117 lines
3 KiB
Text
117 lines
3 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.SMNu.PlusU1.Basic
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import HepLean.AnomalyCancellation.SMNu.PlusU1.FamilyMaps
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/-!
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# B Minus L in SM with RHN.
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Relavent definitions for the SM `B-L`.
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-/
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universe v u
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namespace SMRHN
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namespace PlusU1
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open SMνCharges
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open SMνACCs
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open BigOperators
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variable {n : ℕ}
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/-- $B - L$ in the 1-family case. -/
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@[simps!]
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def BL₁ : (PlusU1 1).Sols where
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val := fun i =>
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match i with
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| (0 : Fin 6) => 1
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| (1 : Fin 6) => -1
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| (2 : Fin 6) => -1
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| (3 : Fin 6) => -3
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| (4 : Fin 6) => 3
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| (5 : Fin 6) => 3
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linearSol := by
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intro i
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simp at i
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match i with
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| 0 => rfl
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| 1 => rfl
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| 2 => rfl
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| 3 => rfl
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quadSol := by
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intro i
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simp at i
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match i with
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| 0 => rfl
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cubicSol := by rfl
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/-- $B - L$ in the $n$-family case. -/
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@[simps!]
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def BL (n : ℕ) : (PlusU1 n).Sols :=
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familyUniversalAF n BL₁
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namespace BL
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variable {n : ℕ}
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lemma on_quadBiLin (S : (PlusU1 n).charges) :
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quadBiLin ((BL n).val, S) = 1/2 * accYY S + 3/2 * accSU2 S - 2 * accSU3 S := by
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erw [familyUniversal_quadBiLin]
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rw [accYY_decomp, accSU2_decomp, accSU3_decomp]
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simp
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ring
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lemma on_quadBiLin_AFL (S : (PlusU1 n).LinSols) : quadBiLin ((BL n).val, S.val) = 0 := by
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rw [on_quadBiLin]
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rw [YYsol S, SU2Sol S, SU3Sol S]
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simp
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lemma add_AFL_quad (S : (PlusU1 n).LinSols) (a b : ℚ) :
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accQuad (a • S.val + b • (BL n).val) = a ^ 2 * accQuad S.val := by
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erw [BiLinearSymm.toHomogeneousQuad_add, quadSol (b • (BL n)).1]
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rw [quadBiLin.map_smul₁, quadBiLin.map_smul₂, quadBiLin.swap, on_quadBiLin_AFL]
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erw [accQuad.map_smul]
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simp
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lemma add_quad (S : (PlusU1 n).QuadSols) (a b : ℚ) :
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accQuad (a • S.val + b • (BL n).val) = 0 := by
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rw [add_AFL_quad, quadSol S]
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simp
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/-- The `QuadSol` obtained by adding $B-L$ to a `QuadSol`. -/
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def addQuad (S : (PlusU1 n).QuadSols) (a b : ℚ) : (PlusU1 n).QuadSols :=
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linearToQuad (a • S.1 + b • (BL n).1.1) (add_quad S a b)
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lemma addQuad_zero (S : (PlusU1 n).QuadSols) (a : ℚ): addQuad S a 0 = a • S := by
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simp [addQuad, linearToQuad]
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rfl
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lemma on_cubeTriLin (S : (PlusU1 n).charges) :
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cubeTriLin ((BL n).val, (BL n).val, S) = 9 * accGrav S - 24 * accSU3 S := by
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erw [familyUniversal_cubeTriLin']
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rw [accGrav_decomp, accSU3_decomp]
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simp
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ring
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lemma on_cubeTriLin_AFL (S : (PlusU1 n).LinSols) :
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cubeTriLin ((BL n).val, (BL n).val, S.val) = 0 := by
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rw [on_cubeTriLin]
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rw [gravSol S, SU3Sol S]
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simp
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lemma add_AFL_cube (S : (PlusU1 n).LinSols) (a b : ℚ) :
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accCube (a • S.val + b • (BL n).val) =
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a ^ 2 * (a * accCube S.val + 3 * b * cubeTriLin (S.val, S.val, (BL n).val)) := by
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erw [TriLinearSymm.toCubic_add, cubeSol (b • (BL n)), accCube.map_smul]
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repeat rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₂, cubeTriLin.map_smul₃]
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rw [on_cubeTriLin_AFL]
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simp
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ring
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end BL
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end PlusU1
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end SMRHN
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