2024-04-18 08:59:48 -04:00
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/-
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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.Basic
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import HepLean.AnomalyCancellation.SMNu.Permutations
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/-!
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# ACC system for SM with RHN (without hypercharge).
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2024-04-18 09:06:16 -04:00
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We define the ACC system for the Standard Model (without hypercharge) with right-handed neutrinos.
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2024-04-18 08:59:48 -04:00
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-/
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universe v u
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namespace SMRHN
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open SMνCharges
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open SMνACCs
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open BigOperators
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/-- The ACC system for the SM plus RHN. -/
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@[simps!]
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def SM (n : ℕ) : ACCSystem where
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numberLinear := 3
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linearACCs := fun i =>
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match i with
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| 0 => @accGrav n
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| 1 => accSU2
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| 2 => accSU3
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numberQuadratic := 0
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quadraticACCs := by
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intro i
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exact Fin.elim0 i
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cubicACC := accCube
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namespace SM
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variable {n : ℕ}
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lemma gravSol (S : (SM n).LinSols) : accGrav S.val = 0 := by
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have hS := S.linearSol
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simp at hS
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exact hS 0
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lemma SU2Sol (S : (SM n).LinSols) : accSU2 S.val = 0 := by
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have hS := S.linearSol
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simp at hS
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exact hS 1
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lemma SU3Sol (S : (SM n).LinSols) : accSU3 S.val = 0 := by
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have hS := S.linearSol
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simp at hS
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exact hS 2
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lemma cubeSol (S : (SM n).Sols) : accCube S.val = 0 := by
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exact S.cubicSol
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/-- An element of `charges` which satisfies the linear ACCs
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gives us a element of `LinSols`. -/
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def chargeToLinear (S : (SM n).charges) (hGrav : accGrav S = 0)
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(hSU2 : accSU2 S = 0) (hSU3 : accSU3 S = 0) : (SM n).LinSols :=
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⟨S, 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 => exact hGrav
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| 1 => exact hSU2
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| 2 => exact hSU3⟩
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/-- An element of `LinSols` which satisfies the quadratic ACCs
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gives us a element of `QuadSols`. -/
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def linearToQuad (S : (SM n).LinSols) : (SM n).QuadSols :=
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⟨S, by
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intro i
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exact Fin.elim0 i⟩
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/-- An element of `QuadSols` which satisfies the quadratic ACCs
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gives us a element of `Sols`. -/
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def quadToAF (S : (SM n).QuadSols) (hc : accCube S.val = 0) :
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(SM n).Sols := ⟨S, hc⟩
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/-- An element of `charges` which satisfies the linear and quadratic ACCs
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gives us a element of `QuadSols`. -/
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def chargeToQuad (S : (SM n).charges) (hGrav : accGrav S = 0)
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(hSU2 : accSU2 S = 0) (hSU3 : accSU3 S = 0) :
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(SM n).QuadSols :=
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linearToQuad $ chargeToLinear S hGrav hSU2 hSU3
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/-- An element of `charges` which satisfies the linear, quadratic and cubic ACCs
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gives us a element of `Sols`. -/
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def chargeToAF (S : (SM n).charges) (hGrav : accGrav S = 0) (hSU2 : accSU2 S = 0)
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(hSU3 : accSU3 S = 0) (hc : accCube S = 0) : (SM n).Sols :=
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quadToAF (chargeToQuad S hGrav hSU2 hSU3) hc
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/-- An element of `LinSols` which satisfies the quadratic and cubic ACCs
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gives us a element of `Sols`. -/
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def linearToAF (S : (SM n).LinSols)
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(hc : accCube S.val = 0) : (SM n).Sols :=
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quadToAF (linearToQuad S) hc
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/-- The permutations acting on the ACC system corresponding to the SM with RHN. -/
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def perm (n : ℕ) : ACCSystemGroupAction (SM n) where
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group := permGroup n
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groupInst := inferInstance
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rep := repCharges
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linearInvariant := 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 => exact accGrav_invariant
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| 1 => exact accSU2_invariant
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| 2 => exact accSU3_invariant
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quadInvariant := by
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intro i
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simp at i
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exact Fin.elim0 i
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cubicInvariant := accCube_invariant
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end SM
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end SMRHN
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