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flexiblesusy-paper.tex
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flexiblesusy-paper.tex
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\documentclass[final,3p,11pt,pdflatex]{elsarticle}
\usepackage[utf8x]{inputenc} % input font encoding
\usepackage{amsmath,amssymb}
\usepackage[T1]{fontenc} % output font encoding
\usepackage{booktabs,tabularx}
\usepackage{rotating} % for sidewaystable
\usepackage{xspace}
\usepackage[usenames]{xcolor}
\usepackage{tikz,tikz-uml}
\usepackage{listings}
\usepackage[absolute]{textpos}
\bibstyle{elsarticle-num}
% source code highlighting
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breakatwhitespace=true,
stepnumber=1,
basicstyle=\ttfamily\footnotesize,
commentstyle=\ttfamily\color{gray},
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showstringspaces=false,
frame=single,
abovecaptionskip=0em,
aboveskip=1.5em,
belowcaptionskip=0.5em,
belowskip=1em,
}
\usepackage[pdftitle={FlexibleSUSY --- A spectrum generator generator for supersymmetric models},
pdfauthor={Peter Athron,Jae-hyeon Park,Dominik Stockinger,Alexander Voigt},
pdfkeywords={FlexibleSUSY,supersymmetry,spectrum,generator,MSSM,NMSSM,E6SSM},
bookmarks=true, linktocpage]{hyperref}
%macros
\newcommand{\sarah}{SARAH\@\xspace}
\newcommand{\fs}{FlexibleSUSY\@\xspace}
\newcommand{\mathematica}{Mathematica\xspace}
\newcommand{\ESSM}{E$_6$SSM\@\xspace}
\newcommand{\code}[1]{\lstinline|#1|} % inline source code
\newcommand{\textoverline}[1]{$\overline{\mbox{#1}}$}
\newcommand{\DRbar}{\textoverline{DR}\xspace}
\newcommand{\MSbar}{\textoverline{MS}\xspace}
\newcommand{\unit}[1]{\,\text{#1}} % units
\newcommand{\userinput}{\text{input}}
\newcommand{\pole}{\text{pole}}
\newcommand{\Lagr}{\mathcal{L}}
\newcommand{\unity}{\mathbf{1}}
\newcommand{\figref}[1]{\figurename~\ref{#1}}
\newcommand{\secref}[1]{Section~\ref{#1}}
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\lstset{%
#1,
numbers=left,
firstnumber=auto,
numberstyle=\tiny\sffamily}%
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\makeatother
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\re}{Re}
\DeclareMathOperator{\im}{Im}
\def\at{\alpha_t}
\def\ab{\alpha_b}
\def\as{\alpha_s}
\def\atau{\alpha_{\tau}}
\def\oat{O(\at)}
\def\oab{O(\ab)}
\def\oatau{O(\atau)}
\def\oatab{O(\at\ab)}
\def\oatas{O(\at\as)}
\def\oabas{O(\ab\as)}
\def\oatababq{O(\at\ab + \ab^2)}
\def\oatqatababq{O(\at^2 + \at\ab + \ab^2)}
\def\oatasatq{O(\at\as + \at^2)}
\def\oatasabas{O(\at\as +\ab\as)}
\def\oatasabasatq{O(\at\as + \at^2 +\ab\as)}
\def\oatq{O(\at^2)}
\def\oabq{O(\ab^2)}
\def\oatauq{O(\atau^2)}
\def\oabatau{O(\ab \atau)}
\def\oas{O(\as)}
\def\oatauqatab{O(\atau^2 +\ab \atau )}
\journal{Computer Physics Communications}
\begin{document}
\begin{frontmatter}
\title{\Large\bf FlexibleSUSY --- A spectrum generator generator for supersymmetric models}
\author[adelaide]{Peter Athron}
\author[valencia]{Jae-hyeon Park}
\author[dresden]{Dominik St\"ockinger}
\author[dresden]{Alexander Voigt}
\address[adelaide]{ARC Centre of Excellence for Particle Physics at
the Tera-scale, School of Chemistry and Physics, University of Adelaide,
Adelaide SA 5005 Australia}
\address[valencia]{Departament de F\'{i}sica Te\`{o}rica and IFIC,
Universitat de Val\`{e}ncia-CSIC,
46100, Burjassot, Spain}
\address[dresden]{Institut f\"ur Kern- und Teilchenphysik,
TU Dresden, Zellescher Weg 19, 01069 Dresden, Germany}
\begin{abstract}
We introduce \fs, a \mathematica and C++ package, which generates a fast,
precise C++ spectrum generator for any SUSY model specified by the
user. The generated code is designed with both speed and
modularity in mind, making it easy to adapt and extend with new
features. The model is specified by supplying the superpotential,
gauge structure and particle content in a \sarah model file;
specific boundary conditions e.g.\ at the GUT, weak or
intermediate scales are defined in a separate \fs model file.
From these model files, \fs generates C++ code for self-energies,
tadpole corrections, renormalization group equations (RGEs) and
electroweak symmetry breaking (EWSB) conditions and combines them
with numerical routines for solving the RGEs and EWSB conditions
simultaneously. The resulting spectrum generator is then able to
solve for the spectrum of the model, including loop-corrected pole
masses, consistent with user specified boundary conditions. The
modular structure of the generated code allows for individual
components to be replaced with an alternative if available. \fs
has been carefully designed to grow as alternative solvers and
calculators are added. Predefined models include the MSSM, NMSSM,
\ESSM, USSM, $R$-symmetric models and models with right-handed
neutrinos.
