feat: Feynman Trees
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3 changed files with 149 additions and 49 deletions
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@ -54,7 +54,7 @@ import HepLean.BeyondTheStandardModel.TwoHDM.GaugeOrbits
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import HepLean.FeynmanDiagrams.Basic
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import HepLean.FeynmanDiagrams.Instances.ComplexScalar
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import HepLean.FeynmanDiagrams.Instances.Phi4
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import HepLean.FeynmanDiagrams.Instances.TwoRealScalar
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import HepLean.FeynmanDiagrams.Instances.TwoComplexScalar
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import HepLean.FeynmanDiagrams.Momentum
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import HepLean.FlavorPhysics.CKMMatrix.Basic
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import HepLean.FlavorPhysics.CKMMatrix.Invariants
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148
HepLean/FeynmanDiagrams/Instances/TwoComplexScalar.lean
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148
HepLean/FeynmanDiagrams/Instances/TwoComplexScalar.lean
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@ -0,0 +1,148 @@
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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 as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.FeynmanDiagrams.Basic
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/-!
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# Feynman rules for a two complex scalar fields
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This file serves as a demonstration of a new approach to Feynman rules.
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-/
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namespace TwoComplexScalar
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open CategoryTheory
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open FeynmanDiagram
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open PreFeynmanRule
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/-- The colors of edges which one can associate with a vertex for a theory
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with two complex scalar fields. -/
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inductive 𝓔 where
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/-- Corresponds to the first complex scalar field flowing out of a vertex. -/
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| complexScalarOut₁ : 𝓔
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/-- Corresponds to the first complex scalar field flowing into a vertex. -/
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| complexScalarIn₁ : 𝓔
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/-- Corresponds to the second complex scalar field flowing out of a vertex. -/
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| complexScalarOut₂ : 𝓔
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/-- Corresponds to the second complex scalar field flowing into a vertex. -/
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| complexScalarIn₂ : 𝓔
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/-- The map taking each color to it's dual, specifying how we can contract edges. -/
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def ξ : 𝓔 → 𝓔
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| 𝓔.complexScalarOut₁ => 𝓔.complexScalarIn₁
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| 𝓔.complexScalarIn₁ => 𝓔.complexScalarOut₁
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| 𝓔.complexScalarOut₂ => 𝓔.complexScalarIn₂
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| 𝓔.complexScalarIn₂ => 𝓔.complexScalarOut₂
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/-- The function `ξ` is an involution. -/
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lemma ξ_involutive : Function.Involutive ξ := by
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intro x
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match x with
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| 𝓔.complexScalarOut₁ => rfl
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| 𝓔.complexScalarIn₁ => rfl
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| 𝓔.complexScalarOut₂ => rfl
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| 𝓔.complexScalarIn₂ => rfl
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/-- The vertices associated with two complex scalars.
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We call this type, the type of vertex colors. -/
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inductive 𝓥 where
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| φ₁φ₁φ₂φ₂ : 𝓥
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| φ₁φ₁φ₁φ₁ : 𝓥
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| φ₂φ₂φ₂φ₂ : 𝓥
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/-- To each vertex, the association of the number of edges. -/
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def 𝓥NoEdges : 𝓥 → ℕ := fun _ => 4
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/-- To each vertex, associates the indexing map of half-edges associated with that edge. -/
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def 𝓥Edges (v : 𝓥) : Fin (𝓥NoEdges v) → 𝓔 :=
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match v with
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| 𝓥.φ₁φ₁φ₂φ₂ => fun i =>
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match i with
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| (0 : Fin 4)=> 𝓔.complexScalarOut₁
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| (1 : Fin 4) => 𝓔.complexScalarIn₁
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| (2 : Fin 4) => 𝓔.complexScalarOut₂
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| (3 : Fin 4) => 𝓔.complexScalarIn₂
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| 𝓥.φ₁φ₁φ₁φ₁ => fun i =>
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match i with
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| (0 : Fin 4)=> 𝓔.complexScalarOut₁
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| (1 : Fin 4) => 𝓔.complexScalarIn₁
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| (2 : Fin 4) => 𝓔.complexScalarOut₁
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| (3 : Fin 4) => 𝓔.complexScalarIn₁
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| 𝓥.φ₂φ₂φ₂φ₂ => fun i =>
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match i with
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| (0 : Fin 4)=> 𝓔.complexScalarOut₂
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| (1 : Fin 4) => 𝓔.complexScalarIn₂
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| (2 : Fin 4) => 𝓔.complexScalarOut₂
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| (3 : Fin 4) => 𝓔.complexScalarIn₂
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/-- A Feynman tree is an similar to a Feynman diagram, except there is an
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order to edges etc. It has a node for each vertex of a Feynman diagram,
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the (disjoint) union of Feynman diagrams, and the joining of two half edges of
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a Feynman diagram.
