forked from Expander/FlexibleSUSY
-
Notifications
You must be signed in to change notification settings - Fork 14
/
flexiblesusy-paper.tex
4643 lines (4328 loc) · 192 KB
/
flexiblesusy-paper.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
\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
\lstset{breaklines=true,
breakatwhitespace=true,
stepnumber=1,
basicstyle=\ttfamily\footnotesize,
commentstyle=\ttfamily\color{gray},
prebreak={\textbackslash},
breakindent=10pt,
breakautoindent=false,
showspaces=false,
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}}
\newcommand{\tabref}[1]{\tablename~\ref{#1}}
\newcommand{\ptitle}[1]{\emph{#1}}
\renewcommand{\ptitle}[1]{}
\newcommand{\Zv}{\mathbf{\backslash}\mkern-11.0mu{Z}}
\makeatletter
\lstnewenvironment{numlstlisting}[1][]{%
\lstset{%
#1,
numbers=left,
firstnumber=auto,
numberstyle=\tiny\sffamily}%
\csname\@lst @SetFirstNumber\endcsname
}{%
\csname \@lst @SaveFirstNumber\endcsname
}
\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\oatplusabsq{O((\at+\ab)^2)}
\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[desy]{Alexander Voigt\corref{cor1}}
\ead{Alexander.Voigt@desy.de}
\cortext[cor1]{Corresponding author}
\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}
\address[desy]{Deutsches Elektronen-Synchrotron (DESY), 22607 Hamburg, 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, which still has scattering in the resulting quantum theory \cite{Coleman:1967ad,Haag:1974qh}. This leads 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), for a review
see \cite{Chung:2003fi}. 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:2004xm, Ellwanger:2005dv,
Ellwanger:2006rn,Ellwanger:2008py, Allanach:2013kza} for the next to
minimal supersymmetric standard model (NMSSM) (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 by raising the tree level
Higgs mass, as can happen in the USSM
\cite{Fayet:1977yc,Cvetic:1997ky,Langacker:2008yv}, \ESSM
\cite{King:2005jy,Athron:2010zz} and similar models
\cite{Nevzorov:2012hs}, or from other gauge extensions
\cite{Batra:2003nj, Bharucha:2013ela}. 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).
The standard input and output of the generated spectrum generator is
the SLHA format \cite{Skands:2003cj,Allanach:2008qq}, which is
intended for the communication between MSSM and NMSSM spectrum
generators and observable calculators. The user has control over the
SLHA block names by editing the SARAH model files and may also add
extra blocks in the \code{FlexibleSUSY.m} model file, as described in
\secref{sec:modfile}.
%
% We use SLHAea \cite{slhaea} to create these SLHA files and with
% simple modifications these can be passed directly to another code
% which uses them as an alternative to writing out the SLHA file.
%
We have tested that the HDECAY $3.4$ \cite{hdecay} and SDECAY $1.4$
\cite{sdecay}, which are shipped with SUSY-HIT \cite{susyhit},
understand the SLHA output of \fs and give matching output to that
from Softsusy.
%
For models which go beyond the MSSM and NMSSM no fixed standard exists
which specifies the spectrum generator input and output format. For
this reason, \fs allows the user to control the SLHA input and output
blocks in order to simplify the process of passing the output of a
custom \fs-generated spectrum generator to any private or new public
tool developed for that model.
\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.
\subsection{Current limitations and future extensions}
Although we try to handle as many models as possible, there are still
some limitations to what can be done. In it's current form \fs assumes
all couplings are real, therefore it is limited to $CP$-conserving
versions of SUSY models. Although it is implicit in the title, we
would like to stress that currently we cannot provide a spectrum
generator for non-SUSY models. This also means that like other
spectrum generators it is difficult to reliably predict the mass
spectrum in extreme cases such as Split-SUSY
\cite{Wells:2003tf,ArkaniHamed:2004fb,Giudice:2004tc}, where the mass
scales of the SUSY model are drastically split, leading to very large
logarithms which are not resummed. In such cases the tower of
effective theories offers the best possibility for a solution.