\end{abstract}
\begin{keyword}
sparticle,
supersymmetry,
Higgs,
renormalization group equations
\PACS 12.60.Jv
\PACS 14.80.Ly
\end{keyword}
\end{frontmatter}
% report numbers
\begin{textblock*}{7em}(\textwidth,1cm)
\noindent\footnotesize
FTUV--14--3904 \\
IFIC--14--40 \\
ADP-14-17/T875
\end{textblock*}
\section{Program Summary}
\noindent{\em Program title:} \fs\\ {\em Program obtainable from:}
{\tt http://flexiblesusy.hepforge.org/}\\ {\em Distribution
format:}\/ tar.gz\\ {\em Programming language:} {\tt
C++, Wolfram/\mathematica, FORTRAN, Bourne shell}\\ {\em Computer:}\/ Personal computer\\ {\em Operating
system:}\/ Tested on Linux 3.x, Mac OS X\\
{\em External routines:}\/ SARAH 4.0.4, Boost library,
Eigen, LAPACK\\ {\em
Typical running time:}\/ 0.06-0.2 seconds per parameter
point.\\ {\em Nature of problem:}\/ Determining the mass
spectrum and mixings for any supersymmetric model. The
generated code must find simultaneous solutions to
constraints which are specified at two or more different
renormalization scales, which are connected by
renormalization group equations forming a large set of
coupled first-order differential equations. \\ {\em Solution method:}\/
Nested iterative algorithm and numerical minimization of the
Higgs potential. \\ {\em Restrictions:} The couplings must
remain perturbative at all scales between the highest and
lowest boundary condition. \fs~ assumes that all couplings
of the model are real (i.e.\ $CP-$conserving). Due to the
modular nature of the generated code adaption and extension
to overcome restrictions in scope is quite straightforward.
\newpage
\section{Introduction}
Supersymmetry (SUSY) provides the only non-trivial way to extend the
space-time symmetries of the Poincar\'e
group \cite{Coleman:1967ad,Haag:1974qh}, leading many to suspect that
SUSY may be realized in nature in some form. In particular
supersymmetric extensions of the standard model (SM) where SUSY is broken
at the TeV scale have been proposed to solve the hierarchy
problem \cite{Weinberg:1975gm, Weinberg:1979bn, Gildener:1976ai,
Susskind:1978ms, 'tHooft:1980xb}, allow gauge coupling
unification \cite{Langacker:1990jh, Ellis:1990wk, Amaldi:1991cn,
Langacker:1991an, Giunti:1991ta} and predict a dark matter candidate
which can fit the observed relic
density \cite{Goldberg:1983nd,Ellis:1983ew}. Such models have also
been used for baryogenesis or leptogensis to solve the
matter-anti-matter asymmetry of the universe and have been considered
as the low energy effective models originating from string
theory.
Detailed phenomenological studies have been carried out for scenarios
within the minimal supersymmetric standard model (MSSM). Such work
has been greatly aided by public spectrum generators for the MSSM
\cite{Allanach:2001kg,Porod:2003um,Djouadi:2002ze,Baer:1993ae,Chowdhury:2011zr},
allowing fast and reliable exploration of the sparticle spectrum,
mixings and couplings, which can be obtained from particular choices
of breaking mechanism inspired boundary conditions and specified
parameters. Beyond the MSSM there are also two public spectrum
generators \cite{Ellwanger:2006rn,Allanach:2013kza} for the next to
minimal supersymmetric standard model (NMSSM) \cite{NMSSM} (or for recent
reviews see \cite{Ellwanger:2009dp,Maniatis:2009re}).
None of the fundamental motivations of supersymmetry require
minimality, and specific alternatives to (or extensions of) the MSSM
are, for example, motivated by the $\mu$-problem of the MSSM
\cite{Kim:1983dt}; explaining the family structure (see
e.g.~\cite{King:2014nza}) or for successful baryogenesis or
leptogenesis (see e.g.~\cite{King:2008qb}). However constructing
specialized tools to study all relevant models would require an
enormous amount of work. So general tools which can automate this
process and produce fast and reliable programs can greatly enhance our
ability to understand and test non-minimal realizations of
supersymmetry.
Recent experimental developments have also increased the relevancy of
such a tool. From the recent $7$ TeV and $8$ TeV runs at the Large
Hadron Collider (LHC) there have been two important developments.
Firstly low energy signatures expected from such models, such as the
classic jets plus missing transverse energy signature, have not been
observed, substantially raising the lower limit on sparticle masses
(see e.g.~\cite{Aad:2013wta,Chatrchyan:2014lfa}). No other signature
of beyond the standard model (BSM) physics has been observed, leaving
the fundamental questions which motivated BSM physics
unanswered. Secondly ATLAS and CMS discovered \cite{ATLAS:2012ae,
Chatrchyan:2012tx} a light Higgs of $125$ GeV, within the mass range
that could be accommodated in the MSSM but requiring stops which are
significantly heavier than both the direct collider limits and
indirect limits that appears in constrained models from the
significantly higher limits on first and second generation squarks.
These developments motivate the exploration of non-minimal SUSY models
which ameliorate the naturalness problems, as can happen in the \ESSM
\cite{King:2005jy,King:2005my,King:2007uj,Athron:2010zz}, USSM
\cite{Fayet:1977yc,Suematsu:1994qm,Cvetic:1995rj,deCarlos:1997yv,Cvetic:1997ky,Demir:1998dk,Langacker:1998tc,Erler:2002pr,Choi:2006fz,Ham:2007wc,Langacker:2008yv,Ham:2008xf,Kalinowski:2008iq} or from other gauge extensions \cite{Batra:2003nj, Medina:2009ey, Bharucha:2013ela},
by raising the tree level Higgs mass. At the same time they can also
motivate models that are developed with a fresh perspective, based on
other considerations. In both cases exploration of such models can be
aided if it is possible to quickly create a fast spectrum generator.
