DRalgo: a package for effective field theory approach for
thermal phase transitions

Andreas<EMAIL_ADDRESS>Philipp
<EMAIL_ADDRESS>and Tuomas V. I.
<EMAIL_ADDRESS>
${}^{\mbox{\scriptsize a}}$Department of Physics and Astronomy, Uppsala
University,
P.O. Box 256, SE-751 05 Uppsala, Sweden

${}^{\mbox{\scriptsize b}}$II. Institute of Theoretical Physics, Universität
Hamburg, D-22761, Hamburg, Germany

${}^{\mbox{\scriptsize c}}$Deutsches Elektronen-Synchrotron DESY, Notkestr.
85, 22607 Hamburg, Germany

${}^{\mbox{\scriptsize d}}$Department of Physics and Helsinki Institute of
Physics,
P.O. Box 64, FI-00014 University of Helsinki, Finland

${}^{\mbox{\scriptsize e}}$Nordita, KTH Royal Institute of Technology and
Stockholm University,
Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden

${}^{\mbox{\scriptsize f}}$Tsung-Dao Lee Institute & School of Physics and
Astronomy,
Shanghai Jiao Tong University, Shanghai 200240, China

${}^{\mbox{\scriptsize g}}$Shanghai Key Laboratory for Particle Physics and
Cosmology, Key Laboratory for Particle Astrophysics and Cosmology (MOE),
Shanghai Jiao Tong University, Shanghai 200240, China

Abstract

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.

###### Contents

  1. 1 Introduction
    1. 1.1 Prologue: dimensional reduction at next-to-leading-order
  2. 2 Installation and running
    1. 2.1 Installation
    2. 2.2 Model implementation
    3. 2.3 Running the dimensional-reduction algorithm
    4. 2.4 Integrating out temporal scalars
    5. 2.5 Two-loop effective potential
  3. 3 Theory behind the scenes: dimensional reduction of a generic model
    1. 3.1 Lagrangian in the 4d fundamental theory
    2. 3.2 Lagrangian in the 3d effective theory
    3. 3.3 Matching
    4. 3.4 Beta functions and anomalous dimensions
  4. 4 Implementing beyond the Standard Model theories
    1. 4.1 Two-Higgs doublet model with fermions
  5. 5 Miscellaneous features
    1. 5.1 User-options and features
    2. 5.2 Model-treatment in the code
    3. 5.3 Frequently asked questions
  6. 6 Outlook
  7. A Interfacing DRalgo output
    1. A.1 From 3d effective theory to thermodynamics

## Program summary

Program title: Dimensional Reduction algorithm (DRalgo)
Version: 1.0
Developer’s repository link: https://github.com/DR-algo/DRalgo
Licensing provisions: GNU General Public License 3 (GPLv3)
Programming languages: Mathematica
External routines/libraries: GroupMath [57]
Nature of problem: Construction of high-temperature effective field theories
for beyond the Standard Model physics.
Solution method: Matching of $n$-point correlation functions using tensor-
notation of couplings [60, 61, 62, 63, 64].
Restrictions: Mathematica version 12 or above.
Operating system: macOS 11 and higher, Linux Ubuntu 18.04.

##  1 Introduction

The origin of Baryon asymmetry [1] in the universe remains obscure. As such,
much powder has been spent throughout past decades to find a sound explanation
of this asymmetry [2, 3, 4, 5]. Amongst the suggested mechanisms, the one
based on the electroweak phase transition – electroweak baryogenesis – stands
out. While the Standard Model (SM) has no strong first-order transition on its
own [6, 7, 8], its extensions can contain myriads of them. For new field
content to trigger a strong first-order phase transition, their masses have to
be around the electroweak scale, and their interaction with the SM Higgs
cannot be too feeble. Beyond-the-Standard-Model (BSM) theories that exhibit
such transitions provide a direct target for many future-generation colliders
[9, 10]. Furthermore, thermal phase transitions can, perchance, generate
gravitational waves that are observable by next-generation space-based
detectors such as LISA [11], DECIGO [12], Taiji [13] and BBO [14]. These waves
might open a fresh window into the early universe – and the underlying quantum
field theory.

The interplay between BSM phenomenology and gravitational waves is among the
most actively studied topics in the high-energy-physics literature [15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25]. This interplay requires solid
understanding of thermodynamic properties of different models. It has been
long known that determining thermodynamics in theories with non-Abelian gauge
fields is challenging because of the Linde problem [26]. In short, there are
non-perturbative effects arising from massless vector-bosons: these infrared
(IR) effects can only be captured by lattice simulations [27]. Despite this,
leading-order perturbation theory is frequently used as an approximation.
However, it has been pointed out that such leading-order studies contain large
theoretical uncertainties, due to slow convergence of perturbation theory [28,
29, 20, 30].

Dimensional reduction [31, 32] offers a way to overcome these challenges. In
this framework, ultraviolet (UV) modes – related to non-zero Matsubara modes
in the imaginary time formalism – are integrated out. The resulting effective
field theory (EFT) [33, 34] describes IR, or long wavelength zero modes,
living in three spatial dimensions (3d) (cf. also [35, 36, 37]). The 3d EFT
can be simulated on a lattice and hence by-pass the Linde problem [27, 38].
Dimensional reduction can also be viewed as a systematic scheme for thermal
resummations, used purely within the realm of perturbation theory [29]. To
leading order, dimensional reduction is equivalent to resummation via thermal
masses. Concretely, the effective potential, that describes the free energy of
thermal plasma, is commonly given in a schematic form [28]

$\displaystyle V_{{\mbox{\scriptsize eff}}}\simeq
V_{T=0}+V_{{\mbox{\tiny\rm{$T$}}}}+V_{{\mbox{\scriptsize daisy}}}\;.$ (1.1)

Here, $V_{T=0}$ is the Coleman-Weinberg zero-temperature contribution, which
is often augmented with the effective potential at one-loop level. Namely,
$V_{{\mbox{\tiny\rm{$T$}}}}$ is the thermal correction at one-loop (commonly
in the high-temperature expansion) and

$\displaystyle V_{{\mbox{\scriptsize
daisy}}}\simeq-\frac{T}{12\pi}\Big{[}(m^{2}+\Pi_{\mbox{\tiny\rm{$T$}}})^{\frac{3}{2}}-(m^{2})^{\frac{3}{2}}\Big{]}\;.$
(1.2)

Here $m^{2}$ represents a field-dependent mass and
$\Pi_{\mbox{\tiny\rm{$T$}}}\sim g^{2}T^{2}$ is a one-loop thermal correction.
It is the first term in eq. (1.2) that implements the resummation.111 The
second $\sim Tm^{3}$ term in eq. (1.2) removes double counting; the same
contribution is included $V_{\mbox{\tiny\rm{$T$}}}$ but with unresummed mass.
This term is related to the Matsubara zero-mode contribution.  This resummed
mass, for zero Matsubara modes arises from thermal corrections due to non-zero
Matsubara modes. Physically, this corresponds to thermal screening: UV modes
in the plasma screen IR modes. In eq. (1.1) all contributions are evaluated at
one-loop order, which means that only masses are resummed. However, at high
temperatures, the perturbative expansion is controlled by $g$, rather than
$g^{2}$ as in a zero-temperature loop expansion. Dimensional reduction makes
it possible to systematically include higher order resummations, in particular
two-loop thermal masses and one-loop resummation of couplings and fields which
are also affected by thermal screening. Hence, dimensional reduction can be
used to include important next-to-leading order thermal effects. In
particular, renormalization-group (RG) improvements require two-loop
computations at high temperatures [28, 30]; these are straightforward to
implement in the 3d EFT approach [29].

To date there are sundry studies using dimensional reduction in electroweak
theories [37, 29, 33, 36, 35, 21, 20, 23, 25, 39, 40, 41, 42]. For similar
studies in hot QCD, see [43, 44]. Still, many models remain to be studied
within the dimensional-reduction framework [45, 46, 47, 48].

Along with this article, we launch the Mathematica package DRalgo to implement
dimensional reduction automatically for user-defined models. This package
calculates parameters in the 3d EFT. For example, DRalgo calculates two-loop
thermal masses, effective couplings, and beta functions. The package also
allows users to integrate out heavy scalars and temporal (Debye) scalars.
Furthermore, DRalgo can be used to compute the two-loop thermal effective
potential within the effective theory. Subsequently, the 3d EFT matching
relations can be implemented to lattice Monte Carlo simulations. The algorithm
1 illustrates this pipeline.

Algorithm 1 DRalgo algorithm outline in alignment with fig. 2 of [49]. The use
of lattice resources is indicated as an optional path. The functions
PerformDRhard[], PerformDRsoft[] and CalculatePotentialUS[] are part of
DRalgo.

Input: Four-dimensional Lagrangian $\mathcal{L}_{{\mbox{\tiny\rm{4d}}}}$ with
parameters $\\{c_{1},\dots,c_{n}\\}$, temperature $T$, physical parameters,
heavy masses $\bm{M}$

Start: Initialize model

Call PerformDRhard[] {

for all $c_{i}\in\\{c_{1},\dots,c_{n}\\}$ do

Compute 4-dimensional $\beta$-functions $\beta(c_{i})$

Compute $c_{i,{\mbox{\scriptsize 3d}}}(T,\bm{M})$ by integrating out non-zero
Matsubara modes

end for

Compute thermal (Debye) masses $m_{{\mbox{\tiny\rm{D}}},i}(T,\bm{M})$

Compute couplings that involve temporal vector fields

}

Output/Input: Three-dimensional soft Lagrangian
$\mathcal{L}_{{\mbox{\tiny\rm{3d}}}}$ with parameters
$\\{c_{1,{\mbox{\scriptsize 3d}}},\dots,c_{n,{\mbox{\scriptsize 3d}}}\\}$

Call PerformDRsoft[] {

for all $c_{i,{\mbox{\scriptsize 3d}}}\in\\{c_{1,{\mbox{\scriptsize
3d}}},\dots,c_{n,{\mbox{\scriptsize 3d}}}\\}$ do

Compute 3-dimensional $\beta$-functions
$\beta_{{\mbox{\tiny\rm{3d}}}}(c_{i,{\mbox{\scriptsize 3d}}})$

Compute $\bar{c}_{i,{\mbox{\scriptsize 3d}}}(T,c_{1,{\mbox{\scriptsize
3d}}},\dots,c_{n,{\mbox{\scriptsize 3d}}})$ by integrating out massive
temporal scalars

end for

}

Output: Three-dimensional ultrasoft Lagrangian
$\bar{\mathcal{L}}_{{\mbox{\tiny\rm{3d}}}}$ with parameters
$\\{\bar{c}_{1,{\mbox{\scriptsize 3d}}},\dots,\bar{c}_{n,{\mbox{\scriptsize
3d}}}\\}$

if Lattice resources then

Compute lattice continuum relations to construct
$\mathcal{L}^{{\mbox{\tiny\rm{lattice}}}}_{{\mbox{\tiny\rm{3d}}}}$

Monte Carlo simulation

else

Call CalculatePotentialUS[] {

Compute effective potential
$V^{{\mbox{\tiny\rm{3d}}}}_{{\mbox{\tiny\rm{eff}}}}(m_{i,{\mbox{\scriptsize
3d}}}^{2}$) up to two-loops

}

end if

Compute thermodynamic parameters $T_{\rm c}$, $L/T_{\rm c}^{4}$

This package can be applied to models previously studied in the electroweak
and dark-sector phase-transition literature.

The remainder of this article is organised as follows. Section 1.1 briefly
introduces dimensional reduction and matching relations. Section 2 explicates
the front-end of the package, its installation, and gives a tutorial based on
the Abelian-Higgs model. Section 3 reviews theory and computations in the
back-end of the package. Section 4 illustrates additional features by
implementing a two-Higgs doublet model. In section 6, we discuss future
prospects and possible updates of the package. Finally, appendix A displays
the computation of thermodynamical observables using the output of DRalgo for
the Abelian-Higgs model.

###  1.1 Prologue: dimensional reduction at next-to-leading-order

At leading order (LO) scalars obtain familiar one-loop thermal masses, while
temporal (longitudinal) scalars get Debye masses. At next-to-leading order
(NLO) couplings are resummed, and scalars receive two-loop thermal masses.
DRalgo also calculates two-loop Debye masses. Corrections at NLO are
particularly important due to large logarithms. The idea with dimensional
reduction is to render some logarithms ($\ln\frac{\bar{\mu}}{T}$) small by
matching at a high energy ($\bar{\mu}\sim T$). The remaining logarithms can
then be resummed by RG-evolution within the EFT.

Phase transitions often occur below the Debye-mass scale. In such cases
temporal scalars can also be integrated out. Thus effectively removing large
logarithms of the form $\ln\frac{m_{\mbox{\tiny\rm{D}}}}{\bar{\mu}}$, where
$m_{\mbox{\tiny\rm{D}}}$ is a Debye mass. This second EFT is said to live at
the ultrasoft scale, which is characterised by energies of
$\mathcal{O}(g^{2}T)$. In summary,222 Some literature [33] interchangeably
refers to the hard scale as superheavy, the soft scale as heavy, and the
ultrasoft scale as light.  the hard scale corresponds to energies $E\sim T$;
the soft scale corresponds to energies $E\sim gT$; and the ultrasoft scale
corresponds to energies $E\sim g^{2}T$.

To illustrate next-to-leading order dimensional reduction, we consider a
schematic model with scalar mass parameter $\mu^{2}$, scalar quartic coupling
$\lambda$, and gauge coupling $g$. Given the power counting $\mu^{2}\sim
g^{2}T^{2}$, $\lambda\sim g^{2}$, the matching of the mass parameter is

$\displaystyle\bar{\mu}^{2}_{3}$
$\displaystyle=\underset{\mathcal{O}(g^{2})}{\framebox{$\hbox{\pagecolor{gray!10}$\overset{\text{tree-
level}}{\mu^{2}}$}+\hbox{\pagecolor{gray!10}$\overset{\text{1-loop}}{\\#g^{2}T^{2}}$}$}}+\underset{\mathcal{O}(g^{4})}{\framebox{$\hbox{\pagecolor{gray!10}$\overset{\text{1-loop}}{\\#g^{2}\mu^{2}}$}+\hbox{\pagecolor{gray!10}$\overset{\text{2-loop}}{\\#g^{4}T^{2}}$}$}}+\mathcal{O}(g^{6})$
$\displaystyle+\underset{\mathcal{O}(g^{3})}{\framebox{$\hbox{\pagecolor{gray!10}$\overset{\text{1-loop}}{\\#g^{2}m_{\mbox{\tiny\rm{D}}}}$}$}}+\underset{\mathcal{O}(g^{4})}{\framebox{$\hbox{\pagecolor{gray!10}$\overset{\text{2-loop}}{\\#g^{4}}$}$}}+\mathcal{O}(g^{5})\;,$
(1.3)

where the first line (with even powers of $g$) results from the first step,
and the second line (with odd power of $g$) from second step of the
dimensional reduction. In practice, full $\mathcal{O}(g^{4})$ contributions
are included. Going to higher orders, requires a three-loop computation for
both steps of the dimensional reduction. The situation is similar for the
coupling:

$\displaystyle\bar{\lambda}_{3}$
$\displaystyle=\underset{\mathcal{O}(g^{2})}{\framebox{$\hbox{\pagecolor{gray!10}$\overset{\text{tree-
level}}{T\lambda\vphantom{g^{4}}}$}$}}+\underset{\mathcal{O}(g^{4})}{\framebox{$\hbox{\pagecolor{gray!10}$\overset{\text{1-loop}}{\\#g^{4}}$}$}}+\mathcal{O}(g^{6})$
$\displaystyle+\underset{\mathcal{O}(g^{3})}{\framebox{$\hbox{\pagecolor{gray!10}$\overset{\text{1-loop}}{\\#\frac{g^{4}}{m_{\mbox{\tiny\rm{D}}}}}$}$}}+\underset{\mathcal{O}(g^{4})}{\framebox{$\hbox{\pagecolor{gray!10}$\overset{\text{2-loop}}{\\#\frac{g^{6}}{m_{\mbox{\tiny\rm{D}}}^{2}}}$}$}}+\mathcal{O}(g^{5})\;.$
(1.4)

In practice for the coupling, $\mathcal{O}(g^{4})$ pieces are neglected since
their numerical effect is small (despite being formally of the same order).
These contributions arise during the second step of the dimensional reduction
at two-loop. Pursuing higher order, requires a two-loop computation for the
first step, and a three-loop computation for the second step of the
dimensional reduction.

