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Sarah

F. Staub

arXiv:0806.0538v6hep-ph

TL;DR

Extending supersymmetric models beyond the MSSM involves substantial work deriving interactions, rotating fields, and preparing calculations and model files. SARAH derives model information and generates SPheno source code; the resulting code calculates spectra and decays without hand-written Fortran.

  • Problem

    Extending a supersymmetric model beyond the well-studied MSSM requires anomaly checks, Lagrangian derivation, mass-basis rotations, tadpole equations, and model files for phenomenological studies.

    Source: Introduction · p. 8

  • Method

    SARAH takes the gauge structure, particle content, and superpotential to derive model information and generate SPheno source code from analytical expressions.

    Source: Introduction · p. 8

  • Results

    The generated SPheno code calculates a new model’s mass spectrum using 2-loop RGEs and 1-loop mass corrections, and includes routines for two- and three-body decays.

    Source: Introduction · p. 8

  • Takeaways & Limitations

    The generated source code makes it possible to implement new models in SPheno without writing Fortran by hand.

    Source: Introduction · p. 8

    Deleting particles can leave supersymmetric interaction relations that are invalid in a non-supersymmetric theory; couplings may need to be redefined.

    Source: §5.5.2 · p. 54

Abstract

from arXiv · show

SARAH is a Mathematica package for building and analyzing supersymmetric models. SARAH just needs the gauge structure, particle content and superpotential to produce all information about the gauge eigenstates of a model. Breaking of gauge symmetries and mixings of particles can easily be in a second step and entire Lagrangian is derived automatically. Also the gauge fixing terms are derived by SARAH in R_Xi gauge, and the corresponding ghost interactions are calculated. Using this information, SARAH can calculate the all mass matrices, tadpole equations and vertices at tree-level for the given model. In addition, the expressions for the 1- and 2-loop renormalization group equations of all parameters can be calculated and an automatic calculation for the 1-loop corrections to self energies and the tadpoles are possible. SARAH can write all information about the model to LaTeX files, or create a model files for FeynArts/FormCalc, WHIZARD/OMEGA and CalcHep/CompHep, which can also be used for dark matter studies using MicrOmegas, and in the UFO format which is supported by MadGraph 5. Beginning with version 3, SARAH is also the first available spectrum-generator-generator: based on the derived, analytical expression it creates source code for SPheno to calculate the mass spectrum as well the SUSY decays with high precision. In that way, it is possible to implement new models in SPheno without the need to write any Fortran code by hand. Already many models beyond the MSSM are included in the public version of SARAH and the implementation of new models is easy and straightforward.

A Model Builder’s Tool

The section gives a software disclaimer and asks users to send suggestions and cite the relevant SARAH publications.

A Model Builder’s Tool

  • SARAH is provided free under the GNU Library Public License, but the authors do not guarantee correct software operation or physical results.The disclaimer also says the software and its results have been tested.Source: A Model Builder’s Tool · p. 2
  • Users are asked to send suggestions about what is missing or could be improved.Source: A Model Builder’s Tool · p. 2
  • Users of SARAH are asked to cite the appropriate references, including works on its UFO format, model implementation, RGEs, and self energies.Source: A Model Builder’s Tool · p. 2

E.2.2.0.1 Corrections to fermion

The supplied material is a contents listing that maps out sections on fermion corrections, SPheno output, and MSSM conventions and implementation.

E.2.2.0.1 Corrections to fermion

F More information about the SPheno output

G The minimal supersymmetric standard model

G.1.3.0.5 Softbreaking terms

G.1.4.0.6 Gauge fixing terms for gauge eigenstates .

Introduction

SARAH addresses the work required to extend supersymmetric models beyond the MSSM by automating model checks and calculations from model-file inputs. It also generates interfaces and SPheno source code for downstream analysis.

Introduction

  • Extending the MSSM requires checking anomalies, deriving and rotating the Lagrangian, finding tadpole equations, and creating model files for phenomenological tools.The introduction also identifies RGEs and loop corrections as needed for further studies.Source: Introduction · p. 8

2.1 Download and installation

  • The package archive includes model and package directories, LaTeX support, and examples for model outputs and SPheno source generation.The manual describes extracting SARAH into Mathematica’s application directory.Source: §2.1 · p. 10

2.2 Run SARAH

  • SARAH is loaded in Mathematica and a model is initialized with Start, using a model name such as MSSM or NMSSM.The NMSSM example specifies the CKM basis.Source: §2.2 · p. 11

2.3 What happens automatically

  • Initialization checks gauge anomalies and charge conservation, derives the tree-level Lagrangian, performs field rotations, and calculates mass matrices and tadpole equations.When numerical parameter values are supplied, SARAH calculates mass-matrix eigenvalues and rotation matrices.Source: §2.3 · p. 11

2.4 Commands

  • The command list includes model checks and commands for calculating vertices, alongside commands for displaying mass matrices.Source: §2.4 · p. 11

Quick start

SARAH’s commands expose model calculations and generate files for external tools. The quick-start listing covers analytical outputs as well as model-file and spectrum-generator generation.

Quick start

  • MassMatrices, MassMatrix, TadpoleEquations, and TadpoleEquation display mass matrices and tadpole equations for specified eigenstates or fields.Source: Quick start · p. 12
  • CalcRGEs and CalcLoopCorrections calculate renormalization-group equations and one-loop one- and two-point functions.Source: Quick start · p. 12
  • MakeVertexList calculates vertices, while ModelOutput creates output for specified eigenstates and options.Source: Quick start · p. 12
  • SARAH can write SPheno source code and export model files for CalcHep/CompHep, FeynArts, WHIZARD/Omega, and UFO, as well as LaTeX files and LHPC plotter steering files.MakeAll generates output for the listed tools and LaTeX.Source: Quick start · p. 12

Working with SARAH

This section describes how to use SARAH to inspect model information and obtain interaction vertices and renormalization-group results, including conventions for interpreting its output.

