# Automated mass spectrum generation for new physics

## Abstract

We describe an extension of the FeynRules package dedicated to the automatic generation of the mass spectrum associated with any Lagrangian-based quantum field theory. After introducing a simplified way to implement particle mixings, we present a new class of FeynRules functions allowing both for the analytical computation of all the model mass matrices and for the generation of a C++ package, dubbed ASperGe. This program can then be further employed for a numerical evaluation of the rotation matrices necessary to diagonalize the field basis. We illustrate these features in the context of the Two-Higgs-Doublet Model, the Minimal Left-Right Symmetric Standard Model and the Minimal Supersymmetric Standard Model.

## Keywords

Minimal Supersymmetric Standard Model Vacuum Expectation Mass Matrice Monte Carlo Event Generator Optional Argument## 1 Introduction

Although the Standard Model of particle physics is very well verified empirically at the current accessible energies, numerous extensions to its Lagrangian are proposed. These extensions describe new or alternative fundamental interactions that typically accommodate possible new physics phenomena at higher energies as well as at the current collider energies. In this top-down approach, the phenomenology of the proposed extensions is to be confronted with experimental observations. In order to obtain the mass spectrum of any new physics model reflected by its Lagrangian, the mixing matrices of the gauge eigenstates into the mass eigenstates are needed. An automated mass spectrum generator, ASperGe ^{1} is developed within the framework of the FeynRules program [1, 2, 3, 4, 5] to determine the mixing matrices numerically. This allows for a study of the direct relation between the parameters of any new physics model and the observable masses of the fundamental particles.

This paper describes in Sects. 2 and 3 the relevant parts of the FeynRules program to introduce the new ASperGe package, detailed in Sect. 4. To illustrate its application several examples are presented in Sect. 5.

## 2 The FeynRules package

The program FeynRules [1, 2, 3, 4, 5] is a Mathematica ^{2} package that allows for the automated extraction of Feynman rules from any Lagrangian describing the dynamics of a perturbative quantum field theory. The Feynman rules, together with general information such as the definitions of the model particles or of the Lagrangian parameters, can subsequently be exported by means of several translation interfaces to matrix-element generators. Up to now, interfaces to CompHep and CalcHep [6, 7, 8, 9], FeynArts and FormCalc [10, 11, 12, 13], MadGraph and MadEvent [14, 15, 16, 17, 18], Sherpa [19, 20] and Whizard [21, 22] have been developed.

In addition, any model can also be converted to a Python library containing classes and objects representing particles, parameters and vertices. This format is dubbed the Universal FeynRules Output (UFO) format [23] and is appropriate to address the implementation of any high-energy physics model into computational tools. Its strength lies in its agnosticism with respect to the allowed Lorentz and/or color structures appearing in the Lagrangian, in contrast to any other more conventional model format for which restrictions are imposed. Presently, the UFO is used by Aloha [24], MadAnalysis 5 [25], and MadGraph 5 [18], and will be used, in the future, by GoSam [26, 27] and Herwig++ [28].

*W*

_{ i }associated to the

*SU*(2)

_{ L }gauge subgroup of the Standard Model could be declared as This set of Mathematica rules defines a vector field (V[1]) represented by the symbol Wi (its ClassName) and carrying a flavor index SU2W associated with the adjoint gauge index of

*SU*(2)

_{ L }. As this field is declared as unphysical

^{3}(Unphysical->True), it must be linked to one or several of the mass eigenstates of the model by means of appropriate mixing relations. As illustrated in the example, these relations are passed through the attribute Definitions of the particle class. They can either be purely numerical, as for the

*W*

_{1}and

*W*

_{2}bosons that are rotated to the charged

*W*

^{+}and

*W*

^{−}bosons, or depend on some of the model parameters, as for the

*W*

_{3}field which is re-expressed in terms of the photon

*A*and the

*Z*boson through a relation depending on the sine and cosine of the electroweak mixing angle (sw and cw).

*Z*-boson field introduced above could hence be declared as

The declaration of the model parameters and gauge group is similar and based on dedicated classes with their own set of attributes. Since only the particle class properties introduced above are sufficient for the understanding of the present work, we omit any further detail and refer the reader to Refs. [1, 4] for more information on particle, parameter and gauge group implementation in FeynRules.