In perturbation theory the definition of 3d EFT parameters are accompanied
with a perturbative computation of the effective potential [29]:

$\displaystyle V^{\text{3d}}_{\text{eff}}=\underbrace{V^{{\mbox{\scriptsize
3d}}}_{{\mbox{\scriptsize
tree}}}}_{\mathcal{O}(g^{2})}+\underbrace{V^{{\mbox{\scriptsize
3d}}}_{{\mbox{\scriptsize
1-loop}}}}_{\mathcal{O}(g^{3})}+\underbrace{V^{{\mbox{\scriptsize
3d}}}_{{\mbox{\scriptsize
2-loop}}}}_{\mathcal{O}(g^{4})}+\mathcal{O}(g^{5})\;.$ (1.5)

Instead of expanding the result in terms of 4d parameters of the parent
theory, resummed couplings and masses are kept along. This improves the
overall convergence as the result is less sensitive to the renormalization
scale [50, 51]. The order $\mathcal{O}(g^{5})$ requires a computation at
three-loop level [52, 53, 54].

#### Dimensional reduction from a zero-temperature EFT perspective

It is instructive to consider the same physics from a standard EFT
perspective. In particular, for equilibrium observables we can, analogous to
Kaluza-Klein theories [55], view thermal corrections as an infinite tower of
heavy particles. To see the connection with dimensional reduction, we consider
a theory with one light and one heavy scalar in Euclidean spacetime:

$\displaystyle\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}+\frac{1}{2}(\partial_{\mu}\Phi)^{2}+\frac{1}{2}m^{2}\phi^{2}+\frac{1}{2}M^{2}\Phi^{2}+\frac{1}{4}\kappa\phi^{2}\Phi^{2}+\frac{1}{4!}\lambda\phi^{4}\;.$
(1.6)

Imagine now that there exists a hierarchy $M^{2}\gg m^{2}$. This is a well-
known situation, and when calculating scattering processes one encounters
large corrections to $m^{2}$ scaling as $\kappa M^{2}$ – analogous to the
hierarchy problem. As well as large logarithms in the form
$\ln\frac{\bar{\mu}^{2}}{M^{2}}$ or $\ln\frac{\bar{\mu}^{2}}{m^{2}}$. One of
these logarithms will then be large regardless the choice of the RG-scale
$\bar{\mu}$.

Using thermal masses is equivalent to using a resummation333 This can also be
viewed as a temperature-dependent renormalization
$\delta_{\mbox{\tiny\rm{$T$}}}m^{2}$ such that
$m^{2}=m^{2}+\delta_{\mbox{\tiny\rm{$T$}}}m^{2}$ [23].  $m^{2}\to
m^{2}+a\kappa M^{2}$. This resummation does, however, not solve our problems
of large logarithms. Therefore, calculations, even with thermal masses, are
sensitive to the RG-scale. The situation is much improved by using an EFT.
Therein, the logarithms $\ln\frac{\bar{\mu}^{2}}{M^{2}}$ are rendered small by
matching at the scale $\bar{\mu}_{\mbox{\tiny\rm{Match}}}\sim M$, and the
remaining logarithms $\ln\frac{\bar{\mu}^{2}}{m^{2}}$ are taken care of by RG-
evolution in the low-energy theory [56].

Once the scalar field $\Phi$ has been integrated out, the resulting EFT is of
the form

$\displaystyle\mathcal{L}_{\text{eff}}=\frac{1}{2}(\partial_{\mu}\phi)^{2}+\frac{1}{2}m^{2}_{\mbox{\scriptsize
eff}}\,\phi^{2}+\frac{1}{4!}\lambda_{\mbox{\scriptsize eff}}\,\phi^{4}\;.$
(1.7)

To leading order $m_{\mbox{\scriptsize eff}}^{2}=m^{2}+a\kappa M^{2}$, and
$\lambda_{\mbox{\scriptsize eff}}=\lambda$. While at NLO

$\displaystyle\delta m_{\text{eff}}^{2}$
$\displaystyle=\kappa^{2}M^{2}\Bigl{(}b+c\ln\frac{\bar{\mu}^{2}_{\mbox{\tiny\rm{Match}}}}{M^{2}}\Bigr{)}\;,$
(1.8) $\displaystyle\delta\lambda_{\mbox{\scriptsize eff}}$
$\displaystyle=\kappa^{2}\Bigl{(}d+e\ln\frac{\bar{\mu}^{2}_{\mbox{\tiny\rm{Match}}}}{M^{2}}\Bigr{)}\;,$
(1.9)

where $b,c,d,e$ are numerical coefficients. It is precisely these kind of
corrections that appear in the dimensionally reduced theory.

Finally, to do calculations in the IR, we should run the couplings within the
EFT from $\bar{\mu}_{\mbox{\tiny\rm{Match}}}\sim M$ down to
$\bar{\mu}_{\mbox{\tiny\rm{IR}}}\sim m_{\mbox{\scriptsize eff}}$. In this way,
all large logarithms are eliminated and perturbative convergence is improved.

In the case of dimensional reduction, we have an infinite tower of heavy
particles with masses $M_{n}^{2}=(2\pi nT)^{2}$ for all integers $n\neq 0$.444
Fermions have masses of $M_{n}=(2n+1)\pi T$.  Hence, the high-temperature
matching scale is $\mu_{\mbox{\tiny\rm{Match}}}\sim T$ and the low-energy
scale is $\mu_{\mbox{\tiny\rm{IR}}}^{2}\sim g^{2}T^{2}$. Since we are
describing equilibrium (static) processes, there is no time dependence – the
degrees of freedom of the EFT are static and live in three dimensions.

##  2 Installation and running

This section explains how to install DRalgo and presents a tutorial based on
the Abelian-Higgs model.

###  2.1 Installation

The current version of DRalgo, 1.0, is installed by placing all the source
files either in the applications folder Mathematica/Applications or running
Mathematica from the package root directory ./DRalgo.

The required source files are outlined in fig. 1.

.1 DRalgo. .2  DRalgo.m package file. .2 Debye.m Creates and organizes
temporal scalars. .2 EffPot.m Computation of the effective potential in the
EFT. .2 HardToSoft.m Matching hard to soft. .2 SoftToUS.m Matching soft to
ultrasoft. .2 ModelCreation.m Creating a model. .2 examples/Model example
library. .3  ah.mAbelian-Higgs model. .3 sm.mStandard Model. .3 xsm.mStandard
Model and one real scalars. .3 2xsm.mStandard Model and two real scalars. .3
cxsm.mStandard Model and one complex scalar. .3  2hdm.mTwo-Higgs Doublet
model. .3 3hdm.mThree-Higgs Doublet model. .3 htm.mStandard Model and one real
SU(2) triplet. .3 SU5.mSU(5) model. .3 LRSymmetric.mLeft-Right symmetric
model. .3 WessZumino.mWess-Zumino model. .3 smZp.mStandard Model and
$Z^{\prime}$-bosons. .2 README.md. .2 LICENSE.

Figure 1: Outline of the structure of DRalgo. The most relevant files for this
article are highlighted. Specific models are discussed in sec. 2 for ah.m and
sec. 4 for 2hdm.m.

To create model files, DRalgo uses functions from GroupMath [57]; see ibid.
for its installation. This tutorial uses GroupMath Version 1.1.2 but GroupMath
is not required for the dimensional reduction itself. If a model file is
already available, it can be loaded independently of GroupMath (cf. sec. Q.1).
Since GroupMath is an external package, any use of the model-creation features
in DRalgo should be accompanied by a citation of [57].

To load DRalgo from the Mathematica applications folder
Mathematica/Applications, the following commands need to be executed:

⬇

SetDirectory[NotebookDirectory[]];

$LoadGroupMath=True;

<<DRalgo@!\backtick@!

Subsequently, we demonstrate the definition of models, the dimensional
reduction, and the calculation of the two-loop effective potential. All steps
are mirrored in the accompanying ./examples/ah.m.

###  2.2 Model implementation

As an example, we consider the Abelian-Higgs model with the Euclidean
Lagrangian

$\displaystyle\mathcal{L}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\left(D^{\mu}\phi\right)^{\dagger}\left(D_{\mu}\phi\right)+V(\phi,\phi^{\dagger})\;,$
(2.1) $\displaystyle
V(\phi,\phi^{\dagger})=m^{2}\phi^{\dagger}\phi+\lambda\bigl{(}\phi^{\dagger}\phi\bigr{)}^{2}\;,$
(2.2)

where the covariant derivative is
$D_{\mu}=\partial_{\mu}-ig_{1}Y_{\phi}A_{\mu}$ and the corresponding field-
strength tensor is $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$.
Here, the field $A_{\mu}$ is an $\mathrm{U}(1)$ gauge field with gauge
coupling $g_{1}$ and $\phi$ is a complex scalar field charged under
$\mathrm{U}(1)$ with general charge $Y_{\phi}$.

The zero, or temporal, component of the vector field receives a thermally
induced mass in the 3d theory. This temporal scalar transforms as a singlet
under $\mathrm{U}(1)$ and has a Lagrangian

$\displaystyle\mathcal{L}_{{\mbox{\scriptsize temporal}}}$
$\displaystyle=\frac{1}{2}(\partial_{i}A_{0})^{2}+\frac{1}{2}{\tt\mu
sqU1}A^{2}_{0}+\frac{1}{4!}\lambda\texttt{VLL[1]}A^{4}_{0}+\frac{1}{2}\lambda\texttt{VL[1]}A^{2}_{0}\phi^{\dagger}\phi\;,$
(2.3)

where indices $i\in\\{1,\dots,d\\}$ are spatial. Here, $\mu$sqU1 is the
temporal-scalar Debye mass (squared), $\lambda$VLL[1] is the self-interaction
coupling, and $\lambda$VL[1] is the portal coupling to $\phi$. The notation
above is aligned with the systematic notation of DRalgo. For earlier studies
of this model in the literature, we refer to [37, 36, 35, 58, 59].

To create a model in DRalgo, we have to specify the gauge group and the scalar
representation. In DRalgo this is done via

⬇

Group={"U1"};

CouplingName={g1};

RepAdjoint={0};

Higgs={{Y$\phi$},"C"};

RepScalar={Higgs};

RepFermion={};

Here, the adjoint (vector) representation is trivial for the Abelian Higgs
model. In the definition of the Higgs scalar `{{Y`$\phi$`},"C<R>"}`,
$Y_{\phi}$ denotes the gauge charge. One additional argument is passed
depending on if the scalar has a real (R) or a complex (C) representation. In
this simple example, fermions are absent and the corresponding bracket is
empty. While not relevant for this model, note that fermions are never part of
the 3d EFT Lagrangian as they are integrated out during the first step of the
dimensional reduction. The model-input from the gauge sector is generated with
the command

⬇

{gvvv,gvff,gvss,$\lambda$1,$\lambda$3,$\lambda$4,$\mu$ij,$\mu$IJ,$\mu$IJC,Ysff,YsffC}=

AllocateTensors[Group,RepAdjoint,CouplingName,RepFermion,RepScalar];

The tensors listed above have the following correspondence

gvvv | structure constants
---|---
gvff | vector-fermion trilinear couplings
gvss | vector-scalar trilinear couplings
$\lambda$1 | scalar tadpole couplings
$\lambda$3 | cubic couplings
$\lambda$4 | quartic couplings
$\mu$ij | scalar-mass matrix
$\mu$IJ, $\mu$IJC | fermion-mass matrices
Ysff, YsffC | Yukawa couplings

and their purpose is described in depth in sec. 3. By default all these
tensors, except for the gauge ones, are empty. While their order on the left
hand side of AllocateTensors is fixed, the above naming of `{gvvv,...,YsffC}`
is arbitrary.

The next task is to specify the scalar potential. Let us start with the mass
matrix. There is only one allowed term: $\phi\phi^{\dagger}$. This term is
selected via

⬇

InputInv={{1,1},{True,False}};

MassTerm1=CreateInvariant[Group,RepScalar,InputInv][[1]]//Simplify;

VMass=msq*MassTerm1; (* corresponds to a term
$\frac{1}{2}m^{2}\phi\phi^{\dagger}$ *)

$\mu$ij=GradMass[VMass];

To understand the syntax, recall that we defined our scalar as
`RepScalar={Higgs}`. Then `{1,1}` specifies that we are after a term with two
(RepScalar[[1]]) $\phi$ fields. For general models, the index 1 can be
replaced by any index in RepScalar as defined by the user. In addition,
`{True,False}` specifies that we are after the $\phi\phi^{\dagger}$ term and
conversely `{True,True}` corresponds to a $\phi\phi$ term. The latter term,
however, is not allowed in this example due to gauge invariance. Since this
example exhibits merely one $\mathrm{U}(1)$ invariant combination of $\phi$
and $\phi^{\dagger}$, we specify [[1]] in the CreateInvariant output.

For the quartic tensor only one possible term arises, namely
$(\phi\phi^{\dagger})^{2}$. We already created a $(\phi\phi^{\dagger})$ term
above, which can be reused to find the quartic:

⬇

VQuartic=$\lambda$*MassTerm1^2;

$\lambda$4=GradQuartic[VQuartic]

To see how quartic and mass terms are composed of scalar-field components, one
can inspect the variables VMass and VQuartic.

This completes the model implementation, and we now have all ingredients to do
the dimensional-reduction step.

###  2.3 Running the dimensional-reduction algorithm

First we need to load the model with the command

⬇

ImportModelDRalgo[Group,gvvv,gvff,gvss,$\lambda$1,$\lambda$3,$\lambda$4,$\mu$ij,$\mu$IJ,$\mu$IJC,Ysff,YsffC,Verbose->False];

Since the gauge group was previously defined, the corresponding Debye masses
are automatically generated and named. The option Verbose->False disables all
progress messages.

