#### A comprehensive approach to new physics simulations

Neil Christensen
1
2
Priscila de Aquino
0
6
Celine Degrande
0
Claude Duhr
0
Benjamin Fuks
5
Michel Herquet
4
Fabio Maltoni
0
Steffen Schumann
3
0
Center for Cosmology, Particle Physics and Phenomenology, Universit Catholique de Louvain
, 1348 Louvain-la-Neuve,
Belgium
1
Department of Physics, University of WisconsinMadison
,
Madison, WI 53706, USA
2
Department of Physics and Astronomy, Michigan State University
, East Lansing,
MI 48824, USA
3
Institut fr Theoretische Physik, Universitt Heidelberg
, Philosophenweg 16, 69120,
Heidelberg, Germany
4
Nikhef Theory Group,
Science Park 105
, 1098XG Amsterdam,
The Netherlands
5
Institut Pluridisciplinaire Hubert Curien/Dpartement Recherche Subatomique, Universit de Strasbourg/CNRS-IN2P3
, 23 Rue du Loess, 67037 Strasbourg,
France
6
Instituut voor Theoretische Fysica,
Katholieke Universiteit Leuven
, Celestijnenlaan 200D,
3001 Leuven, Belgium
We describe a framework to develop, implement and validate any perturbative Lagrangian-based particle physics model for further theoretical, phenomenological and experimental studies. The starting point is FEYNRULES, a MATHEMATICA package that allows to generate Feynman rules for any Lagrangian and then, through dedicated interfaces, automatically pass the corresponding relevant information to any supported Monte Carlo event generator. We prove the power, robustness and flexibility of this approach by presenting a few examples of new physics models (the Hidden Abelian Higgs Model, the general Two-Higgs-Doublet Model, the most general Minimal Supersymmetric Standard Model, the Minimal Higgsless Model, Universal and Large Extra Dimensions, and QCD-inspired effective Lagrangians) and their implementation/validation in FEYNARTS/FORMCALC, CALCHEP, MADGRAPH/MADEVENT, and SHERPA.
1 Introduction
At the Large Hadron Collider (LHC) discoveries most
probably will not be an easy task. The typical final states
produced at this proton-proton collider running at very high
energies will involve a large number of jets, heavy-flavor
quarks, as well as leptons and missing energy, providing an
overwhelming background to many new physics searches.
Complex signal final state signatures will then need a very
careful understanding of the detector and an accurate
modeling of the data themselves. In this process, Monte Carlo
(MC) simulations will play a key role in describing control
data sets and devising robust search strategies.
Already the first step, i.e., establishing an excess over
the Standard Model (SM) background, might be very
difficult, depending on the type of signature involved [1]. At this
stage, matrix-element-based MC (which give reliable
predictions for shapes and can still be tuned to some extent to
the data) will be used to describe backgrounds and
possibly candidates signals. For some specific signals, an
accurate prediction of the background normalization and shapes,
validated via control samples, could be also needed. At the
same time, accurate measurements and comparisons with
the best theoretical predictions (e.g., at the
next-to-nextto-leading order, resummation calculations, . . . ) of a set of
standard-candle observables will also be mandatory to claim
a good understanding and control of physics and detector
effects. Very accurate predictions, possibly including even
weak corrections, and a reliable estimate of errors (such as
those introduced by the parton distribution functions) will
then be needed.
Once the presence of excess(es) is confirmed, model
building activities will be triggered, following both
topdown and bottom-up approaches. In each case, tools that
are able to make predictions for wide classes of Beyond the
Standard Model (BSM) physics, as well as those that help in
building up an effective field theory from the data (such as
the so called OSET method [2]), could be employed. Finally,
as a theoretically consistent picture arises, measurements of
the parameters (masses, spin, charges) will be carried out. In
this case it will be necessary to have at least next-to-leading
order (NLO) predictions (i.e., a reliable normalization) for
the signal processes. As our knowledge about the detector
and the newly discovered physics scenario gets stronger,
more accurate determinations will be possible and
sophisticated analyses tools could be employed, on the very same
lines as current top quark analyses at the Tevatron collider,
e.g., see [3].
As schematically outlined above, Monte Carlo
simulations will play a key, though different role at each stage
of the exploration of the TeV scale, i.e., the discovery and
identification of BSM Physics, and the measurement of its
properties. The realization of the need for better simulation
tools for the LHC has spurred an intense activity over the
last years, that has resulted in several important advances in
the field.
At the matrix-element level, these include the
development of general purpose event generators, such as
COMPHEP/CALCHEP [46], MADGRAPH/MADEVENT [710],
SHERPA [11, 12], as well as high efficiency multiparton
generators which go beyond the usual Feynman diagram
techniques, such as WHIZARD [13], ALPGEN [14], HELAC [15]
and COMIX [16]. As a result, the problem of generating
automatically leading-order matrix elements (and then cross
sections and events) for a very large class of
renormalizable processes has been solved. Quite amazingly, enormous
progress has also been achieved very recently in the
automatization of NLO computations. First the generation of the
real corrections with the appropriate subtractions has been
achieved in an automatic way [1722]. Then several new
algorithms for calculating loop amplitudes numerically have
been proposed (see, e.g., [23] for a review) and some of them
successfully applied to the computation of SM processes of
physical interest [2426].
An accurate simulation of a hadronic collision requires
a careful integration of the matrix-element hard process,
with the full parton showering and hadronization
infrastructure, as efficiently provided by PYTHIA [27, 28],
HERWIG [29, 30] and SHERPA. Here also, significant progress
has been made in the development of matching algorithms
such as that by Catani, Krauss, Kuhn and Webber (CKKW)
[3133], Mangano (MLM) [34] and others [3537], in their
comparison [38, 39] and application to SM [34, 40] and
BSM [41] processes. A breakthrough in merging fixed order
calculations and parton showers was achieved by Frixione,
Webber and Nason [42, 43], who showed how to correctly
interface an NLO computation with a parton shower to avoid
double counting and delivered the first event generator at
NLO, MC@NLO. More recently, a new method along the
same lines, dubbed POWHEG, has been proposed [44] and
applied to Drell-Yan and Higgs production [4548].
The progress in the field of Monte Carlo tools outlined
above shows that we are, or will be soon, able to simulate
all the relevant SM processes at the LHC with a very high
level of accuracy. It is therefore worth considering the status
of the predictions for physics Beyond the Standard Model.
Quite interestingly, the challenges in this case are of a quite
different nature. The main reason is that presently there is
not a leading candidate for BSM, but instead a plethora of
models have been suggested, based on very different ideas
in continuous evolution. The implementation of complex
BSM models in existing general purpose event generators
like those enumerated above remains a long, often
painstaking and error-prone, process. The derivation of the
numerous Feynman rules to describe the new interactions, and
their implementation in codes following conventions is a
very uninteresting and time consuming activity. In addition,
the validation of a given implementation often relies on a
comparison of the obtained analytical and numerical results
with those available in the literature. Again, due to presence
of various conventions, the restricted number of public
results and the lack of a dedicated framework, such a
comparison is often done manually, in a partial and not systematic
way. Finally, besides a handful of officially endorsed and
publicly distributed BSM models (e.g., the Minimal
Supersymmetric Standard Model), many implementations remain
private or only used by a restricted set of theorists and/or
phenomenologists, and never get integrated into the official
chain of simulation tools used by experimental
collaborations. Instead, various home-made or hacked versions
of existing MC softwares are commonly used for specific
BSM process studies, leading to problems in the validation,
traceability and maintenance procedures.
In this work we address the problem of having an
efficient framework where any new physics model can be
developed and its phenomenology can be tested against data.
A first step in the direction of deriving Feynman rules
automatically starting from a model Lagrangian has been made
in the context of the COMPHEP/CALCHEP event generator
with the LANHEP package [49]. Our aim is to go beyond
this scheme and create a general and flexible environment
where communication between theorists and
experimentalists in both directions is fast and robust. The desiderata for
the new physics phenomenological framework linking
theory to experiment and vice versa which we provide a
solution for are:
1. General and flexible environment, where any
perturbative Lagrangian-based model can be developed and
implemented.
2. Modular structure with interfaces to several
multi-purpose MCs and computational tools.
3. Robust, easy-to-validate and easy-to-maintain.
4. Integrable in the experimental software frameworks.
5. Full traceability of event samples.
6. Both top-down and bottom-up approaches are natural.
This paper is organized in five main sections and various
appendices. In Sect. 2, by discussing a simple example, we
expose the strategy which we propose to address the
challenges that model builders, phenomenologists and
experimentalists have to face to study the phenomenology of a new
physics model. This strategy is based on the FEYNRULES
package, and in Sect. 3 we briefly recall how the package
works and present some of the new features recently
implemented. Section 4 contains a brief description of the various
interfaces already available. Section 5 contains the
information about the models that have already been implemented.
In Sect. 6 we present our strategy to validate BSM model
implementations, and illustrate our procedure on the already
implemented models. Finally, in Sect. 7, we discuss the
outlook of our work. In the appendices we collect some
technical information as well as a few representative validation
tables, which constitute the quantitative results of this paper.
2 A simple example: from the standard model to the
Hidden Abelian Higgs model
From the phenomenological point of view, we can
distinguish two classes of BSM models. The first class of models
consists of straightforward extensions of the SM, obtained
by adding one (or more) new particles and interactions to the
SM Lagrangian. In this bottom-up approach, one generally
starts from the SM, and adds a set of new operators
according to some (new) symmetry. The second class of models
are obtained in a top-down approach, where the fundamental
Lagrangian is determined by the underlying global and local
symmetries, and the SM is only recovered in some specific
limit.
In this section we describe in detail our framework
to develop, test, implement and validate any perturbative
Lagrangian-based particle physics model. We concentrate
on the case of the bottom-up models, and show how our
framework allows to easily extend the SM and to go in a
straightforward way from the model building to the study
of the phenomenology. We will comment on the top-down
models in subsequent sections.
2.1 The model
As an illustration, we use the Hidden Abelian Higgs (HAH)
Model, described in [50]. This model can be seen as the
simplest way to consistently add a new gauge interaction to the
SM. More specifically, we consider an SU(3)C SU(2)L
U (1)Y U (1)X gauge theory where all SM particles are
singlets under the new gauge group U (1)X. A new Higgs field
is added that is a singlet under the SM gauge group and
breaks the U (1)X symmetry when it acquires its vacuum
expectation value (vev), = /2. The most general
Lagrangian describing this model is given by
+ LHAH,Yukawa,
1 1 1
LHAH,Gauge = 4 Ga Ga 4 W i Wi 4
LHAH,Higgs = DD + DD + 2
+ ()2 ()2
2
where denotes the SM Higgs field. The covariant
derivative reads
D = igs T aGa ig 2 W ig Y B igXXX, (3)
and we define the field strength tensors,
W i = Wi W i + g ijkWj Wk,
G = G G + gs f abcGbGc .
a a a
gs , g, g and gX denote the four coupling constants
associated with the SU(3)C SU(2)L U (1)Y U (1)X gauge
groups while T , i , Y and X are the corresponding
generators and f abc and ijk represent the totally antisymmetric
tensors of SU(3)C SU(2)L, respectively. We do not write
explicitly the terms in the Lagrangian describing the matter
sector of the theory as they are identical to those of the SM
described in detail in Sect. 5.1,
LHAH,Fermions = LSM,Fermions and
LHAH,Yukawa = LSM,Yukawa.
The kinetic mixing term in LHAH,Gauge induces a mixing
between the two U (1) gauge fields and thus a coupling
between the matter fermions and the new gauge boson. The
kinetic terms for the U (1) fields can be diagonalized via a
GL(2, R) rotation,
After this field redefinition, the gauge sector takes the
diagonal form
1 1 1
LHAH,Gauge = 4 Ga Ga 4 W i Wi 4
where the mixing angles are given by
tan 2 = 1 sw22s2w Z ,
with Z = MX2/MZ20 , where MX and MZ0 denote the
masses of the gauge bosons before the kinetic mixing,
MZ20 = g2 + g 2 v2/4,
and qX denotes the U (1)X charge carried by . The photon
remains massless while the two other states acquire a mass
given by
sh
1We work in unitary gauge.
MZ2 ,Z =
As a result of electroweak symmetry breaking,
nondiagonal mass terms for the Higgs fields appear that can be
diagonalized via an orthogonal transformation,
but the covariant derivative now contains an additional term
coupling the field X to the hypercharge,
D = igs T aGa ig 2 W ig Y B
with = / 1 2. When the Higgs fields acquire their
vev,1
the gauge symmetry gets broken to SU(3)C U (1)EM , and
we obtain a non-diagonal mass matrix for the neutral weak
gauge bosons, which can be diagonalized by an O(3)
rotation,
Table 1 Input parameters for the Hidden Abelian Higgs model,
corresponding to the benchmark point 1 of [51]. Other SM input parameters
are not shown
where the mixing angles and mass eigenvalues are given by
Once the Lagrangian is written down and diagonalized in
terms of mass eigenstates, one can easily identify a minimal
set of parameters the model depends on. Not all the
parameters introduced above are independent, as most of them
are related by some algebraic relations, e.g., the relation
between the mass eigenvalues of the gauge bosons, (13), and
the fundamental parameters appearing in the Lagrangian.