Currently there is only one option for this, a SPheno-like FORTRAN
code which can be generated from \sarah
\cite{Staub:2010ty,Staub:2009bi,Staub:2010jh,Staub:2012pb,Staub:2013tta}.
\fs provides a much needed alternative to this with a structure which
has been freshly designed to accommodate as general range of models as
possible and to be easily adaptable to changing goals and new
ideas. \fs is a \mathematica and C++ package which uses \sarah to create a
fast, modular C++ spectrum generator for a user specified SUSY model.
The generated code structure is designed to be as flexible as possible
to accommodate different types of extensions and due to its modular
nature it is easy to modify, add new features and combine with other
programs. The generated code has been extensively tested against well
known spectrum generators. As well as providing a solution for new
SUSY models, the generated MSSM and NMSSM codes offer a modern and fast
alternative to the existing public spectrum generators.
In \secref{sec:Program} we describe the program in more detail and
explain our design goals. In \secref{sec:download} information on how
to download and compile the code may be found along with details on
how to get started quickly. In \secref{sec:modfile} we describe how
the user can create a new \fs model file. A detailed
description of the structure and features of the generated code is
then given in \secref{sec:SpecGenStruct}. In \secref{sec:Flexible} we
describe the various ways the code can be modified both at the meta
code level by writing model files and at the C++ code level by
modifying the code or adding new modules. Finally in
\secref{sec:comparison} we describe detailed comparisons between our
generated code and existing public spectrum generators as well as
against the SPheno-like FORTRAN code which can be created using SARAH.
\section{Overview of the program and design goals}
\label{sec:Program}
To study the properties of SUSY models programs are needed which
numerically calculate the pole masses and couplings of the SUSY
particles given a set of theory input parameters. The output of these
so-called spectrum generators can be transferred to programs which
calculate further observables such as branching ratios or the dark
matter relic density.
In order to create a spectrum generator the SUSY model must be defined
by specifying the gauge group, the field content and mixings as well
as the superpotential and the soft-breaking terms. From this
information the renormalization group equations, mass matrices,
self-energies, tadpole diagrams and electroweak symmetry breaking
(EWSB) conditions have to be derived. These expressions must then be
combined in a computer program to allow for a numeric calculation of
the mass spectrum. In addition most SUSY models require boundary
conditions for the model parameters at a low and a high scale. For
example in the CMSSM mSUGRA boundary conditions for the soft-breaking
parameters are imposed at the gauge coupling unification scale.
Furthermore, at the $Z$ mass scale the CMSSM is matched to the Standard
Model, which implies conditions for the gauge and Yukawa couplings.
The so defined boundary value problem must be solved numerically until
a set of model parameters has been found consistent with all
user-supplied boundary conditions.
\fs is a \mathematica and C++ package designed to create a fast and easily
adaptable spectrum generator in C++ for any SUSY model.
The user specifies the model by giving the
superfield content, superpotential, gauge symmetries and mass mixings
in form of \sarah model files. The boundary conditions on the model
parameters must be specified in a separate \code{FlexibleSUSY.m.in} file.
Based on this information \fs uses \sarah to obtain tree-level
expressions for the mass matrices and electroweak symmetry breaking
conditions, one-loop self energies, one-loop tadpoles corrections and
two-loop renormalization group equations (RGEs) for the model.
Additional corrections which have been calculated elsewhere, such as
two-loop corrections to the Higgs masses\footnote{By default
\fs has two-loop corrections to the Higgs masses for the
MSSM
\cite{Degrassi:2001yf,Brignole:2001jy,Dedes:2002dy,Brignole:2002bz,Dedes:2003km}
and NMSSM \cite{Degrassi:2009yq} in FORTRAN files supplied by Pietro
Slavich. These are the same corrections which are implemented in
many of the public spectrum generators.} may be added by the user.
%
These algebraic expressions are converted into C++ code and are put
into classes with well-defined interfaces to allow for easy exchange,
extension and reuse of the modules. All of these classes are finally
combined to a complete spectrum generator, which solves the
user-defined boundary value problem. For this task \fs uses some
parts of Softsusy \cite{Allanach:2001kg}, the very fast Eigen library
\cite{eigen}, augmented by LAPACK, as well as the GNU scientific
library and the Boost library to create numerical routines which solve
the RGEs and boundary conditions simultaneously. If a solution has
been found the pole mass spectrum is eventually calculated using full
one-loop self-energies (and leading two-loop Higgs self-energy
contributions for the MSSM and NMSSM).
\subsection*{Design goals}
Since the calculation of the pole mass spectrum in a SUSY model is a
non-trivial task, \fs is designed with the following points in mind:
\paragraph{Modularity}
The large variety of supersymmetric models and potential
investigations makes it likely that the user wants to modify the
generated spectrum generator source code or reuse components in
further programs. \fs offers two levels to influence the code: (i) On
the \mathematica model file level the model itself or GUT/weak scale
boundary conditions as well as input and output parameters can be
controlled (see \secref{sec:quick-start-alternative-models} and
\secref{sec:adapting-model-files} for examples). (ii) In particular
\fs uses C++ object orientation features to modularize the source code
so that it is sharply divided into building blocks performing
distinct duties.
This modular architecture makes it easy
for the user to modify, reuse, replace or
extend the individual components (see \secref{sec:adapting-cpp-code}
for examples). An important application of this concept are the
boundary conditions, for which the C++ level offers a wider range of
possibilities. The boundary conditions solver provides a plugin
mechanism via a common \code{Constraint} interface, which allows a
user to exchange or add boundary conditions at any scale.