The actual dimensional reduction is performed with the command

⬇

PerformDRhard[];

This calculates all thermal masses, effective couplings, $\beta$-functions and
anomalous dimensions. For example, gauge and scalar couplings are given by

⬇

PrintCouplings[]

$\par\Bigl{\\{}\text{g13d}^{2}\to
g^{2}_{1}T-\frac{g^{4}_{1}\text{Lb}TY^{2}_{\phi}}{48\pi^{2}},\;\lambda\text{3d}\to\frac{T(g^{4}_{1}Y^{4}_{\phi}(2-3\text{Lb})+6g^{2}_{1}Y^{2}_{\phi}\lambda\text{Lb}+2\lambda(8\pi^{2}-5\lambda\text{Lb}))}{16\pi^{2}}\Bigr{\\}}$

where the output is a replacement rule. The variables Lb and Lf are dependent
on the 4d-renormalization scale ($\bar{\mu}$) and the temperature ($T$) [33,
29]. They are

$\displaystyle{\tt Lb}$
$\displaystyle=\ln\Bigl{(}\frac{\bar{\mu}^{2}}{T^{2}}\Bigr{)}+2{\gamma_{\mbox{\tiny\rm{E}}}}-2\ln(4\pi)\;,\quad{\tt
Lf}={\tt Lb}+4\ln(2)\;,$ (2.4)

where ${\gamma_{\mbox{\tiny\rm{E}}}}$ is the Euler-Mascheroni constant.
Further information about the functional basis of the effective parameters is
collected in sec. Q.11. The corresponding definitions of constants, along with
other shorthand notations, are shown with the command

⬇

PrintConstants[]

The above RG-scale $\bar{\mu}$ is the hard matching scale, and should be
chosen as $\bar{\mu}\sim T$ to avoid large logarithms.555 In DRalgo the
renormalization scales are denoted by ${\tt\mu}=\bar{\mu}$ for the hard scale
and ${\tt\mu 3}=\bar{\mu}_{3}$ for the soft scale.  Effective mass parameters
are divided into scalar and temporal-scalars. The scalar masses are given by

⬇

PrintScalarMass["LO"]

PrintScalarMass["NLO"]

Here, the LO command prints tree-level masses together with one-loop thermal
contributions; the NLO command prints two-loop thermal masses. By default
these NLO masses contain the running from the matching scale ($\bar{\mu}$) to
an arbitrary 3d scale ($\bar{\mu}_{3}$).666 There are different ways to
express the NLO masses. We refer to sec. Q.11 for further details.  The
resulting Debye masses are displayed by

⬇

PrintDebyeMass["LO"]

PrintDebyeMass["NLO"]

Mixed temporal-scalar couplings (to NLO) are printed with

⬇

PrintTemporalScalarCouplings[]

Such temporal-scalar couplings are denoted as $\lambda$`VLL[a]` for $V^{4}$
couplings, $\lambda$`VVSL[a]` for $V^{2}S$ couplings and $\lambda$`VL[a]` for
$V^{2}S^{2}$ couplings, where $S$ represents scalar fields and $V$ temporal
scalar fields. The index ‘a’ systematically labels all linearly independent
terms.

The coefficient of the unit operator [34], or the hard-mode contribution to
the symmetric-phase pressure, is given by

⬇

PrintPressure["LO"]

PrintPressure["NLO"]

PrintPressure["NNLO"]

Here, LO corresponds to one-loop, NLO to two-loop, and NNLO to three-loop
level. The LO result describes the pressure of an ideal gas, and is given by
$P_{{\mbox{\tiny\rm{LO}}}}(T)=N\frac{\pi^{2}}{90}T^{4}$, where $N$ represents
the number of degrees of freedom. For the Abelian Higgs model $N=4$ with 2
from the photon and 2 from the complex scalar.

DRalgo also provides anomalous dimensions and beta functions in the parent 4d
theory; see sec. 3.4. Beta functions are printed with the command

⬇

BetaFunctions4D[]

Next, to find anomalous dimensions we first need to specify for which
particles we want them. As DRalgo stores everything in tensor form, we need to
find the positions of all particles

⬇

PosScalar=PrintScalarRepPositions[];

To determine the anomalous dimension of e.g. ScalarRep[[1]], we would write

⬇

AnomDim4D["S",{PosScalar[[1]],PosScalar[[1]]}]

See sec. Q.4 for how to obtain anomalous dimensions for vector-bosons and
fermions.

###  2.4 Integrating out temporal scalars

Temporal scalars are often heavy compared to the fields driving the phase
transition, and can be integrated out [33, 29]. It should be stressed that
this is a user-dependent optional step. The resulting EFT is said to describe
ultrasoft physics. The command is

⬇

PerformDRsoft[{}];

Couplings at the ultrasoft scale are given by

⬇

PrintCouplingsUS[];

And effective scalar masses are given by

⬇

PrintScalarMassUS["LO"];

PrintScalarMassUS["NLO"];

Finally, $\beta$-functions for the ultrasoft masses are provided with the
command

⬇

BetaFunctions3DUS[];

In the 3-dimensional theory only scalar masses have non-zero beta functions.

This concludes our tutorial of the dimensional reduction of the Abelian-Higgs
model.

###  2.5 Two-loop effective potential

Once the model is loaded, we can calculate the effective potential. This can
be done by the user, either in the soft or the ultrasoft theory.
Alternatively, DRalgo can calculate the effective potential in the ultrasoft
theory. First, we need to create field-dependent masses which requires to
specify a vacuum expectation value direction:

⬇

DefineNewTensorsUS[$\mu$ij,$\lambda$4,$\lambda$3,gvss,gvvv];

$\phi$vev={0,$\phi$}//SparseArray;

DefineVEVS[$\phi$vev];

Here the vacuum expectation value is in the second, imaginary, Higgs
component.

For the calculation to proceed, all mass matrices must be diagonal. If not, an
error message is printed and it is up for the user to diagonalize the
matrices; see section Q.16 for an example. For the model at hand, the mass
matrix is diagonal, which the user can confirm by printing the field-dependent
masses

⬇

FieldMasses=PrintTensorsVEV[]

The effective potential is calculated via

⬇

CalculatePotentialUS[]

The results are given by

⬇

PrintEffectivePotential["LO"]

PrintEffectivePotential["NLO"]

PrintEffectivePotential["NNLO"]

Here, LO refers to the tree-level, NLO to the one-loop, and NNLO to the two-
loop effective potential. Note that all results are given in Landau gauge. In
the (two-loop) NNLO part, the renormalization scale is denoted by ${\tt\mu
3}=\bar{\mu}_{3}$.

This completes the first tutorial on quick installation and running. Output
from the 3d EFT matching relations can be implemented either to non-
perturbative lattice codes, or perturbative analyses in terms of the effective
potential. While such implementations are left to the user, appendix A
displays how to interface DRalgo output in a Mathematica implementation that
determines selected thermodynamic quantities for the Abelian-Higgs model.

##  3 Theory behind the scenes: dimensional reduction of a generic model

In this section, we dig deeper into the theory and computations in the back-
end of the software. The implementation for dimensional reduction of a generic
model is based on tensor-notation [60, 61, 62, 63, 64] that separates Lorentz
algebra from group algebra. The Lorentz algebra is hard-coded, with
contractions done by FeynCalc [65, 66] and FORM [67, 20], while the group-
structure is provided by the user.

###  3.1 Lagrangian in the 4d fundamental theory

Consider a general Lagrangian in Minkowski spacetime with mostly-plus metric:
$g^{\mu\nu}=\mbox{diag}(-1,1,1,1)$,
$\left\\{\sigma_{\mu},\overline{\sigma}_{\nu}\right\\}=-2g_{\mu\nu}$. In terms
of the functional-integral measure and the action $S$, the partition function
is $\mathcal{Z}=\int\mathcal{D}\,e^{iS}$. Correspondingly, the sigma matrices
are defined as

$\displaystyle\sigma^{\mu}=\left(\mathbbm{1},\sigma^{i}\right),\quad\overline{\sigma}^{\mu}=\left(-\mathbbm{1},\sigma^{i}\right)\;,$
(3.1)

where $\sigma^{i}$ are Pauli matrices.

To write down the Lagrangian, in a flattened form, we employ a basis where all
scalars and vectors are real. In addition, all fermions are composed of two-
component Weyl-spinors [68]. The most general, four dimensional, Lagrangian in
Minkowski space is [60, 61, 34]

$\displaystyle\mathcal{L}$
$\displaystyle=-\frac{1}{2}R_{i}(-\delta_{ij}\partial_{\mu}\partial^{\mu}+\mu_{ij})R_{j}-\frac{1}{4}F_{\mu\nu}^{a}F^{\mu\nu,a}\delta_{ab}-\frac{1}{2\xi_{a}}(\partial_{\mu}A^{a,\mu})^{2}$
$\displaystyle-\partial^{\mu}\overline{\eta}^{a}\partial_{\mu}\eta^{a}+i\psi_{I}^{\dagger}\overline{\sigma}^{\mu}\partial_{\mu}\psi^{I}-\frac{1}{2}(M^{IJ}\psi_{I}\psi_{J}+\text{h.c.})+\mathcal{L}_{{\mbox{\scriptsize
int}}}\;,$ (3.2) $\displaystyle\mathcal{L}_{{\mbox{\scriptsize int}}}$
$\displaystyle=-\lambda_{i}R_{i}-\frac{1}{3!}\lambda_{ijk}R_{i}R_{j}R_{k}-\frac{1}{4!}\lambda_{jklm}R_{i}R_{j}R_{k}R_{m}-\frac{1}{2}(Y^{iIJ}R_{i}\psi_{I}\psi_{J}+h.c)$
$\displaystyle+g_{IJ}^{a}A^{a}_{\mu}\psi_{I}^{\dagger}\overline{\sigma}^{\mu}\psi_{J}-g_{jk}^{a}A^{a}_{\mu}R_{j}\partial^{\mu}R_{k}-\frac{1}{2}g^{a}_{jn}g^{b}_{kn}A_{\mu}^{a}A^{\mu,b}R_{j}R_{k}-g^{abc}A^{\mu,a}A^{\nu,b}\partial_{\mu}A^{c}_{\nu}$
$\displaystyle-\frac{1}{4}g^{abe}g^{cde}A^{\mu a}A^{\nu
b}A_{\mu}^{c}A_{\nu}^{d}+g^{abc}A_{\mu}^{a}\eta^{b}\partial^{\mu}\overline{\eta}^{c}\;,$
(3.3)

where $\mathcal{L}_{{\mbox{\scriptsize int}}}$ is the interaction Lagrangian
and $F_{\mu\nu}=\frac{i}{g}[D_{\mu},D_{\nu}]$ is the field strength tensor
with the corresponding gauge coupling $g$. In this notation, the field $R_{i}$
denotes a real scalar-field with scalar-index $i$. In the Standard Model, $i$
corresponds to the Higgs, neutral Goldstone, and real/imaginary component of
the charged Goldstone. Further, the field $A_{\mu}^{a}$ corresponds to a real
vector-field with vector-index $a$, the field $\eta$ is a ghost-field and the
field $\psi_{I}$ is a Weyl-spinor with fermion-index $I$. In addition,
$\mu_{ij}$ denote (squared) scalar masses, and $M_{IJ}$ fermion masses.
Repeated indices are always summed over irrespective of their vertical
placement.

Above we denoted $Y_{iIJ}$ as Yukawa couplings, $\lambda_{ijkl}$ as scalar
quartic, $\lambda_{ijk}$ as scalar cubic, and $\lambda_{i}$ as scalar tadpole
couplings. The gauge couplings $g^{a}_{jk}$ are (anti-symmetric)
representation matrices. For example, for an adjoint scalar-representation in
$\mathrm{SO}(3)$ $g^{a}_{jk}=\epsilon^{ajk}$. The abbreviation h.c. stands for
the hermitian conjugate. The vertical placement of fermion indices is
important; e.g. $M^{IJ}=M^{*}_{IJ}$. We refer to [68] for further details.

In the dimensional-reduction step, hard modes with masses $\sim\pi T$ are
integrated out. This is done by matching the fundamental Lagrangian above to
the effective Lagrangian living in three-dimensions. To avoid large logarithms
this matching should be performed close to $\bar{\mu}=\pi T$ where $\bar{\mu}$
is the RG-scale. This three-dimensional Lagrangian does not contain fermions.
Moreover, in the dimensional-reduction step the temporal component of vectors
– represented by temporal scalar fields in the EFT – obtain Debye masses, as
well as thermally generated interactions with other scalars.

The matching is straightforward using Euclidean signature. This is achieved by
redefining

$\displaystyle A^{{\mbox{\tiny\rm{M}}},a}_{0}$ $\displaystyle\equiv
iA^{{\mbox{\tiny\rm{E}}},a}_{0}\;,\quad\sigma_{i}^{{\mbox{\tiny\rm{M}}}}=-i\sigma_{i}^{{\mbox{\tiny\rm{E}}}}\;,\quad
t=-i\tau\;,\quad\partial_{0}^{{\mbox{\tiny\rm{M}}}}=i\partial_{0}^{{\mbox{\tiny\rm{E}}}}\;,$
(3.4)
$\displaystyle\left(\partial_{\mu}R\,\partial^{\mu}R\right)_{{\mbox{\tiny\rm{M}}}}$
$\displaystyle=\left(\partial_{\mu}R\,\partial^{\mu}R\right)_{{\mbox{\tiny\rm{E}}}}\;,\quad\,/\\!\\!\\!\\!{D}\,_{{\mbox{\tiny\rm{M}}}}=i\,/\\!\\!\\!\\!{D}\,_{{\mbox{\tiny\rm{E}}}}\;,\quad\left\\{\sigma_{\mu},\overline{\sigma}_{\nu}\right\\}_{{\mbox{\tiny\rm{E}}}}=2\delta_{\mu\nu}\;,$
(3.5)

where quantities are denoted either in Minkowskian ($M$) or Euclidean ($E$)
metric. Henceforth, we suppress these subscripts and implicitly assume an
Euclidean metric.