Our choice of independent input parameters is given in
Table 1. All other parameters appearing in LHAH,Gauge and
LHAH,Higgs can be reexpressed in terms of these
parameters. Let us note, however, that there are strong experimental
constraints from LEP on the masses and the couplings of
additional neutral gauge bosons to fermions, which need to
be taken into account when building the model. In [50] it
was pointed out that = 0.01 is still allowed. In general,
in order to determine a benchmark point that takes into
account the direct and indirect experimental constraints, it is
required to perform (loop) computations for several
physical observables. We will comment more on this in the next
subsection.
Let us note that, although (2) is a very simple extension
of the SM, from a more technical point of view an
implementation of the HAH model in a matrix-element
generator is already not trivial. In this case, it is not sufficient
to start from the existing SM implementation and just add
the vertices contained in LHAH,Higgs, because mixing in the
gauge and scalar sectors implies that all SM vertices
involving a Higgs boson and/or a Z boson need to be modified.
For example, although there is no direct coupling between
the Abelian Higgs field and the matter fermions, all the
Yukawa couplings receive contributions from the two mass
eigenstates h1 and h2, weighted by the mixing angle h,
resulting in an almost complete rewriting of the SM
implementation. In the next subsection we will describe how this
Table 2 Dominant decay
channels of the new particles in
the HAH model
Branching ratios for h1
Branching ratios for h2
Branching ratios for h1
difficulty can be easily overcome and the phenomenology of
the Hidden Abelian Higgs model studied.
2.2 From model building to phenomenology
The starting point of our approach is FEYNRULES (see
Sect. 3). Since in this case we are interested in a simple
extension of the SM, it is very easy to start from the
FEYNRULES implementation of the SM which is included in the
distribution of the package and to extend the model file by
including the new particles and parameters, as well as the
HAH model Lagrangian of (2). Note that, at variance with
the direct implementation into a matrix-element generator
where we need to implement the vertices one at a time, we
can work in FEYNRULES with a Lagrangian written in terms
of the gauge eigenstates and only perform the rotation to the
mass eigenbasis as a second step. This implies that it is not
necessary to modify LHAH,Fermions since the new fields only
enter through LHAH,Gauge and LHAH,Higgs.
Several functions are included in FEYNRULES to
perform sanity checks on the Lagrangian (e.g., hermiticity).
The diagonalization of the mass matrices can be easily
performed directly in MATHEMATICA2 and FEYNRULES
allows us to easily obtain the Feynman rules for the model. As
already mentioned, not only the Feynman rules of the Higgs
and gauge sectors are modified with respect to the Standard
Model, but also the interaction vertices in the fermionic
sector change due to the mixing of the scalars and the
neutral weak bosons. The vertices obtained in this way can
already be used for pen & paper work during the model
building, and to compute simple decay rates and cross sections.
Since FEYNRULES stores the vertices in MATHEMATICA, it
is easy to use them directly for such computations.
After this preliminary study of our model where the mass
spectrum of the theory was obtained and basic sanity checks
have been performed, typically the model is confronted with
2MATHEMATICA is a registered trademark of Wolfram Research, Inc.
all relevant direct and indirect constraints coming from
experiment. This is a necessary step to find areas of
parameters space which are still viable. Once interesting regions in
parameter space are identified, the study of the collider
phenomenology of the model, e.g., at the LHC, can start with the
calculation of cross sections and decay branching ratios. Let
us consider first the calculation of the decay widths of both
SM and new particles. Using the FEYNARTS
implementation of the new model obtained via the FEYNRULES
interface, it becomes a trivial exercise to compute analytically all
tree-level two-body decays for the Higgs bosons and the Z
boson (alternatively, one could calculate them numerically
via e.g., CALCHEP or MADGRAPH/MADEVENT). The
results for the branching ratios of the dominant decay modes
are shown, for the benchmark scenario considered here, in
Table 2. Once decay widths are known, cross sections can
be calculated. However, in many cases it is insufficient to
have only predictions for total cross sections, as a study of
differential distributions, with possibly complicated
multiparticle final states, is necessary to dig the signal out of the
backgrounds. Furthermore, even a parton-level description
of the events might be too simplified and additional
radiation coming from the colored initial and final-state particles,
as well as effects coming from hadronization and underlying
events need to be accounted for. For this reason,
phenomenological studies are in general performed using
generators which include (or are interfaced to) parton shower and
hadronization Monte Carlo codes. The parton level events
for the hard scattering can be generated by the general
purpose matrix-element (ME) program, and those events are
then passed on to the parton shower codes evolving the
parton-level events into physical hadronic final states.
However, similar to FEYNARTS/FORMCALC, for new models
the ME generators require the form of the new vertices, and
different programs use different conventions for the vertices,
making it difficult to export the implementation from one
ME generator to another. To solve this issue, FEYNRULES
includes interfaces to several ME generators that allow to
output the interaction vertices obtained by FEYNRULES
directly in a format that can be read by the external codes. For
Fig. 1 Invariant mass distribution for the four particle final state
bb, both for the gg h2 h1h1 bb signal (plain) and the
main SM backgrounds (dashed) events at the LHC. All simulation
parameters and analysis cuts are identical to those listed in [51]
the moment, such interfaces exist for CALCHEP/COMPHEP,
MADGRAPH/MADEVENT and SHERPA.3 It should be
emphasized that some of these codes have the Lorentz and/or
color structures hardcoded, something that limits the range
of models that can be handled by a given MC. In this
respect (and others) each of MC tools has its own strengths
and weaknesses: having several possibilities available
maximizes the chances that at least one generator is able to
efficiently deal with a given model and the case in which several
MC tools can be used, as most of the examples discussed in
this paper, allows for a detailed comparison and robust
validation of the implementation.
For the sake of illustration, we used the MADGRAPH/
MADEVENT implementation of the HAH model to
generate signal events for the gg h2 h1h1 bb signal
proposed in [51] as a signature of this model at the LHC.
Using the same set of cuts, and the same smearing method
as in [51], we have been able to easily generate final state
invariant mass distributions for both signal and background
events. The result can be seen in Fig. 1, which compares
well to Fig. 5 of [51].
2.3 Validation of new physics models
In the previous subsection we discussed how the
implementation of the model in FEYNRULES allows to go all the
way from model building to phenomenology without
having to deal with the technicalities of the various ME
generators. In this section we argue that our approach does not
only allow to exploit the strength of each ME generator,
3An interface to WHIZARD will be available with the next major
release of FEYNRULES.
but it also has a new power in the validation of BSM
models. Since the various ME generators use different
conventions for the interaction vertices it becomes hence possible
to compare the implementations of the same model into
various different matrix-element generators, and thus the
different tools, among themselves. Furthermore, in many cases
generator specific implementations of BSM models already
exist, at least for restricted classes of models, in which case
the FEYNRULES model can be directly validated against the
existing tool A, and then exported to any other tool B for
which an interface exists. In the same spirit, any BSM model
should be able to reproduce the SM results for observables
which are independent of the new physics.
Let us illustrate this procedure through the example of
the HAH model. We start by implementing our model
into CALCHEP, MADGRAPH/MADEVENT and SHERPA by
means of the corresponding interfaces. We then compute
the cross sections for all interesting two-to-two processes
for this model, and check that we obtain the same numbers
from every ME generator. Note that, since in this case we
have modified the scalar sector of the SM, we pay particular
attention to the unitarity cancellations inherent to the SM in
weak boson scattering, showing in this way that our
implementation does not spoil these cancellations. Since the
different codes used for the computation of the cross-sections
all rely on different conventions for the interaction vertices,
we hence demonstrated that our model is consistent not only
by checking that the cancellations indeed take place in all
the implementations, but we also get the same results for the
total cross section after the cancellation, a strong check very
rarely performed for a general BSM model implementation.
We will comment on the validation of more general models
in subsequent sections.
Finally, let us comment on the fact that a robust
implementation of a BSM model into a code does not only
require the model to be validated to a very high-level, but the
implementation should also be clearly documented in order
to assure its portability and reproducibility. For this reason
FEYNRULES includes an output to TEX which allows to
output the content of the FEYNRULES model files in a
humanreadable format, including the particle content and the
parameters which define the model, as well as the Lagrangian
and the Feynman rules.
2.4 From phenomenology to experiment
Our approach does not only apply to phenomenological
studies at the theory level, but it allows to continue and
pass the model to an experimental collaboration for full
experimental studies. In general, the experimental softwares
used to simulate the detector effects have very strict
requirements regarding sanity checks for new modules (e.g.,
private Monte Carlo programs) to be included in the software,
and a very long and tedious validation procedure is needed.
However, will we be at the point where we have to
discriminate between various competing models at the LHC, this
approach would become extremely inefficient due to the large
number of tools to be validated. In our approach this
validation can be avoided, streamlining in this way the whole
chain all the way from model building to the experimental
studies and vice versa.
Let us again illustrate this statement through the example
of the HAH model. Since this model is now implemented in
FEYNRULES, we can easily pass it into various ME
generators using the translation interfaces, and we demonstrated
the validation power inherent to this approach in the
previous section. These models can then be used in a ME
generator in the same way as any other built-in model, without any
modification to the original code, i.e., without creating a
private version of the ME generator. If the model is validated
it can easily be passed on to the experimental community,
which can then read the FEYNRULES output and use it in
their favorite ME generator already embedded in their
software framework.
By following this procedure tedious validations for each
model implementation in a given MC are avoided. In
addition, the portability and the reproducibility of the
experimentally tested models is guaranteed. As at the origin of
the chain is solely the FEYNRULES model file, all the
information is concentrated in one place, and thus everybody
can reproduce all the results at any stage, from the model
building to the collider signatures, by starting from the very
same file. In addition, since the FEYNRULES model file
contains the Lagrangian of the model, it is very easy to go back
to the model file and understand its physical content, a step
which might be very difficult working with manually created
model files for the ME generators, written in an often rather
cryptic programming language hiding the essential physics.
In our approach it becomes very easy to use later the very
same model information, and to reproduce analyses by just
changing benchmark points or by adding a single new
particle/interaction to the same model.
3 FEYNRULES in a nutshell
FEYNRULES is a MATHEMATICA package which allows
to derive Feynman rules directly from a Lagrangian [52].
The user provides the Lagrangian for the model
(written in MATHEMATICA) as well as all the information
about the particle and parameter content of the model.
This information uniquely defines the model, and hence is
enough to derive all the interaction vertices from the
Lagrangian. FEYNRULES can in principle be used with any
model which fulfills basic quantum field theoretical
requirements (e.g., Lorentz and gauge invariance), the only
current limitation coming from the kinds of fields supported
by FEYNRULES (see below). In particular it can also be
used to obtain Feynman rules for effective theories
involving higher-dimensional operators. In a second step, the
interaction vertices obtained by FEYNRULES can be
exported by the user to various matrix-element generators
by means of a set of translation interfaces included in the
package. In this way the user can directly obtain an
implementation of his/her model into these various tools,
making it straightforward to go from model building to
phenomenology. Presently, interfaces to CALCHEP
/COMPHEP, FEYNARTS/FORMCALC, MADGRAPH/MADEVENT
and SHERPA are available. In the following we briefly
describe the basic features of the package and the model files,
the interfaces to the matrix-element generators being
described in Sect. 4. For more details on both the FEYNRULES
package as well as the interfaces, we refer the reader to
the FEYNRULES manual and to the FEYNRULES website
[52, 53].
3.1 Model description
The FEYNRULES model definition is an extension of the
FEYNARTS model file format and consists of the definitions
of the particles, parameters and gauge groups that
characterize the model and the Lagrangian. This information can be
placed in a text file or in a MATHEMATICA notebook or a
combination of the two as convenient for the user.
Let us start with the particle definitions. Following the
original FEYNARTS convention, particles are grouped into
classes describing multiplets having the same quantum
numbers, but possibly different masses. Each particle class
is defined in terms of a set of class properties, given as a
MATHEMATICA replacement list. For example, the up-type
quarks could be written as
-> q,
-> {u, c, t},
-> False,
-> {Index[Generation],
Index[Colour]},
FlavorIndex -> Generation,
Mass -> {Mq, 0, 0, {MT, 174.3}},
Width -> {Wq, 0, 0, {WT, 1.508}},
QuantumNumbers -> {Q -> 2/3},
PDG -> {2, 4, 6}}
This defines a Dirac fermion (F) represented by the
symbol q. Note that the antiparticles are automatically declared
and represented by the symbol qbar. The class has three
members, u, c, and t (ubar, cbar, and tbar for the
antiparticles, respectively), distinguished by a generation
index (whose range is defined at the beginning of the model
definition) and the fields carry an additional index labelled
Colour. The complete set of allowed particle classes is
given in Table 3. Additional information, like the mass and
width of the particles, as well as the U (1) quantum numbers
Table 3 Particle classes supported by FEYNRULES
carried by the fields can also be included. Finally, some more
specific information not directly needed by FEYNRULES but
required by some of the matrix-element generators (e.g., the
Particle Data Group (PDG) codes [54]) can also be defined.
A complete description of the particle classes and properties
can be found in the FEYNRULES manual.
