146 lines
3.7 KiB
Text
146 lines
3.7 KiB
Text
/-
|
||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||
Released under Apache 2.0 license.
|
||
Authors: Joseph Tooby-Smith
|
||
-/
|
||
import HepLean.AnomalyCancellation.SMNu.PlusU1.Basic
|
||
import HepLean.AnomalyCancellation.SMNu.PlusU1.FamilyMaps
|
||
/-!
|
||
# Hypercharge in SM with RHN.
|
||
|
||
Relavent definitions for the SM hypercharge.
|
||
|
||
-/
|
||
universe v u
|
||
|
||
namespace SMRHN
|
||
namespace PlusU1
|
||
|
||
open SMνCharges
|
||
open SMνACCs
|
||
open BigOperators
|
||
|
||
/-- The hypercharge for 1 family. -/
|
||
@[simps!]
|
||
def Y₁ : (PlusU1 1).Sols where
|
||
val := fun i =>
|
||
match i with
|
||
| (0 : Fin 6) => 1
|
||
| (1 : Fin 6) => -4
|
||
| (2 : Fin 6) => 2
|
||
| (3 : Fin 6) => -3
|
||
| (4 : Fin 6) => 6
|
||
| (5 : Fin 6) => 0
|
||
linearSol := by
|
||
intro i
|
||
simp at i
|
||
match i with
|
||
| 0 => rfl
|
||
| 1 => rfl
|
||
| 2 => rfl
|
||
| 3 => rfl
|
||
quadSol := by
|
||
intro i
|
||
simp at i
|
||
match i with
|
||
| 0 => rfl
|
||
cubicSol := by rfl
|
||
|
||
/-- The hypercharge for `n` family. -/
|
||
@[simps!]
|
||
def Y (n : ℕ) : (PlusU1 n).Sols :=
|
||
familyUniversalAF n Y₁
|
||
|
||
namespace Y
|
||
|
||
variable {n : ℕ}
|
||
|
||
lemma on_quadBiLin (S : (PlusU1 n).charges) :
|
||
quadBiLin ((Y n).val, S) = accYY S := by
|
||
erw [familyUniversal_quadBiLin]
|
||
rw [accYY_decomp]
|
||
simp
|
||
ring_nf
|
||
simp
|
||
|
||
lemma on_quadBiLin_AFL (S : (PlusU1 n).LinSols) : quadBiLin ((Y n).val, S.val) = 0 := by
|
||
rw [on_quadBiLin]
|
||
rw [YYsol S]
|
||
|
||
|
||
lemma add_AFL_quad (S : (PlusU1 n).LinSols) (a b : ℚ) :
|
||
accQuad (a • S.val + b • (Y n).val) = a ^ 2 * accQuad S.val := by
|
||
erw [BiLinearSymm.toHomogeneousQuad_add, quadSol (b • (Y n)).1]
|
||
rw [quadBiLin.map_smul₁, quadBiLin.map_smul₂, quadBiLin.swap, on_quadBiLin_AFL]
|
||
erw [accQuad.map_smul]
|
||
simp
|
||
|
||
lemma add_quad (S : (PlusU1 n).QuadSols) (a b : ℚ) :
|
||
accQuad (a • S.val + b • (Y n).val) = 0 := by
|
||
rw [add_AFL_quad, quadSol S]
|
||
simp
|
||
|
||
/-- The `QuadSol` obtained by adding hypercharge to a `QuadSol`. -/
|
||
def addQuad (S : (PlusU1 n).QuadSols) (a b : ℚ) : (PlusU1 n).QuadSols :=
|
||
linearToQuad (a • S.1 + b • (Y n).1.1) (add_quad S a b)
|
||
|
||
lemma addQuad_zero (S : (PlusU1 n).QuadSols) (a : ℚ): addQuad S a 0 = a • S := by
|
||
simp [addQuad, linearToQuad]
|
||
rfl
|
||
|
||
lemma on_cubeTriLin (S : (PlusU1 n).charges) :
|
||
cubeTriLin ((Y n).val, (Y n).val, S) = 6 * accYY S := by
|
||
erw [familyUniversal_cubeTriLin']
|
||
rw [accYY_decomp]
|
||
simp
|
||
ring
|
||
|
||
lemma on_cubeTriLin_AFL (S : (PlusU1 n).LinSols) :
|
||
cubeTriLin ((Y n).val, (Y n).val, S.val) = 0 := by
|
||
rw [on_cubeTriLin]
|
||
rw [YYsol S]
|
||
simp
|
||
|
||
lemma on_cubeTriLin' (S : (PlusU1 n).charges) :
|
||
cubeTriLin ((Y n).val, S, S) = 6 * accQuad S := by
|
||
erw [familyUniversal_cubeTriLin]
|
||
rw [accQuad_decomp]
|
||
simp
|
||
ring_nf
|
||
|
||
lemma on_cubeTriLin'_ALQ (S : (PlusU1 n).QuadSols) :
|
||
cubeTriLin ((Y n).val, S.val, S.val) = 0 := by
|
||
rw [on_cubeTriLin']
|
||
rw [quadSol S]
|
||
simp
|
||
|
||
lemma add_AFL_cube (S : (PlusU1 n).LinSols) (a b : ℚ) :
|
||
accCube (a • S.val + b • (Y n).val) =
|
||
a ^ 2 * (a * accCube S.val + 3 * b * cubeTriLin (S.val, S.val, (Y n).val)) := by
|
||
erw [TriLinearSymm.toCubic_add, cubeSol (b • (Y n)), accCube.map_smul]
|
||
repeat rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₂, cubeTriLin.map_smul₃]
|
||
rw [on_cubeTriLin_AFL]
|
||
simp
|
||
ring
|
||
|
||
lemma add_AFQ_cube (S : (PlusU1 n).QuadSols) (a b : ℚ) :
|
||
accCube (a • S.val + b • (Y n).val) = a ^ 3 * accCube S.val := by
|
||
rw [add_AFL_cube]
|
||
rw [cubeTriLin.swap₃]
|
||
rw [on_cubeTriLin'_ALQ]
|
||
ring
|
||
|
||
lemma add_AF_cube (S : (PlusU1 n).Sols) (a b : ℚ) :
|
||
accCube (a • S.val + b • (Y n).val) = 0 := by
|
||
rw [add_AFQ_cube]
|
||
rw [cubeSol S]
|
||
simp
|
||
|
||
/-- The `Sol` obtained by adding hypercharge to a `Sol`. -/
|
||
def addCube (S : (PlusU1 n).Sols) (a b : ℚ) : (PlusU1 n).Sols :=
|
||
quadToAF (addQuad S.1 a b) (add_AF_cube S a b)
|
||
|
||
|
||
end Y
|
||
end PlusU1
|
||
end SMRHN
|