The last key ingredient to achieve a model implementation consists of its Lagrangian. It is provided using standard Mathematica commands, augmented by some special symbols representing objects such as Dirac matrices, vector field strength tensors or covariant derivatives. The user has then the possibility to perform basic checks on the Lagrangian, such as verifying its hermiticity, the normalization of kinetic terms, *etc.* We again refer to the FeynRules manual for more information [1].

*etc.*, according to the convenience of the user. It can also be directly implemented within the Mathematica session so that the function LoadModel is called, in that case, without any argument.

In the context of supersymmetric theories, the most natural and convenient way to construct a Lagrangian consists of employing the superspace formalism. Therefore, the FeynRules package includes a module allowing for superfield declarations and Lagrangian implementation in terms of superfields. Dedicated functions are then provided to convert superfield expressions into a form more suitable with respect to the requirements of the interfaces to the Monte Carlo event generators [4].

## 3 Implementing mixings in FeynRules

In this work, we propose an extension of the FeynRules package aiming to simplify the declaration of the mixing relations linking the unphysical degrees of freedom of the theory to the physical fields. This new module allows to automatically fill the Definitions attribute of the fields, where relevant, and declare the mixing matrices as parameters. In addition, we have developed an interface generating a C++ code dedicated to the diagonalization of the mass matrices of the model (see Sect. 4) after having implemented in FeynRules a function allowing for their analytical extraction from the Lagrangian. In this way, the values of all the mixing parameters are derived numerically and can be re-imported into FeynRules.

### 3.1 Mixing declarations

*etc.*) with a set of Mathematica replacement rules defining the mixing properties (options1, options2,

*etc.*).

*SU*(2)

_{ L }gauge bosons. We start by implementing the mixing of the

*W*

_{1}and

*W*

_{2}gauge fields,

*SU*(2)

_{ L }in the adjoint representation. As Eq. (1) is purely numerical, i.e., it does not involve any model parameter, it can be declared in the model file in a very compact form, The command above declares a mixing relation, dubbed Wmix, that can be schematically written as where the dot product stands for the usual matrix product. The information on the gauge basis is provided as the value of the attribute GaugeBasis which refers here to the unphysical fields

*W*

_{1}(Wi[1]) and

*W*

_{2}(Wi[2]). Similarly, the MassBasis attribute refers to the mass basis, containing here the symbols associated with the

*W*

^{+}(W) and

*W*

^{−}(Wbar) bosons. Finally, the mixing matrix is given under a numerical form as the argument of the attribute Value.

*i*being a fundamental

*SU*(2)

_{ L }index, the index

*f*a flavor index and the index

*c*a fundamental color index. The corresponding declaration reads where the mixing matrix is the identity. The underscore reflects that the same color index is carried by all fields.

*W*

_{3}and the hypercharge gauge boson

*B*to the photon and

*Z*-boson states,

*a priori*unknown weak mixing matrix

*U*

_{ w }

^{4}. The computation of the numerical values of its matrix elements is addressed by means of the C++ package generated by FeynRules (see Sect. 4) and is only possible if the mixing is declared according to the syntax The declaration of the gauge and mass bases is similar to the case of the charged

*W*bosons, while the attribute Value has been removed as the numerical value of the mixing matrix is not known. The user provides instead the symbol referring to the mixing matrix (UW) by means of the MixingMatrix attribute,

*without*declaring it as one of the model parameters. This last task is internally handled by FeynRules which assumes that the mixing matrix is complex and which creates two external tensorial parameters, one for the real part and one for the imaginary part of the matrix, together with one internal tensorial parameter being the matrix itself.

When a symbol for a mixing matrix is provided, it is mandatory to specify, in addition, the name of a Les Houches block which will contain the numerical values associated with the elements of the matrix. We indeed recall that both FeynRules and most of the interfaced Monte Carlo event generators order the model parameters according to a structure inspired by the Supersymmetry Les Houches Accord (SLHA) [30, 31]. In our example, we impose the real part of the elements of *U* _{ w } to be stored in a Les Houches block WEAKMIX and their imaginary part in an automatically created block IMWEAKMIX, i.e., a block of the same name with the prefix IM appended.

Implementing model Lagrangians might require to explicitly use one or several of the mixing matrices for some of the model interactions, as for the Minimal Supersymmetric Standard Model where the CKM matrix is employed in the superpotential [4]. In this case, the matrices must be declared according to the standard syntax presented in the FeynRules manual, numerical values being provided as inputs. This subsequently renders the attribute BlockName of the mixing class obsolete and ignored by FeynRules. Contrary, mixing matrices automatically declared through a mixing declaration cannot be employed in Lagrangians.