With the above redefinitions, the partition function is defined as
$\mathcal{Z}_{{\mbox{\tiny\rm{E}}}}=\int\mathcal{D}\,e^{-S_{{\mbox{\tiny\rm{E}}}}}$
with the most general tree-level Lagrangian in Euclidean signature

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2}R_{i}(\delta_{ij}\partial_{\mu}\partial^{\mu}+\mu_{ij})R_{j}+\frac{1}{4}F_{\mu\nu}^{a}F^{\mu\nu,a}\delta_{ab}+\frac{1}{2\xi_{a}}(\partial_{\mu}A^{a,\mu})^{2}$
$\displaystyle+\partial^{\mu}\overline{\eta}^{a}\partial_{\mu}\eta^{a}+\psi_{I}^{\dagger}\overline{\sigma}^{\mu}\mathcal{\partial}_{\mu}\psi^{I}+\frac{1}{2}(M^{IJ}\psi_{I}\psi_{J}+\mathrm{h.c.})+\mathcal{L}_{{\mbox{\scriptsize
int}}}\;,$ (3.6) $\displaystyle\mathcal{L}_{{\mbox{\scriptsize int}}}$
$\displaystyle=\lambda_{i}R_{i}+\frac{1}{3!}\lambda_{ijk}R_{i}R_{j}R_{k}+\frac{1}{4!}\lambda_{jklm}R_{i}R_{j}R_{k}R_{m}+\frac{1}{2}(Y^{iIJ}R_{i}\psi_{I}\psi_{J}+\mathrm{h.c.})$
$\displaystyle+ig_{IJ}^{a}A^{a}_{\mu}\psi_{I}^{\dagger}\overline{\sigma}^{\mu}\psi_{J}+g_{jk}^{a}A^{a}_{\mu}R_{j}\partial^{\mu}R_{k}+\frac{1}{2}g^{a}_{jn}g^{b}_{kn}A_{\mu}^{a}A^{\mu,b}R_{j}R_{k}+g^{abc}A^{\mu,a}A^{\nu,b}\partial_{\mu}A^{c}_{\nu}$
$\displaystyle+\frac{1}{4}g^{abe}g^{cde}A^{\mu a}A^{\nu
b}A_{\mu}^{c}A_{\nu}^{d}-g^{abc}A_{\mu}^{a}\eta^{b}\partial^{\mu}\overline{\eta}^{c}\;.$
(3.7)

All Lorentz indices are contracted using the Euclidean metric
$\delta_{\mu\nu}=\mbox{diag}(+1,+1,+1,+1)$. The Feynman rules for the vertices
are

$\displaystyle R_{i}R_{j}R_{k}R_{l}$ $\displaystyle:-\lambda_{ijkl}\;,$ (3.8)
$\displaystyle R_{i}R_{j}R_{k}$ $\displaystyle:-\lambda_{ijk}\;,$ (3.9)
$\displaystyle R_{i}$ $\displaystyle:-\lambda_{i}\;,$ (3.10) $\displaystyle
R_{i}\psi_{I}\psi_{J}$ $\displaystyle:-Y_{iIJ}\;,$ (3.11)
$\displaystyle\psi_{I}\psi_{J}A^{a}_{\mu}$
$\displaystyle:-ig^{a}_{IJ}\overline{\sigma}_{\mu}\;,$ (3.12) $\displaystyle
R_{i}R_{j}A^{a}_{\mu}$ $\displaystyle:ig_{ij}^{a}(p+q)_{\mu}\;,$ (3.13)
$\displaystyle A^{a}_{\mu}A^{b}_{\nu}A^{c}_{\rho}$
$\displaystyle:ig^{abc}T_{\mu\nu\rho}\;,$ (3.14) $\displaystyle
A^{a}_{\mu}A^{b}_{\nu}A^{c}_{\rho}A_{\sigma}^{d}$
$\displaystyle:-G_{\mu\nu\rho\sigma}^{abcd}\;,$ (3.15)
$\displaystyle\overline{\eta}^{a}\eta^{b}A^{c}_{\mu}$
$\displaystyle:ig^{abc}p_{\mu}\;,$ (3.16) $\displaystyle
R_{i}R_{j}A^{b}_{\mu}A^{b}_{\nu}$
$\displaystyle:\delta_{\mu\nu}\bigl{(}g_{in}^{a}g_{nj}^{b}+g_{jn}^{a}g_{ni}^{b}\bigr{)}\equiv\delta_{\mu\nu}H^{ab}_{ij}\;,$
(3.17)

where $G_{\mu\nu\rho\sigma}^{abcd}$ and $T_{\mu\nu\rho}$ are defined in [61].
Since we only focus on the matching, only the symmetric phase is relevant and
vacuum expectation values are not introduced. The propagators are

$\displaystyle\langle R_{i}(p)R_{j}(q)\rangle$
$\displaystyle=\frac{\delta_{ij}\delta(p+q)}{p^{2}}\;,$ $\displaystyle\langle
A^{a}_{\mu}(p)A^{b}_{\nu}(q)\rangle$
$\displaystyle=\frac{\delta^{ab}\delta(p+q)}{p^{2}}P_{\mu\nu}(p)\;,$
$\displaystyle\langle\overline{\eta}^{a}(p)\eta^{b}(q)\rangle$
$\displaystyle=\frac{\delta^{ab}\delta(p-q)}{p^{2}}\;,$
$\displaystyle\langle\psi_{I}(p)\psi_{J}(q)\rangle$
$\displaystyle=\frac{\delta_{IJ}\delta(p-q)ip_{\mu}\sigma_{\mu}}{p^{2}}\;,$
(3.18)

where all momenta are incoming by convention and

$\displaystyle
P_{\mu\nu}(p)=\delta_{\mu\nu}-(1-\xi)\frac{p_{\mu}p_{\nu}}{p^{2}}\;,$ (3.19)

where $\xi$ is the corresponding gauge parameter which for our investigations
in Landau gauge is set to $\xi=0$. All the Lorentz structure is contained in
the vertices, and the generalized coupling constants take care of the group
structure.

To do the matching we need to renormalize our theory. The matching can, and
will, introduce kinetic mixing. To allow for this, we express the bare $(b)$
fields and scalar masses as

$\displaystyle
R_{i(b)}=Z^{1/2}_{ij}R_{j}\;,\quad\mu_{ij(b)}=Z^{\mu}_{ij}=\mu_{ij}+\delta\mu_{ij}\;,\quad{A}^{a}_{\mu(b)}=Z^{1/2}_{ab}A^{b}_{\mu}\;,\quad\psi_{I(b)}=Z^{1/2}_{IJ}\psi_{J}\;.$
(3.20)

By construction all propagators should be diagonal at tree-level, to wit

$\displaystyle Z^{1/2}_{ik}Z^{1/2}_{kj}=\delta_{ij}+\delta Z_{ij}\;,\quad
Z^{1/2}_{ac}Z^{1/2}_{cb}=\delta^{ab}+\delta Z^{ab}\;.$ (3.21)

#### Assumed size of masses and couplings

For the current problem there are three energy scales: $\pi T$, $gT$, and
$g^{2}T$. Denoted as the hard, soft, and ultrasoft scale, respectively. In
DRalgo it is assumed that all scalar (fermion) masses scale as $gT$ or
$g^{2}T$. If, on the contrary, some masses would be hard, then they should
formally be integrated out together with high-temperature modes.777 This will
be implemented in future versions of DRalgo.  After hard modes are integrated
out, it is up to the user whether soft-scale ($gT$) fields are integrated out
as well. By default all temporal scalars have masses of this order, but some
models might also have additional scalars with soft masses. For the couplings,
DRalgo assumes that all quartics and cubics scale as $g^{2}$. This scaling is
chosen so that scalar masses of $\mathcal{O}(gT)$ can be integrated out safely
[23].

In summary, DRalgo assumes the following scaling

$\displaystyle\lambda_{ijkl}$ $\displaystyle\sim g^{2}\;,$
$\displaystyle\lambda_{ijk}$ $\displaystyle\sim g^{2}\;,$
$\displaystyle\lambda_{i}$ $\displaystyle\sim g^{2}\;,$
$\displaystyle\mu_{ij}$ $\displaystyle\sim g^{2}T^{2}\;,$ $\displaystyle
M_{IJ}$ $\displaystyle\sim gT\;,$ $\displaystyle Y_{iIJ}$ $\displaystyle\sim
g\;,$ $\displaystyle g^{a}_{IJ}$ $\displaystyle\sim g^{a}_{ij}\sim g^{abc}\sim
g\;.$ (3.22)

It is of course possible to consider smaller couplings and masses than those
above.

###  3.2 Lagrangian in the 3d effective theory

The three-dimensional theory is of the form

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2}R_{i}(\delta_{ij}\partial_{\mu}\partial^{\mu}+\mu_{ij})R_{j}+\frac{1}{4}F_{\mu\nu}^{a}F^{\mu\nu,a}\delta_{ab}+\frac{1}{2\xi_{a}}(\partial_{\mu}A^{a,\mu})^{2}$
$\displaystyle+\partial^{\mu}\overline{\eta}^{a}\partial_{\mu}\eta^{a}+\frac{1}{2}\partial_{\mu}A_{0}^{a}\partial^{\mu}A_{0}^{a}+\frac{1}{2}\mu_{{\mbox{\tiny\rm{D}}}}^{ab}A_{0}^{a}A_{0}^{b}+\mathcal{L}_{\text{int}}\;,$
(3.23) $\displaystyle\mathcal{L}_{\text{int}}$
$\displaystyle=h_{i}R_{i}+\frac{1}{3!}h_{ijk}R_{i}R_{j}R_{k}+\frac{1}{4!}h_{jklm}R_{i}R_{j}R_{k}R_{m}$
$\displaystyle+g_{jk}^{a}A^{a}_{\mu}R_{j}\partial^{\mu}R_{k}+\frac{1}{2}A_{\mu}^{a}A^{\mu,b}R_{j}R_{k}+g^{abc}A^{\mu,a}A^{\nu,b}\partial_{\mu}A^{c}_{\nu}$
$\displaystyle+\frac{1}{4}g^{abe}g^{cde}A^{\mu a}A^{\nu
b}A_{\mu}^{c}A_{\nu}^{d}-g^{abc}A_{\mu}^{a}\eta^{b}\partial^{\mu}\overline{\eta}^{c}$
$\displaystyle+\lambda_{A}^{abcd}A_{0}^{a}A_{0}^{b}A_{0}^{c}A_{0}^{d}+{g_{0}}_{ij}^{ab}A_{0}^{a}A_{0}^{b}R_{i}R_{j}+g_{0}^{abc}A_{\mu}^{a}A_{0}^{b}\partial^{\mu}A_{0}^{c}\;,$
(3.24)

where Lorentz indices range between $\mu,\nu=\\{1,\dots,3\\}$. In
renormalizing the fields, we define

$\displaystyle Z^{1/2}_{ik}Z^{1/2}_{kj}=\delta_{ij}+\delta Z_{3,ij}\;,\quad
Z^{1/2}_{ac}Z^{1/2}_{cb}=\delta^{ab}+\delta Z^{ab}_{3}\;,\quad
Z^{1/2}_{ac,{\mbox{\tiny\rm{L}}}}Z^{1/2}_{cb,{\mbox{\tiny\rm{L}}}}=\delta^{ab}+\delta
Z^{ab,{\mbox{\tiny\rm{L}}}}_{3}\;.$ (3.25)

Here, the term $\delta Z^{ab,{\mbox{\tiny\rm{L}}}}_{3}$ corresponds to
temporal vectors. The propagators of this theory are as in eq. (3.1) in three
dimensions with the absence of the fermionic propagator and the inclusion of
the temporal-vector propagator

$\displaystyle\langle A_{0}^{a}(p)A_{0}^{b}(q)\rangle$
$\displaystyle=\frac{\delta^{ab}\delta(p+q)}{p^{2}}\;.$ (3.26)

The counterterm Feynman rules are the same as above, barring the new temporal-
vector diagram

$\displaystyle A_{0}^{a}A_{0}^{b}:-p^{2}\delta
Z^{ab,{\mbox{\tiny\rm{L}}}}_{3}-\delta\mu_{{\mbox{\tiny\rm{D}}}}^{ab}.$ (3.27)

###  3.3 Matching

Self-energies and general $n$-point correlations are calculated both in the
original 4d theory and the effective 3d theory. The 3-dimensional parameters
are chosen, or matched, to give the same correlators as the original theory
[29, 33].

#### One-loop thermal scalar masses

We start with the matching for scalar masses.

missing

$\scriptstyle i$

missing

$\scriptstyle j$

missing

$\scriptstyle n$

missing

$\scriptstyle i$

missing

$\scriptstyle j$

missing

$\scriptstyle a$

missing

$\scriptstyle i$

missing

$\scriptstyle j$

missing

$\scriptstyle n$

missing

$\scriptstyle l$

missing

$\scriptstyle i$

missing

$\scriptstyle j$

missing

$\scriptstyle n$

missing

$\scriptstyle a$

missing

$\scriptstyle i$

missing

$\scriptstyle j$

missing

$\scriptstyle I$

missing

$\scriptstyle J$

Figure 2: Diagrams contributing to the thermal scalar masses. A wiggly line
denotes gauge bosons, a black line scalar particles, and a directed black line
fermions. A black dot represents corresponding counterterms.

For the matching we assume to be in a regime of soft or ultrasoft external
momenta $p\sim gT$ or $p\sim g^{2}T$. The corresponding diagrams at one-loop
level are illustrated in fig. 2. In their computation, we need the master
integrals [69]

$\displaystyle I^{4b}_{\alpha}$
$\displaystyle={\hbox{$\sum$}\\!\\!\\!\\!\\!\\!\\!\int\,}_{\\!\\!\\!\\!\raise-2.71246pt\hbox{$\scriptstyle{Q}$}}\frac{1}{[Q^{2}]^{\alpha}}=\left(\frac{\bar{\mu}^{2}e^{\gamma_{\mbox{\tiny\rm{E}}}}}{4\pi}\right)^{\epsilon}2T\frac{[2\pi
T]^{d-2\alpha}}{(4\pi)^{\frac{d}{2}}}\frac{\Gamma\left(\alpha-\frac{d}{2}\right)}{\Gamma(\alpha)}\zeta_{2\alpha-d}\;,$
(3.28) $\displaystyle I^{4f}_{\alpha}$
$\displaystyle={\hbox{$\sum$}\\!\\!\\!\\!\\!\\!\\!\int\,}_{\\!\\!\\!\\!\raise-2.71246pt\hbox{$\scriptstyle{\\{Q\\}}$}}\frac{1}{[Q^{2}]^{\alpha}}=\left(\frac{\bar{\mu}^{2}e^{\gamma_{\mbox{\tiny\rm{E}}}}}{4\pi}\right)^{\epsilon}2T\frac{[2\pi
T]^{d-2\alpha}}{(4\pi)^{\frac{d}{2}}}\frac{\Gamma\left(\alpha-\frac{d}{2}\right)}{\Gamma(\alpha)}\bigl{(}1-2^{d-2\alpha}\bigr{)}\zeta_{2\alpha-d}\;,$
(3.29)

where $\zeta_{s}=\zeta(s)$ is the Riemann zeta function. The $d$-dimensional
bosonic integral measure is defined as

$\displaystyle{\hbox{$\sum$}\\!\\!\\!\\!\\!\\!\\!\int\,}_{\\!\\!\\!\\!\raise-2.71246pt\hbox{$\scriptstyle{Q}$}}\equiv
T\sum_{q_{n}}\left(\frac{\bar{\mu}^{2}e^{\gamma_{\mbox{\tiny\rm{E}}}}}{4\pi}\right)^{\epsilon}\int\frac{{\rm
d}^{d}{\bf q}}{(2\pi)^{d}}\;,$ (3.30)

while for fermions the summation is written as ${{\Sigma}\\!\\!\\!\\!\raise
1.29167pt\hbox{$\int$}_{\mbox{\tiny\rm{${\\{Q\\}}$}}}}$. We set
$d=3-2\epsilon$ with Euclidean four-momenta $Q^{2}=q_{n}^{2}+{\bf
q}^{2}=\bigl{[}(2n+\sigma)\pi T\bigr{]}^{2}+{\bf q}^{2}$, and $\sigma=0(1)$
for bosons(fermions).

The first diagram in fig. 2 gives

$\displaystyle D_{1}^{ij}$
$\displaystyle=-\frac{1}{2}\lambda_{ijnn}I^{4b}_{1}\;,$ (3.31)

where the factor of $\frac{1}{2}$ is the symmetry factor, $-\lambda_{ijnn}$ is
the Feynman rule, and

$\displaystyle I^{4b}_{1}$
$\displaystyle=\frac{T^{2}}{12}+\mathcal{O}(\epsilon)\;.$ (3.32)

The zero Matsubara mode is here ignored. The reason for this is that the zero
Matsubara mode corresponds to soft momenta, and should not be integrated out.
Conversely, since the $n=0$ mode exists both in the EFT and the parent theory,
this contribution cancels in the matching. The second diagram is likewise
(neglecting higher $\epsilon$ terms)

$\displaystyle D_{2}^{ij}$
$\displaystyle=\frac{1}{2}H^{aa}_{ij}dI_{1}^{4b}\;,\quad
dI_{1}^{4b}=\frac{T^{2}}{4}\;,$ (3.33)

where we used the shorthand notation for $H^{ab}_{ij}$ from eq. (3.17).