3.1 Definition of particles

  • Particle names encode field type, underlying superfield basis, and particle indices; vector bosons retain the V prefix after rotations so ghosts can be added automatically.Indices can denote generation, Lorentz, or gauge-group representations.Source: §3.1 · pp. 14–15

3.1.1 Antiparticles

  • SARAH uses conj for scalars, vector bosons, and Weyl spinors, and bar for Dirac fermions and ghosts; it also tracks real particles and Majorana fermions.bar can denote Hermitian or complex conjugation depending on its position in a fermion chain, while conj is retained partly for Mathematica compatibility and performance.Source: §3.1.1 · pp. 15–16

3.2 Different eigenstates of one model

  • The package’s standard eigenstate names are GaugeES for unrotated gauge eigenstates, EWSB after electroweak symmetry breaking, SCKM for the Super-CKM basis, and TEMP for auxiliary states.Users can define eigenstate names in the model file.Source: §3.2 · p. 16

3.3.1 Particle content; 3.3.6 Writing all information about particles and parameters in an external file

  • Particles[Eigenstates] lists particle names, types, generation ranges, and indices, while ParticleInfo.m and ParameterInfo.m export particle and parameter information to files.ParticleInfo.m includes whether particles are self-conjugated; ParameterInfo.m includes parameter dependence and LesHouches input positions.Source: §3.3.1 · p. 16, §3.3.6 · p. 19

3.3.2 Masses of particles; 3.3.3 Mass Matrices

  • Masses[$EIGENSTATES] stores automatically calculated tree-level masses, with values represented as model expressions, supplied numerical values, or values read from LesHouches input.MixBasis[$EIGENSTATES] shows rotation bases, and MassMatrices[$EIGENSTATES] and MassMatricesFull[$EIGENSTATES] store matrices with generation indices implicit or explicit; MassMatrix[Field] retrieves a specific matrix.Source: §3.3.2 · p. 17, §3.3.3 · p. 17

3.3.4 Tadpole Equations

  • SARAH saves the tadpole equations—the conditions on vacuum expectation values that minimize the scalar potential—in TadpoleEquations[$EIGENSTATES], with an option to retrieve an equation for a specific VEV or state.The equations’ array order follows the VEV definitions in the model file.Source: §3.3.4 · p. 18

3.3.5 Parts of the Lagrangian

  • SARAH derives the full Lagrangian from the superpotential and gauge sector, saves it for each eigenstate, and splits it into parts to aid calculations and interaction analysis.Results from intermediate gauge-eigenstate calculation steps, including F-terms, D-terms, and kinetic parts, are also saved.Source: §3.3.5 · p. 19

3.4 Calculating Vertices; 3.4.1 Calculating specific vertices

  • Vertices are not calculated automatically by default because computing all of them can take minutes; Vertex[ParticleList,Options] calculates selected vertices, with up to six fields for effective theories.The Eigenstates option selects the eigenstate set and defaults to the last entry in NameOfState.Source: §3.4 · p. 20, §3.4.1 · p. 20

3.4.1 Calculating specific vertices

  • Specific vertices return the involved particles and their coupling coefficients alongside one or more Lorentz structures.For fermion interactions, structures can include PL and PR; effective operators can also represent two-fermion, two-scalar interactions, with SARAH checking eight possible operator forms.Source: §3.4.1 · pp. 21–23
  • Vertex calculations can use Weyl fermions or gauge eigenstates, apply defined parameter dependences, and accept fixed particle-generation indices.For example, UseDependences -> True replaces g1 and g2 with the electric charge e when the corresponding relation is defined in the parameters file.Source: §3.4.1 · pp. 21–22

3.4.2 Calculating all vertices

  • MakeVertexList[Eigenstates,Options] calculates all vertices, with options for higher-dimensional operators, six-point interactions, or selected generic interaction types.Higher-dimensional operators default to four-point interactions, while GenericClasses defaults to All.Source: §3.4.2 · p. 23

3.5 Renormalization Group Equations

  • SARAH calculates one- and two-loop RGEs for superpotential parameters, soft-breaking terms, and gauge couplings using generic formulas extended for several Abelian gauge groups and Dirac gaugino masses.Calculation starts with CalcRGEs[Options].Source: §3.5 · p. 23
  • RGE options allow generation counts of selected superfields to remain variable, selected parameters to be omitted at two loops, and generation-index sums to be expressed as matrix multiplications.These choices support theories with heavy superfields and can reduce the calculation of two-loop RGEs for models with many new interactions.Source: §3.5 · pp. 24–25
  • The output defines trace, adjoint, and transpose operations as trace[A,B,C,...], Adj[M] = M†, and Tp[M] = Mᵀ.trace[A,B,C,...] denotes Tr(ABC...).Source: §3.5 · p. 25
  • Generation indices use Kronecker[i,j] rather than Delta[i,j], and scalar-mass results use trace abbreviations defined in TraceAbbr (see Ref.).The abbreviations simplify frequently appearing traces.Source: §3.5 · p. 25
  • For parameters with three indices, matrix multiplication is disabled, results are summed over involved indices, and three-index parameters are replaced by sums of two-index parameters.These beta functions are saved in NAME <> 3I, where NAME is the standard RGE array.Source: §3.5 · p. 25

Examples

The examples show how SARAH exposes one-loop beta functions, trace abbreviations, and options for varying generations or expression form. Loop corrections are calculated under stated conventions and saved as both individual contributions and sums.