A Lagrangian is not only defined by its particle content,
but also by the local and global symmetries defining the
model. FEYNRULES allows to define gauge group classes
in a way similar to the particle classes. As an example, the
definition of the QCD gauge group can be written
SU3C == { Abelian
GaugeBoson
CouplingConstant
StructureConstant
Representations
where the gluon field G is defined together with the quark
field during the particle declaration. The declaration of
Abelian gauge groups is analogous. FEYNRULES uses this
information to construct the covariant derivative and field
strength tensor which the user can use in the Lagrangian.
The third main ingredient to define a model is the set
of parameters which it depends on. In general, not all the
parameters appearing in a Lagrangian are independent, but
they are related through certain algebraic relations specific
to each model, e.g., the relation cos w = MW /MZ relating
at tree-level the masses of the weak gauge bosons to the
electroweak mixing angle. FEYNRULES therefore distinguishes
between external and internal parameters. External
parameters denote independent parameters which are given as
numerical inputs to the model. An example of a declaration of
an external parameter reads
aS == {ParameterType
Value
-> External,
-> 0.118}
defining an external parameter aS with numerical value
0.118. Several other properties representing additional
information needed by matrix-element generators are also
available, and we refer to the FEYNRULES manual for an
extensive list of parameter class properties. Internal parameters
are defined in a similar way, except that the Value is given
by an algebraic expression linking the parameter to other
external and/or internal parameters. For example, the cos w
parameter definition could read
cw == {ParameterType
Value
-> Internal,
-> MW/MZ}
Note that it is also possible to define tensors as parameters
in exactly the same way, as described in more detail in the
manual.
At this point, we need to make a comment about the
conventions used for the different particle and parameter
names inside FEYNRULES. In principle, the user is free to
choose the names for the gauge groups, particles and
parameters at his/her convenience, without any restriction. The
matrix-element generators however have certain information
hardcoded (e.g., reference to the strong coupling constant
or electroweak input parameters). For this reason,
conventions regarding the implementation of certain SM
parameters have been established to ensure the proper translation
to the matrix-element generator. These are detailed in the
manual and recalled in Appendix A.
In complicated models with a large parameter space, it is
sometimes preferable to restrict the model to certain slices
of that parameter space. This can be done as usual by
adjusting the parameters to lie at that particular point in
parameter space and doing the desired calculation. However,
sometimes these slices of parameter space are special in that
many vertices become identically zero and including them
in the Feynman diagram calculation can be very inefficient.
In order to allow both the general parameter space and a
restricted parameter space, we introduce the model restriction.
A model restriction is a MATHEMATICA list containing
replacements to the parameters which simplify the model. For
example, in the SM the CKM matrix has non-zero matrix
elements, but it is sometimes useful to restrict a calculation
to a purely diagonal CKM matrix. Rather than creating a
new implementation of the SM with a diagonal CKM
matrix, a restriction can be created and used when desired. The
following statement restricts the SM to a diagonal CKM
matrix
CKM[i_, i_] :> 1,
CKM[i_, j_] :> 0 /; i != j,
When this restriction is applied, all vertices containing off
diagonal CKM elements vanish identically and are removed
before passing it on to a matrix-element generator. The result
is that these vertices never appear in Feynman diagrams and
the calculation is more efficient. Several restriction files can
be created corresponding to various different slices of
parameter space. The user then selects the restriction file that
they are interested in and applies it to the model before
running the translation interfaces.
3.2 Running FEYNRULES After having loaded the FEYNRULES package into MATHEMATICA, the user can load the model and the model restrictions via the commands
LoadModel[ < model file 1 > ,
< model file 2 > , ... ];
LoadRestriction[ < restriction file > ];
where the model can be implemented in as many files as
convenient or it can be implemented directly in the
MATHEMATICA notebook in which case the list of files would be
empty. The restriction definitions can also be placed in a file
or directly in the MATHEMATICA notebook. The Lagrangian
can now be entered directly into the notebook4 using
standard MATHEMATICA commands, augmented by some
special symbols representing specific objects like Dirac
matrices. As an example, we show the QCD Lagrangian,
L = -1/4 FS[G, mu, nu, a] FS[G, mu, nu, a]
+ I qbar . Ga[mu]. DC[q, mu];
where FS[G, mu, nu, a] and DC[q, mu] denote the
SU(3)C field strength tensors and covariant derivatives
automatically defined by FEYNRULES. At this stage, the user
can perform a set of basic checks on the Lagrangian
(hermiticity, normalization of kinetic terms, . . . ), or directly
proceed to the derivation of the Feynman rules via the command
verts = FeynmanRules[ L ];
FEYNRULES then computes all the interaction vertices
associated with the Lagrangian L and stores them in the variable
verts. The vertices can be used for further computations
within MATHEMATICA, or they can be exported to one of
the various matrix-element generators for further
phenomenological studies of the model. The translation interfaces
can be directly called from within the notebook, e.g., for the
FEYNARTS interface,
WriteFeynArtsOutput[ L ];
This will produce a file formatted for use in FEYNARTS.
All other interfaces are called in a similar way. As already
mentioned, let us note that, even if FEYNRULES is not
restricted and can derive Feynman rules for any Lagrangian,
the matrix-element generators usually have some
information on the Lorentz and color structures hardcoded, and
therefore they are much more limited in the set of
vertices they can handle. Each matrix-element generator has its
own strengths, and in the FEYNRULES approach the same
model can be easily exported to various codes, exploiting
in this way the strength of each individual tool. In practice,
the interfaces check whether all the vertices are compliant
with the structures supported by the corresponding
matrixelement generator. If not, a warning is printed and the vertex
is discarded. Each interface produces at the end a (set of)
4Alternatively, the Lagrangian can also be included in the model file,
in which case it is directly loaded together with the model file.
text file(s), often consistently organized in a single
directory, which can be read into the matrix-element generator at
runtime and allows to use the new model in a way similar
to all other built-in models. For more details on the various
interfaces, we refer to Sect. 4 and to the manual.
4 Interfaces
In this section we provide a concise description of the
FEYNRULES interfaces to several matrix-element
generators and symbolic tools available to perform
simulations/calculations from Lagrangian-based theories. The
most important features of the general structure of the new
physics models in the codes and their limitations are
emphasized. Complete description of the options and more
technical details can be found in the FEYNRULES users manual,
available on the FEYNRULES website. Interfaces to other
codes, once available, will be included in the main release
of the package and documented in the users manual.
4.1 FEYNARTS/FORMCALC
FEYNARTS is a MATHEMATICA package for generating,
computing and visualizing Feynman diagrams, both at
treelevel and beyond [55]. For a given process in a specific
model, FEYNARTS starts by generating all the possible
topologies, taking into account the number of external legs
and internal loops associated to the considered case, together
with the set of constraints required by the user, such as the
exclusion of one-particle reducible topologies. This stage is
purely topological and does not require any physical input.
Based on a pre-defined library containing topologies
without any external leg for tree-level, one-loop, two-loop and
three-loop calculations, the algorithm successively adds the
desired number of external legs. Then, the particles present
in the model must be distributed over the obtained
topologies in such a way that the resulting diagrams contain the
external fields corresponding to the considered process and
only vertices allowed by the model. Finally, a
MATHEMATICA expression for the sum over all Feynman diagrams is
created.
The second step in the perturbative calculation consists in
the evaluation of the amplitudes generated by FEYNARTS.
This can be handled with the help of the
MATHEMATICA package FORMCALC which simplifies the symbolic
expressions previously obtained in such a way that the output
can be directly used in a numerical program [56].
FORMCALC first prepares an input file that is read by the program
FORM which performs most of the calculation [5761]. The
Lorentz indices are contracted, the fermion traces are
evaluated, the color structures are simplified using the SU(3)
algebra, and the tensor reduction is performed.5 The results
are expressed in a compact form through abbreviations and
then read back into MATHEMATICA where they can be used
for further processing. This allows to combine the speed of
FORM with the powerful instruction set of MATHEMATICA.
4.1.1 Model framework
The FEYNARTS models have a very simple structure which
can be easily extended to include BSM models. In
particular, the current distribution of FEYNARTS contains already
several models, including a complete implementation of the
Standard Model, as well as a Two-Higgs-Doublet Model and
a completely generic implementation of the Minimal
Supersymmetric Standard Model.
The FEYNARTS models are separated into two files:
The generic model file: This file is not specific to any
model, but it contains the expressions for the
propagators and the Lorentz structures of the vertices for generic
scalar, fermion and vector fields. Note that since this file
is not specific to any model, different BSM models can be
related to the same generic model file.
The classes model file: This file is dedicated to a
specific model, and contains the declarations of the
particles and the analytic expressions of the couplings between
the different fields. This information is stored in the two
lists M$ClassesDescription for the particle
declarations and M$ClassesCouplings for the couplings.
FEYNARTS requires all the particles to be grouped into
classes, and as a consequence also all the classes couplings
must be given at the level of the particle classes. If this
is the case, the number of Feynman diagrams generated at
runtime is much smaller, which speeds up the code. Since
the FEYNRULES model files are an extension of the
FEYNARTS classes model files, the explicit structure of the
particle class definitions is very similar to the FEYNRULES
particle classes discussed in Sect. 3.
4.1.2 FEYNRULES interface
FEYNRULES includes an interface that allows to
output the interaction vertices derived from the Lagrangian
as a FEYNARTS model file. Note however that at the
present stage only the classes model file is generated by
FEYNRULES, the generic model file is hardcoded. The
generic model file used by FEYNRULES generated models,
feynrules.gen, is included in the FEYNRULES
distribution in the Interfaces/FeynArts subdirectory and
5Let us note that, in order to function correctly, FORMCALC requires
the amplitude to be given in Feynman gauge.
needs to be copied once and for all into the Models
directory of FEYNARTS. feynrules.gen is based on the
corresponding Lorentz.gen file included in FEYNARTS,
with some extensions to higher-dimensional scalar
couplings as those appearing in non-linear sigma models (see
Sect. 5.6).
The FEYNRULES interface to FEYNARTS can be called
within a MATHEMATICA notebook via the command
WriteFeynArtsOutput[ L ];
FEYNRULES then computes all the vertices associated with
the Lagrangian L, and checks whether they all have Lorentz
structures compatible with the generic couplings in the
generic coupling file, and if so, it extracts the corresponding
classes coupling. If not, a message is printed on the screen
and the vertex is discarded. At this point we should
emphasize that in order to obtain FEYNARTS couplings at the level
of the particle classes, it is necessary that the Lagrangian
is also given completely in terms of particle classes. If the
interface encounters a Lagrangian term which violates this
rule, it stops and redefines all the classes such that all
particles live in their own class. It then starts over and recomputes
all the interaction vertices, this time for a Lagrangian where
all particle classes are expanded out explicitly. In this way a
consistent FEYNARTS model file is obtained which can be
used with FEYNARTS. It should however be noted that the
generation of the diagrams can be considerably slower in
this case, which makes it desirable to write the Lagrangian
in terms of particle classes whenever possible.
The model file produced by FEYNRULES has the usual
FEYNARTS structure. Besides the lists which contain the
definitions of the particle classes and the couplings, the
FEYNRULES generated model files contain some more
information, which can be useful at various stages during the
computation:
-- M$ClassesDescription: This is in general a copy
of the corresponding list in the original FEYNRULES
model file.
-- M$ClassesCouplings: Each entry in the list
represents a given interaction between the particle classes,
together with the associated coupling constant, represented
by an alias gcxx, xx being an integer. Let us note that
currently FEYNRULES does not compute the
counterterms necessary for loop calculations, and they should be
added by the user by hand.
-- M$FACouplings: A replacement list, containing the
definition of the couplings gcxx in terms of the
parameters of the model.
Furthermore, several other replacement lists
(M$ExtParams, M$IntParams, M$Masses) are
included, containing the values of the parameters of the model,
as well as the masses and widths of all the particles.
4.2 CALCHEP/COMPHEP
The CALCHEP [4, 6] and COMPHEP [4, 5] packages
automate the tree-level Feynman diagram calculation and
production of partonic level collider events. Models with very
general Lorentz structures are allowed and general color
structures can be incorporated via auxiliary fields. Vertices
with more than four particles are not supported at this time.
In this subsection, we will describe the model file
structure and how the FEYNRULES interface to CALCHEP and
COMPHEP works.
4.2.1 Model framework
Models in CALCHEP and COMPHEP are essentially
comprised of four files:
prtclsN.mdl: a list of all the particles in the model
along with information about the particles that is
necessary for calculation of Feynman diagrams.
varsN.mdl: a list of the independent (external)
parameters in the model along with their numerical value.
funcN.mdl: a list of the dependent (internal)
parameters of the model along with their functional definition.
These definitions can contain any standard mathematical
functions defined in the C code.
lgrngN.mdl: a list of all the vertices in the model. It
includes the specification of the particles involved in the
vertex, an overall constant to multiply the vertex with and
the Lorentz form of the vertex.
Note that the letter N in the names of the files is an integer
which refers to the number of the model.
4.2.2 FEYNRULES interface
The CALCHEP/COMPHEP interface can be invoked with the
command
WriteCHOutput[ L ]
where L denotes the Lagrangian.
When invoked, this interface creates the directory
M$ModelName with -CH appended if it does not already
exist and then the files prtclsN.mdl, varsN.mdl,
funcN.mdl and lgrngN.mdl. Feynman rules with four
particles or less are generated and written to lgrngN.mdl.