To realize this, all (derived) constraint objects are intentionally
kept outside the solver. Despite being independent of one another,
they can fit together with the aid of class inheritance. An
elaborate example of a tower of effective field theories and multiple
matching scales is presented in \secref{sec:tower construction}.
Alternatively, the modular structure makes it straightforward to take
\fs generated code for e.g.\ RGEs or self-energies and reuse it in an
existing code for some other purpose. Conversely, it is also easy to
include code from elsewhere into the spectrum generator. For an
example see \secref{sec:integrating-custom-built}.
\paragraph{Speed}
Exploring the parameter space of supersymmetric models with a high
number of free parameters is quite time consuming. Therefore \fs aims
to produce spectrum generators with a short run-time. The two most
time consuming parts of a SUSY spectrum generator are usually the
calculation of the two-loop $\beta$-functions and the pole masses of
mixed particles:
%
\begin{itemize}
\item \emph{Calculation of the $\beta$-functions:} The RG solving
algorithms usually need $O(10)$ iterations between the high and the
low scale to find a set of parameters consistent will all boundary
conditions with a $0.01\%$ precision goal. During each iteration
the Runge-Kutta algorithm needs to calculate all $\beta$-functions
$O(50)$ times. Most two-loop $\beta$-functions involve $O(50)$
matrix multiplications and additions. All together one arrives at
$O(10^4)$ matrix operations. To optimize these, \fs uses the fast
linear algebra package \href{Eigen}{http://eigen.tuxfamily.org}.
Eigen uses C++ expression templates to remove temporary objects and
enable lazy evaluation of the expressions. It supports explicit
vectorization, and provides fixed-size matrices to avoid dynamic
memory allocation. All of these features in combination result in
very fast code for the calculation of the $\beta$-functions in \fs.
%
\item \emph{Calculation of the pole masses:} The second most time
consuming part is the precise calculation of the pole masses for
mixed particles. For each particle $\psi_k$ in a multiplet the full
self-energy matrix $\Sigma^\psi_{ij}(p=m^\text{tree}_{\psi_k})$ has
to be evaluated. Each self-energy matrix entry again involves the
calculation of $O(50)$ Feynman diagrams, each involving the
calculation of vertices and a loop-function. All in all, one
arrives at $O(500)$ Feynman diagrams and $O(10^4)$ loop function
evaluations. To speed up the calculation of the pole masses \fs
makes use of multi-threading, where each pole mass is calculated in
a separate thread. This allows the operating system to distribute
these calculations among different CPU cores. With this technique
one can gain a speed-up of $20$--$30\%$.
\end{itemize}
\paragraph{Alternative boundary value problem solvers}
Furthermore, the standard algorithm which solves the user-defined
boundary value problem via a fixed-point iteration is not guaranteed
to converge in all regions of the model parameter space. Therefore,
\fs has been intentionally designed to allow for alternative solvers
to search for solutions in such critical parameter regions. A
subsequent release with an alternative solver is already planned.
\paragraph{Towers of effective theories}
In \fs the standard fixed-point iteration solver has been generalized
to handle towers of models (effective theories), which are matched at
intermediate scales. An example of such a tower construction will be
given in \secref{sec:tower construction}, where right-handed neutrinos
are integrated out at the see-saw scale, between the SUSY and the GUT
scale.
\section{Quick start}
\label{sec:download}
\subsection{Requirements}
\fs can be downloaded from \url{http://flexiblesusy.hepforge.org}. To
create a custom spectrum generator the following requirements are
necessary:
%
\begin{itemize}
\item \mathematica, version 7 or higher
\item SARAH, version 4.0.4 or higher \url{http://sarah.hepforge.org}
\item C++11 compatible compiler (g++ 4.4.7 or higher, clang++ 3.1 or
higher, icpc 12.1 or higher)
\item FORTRAN compiler (gfortran, ifort etc.)
\item Eigen library, version 3.1 or higher
\url{http://eigen.tuxfamily.org}
\item Boost library, version 1.36.0 or higher
\url{http://www.boost.org}
\item GNU scientific library \url{http://www.gnu.org/software/gsl}
\item an implementation of LAPACK \url{http://www.netlib.org/lapack}
such as ATLAS \url{http://math-atlas.sourceforge.net} or
Intel Math Kernel Library \url{http://software.intel.com/intel-mkl}
\end{itemize}
%
Optional:
%
\begin{itemize}
\item Looptools, version 2.8 or higher
\url{http://www.feynarts.de/looptools}
\end{itemize}
\subsection{Downloading \fs and generating a first spectrum generator}
\label{sec:quick-start-cmssm}
\fs can be downloaded as a gzipped tar file from
\url{http://flexiblesusy.hepforge.org}. To download and install
version 1.0.0 run:
%
\begin{lstlisting}[language=bash]
$ wget https://www.hepforge.org/archive/flexiblesusy/FlexibleSUSY-1.0.0.tar.gz
$ tar -xf FlexibleSUSY-1.0.0.tar.gz
$ cd FlexibleSUSY-1.0.0
\end{lstlisting}%% $