Examples

  • The electron Yukawa beta function is stored in BetaYijk, with its one-loop expression shown and the two-loop expression omitted for length.The displayed one-loop terms include gauge couplings, Yukawa traces, and a matrix product.Source: Examples · pp. 25–26
  • The selectron soft-mass beta function is accessed through Betam2ij, with its one-loop expression and the Tr1[1] trace abbreviation shown.The example also gives the trace abbreviation's definition in terms of soft masses.Source: Examples · p. 26
  • CalcRGEs[VariableGenerations -> {q}] treats the left-quark superfield generation count as a variable in the hypercharge beta function.The displayed expression includes NumberGenerations[q].Source: Examples · p. 26
  • Options can expand matrix products into sums and omit selected parameters from two-loop RGEs, as illustrated for the MSSM with trilinear RpV.The examples show an expanded Yukawa beta function and exclusion of λ, λ′, and their soft-breaking equivalents.Source: Examples · p. 26

3.6 Loop Corrections

  • SARAH calculates analytical one-loop tadpole and self-energy corrections in the DR scheme and ’t Hooft gauge.The calculation can be restricted to corrections involving specified internal particles; vertices are calculated first if needed.Source: §3.6 · p. 27

Conventions

  • Loop contributions account for charge and symmetry factors; indistinguishable loop particles receive half the distinguishable-particle weight.Two possible charge flows double the diagram weight; the absolute factor depends on diagram type.Source: Conventions · p. 27

One Loop Tadpoles

  • One-loop tadpole sums list each VEV with its correction, while a separate list records contributions and their symmetry and charge factors.Source: One Loop Tadpoles · p. 28

One Loop Self Energies

  • One-loop self-energy sums list each particle with a field-type-dependent correction, and individual contributions are also available.Source: One Loop Self Energies · p. 28

Examples

This section uses examples and implementation rules to document how SARAH specifies model parameters, particles, gauge groups, and chiral-superfield soft parameters. It also describes how these settings shape model calculations and outputs, including loop corrections and output-file configuration.

Examples

  • Examples represent one-loop corrections with diagram descriptors and summed expressions, including a chargino-loop tadpole and a pseudoscalar-Higgs contribution to the down-squark self-energy.The self-energy example arises from a four-point interaction.Source: Examples · pp. 28–29

3.7 Output for LATEXand diagram calculators

  • Output options generate model information in LaTeX and formats for FeynArts, CompHep/CalcHep, WHIZARD, and UFO, with switches for diagrams, interactions, RGEs, and loop corrections.Higher-dimensional operators and six-point interactions can also be enabled; two-loop RGEs require IncludeRGEs=True.Source: §3.7 · p. 29

The model files

  • A model directory contains ModelName.m, particles.m, and parameters.m; the model file is required for calculating the Lagrangian, while the other files support property definitions and appropriate output.The directory and model file must share the model's name.Source: The model files · p. 31

4.1 The model file

  • The model file is the heart of SARAH and specifies the complete model.The text introduces the file's structure and the functions that define its physical content.Source: §4.1 · p. 31

4.1.1 Description

  • A model file begins with gauge structure, particle content, and the superpotential, then defines eigenstates and their properties, particle removals, Dirac spinors, spectrum files, and output choices.Eigenstate properties include VEV decompositions, matter and gauge rotations, flavor decompositions, and additional or redefined couplings.Source: §4.1.1 · p. 31

4.1.2 Schematic Structure

  • The schematic structure lists gauge and matter fields and the superpotential, followed by eigenstate names and definitions, then general information such as particle removal and spectrum-file assignment.Eigenstate definitions include VEVs, gauge and matter rotations, flavor expansion, gauge fixing, additional couplings, and Dirac spinors.Source: §4.1.2 · p. 32

4.2.1 General

  • The parameter file's ParameterDefinition list pairs parameter names with properties for descriptions, reality, form and symmetry, LaTeX names, dependencies, matrix products, LesHouches input, or numerical values.The file supports input and output routines and simplifying parameter assumptions.Source: §4.2.1 · p. 33

4.2.2 Simplifying assumptions

  • Parameter assumptions can make parameters real and constrain tensor forms, including diagonal, symmetric, antisymmetric, and Hermitian structures.SARAH otherwise generally treats parameters as complex and tensor entries as potentially off-diagonal; gauge couplings default to real.Source: §4.2.2 · pp. 33–34

Examples

  • Examples specify CP- and flavor-conserving Yukawa matrices and parametrize charged-Higgs mixing in the MSSM by the angle β.The charged-Higgs mixing matrix is defined through a Dependence relation for ZP.Source: Examples · pp. 34–35

4.2.3.1 Dependences

  • Dependence statements encode relations that apply always, only when requested, or only for numerical evaluation; MatrixProduct also defines matrix relations such as those used for CKM.DependenceOptional can be enabled for specific commands, such as vertex calculations; DependenceNum substitutions are used when a parameter’s numerical value is calculated.Source: §4.2.3.1 · p. 34

4.2.3.2 Numerical values

  • Parameters without other dependencies can receive numerical values directly or through LesHouches entries, with optional block-occurrence indices selecting, for example, SUSY-scale values.The example distinguishes the first gauge block, normally holding GUT-scale values, from the second block, which holds SUSY-scale values.Source: §4.2.3.2 · p. 35

Example

  • The CKM example uses a numerical Wolfenstein parametrization, with real Wolfenstein parameters and an experimental input value for η of 0.341.The definition also specifies the CKM matrix product, LesHouches block, and SPheno output expression.Source: Example · p. 36

4.3 Particles File

  • The particle file records eigenstate metadata and supports numerical, automatically derived tree-level, or LesHouches-read masses.Metadata can include descriptions, R-parity, PDG identifiers, electric charge, widths, output names, and formatting or tool-specific fields.Source: §4.3 · p. 36

4.4 Global definitions

  • Global definitions apply particle or parameter properties across model files, while local entries can override selected global properties.A unique Description identifier lets model files refer to a shared definition.Source: §4.4 · pp. 37–38

5.1.1 Vector superfields

  • Gauge entries define vector superfield names, group dimensions and names, couplings, and whether index sums are expanded during Lagrangian calculation.SARAH also adds a soft-breaking gaugino mass for each vector superfield.Source: §5.1.1 · p. 39