The vertex list is simplified by renaming the vertex
couplings as x1, x2, x3, etc. and the definitions of these
couplings written in funcN.mdl along with the other internal
parameters.
Although CALCHEP and COMPHEP can calculate
diagrams in both Feynman and unitary gauge, they are much
faster in Feynman gauge and it is highly recommended
to implement a new model in Feynman gauge.
However, if a user decides to implement the model in unitary
gauge, he/she should remember that according to the way
CALCHEP/COMPHEP were written, the ghosts of the
massless non-Abelian gauge bosons must still be implemented.
In particular, the gluonic ghosts must be implemented in
either gauge for this interface.
One major constraint of the CALCHEP/COMPHEP
system is that the color structure is implicit. For many vertices
(e.g., quark-quark-gluon), this is not a problem. However,
for more complicated vertices, there may be an ambiguity.
For this reason, the writers of CALCHEP/COMPHEP chose
to split them up using auxiliary fields. Although this can be
done for very general vertices, it is not yet fully
automatized in FEYNRULES. Currently, only the gluon four-point
vertex and squark-squark-gluon-gluon vertices are
automatically split up in this way.
The model files are ready to be used and can be
directly copied to the CALCHEP/COMPHEP model
directories. The default format for this interface is the CALCHEP
format. A user can direct this interface to write the files in
the COMPHEP format by use of the CompHep option. The
user who writes COMPHEP model files should note one
subtlety. If the model is written to the COMPHEP directory and
if the user edits the model inside COMPHEP and tries to save
it, COMPHEP will complain about any C math library
functions in the model. Nevertheless, it does understand them.
We have checked that if the model works in CALCHEP, it
will work in COMPHEP and give the same results.
CALCHEP has the ability to calculate the widths of the
particles on the fly. By default, this interface will write
model files configured for automatic widths. This can be
turned off by setting the option CHAutoWidths to False.
This option is set to False if COMPHEP is set to True.
This interface also contains a set of functions that read
and write the external parameters from and to the CALCHEP
variable files (varsN.mdl). After loading the model into
FEYNRULES, the external parameters can be updated by
running
ReadCHExtVars[ Input -> < file > ]
This function accepts all the options of the CALCHEP
interface plus the option Input which instructs FEYNRULES
where to find the CALCHEP variable file. The default is
varsN.mdl in the current working directory. If reading a
COMPHEP variable file, then the option CompHep should
be set to true. After reading in the values of the variables in
the CALCHEP file, it will update the values in FEYNRULES
accordingly.
The current values of the external parameters in
FEYNRULES can also be written to a CALCHEP external variable
file (varsN.mdl) using
WriteCHExtVars[ Output -> < file >]
This can be done to bypass writing out the entire model if
only the model parameters are changed.
The MADGRAPH/MADEVENT V4.4 software [710]
allows users to generate tree-level amplitudes and parton-level
events for any process (with up to nine external particles).
It uses the HELAS library [62, 63] to calculate matrix
elements using the helicity formalism in the unitary gauge.
Starting from version 4, users have the possibility to use
several pre-defined BSM models, including the most generic
Two-Higgs-Doublet Model and the Minimal
Supersymmetric Standard Model, but can also take advantage of the
USRMOD interface to implement simple Standard Model
extensions.
The existing scheme for new model implementations in
MADGRAPH/MADEVENT has two major drawbacks. First,
users need to explicitly provide algebraic expressions for the
coupling used by MADGRAPH to calculate amplitudes.
Second, the first version of the USRMOD interface only works
for models extending the existing Standard Model by adding
a limited set of new particles and/or interactions. This
renders difficult any attempt to modify existing BSM models,
or to generalize models previously implemented with this
method.
The current version of MADGRAPH relies on a new
clearly defined structure for all model libraries generated via
the corresponding interface to FEYNRULES, which
generates all the required code files automatically. Finally, a new
version of the USRMOD scripts exists which can be used
complementary to FEYNRULES for simple extensions of
models relying on this structure. All these three new
frameworks are introduced and described in the present section.
4.3.1 Model framework
All model libraries supported in the latest versions of
MADGRAPH/MADEVENT now have the same structure. They are
composed of a set of text and FORTRAN files grouped in a
single directory, stored in the Models subdirectory of the
root MADGRAPH/MADEVENT installation:
particles.dat: a text file containing a list of all
particles entering the model and the corresponding properties
(name, spin, mass, width, color representation, PDG code,
. . . )
param_card.dat: a text file containing the numerical
values of the necessary external parameters for a specific
model. The parameter card has a format compliant with
the SUSY Les Houches Accord (SLHA) and is
dependent on the physics model. One should pay attention to the
fact that some of these parameters are related one to each
other (e.g., the masses and the widths are generally related
to more fundamental Lagrangian parameters). If possible,
this file should also contain (preferably at the end) a list of
Les Houches QNUMBERS blocks describing properties
of non-SM particles to facilitate the interface of
matrixelement and parton-shower based generators, as proposed
in [64].
intparam_definition.inc: a text file containing
all the algebraic expressions relating internal parameters
to external and/or internal parameters. There are two
different kinds of internal parameters. Indeed, most of the
expressions can be computed once and for all, but in some
cases where the parameter depends on the scale of the
process (e.g., the strong coupling constant), it might be
desirable to re-evaluate it at an event-by-event basis.
interactions.dat: a text file containing a list of all
interactions entering the model. Each interaction is
characterized by an ordered list of the involved particles, the
name of the corresponding coupling, the corresponding
type of coupling (for coupling order restrictions) and
possible additional switches to select particular HELAS
routines.
couplingsXX.f (where XX can be any integer
number): these files contain the algebraic expressions for the
couplings, expressed as FORTRAN formulas. By
convention, the file couplings1.f contains all expressions
which should be re-evaluated when an external parameter
(e.g., the renormalization scale) is modified on an
eventby-event basis. The files couplingsXX.f where XX
is greater than 1 contain all expressions which should be
only re-evaluated if the default external parameter values
are explicitly read from the LHA param_card.dat
parameter card. The actual number of these files may
vary, but a single file should be small enough to be
compiled using standard FORTRAN compilers. The full list of
these files should be included in the makefile through the
makeinc.inc include file.
input.inc and coupl.inc: FORTRAN files
containing all the necessary variable declarations. All parameters
and couplings can be printed on screen or written to file
using the routines defined in param_write.inc and
coupl_ write.inc, respectively. If needed, the
latter can also be printed in a stricter format using routines
defined in helas_ couplings.f, so they can be used
by external tools (e.g., BRIDGE [65]).
Additional FORTRAN files, which are not model dependent,
should also be provided in order to build the full library.
Most of them simply include one or more of the above files,
except lha_read.f which contains all the routines
required to read the LHA format. A makefile allows the
user to easily compile the whole package, to produce a
library or a test program called testprog which can be
used to quickly check the library behavior by producing a
standard output.
The USRMOD V2 framework
The USRMOD V2 framework has been designed as
the successor of the widely-used original USRMOD
template described in [9]. Taking advantage of the fixed
structure that we have just defined, it provides the user with
two new possibilities. First, any pre-existing model can
be used as a starting point. This of course includes all
models described in the present paper and soon part of
the MADGRAPH/MADEVENT distribution, but also all
future models. This gives a natural framework for building
simple extensions following a bottom-up approach, i.e.,
by adding successively new particles and new interactions
and testing their implications at each step. Second, the
possible modifications are no longer restricted to the
addition of new particles/interactions, but any alteration of
the model content (including particle removal,
modification of existing properties, . . . ) allowed in the context of
MADGRAPH/MADEVENT is supported in an user-friendly
way.
The USRMOD V2 approach can advantageously replace
the full FEYNRULES package when only minor
modifications to an existing MADGRAPH/MADEVENT model are
necessary, e.g., in order to study the phenomenology of a
specific new particle and/or interaction, or when the use of
the FEYNRULES machinery is not possible, e.g., if a
MATHEMATICA license is not available. However, when possible,
we believe the use of FEYNRULES should be favored over
the USRMOD V2 for the consistent implementation of full
new models in MADGRAPH/MADEVENT, especially due to
the extended validation possibilities available in this
context.
From the implementation point of view, the USRMOD V2
package consists in various PYTHON scripts (with one single
main script) and works on any platform offering support for
this programming language. The actual implementation of a
new model is decomposed into four distinct phases:
1. Saving: the model directory used as a starting point
should be copied to a new location. The USRMOD script
should be run a first time to create a content archive used
as a reference to identify the forthcoming modifications.
2. Modifying: the particles.dat, interactions.
dat and ident_card.dat files can be modified to
arbitrarily add, remove or modify the particle, interaction
and parameter content.
3. Creating: the USRMOD script should be then run a second
time to actually modify all the model files to consistently
reflect the changes applied in the previous phase.
4. Adjusting: the couplingsXX.f file(s) can finally be
edited, if necessary, to add or modify the relevant
coupling expressions. The param_card.dat file can also
be edited to modify default values of external parameters.
At any time, the archive file created during the first phase
can be used to restore the initial content of all model files.
Several archive files can also be simultaneously saved into
the same directory to reflect, for example, the successive
versions of a single model. Finally, the intrinsic structure of the
USRMOD V2 package favors various technical (not
physical) consistency checks in the output files to minimize as
much as possible the compilation and runtime errors.
4.3.2 FEYNRULES interface
The MADGRAPH/MADEVENT interface can be called from
the FEYNRULES package using the
WriteMGOutput[ L ]
routine described in the FEYNRULES documentation, where
L is the name of the model Lagrangian. Since
MADGRAPH/MADEVENT currently only supports calculations
in unitary gauge, all the Goldstone and ghost fields are
discarded in the particles.dat output, which is
directly generated from the model description. After
expanding all possible field indices (e.g., associated to
flavor), an exhaustive list of non-zero vertices is generated
and output as interactions.dat. If possible, the
relevant coupling is extracted and, in case it does not
already exist, stored in a new coupling variable of the form
MGVXXX in a couplingsXX.f file. All the other
required model-dependent files are finally generated,
including the param_card.dat where the default values
(which are also the default values for the reading routines
in param_read.inc) are set as specified in the
FEYNRULES model file, and where the QNUMBERS blocks
correctly reflect the new particle content. All the produced files,
together with the relevant model independent files are stored
in a local directory model_name_MG, ready to be used
in MADGRAPH/MADEVENT. As mentioned previously, the
testprog test program can be compiled and run to check
the consistency of the created library.
The two main restrictions of the MADGRAPH/
MADEVENT interface are related to the allowed Lorentz and
color structures of the vertices. As already mentioned, even
though FEYNRULES itself can deal with basically any
interaction involving scalars, fermions, vectors and spin-two
tensors, the HELAS library, used by MADGRAPH/MADEVENT
to build and evaluate amplitudes, is more restricted. In the
case no correspondence is found for a specific interaction,
a warning message is displayed by the interface and the
corresponding vertex is discarded. If this particular vertex is
required for a given application, the user has still the
possibility to implement it manually following the HELAS library
conventions and to slightly modify the interface files to deal
with this addition. When the vertex structure is not present in
MADGRAPH/MADEVENT, a more involved manual
modification of the code is also required. The second limitation
of the present interface comes from the fact that the color
factor calculations are currently hardcoded internally within
MADGRAPH. While FEYNRULES can deal with fields in
any representation of the QCD color group, MADGRAPH
itself is basically limited to the color representations
appearing in the Standard Model and the Minimal Supersymmetric
Standard Model, e.g., a color sextet is not supported.
Let us mention that work to alleviate both limitations is
already in progress. The FEYNRULES package could, for
example, be used to generate automatically missing HELAS
routines, while a more open version of the MADGRAPH
matrix-element generator, e.g., taking advantage of a
highlevel programming environment, could advantageously deal
with arbitrary color structures
4.4 SHERPA
SHERPA [11, 12] is a general-purpose Monte Carlo event
generator aiming at the complete simulation of physical
events at lepton and hadron colliders. It is entirely written
in C++ featuring a modular structure where dedicated
modules encapsulate the simulation of certain physical aspects
of the collisions.
The central part is formed by the hard interaction,
described using perturbative methods. The respective
generator for matrix elements and phase-space integration is
AMEGIC++ [66], which employs the spinor helicity
formalism [67, 68] in a fully automated approach to
generate processes for a variety of implemented physics models,
see Sec. 4.4.1. Phase-space integration is accomplished
using self-adaptive Monte Carlo integration methods [6972].
Note that since version SHERPA-1.2 features a second
matrix-element generator called COMIX [16]. However, at
present COMIX is restricted to Standard Model processes
and does not yet support inputs from FEYNRULES.
The QCD evolution of partons originating from the hard
interaction down to the hadronization scale is simulated by
a parton-shower algorithm based on CataniSeymour
dipole subtraction [73]. It accounts for parton emissions off
all colored particles present in the Standard Model and the
Minimal Supersymmetric Standard Model. This shower
algorithm, replacing SHERPAs old APACIC++ shower [74],
accounts for QCD coherence and kinematic effects in a
way consistent with NLO subtraction schemes and is well
suited to accomplish the merging of leading and
next-toleading order matrix-element calculations with parton
showers [37, 75, 76].