The remaining diagrams give

$\displaystyle
D_{3}^{ij}=i^{2}g^{a}_{in}g^{a}_{nj}\frac{3}{(4\pi)^{2}\epsilon_{b}}p^{2}\;,\quad
D_{4}^{ij}=\frac{1}{2}\lambda_{ijnl}\,\mu_{nl}\frac{1}{(4\pi)^{2}\epsilon_{b}}\;.$
(3.34)

The first diagram contains the external momentum is $p$, and the last diagram
is the contribution from a scalar-mass insertion. With the assumed power-
counting (3.1), this contribution is formally of higher order. Finally, there
is the fermion loop

$\displaystyle D_{5}^{ij}$
$\displaystyle=\frac{1}{2}\bigl{(}Y^{iIJ}Y_{jIJ}+Y_{iIJ}Y^{jIJ}\bigr{)}\biggl{[}-\frac{T^{2}}{12}-p^{2}\frac{1}{(4\pi)^{2}\epsilon_{f}}\biggr{]}\;.$
(3.35)

The second coupling-constant term above is the hermitian conjugate. Also,
above we use $\epsilon_{b}$ and $\epsilon_{f}$ to denote $\epsilon$ poles,
albeit with some additional factors. They are defined in [33, 29], and are

$\displaystyle\frac{1}{\epsilon_{b}}=\frac{1}{\epsilon}+\text{\tt
Lb},\quad\frac{1}{\epsilon_{f}}=\frac{1}{\epsilon}+\text{\tt Lf},$ (3.36)

where Lb and Lf are given in eq. (2.4).

To perform the matching, we demand

$\displaystyle\int{\rm d}^{4}x\langle
R_{i}R_{j}\rangle_{\mbox{\tiny\rm{4d}}}=T^{-1}\int{\rm d}^{3}x\langle
R_{i}R_{j}\rangle_{\mbox{\tiny\rm{4d}}}=\int{\rm d}^{3}x\langle
R_{i}R_{j}\rangle_{\mbox{\tiny\rm{3d}}}\;.$ (3.37)

At leading order, 3d fields are rescaled by a factor $T^{-1/2}$. Next,
consider the self-energies. One finds

$\displaystyle\bigl{(}-\delta
Z_{ij}p^{2}-\delta\mu_{ij}-\mu_{ij}+\Pi(0)_{ij}+p^{2}\Pi^{\prime}(0)_{ij}\bigr{)}_{\mbox{\tiny\rm{4d}}}=\bigl{(}-\delta
Z_{3,ij}p^{2}-\delta\mu_{3,ij}-\mu_{3,ij}\bigr{)}_{\mbox{\tiny\rm{3d}}}\;,$
(3.38)

where $\Pi_{ij}(p)=(D_{1}+\dots+D_{5})_{ij}$. To avoid confusion we added a
subscript $3$ to three-dimensional quantities.

Here $-\delta Z_{ij}p^{2}-\delta\mu_{ij}$ are $4$d counterterms, and cancel
all $\epsilon$ poles. The matching gives

$\displaystyle\mu_{3,ij}=\mu_{ij}-\Pi(0)_{ij}\;,\quad\delta Z_{3,ij}=\delta
Z_{ij}-\Pi^{\prime}(0)_{ij}\;.$ (3.39)

In this step all the Lorentz structure is stripped away, leaving mere group-
theory factors.

#### One-loop scalar quartics

Next, we consider scalar quartics. Before calculating the actual diagrams, it
is useful to look at what happens on the 3d side. The relevant terms in the
Lagrangian are

$\displaystyle R_{i}R_{j}R_{k}R_{l}\,h_{ijkl}\;.$ (3.40)

There are two contributions to the matching. First, the coupling constant,
$h$, receives contributions from different loop orders

$\displaystyle h_{ijkl}=h^{0}_{ijkl}+h^{1}_{ijkl}+\ldots$ (3.41)

Second, as shown in the previous section, $3$d fields are renormalized. In
terms of the scalar counterterm, $\delta Z_{3,ij}$, this renormalization is
(to leading order) $R_{i(b)}=R_{i}+\frac{1}{2}\delta Z_{3,ij}R_{j}$. Combined,
these considerations imply that the contribution from the 3d side is

$\displaystyle\langle R_{i}R_{j}R_{k}R_{l}\rangle_{\mbox{\tiny\rm{3d}}}$
$\displaystyle=-h_{ijkl}^{0}-h_{ijkl}^{1}$
$\displaystyle-\frac{1}{2}\left(\delta Z_{3,im}h_{mjkl}^{0}+\delta
Z_{3,jm}h_{imkl}^{0}+\delta Z_{3,km}h_{ijml}^{0}+\delta
Z_{3,lm}h_{ijkm}^{0}\right)\;.$ (3.42)

missing

$\scriptstyle i$

missing

$\scriptstyle j$

missing

$\scriptstyle k$

missing

$\scriptstyle l$

missing

$\scriptstyle n$

missing

$\scriptstyle m$

missing

$\scriptstyle i$

missing

$\scriptstyle j$

missing

$\scriptstyle k$

missing

$\scriptstyle l$

missing

$\scriptstyle a$

missing

$\scriptstyle b$

missing

$\scriptstyle i$

missing

$\scriptstyle j$

missing

$\scriptstyle k$

missing

$\scriptstyle l$

missing

$\scriptstyle I$

missing

$\scriptstyle J$

missing

$\scriptstyle K$

missing

$\scriptstyle L$

Figure 3: Diagrams contributing to the thermal corrections to the scalar
quartics. Indices correspond to the line notation in fig. 2.

In the $4$d theory, the first and second diagrams give

$\displaystyle D_{1}^{ijkl}$
$\displaystyle=\frac{1}{2}\Bigl{(}\lambda_{ijnm}\lambda_{nmkl}+\lambda_{iknm}\lambda_{nmjl}+\lambda_{ilnm}\lambda_{nmkj}\Bigr{)}\frac{1}{16\pi^{2}\epsilon_{b}}\;,$
(3.43) $\displaystyle D_{2}^{ijkl}$
$\displaystyle=\frac{1}{2}\Bigl{(}H^{ab}_{ij}H^{ab}_{kl}+H^{ab}_{ik}H^{ab}_{jl}+H^{ab}_{il}H^{ab}_{kj}\Bigr{)}\biggl{[}\frac{3}{(4\pi)^{2}\epsilon_{b}}-\frac{1}{8\pi}\biggr{]}\;.$
(3.44)

Finally, the fermion contribution is

$\displaystyle D_{3}^{ijkl}$
$\displaystyle=\frac{1}{2}\bigl{(}Y_{iIJ}Y^{jJK}Y_{kKL}Y^{lLI}+(jkl)\bigr{)}\frac{1}{(4\pi)^{2}\epsilon_{f}}+\mathrm{h.c.}$
(3.45)

where the terms $(jkl)$ contain all permutations over indices $j,k,l$.

The matching is performed by demanding

$\displaystyle\int{\rm d}^{4}x\langle
R_{i}R_{j}R_{k}R_{l}\rangle_{{\mbox{\tiny\rm{4d}}}}=T^{-1}\int{\rm
d}^{3}x\langle R_{i}R_{j}R_{k}R_{l}\rangle_{{\mbox{\tiny\rm{4d}}}}=\int{\rm
d}^{3}x\langle R_{i}R_{j}R_{k}R_{l}\rangle_{{\mbox{\tiny\rm{3d}}}}\;.$ (3.46)

In the light of the discussion at the beginning of this section, this implies

$\displaystyle T\Bigl{(}-\lambda_{ijkl}+\Lambda_{ijkl}\Bigr{)}$
$\displaystyle=-h_{ijkl}^{0}-h_{ijkl}^{1}$
$\displaystyle-\frac{1}{2}\left(\delta Z_{3,im}h_{mjkl}^{0}+\delta
Z_{3,jm}h_{imkl}^{0}+\delta Z_{3,km}h_{ijml}^{0}+\delta
Z_{3,lm}h_{ijkm}^{0}\right)\;.$ (3.47)

The renormalized sum of the diagrams above was denoted as
$\Lambda_{ijkl}=(D_{1}+D_{2}+D_{3})_{ijkl}$ and one finds

$\displaystyle h_{ijkl}^{0}$ $\displaystyle=T\lambda_{ijkl},$ (3.48)
$\displaystyle h_{ijkl}^{1}T^{-1}$ $\displaystyle=-\frac{1}{2}\Bigl{(}\delta
Z_{3,im}\lambda_{mjkl}+\delta Z_{3,jm}\lambda_{imkl}+\delta
Z_{3,km}\lambda_{ijml}+\delta
Z_{3,lm}\lambda_{ijkm}\Bigr{)}-\Lambda_{ijkl}\;,$ (3.49)

where $\delta Z_{3,km}$ was given in the previous section.

#### Higher loop levels

When performing dimensional reduction at NLO, two-loop contributions are
required for scalar thermal masses. Here, we do not show details of the two-
loop computation, as they are analogous to the one-loop computation presented
above. Naturally the number of required diagrams is larger. Due to the use of
integration-by-parts identities (IBP) [70, 71, 72] it can be shown that to the
given order, the two-loop master sum-integrals factorize into the one-loop
master integrals given in eqs. (3.28) and (3.29). While this streamlines the
computation at NLO significantly, contributions at higher orders are non-
factorizable. Nonetheless, DRalgo also implements a generic computation of
two-loop thermal masses and other 3d parameters at NLO.

###  3.4 Beta functions and anomalous dimensions

DRalgo calculates beta functions and anomalous dimensions by renormalizing 4d
parameters (cf. [64, 63, 62]). As an example, consider the scalar sector. We
have renormalized our fields and couplings as

$\displaystyle R_{i(b)}=Z^{1/2}_{ij}R_{j}$
$\displaystyle=\Bigl{(}\delta_{ij}+\frac{1}{2}\delta Z_{ij}\Bigr{)}R_{j}\;,$
(3.50) $\displaystyle R_{i(b)}R_{j(b)}R_{k(b)}R_{l(b)}\lambda_{ijkl(b)}$
$\displaystyle=R_{i}R_{j}R_{k}R_{l}\bigl{(}\lambda_{ijkl}+\delta\lambda_{ijkl}\bigr{)}\bar{\mu}^{2\epsilon}.\;$
(3.51)

On the 4d side, we assume the $\overline{\text{MS}}$ scheme, so $\delta
Z_{ij}$, $\delta\mu_{ij},$ and $\delta\lambda_{ijkl}$ are chosen to only
cancel $\epsilon$ poles.

We demand that bare parameters are independent of the RG-scale $\bar{\mu}$. To
leading order in $\epsilon$, we find ($t=\log\bar{\mu}$)

$\displaystyle\partial_{t}\lambda_{ijkl}=-2\epsilon\lambda_{ijkl}\;,\quad\partial_{t}\delta
Z_{ij}=-2\epsilon\delta Z_{ij}.$ (3.52)

Using this we can find the anomalous dimensions by demanding that
$\partial_{t}R_{i(b)}=0$:

$\displaystyle\implies\partial_{t}{R}_{i}=\epsilon
Z_{ij}R_{j}\equiv\gamma_{ij}R_{j}\;,$ (3.53)

where we have introduced the anomalous dimension $\gamma_{ij}$. Then, the
quartic beta function is determined by

$\displaystyle 0=\bar{\mu}^{-2\epsilon}\partial_{t}\lambda_{ijkl(b)}$
$\displaystyle=2\epsilon\bigl{(}\lambda_{ijkl}+\delta\lambda_{ijkl}\bigr{)}+\partial_{t}\delta\lambda_{ijkl}+\partial_{t}\lambda_{ijkl}\;,$
$\displaystyle\implies\partial_{t}\lambda_{ijkl}$
$\displaystyle=-2\epsilon\lambda_{ijkl}+2\epsilon\delta\lambda_{ijkl}\;.$
(3.54)

Where we have used
$\partial_{t}\delta\lambda_{ijkl}=-4\epsilon\delta\lambda_{ijkl}$; which
follows from counting powers of couplings in $\delta\lambda_{ijkl}$. We
identify $2\epsilon\delta\lambda_{ijkl}$ with the beta function for
$\lambda_{ijkl}$.

This beta function can be written in an equivalent form by writing the
counterterm, for $\lambda$, as the sum of a bare term and field
renormalization:

$\displaystyle\delta\lambda_{ijkl}=\bar{\lambda}_{ijkl}+\frac{1}{2}\left(\delta
Z_{im}\lambda_{mjkl}+\delta Z_{jm}\lambda_{imkl}+\delta
Z_{km}\lambda_{ijml}+\delta Z_{lm}\lambda_{ijkm}\right).$ (3.55)

Using this one finds

$\displaystyle\partial_{t}{\lambda}_{ijkl}=-2\epsilon\lambda_{ijkl}+2\epsilon\bar{\lambda}_{ijkl}-(\gamma\lambda)_{(ijkl)}\;,$
(3.56)

where the last term denotes all possible contractions between one index of
$\lambda_{ijkl}$ and one index of $\gamma_{ij}$. Other beta functions are
calculated analogously.

All gauge beta functions in DRalgo are defined with couplings squared, viz.
$\partial_{t}g^{2}$. Conversely beta functions for masses, scalar couplings,
and Yukawa couplings are defined as linear in the parameters, i.e.
$\partial_{t}\lambda$ and so forth. These conventions are also shown in the
DRalgo output.

In the 3-dimensional theory only scalar masses have non-zero beta functions.

##  4 Implementing beyond the Standard Model theories

In this section, we demonstrate the use of DRalgo for BSM theories based on
the example of the Two-Higgs doublet model (2HDM).

###  4.1 Two-Higgs doublet model with fermions

Consider the 2HDM potential [73, 74]

$\displaystyle V(\phi_{1},\phi_{2})$
$\displaystyle=m_{1}^{2}\phi_{1}\phi_{1}^{\dagger}+m_{2}^{2}\phi_{2}\phi_{2}^{\dagger}-(m_{12}^{2}\phi_{1}\phi_{2}^{\dagger}+\text{h.c.})+\lambda_{1}(\phi_{1}\phi_{1}^{\dagger})^{2}+\lambda_{2}(\phi_{2}\phi_{2}^{\dagger})^{2}$
$\displaystyle+\lambda_{3}(\phi_{1}\phi_{1}^{\dagger})(\phi_{2}\phi_{2}^{\dagger})+\lambda_{4}(\phi_{1}\phi_{2}^{\dagger})(\phi_{2}\phi_{1}^{\dagger})+\frac{\lambda_{5}}{2}\left[(\phi_{1}\phi_{2}^{\dagger})^{2}+(\phi_{2}\phi_{1}^{\dagger})^{2}\right]$
$\displaystyle+\left\\{\left[\lambda_{6}(\phi_{1}\phi_{1}^{\dagger})+\lambda_{7}(\phi_{2}\phi_{2}^{\dagger})\right](\phi_{1}^{\dagger}\phi_{2})+\mathrm{h.c.}\right\\}\;.$
(4.1)

The scalars are $\mathrm{SU}(2)$ doublets with hypercharge $Y_{\phi}=1$. This
means that the covariant derivative is given by888 We use the convention that
$(\phi_{1}\phi_{1}^{\dagger})=(\phi_{1}^{\dagger}\phi_{1})=\phi_{1}^{I}\phi_{1}^{\dagger,I}$,
where $I=1,2$.