Example: Standard model color group

  • In the Standard Model color-group example, SARAH uses SU(3) generators, names the strong coupling g3, and leaves color-index sums unevaluated.The example also names the gluon VG, gluino fG, color indices colX, and adjoint color indices acolX.Source: Example: Standard model color group · p. 40
  • For models with several Abelian groups, SARAH generates off-diagonal gauge couplings and gaugino masses, with RGE mixing changes relative to.The off-diagonal couplings are named gA<>gB, and the gaugino masses Mass<>A<>B; A and B refer to vector superfields defined in Gauge.Source: Example: Standard model color group · p. 40

5.1.2 Chiral Superfields

  • Chiral superfields are specified by components, generation count, superfield name, and gauge transformations, with Dynkin labels resolving non-unique representation dimensions.Components are one-dimensional for trivial transformations under expanded-index groups and vectors or higher-dimensional tensors for nontrivial transformations; larger-than-fundamental representations use tensor products.Source: §5.1.2 · p. 41
  • SARAH automatically adds soft-breaking squared masses for chiral superfields and gauge-invariant mixed soft parameters allowed by R-parity.The R-parity check can be disabled in the model file.Source: §5.1.2 · p. 41

Examples

This section illustrates how SARAH model files define fields, superpotential terms, eigenstates, and particle rotations, and how SARAH handles Lagrangian construction, term controls, gauge-sector mixing, gauge fixing, and ghost interactions.

Examples

  • MSSM field examples distinguish expanded-index left-quark fields, with generation and representation data, from an unexpanded right down-quark superfield.The examples identify left up- and down-squarks and quarks as SuL/FuL and SdL/FdL, and right down states as SdR/FdR.Source: Examples · p. 42
  • Because an SU(5) 10 representation is not unique, its specification requires adding the appropriate Dynkin labels.Source: Examples · p. 42

5.2 Superpotential

  • Superpotential terms use coefficient-and-parameter and superfield lists, allow optional index contractions and up to four fields, and automatically add corresponding soft-breaking couplings.Soft-breaking couplings carry the same generation indices as their superpotential couplings; simplifying assumptions for them can be specified independently.Source: §5.2 · p. 43

5.3 Excluding terms or adding Dirac gaugino mass terms

  • SARAH derives the gauge-eigenstate Lagrangian from vector and chiral superfield information, while model-file flags can exclude selected soft-breaking, D-term, or F-term contributions.The listed controls cover trilinear, bilinear, linear, scalar-mass, and gaugino-mass terms, as well as all soft-breaking terms.Source: §5.3 · p. 44
  • Dirac gaugino mass terms are not automatic: explicitly adding them makes SARAH generate allowed chiral-vector mass terms and corresponding D-terms, accounting for Abelian kinetic mixing.Parameters can be set to zero to remove selected terms when multiple adjoint fields are present.Source: §5.3 · p. 44

5.4 Properties of different eigenstates

  • Eigenstate properties are defined after fixing eigenstate names and can include VEVs, gauge and matter sectors, phases, flavors, and additional properties.Source: §5.4 · p. 45

5.4.1.1 Introduction

  • After symmetry breaking, a field receiving a VEV is parametrized by its VEV and scalar and pseudoscalar components, which SARAH treats as real parameters; the phase is optional.The MSSM example assigns vd and vu to the neutral Higgs fields and shows how to add a relative phase.Source: §5.4.1.1 · p. 45, §5.4.1.2 · p. 46

5.4.2.1 Introduction

  • For flavor-specific names, SARAH expands a field into as many names as it has generations, as illustrated by separate down-quark names for left- and right-handed states.Source: §5.4.2.1 · p. 46, §5.4.2.2 · p. 47

5.4.3.2 Implementation in SARAH

  • SARAH defines vector-boson rotations in model parameter files, treats unbroken-gauge vector bosons as real, and uses degrees of freedom to determine whether mass eigenstates are real or complex.In electroweak symmetry breaking, W+ cannot be used as a new field because it would not be related to W− and would change the theory’s degrees of freedom; winos use separate fWm and fWp fields after mixing.Source: §5.4.3.2 · p. 48

5.4.4 Particle Mixings

  • Symmetry breaking or bilinear superpotential terms rotate gauge eigenstates into mass eigenstates, with the rotation prescription determined by whether the mass matrix is hermitian.Source: §5.4.4 · p. 48

5.4.4.1 Introduction

  • Hermitian mass matrices use one diagonalizing matrix whose eigenvalues give the mass eigenstates’ masses; non-hermitian matrices require two mixing matrices that diagonalize MM^T and M^T M.Source: §5.4.4.1 · p. 48

5.4.4.2 Implementation in SARAH

  • SARAH distinguishes hermitian from non-hermitian mixings by their definition form and lists all generations of each basis particle before the next particle.Source: §5.4.4.2 · p. 49

5.4.4.2.1 Hermitian mixings in SARAH

  • Hermitian rotations specify old states, new states, and a mixing matrix; the down-squark example combines three left and three right states into six, and reducible matrices are automatically block-reduced.Source: §5.4.4.2.1 · p. 49

5.4.4.2.2 Non-hermitian mixings in SARAH

  • Non-hermitian rotations specify two bases, states, and matrices; in the MSSM chargino example, fWm and FHdm form negative states while fWp and FHup form positive states.The negative and positive states are connected to the original states by the respective mixing matrices Um and Up.Source: §5.4.4.2.2 · p. 49

5.4.5.1 Introduction

  • The general Rξ-gauge fixing term is expressed using gauge-fixing functions f_a.Source: §5.4.5.1 · p. 50

5.4.5.2 Implementation in SARAH

  • Since version 3.2, SARAH derives gauge-fixing terms from kinetic terms by requiring scalar–vector mixing to vanish, then uses them to calculate ghost interactions.In extended gauge sectors, each massive vector boson requires its corresponding Goldstone boson; users can inspect the generated terms in DEFINITION[$EIGENSTATES][GeneratedGaugeFixing].Source: §5.4.5.2 · p. 50

Examples

SARAH illustrates automatic gauge fixing and supports adding or changing Lagrangian interactions when the model’s terms cannot be derived from its superpotential or kinetic interactions. These manually defined terms are incorporated into the Lagrangian and treated through field rotations and replacements.