%
A CMSSM spectrum generator can be created with the following three
commands:
%
\begin{lstlisting}[language=bash]
$ ./createmodel --name=MSSM
$ ./configure --with-models=MSSM
$ make
\end{lstlisting}%% $
%
The first command creates the model directory \code{models/MSSM/}
together with a CMSSM model file. The \code{configure} script checks
the system requirements and creates the \code{Makefile}. See
\code{./configure --help} for more options. Executing \code{make}
will start \mathematica to generate the spectrum generator and compile
it. The resulting executable can be run like this:
%
\begin{lstlisting}[language=bash]
$ cd models/MSSM
$ ./run_MSSM.x --slha-input-file=LesHouches.in.MSSM
\end{lstlisting}
%
When executed, the spectrum generator tries to find a set of \DRbar\
model parameters consistent with all CMSSM boundary conditions for the
parameter point given in the SLHA input file
\code{model_files/MSSM/LesHouches.in.MSSM}. Afterwards, the pole mass
spectrum and mixing matrices are calculated and written to the standard
output in SLHA format \cite{Skands:2003cj,Allanach:2008qq}. For the
parameter point given in the above example the calculated pole mass
spectrum reads
%
\begin{lstlisting}
Block MASS
1000021 1.15236966E+03 # Glu
1000024 3.85774334E+02 # Cha_1
1000037 6.50460073E+02 # Cha_2
25 1.14766149E+02 # hh_1
35 7.06792640E+02 # hh_2
37 7.11388516E+02 # Hpm_2
36 7.06523105E+02 # Ah_2
1000012 3.51856376E+02 # Sv_1
1000014 3.53042556E+02 # Sv_2
1000016 3.53046504E+02 # Sv_3
1000022 2.03889780E+02 # Chi_1
1000023 3.85760714E+02 # Chi_2
1000025 6.36544884E+02 # Chi_3
1000035 6.50133768E+02 # Chi_4
1000001 9.66656018E+02 # Sd_1
1000003 1.00983181E+03 # Sd_2
1000005 1.01651873E+03 # Sd_3
2000001 1.01653005E+03 # Sd_4
2000003 1.06089534E+03 # Sd_5
2000005 1.06090238E+03 # Sd_6
1000011 2.22570305E+02 # Se_1
1000013 2.29864536E+02 # Se_2
1000015 2.29888846E+02 # Se_3
2000011 3.61946671E+02 # Se_4
2000013 3.61950866E+02 # Se_5
2000015 3.63136031E+02 # Se_6
1000002 8.09787818E+02 # Su_1
1000004 1.01454197E+03 # Su_2
1000006 1.01981109E+03 # Su_3
2000002 1.02015269E+03 # Su_4
2000004 1.05807759E+03 # Su_5
2000006 1.05808168E+03 # Su_6
\end{lstlisting}
\subsection{Spectrum generators for alternative models}
\label{sec:quick-start-alternative-models}
\fs already comes with plenty of predefined models: the CMSSM (simply
called \code{MSSM}), the NMSSM \cite{NMSSM} in it's $Z_3$-symmetric form (called \code{NMSSM}), $Z_3$-violating NMSSM (\code{SMSSM}), the USSM (\code{UMSSM})
\cite{Fayet:1977yc,Suematsu:1994qm,Cvetic:1995rj,deCarlos:1997yv,Cvetic:1997ky,Demir:1998dk,Langacker:1998tc,Erler:2002pr,Choi:2006fz,Ham:2007wc,Langacker:2008yv,Ham:2008xf,Kalinowski:2008iq},
the NUHM \ESSM (\code{E6SSM}) \cite{Athron:2007en}, the
right-handed neutrino extended MSSM (\code{MSSMRHN}), the NUHM-MSSM
(\code{NUHMSSM}) and the $R$-symmetric MSSM (\code{MRSSM})
\cite{Kribs:2007ac}. See the content of \code{model_files/} for all
predefined model files. For all these models spectrum generators can
be generated easily like for the CMSSM in
\secref{sec:quick-start-cmssm}. The spectrum generator for the
$Z_3$-symmetric NMSSM for example can be generated like this:
%
\begin{lstlisting}[language=bash]
$ ./createmodel --name=NMSSM
$ ./configure --with-models=NMSSM
$ make
\end{lstlisting}%% $
%
One of the design goals is modularity and the possibility to easily
construct custom spectrum generators. The details of the
customization can be found in Sections
\ref{sec:modfile}--\ref{sec:Flexible}. As a simple example consider
the NMSSM. The NMSSM variant above unifies all soft-breaking
trilinear scalar couplings at the GUT scale. In order to relax this
constraint and use a separate value for $A_\lambda$ at the GUT scale
one can edit the model file \code{model_files/NMSSM/FlexibleSUSY.m.in} and
change the lines
%
\begin{lstlisting}[language=Mathematica]
EXTPAR = { {61, LambdaInput} };
HighScaleInput = {
...
{T[\[Lambda]], Azero LambdaInput},
...
};
\end{lstlisting}
%
into
%
\begin{lstlisting}[language=Mathematica]
EXTPAR = { {61, LambdaInput},
{63, ALambdaInput} };
HighScaleInput = {
...
{T[\[Lambda]], ALambdaInput LambdaInput},
...
};
\end{lstlisting}
%
The value of $A_\lambda$ at the GUT scale can then be set in the SLHA
input file in the \code{EXTPAR} block entry $63$ via
%
\begin{lstlisting}
Block EXTPAR
61 0.1 # LambdaInput
63 -100 # ALambdaInput
\end{lstlisting}
\section{Setting up a FlexibleSUSY model}
\label{sec:modfile}
A general (non-constrained) softly broken SUSY model is defined by the
gauge group, the field content and mixings as well as the
superpotential and the soft-breaking Lagrangian. In order to create a
spectrum generator for such a SUSY model with \fs, the aforementioned
model properties have to be defined in a SARAH model file. The SARAH
model file can be put into the \code{sarah/<model>/} directory. See
the SARAH manual \cite{Staub:2008uz,Staub:2013tta} for a detailed
explanation of how to write such a model file. Note that SARAH
already is distributed with a lot of predefined models, which can be used with
\fs immediately.