*d*

_{ L }as

### 3.2 More advanced cases

#### 3.2.1 Scalar/pseudoscalar splittings

#### 3.2.2 Dirac and Weyl fermion mixings

Several options are left to the user concerning the implementation of Dirac fermion mixings. Either one single gauge basis is employed, so that FeynRules internally takes care of the chirality projectors that appear in the related mass terms, or different particle classes can be used for the left-handed and right-handed components of the fermions. In this case, the GaugeBasis attribute refers to a list of two gauge bases instead of to a single basis. For both options, the arguments of the attributes Value, BlockName, MixingMatrix and Inverse consist of lists, the first component being related to the mixing of the left-handed fermions and the second one to the mixing of the right-handed fermions. As for neutral scalar mixing, underscores are used for irrelevant list elements.

*M*stands for the mass matrix and

*ψ*

_{ i }and

*χ*

_{ i }are Weyl fermions which have been assigned an electric charge of ±1 for the sake of the example. The diagonalization of the matrix

*M*proceeds through two unitary rotations

*U*and

*V*,

### 3.3 Vacuum expectation value declarations

### 3.4 User functions

*M*defined by the Lagrangian

*M*is achieved by issuing where the symbols b1 and b2 are associated with the bases \(\mathcal{B}_{1}\) and \(\mathcal{B}_{2}\) and refer to lists of fields. In this case, the printing functions introduced above are not available.

## 4 Automated spectrum generation

### 4.1 The ASperGe package

The computation of the unknown mixing matrices necessary for diagonalizing all the model mass matrices can in general only be achieved numerically. To this end, we have developed the C++ program ASperGe. It includes a set of C++ source files (stored in the subdirectory src), coming together with the related header files (stored in the subdirectory inc), that can be split into model-independent and model-dependent files. For an efficient use of the ASperGe program, it has been entirely embedded within the FeynRules package. Therefore, only a brief discussion of the structure of the code is presented in this paper. More information, such as a doxygen documentation, can be found on the ASperGe webpage [32].

The set of model-independent files contains, on the one hand, several tools dedicated to matrices and their diagonalization (MassMatrix.cpp, MassMatrix.hpp as well as Matrix.hpp). On the other hand, the ASperGe code is based on an internal format for parameters, defined in the source files Par.cpp, CPar.cpp, RPar.cpp and in the associated header files. This format is inspired from a SLHA structure and the corresponding mapping is encoded into the files ParSLHA.cpp, SLHABlock.cpp, and in the associated header files. Finally, printing and string manipulation routines are included in the files tools.cpp and tools.hpp and the program comes with a makefile.

All the model dependency is included in the two files Parameters.cpp and Parameters.hpp as well as in the core program implemented in the main.cpp file.

The information encompassed in the two parameter files is threefold. First, the SLHA structure ordering the external parameters is encoded in terms of blocks and counters. Next, the definitions of the internal parameters as functions of the other model parameters are implemented, where a proper running of the ASperGe program is only guaranteed if the parameters do not depend on the masses and mixing matrices to be computed. Finally, the analytical formulas of the mass matrices to diagonalize are included.

The main program (main.cpp) starts with the declaration of the different mass matrices of the model. Links to the relevant elements of the mass basis are then implemented by means of the associated PDG codes, which allow to assign the mass eigenvalues to each of the physical particles, the ordering of the PDG codes following the mass ordering.

### 4.2 Interfacing the ASperGe package to FeynRules

The interface first extracts all the relevant mass matrices from the Lagrangian Lag by means of the function ComputeMassMatrix introduced in Sect. 3.4. It then writes, in addition to model-independent files described in Sect. 4.1, the three model-dependent files main.cpp, Parameters.cpp and Parameters.hpp, together with one data file Externals.dat (stored in the subdirectory input). This last file contains the numerical values of the external parameters of the model, necessary for the numerical evaluation of the mass matrices. When running the code (see Sect. 4.3), the user can update this file or even employ a different file according to his needs.

The numerical matrix diagonalization performed by ASperGe is based on Gsl functions relying on the hermiticity of the mass matrices which employs symmetric bi-diagonalization followed by QR reduction. This contrasts with existing diagonalization packages developed in the framework of FeynArts [33] and CalcHep [34] that are based on Jacobi-type iterative algorithms. A hermiticity check is therefore performed by the interface before writing down the output. Since the mass matrix *M* related to charged fermions is by construction non-hermitian, the matrices *M* ^{†} *M* and *MM* ^{†} are employed instead, which allows to obtain left-handed and right-handed fermion mixing matrices separately.