Examples

  • The examples show SARAH deriving Rξ gauge-fixing terms for the color group and the Z boson after electroweak symmetry breaking.The Z-boson expression uses the γ–Z mixing matrix, which can be expressed through the Weinberg angle.Source: Examples · pp. 50–51

5.4.6.1 Introduction

  • After integrating out particles, SUSY relations may no longer hold, leaving interactions such as the Standard Model Higgs self-coupling as free parameters.The passage contrasts this with supersymmetry, where that coupling is fixed by gauge and Yukawa couplings.Source: §5.4.6.1 · p. 51

5.4.6.2 Implementation in SARAH

  • SARAH lets users add or change Lagrangian terms separately for each eigenstate, with manually defined terms subject to rotations and replacements.The added Lagrangian must have mass dimension 4; fields are dot-separated, Weyl fermions are used, and indices are contracted automatically.Source: §5.4.6.2 · pp. 51–52
  • The AddHC and Overwrite options control whether a term’s hermitian conjugate is added and whether existing couplings involving the same fields are replaced.New couplings receive automatically added tensor indices and are assumed complex by default.Source: §5.4.6.2 · p. 52

Example

SARAH supports adding terms to a Lagrangian, including additions to or replacements of existing Higgs self-couplings. It can also form effective theories by integrating out heavy particles or create non-supersymmetric limits by deleting particles.

Example

  • SARAH’s examples demonstrate adding a mixed soft-breaking term and adding an additional contribution to existing Higgs self-couplings.An example also shows how to overwrite existing Higgs self-interaction expressions.Source: Example · p. 52
  • The displayed Higgs self-interaction example replaces the prior expressions with a new set of terms using Overwrite->True.The example includes Higgs self-couplings involving hh and Ah fields.Source: Example · p. 52

5.5 Effective or non-supersymmetric theories

  • SARAH can integrate out heavy particles or delete them to obtain an effective theory or a non-supersymmetric limit of a model.Integrating out heavy states can produce higher-dimensional operators, such as four-fermion interactions.Source: §5.5 · p. 53, §5.5.1.1 · p. 53

5.5.1.1 General

  • When heavy particles are absent below the energy scale of their mass, an effective theory for the lighter fields can be derived by integrating them out.The procedure can generate higher-dimensional operators.Source: §5.5.1.1 · p. 53

5.5.1.2 Implementation in SARAH

  • The IntegrateOut list can specify particles by component or superfield and can select particular generations for removal.Listed particles may belong to different eigenstates, and each is integrated out the first time it appears.Source: §5.5.1.2 · p. 53

Example

SARAH documents procedures for deleting particles, defining Dirac spinors, checking model files, and exporting model information to LaTeX and external tools. The section also describes options for FormCalc output and generating PDFs from the LaTeX files.

5.5.2 Deleting particles

  • Deleting particles is faster than integrating them out when higher-dimensional operators are not needed, and the section gives gluino removal as an example.Deleting SUSY particles can produce a non-supersymmetric limit, but remaining SUSY-based relations may need to be redefined.Source: Example · p. 53, §5.5.2 · p. 54

5.6 Definition of Dirac spinors

  • SARAH defines Dirac spinors by specifying the Weyl spinors that form their left and right components; Majorana neutralinos use a Weyl spinor and its conjugate.The electron is given as an example of a Dirac spinor assembled from FeL and FeR.Source: §5.6 · p. 54

5.7 Checking the model files

  • Model checks cover anomalies, charge and R-parity conservation, possible superpotential terms, mixings, mass matrices, and particle and parameter definitions.The anomaly check is supported only for supersymmetric models.Source: §5.7 · pp. 54–55

6.1 Writing a LATEX file

  • LaTeX output can include fields, Lagrangian terms, mass matrices, tadpole equations, renormalization group equations, and one-loop self-energies and tadpoles.The command first calculates interactions for the requested eigenstates if they have not already been calculated.Source: §6.1 · p. 56

6.2 Options

  • LaTeX options control diagram output, inclusion of effective operators, six-particle interactions, and use of shorter vertex notation.Effective operators are included by default, while six-particle interactions and ShortForm are off by default.Source: §6.2 · pp. 56–57

6.3 Creating the pdf File

  • SARAH writes a shell script that compiles the LaTeX file, processes diagram files with mpost, and recompiles the document to create a PDF.On Linux, the script may first need to be made executable.Source: §6.3 · p. 57

7.1 Generate model files for FeynArts

  • The FeynArts export creates a model file listing particles, indices, and interactions, with labels derived from the defined LaTeX names.For multiple unbroken non-Abelian gauge groups, generators appear in vertices; SU(3) generators use SUNT and are simplified by FormCalc.Source: §7.1 · p. 58

7.2 Dependences, numerical values and special abbreviations for FormCalc

  • A companion substitutions file provides parameter dependences, mass definitions, numerical values, and FormCalc abbreviations.The abbreviations include replacements for complex conjugation, squares, and soft-breaking coupling names, and are recommended to speed FormCalc calculations.Source: §7.2 · p. 59

Output for CalcHep/CompHep

This section describes SARAH’s generated model files and source code, along with setup and input options for using them with external tools and for calculating spectra, decays, and electroweak-scale quantities.