An important aspect of SHERPA is its
implementation of a generalized version of the CKKW algorithm for
merging higher-order matrix elements and parton showers
[31, 32, 37]. It has been validated in a variety of processes
[40, 7780] and proved to yield reliable results in
comparison with other generators [38, 39].
SHERPA features an implementation of the model for
multiple-parton interactions presented in [81], which was
modified to allow for merging with hard processes of
arbitrary final-state multiplicity and eventually including
Fig. 2 Schematic view of the SHERPAs MODEL module, hosting the
particle and parameter definitions of physics models as well as
corresponding interaction vertices
CKKW merging [82]. Furthermore SHERPA provides an
implementation of a cluster-fragmentation model [83],
a hadron and tau decay package including the simulation
of mixing effects for neutral mesons [84], and an
implementation of the YFS formalism to simulate soft-photon
radiation [85].
4.4.1 Model framework
Physics model definitions within SHERPA are hosted by the
module MODEL. Here the particle content and the
parameters of any model get defined and are made accessible for use
within the SHERPA framework. This task is accomplished
by instances of the basic class Model_Base. Furthermore
the interaction vertices of various models are defined here
that in turn can be used by AMEGIC++ to construct
Feynman diagrams and corresponding helicity amplitudes.6 The
corresponding base class from which all interaction models
are derived is called Interaction_Model. A schematic
overview of the MODEL module is given in Fig. 2.
The list of currently implemented physics models reads:
the Standard Model including effective couplings of the
Higgs boson to gluons and photons [86], an extension of
the SM by a general set of anomalous triple- and quartic
gauge couplings [87, 88], the extension of the SM through
a single complex scalar [89], the extension of the Standard
Model by a fourth lepton generation, the SM plus an
axigluon [90], the Two-Higgs-Doublet Model, the Minimal
Supersymmetric Standard Model, and the
Arkani-HamedDimopoulos-Dvali (ADD) model of large extra dimensions
[91, 92], for details see [12]. Besides routines to set up the
spectra and Feynman rules of the models listed above
corresponding helicity-amplitude building blocks are provided
within AMEGIC++ that enable the evaluation of production
and decay processes within the supported models. In
particular this includes all the generic three- and four-point
interactions of scalar, fermionic and vector particles present in
6Note that within SHERPA Feynman rules are always considered in
unitary gauge.
the SM and Minimal Supersymmetric Standard Model plus
the effective operators for the loop-induced Higgs couplings
and the anomalous gauge couplings. The implementation of
the ADD model necessitated the extension of the helicity
formalism to interaction vertices involving spin-two
particles [93].
A necessary ingredient when dealing with the Minimal
Supersymmetric Standard Model are specific Feynman rules
for Majorana fermions or fermion number violating
interactions. To unambiguously fix the relative signs amongst
Feynman diagrams involving Majorana spinors the
algorithm described in [94] is used. Accordingly, the explicit
occurrence of charge-conjugation matrices in the
Feynman rules is avoided and instead a generalized fermion
flow is employed that assigns an orientation to complete
fermion chains. This uniquely determines the external
spinors, fermion propagators and interaction vertices
involving fermions.
The implementation of new models in SHERPA in the
traditional way is rather straight-forward and besides the public
model implementations shipped with the SHERPA code there
exist further private implementations that were used for
phenomenological studies, cf. [9597]. From version
SHERPA1.2 onwards SHERPA supports model implementations from
FEYNRULES outputsfacilitating the incorporation of new
models in SHERPA further.
4.4.2 FEYNRULES interface
To generate FEYNRULES output to be read by SHERPA, the
tailor-made FEYNRULES routine
WriteSHOutput[ L ]
has to be called, resulting in a set of ASCII files that
represent the considered model through its particle data, model
parameters and interaction vertices.7
To allow for an on-the-flight model implementation from
the FEYNRULES outputs, instances of the two basic classes
Model_Base and Interaction_Model_Base are
provided dealing with the proper initialization of all the
particles and parameters, and the interaction vertices of the new
model, respectively. The actual C++ classes for these tasks
are called FeynRules_Model and Interaction_
Model_FeynRules, see Fig. 2.
The master switch to use a FEYNRULES generated model
within SHERPA is
MODEL = FeynRules
to be set either in the (model) section of the SHERPA run
card or on the command line once the SHERPA executable
is called. Furthermore the keywords FR_PARTICLES,
7Note again that Feynman rules have to be considered in unitary gauge.
FR_IDENTFILE, FR_PARAMCARD, FR_PARAMDEF and
FR_INTERACTIONS, specifying the names of
corresponding input files, need to be set. The actual format and assumed
default names of these input cards will be discussed in the
following:
FR_PARTICLES specifies the name of the input file
listing all the particles of the theory including their SM
quantum numbers and default values for their masses and
widths, default name is Particle.dat. An actual
particle definition, e.g., for the gluon, looks like
Width 3*e Y SU(3) 2*Spin
.0 0 0 8 2
maj on stbl m_on IDName TeXName
-1 1 1 0 G G
Hereby kf defines the code the particle is referred to
internally and externally, typically its PDG number [54].
The values for Mass and Width need to be given in units
of GeV. The columns 3*e and Y specify three times the
electric charge and twice the weak-isospin. SU(3)
defines if the particle acts as a singlet (0), triplet (3) or
octet (8) under SU(3)C . 2*Spin gives twice the
particles spin and maj indicates if the particle is charged (0),
self-adjoint (-1) or a Majorana fermion (1). The flags
on, stbl and m_on are internal basically and define
if a particle is considered/excluded, considered stable,
and if its kinematical mass is taken into account in the
matrix-element evaluation. IDName and TeXName
indicate names used for screen outputs and potential LATEX
outputs, respectively.
In FR_IDENTFILE all the external parameters of the
model get defined, default file name is ident_card.
dat. Names and counters of corresponding parameter
blocks to be read from FR_PARAMCARD are listed and
completed by the actual variable names and their
numerical types, i.e., real R or complex C. Besides, variable
names for all particle masses and widths are defined here.
To give an example, the section defining the electroweak
inputs of the SM may look like
SMINPUTS 1 aEWM1 R
SMINPUTS 2 Gf R
SMINPUTS 3 aS R
CKMBLOCK 1 cabi R
In the file specified through FR_PARAMCARD the
numerical values of all elementary parameters, particle masses
and decay widths are given, default file is param_card.
dat. Following the example above the electroweak inputs
of the SM can be set through:
Block SMINPUTS
1 1.2790000000000E+02
2 1.1663900000000E-05
3 1.1800000000000E-01
Block CKMBLOCK
1 2.2773600000000E-01
# cabi
FR_PARAMDEF gives the file name where all sorts
of internal parameters are defined, default param_
definition.dat. Such variables can be functions of
the external parameters and subsequently other derived
quantities. A few examples for the case of the SM again
might read:
aEW = pow(aEWM1,-1.) R
! Electroweak coupling constant
G = 2.*sqrt(aS)*sqrt(M_PI) R
! Strong coupling constant
CKM11 = cos(cabi) C
! CKM-Matrix ( CKM11 )
The parameter definitions get interpreted using an
internal algebra interpreter, no additional compilation is
needed for this task. All standard C++ mathematical
functions are supported, e.g., sqr, log, exp, abs. For
complex valued parameters, e.g., CKM11, the real and
imaginary part can be accessed through Real(CKM11)
and Imag(CKM11), the complex conjugate is obtained
through Conjugate(CKM11).
The keyword FR_INTERACTIONS (Interactions.
dat) specifies the input file containing all the vertices of
the considered model in a very simple format:
3 F[1,2,3]
# G G G
# right-handed coupling
# left-handed coupling
# colour structure
# Lorentz structure
The keyword VERTEX signals the start of a new
Feynman rule followed by the PDG codes of the involved
particles. Note, the first particle is always considered incoming
the others outgoing. Counters number 1 and 2 indicate the
right and left-handed coupling of the vertex rule, the right
and left-hand projector being given by PR/L = 21 (1 5),
respectively. Couplings are given in terms of the
elementary and derived parameters. Counter number 3
explicitly gives the color structure of the interaction in terms
of the SU(3)C structure constants or generators. The spin
structure of the vertex is given under 4, identified through
a keyword used by SHERPA to relate a corresponding
sub-amplitude to the correct helicity-amplitude building
block.
SHERPAs interface to FEYNRULES is designed to be as
general as possible, it is, however, by construction restricted
in two ways. First, the functional form of the model
parameters, and respectively the couplings, is limited by the
capabilities of the algebra interpreter that has to construct
them. This limitation, however, might be overcome by using
an external code to calculate the needed variables and
redefining them as external giving their numerical values in
FR_PARAMCARD. Second, a more severe limitation
originates from the restricted ability of SHERPA/AMEGIC++
to handle new types of interactions. Only three and
fourpoint functions can be incorporated. For the color structures
only the SU(3)C objects 1, ij , ab, Tiaj , f abc and products
of those, e.g., f abcf cde , are supported. Lorentz structures
not present in the SM or the Minimal Supersymmetric
Standard Model are currently not supported by the interface.
Furthermore, SHERPA cannot handle spin-3/2 particles. QCD
parton showers are only invoked for the colored particles
present in the SM and the Minimal Supersymmetric
Standard Model. Hadronization of new colored states is not
accomplished, they have to be decayed before entering the
stage of primary hadron generation.
5 Models
In this section we briefly present the implementation of the
Standard Model and several other important New Physics
models in FEYNRULES. Our aim is to show that very
complete and sophisticated implementations are possible
and that FEYNRULES offers a very natural and
convenient framework where models can be first developed (from
the theoretical point of view) and then tested/constrained
against experimental data. Since the main focus is the
implementation procedure, the actual model descriptions, as well
as the information about values of parameters, are kept to
a minimum. More exhaustive information about each of the
following models, all of which are publicly available, can be
found on the FEYNRULES website.
5.1 The standard model
5.1.1 Model description
As it serves as basis to any new bottom-up
implementation, we briefly describe here the Standard Model
implementation. The SM of particle physics is described by an
SU(3)C SU (2)L U (1)Y gauge theory, where the
electroweak symmetry is spontaneously broken so that the
fundamental fermions and the weak gauge bosons acquire a
mass. The particle content of the SM is summarized in
Table 4. The Lagrangian can be written as a sum of four parts,
Table 4 The SM fields and their representations under the Standard
Model gauge groups SU(3)C SU (2)L U (1)Y
The pure gauge sector of the theory reads
1 1 1
LSM,Gauge = 4 Ga Ga 4 W i Wi 4 B B , (17)
where the SM field strength tensors are defined following
the conventions introduced in (4). The Lagrangian
describing the matter fermions can be written as
+ dR0i iD/ dR0i + lR0i iD/ lR0i ,
where D denotes the SU(3)C SU(2)L U (1)Y covariant
derivative, and we use the conventions of (3). The
superscript 0 refers to the gauge eigenstates. Note in particular
that explicit mass terms for the matter fermions are
forbidden by gauge symmetry. The Higgs field is described by the
Lagrangian
LSM,Higgs = DD 2 2.
If 2 < 0, then the Higgs field acquires a vacuum
expectation value that breaks the electroweak symmetry
spontaneously. Expanding the Higgs field around its vev,
we generate mass terms for the Higgs boson H and the
electroweak gauge fields. The mass eigenstates for the gauge
bosons are the W and Z bosons, as well as the photon, which
remains massless. The relations between those fields and the
original SU(2)L U (1)Y gauge fields are
where we have introduced the weak mixing angle
The interactions between the fermions and the Higgs field
are described by the Yukawa interactions
LSM,Yukawa = ui0Ryiuj Qj0 di0Ryidj Qj0
where = i 2. After electroweak symmetry breaking
the Yukawa interactions generate non-diagonal mass terms
for the fermions that need to be diagonalized by unitary
rotations on the left and right-handed fields. Since there is no
right-handed neutrino, we can always rotate the leptons such
that the mass matrix for the charged leptons becomes
diagonal and lepton flavor is still conserved. For the quarks
however, the diagonalization of the mass matrices introduces
flavor mixing in the charged current interactions, described by
the well-known CKM matrix.
5.1.2 FEYNRULES implementation
The SM implementation in FEYNRULES is divided into the
four Lagrangians described in the previous section. In
particular, one can use the dedicated functions for the field
strength tensors and the covariant derivative acting on the
left and right-handed fermions. Matrix-element generators
however need as an input the mass eigenstates of the
particles, and therefore it is mandatory to rotate all the gauge
eigenstates into mass eigenstates according to the
prescriptions discussed in the previous section. This can be done
very easily in FEYNRULES by writing the Lagrangian in the
gauge eigenbasis, and then letting FEYNRULES perform the
rotation into the mass eigenstates (note, at this point, that
FEYNRULES does not diagonalize the mass matrices
automatically, but this information has to be provided by the
user). However, as the SM Lagrangian is the starting point
for many bottom up extensions, the actual implementation
was performed directly in terms of the fermion mass
eigenstates. The benefit is a slight speed gain due to the rotations
in the fermion sector. The default values of the external
parameters in the model file are given in Table 7, in
Appendix B.1.
Three restriction files for the SM implementation are
provided with the default model distribution:
Massless.rst: the electron and the muon, as well as
the light quarks (u, d , s) are massless.
DiagonalCKM.rst: the CKM matrix is diagonal.
Cabibbo.rst: the CKM matrix only contains Cabibbo
mixing.