$\displaystyle\mathcal{D}_{\mu}\phi_{1,2}=\Bigl{(}\partial_{\mu}-ig_{1}\frac{Y_{\phi}}{2}B_{\mu}-ig_{2}\frac{\tau^{a}}{2}A^{a}_{\mu}\Bigr{)}\phi_{1,2}\;,$
(4.2)

where $\tau^{a}$ are Pauli matrices. We further assume that all Standard-Model
fermions are present, and consider Yukawa interactions of the form

$\displaystyle\mathcal{L}_{\mbox{\scriptsize
Yuk}}=y_{i}\overline{q}_{L}\phi^{\dagger}_{i}q_{R}+\mathrm{h.c.}$ (4.3)

We define the group structure and scalar representations via

⬇

Group={"SU3","SU2","U1"};

RepAdjoint={{1,1},{2},0};

HiggsDoublet1={{{0,0},{1},1/2},"C"};

HiggsDoublet2={{{0,0},{1},1/2},"C"};

RepScalar={HiggsDoublet1,HiggsDoublet2};

CouplingName={g3,g2,g1};

All representations are specified according to their Dynkin coefficients (see
e.g. [75, 76, 57]). For example, gauge bosons transforms as $(1,1)$, or an
octet, under $\mathrm{SU}(3)$. Next we need to create the fermions. Let us
start by creating a single generation

⬇

Rep1={{{1,0},{1}, 1/6},"L"}; (*$q_{\mbox{\tiny\rm{L}}}$*)

Rep2={{{1,0},{0}, 2/3},"R"}; (*$u_{\mbox{\tiny\rm{R}}}$*)

Rep3={{{1,0},{0},-1/3},"R"}; (*$d_{\mbox{\tiny\rm{R}}}$*)

Rep4={{{0,0},{1},-1/2},"L"}; (*$\ell_{\mbox{\tiny\rm{L}}}$*)

Rep5={{{0,0},{0},-1},"R"}; (*$e_{\mbox{\tiny\rm{R}}}$*)

RepFermion1Gen={Rep1,Rep2,Rep3,Rep4,Rep5};

In the definition of a fermion representation `{{{1,0},{1},Y/2},"L<R>"}` the
last argument depends on if the fermion left-handed (L) or right-handed (R)
and Y is the corresponding hypercharge. For the Standard Model, we have

$\displaystyle Y_{q}=\frac{1}{3}\;,\quad Y_{u}=\frac{4}{3}\;,\quad
Y_{d}=-\frac{2}{3}\;,\quad Y_{\ell}=-1\;,\quad Y_{e}=-2\;.$ (4.4)

The above notation identifies Rep1 as the left-handed quark doublet, Rep2 as
the right-handed up-quark and so forth. The extra factor of $\frac{1}{2}$ for
hypercharges in the definition of e.g. Rep1 arises from the definition of the
covariant derivative similar to eq. (4.2).

Additional fermion-families can be added by grouping multiple instances of
RepFermion1Gen together:

⬇

RepFermion3Gen={RepFermion1Gen,RepFermion1Gen,RepFermion1Gen}//Flatten[#,1]&;

where the number of generations is chosen to be $n_{\rm f}=3$. When creating
Yukawa interactions the user decides which index contains which family. For
example, above we stacked all generations after each other in RepFermion3Gen.
Hence, in RepFermion3Gen indices 1–5 correspond to the first generation,
indices 6–10 to the second, and indices 11–15 to the last. Thus the user could
use index 1, 6, or 11 as the top quark. An alternative way to add an arbitrary
number of fermion families $n_{\rm f}$ is given in sec. Q.9.

Next, to create the tensors, we write

⬇

{gvvv,gvff,gvss,$\lambda$1,$\lambda$3,$\lambda$4,$\mu$ij,$\mu$IJ,$\mu$IJC,Ysff,YsffC}=

AllocateTensors[Group,RepAdjoint,CouplingName,RepFermion3Gen,RepScalar];

The mass terms in the potential in eq. (4.1) are

$\displaystyle V(\phi_{1},\phi_{2})$ $\displaystyle\supset
m_{1}^{2}\phi_{1}\phi_{1}^{\dagger}+m_{2}^{2}\phi_{2}\phi_{2}^{\dagger}-(m_{12}^{2}\phi_{1}\phi_{2}^{\dagger}+\mathrm{h.c.})\;,$
(4.5)

and due to the presence of two Higgs doublets, we need to specify the doublet
for each term:

⬇

InputInv={{1,1},{True,False}}; (*$\phi_{1}\phi_{1}^{\dagger}$*)

MassTerm1=CreateInvariant[Group,RepScalar,InputInv][[1]]//Simplify;

InputInv={{2,2},{True,False}}; (*$\phi_{2}\phi_{2}^{\dagger}$*)

MassTerm2=CreateInvariant[Group,RepScalar,InputInv][[1]]//Simplify;

InputInv={{1,2},{True,False}}; (*$\phi_{1}\phi_{2}^{\dagger}$*)

MassTerm3=CreateInvariant[Group,RepScalar,InputInv][[1]]//Simplify;

InputInv={{2,1},{True,False}}; (*$\phi_{2}\phi_{1}^{\dagger}$*)

MassTerm4=CreateInvariant[Group,RepScalar,InputInv][[1]]//Simplify;

The mass matrix is then generated by the GradMass[] command:

⬇

VMass=(

+m1*MassTerm1

+m2*MassTerm2

-(m12R+I*m12I)(MassTerm3)

-(m12R-I*m12I)(MassTerm4)

);

$\mu$ij=GradMass[VMass]//Simplify;

We allowed $m_{12}^{2}$ to be complex, with real part m12R, and imaginary part
m12I. For the scalar quartics in eq. (4.1) the corresponding part of the
potential is

$\displaystyle V(\phi_{1},\phi_{2})$
$\displaystyle=\lambda_{1}(\phi_{1}\phi_{1}^{\dagger})^{2}+\lambda_{2}(\phi_{2}\phi_{2}^{\dagger})^{2}+\lambda_{3}(\phi_{1}\phi_{1}^{\dagger})(\phi_{2}\phi_{2}^{\dagger})$
$\displaystyle+\lambda_{4}(\phi_{1}\phi_{2}^{\dagger})(\phi_{2}\phi_{1}^{\dagger})+\frac{\lambda_{5}}{2}\bigl{[}(\phi_{1}\phi_{2}^{\dagger})^{2}+(\phi_{2}\phi_{1}^{\dagger})^{2}\bigr{]}$
$\displaystyle+\left\\{\bigl{[}\lambda_{6}(\phi_{1}\phi_{1}^{\dagger})+\lambda_{7}(\phi_{2}\phi_{2}^{\dagger})\bigr{]}(\phi_{1}^{\dagger}\phi_{2})+\mathrm{h.c.}\right\\}\;,$
(4.6)

for which we already created all the building blocks. Thus, the quartics can
be constructed as

⬇

QuarticTerm1=MassTerm1^2; (*$(\phi_{1}\phi_{1}^{\dagger})^{2}$*)

QuarticTerm2=MassTerm2^2; (*$(\phi_{2}\phi_{2}^{\dagger})^{2}$*)

QuarticTerm3=MassTerm1*MassTerm2;
(*$(\phi_{1}\phi_{1}^{\dagger})(\phi_{2}\phi_{2}^{\dagger})$*)

QuarticTerm4=MassTerm3*MassTerm4;
(*$(\phi_{1}\phi_{2}^{\dagger})(\phi_{2}\phi_{1}^{\dagger})$*)

QuarticTerm5=(MassTerm3^2+MassTerm4^2);
(*$(\phi_{1}\phi_{2}^{\dagger})^{2}+(\phi_{2}\phi_{1}^{\dagger})^{2}$*)

QuarticTerm6=MassTerm1*MassTerm3+MassTerm1*MassTerm4;
(*$(\phi_{1}\phi_{1}^{\dagger})\bigl{[}(\phi_{1}\phi_{2}^{\dagger})+(\phi_{2}\phi_{1}^{\dagger})\bigr{]}$*)

QuarticTerm7=MassTerm2*MassTerm3+MassTerm2*MassTerm4;
(*$(\phi_{2}\phi_{2}^{\dagger})\bigl{[}(\phi_{1}\phi_{2}^{\dagger})+(\phi_{2}\phi_{1}^{\dagger})\bigr{]}$*)

Consequently, the quartic tensor itself is defined as

⬇

VQuartic=(

+$\lambda$1H*QuarticTerm1

+$\lambda$2H*QuarticTerm2

+$\lambda$3H*QuarticTerm3

+$\lambda$4H*QuarticTerm4

+$\lambda$5H/2*QuarticTerm5

+$\lambda$6H*QuarticTerm6

+$\lambda$7H*QuarticTerm7

);

$\lambda$4=GradQuartic[VQuartic];

For simplicity, we assumed above that $\lambda$5H, $\lambda$6H, and
$\lambda$7H are real. One can allow for complex couplings by adding them, and
their conjugates, directly in QuarticTerm6 and QuarticTerm7. To create a
complex coupling $\lambda$6H, we would write

⬇

QuarticTerm6=($\lambda$6HR+I*$\lambda$6HI)*MassTerm1*MassTerm3+($\lambda$6HR-I*$\lambda$6HI)*MassTerm1*MassTerm4;

For Yukawa couplings, we only consider the top-quark coupling. If we choose
the first family to contain the top-quark, then a Yukawa term
$\sim\phi^{\dagger}\overline{q}_{L}u_{R}$ would involve fermion
representations number 1 and 2. When defining a Yukawa interaction,
representations should be specified in the order: scalar, first fermion,
second fermion. With this in mind, the Yukawa coupling of the first Higgs
doublet is999 Here we assume that the top quark resides in the first
generation. This has nothing to do with how the top-quark is usually placed in
the third generation. Rather, we here place it in the first for simplicity as
normally only the top-quark has a sizeable Yukawa coupling.

⬇

InputInv={{1,1,2},{False,False,True}};
(*$\phi_{1}^{\dagger}\overline{q}_{L}u_{R}$*)

YukawaDoublet1=CreateInvariantYukawa[Group,RepScalar,RepFermion3Gen,InputInv][[1]]//Simplify;

Here, `{1,1,2}` specifies

$\text{First Higgs doublet}\times\text{Left-handed top-
quark}\times\text{Right-handed top-quark}\,,$

and `{False,False,True}` specifies that the Higgs and left-handed quarks are
conjugated. The coupling to the second Higgs doublet is

⬇

InputInv={{2,1,2},{False,False,True}};
(*$\phi_{2}^{\dagger}\overline{q}_{L}u_{R}$*)

YukawaDoublet2=CreateInvariantYukawa[Group,RepScalar,RepFermion3Gen,InputInv][[1]]//Simplify;

The complete Yukawa tensor is

⬇

Ysff=-GradYukawa[yt1*YukawaDoublet1+yt2*YukawaDoublet2];

For the sake of generality, we assumed that both Higgs doublets couple to the
top-quark. Note that this is not allowed for a realistic model due to flavor-
changing-neutral-current constraints.

Finally, assuming that the Yukawa couplings are real, we can define

⬇

YsffC=Simplify[Conjugate[Ysff//Normal],Assumptions->{yt1>0,yt2>0}]//SparseArray;

Above we did not consider Yukawa couplings between different generations but
such couplings can be added. To wit, when we defined RepFermion3Gen we put the
first generation on indices 1–5; the second on indices 6–10; and the third on
indices 11–15. To define a coupling between, e.g. the left-handed top quark
(assumed to reside in generation 1) and the right-handed charm quark (assumed
to reside in generation 2), we would write

⬇

InputInv={{1,1,7},{False,False,True}};

YukawaTopCharm=CreateInvariantYukawa[Group,RepScalar,RepFermion3Gen,InputInv]//Simplify;

With the model implementation complete, all dimensional-reduction commands are
identical to those of the Abelian-Higgs model. We refer to [77, 78, 79] for
earlier results in the literature.

#### Integrating out temporal scalars

For the Abelian-Higgs case in sec. 2 we only had one scalar field, whereas
with the 2HDM we have two. When integrating out temporal scalars we have two
options. First, all scalars are light and we have two active, dynamical
doublets. Second, one doublet is heavy and is integrated out when going from
the soft to the ultrasoft scale.

For the first case, the corresponding command is

⬇

PerformDRsoft[{}];

For the second case, we assume that the second doublet is heavy. It can then
be integrated out via

⬇

PerformDRsoft[{5,6,7,8}];

To understand the above syntax, note that the indices of all representations
are given by

⬇

PosScalars=PrintScalarRepPositions[];

So the second Higgs doublet resides at PosScalars[[2]]=5;;8.

There is one complication when integrating out one of the Higgs doublets.
Namely, generally the two doublets mix through the $m_{12}^{2}$ term. For a
complete treatment the mass matrix needs to be diagonalized before the heavy
doublet can be integrated out. This lies beyond DRalgo and is an optional step
for the user. Instead, the code by default assumes that $m_{12}^{2}$ is small
– of $\mathcal{O}(g^{2}T)$ in power counting – and performs the
diagonalization perturbatively to first order in $m_{12}^{2}$.

##  5 Miscellaneous features

In this section, we discuss specific features of the software on a case-by-
case basis.

###  5.1 User-options and features

#### Lower-order dimensional reduction for speed

The user has several options to control what is calculated and how the code
operates. For example, if the model has many degrees of freedom, the user
might wish to save running-time and only calculate one-loop thermal masses and
couplings. This works by specifying Mode->1 when loading the model

⬇

Group={"U1"};

ImportModelDRalgo[Group,gvvv,gvff,gvss,$\lambda$1,$\lambda$3,$\lambda$4,$\mu$ij,$\mu$IJ,$\mu$IJC,Ysff,YsffC,Mode->1];

The default is Mode->2, in which case everything is calculated to NLO. And for
the most bare-bone/fast option select Mode->0; in this case only one-loop
thermal masses are calculated.

###  5.2 Model-treatment in the code

DRalgo works by factorizing all group and Lorentz algebra. The Lorentz algebra
is hard-coded while the group algebra is supplied by the user. All particles
are indexed by the order they appear. If the user has an
$\mathrm{SU}(3)\times\mathrm{SU}(2)$ model with gauge bosons, the vectors in
the $\mathrm{SU}(3)$ group have 8 components, while those of the
$\mathrm{SU}(2)$ group have 3. The code indexes these components by
$a=\bigl{(}8^{\alpha}_{{\mbox{\tiny\rm{SU}}}(3)},3^{\beta}_{{\mbox{\tiny\rm{SU}}}(2)}\bigr{)}$.
Therein, $\alpha$ runs from 1–8, $\beta$ from 1–3, and $a$ from 1–11. For
example, the Debye mass tensor $\mu_{{\mbox{\tiny\rm{D}}}}^{ab}$ stores the
information of both groups. For most cases this mass tensor is diagonal.