However, for this to work \fs must be extended to include non-SUSY
models. Finally, the gauge group of the model is currently restricted
to be semi-simple and to contain the Standard Model gauge group as
factor, so that the SUSY model can be directly matched to the SM at
low energies.
%
Future releases which extend \fs beyond each of these limitations are
already planned.
\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 1.0.1 run:
%
\begin{lstlisting}[language=bash]
$ wget https://www.hepforge.org/archive/flexiblesusy/FlexibleSUSY-1.0.1.tar.gz
$ tar -xf FlexibleSUSY-1.0.1.tar.gz
$ cd FlexibleSUSY-1.0.1
\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 accompanied by
a specimen SLHA input 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{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 in its $Z_3$-symmetric form (called \code{NMSSM}), $Z_3$-violating NMSSM (\code{SMSSM}), the USSM (\code{UMSSM}),
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[Product[M[Su[i]]^(Abs[ZU[i,3]]^2 + Abs[ZU[i,6]]^2), {i,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 = {};
(* optional *)
ExtraSLHAOutputBlocks = {
{ALPHA, {{ArcCos[Pole[ZH[1,2]]]}}},
{HMIX , {{1, \[Mu]},
{2, vu / vd},
{3, Sqrt[vu^2 + vd^2]},
{4, M[Ah[2]]^2},
{101, B[\[Mu]]},
{102, vd},
{103, vu} } },
{Au, {{1, 1, T[Yu][1,1] / Yu[1,1]},
{2, 2, T[Yu][2,2] / Yu[2,2]},
{3, 3, T[Yu][3,3] / Yu[3,3]} } },
{Ad, {{1, 1, T[Yd][1,1] / Yd[1,1]},
{2, 2, T[Yd][2,2] / Yd[2,2]},
{3, 3, T[Yd][3,3] / Yd[3,3]} } },
{Ae, {{1, 1, T[Ye][1,1] / Ye[1,1]},
{2, 2, T[Ye][2,2] / Ye[2,2]},
{3, 3, T[Ye][3,3] / Ye[3,3]} } },
{MSOFT, {{1, MassB},
{2, MassWB},
{3, MassG},
{21, mHd2},
{22, mHu2},
{31, Sqrt[ml2[1,1]]},
{32, Sqrt[ml2[2,2]]},
{33, Sqrt[ml2[3,3]]},
{34, Sqrt[me2[1,1]]},
{35, Sqrt[me2[2,2]]},
{36, Sqrt[me2[3,3]]},
{41, Sqrt[mq2[1,1]]},
{42, Sqrt[mq2[2,2]]},
{43, Sqrt[mq2[3,3]]},
{44, Sqrt[mu2[1,1]]},
{45, Sqrt[mu2[2,2]]},
{46, Sqrt[mu2[3,3]]},
{47, Sqrt[md2[1,1]]},
{48, Sqrt[md2[2,2]]},
{49, Sqrt[md2[3,3]]} } }
};
\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 defined as
\begin{align}
M_S = \sqrt{\prod_{i=1}^6 m_{\tilde{u}_i}^{|(Z_u)_{i3}|^2 + |(Z_u)_{i6}|^2}} ,
\label{eq:definition_of_MS}
\end{align}
where $m_{\tilde{u}_i}$ is the \DRbar\ mass of the $i$th up-type
squark and $Z_u$ is the up-type squark mixing matrix. The
definition \eqref{eq:definition_of_MS} is equivalent to the usual
choice $M_S = \sqrt{m_{\tilde{t}_1}m_{\tilde{t}_2}}$ without squark
flavour mixing.
%
\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 the user to add leading two-loop contributions to the
$CP$-even and $CP$-odd Higgs self-energies as well as to the $CP$-even Higgs
tadpoles. For MSSM-like models (with two $CP$-even Higgs bosons, one
$CP$-odd Higgs boson, one neutral Goldstone boson) routines dor
caluclating these corrections will be generated by setting
\code{UseHiggs2LoopMSSM = True} in the model file and by defining the
effective $\mu$-term \code{EffectiveMu = \\[Mu]}. This will add the