Another particularity of the SM implementation is that
it was performed both in unitary and in Feynman gauge.
The model file contains a switch FeynmanGauge, which,
if turned to False, puts to zero all the terms involving ghost
and Goldstone fields. The default value is False.
Possible extensions The SM is at the basis of almost all
BSM models, and thus the number of possible extensions
of the SM implementation is basically unlimited. A first
extension of this model was presented in Sect. 2 with the HAH
model, based on the simplest possible extension of the gauge
sector of the SM. Other possibilities are the addition of
higher-dimensional operators compatible with the SM
symmetries (see Sect. 5.6) or the inclusion of right-handed
neutrinos via see-saw models.
The Two-Higgs-Doublet Model (2HDM) has been
extensively studied for more than twenty years, even though
it has often been only considered as the scalar sector
of some larger model, like the Minimal Supersymmetric
Standard Model or some Little Higgs models for
example. The general 2HDM considered here already displays,
by itself, an interesting phenomenology that justifies its
study. For example, new sources of CP violation in
scalarscalar interactions, tree-level flavor changing neutral
currents (FCNCs) due to non-diagonal Yukawa interactions,
or a light pseudoscalar state and unusual Higgs decays
(see [98]).
5.2.1 Model description
+ L2HDM,Yukawa.
The gauge and fermion sectors of the model are identical to
the SM,
L2HDM,Gauge = LSM,Gauge and
L2HDM,Fermions = LSM,Fermions.
The Lagrangian of the Higgs sector differs from the SM, and
can be written
L2HDM,Higgs = D1D1 + D2D2 V (1, 2),
and the scalar potential reads, in the notation of [99],
V (1, 2) = 11 1 + 22 2 + 31 2 + h.c.
where 1,2 and 1,2,3,4 are real parameters while 3 and
5,6,7 are a priori complex. We assume that the
electromagnetic gauge symmetry is preserved, i.e., that the vevs of 1
and 2 are aligned in the SU(2)L space in such a way that a
single SU(2)L gauge transformation suffices to rotate them
both to their neutral components,
with v1 and v2 two real parameters such that v12 + v22 v2 =
(2GF )1 and v2/v1 tan .
The most general form for the Yukawa interactions of the
two doublets reads
L2HDM,Yukawa =
with i i 2i and where the 3 3 complex Yukawa
coupling matrices i and i are expressed in the fermion
physical basis, i.e., in the basis where the fermion mass matrices
are diagonal. We choose as free parameters the i matrices,
while the other Yukawa couplings, the i matrices, are
deduced from the matching with the observed fermion masses.
Conventionally, the two indices a and b of the elements of
the Yukawa matrices (i )ab and (i )ab refer to the
generations of the SU(2)L doublet and singlet, respectively.
5.2.2 FEYNRULES implementation
The 2HDM Lagrangian implemented in FEYNRULES is
composed of (27) and (29), together with the canonically
normalized kinetic energy terms for the two doublets and
the other SM terms. An important feature of this model is
the freedom to redefine the two scalar fields 1 and 2 using
arbitrary U (2) transformations
U
U U (2)
since this transformation leaves the gauge-covariant kinetic
energy terms invariant. This notion of basis invariance has
been emphasized in [99] and considered in great detail more
recently in [100102]. Since a given set of Lagrangian
parameter values is only meaningful for a given basis, let us
take advantage of this invariance property to select the Higgs
basis (by defining the additional file HiggsBasis.fr)
where only one of the two Higgs fields acquires a non-zero
vev, i.e.,
H20 = 0.
Let us note that the Higgs basis is not defined unambigously
since the reparametrization H2 eiH2 leaves the
condition (31) invariant, so that the phase of H2 can be fixed in
such a way that 5 becomes real. Other basis choices can in
principle be easily implemented as different extension files
for the main Lagrangian file Lag.fr.
The minimization conditions for the potential of (27) read
(in the basis defined in (31))
which reduces the number of free parameters in the most
general 2HDM to ten (seven real parameters, three
complex ones and three minimization conditions). Besides the
usual three massless would-be Goldstone bosons, the
physical spectrum contains a pair of charged Higgs bosons with
mass
m2H =
and three neutral states with the squared mass matrix
The symmetric matrix M is diagonalized by an orthogonal
matrix T . The diagonalization yields masses mi for the three
physical neutral scalars Si of the model (where the index i
refers to mass ordering),
2 T diag m12, m22, m32 T T .
The doublet components are related to these physical states
through
The Yukawa couplings of the model are fully determined by
the i matrices in (29), since the i are, by definition, fixed
to the diagonal fermion mass matrices in the Higgs basis.
In the current implementation of the 2HDM into
FEYNRULES, the user has to provide numerical values for all the
i parameters in the basis of (31), together with the charged
Higgs mass mH . The other parameters of the potential,
such as the i , are then deduced using (32) and (33). As
a consequence, the orthogonal matrix T must be calculated
externally. This, together with the change of basis required if
the user wants to provide potential parameters and Yukawa
coupling values in bases different from this of (31), can
be done using the TWOHIGGSCALC calculator introduced
in [9] which has been modified to produce a parameter file
compatible with the present implementation. This
calculator can also be used to calculate the required Higgs boson
tree-level decay widths.
5.3 The most general minimal supersymmetric standard
model
Most present supersymmetric models are based on the
fourdimensional supersymmetric field theory of Wess and
Zumino [103]. The simplest model is the straightforward
supersymmetrization of the Standard Model, with the same
gauge interactions, including R-parity conservation, and is
called the MSSM [104, 105]. Its main features are to link
bosons with fermions and unify internal and external
symmetries. Moreover, it allows for a stabilization of the gap
between the Planck and the electroweak scale and for gauge
coupling unification at high energies, provides the
lightest supersymmetric particle as a dark matter candidate and
appears naturally in string theories. However, since
supersymmetric particles have not yet been discovered,
supersymmetry must be broken at low energies, which makes the
superpartners heavy in comparison to their Standard Model
counterparts.
Supersymmetric phenomenology at colliders has been
extensively investigated for basic processes at leading order
[106117] and next-to-leading order [118124] of
perturbative QCD. More recently, for some processes, soft-gluon
radiation has been resummed to all orders in the strong
coupling constant and the results have been matched with
the next-to-leading order calculations [125130]. However,
even if those calculations are useful for inclusive enough
analyses, they are not suitable if we are interested in a proper
description of the full collider environment, for which
Monte Carlo event generators are needed. For a couple of
years, all the multi-purpose generators already mentioned
contain a built-in version of the MSSM. The model files
for FEYNARTS/FORMCALC are described in [131, 132],
for CALCHEP in [133], for MADGRAPH/MADEVENT in
[134], and for SHERPA in [135]. The SHERPA and
FEYNARTS/FORMCALC implementations keep generic mixing in
Table 5 The MSSM fields and
their representations under the
Standard Model gauge groups
SU(3)C SU (2)L U (1)Y .
The quarks and leptons are
denoted in terms of
two-component Weyl spinors
B-boson, bino
W -bosons, winos
(Hd0Hd)T
(W 1W 2W 3)T
(3, 1, 2/3)
(1, 2, 1/2)
(1, 2, 1/2)
the scalar sector while the other generators rely on a
simplified model with only helicity mixing for third generation
sfermions.
Our MSSM implementation in FEYNRULES is the most
general one in a sense that it is keeping all the
flavorviolating and helicity-mixing terms in the Lagrangian and
also all the possible additional CP-violating phases. This
yields thus 105 new free parameters [136], and in order to
deal in a transparent way with all of those, our
implementation will follow the commonly used universal set of
conventions provided by the Supersymmetry Les Houches Accord
(SLHA) [137, 138], except for some minor points. We will
dedicate a complementary paper to a complete description
of the model [139].
5.3.1 Model description
Field content Each of the Standard Model quarks and
leptons is represented by a four-component Dirac spinor fi0,
where i stands for the generation index and the
superscript 0 denotes interaction eigenstates. It has two associated
scalar superpartners, the sfermion fL0i and the antisfermion
f0i, being related to the two-component holomorphic Weyl
R
fermion f i and antifermion fi , respectively. Let us recall
that we relate the Dirac fermion representations to the Weyl
ones by
fi0 =
Hu =
Hd =
Finally, the spin-one vector bosons of the Standard Model
will be associated to Majorana fermions, the gauginos B ,
W k and g . The names and representations under the
Standard Model gauge groups SU(3)C SU (2)L U (1)Y of the
various fields are summarized in Table 5.
The full MSSM Lagrangian can we written as
+ LMSSM,Scalar kinetic + LMSSM,Scalar FDW
+ LMSSM,Ino kinetic + LMSSM,Ino Yukawa
+ LMSSM,Ino mix + LMSSM,Soft.
Starting from the expression of the Lagrangian in the
gaugeeigenstate basis of fields given above, we diagonalize the
non-diagonal mass matrices arising after electroweak
symmetry breaking and provide transformation rules allowing to
re-express the Lagrangian in the physical basis.
Supersymmetry-conserving Lagrangian In order to have
more compact notations, we introduce the SU(2)L-doublets
of left-handed fermions and sfermions,
For clarity, we will use in the following the left-handed
component fL0i and the right-handed component fR0i of the Dirac
fermion fi0 and not the Weyl fermions f i and fi . To
preserve the electroweak symmetry from gauge anomaly and in
order to give masses to both up-type and down-type
fermions, the MSSM contains two Higgs doublets Hi , together
Qi0 =
Q i0 =
Li0 =
L i0 =
only the terms quadratic in the cutoff have been
computed. More precisely, we computed the one-loop
corrections to the two-point functions , , and
. The momentum independent part of each of these
amplitudes, but the last one, corresponds to mass
corrections and is b-independent as it should. For example, the
renormalization of the pion wave function is
Z = 6(1 8b)
where R 1 + Z,
while its mass correction vanishes.11 The decay
amplitude has also been computed at tree-level and at
oneloop. In this last result, also the c-dependence cancels.
Eventually, a total of about 60 diagrams were computed with both
methods and perfect agreement was found.
SILH model The check of the SILH model consists in an
analytic comparison between the decay widths computed
in [174] and computations based on the vertices given by
FEYNRULES. We used the same simplifications, i.e., cT = 0
and (cW + cB )MW2 /M2 = 0 and neglect Lvect. For tree-level
decay widths of the Higgs into two fermions, both
implementations leads to exactly the same results. For decays into
a gauge boson pair, the contribution to the Feynman rules
from higher-dimensional operators read, e.g., for the hW W
vertex,
2
+ i cW g 2 2 2 p32 p3
2v g2 2,3 p2 + p3 p22 p3 3
+ i cH W g2
2v (4 )2 2,3 p12 p12 p13 ,
where p1, p2 and p3 denote the momenta of the Higgs boson
and the two W bosons respectively. As a consequence, the
corrections are not just proportional to the SM decay widths
since the vertices have a more complicated kinematic
structure. We find
g2
= H W +W SM 1 cH g2 cW
1 4
11All the other results can be found in [188].
g2
= (H ZZ)SM 1 cH g2 cW + tan2 wcB
g2 mH 2
+ cH W + tan2 wcH B (4 )2 16 v2 mH + 2MZ2
1 4
in agreement with [189]. In the situation where we have
the hierachy of couplings g < g < 4 , the second term in
(131) and (132) is suppressed parametrically with respect to
the first one and could thus be neglected, leading to just a
rescaling of the SM decay widths [174]. Let us note
however that in the case where the ratio g2/(4 )2 is not too
small, this additional term could have a numerical impact
on the decay rates.
7 Outlook
We have described a new framework where BSM physics
scenarios can be developed, studied and automatically
implemented in Monte Carlo or symbolic calculation tools for
theoretical and experimental investigations. The main
purpose of this work has been to contribute to streamlining the
communication (in both directions) between the theoretical
and the experimental HEP communities.
The cornerstone of our approach is the
MATHEMATICA package FEYNRULES where any perturbative quantum
field theory Lagrangian, renormalizable or not, can be
written in a straightforward way and the corresponding
Feynman rules obtained automatically. All the relevant
information can then be passed through dedicated interfaces to
matrix-element generators for Feynman diagram
calculations at tree level or at higher orders. The scheme itself
looks very simple and is in fact not a new idea. The
novelty, however, lies in several technical and design aspects
which, we believe, constitute a significant improvement over
the past.
First, the use of MATHEMATICA as a working
environment for the model development gives all the
flexibility that is needed for symbolic manipulations. The many
built-in features of MATHEMATICA, such as matrix
diagonalization and pattern recognition functions, play an
important role in building not only robust interfaces for
very different codes but also to open up new
possibilities. For instance, besides implementing BSM models, new
high-level functionalities/applications can be easily
developed by the users themselves, made public and possibly
included in subsequent official releases. In other words,
the code is naturally very open to community
contributions. A typical application that could take advantage of
this open structure is the (semi-)automatic development
of model calculators inside the FEYNRULES package
itself, including mass spectrum and decay width
calculations.