The scalars are treated differently since DRalgo only deals with real scalar
components. For example, a complex scalar $\Phi$ is rewritten
$\Phi=\frac{1}{\sqrt{2}}\left(\phi+i\psi\right)$. Hence the scalar indices
would in this case be $i=(\phi,\psi)$. Further, the scalar-mass matrix is
stored as $\mu_{S,ij}$. For example, a $m^{2}\phi^{2}$ mass term resides at
$\mu_{S,11}$. Consider now a complex representation with $n$ scalar components
labelled by $I,J$. In a complex basis vector-scalar-scalar trilinear couplings
are of the form
$A_{\mu}\left(G_{IJ}^{a}\partial^{\mu}\Phi_{I}\Phi_{J}^{*}+G^{a,*}_{IJ}\partial^{\mu}\Phi_{I}^{*}\Phi_{J}\right)$,
where $G^{a}_{IJ}$ are representation matrices. To convert to a real basis
each scalar component is split as
$\Phi_{I}=\frac{1}{\sqrt{2}}\left(\phi_{I}+i\psi_{I}\right)$. The components
are then reordered with the real components first: $i=(\phi_{I},\psi_{I})$, so
$i=1,\ldots,2n$. If we call the scalars in the real basis $\varphi_{i}$, our
vector-scalar-scalar trilinear couplings become

$\displaystyle A_{\mu}g^{a}_{ij}\partial^{\mu}\varphi_{i}\varphi_{j}\;,\qquad
g_{ij}^{a}=\begin{pmatrix}\mathop{\mbox{Im}}(G^{a}_{IJ})&\mathop{\mbox{Re}}(G^{a}_{IJ})\\\
-\mathop{\mbox{Re}}(G^{a}_{IJ})&\mathop{\mbox{Im}}(G^{a}_{IJ})\end{pmatrix}\;.$
(5.1)

Thus, $g^{a}_{ij}$ is automatically antisymmetric under $i\leftrightarrow j$.

As an example, consider a $\mathrm{U}(1)$ theory with a complex scalar. The
vector-scalar-scalar coupling is
$igA_{\mu}\left(\partial^{\mu}\Phi\Phi^{*}-\partial^{\mu}\Phi^{*}\Phi\right)$.
In a real basis $\Phi=\frac{1}{\sqrt{2}}\left(\phi+i\psi\right)$, this becomes

$\displaystyle
gA_{\mu}\partial^{\mu}\left(\phi,\psi\right)\begin{pmatrix}0&\frac{1}{2}\\\
-\frac{1}{2}&0\end{pmatrix}\begin{pmatrix}\phi\\\
\psi\end{pmatrix}\;,\quad\text{and}\quad
g_{ij}=g\begin{pmatrix}0&\frac{1}{2}\\\ -\frac{1}{2}&0\end{pmatrix}\;.$ (5.2)

###  5.3 Frequently asked questions

Below, we discuss various questions and problems the user might have
encountered.

####  Q.1 How to save and load my model?

Saving and loading a model is straightforward with DRalgo built-in functions.
Once a model is created, and loaded with ImportModelDRalgo (see the .m files
in the example folder), it can be saved by writing

⬇

SaveModelDRalgo[ModelInfo,"<modelname>.txt"],

Here, ModelInfo is a string and should contain information about the authors,
DRalgo version, and citations to relevant articles. See ./examples/ah.m or
./examples/2hdm.m for some examples. To load the model write

⬇

{Group,gvvv,gvff,gvss,$\lambda$1,$\lambda$3,$\lambda$4,$\mu$ij,$\mu$IJ,$\mu$IJC,Ysff,YsffC}=LoadModelDRalgo["<modelname>.txt"];

The loaded model can be your own, or perhaps one of the provided DRalgo model
repository. We also encourage you to make your own models available for the
community which is possible by submitting the model file via the Issue Tracker
on https://github.com/DR-algo/DRalgo. The model will then be verified and
added to the model repository. When submitting a model, please refer to a
paper or explicitly write out the Lagrangian in an accompanying notebook.

####  Q.2 How do I order semi-simple groups?

If the user-defined model contains multiple groups, the groups and
corresponding representations must be ordered as: ${\rm SO}(n)$, ${\rm
SU}(n)$, ${\rm SP}(n)$, ${\rm G}_{2}$, ${\rm F}_{4}$ ${\rm E}_{6}$, ${\rm
E}_{7}$, ${\rm E}_{8}$, ${\rm U}(1)$. For example, if the user has a model
with gauge group $\mathrm{G}=\mathrm{SU}(5)\otimes\mathrm{SO}(10)\otimes
E_{6}\otimes\mathrm{U}(1)$, the model-input should be defined as

⬇

Group={"SO10","SU5","E6","U1"};

The internal order of groups is not restricted. Hence, for a group such as
$\mathrm{G}=\mathrm{SU}(3)\otimes\mathrm{SU}(2)$, the following definitions
are equivalent

⬇

Group={"SU3","SU2"};

Group={"SU2","SU3"};

####  Q.3 How do I check that my model is anomaly free?

Once you have defined your model, the anomaly-free condition is that

⬇

Table[Tr[(a.b+b.a).c],{a,gvff},{b,gvff},{c,gvff}]

vanishes identically. Since this condition is not automatically fulfilled for
general, non-numeric, $\mathrm{U}(1)$ charges, it is the responsibility of the
user to choose $\mathrm{U}(1)$ charges such that all anomalies cancel.

####  Q.4 How do I calculate anomalous dimensions?

First find the position of all representations:

⬇

PosScalar=PrintScalarRepPositions[];

PosVector=PrintGaugeRepPositions[];

PosFermion=PrintFermionRepPositions[];

All anomalous dimensions are then found via

⬇

Table[AnomDim4D["S",{i,j}],{i,PosScalar},{j,PosScalar}]

Table[AnomDim4D["V",{i,j}],{i,PosVector},{j,PosVector}]

Table[AnomDim4D["F",{i,j}],{i,PosFermion},{j,PosFermion}]

For more details regarding anomalous dimensions and beta functions cf. sec.
3.4.

####  Q.5 Temporal scalar mixing

In most models the temporal-scalar masses are diagonal. However, in cases with
multiple $\mathrm{U}(1)$ groups there can be mixing. In such a case, the code
automatically recognizes this. For example, if the group is

⬇

Group={"U1","U1","U1"};

the code creates the mixed U(1) masses with the convention that $\mu$U1Mix1 is
the mixing between groups 1 and 2, $\mu$U1Mix2 is the mixing between groups 1
and 3, and $\mu$U1Mix3 is the mixing between groups 2 and 3. These masses are
displayed with the PrintDebyeMass command. See the ./examples/SMZp.m for an
example.

####  Q.6 What if I miss some couplings?

By default DRalgo assumes that the user has defined all couplings allowed by
symmetry in their 4d theory. The code runs even though some couplings are
missed. For example, consider the 2HDM with scalar potential as in eq. (4.1)
but without $\lambda_{6}$ and $\lambda_{7}$ couplings. For a
$\phi_{1}\leftrightarrow\phi_{2}$ symmetric model no
$\lambda_{6}$/$\lambda_{7}$-type scalar quartic couplings are induced.
However, if the model breaks this symmetry by e.g. a Yukawa sector, then
$\lambda_{7}$ and $\lambda_{6}$ couplings are generated at one-loop. In this
case DRalgo would still calculate these induced couplings; with the crux that
only the $\lambda_{1},\ldots,\lambda_{5}$ couplings are printed by the
PrintCouplings[] command, while the induced $\lambda_{7}$ and $\lambda_{6}$
couplings must be found manually via the PrintTensorDRalgo[] command. In
addition, DRalgo automatically alerts the user with a message if some new
couplings are generated at one-loop. This is a cross-check that no couplings
are forgotten.

####  Q.7 Can I run DRalgo without specifying the representation?

For general groups no. However, it is possible for the user to specify
arbitrary (non-numeric) $\mathrm{U}(1)$ charges. With non-numeric charges the
code does not check that various quartic or Yukawa terms are allowed. Hence,
it is the responsibility of the user to ensure gauge invariance.

####  Q.8 How large representations can I use?

In principle any group and representations can be used. In practice the code
slows down for huge representations. The code has been tested on general
models with $\sim$100–200 components in a given representation. For example an
$\mathrm{SO}(10)$ model with a $120$-dimensional scalar. Since DRalgo was
purposely written to deal with any quartic and Yukawa sector, these are often
the bottlenecks. If the user wants to omit two-loop contributions,
significantly larger representations are possible. Further still, if the user
only wants one-loop thermal masses, almost any model (within reason) can be
run in quick order.

####  Q.9 Can I include an arbitrary number of fermion generations?

Yes, take for example the 2HDM. As described in sec. 4.1, a single family of
SM fermions is defined via

⬇

Rep1={{{1,0},{1}, 1/6},"L"}; (*$q_{\mbox{\tiny\rm{L}}}$*)

Rep2={{{1,0},{0}, 2/3},"R"}; (*$u_{\mbox{\tiny\rm{R}}}$*)

Rep3={{{1,0},{0},-1/3},"R"}; (*$d_{\mbox{\tiny\rm{R}}}$*)

Rep4={{{0,0},{1},-1/2},"L"}; (*$\ell_{\mbox{\tiny\rm{L}}}$*)

Rep5={{{0,0},{0},-1},"R"}; (*$e_{\mbox{\tiny\rm{R}}}$*)

RepFermion1Gen={Rep1,Rep2,Rep3,Rep4,Rep5};

Defining Yukawa and scalar couplings proceed as before, and the model is
loaded with

⬇

ImportModelDRalgo[Group,gvvv,gvff,gvss,$\lambda$1,$\lambda$3,$\lambda$4,$\mu$ij,$\mu$IJ,$\mu$IJC,Ysff,YsffC,Verbose->False];

The user can then define $n_{\rm f}={\tt nF}$ fermion families by
writing101010 This adds nF copies of all fermions.

⬇

PosFermion=PrintFermionRepPositions[];

FermionMat=Table[{nF,i},{i,PosFermion}];

DefineNF[FermionMat]

These commands should be used before running

⬇

PerformDRhard[];

Adding $n_{\rm f}$ fermion families in this way does not add new Yukawa or
scalar couplings as the extra $(n_{\rm f}-1)$ families only have gauge
interactions. To define cross-family Yukawa couplings, one should follow the
procedure in sec. 4.1.

####  Q.10 How do I compare with the literature?

Most outputs from DRalgo can directly be compared with the literature. For
temporal scalars this requires some care. For example, if the original 4d
theory has a $\mathrm{SU}(2)$ triplet, then two types of $A_{0}^{2}\Sigma^{2}$
couplings are allowed:

$\displaystyle\mathcal{L}_{{\mbox{\scriptsize
3d}}}\supset\kappa_{1}(A_{0}^{a}A_{0}^{a})(\Sigma^{a}\Sigma^{a})+\kappa_{2}(A_{0}^{a}\Sigma^{a})^{2}\;.$
(5.3)

DRalgo does not distinguish between these two terms since its output is stored
in the form
$\frac{1}{4!}A_{0}^{a}A_{0}^{b}\Sigma^{c}\Sigma^{d}\lambda_{K}^{abcd}$. To
compare with the two parametrizations, the user can rewrite the
$\kappa_{1}$/$\kappa_{2}$ basis in tensor form. To this end, it is necessary
to additionally define the relevant terms of the effective model and compare
to its Lagrangian. This procedure is analogous to creating the fundamental
model where invariants can be compared using the command CompareInvariants[].
See the htm.m file for a worked example of the Higgs triplet model (HTM).

The matching of all possible allowed operators in the EFT is automatic in
DRalgo. This way also previously disregarded effective coefficients can be
determined. One example is a $\mathcal{L}_{{\mbox{\scriptsize
3d}}}\supset\kappa{\rm Tr\,}C_{0}^{3}B_{0}$ operator in the Standard Model.
Here, $C_{0}$ is the gluon (temporal) field, and $B_{0}$ is the temporal
hypercharge field. The output from DRalgo gives
$\kappa\propto(2Y_{q}+Y_{d}+Y_{u})$. See eq. (4.4) for the corresponding
hypercharges and the 2hdm.m example file for the full expression

####  Q.11 What is the functional basis used in the matching?

In addition to the variables Lb and Lf defined in eq. (2.4), two-loop diagrams
also contain factors of

$\displaystyle{\tt c}$
$\displaystyle=\frac{1}{2}\Bigl{(}\ln\Big{(}\frac{8\pi}{9}\Big{)}+(\ln\zeta_{2})^{\prime}-2{\gamma_{\mbox{\tiny\rm{E}}}}\Bigr{)}\;,$
(5.4)

where $\zeta_{s}=\zeta(s)$ is the Riemann zeta function and
$(\ln\zeta_{s})^{\prime}=\zeta^{\prime}(s)/\zeta(s)$. By default DRalgo uses
the relations
$\ln(2\pi)-(\ln\zeta_{2})^{\prime}=1-{\gamma_{\mbox{\tiny\rm{E}}}}+(\ln\zeta_{-1})^{\prime}$
and $1+(\ln\zeta_{-1})^{\prime}=12\ln A$, where $A$ is the Glaisher-Kinkelin
constant. It is possible to convert the output of DRalgo to the conventions of
[33] by using the replacement rule Log[Glaisher]->-1/12(Lb+2cplus-EulerGamma),
where cplus=(c+Log[3T/$\mu$]). This rule is implemented as PrintGenericBasis[]
in DRalgo.

####  Q.12 How do I define scalar cubic operators?

Scalar cubics are created analogously to quartics. Consider for example a
$\mathrm{SU}(2)$ theory with a scalar doublet $\phi$ and a singlet $S$. We can
then create the cubic operator $(\phi\phi^{\dagger})S$ via

⬇

InputInv={{1,1,2},{True,False,True}}; (*$\phi\phi^{\dagger}S$*)

CubicTerm1=CreateInvariant[Group,RepScalar,InputInv][[1]]//Simplify;

VCubic=$\lambda$C*CubicTerm1;

The corresponding tensor is defined with the command

⬇

$\lambda$3=GradCubic[VCubic];

Tadpoles are defined analogously. See the 2xsm.m file for a worked example
with two real scalars.

####  Q.13 Do I need to define tadpoles?

If tadpoles are allowed by symmetry we encourage the user to define them –
even if they are absent at tree-level.

####  Q.14 How do I define Dirac and Majorana masses?

Fermion masses are defined analogously to scalar masses. We stress that DRalgo
only works with Weyl fermions. For example, assume a theory with two Weyl
fermions $\psi_{1}$ and $\psi_{2}$. The following terms are then possible

$\displaystyle
m_{1}\psi_{1}\psi_{1}+m_{2}\psi_{2}\psi_{2}+m_{\mbox{\tiny\rm{D}}}\psi_{1}\psi_{2}+\mathrm{h.c.}$
(5.5)

Where the first two terms are Majorana masses, and the third is a Dirac-type
mass. These masses can be defined in DRalgo via

⬇

InputInv={{1,1},{True,True}}; (*$\psi_{1}\psi_{1}$*)

MassTerm1=CreateInvariantFermion[Group,RepFermion,InputInv][[1]]//Simplify;

InputInv={{2,2},{True,True}}; (*$\psi_{2}\psi_{2}$*)

MassTerm2=CreateInvariantFermion[Group,RepFermion,InputInv][[1]]//Simplify;

InputInv={{1,2},{True,True}}; (*$\psi_{1}\psi_{2}$*)

MassTerm3=CreateInvariantFermion[Group,RepFermion,InputInv][[1]]//Simplify;

The mass matrix is given by

⬇

FermionMasses=1/2*m1*MassTerm1+1/2*m2*MassTerm2+mD MassTerm3;

$\mu$IJ=GradMassFermion[FermionMasses];

$\mu$IJC=SparseArray[Simplify[Conjugate[$\mu$IJ]//Normal,Assumptions->{m1>0,m2>0,mD>0}]];

See the WessZumino.m notebook for a worked example.