### 4.3 Running ASperGe

Since the ASperGe package is based on Gsl functions, it is mandatory to have the Gsl libraries installed on the system. Then, if the g++ compiler is available, the makefile generated by FeynRules can be employed directly. Otherwise, it must be first edited accordingly to include proper compiler information.

*e.g.*, to generate a UFO model. Information about the run of ASperGe can be found in the file ASperGe.log stored in the same folder as the executable.

*etc.*, are the names of the mixing matrices to be computed.

## 5 Illustrative examples

In this section, we illustrate the features of the ASperGe program and its interface to FeynRules by choosing three extensions of the Standard Model with non-trivial mixing relations, i.e., the Two-Higgs-Doublet Model (2HDM), the Minimal Left-Right Symmetric Standard Model (LRSM) and the Minimal Supersymmetric Standard Model (MSSM). We modify their original FeynRules implementations^{5} [2, 4] to accommodate for the mixings as described in Sect. 3. We then employ the ASperGe program (see Sect. 4) to numerically calculate some of the mass and mixing matrices of these models.

### 5.1 The general two-Higgs-doublet model

*ϕ*

_{1}and

*ϕ*

_{2}carry the same hypercharge so that they can always be redefined by means of

*U*(2) transformations [35, 36, 37, 38]. Adopting the so-called Higgs-basis, the two doublets read

*ϕ*

_{1}field acquires a vacuum expectation value

*v*. Moreover, the Goldstone bosons

*G*

^{±}and

*G*

^{0}as well as the charged Higgs field

*H*

^{±}are not required to be further rotated, so that only the mass matrix of the neutral fields

*H*

^{0},

*R*

^{0}and

*I*

^{0}must still be diagonalized.

*ϕ*

_{1}field, represented by the symbol phi1, acquires a non-vanishing vacuum expectation value

*v*labeled by the symbol vev, as shown in Sect. 3.3. Then, we choose to implement the mixing of the

*H*

^{0},

*R*

^{0}and

*I*

^{0}fields to the physical

*h*

_{1},

*h*

_{2}and

*h*

_{3}fields in a two-step manner. In a first stage, the gauge eigenstates are split into their scalar and pseudoscalar components, following the syntax introduced in Sects. 3.1 and 3.2 and making use of the self-explained symbols H0 and R0 (G0 and I0) for representing the (pseudo)scalar degrees of freedom. Similarly, we can employ the mixing infrastructure to map the charged components of

*ϕ*

_{1}and

*ϕ*

_{2}to the physical fields

*G*

^{+}and

*H*

^{+}by means of a 1×1 identity matrix. Since this procedure is trivial, we omit any further details from the present manuscript and refer to the model implementation [32]. In a second stage, the rotation to the physical fields, represented by the symbols h1, h2 and h3, is declared as where we associate the symbol NH to the corresponding mixing matrix and assign the Les Houches block (IM)NHMIX to the numerical value of its elements. The neutral squared mass matrix \(\mathcal{M}^{2}\) can then be derived from the model Lagrangian (represented by the symbol L2HDM) by typing, in the Mathematica session, As a result, one recovers the well-known expression depending on the most general scalar potential parameters

*λ*

_{ i }and

*μ*

_{ i }(see Ref. [2] for further information),

*μ*-parameters by means of the potential minimization conditions,

*μ*

_{1}=−

*λ*

_{1}

*v*

^{2}and

*μ*

_{3}=−1/2

*λ*

_{6}

*v*

^{2}.

*U*diagonalizing \(\mathcal {M}^{2}\) is obtained by generating and making use of the ASperGe package, as shown in Sect. 4. We fix, adopting a representative benchmark scenario, the Higgs potential parameters to

*λ*

_{1}=

*λ*

_{2}=

*λ*

_{3}=1.0,

*λ*

_{4}=0.5,

*λ*

_{5}=0.4,

*λ*

_{6}=0.3,

*λ*

_{7}=0.2 and

*μ*

_{2}=6⋅10

^{4}GeV. This leads to the three mass eigenvalues

### 5.2 The minimal left-right symmetric standard model

*SU*(3)

_{ c }×

*SU*(2)

_{ L }×

*SU*(2)

_{ R }×

*U*(1)

_{ B−L }gauge symmetry. In this model, the fermionic degrees of freedom of the Standard Model lying in the trivial representation of

*SU*(2)

_{ L }are collected into

*SU*(2)

_{ R }doublets, as shown in the first part of Table 1 where the model matter field content is presented together with the associated quantum numbers. In addition, the symmetry-breaking mechanism down to electromagnetism is also more involved, relying on an enriched Higgs sector (see the second part of the table).