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To each Feynman tree is associated a Feynman diagram. But more then
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one distinct Feynman tree can lead to the same Feynman diagram. -/
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inductive FeynmanTree : {n : ℕ} → (c : Fin n → 𝓔) → Type where
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| vertex (v : 𝓥) : FeynmanTree (𝓥Edges v)
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| union {n m : ℕ} {c : Fin n → 𝓔} {c1 : Fin m → 𝓔} (t : FeynmanTree c) (t1 : FeynmanTree c1) :
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FeynmanTree (Sum.elim c c1 ∘ finSumFinEquiv.symm)
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| join {n : ℕ} {c : Fin n.succ.succ → 𝓔} : (i : Fin n.succ.succ) →
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(j : Fin n.succ) → (h : c (i.succAbove j) = ξ (c i)) → FeynmanTree c →
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FeynmanTree (c ∘ i.succAbove ∘ j.succAbove)
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namespace FeynmanTree
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/-- The number of nodes in a feynman tree. -/
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def size {n : ℕ} {c : Fin n → 𝓔} : FeynmanTree c → ℕ := fun
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| vertex _ => 1
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| union t1 t2 => t1.size + t2.size + 1
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| join _ _ _ t => t.size + 1
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/-- The number of half-edges associated with a Feynman tree. -/
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def numHalfEdges {n : ℕ} {c : Fin n → 𝓔} : FeynmanTree c → ℕ := fun
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| vertex v => 𝓥NoEdges v
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| union t1 t2 => t1.numHalfEdges + t2.numHalfEdges
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| join _ _ _ t => t.numHalfEdges
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/-- The type of vertices of a Feynman tree. -/
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def vertexType {n : ℕ} {c : Fin n → 𝓔} : FeynmanTree c → Type := fun
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| vertex v => Unit
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| union t1 t2 => Sum t1.vertexType t2.vertexType
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| join _ _ _ t => t.vertexType
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/-- Maps `vertexType` to `𝓥` taking each vertex to it's vertex color. -/
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def vertexTo𝓥 {n : ℕ} {c : Fin n → 𝓔} : (T : FeynmanTree c) → T.vertexType → 𝓥 := fun
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| vertex v => fun _ => v
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| union t1 t2 => Sum.elim t1.vertexTo𝓥 t2.vertexTo𝓥
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| join _ _ _ t => t.vertexTo𝓥
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/-- The type of half edges of a tensor tree. -/
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def halfEdgeType {n : ℕ} {c : Fin n → 𝓔} : FeynmanTree c → Type := fun
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| vertex v => Fin (𝓥NoEdges v)
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| union t1 t2 => Sum t1.halfEdgeType t2.halfEdgeType
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| join _ _ _ t => t.halfEdgeType
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/-- The map taking each half-edge to it's associated vertex. -/
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def halfEdgeToVertex {n : ℕ} {c : Fin n → 𝓔} : (T : FeynmanTree c) → T.halfEdgeType → T.vertexType := fun
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| vertex v => fun _ => ()
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| union t1 t2 => Sum.map t1.halfEdgeToVertex t2.halfEdgeToVertex
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| join _ _ _ t => t.halfEdgeToVertex
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/-- The inclusion of external half-edges into all half-edges. -/
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def inclExternalEdges {n : ℕ} {c : Fin n → 𝓔} : (T : FeynmanTree c) →
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Fin n → T.halfEdgeType := fun
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| vertex v => fun i => i
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| union t1 t2 => Sum.map t1.inclExternalEdges t2.inclExternalEdges ∘ finSumFinEquiv.symm
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| join i j _ t => t.inclExternalEdges ∘ i.succAbove ∘ j.succAbove
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/-- The type of edges of a Feynman tree. -/
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def edgeType {n : ℕ} {c : Fin n → 𝓔} : FeynmanTree c → Type := fun
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| vertex v => Empty
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| union t1 t2 => Sum t1.edgeType t2.edgeType
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| join _ _ _ t => Sum t.edgeType Unit
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end FeynmanTree
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end TwoComplexScalar
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@ -1,48 +0,0 @@
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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 as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.FeynmanDiagrams.Basic
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/-!
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# Feynman rules for a two complex scalar fields
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This file serves as a demonstration of a new approach to Feynman rules.
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-/
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namespace TwoComplexScalar
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open CategoryTheory
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open FeynmanDiagram
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open PreFeynmanRule
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/-- The colors of edges which one can associate with a vertex for a theory
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with two complex scalar fields. -/
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inductive 𝓔 where
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/-- Corresponds to the first complex scalar field flowing out of a vertex. -/
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| complexScalarOut₁ : 𝓔
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/-- Corresponds to the first complex scalar field flowing into a vertex. -/
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| complexScalarIn₁ : 𝓔
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/-- Corresponds to the second complex scalar field flowing out of a vertex. -/
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| complexScalarOut₂ : 𝓔
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/-- Corresponds to the second complex scalar field flowing into a vertex. -/
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| complexScalarIn₂ : 𝓔
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/-- The map taking each color to it's dual, specifying how we can contract edges. -/
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def ξ : 𝓔 → 𝓔
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| 𝓔.complexScalarOut₁ => 𝓔.complexScalarIn₁
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| 𝓔.complexScalarIn₁ => 𝓔.complexScalarOut₁
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| 𝓔.complexScalarOut₂ => 𝓔.complexScalarIn₂
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| 𝓔.complexScalarIn₂ => 𝓔.complexScalarOut₂
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/-- The function `ξ` is an involution. -/
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lemma ξ_involutive : Function.Involutive ξ := by
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intro x
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match x with
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| 𝓔.complexScalarOut₁ => rfl
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| 𝓔.complexScalarIn₁ => rfl
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| 𝓔.complexScalarOut₂ => rfl
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| 𝓔.complexScalarIn₂ => rfl
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end TwoComplexScalar
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