####  Q.15 What if a model has $\overline{\Psi}_{1}\gamma_{5}\Psi_{2}$ terms?

Since DRalgo only uses Weyl fermions, the inclusion of other fermionic
bilinear operators requires some extra work. One example is the operator
$\overline{\Psi}_{1}\gamma_{5}\Psi_{2}$ which can be expanded in Weyl spinors

$\displaystyle\Psi_{1}=\begin{pmatrix}\psi_{1}^{L}\\\
\psi_{1}^{R}\end{pmatrix},\quad\Psi_{2}=\begin{pmatrix}\psi_{2}^{L}\\\
\psi_{2}^{R}\end{pmatrix}\;,$ (5.6)

such that

$\displaystyle\overline{\Psi}_{1}\Psi_{2}$
$\displaystyle=\psi_{1}^{R}\psi_{2}^{L}+\psi_{1}^{L}\psi_{2}^{R}\;,$ (5.7)
$\displaystyle\overline{\Psi}_{1}\gamma_{5}\Psi_{2}$
$\displaystyle=-\psi_{1}^{R}\psi_{2}^{L}+\psi_{1}^{L}\psi_{2}^{R}\;.$ (5.8)

These terms can then be implemented in DRalgo as explained in earlier
examples.

####  Q.16 My mass matrix is not diagonal. Is the effective potential still
calculable?

Yes, it is still possible to calculate the effective potential, just not in
general. It is up to the user to diagonalize the mass matrix. But once done,
the user has two choices. First, diagonalization-matrices can be given to
DRalgo. Second, the user can do all the diagonalization themselves and reload
the model. The first option is quick, but becomes protracted if the
diagonalization is complicated or needs to be done perturbatively.

Consider for example the Standard Model. As before, field-dependent masses can
be created via

⬇

DefineNewTensorsUS[$\mu$ij,$\lambda$4,$\lambda$3,gvss,gvvv];

$\phi$vev={0,0,0,$\phi$}//SparseArray;

DefineVEVS[$\phi$vev];

PrintTensorsVEV[];

To diagonalize the vector-bosons, we first need to extract the field-dependent
mass-tensor:

⬇

MassMatrix=PrintTensorsVEV[];

VectorMass=MassMatrix[[2]]//Normal;

VectorEigenvectors=FullSimplify[

Transpose[Normalize/@Eigenvectors[VectorMass[[11;;12,11;;12]]]],

Assumptions->{g1>0,g2>0,$\phi$>0}];

DVRot={{IdentityMatrix[10],0},{0,VectorEigenvectors}}//ArrayFlatten;

DSRot=IdentityMatrix[4];

RotateTensorsUSPostVEV[DSRot,DVRot];

After this the effective-potential calculation proceeds as before.

####  Q.17 Can DRalgo handle non-renormalizable operators?

Currently no. This features will be implemented in future versions.

####  Q.18 How do I define complicated scalar potentials?

For most cases, scalar quartics can be easily defined using the examples
above. For non-standard representations, the model-building tools in DRalgo
might differ by a basis change from other conventions in the literature. Let
us take an example, a $\mathrm{SU}(5)$ model with an adjoint scalar. Commonly
this scalar is written as $\Phi=\Phi^{a}T^{a}$ where $T^{a}$ are traceless
hermitian matrices satisfying ${\rm Tr\,}T^{a}T^{b}=\frac{1}{2}\delta^{ab}$.
The most general (quartic) potential is

$\displaystyle V_{1}=\delta_{1}({\rm Tr\,}\Phi^{2})^{2}+\delta_{2}{\rm
Tr\,}\Phi^{4}\;.$ (5.9)

By default DRalgo uses GroupMath, which defines its invariants differently to
those above. In fact, the two scalar quartic operators given in GroupMath can
be linear combinations of those above. Thus, the result from DRalgo would be
in the form

$\displaystyle
V_{2}=\lambda_{H}\text{inv}_{1}(\Phi^{4})+\lambda_{S}\text{inv}_{2}(\Phi^{4})\;,$
(5.10)

where the invariants $\text{inv}_{1,2}$ are the output from the
CreateInvariant command. Fortunately, it is quite easy to find the relations
between the $V_{1}$ and the $V_{2}$ basis. To do so, first, rewrite everything
in tensor form

$\displaystyle V_{1}=1/4!\lambda_{1,ijkl}\Phi_{i}\Phi_{j}\Phi_{k}\Phi_{l}\;,$
(5.11)

and likewise for the DRalgo output

$\displaystyle V_{2}=1/4!\lambda_{2,ijkl}\Phi_{i}\Phi_{j}\Phi_{k}\Phi_{l}\;.$
(5.12)

Since $\lambda_{1}$ and $\lambda_{2}$ are defined in different bases, we want
to compare invariants. Here we only need three:

$\displaystyle\lambda_{ijkl}\delta_{ij}\delta_{kl}\;,\quad\lambda_{ijkl}\lambda_{ijkl}\;,\quad\lambda_{ijnm}\lambda_{nmkl}\lambda_{klij}\;.$
(5.13)

Comparing the invariants one finds

$\displaystyle\lambda_{H}=\frac{1}{960}\sqrt{\frac{23}{14}}(130\delta_{1}+47\delta_{2})\;,\quad\lambda_{S}=-\frac{5}{64}\sqrt{\frac{13}{42}}(2\delta_{1}-\delta_{2})\;.$
(5.14)

The above procedure works for any representation as long as the user can write
the quartic operator from DRalgo in their preferred form. See SU5.m for a
concrete example.

####  Q.19 Can I use dimensionally reduced theory for calculating nucleation
rates?

Certainly. The user can use the effective theory and directly find the bounce
in the 3d. To first approximation the nucleation rate is
$e^{-S_{\mbox{\scriptsize 3d}}}$, where $S_{\mbox{\scriptsize 3d}}$ is the
bounce action in the dimensionally reduced theory. See [80] for the original
work, and [81, 82, 58, 83] for recent examples.

####  Q.20 Can I study GUT models with DRalgo?

Most GUT models can be readily studied with DRalgo. However, for models with
large representations the running-time rises rather rapidly. The bottleneck is
not the complexity of e.g. the scalar potential but rather the size of the
representations of particles. For example, DRalgo rapidly handles the general
Pati-Salam, $\mathrm{SU}(5)$, 3HDM, or left-right symmetric model – even with
20–30 free parameters in the potential. See SU5.m, 3hdm.m, and LRSymmetric.m
for explicit implementations.

####  Q.21 How much RAM does DRalgo require?

The required RAM is negligible for most models. However, for models with
50–100 dimensional representations, the requirements start to rise. And for
said scenario around 1 GB of RAM is required. If the package drains too much
RAM, it is recommended to not include two-loop contributions.

####  Q.22 How should I report possible bugs?

We are grateful if any bugs are reported. We kindly ask users to supply both a
short description and a Mathematica file describing the error. These reports
and attached files can provided via the Issue Tracker on
https://github.com/DR-algo/DRalgo.

##  6 Outlook

High-temperature field theory is pestered by large radiative corrections,
which compromises perturbative calculations. In effect, perturbation theory
needs to be reorganized in terms of thermal resummations. While at leading
order only thermally corrected masses are required, new contributions to
couplings arise at higher orders. In addition, there are currently several
different schemes for incorporating thermal masses – all depending
substantially on the renormalization scale.

As an effective field theory, dimensional reduction by-passes these issues.
The ultraviolet sector of the theory is controlled by a matching computation
and the infrared sector is resummed to all orders. Indeed, as discussed in
this paper, using a three dimensional EFT unambiguously resums masses and
couplings. Using the framework is simple from a practical standpoint since
three-dimensional theories are super renormalizable. Concretely, only mass
parameters are RG-scale dependent and their beta functions are exact at two-
loop level. This property not only allows for a straightforward perturbative
treatment, but also provides an attractive framework to a non-perturbative
lattice study of the 3d EFT since in the effective theory relations between
continuum and lattice parameters are exact at two-loop level [84].

Hitherto, dimensional reduction has been used sparsely. With this paper and
the associated software, we aim to change this by automating the EFT
construction. In summary, DRalgo calculates all effective couplings and masses
in the effective theory, the leading-order beta functions both in the 4d
parent and 3d effective theory, as well as the effective potential within the
EFT. This facilitates studying models with dimensional reduction and requires
merely three-dimensional calculations that are analogous to zero-temperature
computations. Since the effective theory is fully bosonic, the perturbative
calculation can be compared with lattice simulations. This has one clear
benefit as a large number of fundamental 4d theories can map into the same
effective theory given the mass hierarchy of the additional scalars. Not only
does the lattice provide an invaluable cross-check, but pre-existing
simulations can be reused and applied to new BSM theories.

In conclusion, gravitational waves have opened up a new gateway to the early
universe, and particle physics stands at its threshold. Upcoming experiments
are fast approaching both at future particle colliders and gravitational wave
observatories. It is therefore important to control theoretical uncertainties
at unprecedented precision. In this venture, dimensional reduction is the tool
of choice, and DRalgo its harbinger.

## Acknowledgements

We thank Renato Fonseca and Johan Löfgren for illuminating discussions during
the inception of this project. We express our gratitude to Lauri Niemi and
Jorinde van de Vis for testing the alpha version of the software and providing
invaluable feedback. We also thank Oliver Gould, Joonas Hirvonen, Johan
Löfgren, and Juuso Österman for interesting discussions. The work of Andreas
Ekstedt has been supported by the Deutsche Forschungsgemeinschaft under the
Germany Excellence Strategy - EXC $2121$ Quantum Universe - $390833306$; and
by the Swedish Research Council, project number VR:$2021$-$00363$. Philipp
Schicho was supported by the European Research Council, grant no. 725369, and
by the Academy of Finland, grant no. 1322507. The work of Tuomas V. I.
Tenkanen has been supported in part by the National Science Foundation of
China grant no. 19Z103010239.

##  Appendix A Interfacing DRalgo output

In this appendix, we collect additional material that is not part of the
package itself.

###  A.1 From 3d effective theory to thermodynamics

A possible interface between the output of DRalgo, in the 3-dimensional
theory, and the determination of thermodynamic quantities is implemented in
./examples/ah-thermo.m. This implementation comes with a disclaimer: we
present a simplified algorithm suitable for the simple case of the Abelian-
Higgs model (cf. eq. (2.1)) but this is not part of the package itself. We
encourage users to implement their own numerical minimisation routines for
thermodynamics optimized for individual models. The algorithm given here is
not gauge invariant and should be used with discretion.

The algorithm 2 illustrates our implementation.

Algorithm 2 The DRalgo algorithm output is interfaced in ./examples/ah-
thermo.m with functions solveBetas[], DRstep[] and Veff3d[], which are called
by findThermo[].

Input: Four-dimensional theory parameters $c_{i}\in\\{c_{1},\dots,c_{n}\\}$
and scale factor $X$

for all $c_{i}\in\\{c_{1},\dots,c_{n}\\}$ do

Call findThermo[$X$,$c_{i}$] {

Call $c_{i}(\bar{\mu})$ = solveBetas[$c_{i}$]

for all $T_{i}\in\\{T_{{\mbox{\scriptsize min}}},\dots,T_{{\mbox{\scriptsize
max}}}\\}$ do // e.g. binary search

Fix $T$-dependent RG-scale: $\bar{\mu}=X\pi T$

Call $c_{i,{\mbox{\scriptsize 3d}}}$ = DRstep[$T$,$\bar{\mu}$,$c_{i}$]

Minimize the effective potential by
NMinimize[Re[Veff3d[$\phi$,$c_{i,{\mbox{\scriptsize 3d}}}$]]]

end for

Return {$T_{\rm c}$, $\phi_{{\mbox{\scriptsize c}}}/T_{\rm c}$, $L/T_{\rm
c}^{4}$}, based on degenerate minima.

}

end for

Output: Thermodynamics as function of $\\{c_{1},\dots,c_{n}\\}$

Export the data

Plot by Python: .examples/ah-thermo-python-plots/plot.py

In the simplified algorithm ah-thermo.m the loop over 4-dimensional variables
$c_{i}$ contains only a single $\lambda$ and for simplicity and faster running
time $M=100$ and $g^{2}=0.42$ are fixed. Here, $M$ is the Higgs mass in
arbitrary units related to the $\overline{\rm MS}$ mass parameter by the tree-
level relation $\mu^{2}=-M^{2}/2$. The critical temperature $T_{\rm c}$ is
defined from the condition that the effective potential at the broken minimum
vanishes since the potential at the symmetric minimum at the origin is
normalized to zero. To find this condition, we use an elementary binary
search.

As an indicator of the transition strength, we also output the value of the
background field at the critical temperature, $\phi_{\mbox{\scriptsize
c}}/T_{\rm c}$, that describes the discontinuity of the minima at $T_{\rm c}$.
Since $\phi_{\mbox{\scriptsize c}}/T_{\rm c}$ is gauge-dependent, it should
not be given physical meaning but used as a rough indicator that correlates
positively with the phase transition strength. This strength can be defined in
terms of the released latent heat at the critical temperature $T_{\rm c}$

$L=T\Delta p^{\prime}=T\Delta\frac{{\rm d}V^{{\mbox{\scriptsize
4d}}}_{{\mbox{\scriptsize eff}}}}{{\rm d}T}=T^{2}\Delta\frac{{\rm
d}V^{{\mbox{\scriptsize 3d}}}_{{\mbox{\scriptsize eff}}}}{{\rm d}T}\;,$ (A.1)

where $p^{\prime}$ is the derivative of the pressure with respect to
temperature and $\Delta\equiv(\dots)_{{\mbox{\scriptsize
low-}}T}-(\dots)_{{\mbox{\scriptsize high-}}T}$ denotes the difference between
the broken and symmetric phase. The $T$-derivative is approximated as a finite
difference ${\rm d}T=0.1$. The symmetric-phase part does not contribute here
since we normalized the effective potential to zero at all $T$ at the origin.
For a commented documentation of the technicalities of this implementation see
directly the source ah-thermo.m.

The described perturbative determination is done in Landau gauge with $\xi=0$
in eq. (3.19) and the results for $T_{\rm c}$ and $L/T_{\rm c}^{4}$ are gauge-
dependent. Despite this simple user-friendly example, a recipe for a more
sophisticated, gauge-invariant determination can be found in e.g. [85] (cf.
refs. therein). The final output data for thermodynamics is
stored in ./examples/ah-thermo-python-plots/*.dat,
plotted in ./examples/ah-thermo-python-plots/plot.py, and shown in fig. 4.

Figure 4: Thermodynamics of the Abelian-Higgs model as function of $\lambda$
(cf. eq. (2.1)) at fixed $g^{2}=0.42$ and $M=100$ as obtained in
./examples/ah-thermo.m. Both $M$ and $T$ are in arbitrary units of mass. The
RG-scale is varied at the edges of the interval $\bar{\mu}=(0.5\dots 2.0)\pi
T$. While the curves are barely discernible, this showcases that the scale-
dependence is indeed a higher-order effect once at full NLO dimensional
reduction. The additional RG-scale of the 3d EFT is not varied in the plots.

We encourage DRalgo users to develop and optimize their individual
implementations for algorithms to determine thermodynamic properties. While
including efficient algorithms for determining thermodynamics is conceivable
for future versions of DRalgo, in the current version 1.0 these features are
not implemented in the package itself.

## References

  * [1] L. Canetti, M. Drewes, and M. Shaposhnikov, Matter and Antimatter in the Universe, New J. Phys. 14 (2012) 095012 [1204.4186].
  * [2] G. Elor et al., New Ideas in Baryogenesis: A Snowmass White Paper, in 2022 Snowmass Summer Study, 3, 2022 [2203.05010].
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