8.1 Generate model files for CalcHep and CompHep

  • ModelOutput[$EIGENSTATES, WriteCHep->True] generates the files, with options controlling Goldstone interactions, CP violation, CompHep format, and auxiliary-field splitting.Other options include running strong coupling, SLHA input, CalcHep mass calculations, SPheno calls from CalcHep, and loop-induced Higgs-to-photon or gluon vertices.Source: §8.1 · p. 60

8.1.1 Model Files

  • The output consists of prtclX.mdl for particles, lgrngX.mdl for interactions, varsX.mdl for numerical variables, and funcX.mdl for parameter dependences.The integer X is selected with ModelNr.Source: §8.1.1 · p. 61

8.1.1.1 Particles

  • CalcHep names use each particle’s OutputName, adjust the first letter for non-self-conjugate antiparticles, append generation numbers, and prefix R-parity -1 particles with ~.This naming scheme permits MicrOmegas use without an additional list of SUSY particles; CalcHep’s four-letter naming limit and lack of indices prevent direct use of SARAH’s internal particle definitions.Source: §8.1.1.1 · p. 61

8.1.2 Auxiliary fields in CalcHep/CompHep

  • Because CalcHep/CompHep color structure is implicit, SARAH splits four-colored-particle, two-colored-particle-plus-two-gluon, four-scalar, and squark-gluon interactions into three-particle vertices using auxiliary fields.NoSplittingWith and NoSplittingOnly can suppress splitting for selected vertices.Source: §8.1 · p. 60, §8.1.2 · p. 62

8.1.3 Vertex functions

  • SARAH derives vertex variables from its calculated interactions, evaluates tensor-index sums, truncates variable names longer than six letters, and assumes parameters are real by default.Setting CPViolation->True splits parameters not explicitly declared real into real and imaginary parts, which are written separately for each interaction.Source: §8.1.3 · p. 62, §8.1.4 · p. 63

8.1.5 SLHA input

  • CalcHep can read LesHouches input through definitions SARAH writes to its functions file, assuming an SPheno.spc.[MODEL] file in the same directory by default.The assumed file location can be adjusted manually.Source: §8.1.5 · p. 63

8.1.6 What can be a problem...

  • CalcHep/CompHep prohibit non-auxiliary particles with PDG number 0, higher-dimensional operators, and unbroken non-color non-Abelian gauge groups; case-insensitive variable names and large flavor-index combinations also cause problems.Writing can take a long time when many generations and non-reducible mixing matrices require all index combinations to be listed.Source: §8.1.6 · p. 63

8.2 micrOMEGAs

  • SARAH writes micrOMEGAs main files for relic-density calculations and direct-detection signals, which use CalcHep to calculate annihilation and co-annihilation processes.SLHA+ also permits micrOMEGAs calculations by copying the SPheno spectrum file into its directory and starting the calculation.Source: §8.2 · pp. 63–64

(in collaboration with C. Speckner)

  • For WHIZARD 2.x, compile and install generated files from the output directory; WHIZARD 1.x instead requires injecting them into its source tree.Installation paths can be adjusted with --prefix and --localprefix, and WO_CONFIG can locate WHIZARD binaries.Source: (in collaboration with C. Speckner) · p. 66

Output in UFO format

  • The UFO output includes files defining model particles, parameters, Lorentz structures, vertices, couplings, and coupling-order hierarchies.SARAH writes these files to the model’s UFO output directory.Source: Output in UFO format · p. 67

Using the UFO model files of SARAH with Madgraph

  • To use SARAH’s UFO model with MadGraph, copy its files into a model directory and import it; an option avoids particle-name clashes.A SPheno-generated SLHA spectrum file can provide the numerical masses and parameters needed for cross-section calculations.Source: Using the UFO model files of SARAH… · p. 68

11.1 Introduction

  • SARAH generates SPheno source code that uses model-independent routines for numerical RGE evaluation, 2- and 3-body decay phase space, and Passarino–Veltman integrals.Generation starts with MakeSPheno[Options].Source: §11.1 · p. 69

11.2 Input file

  • The SPheno.m input file specifies parameters read from Les Houches input and parameters fixed by SARAH-derived tadpole equations.SARAH solves these equations with Mathematica; for non-unique solutions, it solves for the absolute squared and expects the phase through SignumX.Source: §11.2 · pp. 70–71
  • Boundary conditions can be defined at the electroweak, SUSY, and GUT scales and applied while running up and down with RGEs.The input file also allows threshold-scale and low-scale-input boundary conditions.Source: §11.2 · p. 72
  • Boundary conditions can be overridden in the LesHouches input, and nested boundary-condition sets can be selected for a run with the MODSEL flag.The default set is 1; the passages also give an example of forcing Higgs soft masses to specific values.Source: §11.2 · pp. 73–74
  • Decay-width inputs select particles for two-body decays and fermions for three-body decays; Automatic calculates widths for non-standard-model particles and, for two-body decays, the top quark.Automatically generated three-body decay files use the particle name followed by .f90.Source: §11.2 · p. 74
  • Input flags can disable loops containing specified fields in both one-loop mass corrections and low-energy-observable calculations.The corresponding entries are added to the SPhenoInput block and listed in the SARAH-generated template.Source: §11.2 · p. 74
  • Low-scale input defines model parameters at a specified scale without RGE running, then uses them to calculate the loop-corrected mass spectrum and decays.The example sets the renormalization scale to 1 TeV when no explicit scale is defined.Source: §11.2 · p. 75

Information about particles and parameters

  • Particle and parameter descriptions specify physical meanings used to calculate electroweak-scale gauge and Yukawa couplings, the CKM matrix, and on-shell masses.The passages state that the definitions are made in the parameters and particles file.Source: Information about particles and… · p. 75

Mixing angles in gauge sector

This section describes how SARAH and generated SPheno modules handle gauge mixing angles, scale-dependent running and model constraints, alongside calculations of mass eigenstates, mixing matrices, numerical expressions, group factors, and supersymmetric Lagrangian interactions.

