ABSTRACT (original article): DRalgo is an algorithmic implementation that constructs an effective, dimensionally reduced, high-temperature field theory for generic models. The corresponding Mathematica package automatically performs the matching to next-to-leading order. This includes two-loop thermal corrections to scalar and Debye masses as well as one-loop thermal corrections to couplings. DRalgo also allows for integrating out additional heavy scalars. Along the way, the package provides leading-order beta functions for general gauge-charges and fermion-families; both in the fundamental and in the effective theory. Finally, the package computes the finite-temperature effective potential within the effective theory. The article explains the theory of the underlying algorithm while introducing the software on a pedagogical level. CITATION (original article): Ekstedt , A , Schicho , P & Tenkanen , T V I (2023) , DRalgo : A package for effective field theory approach for thermal phase transitions, Computer Physics Communications , vol. 288 , 108725. https://doi.org/10.1016/j.cpc.2023.108725
Paclet page: https://resources.wolframcloud.com/PacletRepository/resources/DRalgo/DRalgo/
Paclet page: https://resources.wolframcloud.com/PacletRepository/resources/DRalgo/DRalgo/
Installation
Installation
Please first make sure GroupMath package is installed using the instructions here: https://renatofonseca.net/groupmath
In[]:=
<<GroupMath`
To install this paclet in your Wolfram Language environment, evaluate this code:
In[]:=
PacletInstall["DRalgo/DRalgo"]
Out[]=
PacletObject
To load the code after installation, evaluate this code:
In[]:=
Needs["DRalgo`DRalgo`"]
Basic Example of the Abelian Higgs Model
Basic Example of the Abelian Higgs Model
Model
Model
We consider a simple model with a U(1) gauge group. The adjoint representation is trivial for abelian groups and has Dynkin index {0}. We also include a single complex scalar in the fundamental representation (R= 1).
In[]:=
Group={"U1"};RepAdjoint={0};(*Dynkinindexoftheadjoint*)CouplingName={g1};(*Gaugecoupling*)
One complex scalar (interpreted as a Higgs-like field) is present:
In[]:=
Higgs={{Yϕ},"C"};(*ComplexscalarwithHyperchargeYϕ*)RepScalar={Higgs};(*Representationofthescalarfield*)
Fermions are treated as Weyl spinors in DRalgo. Therefore, to define a Dirac fermion, one must include both a left-handed and a right-handed Weyl spinor. In this minimal model, no fermions are included:
In[]:=
RepFermion={};(*Nofermionsinthemodel*)
We allocate all necessary coupling tensors for the model:
In[]:=
{gvvv,gvff,gvss,λ1,λ3,λ4,μij,μIJ,μIJC,Ysff,YsffC}=AllocateTensors[Group,RepAdjoint,CouplingName,RepFermion,RepScalar];
To build the Lagrangian, we construct the mass term ϕ :
2
m
†
ϕ
In[]:=
InputInv={{1,1},{True,False}};(*Indicatesaϕ†ϕ-typeinvariant*)MassTerm1=CreateInvariant[Group,RepScalar,InputInv]//Simplify//FullSimplify;VMass=msq*MassTerm1[[1]];(*Component-levelexpressionforϕ†ϕ*)μij=GradMass[VMass]//Simplify//SparseArray;
We add the quartic interaction of the complex scalar:
In[]:=
QuarticTerm1=MassTerm1[[1]]^2;(*BecauseMassTerm1=ϕ^+ϕ,wecanwrite(ϕ^+ϕ)^2=MassTerm1^2*)VQuartic=λ*QuarticTerm1;λ4=GradQuartic[VQuartic];
To invoke the model in DRalgo, it needs to be imported:
In[]:=
ImportModelDRalgo[Group,gvvv,gvff,gvss,λ1,λ3,λ4,μij,μIJ,μIJC,Ysff,YsffC,Verbose->True];
Creating Help Tensors
Dimensional reduction from hard to soft scale
Dimensional reduction from hard to soft scale
Integrating out the hard-scale modes in verbose mode:
In[]:=
PerformDRhard[]
Calculating 1-Loop Scalar Mass
Calculating 1-Loop Vector Mass
Calculating 1-Loop Tadpoles
Calculating Scalar-Field Renormalization
Calculating Vector-Field Renormalization
Calculating Scalar-Cubic Couplings
Calculating Scalar Quartic
Calculating Transverse-Vector Couplings
Calculating Scalar-Temporal-Vector Couplings
Calculating Temporal-Vector Quartics
Calculating Temporal-Vector-Scalar Cubics
Calculating CounterTerms
Calculating 2-Loop Debye Mass
Calculating 2-Loop Scalar Mass
Calculating 2-Loop Tadpoles
Calculating Leading-Order Terms
4
T
Calculating NLO Terms
4
T
Calculating NNLO Terms
4
T
Calculating Quartic Tensor
Calculating Cubic Tensor