Field content of the LRSM, given together with their representation under the *SU*(3)_{ c }×*SU*(2)_{ L }×*SU*(2)_{ R }×*U*(1)_{ B−L } gauge group. The *SU*(2)_{ L } (*i*,*j*=1,2) and *SU*(2)_{ R } (*i*′,*j*′=1,2) fundamental index structure is explicitly indicated

Field | Components | Representation |
---|---|---|

\(Q_{L}^{i}\) | \(\begin{pmatrix} u_{L}\\ d_{L} \end{pmatrix} \) | |

| \(\begin{pmatrix} u_{R}^{c} & d_{R}^{c} \end{pmatrix} \) | |

\(L_{L}^{i}\) | \(\begin{pmatrix} \nu_{L}\\ \ell_{L} \end{pmatrix} \) | |

| \(\begin{pmatrix} \nu_{R}^{c} & \ell_{R}^{c} \end{pmatrix} \) | |

| \(\begin{pmatrix} \varPhi^{0} & \varPhi^{+} \\ \varPhi^{'\prime-} & \varPhi'{}^{0} \end{pmatrix} \) | |

Δ | \(\begin{pmatrix} \frac{1}{\sqrt{2}} \Delta_{L}^{+} & \Delta_{L}^{++} \\ \Delta_{L}^{0}& -\frac{1}{\sqrt{2}} \Delta_{L}^{+} \end{pmatrix} \) | |

Δ | \(\begin{pmatrix} \frac{1}{\sqrt{2}} \Delta_{R}^{+} & \Delta_{R}^{++} \\ \Delta_{R}^{0}& -\frac{1}{\sqrt{2}} \Delta_{R}^{+} \end{pmatrix} \) |

*c*indicates charge conjugation.

^{6}Moreover, gauge invariance is ensured by the introduction of the hatted fields

*ϵ*

_{12}=−

*ϵ*

^{12}=1. Introducing Higgs mass parameters

*μ*

_{ i }and quartic interaction strengths

*λ*

_{ i },

*ρ*

_{ i }and

*α*

_{ i }, the scalar potential reads Since the corresponding FeynRules model description is standard, we refer to the FeynRules manual [1] and leave all implementation details out of this work.

In the LRSM, the symmetry-breaking mechanism is performed in two steps. At high energy, the *SU*(2)_{ L }×*SU*(2)_{ R }×*U*(1)_{ B−L } gauge symmetry is spontaneously broken to the electroweak symmetry, the latter being subsequently broken to electromagnetism at a lower scale. Consequently, the neutral components of the scalar fields get vacuum expectation values at the minimum of the potential, \(\langle\varPhi^{0}\rangle= v/\sqrt{2}\), \(\langle\varPhi^{\prime 0}\rangle= v'/\sqrt{2}\) and \(\langle\Delta_{L,R}^{0}\rangle= v_{L,R}/\sqrt{2}\), by which they are shifted. In the rest of this section, we focus on the mixing of the neutral Higgs fields and illustrate the way to implement a two-stage field rotation. For all the other mixing relations of the LRSM, we refer to the implementation [32].

*v*

_{ L }=

*v*′≈0. Next, we implement the rotation associated with the diagonalization of the third generator of

*SU*(2) in the adjoint representation as depicting the example of the

*SU*(2)

_{ L }Higgs triplet. These replacement rules translate the rotation of the \(\Delta_{L}^{1}\) and \(\Delta_{L}^{2}\) fields, represented by the DL[1] and DL[2] symbols, to the Δ

^{0}and Δ

^{++}states labeled by DL0 and DLpp. Then, the neutral fields \(\Delta_{L}^{0}\), \(\Delta_{R}^{0}\),

*Φ*

^{0}and

*Φ*

^{′0}, represented by the symbols DL0, DR0, phi[1,1] and phi[2,2], mix to four scalar degrees of freedom \(h^{0}_{1}\), \(h^{0}_{2}\), \(h^{0}_{3}\) and \(h^{0}_{4}\), two physical pseudoscalar Higgs bosons \(a^{0}_{1}\) and \(a^{0}_{2}\) and two Goldstone bosons \(G^{0}_{1}\) and \(G^{0}_{2}\) to be eaten by the

*Z*and

*Z*′ vector fields when getting massive. Introducing the corresponding symbols h01, h02, h03, h04, a01, a02, G01 and G02, these rotations are implemented as The two symbols UHN and UAN respectively denote the scalar and pseudoscalar mixing matrices, the numerical value of their elements being included in the two Les Houches blocks (IM)HMIX and (IM)AMIX.