Mixing angles in gauge sector

  • SPheno obtains gauge-boson rotation matrices by diagonalizing their mass matrix, and SARAH lets users define mixing-angle relations through DependenceSPheno.The example defines the Weinberg angle as ArcCos[Abs[ZZ[1,1]]].Source: Mixing angles in gauge sector · p. 76

11.2.1.1 Thresholds without gauge symmetry breaking

  • For thresholds without gauge-symmetry breaking, SARAH derives lower-scale RGEs from the highest-scale RGEs by adjusting heavy-field generation counts and removing their couplings as thresholds are crossed.The setup assumes equal scalar and fermion masses much larger than soft-breaking masses, with heavy masses from bilinear superpotential terms; 2-loop RGEs require 1-loop threshold corrections to gauge couplings and gaugino masses, with opposite signs for upward and downward running.Source: §11.2.1.1 · pp. 76–77

11.2.1.2 Thresholds with gauge symmetry breaking

  • When gauge structures differ across scales, SARAH calculates each regime’s RGEs separately and combines them in SPheno code that includes finite shifts in gauge couplings and gaugino masses.Each regime needs its own SARAH model file, while the higher-scale SPheno input specifies integrated-out fields, the gauge sector below each threshold, and the fields’ quantum numbers.Source: §11.2.1.2 · p. 78

11.2.2 Supported models and known issues

  • Full SPheno output requires Yukawa couplings, two Higgs VEVs, three SM gauge couplings, and SM particle content for low-energy fits; flavor decomposition and NoFlavorMixing matrices are unsupported.Users can instead supply model parameters manually, skipping RGE evaluation and electroweak-data fitting while retaining one-loop corrected masses, decay widths, and branching ratios.Source: §11.2.2 · p. 79

11.2.3 Low energy SPheno version

  • A low-energy-only SPheno version skips RGEs and electroweak-data fitting but solves tadpoles and calculates tree- and one-loop masses, decay widths, and branching ratios.It supports a larger model set, including non-SUSY models; when the SM couplings and Higgs VEVs are present, SARAH calculates running gauge and Yukawa couplings at the SUSY scale using DR masses from and 2-loop SM RGEs.Source: §11.2.3 · pp. 79–80

11.2.4 Models with another gauge group at the SUSY scale

  • For models whose SUSY-scale gauge group lacks the SM hypercharge factor, SARAH introduces an auxiliary hypercharge coupling and requires the user to define its relation to existing gauge couplings.The text gives left-right models with U(1)R × U(1)B−L as an example and specifies that gauge-coupling boundary conditions must use non-GUT-normalized values.Source: §11.2.4 · p. 80

11.3 Writing input files for HiggsBounds with SPheno

  • SARAH-generated SPheno modules can write HiggsBounds input files using effective couplings, but omit H → γZ and H → ggZ couplings and e+e− cross sections, setting unavailable values to zero.The LesHouches interface is not used because it supports at most five neutral Higgs fields; charged-Higgs LEP production cross sections are also set to zero.Source: §11.3 · p. 81

12.1 LHPC spectrum plotter; 12.2 All at once; Other output; Numerical values

  • SARAH provides an LHPC spectrum-plotter control file and can generate SPheno, CalcHep, WHIZARD, FeynArts, UFO, and LaTeX outputs, with selectable output flags and numerical routines for masses and mixing matrices.The LHPC control file specifies particle colors and columns; the numerical routines also work with parameter values.Source: §12.1 · p. 83, §12.2 · p. 83, Other output · p. 84

13.1 Adding numerical values

  • SARAH accepts parameter values from parameter files or Les Houches spectra, and users can set or delete values during a session.A spectrum can be read automatically at Start when it is in the model-file directory, or later with ReadSpectrum["Spectrum File"].Source: §13.1 · p. 85

13.2 Calculate Mixing Matrices

  • After free-parameter values are defined, CalcMatrices calculates mass eigenstates and mixing matrices, with negative scalar squared masses triggering a warning and setting warning to True.For example, the down-squark mixing matrix is stored in MDNum and the corresponding masses in MassSd.Source: §13.2 · pp. 85–86

13.3 Calculate Numerical Values

  • NumericalValue[x] returns a numerical value for an expression such as a mass-matrix entry or a vertex.The examples evaluate an entry of the Higgs mass matrix and a photon–down-squark interaction.Source: §13.3 · p. 86

Examples

  • The examples apply NumericalValue to a Higgs mass-matrix entry and to a photon–down-squark vertex.The expressions shown are NumericalValue[MassMatricesFullEWSB[[4]]] and NumericalValue[Vertex[{VP,Sd,conj[Sd]}][[2,1]]].Source: Examples · p. 86

Calculation of Group Factors

  • For SU(N) irreducible representations, SARAH derives generators, quadratic Casimirs, and Dynkin indices using Young tableaux and representation weights.The described procedure extracts the highest weight, computes C2(r) with the Weyl formula, and obtains I(r) from C2(r), normalizing the fundamental representation to 1/2.Source: Calculation of Group Factors · p. 87

B.1 The supersymmetric Lagrangian

  • The supersymmetric Lagrangian section derives matter interactions from the superpotential, including fermion–fermion–scalar interactions and four-scalar F-terms.It also introduces the vector-superfield components and covariant derivative.Source: §B.1 · p. 89

Calculation of the Lagrangian of supersymmetric models and deriving the vertices

This section describes how SARAH derives interactions and vertices for supersymmetric models, including treatments of multiple U(1) groups, loop calculations, and renormalization group equations. It also covers generated SPheno routines and model-file construction, with input, boundary, and threshold settings for examples including the MSSM, NMSSM, and seesaw models.