Calculating Vector-Scalar Interaction Tensor
Calculating Temporal-Scalar Quartics
RG-evolving from μ to μ3
After the reduction completes, the effective couplings are given by:
In[]:=
PrintCouplings[]
Printing 3D vector and quartic couplings in terms of 4D couplings
Out[]=
T-LbT,λ3d
2
g13d
2
g1
4
g1
2
Yϕ
48
2
π
T(2-3Lb)+6Lbλ+2λ(8-5Lbλ)
4
g1
4
Yϕ
2
g1
2
Yϕ
2
π
16
2
π
Internally, a list of constants is used and they can be printed via:
In[]:=
PrintConstants[]
Out[]=
Lb2EulerGamma-2Log[4π]+Log,Lf2EulerGamma+4Log[2]-2Log[4π]+Log
2
μ
2
T
2
μ
2
T
Both the leading order (LO) and next-to-leading order (NLO) scalar masses are given by:
In[]:=
PrintScalarMass["LO"]PrintScalarMass["NLO"]
Printing Scalar Masses
Out[]=
msq3dmsq+3+4λ
1
12
2
T
2
g1
2
Yϕ
Printing Scalar Masses
Out[]=
msq3d12Lb(9msq-6λ)+2λ(1+6EulerGamma-72Log[Glaisher])+24λLb(-6msq+λ)-6λ(EulerGamma-12Log[Glaisher])+(-8-108EulerGamma+69Lb+1296Log[Glaisher])+18Log8-16λ3d+16+
1
576
2
π
2
g1
2
Yϕ
2
T
2
T
2
T
2
T
4
g1
2
T
4
Yϕ
μ3
μ
4
g13d
4
Yϕ
2
g13d
2
Yϕ
2
λ3d
2
λVL[1]
Similarly, the Debye masses are given by:
In[]:=
PrintDebyeMass["LO"]PrintDebyeMass["NLO"]
Printing Debye Masses
Out[]=
μsqU1
1
3
2
g1
2
T
2
Yϕ
Printing Debye Masses
Out[]=
μsqU136msq+7-2EulerGamma+12λ+2Log[4πT]-2Log[μ]
1
144
2
π
2
g1
2
Yϕ
2
g1
2
T
2
Yϕ
2
g1
2
T
2
Yϕ
2
T
2
g1
2
T
2
Yϕ
2
g1
2
T
2
Yϕ
Often one needs the explicit coupling tensors of DRalgo whose indices can be accessed via:
In[]:=
PrintTensorDRalgo[];
Order of Tensors: (1) Scalar Quartic, (2) Vector-Scalar Couplings, (3) Temporal-Vector Scalar Couplings, (4) Temporal-Vector Quartics, (5) Temporal-Vector Mass, (6) Scalar Mass, (7) Scalar Cubics, (8) Temporal Vector-Scalar Cubics, (9) Tadpoles
For a specific tensor, e.g. for the scalars quartics with index "1", we can print it directly:
In[]:=
PrintTensorDRalgo[1]//MatrixForm
Out[]//MatrixForm=
|
| ||||||||
|
|
The hard-scale pressure contribution in the high-temperature expansion is obtained by:
Finally, the β-functions of the model are:
Dimensional reduction from hard to ''ultrasoft'' scale
Dimensional reduction from hard to ''ultrasoft'' scale
Integrate out the soft-scale modes in verbose mode:
After the reduction completes, the effective couplings are given by:
Both the leading order (LO) and next-to-leading order (NLO) scalar masses are given by:
The soft-scale pressure contribution in the high-temperature expansion is obtained by:
The NNLO soft-scale pressure is not yet included in DRalgo:
Finally, the β-functions of the model are:
Effective potential directly
Effective potential directly
First, the theory scale is chosen:
Then, a constant background is introduced to the scalars and the potential is calculated:
Finally, the different orders of the potential can be computed:
Effective potential directly using HEFT
Effective potential directly using HEFT
In this example, we assume that the temporal and vector degrees of freedom are heavy compared to the scalar sector. Thus, we integrate them out starting from the soft theory, ultimately constructing a High-Energy Effective Theory (HEFT).
Start from the soft theory:
We introduce a background field, φ, for the scalars and calculate the effective potential:
Before computing the final potential, we must identify which temporal scalars and gauge bosons are heavy and are to be integrated out. Use PrepareHET[{scalarIndices}, {vectorIndices}] to specify them:
This functions similar to PerformDRsoft[{}], and the indices can be found with the commands
After, calculating the effective potential in the HEFT setup,
finally, the different orders of the potential can be computed :
The U(1) gauge bosons modify the scalar kinetic terms. The renormalization (wavefunction normalization) is encoded in the Z-factor:
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
DRalgo: A package for effective field theory approach for thermal phase transitions
by Andreas Ekstedt, Philipp Schicho & Tuomas V.I. Tenkanen
Wolfram Community, STAFF PICKS, May 27, 2025
https://community.wolfram.com/groups/-/m/t/3477741
by Andreas Ekstedt, Philipp Schicho & Tuomas V.I. Tenkanen
Wolfram Community, STAFF PICKS, May 27, 2025
https://community.wolfram.com/groups/-/m/t/3477741