117 lines
3.4 KiB
Text
117 lines
3.4 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.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 only [PlusU1_numberLinear] 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 only [PlusU1_numberQuadratic] 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 only [Fin.isValue, BL₁_val, SMνSpecies_numberCharges, toSpecies_apply, one_mul, mul_neg,
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mul_one, neg_mul, sub_neg_eq_add, one_div]
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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, 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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exact Rat.mul_zero (a ^ 2)
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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 only [addQuad, linearToQuad, zero_smul, add_zero]
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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 only [Fin.isValue, BL₁_val, mul_one, SMνSpecies_numberCharges, toSpecies_apply, mul_neg,
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neg_neg, neg_mul]
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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, 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 only [HomogeneousCubic, accCube, TriLinearSymm.toCubic_apply, Fin.isValue,
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add_zero, BL_val, mul_zero]
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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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