GroupMath

Quit[];

<<GroupMath`

RepName[SU2,{2}]
RepName[SU2,{1}]

Initialization

Quit[];

SetDirectory[NotebookDirectory[]];
$LoadGroupMath=True;
$GroupMathMultipleModels=True; (*Put this if you want to create multiple model-files with the same kernel*)
<<DRalgo`

SU2 Higgs Model

(*See 1609.00474 [hep-ph]*)

Model

Using SU(2) in the adjoint representation with Dynkin index {2} and representation R= 3:

Group={"SU2"};
RepAdjoint={{2}};
CouplingName={gw};

Scalars are present in the theory:

HiggsDoublet={{{1}},"C"};
RepScalar={HiggsDoublet};

The input for the gauge interactions to DRalgo are then given by

{gvvv,gvff,gvss,λ1,λ3,λ4,μij,μIJ,μIJC,Ysff,YsffC}=AllocateTensors[Group,RepAdjoint,CouplingName,RepFermion3Gen,RepScalar];

Create group invariant for the scalars

InputInv={{1,1},{True,False}};
MassTerm1=CreateInvariant[Group,RepScalar,InputInv][[1]]//Simplify//FullSimplify

Create mass term for the scalars

VMass=(
	+μSq*MassTerm1
	);

μij=GradMass[VMass]//Simplify//SparseArray;
μij//MatrixForm;

Create quartic term for the scalars

QuarticTerm1=MassTerm1^2; (*[(S*S^+)]^2*)

VQuartic=(
	+λ*QuarticTerm1
	);

λ4=GradQuartic[VQuartic];
λ4//MatrixForm;

Dimensional Reduction

To invoke the model, it needs to be imported:

ImportModelDRalgo[Group,gvvv,gvff,gvss,λ1,λ3,λ4,μij,μIJ,μIJC,Ysff,YsffC,Verbose->True];

Dimensional reduction hard to soft

PerformDRhard[]

PrintCouplings[]
BetaFunctions4D[]

Scalar thermal masses

PrintScalarMass["LO"]
PrintScalarMass["NLO"]

Debye masses

PrintDebyeMass["LO"]
PrintDebyeMass["NLO"]

PrintTemporalScalarCouplings[]

Soft to softer

Dimensional reduction soft to softer

PerformDRsoft[{}];

Scalar effective couplings

PrintCouplingsUS[]//Expand

Scalar effective masses

PrintScalarMassUS["LO"]//Expand
PrintScalarMassUS["NLO"]//Expand

Integrating out SU(2) bosons: Softer to super-soft

Pick theory

(*
Computations are done in the ultrasoft theory
*)
UseUltraSoftTheory[];
(*UseSoftTheory[];*)

Choose format of VEV and diagonalize mass matrix

φvev={0,sphi,0,0}//SparseArray;
DefineVEVS[φvev];
PrintTensorsVEV[];
ScalarMass=PrintTensorsVEV[1]//Normal
VectorMass=PrintTensorsVEV[2]//Normal

ScalarMass//MatrixForm
VectorMass//MatrixForm

Effective potential

Calculate effective potential.
Vector bosons are counted at LO

PrepareHET[{Scalar_indices},{vector_indices}]
indicates which particles are integrated out.
This functions similar to PerformDRsoft[{}], and the indices
can be found with the commands
PrintScalarRepPositions[] and 
PrintGaugeRepPositions[]

PrepareHET[{},{1,2,3}]
CalculatePotentialHET[]

Veff

PrintActionHET["LO"]
PrintActionHET["NLO"]

Kinetic terms

The U(1) gauge bosons also change the scalar fields' kinetic term,
printed below is the Z factor

that is PrintScalarKineticHET[][[i,j]]=δZ^ij; (δ^ij+2 δZ^ij)(Subscript[∇, μ])R[i] ∇^μR[j]

Notice that the kinetic term is different for the Higgs components;
this reflects that φ!=0 breaks the symmetry;
so the Higgs and Goldstone terms become different

PrintScalarKineticHET[]//Normal//MatrixForm

Integrating out SU(2) vector and temporal bosons

Pick theory

(*
Computations are done in the ultrasoft theory
*)
UseSoftTheory[];

φvev={0,0,0,sphi,0,0,0}//SparseArray;
DefineVEVS[φvev];
PrintTensorsVEV[];
ScalarMass=PrintTensorsVEV[1]//Normal
VectorMass=PrintTensorsVEV[2]//Normal

ScalarMass//MatrixForm
VectorMass//MatrixForm

Effective potential

PrepareHET[{5,6,7},{1,2,3}]
CalculatePotentialHET[]

3/12(1)^(3/2)

Veff

PrintActionHET["LO"]
PrintActionHET["NLO"]

Kinetic terms

The U(1) gauge bosons also change the scalar fields' kinetic term,
printed below is the Z factor

that is PrintScalarKineticHET[][[i,j]]=δZ^ij; (δ^ij+2 δZ^ij)(Subscript[∇, μ])R[i] ∇^μR[j]

Notice that the kinetic term is different for the Higgs components;
this reflects that φ!=0 breaks the symmetry;
so the Higgs and Goldstone terms become different

PrintScalarKineticHET[]//Normal//MatrixForm