Second, the interfaces to MC codes, all of them quite
different both in philosophy, architecture and aim, offer
the possibility of testing and validating model
implementations at a very high level of accuracy. It also maximizes
the probability that a given model might be dealt with by
at least one matrix element generator. For example, purely
symbolic generators such as FEYNARTS/FORMCALC can
be used for tree-level (or even loop) calculations which
can then be compared or extended to numerical results
from CALCHEP, MADGRAPH/MADEVENT or SHERPA. In
this respect we note that one of the current major and
common limitations of the matrix-element generators (but
not of FEYNRULES) is connected to use of a fixed
library of Lorentz and/or gauge structures for the vertices.
These libraries are in principle extendable, but at present
this is done by hand and it entails a tedious and often
quite long work. The automatization of this part (and
possibly the reduction of higher-dimensional operators to
renormalizable ones) through FEYNRULES would be the final
step towards full automatization of any Lagrangian into
Monte Carlo codes. Work in this direction is already in
progress.
With such a framework in place, we hope that several
of the current problems and drawbacks in new physics
simulations faced by the experimental groups will be
alleviated if not completely solved. First the need of dedicated
codes for specific models, which then call for long and
tedious validations both from the physics point of view
as well as in their interplay with collaboration softwares,
will be greatly reduced. General purpose tools, from
HERWIG and PYTHIA to SHERPA, MADGRAPH/MADEVENT,
WHIZARD, CALCHEP/COMPHEP (and potentially also
HELAC and ALPGEN), several of which have been
successfully embedded in frameworks such as ATHENA (ATLAS)
and CMSSW (CMS), offer several ready-to-go solutions
for any FEYNRULES based models. Reducing the
proliferation of highly dedicated tools will greatly simplify the
maintenance and reliability of the software and more
importantly the reproducibility of the MC samples in the mid
and long terms. In addition, we believe an effort towards
making new Lagrangians in FEYNRULES and the
corresponding benchmark points publicly available (for
example by the proponents of the model themselves at the same
time of the release of the corresponding publication) would
certainly be a great advantage for the whole HEP
community.
It is our hope that FEYNRULES will effectively facilitate
interactions between theorists and experimentalists. Until
now there has not been a preferred way to link the two
communities. Various solutions have been proposed and used,
most of them plagued by significant limitations. The best
available proposal so far has been to use parton-level events
in the Les Houches Event File (LHEF) format [190]. This is
a natural place to cut a line given that theorists and
phenomenologists can generate events through various private or
public tools, and then pass them for showering and
hadronization to codes such as PYTHIA or HERWIG. These codes
are not only already embedded in the experimental software
(for the following detector simulations), but also have been
(or will be) tuned carefully to reproduce control data
samples.
While we think this is still a useful approach that should
be certainly left open and supported, we are convinced the
framework we propose is a promising extension. The deep
reason is that, in our approach, there is no definite line
between theory and experiment. On the contrary it creates a
very extended region where the two overlap and work can
be done in the same common framework. This leaves much
more freedom in where exactly the two ends meet and what
kind of checks and information can be exchanged. As a
result there are several practical advantages that come for
free.
As an example, we remind that the LHEF format only
standardizes the information on the events themselves and
some very basic global properties, but any information on
the physical model (i.e., the explicit form of the Lagrangian)
or the parameter choices is in general absent. This is of
course due to implementation differences among various
codes which severely compromise any standardization
attempt at this level. It is clear that, in the long run, this
might lead to serious problems of traceability of the MC
samples and various ambiguities in understanding
experimental analyses (such as placing of exclusion limits). This
problem is of course completely overcome by the approach
advocated here, since models are now fully and uniquely
defined. In this sense FEYNRULES itself offers the sought for
standardization.
Another, and maybe even more striking example is that,
within FEYNRULES, model building and/or refinement can
in principle be done in realtime together with the related
experimental analyses. One could imagine, for example, that
if the TeV world is as rich as we hope and as data start
showing hints for new particles or effects, a large number of
competing Lagrangians (and not only benchmark points as
used in the typical top-down SUSY analyses) could be
easily and quickly implemented and readily confronted to data.
This could be done in a virtuous loop, where theorists and
experimentalists meet at a convenient point of the
simulation chain. In other words, various top-down and
bottomup studies can fit naturally in this framework, partially
addressing often reported worries about the actual possibilities
to extract precise information on BSM physics from LHC
data.
Finally, let us also mention a more long-term
advantage of the proposed framework. Automatic NLO
calculations for SM processes are now clearly in sight, and,
in this context, it might be reasonable to ask whether
those developments can be extended to BSM processes.
We believe the answer is yes, and, since this
generalization will probably rely on the simultaneous implementation
of the model characteristic in different codes (e.g.,
dealing with different parts of the calculation like real and
virtual corrections, or analytic and numerical results) our
approach might also naturally play a crucial role in this
context.
Acknowledgements We are particularly thankful to the
organizers and participants of the MC4BSM series of workshops for the
many stimulating discussions that have helped in shaping the
approach presented in this paper. Several of us are grateful to the
members of their respective MC development teams (CALCHEP,
MADGRAPH/MADEVENT and SHERPA) and to Thomas Hahn for
FEYNARTS/FORMCALC for their support and help in developing the
interfaces to FEYNRULES. The authors are also grateful to A. Datta and
C. Grojean for valuable discussions on the MUED and SILH models.
C. Degrande and C. Duhr are research fellows of the Fonds National
de la Recherche Scientifique, Belgium. This work was partially
supported by the Institut Interuniversitaire des Sciences Nuclaires, by
the FWO -Vlaanderen, project G.0235.05 and by the Belgian
Federal Office for Scientific, Technical and Cultural Affairs through the
Interuniversity Attraction Pole P6/11. It is also part of the research
program of the Stichting voor Fundamenteel Onderzoek der Materie
(FOM), which is financially supported by the Nederlandse
organisatie voor Wetenschappelijke Onderzoek (NWO). N.D.C. was
supported by the U.S. Department of Energy under Grant No.
DE-FG0206ER41418 and the National Science Foundation under grant number
PHY-0705682. The authors of FEYNRULES are grateful to R.
Franceschini and K. Kannike for their valuables comments on the code, and
especially to M. Wiebusch for implementing the covariant
derivative.
Open Access This article is distributed under the terms of the
Creative Commons Attribution Noncommercial License which permits
any noncommercial use, distribution, and reproduction in any medium,
provided the original author(s) and source are credited.
Appendix A: The FEYNRULES convention for standard
model inputs
There are several parameters and particles that have
special significance in Feynman diagram calculators. Examples
of these are the strong and electromagnetic couplings, the
names of the fundamental representation and the structure
constants of the strong gauge group and so on. For this
reason, the names of these objects are fixed at the
FEYNRULES level. Adherence to these standards will increase the
chances of successful translation.
The strong gauge group has special significance in many
Feynman diagram calculators. A user who implements the
strong gauge group should adhere to the following rules.
The indices for the fundamental and adjoint representations
of this gauge group should be called Colour and Gluon
respectively. Furthermore, the names of the QCD gauge
boson, coupling constant, structure constant, totally symmetric
tensor and fundamental representation should be given by G,
gs, f, dSU3 and T as in the following example:
SU3C == {
Abelian -> False,
GaugeBoson -> G,
StructureConstant -> f,
SymmetricTensor -> dSU3,
Representations -> {T, Colour},
CouplingConstant -> gs
}
In addition, the strong coupling constant and its square over
4 should be declared in the parameter section in the
following form:
\[Alpha]S == {
ParameterType -> External,
Value -> 0.118,
ParameterName -> aS,
BlockName -> SMINPUTS,
InteractionOrder -> {QCD, 2},
Description -> "Strong coupling
at the Z pole."
},
gs == {
ParameterType -> Internal,
Value -> Sqrt[4 Pi \[Alpha]S],
ParameterName -> G,
InteractionOrder -> {QCD, 1}
Note that S is given as the external parameter and gS as
the internal parameter. The description of S may be edited,
but it should be remembered that, for the Monte Carlo
programs that run the strong coupling constant, the value of S
should be set at the Z pole. For calculation programs that do
not run the strong coupling, on the other hand, it should be
set according to the scale of the interaction. A description
may also be added to the parameter gS .
The electromagnetic interaction also has special
significance in many Feynman diagram calculators and we
outline the following standard definitions. The electric coupling
constant should be called ee, the electric charge should be
called Q. The declaration of the electric charge should follow
the following conventions for naming:
\[Alpha]EWM1 == {
ParameterType -> External,
Value -> 127.9,
ParameterName -> aEWM1,
BlockName -> SMINPUTS,
InteractionOrder -> {QED, -2},
-> "alpha_EM inverse
at the Z pole."
},
\[Alpha]EW == {
ParameterType -> Internal,
Value -> 1/\[Alpha]EWM1,
InteractionOrder -> {QED, 2},
ParameterName -> aEW,
},
ee == {
ParameterType -> Internal,
Value -> Sqrt[4 Pi \[Alpha]EW ],
InteractionOrder -> {QED, 1}
}
As for the strong coupling, the description of EW1 may
be edited,12 but it should be remembered that for
calculation programs that run the electric coupling, it should be set
at the Z pole. For programs which do not run it, the electric
coupling should be set at the interaction scale. Again, a
description may be added to the definition of \[Alpha]EW
and ee.
The Fermi constant and the Z pole mass are very
precisely known and are often used in calculators to define
coupling constants and the scale where couplings are run from.
They should be included in the SMINPUTS block of the Les
Houches Accord and should be defined by at least the
following:
Gf == {
ParameterType -> External,
Value -> 1.16639 * 10^({-}5),
BlockName -> SMINPUTS,
InteractionOrder -> {QED, 2},
Description -> "Fermi constant"
},
MZ == {
ParameterType -> External,
Value -> 91.188,
BlockName -> SMINPUTS,
Description -> "Z pole mass"
}
Moreover, the weak coupling constant name gw and the
hypercharge symbol Y are used by some calculators and
the user is encouraged to use these names where
appropriate. The masses and widths of particles should be assigned
whenever possible. If left out, FEYNRULES will assign the
value 1 to each. Finally, particles are also identified by a
PDG number. The user is strongly encouraged to use
existing PDG codes in their model wherever possible. If not
included, a PDG code will be automatically assigned by
FEYNRULES beginning at 9000001.
12The reason for choosing EW1 as the external input parameter, and
not EW itself is only to be compliant with the Les Houches Accord.
Appendix B: Validation tables
In this Appendix we report the main results of our work,
i.e., the validation tables. In general, the tables list
quantities (such as decay widths or cross sections) that have
no direct phenomenological interest but they are physical,
easily reproducible and provide an exhaustive check of the
(complex) values of all the couplings of the model. In
several instances, other powerful checks (such as gauge
invariance, unitarity cancellation at high energy, and so on) have
been performed and are not presented here. Whenever
possible, comparisons between the so-called stock
implementations, i.e., implementations already available in the Monte
Carlo tools, have been made as well as comparisons between
different Monte Carlos also in different gauges. All
numbers quoted in this section are expressed in picobarns and
correspond to a collision in the center-of-mass frame.
Furthermore, in all cases the generators were set up such as to
achieve numerical results within an accuracy below the
permille level, and we observe that with this setup the
differences between the different generators as well as between
the FEYNRULES and the stock implementations is always
significantly below the 1% level.13
B.1 The standard model
In this section we give the results for the 35 cross
sections tested for the SM between CALCHEP, MADGRAPH/
MADEVENT and SHERPA for a total center of mass energy
of 550 GeV. A pT cut of 20 GeV was applied to each
final state particle. The set of external parameters used for the
test is given in Table 7. A selection of processes is shown in
Tables 8 and 9.
B.2 The general two-Higgs-doublet model
In this section we give a selection of the results for the
185 cross sections tested for the 2HDM between CALCHEP,
MADGRAPH/MADEVENT and SHERPA for a total
centerof-mass energy of 800 GeV. A pT cut of 20 GeV was
applied to each final state particle. The set of external
parameters used for the test is given in Table 10. A selection of
processes is shown in Tables 11, 12 and 13.
B.3 The most general minimal supersymmetric standard
model
In order to fully determine the MSSM Lagrangian at low
energy scale, it is sufficient to fix the SM sector and the
13A more accurate validation can be achieved by comparing
individual phase-space points rather than integrated cross sections. CALCHEP
and MADGRAPH/MADEVENT allow for the numerical evaluation of
single points in phase space, and we compared the matrix elements
between the two generators at various randomly chosen points and
observe agreement to a much higher accuracy in all cases.
Table 8 Cross sections for a
selection of SM production
processes. The built-in SM
implementation in MADGRAPH
and CALCHEP and SHERPA are
denoted MG-ST, CH-ST and
SH-ST, respectively, while the
FEYNRULES-generated ones
MG-FR, CH-FR, SH-FR. The
center-of-mass energy is fixed to
550 GeV, and a pT cut of
20 GeV is applied to each final
state particle
Table 9 Cross sections for a
selection of SM production
processes. The built-in SM
implementation in
MADGRAPH, CALCHEP and
SHERPA are denoted MG-ST,
CH-ST and SH-ST, respectively,
while the
FEYNRULES-generated ones for
MADGRAPH/MADEVENT,
CALCHEP and SHERPA are
MG-FR, CH-FR and SH-FR.