The model boundary conditions are defined in the \fs model file
\code{FlexibleSUSY.m}, which has to be located in the model directory
\code{models/<model>/}. To add this the user should create a
\code{FlexibleSUSY.m.in} file in the directory
\code{model_files/<model>/}. When the \code{./createmodel} script is
executed, the \code{FlexibleSUSY.m} file is created from the
\code{model_files/<model-file-name>/FlexibleSUSY.m.in} file, where the
directory \code{<model-file-name>} is the specified by the
\code{--model-file=<model-file-name>} option. If no such option is
given the directory matching the \code{--name=<model>} option is used.
In either case the \code{FlexibleSUSY.m} file which is created is then
automatically placed in the directory \code{models/<model>/}. Note
that many predefined example model files can already be found in
\code{model_files/}.
In the following it is explained how the
boundary conditions can be defined on the basis of the CMSSM. The
application to other models is straightforward. The CMSSM model file
reads:
%
\begin{lstlisting}[language=Mathematica]
FSModelName = "@CLASSNAME@";
MINPAR = {
{1, m0},
{2, m12},
{3, TanBeta},
{4, Sign[\[Mu]]},
{5, Azero}
};
EWSBOutputParameters = { B[\[Mu]], \[Mu] };
HighScale = g1 == g2;
HighScaleFirstGuess = 2.0 10^16;
HighScaleMinimum = 1.0 10^10; (* optional *)
HighScaleMaximum = 1.0 10^18; (* optional *)
HighScaleInput = {
{T[Ye], Azero*Ye},
{T[Yd], Azero*Yd},
{T[Yu], Azero*Yu},
{mHd2, m0^2},
{mHu2, m0^2},
{mq2, UNITMATRIX[3] m0^2},
{ml2, UNITMATRIX[3] m0^2},
{md2, UNITMATRIX[3] m0^2},
{mu2, UNITMATRIX[3] m0^2},
{me2, UNITMATRIX[3] m0^2},
{MassB, m12},
{MassWB, m12},
{MassG, m12}
};
SUSYScale = Sqrt[M[Su[1]]*M[Su[6]]];
SUSYScaleFirstGuess = Sqrt[m0^2 + 4 m12^2];
SUSYScaleInput = {};
LowScale = SM[MZ];
LowScaleFirstGuess = SM[MZ];
LowScaleInput = {
{Yu, Automatic},
{Yd, Automatic},
{Ye, Automatic},
{vd, 2 MZDRbar / Sqrt[GUTNormalization[g1]^2 g1^2 + g2^2]
Cos[ArcTan[TanBeta]]},
{vu, 2 MZDRbar / Sqrt[GUTNormalization[g1]^2 g1^2 + g2^2]
Sin[ArcTan[TanBeta]]}
};
InitialGuessAtLowScale = {
{vd, SM[vev] Cos[ArcTan[TanBeta]]},
{vu, SM[vev] Sin[ArcTan[TanBeta]]},
{Yu, Automatic},
{Yd, Automatic},
{Ye, Automatic}
};
InitialGuessAtHighScale = {
{\[Mu] , 1.0},
{B[\[Mu]], 0.0}
};
UseHiggs2LoopMSSM = True;
EffectiveMu = \[Mu];
OnlyLowEnergyFlexibleSUSY = False; (* default *)
PotentialLSPParticles = { Chi, Cha, Glu, Sv, Su, Sd, Se };
DefaultPoleMassPrecision = MediumPrecision;
HighPoleMassPrecision = {hh, Ah, Hpm};
MediumPoleMassPrecision = {};
LowPoleMassPrecision = {};
\end{lstlisting}
%
The first line \code{FSModelName = "@CLASSNAME@";} will be replaced
with \code{FSModelName = "<model>";} in the generated
\code{FlexibleSUSY.m} file, where \code{<model>} is specified by the
\code{--name=<model>} option for the \code{./createmodel} script. So
the variable \code{FSModelName} then contains the name of the \fs
model.
All non-Standard Model input variables must be specified in the lists
\code{MINPAR} and \code{EXTPAR}. These two variables refer to the
MINPAR and EXTPAR blocks in a SLHA input file \cite{Skands:2003cj}.
The list elements are two-component lists where the first entry is the
SLHA index in the MINPAR or EXTPAR block, respectively, and the second
entry is the name of the input parameter. In the above example the
input parameters are the universal soft-breaking parameters $m_0$,
$M_{1/2}$, $A_0$ as well as $\tan\beta$ and $\sign\mu$.
Using the variable \code{EWSBOutputParameters} the user can
specify the model parameters that are output of the electroweak
symmetry breaking consistency conditions. When imposing the EWSB, \fs
will adjust these parameters until the EWSB conditions are fulfilled.
In the CMSSM example above these are the superpotential parameter
$\mu$ and its corresponding soft-breaking parameter $B\mu$. In the
NMSSM the parameters $\kappa$, $|v_s|$ and $m_s^2$ are usually chosen
for this purpose.
Furthermore, the user has to specify three model constraints:
low-scale, SUSY-scale and high-scale. In \fs they are named as
\code{LowScale}, \code{SUSYScale} and \code{HighScale}. For each
constraint there is (i) a scale definition (named after the
constraint), (ii) an initial guess for the scale (concatenation of the
constraint name and \code{FirstGuess}) and (iii) a list of parameter
settings to be applied at the scale (concatenation of the constraint
name and \code{Input}). Optionally a minimum and a maximum value for
the scale can be given (concatenation of the constraint name and
\code{Minimum} or \code{Maximum}, respectively). The latter avoids
underflows or overflows of the scale value during the iteration. This
is especially useful in models where the iteration is very unstable
and the value of the scale is very sensitive to the model parameters.