*U*diagonalizing \(\tilde{\mathcal{M}}^{2}\), we use the ASperGe program, generated from FeynRules by issuing after fixing the external Lagrangian parameters of the model to

*λ*

_{1}=

*λ*

_{2}=

*λ*

_{3}=

*λ*

_{4}=

*λ*

_{6}=0.1,

*α*

_{1}=0.1,

*α*

_{2}=0.3,

*α*

_{3}=0.1,

*ρ*

_{1}=0.1,

*ρ*

_{2}=0 and

*ρ*

_{3}=0.5. In addition, the

*μ*-terms are deduced from the minimization conditions of the scalar potential and the vacuum expectation values are taken as

*v*=248 GeV and

*v*

_{ R }=6000 GeV. It should be noted that the relevance of such numerical values is going beyond the scope of this paper, and could be addressed by means of external packages such as the one presented in Ref. [47]. Once ASperGe is executed, one obtains the mass eigenvalues being

### 5.3 The minimal supersymmetric standard model

In supersymmetric extensions of the Standard Model, each of the model’s degrees of freedom comes accompanied by a superpartner with opposite statistics. The minimal version of such theories, the so-called MSSM [48, 49], has been originally implemented in FeynRules by making use of its superspace module [4]. We hence refer, on the one hand, to Ref. [4] for notations and conventions and, on the other hand, to the new model implementation [32] for more information on the way in which particle mixings have been implemented. In the rest of this subsection, we employ the MSSM implementation to illustrate fermion mixing declaration.

*χ*

^{±}. Introducing the two rotation matrices

*U*and

*V*(labeled by the symbols UU and VV), this mixing is declared through an instance of the mixing class, In this list of replacement rules, wowm (hdw) and wowp (huw) are the labels of the negatively and positively charged wino (higgsino) states, the related mass eigenstates being represented by the symbols chmw and chpw. In addition, we link the mixing matrices

*U*and

*V*to the Les Houches blocks UMIX and VMIX.

*m*

_{ W }denotes the

*W*-boson mass,

*μ*the superpotential Higgs mixing parameter,

*M*

_{2}the supersymmetry-breaking wino mass and tan

*β*is defined as the ratio of the two neutral Higgs field vacuum expectation values \(\tan\beta= \langle H_{u}^{0}/H_{d}^{0}\rangle\).

*U*and

*V*, as well as the corresponding mass eigenvalues

## 6 Summary

In this paper, we have presented an extension of the FeynRules package dedicated to the automated generation of the particle mass spectrum and mixing structure associated to any Lagrangian-based quantum field theory. The new module is based on the introduction of a new structure for particle mixing declaration allowing, on the one hand, for the analytical computation of all the model mass matrices, and, on the other hand, for the generation of a C++ program dubbed ASperGe, yielding the numerical evaluation of the associated rotation matrices. We illustrate the strength of this new module in the context of the Two-Higgs-Doublet Model, the Minimal Left-Right Symmetric Standard Model and the Minimal Supersymmetric Standard Model.

## Footnotes

- 1.
The acronym ASperGe stands for

**A**utomated**Spe**ct**r**um**Ge**neration. - 2.
Mathematica is a registered trademark of Wolfram Research, Inc.

- 3.
In this work, we denote as unphysical any field that is not a mass eigenstate of the theory.

- 4.
- 5.
Since no previous implementation of the LRSM exists, we take the opportunity to provide the relevant details in Sect. 5.2.

- 6.
We recall that the components of the field

*Q*_{ R }are charge conjugate (see Table 1).

