.toCubic.map_smul]
rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₂]
rw [cubeTriLin.map_add₃, cubeTriLin.map_smul₃, cubeTriLin.map_smul₃]
rw [show (TriLinearSymm.toCubic cubeTriLin) R.val = cubeTriLin (R.val, R.val, R.val) by rfl]
ring
/-- The line in the plane spanned by $Y_3$, $B_3$ and $R$ which is in the quadratic,
as `LinSols`. -/
def lineQuadAFL (R : MSSMACC.AnomalyFreePerp) (c1 c2 c3 : ℚ) : MSSMACC.LinSols :=
planeY₃B₃ R (c2 * quadBiLin (R.val, R.val) - 2 * c3 * quadBiLin (B₃.val, R.val))
(2 * c3 * quadBiLin (Y₃.val, R.val) - c1 * quadBiLin (R.val, R.val))
(2 * c1 * quadBiLin (B₃.val, R.val) - 2 * c2 * quadBiLin (Y₃.val, R.val))
lemma lineQuadAFL_quad (R : MSSMACC.AnomalyFreePerp) (c1 c2 c3 : ℚ) :
accQuad (lineQuadAFL R c1 c2 c3).val = 0 := by
erw [planeY₃B₃_quad]
rw [mul_eq_zero]
apply Or.inr
ring
/-- The line in the plane spanned by $Y_3$, $B_3$ and $R$ which is in the quadratic. -/
def lineQuad (R : MSSMACC.AnomalyFreePerp) (c1 c2 c3 : ℚ) : MSSMACC.QuadSols :=
AnomalyFreeQuadMk' (lineQuadAFL R c1 c2 c3) (lineQuadAFL_quad R c1 c2 c3)
lemma lineQuad_val (R : MSSMACC.AnomalyFreePerp) (c1 c2 c3 : ℚ) :
(lineQuad R c1 c2 c3).val = (planeY₃B₃ R
(c2 * quadBiLin (R.val, R.val) - 2 * c3 * quadBiLin (B₃.val, R.val))
(2 * c3 * quadBiLin (Y₃.val, R.val) - c1 * quadBiLin (R.val, R.val))
(2 * c1 * quadBiLin (B₃.val, R.val) - 2 * c2 * quadBiLin (Y₃.val, R.val))).val := by
rfl
lemma lineQuad_smul (R : MSSMACC.AnomalyFreePerp) (a b c d : ℚ) :
lineQuad R (d * a) (d * b) (d * c) = d • lineQuad R a b c := by
apply ACCSystemQuad.QuadSols.ext
change _ = (d • planeY₃B₃ R _ _ _).val
rw [← planeY₃B₃_smul]
rw [lineQuad_val]
congr 2
ring_nf
/-- A helper function to simplify following expressions. -/
def α₁ (T : MSSMACC.AnomalyFreePerp) : ℚ :=
(3 * cubeTriLin (T.val, T.val, B₃.val) * quadBiLin (T.val, T.val) -
2 * cubeTriLin (T.val, T.val, T.val) * quadBiLin (B₃.val, T.val))
/-- A helper function to simplify following expressions. -/
def α₂ (T : MSSMACC.AnomalyFreePerp) : ℚ :=
(2 * cubeTriLin (T.val, T.val, T.val) * quadBiLin (Y₃.val, T.val) -
3 * cubeTriLin (T.val, T.val, Y₃.val) * quadBiLin (T.val, T.val))
/-- A helper function to simplify following expressions. -/
def α₃ (T : MSSMACC.AnomalyFreePerp) : ℚ :=
6 * ((cubeTriLin (T.val, T.val, Y₃.val)) * quadBiLin (B₃.val, T.val) -
(cubeTriLin (T.val, T.val, B₃.val)) * quadBiLin (Y₃.val, T.val))
lemma lineQuad_cube (R : MSSMACC.AnomalyFreePerp) (c₁ c₂ c₃ : ℚ) :