The meaning of the three constraints is the following:
%
\begin{itemize}
\item \emph{High-scale constraint:} The high-scale constraint is
usually the GUT-scale constraint, imposed at the scale where the
gauge couplings $g_1$ and $g_2$ unify. The high-scale can be
defined by an equation of the form \code{g1 == g2} or by a fixed
numerical value. Note that \fs GUT-normalizes all gauge couplings.
Thus, the high-scale definition takes the simple form \code{g1 ==
g2}. As a consequence in the calculation of the VEVs $v_u$ and
$v_d$ from $M_Z$ and $\tan\beta$ at the low-scale the
GUT-normalization has to be taken into account, see the example
above.
%
\item \emph{SUSY-scale constraint:} The SUSY-scale is the typical mass
scale of the SUSY particle spectrum. At this scale \fs imposes the
EWSB conditions and calculates the pole mass spectrum. The
SUSY-scale, $M_S$, is usually defined as $M_S =
\sqrt{m_{\tilde{t}_1}m_{\tilde{t}_2}}$. However, in the example
above, where sfermion flavour violation is enabled, it has the value
$M_S = \sqrt{m_{\tilde{u}_1}m_{\tilde{u}_6}}$, where
$m_{\tilde{u}_1}$ and $m_{\tilde{u}_6}$ are the \DRbar\ masses of
the lightest and heaviest up-type squark, respectively.
%
\item \emph{Low-scale constraint:} The low-scale constraint is the
constraint where the SUSY model is matched to the Standard Model.
This is done by automatically calculating the gauge couplings $g_i$
($i=1,2,3$) of the SUSY model from the known Standard Model
quantities $\alpha_{\text{e.m.}}(M_Z)$, $\alpha_{s}(M_Z)$, $M_Z$,
$M_W$. The details of this calculation are explained in
\secref{sec:calculation-of-gauge-couplings}. Currently this scale
is fixed to be the $Z$ pole mass scale $M_Z$. Optionally the Yukawa
couplings $y_f$ ($f=u,d,e$) can be calculated automatically from the
known Standard Model fermion masses $m_f$ by setting their values to
\code{Automatic}. This automatic calculation is explained in
\secref{sec:calculation-of-yukawa-couplings}.
\end{itemize}
%
The variables \code{LowScaleInput}, \code{SUSYScaleInput}
\code{HighScaleInput}, which list the parameter settings for imposing
the constraints can contain as elements any of the following:
%
\begin{itemize}
\item Two-component lists of the form \code{\{parameter, value\}},
which indicates that the \code{parameter} is set to \code{value} at
the defined scale. If the \code{value} should be read from the SLHA
input file, it must be written as \code{LHInput[value]}. Example:
%
\begin{lstlisting}
SUSYScaleInput = {
{mHd2, m0^2},
{mHu2, LHInput[mHu2]}
};
\end{lstlisting}
%
In this example the parameter \code{mHd2} is set to the value of
\code{m0^2}, and \code{mHu2} is set to the value given in the SLHA
input file in block \code{MSOFTIN}, entry 22 at the SUSY scale. The
SLHA block names and keys for the MSSM and NMSSM are defined in
SARAH's \code{parameters.m} file, see the SARAH manual or
\cite{Staub:2010jh}. For the Standard Model Yukawa couplings
\code{Yu}, \code{Yd}, \code{Ye} the value \code{Automatic} is
allowed, which triggers their automatic determination from the known
Standard Model quark and lepton masses, see
\secref{sec:calculation-of-yukawa-couplings}.
\item The function \code{FSMinimize[parameters, function]} can be
given, where \code{parameters} is a list of model parameters and
\code{function} is a function of these parameters.
\code{FSMinimize[parameters, function]} will numerically vary the
\code{parameters} until the \code{function} is minimized. Example:
%
\begin{lstlisting}
FSMinimize[{vd,vu},
(SM[MZ] - Pole[M[VZ]])^2 / STANDARDDEVIATION[MZ]^2 +
(SM[MH] - Pole[M[hh[1]]])^2 / STANDARDDEVIATION[MH]^2]
\end{lstlisting}
%
Here, the parameters \code{vu} and \code{vd} are varied until the
function
%
\begin{align}
\chi^2(v_d,v_u) =
\frac{(\texttt{SM[MZ]}-m_Z^\pole)^2}{\sigma_{m_Z}^2} +
\frac{(\texttt{SM[MH]}-m_{h_1}^\pole)^2}{\sigma_{m_h}^2}
\end{align}
%
is minimal. The constants \code{SM[MZ]}, \code{SM[MH]},
$\sigma_{m_Z}$ and $\sigma_{m_h}$ are defined in
\code{src/ew_input.hpp} to be
%
\begin{align}
\texttt{SM[MZ]} &= 91.1876, &
\texttt{SM[MH]} &= 125.9, \\
\sigma_{m_Z} &= 0.0021, &
\sigma_{m_h} &= 0.4 .
\end{align}
\item The function \code{FSFindRoot[parameters, functions]} can be
given, where \code{parameters} is a list of model parameters and
\code{functions} is a list of functions of these parameters.
\code{FSFindRoot[parameters, functions]} will numerical vary the
\code{parameters} until the \code{functions} are zero. Example:
%
\begin{lstlisting}
FSFindRoot[{vd,vu},
{SM[MZ] - Pole[M[VZ]], SM[MH] - Pole[M[hh[1]]]}]
\end{lstlisting}
%
Here, the parameters \code{vu} and \code{vd} are varied until the
vector-valued function
%
\begin{align}
f(v_d,v_u) =
\begin{pmatrix}
\texttt{SM[MZ]} - m_Z^\pole \\
\texttt{SM[MH]} - m_{h_1}^\pole
\end{pmatrix}
\end{align}
%
is zero.