## Notes

### Acknowledgements

The authors are grateful to N.D. Christensen, C. Degrande and C. Duhr for useful discussions on the project. This work has been partially supported by a Ph.D. fellowship of the French ministry for education and research, by the Theory-LHC France-initiative of the CNRS/IN2P3, by the French ANR 12 JS05 002 01 BATS@LHC, by the Concerted Research action ‘Supersymmetric Models and their Signatures at the Large Hadron Collider’ and the Strategic Research Program ‘High Energy Physics’ of the Vrije Universiteit Brussel (VUB), by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole IAP VI/11 and P7/37 and by a ‘FWO-Vlaanderen’ aspirant fellowship.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

## References

- 1.N.D. Christensen, C. Duhr, Comput. Phys. Commun.
**180**, 1614 (2009) ADSCrossRefGoogle Scholar - 2.N.D. Christensen, P. de Aquino, C. Degrande, C. Duhr, B. Fuks, M. Herquet, F. Maltoni, S. Schumann, Eur. Phys. J. C
**71**, 1541 (2011) ADSCrossRefGoogle Scholar - 3.N.D. Christensen, C. Duhr, B. Fuks, J. Reuter, C. Speckner, Eur. Phys. J. C
**72**, 1990 (2012) ADSCrossRefGoogle Scholar - 4.C. Duhr, B. Fuks, Comput. Phys. Commun.
**182**, 2404 (2011) ADSMATHCrossRefGoogle Scholar - 5.B. Fuks, Int. J. Mod. Phys. A
**27**, 1230007 (2012) ADSCrossRefGoogle Scholar - 6.A. Pukhov, E. Boos, M. Dubinin, V. Edneral, V. Ilyin, D. Kovalenko, A. Kryukov, V. Savrin et al., hep-ph/9908288
- 7.E. Boos et al., Nucl. Instrum. Methods A
**534**, 250 (2004) ADSCrossRefGoogle Scholar - 8.A. Pukhov, hep-ph/0412191
- 9.A. Belyaev, N.D. Christensen, A. Pukhov, arXiv:1207.6082 [hep-ph]
- 10.T. Hahn, M. Perez-Victoria, Comput. Phys. Commun.
**118**, 153 (1999) ADSCrossRefGoogle Scholar - 11.T. Hahn, Comput. Phys. Commun.
**140**, 418 (2001) ADSMATHCrossRefGoogle Scholar - 12.T. Hahn, PoS ACAT
**08**, 121 (2008) Google Scholar - 13.S. Agrawal, T. Hahn, E. Mirabella, J. Phys. Conf. Ser.
**368**, 012054 (2012) ADSCrossRefGoogle Scholar - 14.T. Stelzer, W.F. Long, Comput. Phys. Commun.
**81**, 357 (1994) ADSCrossRefGoogle Scholar - 15.F. Maltoni, T. Stelzer, J. High Energy Phys.
**0302**, 027 (2003) ADSCrossRefGoogle Scholar - 16.J. Alwall, P. Demin, S. de Visscher, R. Frederix, M. Herquet, F. Maltoni, T. Plehn, D.L. Rainwater et al., J. High Energy Phys.
**0709**, 028 (2007) ADSCrossRefGoogle Scholar - 17.J. Alwall, P. Artoisenet, S. de Visscher, C. Duhr, R. Frederix, M. Herquet, O. Mattelaer, AIP Conf. Proc.
**1078**, 84 (2009) ADSGoogle Scholar - 18.J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer, T. Stelzer, J. High Energy Phys.
**1106**, 128 (2011) ADSCrossRefGoogle Scholar - 19.T. Gleisberg, S. Hoeche, F. Krauss, A. Schalicke, S. Schumann, J.-C. Winter, J. High Energy Phys.
**0402**, 056 (2004) ADSCrossRefGoogle Scholar - 20.T. Gleisberg, S. Hoeche, F. Krauss, M. Schonherr, S. Schumann, F. Siegert, J. Winter, J. High Energy Phys.
**0902**, 007 (2009) ADSCrossRefGoogle Scholar - 21.M. Moretti, T. Ohl, J. Reuter, hep-ph/0102195
- 22.W. Kilian, T. Ohl, J. Reuter, Eur. Phys. J. C
