feat: Informal lemmas about Higgs bosons

HepLean/BeyondTheStandardModel/TwoHDM/Basic.lean

@@ -142,7 +142,8 @@ lemma left_eq_neg_right : P.toFun Φ1 (- Φ1) =
 -/
 
 /-! TODO: Prove bounded properties of the 2HDM potential.
-  See e.g. https://inspirehep.net/literature/201299. -/
+  See e.g. https://inspirehep.net/literature/201299 and
+  https://arxiv.org/pdf/hep-ph/0605184. -/
 
 /-- The proposition on the coefficents for a potential to be bounded. -/
 def IsBounded : Prop :=

HepLean/BeyondTheStandardModel/TwoHDM/GaugeOrbits.lean

@@ -0,0 +1,35 @@
+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.BeyondTheStandardModel.TwoHDM.Basic
+/-!
+
+# Gauge orbits for the 2HDM
+
+The main reference for material in this section is https://arxiv.org/pdf/hep-ph/0605184.
+
+-/
+
+namespace TwoHDM
+
+informal_definition prodMatrix where
+  math :≈ "For two Higgs fields `Φ₁` and `Φ₂`, the map from space time to 2 x 2 complex matrices
+  defined by ((Φ₁^†Φ₁, Φ₂^†Φ₁), (Φ₁^†Φ₂, Φ₂^†Φ₂)). "
+  ref :≈ "https://arxiv.org/pdf/hep-ph/0605184 eq 3.8."
+  deps :≈ [``StandardModel.HiggsVec, ``SpaceTime]
+
+informal_lemma prodMatrix_positive_semidefinite where
+  math :≈ "For all x in ``SpaceTime, ``prodMatrix at `x` is positive semidefinite."
+  deps :≈ [``prodMatrix, ``SpaceTime]
+
+informal_lemma prodMatrix_hermitian where
+  math :≈ "For all x in ``SpaceTime, ``prodMatrix at `x` is hermitian."
+  deps :≈ [``prodMatrix, ``SpaceTime]
+
+informal_lemma prodMatrix_smooth where
+  math :≈ "The map ``prodMatrix is a smooth function on spacetime."
+  deps :≈ [``prodMatrix]
+
+end TwoHDM