\end{itemize}
%
Finally, the user can set an initial guess for the model parameters at
the low- and high-scale using the variables
\code{InitialGuessAtLowScale} and \code{InitialGuessAtHighScale},
respectively. The gauge couplings will be guessed automatically at
the low-scale from the known Standard Model parameters.
\fs allows to add leading two-loop contributions to the CP-even Higgs
tadpoles and self-energies. For MSSM-like models (with two CP-even
Higgs bosons, one CP-odd Higgs boson, one neutral Goldstone boson)
these corrections can be enabled by setting \code{UseHiggs2LoopMSSM =
True} in the model file and by defining the effective $\mu$-term
\code{EffectiveMu = \\[Mu]}. This will add the zero-momentum
corrections of the order $O(y_t^4 + y_b^2 y_t^2 + y_b^4)$, $O(y_t^2
g_3^2)$, $O(y_b^2 g_3^2)$, $O(y_\tau^4)$ from
\cite{Degrassi:2001yf,Brignole:2001jy,Dedes:2002dy,Brignole:2002bz,Dedes:2003km}.
For NMSSM-like models (with three CP-even Higgs bosons, two CP-odd
Higgs bosons, one neutral Goldstone boson) the two-loop contributions
are enabled by setting \code{UseHiggs2LoopNMSSM = True} and by
defining the effective $\mu$-term like \code{EffectiveMu = \\[Lambda]
vS / Sqrt[2]}, for example. This will add the zero-momentum
corrections of the order $O(y_t^2 g_3^2)$, $O(y_b^2 g_3^2)$ from
\cite{Degrassi:2009yq}, plus MSSM-like contributions of the order
$O(y_\tau^4)$, $O(y_t^4 + y_t^2 y_b^2 + y_b^4)$
\cite{Brignole:2001jy,Dedes:2003km}.
One can create a pure low-energy model by setting
\code{OnlyLowEnergyFlexibleSUSY = True}. In this case the high-scale
constraint is ignored and only the low-scale and SUSY-scale
constraints are used. All model parameters which are not specified in
\code{MINPAR} or \code{EXTPAR} will then be read from the
corresponding input blocks in the SLHA input file and will be set at
the SUSY-scale. An example of such a pure low-energy model is the
MRSSM, where the three gauge couplings do not unify at a common scale.
\fs can create the helper function \code{get_lsp()}, which finds the
lightest supersymmetric particle (LSP). To have this function be
created the model file variable \code{PotentialLSPParticles} must be
set to a list of SUSY particles which are potential LSPs. In the
model file example above, the particles \code{Chi}, \code{Cha},
\code{Glu}, \code{Sv}, \code{Su}, \code{Sd}, \code{Se} (neutralino,
chargino, gluino, sneutrino, up-type squark, down-type squark,
selectron) are considered to be LSP candidates.
\section{Structure of the spectrum generator}
\label{sec:SpecGenStruct}
In this section we explain the internals of \fs's automatically
generated spectrum generator.
As mentioned in \secref{sec:Program}, \fs uses SARAH-generated
expressions for the $\beta$-functions, mass matrices, self-energies
and EWSB conditions plus the user-defined parameter boundary
conditions to create a spectrum generator in C++. This program takes
the Standard Model and user-defined input parameters and numerically
solves the boundary value problem, which is defined by the RG
equations and the boundary conditions. If a solution is found the
pole mass spectrum is calculated.
In the following it is explained how this procedure is realized in
\fs. As mentioned in \secref{sec:Program} one of \fs's design goals
is to create modular C++ code to allow for an easy exchange, extension
and reuse of the generated modules. For this reason
\secref{sec:ModelParametersAndRGEs} first of all briefly describes the
so-called C++ ``model class'' hierarchy, which contains the general
model information, such as parameters, $\beta$-functions, \DRbar\ mass
spectrum, EWSB, self-energies, and the pole mass spectrum.
%
\secref{sec:boundary-conditions} describes how boundary conditions on
the model parameters are implemented in general at the C++ level.
Subsections \ref{sec:calculation-of-gauge-couplings}--\ref{sec:ewsb}
then show the two concrete boundary conditions, which are always
imposed: The matching of the model parameters to the Standard Model
and the electroweak symmetry breaking.
%
In \secref{sec:TreeLevelSpectrum} we describe the conventions used to
calculate the \DRbar\ mass spectrum given a set of \DRbar\ model
parameters.
%
Afterwards, in \secref{sec:TwoScaleFixedPointIteration} the algorithm,
which solves the user-defined boundary value problem is described on
the basis of the CMSSM example given in \secref{sec:modfile}.
%
Finally, \secref{sec:PoleMasses} explains how the pole mass spectrum
is obtained from the \DRbar\ model parameters after a solution to the
boundary value problem has been found.
\subsection{Model parameters and RGEs}
\label{sec:ModelParametersAndRGEs}
The parameters of the model together with their RGEs, mass
matrices, self-energies and EWSB equations are stored at the C++ level
in the model class hierarchy, which is shown in the UML diagram in
\figref{fig:parameter-classes}.
%
\begin{figure}
\centering
\tikzumlset{fill class=white}
\begin{tikzpicture}
\umlclass[x=0, y=8, type=abstract]{Beta\_function}{
}{
+ \umlvirt{get()}\\
+ \umlvirt{set()}\\
+ \umlvirt{beta()}\\
+ run\_to()
}
\umlclass[x=0, y=4]{<model>\_susy\_parameters}{
-- susy parameters
}{
+ get()\\
+ set()\\
+ beta()\\
+ run\_to()
}
\umlclass[x=0, y=0]{<model>\_soft\_parameters}{
-- soft-breaking parameters
}{