**71**, 1742 (2011) ADSCrossRefGoogle Scholar - 23.C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer, T. Reiter, Comput. Phys. Commun.
**183**, 1201 (2012) ADSCrossRefGoogle Scholar - 24.P. de Aquino, W. Link, F. Maltoni, O. Mattelaer, T. Stelzer, Comput. Phys. Commun.
**183**, 2254 (2012) ADSCrossRefGoogle Scholar - 25.E. Conte, B. Fuks, G. Serret, Comput. Phys. Commun.
**184**, 222 (2013) MathSciNetADSCrossRefGoogle Scholar - 26.G. Cullen, N. Greiner, G. Heinrich, G. Luisoni, P. Mastrolia, G. Ossola, T. Reiter, F. Tramontano, Eur. Phys. J. C
**72**, 1889 (2012) ADSCrossRefGoogle Scholar - 27.G. Cullen, N. Greiner, G. Heinrich, G. Luisoni, P. Mastrolia, G. Ossola, T. Reiter, F. Tramontano, J. Phys. Conf. Ser.
**368**, 012056 (2012) ADSCrossRefGoogle Scholar - 28.M. Bahr, S. Gieseke, M.A. Gigg, D. Grellscheid, K. Hamilton, O. Latunde-Dada, S. Platzer, P. Richardson et al., Eur. Phys. J. C
**58**, 639 (2008) ADSCrossRefGoogle Scholar - 29.J. Beringer et al. (Particle Data Group Collaboration), Phys. Rev. D
**86**, 010001 (2012) ADSCrossRefGoogle Scholar - 30.P.Z. Skands, B.C. Allanach, H. Baer, C. Balazs, G. Belanger, F. Boudjema, A. Djouadi, R. Godbole et al., J. High Energy Phys.
**0407**, 036 (2004) ADSCrossRefGoogle Scholar - 31.B.C. Allanach, C. Balazs, G. Belanger, M. Bernhardt, F. Boudjema, D. Choudhury, K. Desch, U. Ellwanger et al., Comput. Phys. Commun.
**180**, 8 (2009) ADSCrossRefGoogle Scholar - 32.
- 33.T. Hahn, physics/0607103
- 34.G. Belanger, N.D. Christensen, A. Pukhov, A. Semenov, Comput. Phys. Commun.
**182**, 763 (2011) ADSMATHCrossRefGoogle Scholar - 35.G.C. Branco, L. Lavoura, J.P. Silva, Int. Ser. Monogr. Phys.
**103**, 1 (1999) Google Scholar - 36.I.F. Ginzburg, M. Krawczyk, Phys. Rev. D
**72**, 115013 (2005) ADSCrossRefGoogle Scholar - 37.S. Davidson, H.E. Haber, Phys. Rev. D
**72**, 035004 (2005). Erratum-ibid. D**72**, 099902 (2005) ADSCrossRefGoogle Scholar - 38.H.E. Haber, D. O’Neil, Phys. Rev. D
**74**, 015018 (2006) ADSCrossRefGoogle Scholar - 39.J.C. Pati, A. Salam, Phys. Rev. D
**10**, 275 (1974). Erratum-ibid. D**11**, 703 (1975) ADSCrossRefGoogle Scholar - 40.R.N. Mohapatra, J.C. Pati, Phys. Rev. D
**11**, 566 (1975) ADSCrossRefGoogle Scholar - 41.R.N. Mohapatra, J.C. Pati, Phys. Rev. D
**11**, 2558 (1975) ADSCrossRefGoogle Scholar - 42.G. Senjanovic, R.N. Mohapatra, Phys. Rev. D
**12**, 1502 (1975) ADSCrossRefGoogle Scholar - 43.R.N. Mohapatra, F.E. Paige, D.P. Sidhu, Phys. Rev. D
**17**, 2462 (1978) ADSCrossRefGoogle Scholar - 44.G. Senjanovic, Nucl. Phys. B
**153**, 334 (1979) MathSciNetADSCrossRefGoogle Scholar - 45.C.S. Lim, T. Inami, Prog. Theor. Phys.
**67**, 1569 (1982) ADSCrossRefGoogle Scholar - 46.R.N. Mohapatra, G. Senjanovic, Phys. Rev. D
**23**, 165 (1981) ADSCrossRefGoogle Scholar - 47.R. Coimbra, M.O.P. Sampaio, R. Santos, arXiv:1301.2599 [hep-ph]
- 48.H.P. Nilles, Phys. Rep.
**110**, 1 (1984) ADSCrossRefGoogle Scholar - 49.H.E. Haber, G.L. Kane, Phys. Rep.
**117**, 75 (1985) ADSCrossRefGoogle Scholar - 50.B.C. Allanach, M. Battaglia, G.A. Blair, M.S. Carena, A. De Roeck, A. Dedes, A. Djouadi, D. Gerdes et al., Eur. Phys. J. C
**25**, 113 (2002) ADSCrossRefGoogle Scholar