The center-of-mass energy is
fixed to 550 GeV, and a pT cut
of 20 GeV is applied to each
final state particle
Lepton and weak boson processes in the standard model
Process MG-FR MG-ST CH-FR
Quark and gluon processes in the standard model
H H ZZ
H H W +W
GG GG
4.7 GeV
Table 10 Input parameters for
the 2HDM. All parameters that
are not quoted have a zero value
Charged Higgs mass
Yukawa parameters (real)
4.2 GeV
3 GeV
Table 11 Cross sections for a
selection of + initiated
processes in the 2HDM. The
built-in 2HDM implementation
in MADGRAPH is denoted
MG-ST, while the
FEYNRULES-generated ones for
MADGRAPH/MADEVENT,
CALCHEP and SHERPA are
MG-FR, CH-FR and SH-FR.
The center-of-mass energy is
fixed to 800 GeV, and a pT cut
of 20 GeV is applied to each
final state particle
Lepton processes in the two-Higgs-doublet model
Process MG-FR
Table 12 Cross sections for a
selection of + initiated
processes in the 2HDM. The
built-in 2HDM implementation
in MADGRAPH is denoted
MG-ST, while the
FEYNRULES-generated ones for
MADGRAPH/MADEVENT,
CALCHEP and SHERPA are
MG-FR, CH-FR and SH-FR.
The center-of-mass energy is
fixed to 800 GeV, and a pT cut
of 20 GeV is applied to each
final state particle
Neutrino processes in the two-Higgs-doublet model
Process MG-FR MG-ST
Table 13 Cross sections for a
selection of processes in the
2HDM with two weak bosons in
the initial state. The built-in
2HDM implementation in
MADGRAPH is denoted
MG-ST, while the
FEYNRULES-generated ones are
MG-FR, CH-FR and SH-FR.
The center-of-mass energy is
fixed to 800 GeV, and a pT cut
of 20 GeV is applied to each
final state particle
Weak boson processes in the two-Higgs-doublet model
Process MG-FR MG-ST
Table 14 Input parameters for
the SPS 1a benchmark point for
the MSSM
Inverse of the electromagnetic
b quark mass
t quark mass
Universal scalar mass
Universal gaugino mass
Universal trilinear coupling
Ratio of the two vevs
Off-diagonal Higgs mixing
1.16637e5 GeV2
91.1876 GeV
100 GeV
supersymmetry-breaking scenario. We have chosen the
typical minimal supergravity point SPS 1a [185] which is
completely defined once we fix the values of four parameters at
the gauge coupling unification scale and one sign. The
complete set of input parameters is shown in Table 14. We then
use the numerical program SOFTSUSY [191] to solve the
renormalization group equations linking this restricted set
of supersymmetry-breaking parameters at high-energy scale
to the complete set of masses, mixing matrices and
parameters appearing in LMSSM at the weak scale. The output is
encoded in a file following the SLHA conventions, readable
by FEYNRULES after the use of an additional translation
interface taking into account the small differences between the
SLHA2 format and the one of our implementation. This
interface is available on the FEYNRULES website, as well as
the corresponding SLHA2 output file.
In Tables 15, 16, 17 and 18, we give some examples
of the numerical checks which we have performed in
order to validate the implementation of the MSSM in
FEYNRULES. We recall that the built-in MSSM implementation in
MADGRAPH and CALCHEP are denoted MG-ST and
CHST, respectively, while the FEYNRULES-generated ones for
CALCHEP, MADGRAPH/MADEVENT are CH-FR and
MGFR. The full list of results can be found on the FEYNRULES
webpage.
B.4 The minimal Higgsless model
The external parameters used for the validation of the
Minimal Higgsless Model are shown in Table 19, and all the
widths are set to zero. We employ a center-of-mass energy
of 600 GeV and a pT cut of 20 GeV if only SM particles
are present, a center-of-mass energy of 1200 GeV and a pT
cut of 200 GeV if heavy vector bosons are present but heavy
fermions not, and a center-of-mass energy of 10000 GeV
and a pT cut of 2000 GeV if heavy fermions are present.
B.5 Extra dimensional models
The benchmark point which we have used for our numerical
comparison of the MUED implementation in FEYNRULES
to the existing one in CALCHEP is defined through the
various external parameters shown in Table 23. The masses of
the first Kaluza-Klein excitations are computed via one-loop
calculations [172, 192], and for s (MZ) = 0.1172, the
spectrum agrees with the one given in [192, 193]. We obtain for
the excitations of the quarks at s = 1400 GeV,
mu1D = mdD1 = mcD1 = msD1 = 573.3793 GeV,
mtD1 = 560.4622 GeV
mbD1 = 558.9203 GeV,
mu1S = mcS1 = 562.0523 GeV,
mdS1 = msS1 = 560.2356 GeV,
mtS1 = 586.2638 GeV
Table 15 Widths (in GeV) of
some of the allowed decay
channels in the SPS 1a scenario.
MG-FR amd MG-ST denote the
FEYNRULES-generated and
built-in MADGRAPH
implementations
Table 16 Cross sections for a
selection of Higgs production
processes in the MSSM scenario
SPS 1a. The built-in MSSM
implementation in MADGRAPH
and CALCHEP are denoted
MG-ST and CH-ST,
respectively, while the
FEYNRULES-generated ones for
CALCHEP and
MADGRAPH/MADEVENT are
CH-FR and MG-FR. The
center-of-mass energy is fixed to
1200 GeV
Higgs production in the MSSM
Table 17 Cross sections for a
selection of supersymmetric
particle pair production
processes in the MSSM scenario
SPS 1a. The built-in MSSM
implementation in MADGRAPH
and CALCHEP are denoted
MG-ST and CH-ST,
respectively, while the
FEYNRULES-generated ones for
CALCHEP and
MADGRAPH/MADEVENT are
CH-FR and MG-FR. The
center-of-mass energy is fixed to
1200 GeV
Supersymmetric particle production from fermions
Process MG-FR MG-ST
Table 18 Cross sections for a
selection of supersymmetric
particle pair production
processes in the MSSM scenario
SPS 1a. The built-in MSSM
implementation in MADGRAPH
and CALCHEP are denoted
MG-ST and CH-ST,
respectively, while the
FEYNRULES-generated ones for
CALCHEP and
MADGRAPH/MADEVENT are
CH-FR and MG-FR. The
center-of-mass energy is fixed to
1200 GeV. The differences with
respect to CH-ST are explained
in Sect. 6.4
for those of the leptons
and for those of the gauge bosons
= 514.9604 GeV,
me1 = m1 = m 1 = me1 = m1 = m1
D D D
me1 = m1 = m 1 = 505.4502 GeV,
S S S
mG1 = 603.3141 GeV,
mB1 = 500.8931 GeV.
mZ1 = 535.4923 GeV,
mW 1 = 500.8931 GeV
Supersymmetric particle production from gauge bosons
W +W l1l1+
W +W u1u6
ZZ d5d5
W Z d6u4
gZ d4d4
gW d5u5
gW + d2u6
gg u3u3
Inverse of the electromagnetic
Heavy fermion mass
2.289e3
5.561e2
5.338e2
7.678e2
4.693e2
3.283e2
1.712e2
3.952e2
2.690e2
3.618e4
1.129e1
4.637e3
2.569e1
2.208e2
1.865e1
6.514e1
6.250e1
4.738e1
5.235e2
2.290e3
5.564e2
5.349e2
7.686e2
4.695e2
3.285e2
1.711e2
3.941e2
2.689e2
3.618e4
1.129e1
4.633e3
2.566e1
2.206e2
1.865e1
6.515e1
6.263e1
4.751e1
5.235e2
2.289e3
5.562e2
5.344e2
7.686e2
4.693e2
3.286e2
1.712e2
3.950e2
2.690e2
3.618e4
1.129e1
4.634e3
2.566e1
2.206e2
1.866e1
6.515e1
6.257e1
4.746e1
5.236e2
1.682e2
5.562e2
9.183e1
7.686e2
4.693e2
3.286e2
1.712e2
3.950e2
2.690e2
3.618e4
1.129e1
4.634e3
2.566e1
2.206e2
1.866e1
6.515e1
6.257e1
4.746e1
5.236e2
1.16637e5 GeV2
91.1876 GeV
Table 20 Cross sections for a selection of strong processes in the
Minimal Higgsless Model. The LANHEP-generated MHM implementation
in CALCHEP is denoted CH-LH, while the FEYNRULES-generated
ones for MADGRAPH/MADEVENT, CALCHEP are MG-FR, CH-FR
and SH-FR. F means the calculation was done in Feynman gauge while
U means it was done in unitary gauge The center-of-mass energy is
Strong processes in the minimal Higgsless model
fixed to 600 GeV and a pT cut of 20 GeV is applied to each final state
particle, if only SM particles are present. A center-of-mass energy of
1200 GeV and a pT cut of 200 GeV is used if heavy vector bosons
are present but heavy fermions not, and a center-of-mass energy of
10000 GeV and a pT cut of 2000 GeV if heavy fermions are present
Charged electroweak processes in the minimal Higgsless model
G, G G, G
Table 21 Cross sections for a
selection of charged
electroweak processes in the
Minimal Higgsless Model. The
LANHEP-generated MHM
implementation in CALCHEP is
denoted CH-LH, while the
FEYNRULES-generated ones for
MADGRAPH/MADEVENT,
CALCHEP and SHERPA are
MG-FR, CH-FR and SH-FR.
F means the calculation was
done in Feynman gauge while U
means it was done in unitary
gauge The center-of-mass
energy is fixed to 600 GeV and
a pT cut of 20 GeV is applied to
each final state particle, if only
SM particles are present.
A center-of-mass energy of
1200 GeV and a pT cut of
200 GeV is used if heavy vector
bosons are present but heavy
fermions not, and a
center-of-mass energy of
10000 GeV and a pT cut of
2000 GeV if heavy fermions are
present
Z, W + W +, Z
W +, Z W +, Z
Z , W + W +, Z
W +, Z W +, Z
E, N1 B, t
E, N1 B, T
MG-FR-U
Table 22 Cross sections for a
selection of neutral electroweak
processes in the Minimal
Higgsless Model. The
LANHEP-generated MHM
implementation in CALCHEP is
denoted CH-LH, while the
FEYNRULES-generated ones for
MADGRAPH/MADEVENT,
CALCHEP and SHERPA are
MG-FR, CH-FR and SH-FR.
F means the calculation was
done in Feynman gauge while
U means it was done in unitary
gauge. The center-of-mass
energy is fixed to 600 GeV and
a pT cut of 20 GeV is applied to
each final state particle, if only
SM particles are present.
A center-of-mass energy of
1200 GeV and a pT cut of
200 GeV is used if heavy vector
bosons are present but heavy
fermions not, and a
center-of-mass energy of
10000 GeV and a pT cut of
2000 GeV if heavy fermions are
present
Neutral electroweak processes in the minimal Higgsless model
Process CH-LH-F CH-FR-F
MG-FR-U
Table 23 Input parameters for
our MUED benchmark scenario
Table 24 Cross sections for a
selection of processes in MUED
with two gauge bosons in the
initial state. The existing MUED
implementation in CALCHEP is
denoted CH-ST, while the
FEYNRULES-generated ones in
MADGRAPH, CALCHEP and
SHERPA are denoted MG-FR,
CH-FR and SH-FR. The
center-of-mass energy is fixed to
1400 GeV, and a pT cut of
20 GeV is applied to each final
state particle. Note that the
FEYNRULES interface to
CALCHEP cannot handle the
four-point interaction involving
two gluons and two KK gluons
MUED processes with gauge boson excitations
Process MG-FR
4.2 GeV
Table 25 Cross sections for a
selection of fermionic processes
in MUED. The existing MUED
implementation in CALCHEP is
denoted CH-ST, while the
FEYNRULES-generated ones in
MADGRAPH, CALCHEP and
SHERPA are denoted MG-FR,
CH-FR and SH-FR. The
center-of-mass energy is fixed to
1400 GeV, and a pT cut of
20 GeV is applied to each final
state particle
MUED processes with fermion excitations
eS1eS1+ uu
eS1eS1 ee
eD1eD1+ uu
eD1eD1+ ee+
u1Du1D uu
1.109e1
4.766e1
2.078e1
1.635e1
5.905e1
2.298e1
2.496e1
6.399e1
1.110e1
2.553e1
6.585e1
4.765e1
1.502e1
1.634e1
4.141e1
1.426e1
6.557e1
1.638e1
1.109e1
4.768e1
2.079e1
1.635e1
5.901e1
2.298e1
2.498e1
6.400e1
1.109e1
2.554e1
6.582e1
4.768e1
1.504e1
1.635e1
4.135e1
1.427e1
6.560e1
1.639e1
1.109e1
4.768e1
2.079e1
1.635e1
5.901e1
2.298e1
2.498e1
6.400e1
1.109e1
2.554e1
6.582e1
4.768e1
1.504e1
1.635e1
4.135e1
1.427e1
6.560e1
1.639e1
1.110e1
4.768e1
2.079e1
1.636e1
5.897e1
2.299e1
2.498e1
6.403e1
1.110e1
2.553e1
6.582e1
4.768e1
1.504e1
1.636e1
4.133e1
1.428e1
6.563e1
1.639e1
Some examples of the obtained results for the calculation
of cross sections of some 2 2 processes relative to the
production of two Standard Model particles or two
KaluzaKlein excitations are shown in Tables 24 and 25. We set the
center-of-mass energy to 1400 GeV and we applied a pT of
20 GeV on each final state particle.