accCube (lineQuad R c₁ c₂ c₃).val =
- 4 * ( c₁ * quadBiLin (B₃.val, R.val) - c₂ * quadBiLin (Y₃.val, R.val)) ^ 2 *
( α₁ R * c₁ + α₂ R * c₂ + α₃ R * c₃ ) := by
rw [lineQuad_val]
rw [planeY₃B₃_cubic, α₁, α₂, α₃]
ring
/-- The line in the plane spanned by $Y_3$, $B_3$ and $R$ which is in the cubic. -/
def lineCube (R : MSSMACC.AnomalyFreePerp) (a₁ a₂ a₃ : ℚ) :
MSSMACC.LinSols :=
planeY₃B₃ R
(a₂ * cubeTriLin (R.val, R.val, R.val) - 3 * a₃ * cubeTriLin (R.val, R.val, B₃.val))
(3 * a₃ * cubeTriLin (R.val, R.val, Y₃.val) - a₁ * cubeTriLin (R.val, R.val, R.val))
(3 * (a₁ * cubeTriLin (R.val, R.val, B₃.val) - a₂ * cubeTriLin (R.val, R.val, Y₃.val)))
lemma lineCube_smul (R : MSSMACC.AnomalyFreePerp) (a b c d : ℚ) :
lineCube R (d * a) (d * b) (d * c) = d • lineCube R a b c := by
apply ACCSystemLinear.LinSols.ext
change _ = (d • planeY₃B₃ R _ _ _).val
rw [← planeY₃B₃_smul]
change (planeY₃B₃ R _ _ _).val = (planeY₃B₃ R _ _ _).val
congr 2
ring_nf
lemma lineCube_cube (R : MSSMACC.AnomalyFreePerp) (a₁ a₂ a₃ : ℚ) :
accCube (lineCube R a₁ a₂ a₃).val = 0 := by
change accCube (planeY₃B₃ R _ _ _).val = 0
rw [planeY₃B₃_cubic]
ring_nf
lemma lineCube_quad (R : MSSMACC.AnomalyFreePerp) (a₁ a₂ a₃ : ℚ) :
accQuad (lineCube R a₁ a₂ a₃).val =
3 * (a₁ * cubeTriLin (R.val, R.val, B₃.val) - a₂ * cubeTriLin (R.val, R.val, Y₃.val)) *
(α₁ R * a₁ + α₂ R * a₂ + α₃ R * a₃) := by
erw [planeY₃B₃_quad]
rw [α₁, α₂, α₃]
ring
section proj
lemma α₃_proj (T : MSSMACC.Sols) : α₃ (proj T.1.1) =
6 * dot (Y₃.val, B₃.val) ^ 3 * (
cubeTriLin (T.val, T.val, Y₃.val) * quadBiLin (B₃.val, T.val) -
cubeTriLin (T.val, T.val, B₃.val) * quadBiLin (Y₃.val, T.val)) := by
rw [α₃]
rw [cube_proj_proj_Y₃, cube_proj_proj_B₃, quad_B₃_proj, quad_Y₃_proj]
ring
lemma α₂_proj (T : MSSMACC.Sols) : α₂ (proj T.1.1) =
- α₃ (proj T.1.1) * (dot (Y₃.val, T.val) - 2 * dot (B₃.val, T.val)) := by
rw [α₃_proj, α₂]
rw [cube_proj_proj_Y₃, quad_Y₃_proj, quad_proj, cube_proj]
ring
lemma α₁_proj (T : MSSMACC.Sols) : α₁ (proj T.1.1) =
- α₃ (proj T.1.1) * (dot (B₃.val, T.val) - dot (Y₃.val, T.val)) := by
rw [α₃_proj, α₁]
rw [cube_proj_proj_B₃, quad_B₃_proj, quad_proj, cube_proj]
ring
lemma α₁_proj_zero (T : MSSMACC.Sols) (h1 : α₃ (proj T.1.1) = 0) :
α₁ (proj T.1.1) = 0 := by
rw [α₁_proj, h1]
simp
lemma α₂_proj_zero (T : MSSMACC.Sols) (h1 : α₃ (proj T.1.1) = 0) :
α₂ (proj T.1.1) = 0 := by
rw [α₂_proj, h1]
simp
end proj
end MSSMACC
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