Calculation of the Lagrangian of supersymmetric models and deriving the vertices

  • The Lagrangian includes gauge self-interactions, vector–gaugino interactions, chiral–vector kinetic terms, gaugino–matter interactions, and soft-breaking mass and interaction terms.The soft-breaking terms include scalar and gaugino masses and interactions corresponding to superpotential terms.Source: Calculation of the Lagrangian of… · p. 90
  • Dirac gaugino models are supported from SARAH 3.2.0, including their additional D-term contribution and two-loop RGEs.The component expansion of a Dirac mass term between a vector and an adjoint chiral superfield yields a gaugino Dirac mass and an additional D-term contribution.Source: Calculation of the Lagrangian of… · p. 90

Gauge fixing terms and ghost interactions

  • Gauge fixing breaks gauge invariance for quantization, while broken-symmetry terms remove vector–scalar mixing and introduce corresponding ghost interactions.For unbroken symmetries SARAH uses Rξ gauge; for the Standard Model gauge sector after electroweak symmetry breaking, the fixing terms involve Goldstone bosons.Source: Gauge fixing terms and ghost… · p. 91

B.2 Deriving the vertices

  • SARAH derives vertices by differentiating with respect to external fields and then applying the vacuum conditions.The interaction coefficient is i when one or more scalars are involved and 1 otherwise; the method also specifies a momentum flow.Source: §B.2 · p. 91

Appendix C Parts of the Lagrangian in SARAH

  • SARAH provides flags to suppress selected Lagrangian terms and stores calculated components separately for mass and gauge eigenstates.The flags cover soft terms, Dirac gaugino masses, and D- and F-terms; additional interactions and redefinitions are saved in LagRedefinition.Source: Appendix C · pp. 92–93

Gauge anomalies

  • Before calculating the Lagrangian, SARAH checks triangle gauge anomalies and the Witten anomaly, warning when a condition fails but still allowing model evaluation.The Witten-anomaly condition requires an even number of SU(2) doublets.Source: Gauge anomalies · p. 94

E.1.1 Generic form of β functions

  • SARAH summarizes generic one- and two-loop RGE equations for parameters of N = 1 supersymmetric gauge theories, including soft-breaking terms and anomalous dimensions.The equations cover superpotential parameters and trilinear, bilinear, and linear soft-breaking parameters.Source: §E.1.1 · pp. 95–96

E.1.2 Direct product of gauge groups

  • For product gauge groups, SARAH uses substitution rules, while multiple Abelian groups require generalized terms and charges.The complete two-loop RGEs for several U(1) groups are cited as published in.Source: §E.1.2 · p. 98, §E.1.3 · p. 99

E.1.3 Several U(1) gauge groups

  • For multiple U(1) groups, SARAH modifies the Abelian gauge-coupling and gaugino terms in the RGEs, while non-Abelian terms remain unchanged.The expressions cover only Abelian-group contributions, with generalized Abelian indices and charges defined for the chiral superfields.Source: §E.1.3 · pp. 99–100

E.2 One-loop amplitudes for one- and two-point functions

  • One-loop amplitudes use DR-scheme scalar integrals and diagram coefficients for symmetry, charge, and real-field or Majorana-loop factors.The passage defines the basic integrals and notes that other necessary functions can be expressed through A0 and B0.Source: §E.2 · pp. 101–102

E.2.1 One-loop tadpoles

  • The one-loop tadpole expressions include fermion, scalar, and vector-boson loops, identified in SARAH by the generic names FFS, SSS, and SVV.These labels distinguish the three loop contributions to tadpoles.Source: §E.2.1 · p. 103

E.2.2.0.1 Corrections to fermion

  • Fermion self-energies include fermion-scalar and fermion-vector-boson loops; scalar self-energies include fermion, scalar, and vector-boson loop diagrams.For scalar corrections, the listed diagrams include three- and four-point interactions; vector-boson corrections use three-point interactions because gauge invariance relates the four-point terms.Source: §E.2.2.0.1 · p. 103, §E.2.2.0.2 · p. 103, §E.2.2.0.3 · p. 104

E.2.2.0.3 Corrections to vector boson

  • Vector-boson self-energies comprise fermion, scalar, vector-boson, and vector-scalar loop contributions.The passage specifies that only diagrams involving three-point interactions are needed, since gauge invariance relates the four-point interactions to them.Source: §E.2.2.0.3 · p. 104

E.2.3 One-loop corrections to masses

  • One-loop self-energies yield corrected masses and mass matrices, with pole equations solved iteratively for eigenvalues and corrections specified for real and complex scalars and Majorana and Dirac fermions.For parameters fixed by tadpole equations, the one-loop corrected values are used with the tree-level mass matrix.Source: §E.2.3 · p. 104

F.1 Generated Files for SPheno

  • Generated SPheno files provide branching ratios, mass-eigenstate and loop-calculation couplings, input/output routines, and one-loop tadpole and self-energy calculations.LoopMasses routines calculate corrected masses iteratively; calculations are performed in ’t Hooft gauge, and SM pole masses are used in decays and output.Source: §F.1.1 · p. 106, §F.1.2 · p. 106, §F.1.4 · p. 107

F.1.7 LowEnergy_[Model].f90

  • Other generated SPheno routines cover low-energy observables, model data, and RGE running across GUT, SUSY, and electroweak scales.The routines include observables such as lepton anomalous magnetic moments, electric dipole moments, and muon-to-electron conversion; RGE files contain beta functions for the different running ranges.Source: §F.1.7 · p. 108, §F.1.8 · p. 109, §F.1.9 · p. 109

F.1.9 RGEs_[Model].f90

  • The generated RGE routines precompute matrix products and parameter powers as constants, while threshold-shift routines are created only for models with gauge-symmetry-breaking threshold scales.Those threshold routines calculate finite shifts for gauge couplings and gaugino masses.Source: §F.1.9 · p. 109, §F.1.10 · p. 110

F.1.12 SugraRuns_[Model].f90

  • The SPheno main program calls input, spectrum, decay-width, and low-energy-observable routines, while SugraRuns iterates the spectrum from a first guess and applies scale-dependent boundary conditions.Boundary routines use electroweak data for gauge and Yukawa couplings and set conditions at GUT, SUSY, and threshold scales.Source: §F.1.11 · p. 110, §F.1.12 · p. 110

F.1.14 SusyMasses_[Model].f90

  • SusyMasses calculates running masses at tree level by solving each particle’s mass-matrix eigensystem; Goldstone-containing matrices use ’t Hooft gauge when Goldstones are associated with vector bosons.TreeMasses calls the other subroutines, including CalculateX for the mass and mixing matrices.Source: §F.1.14 · p. 110

F.1.15 Three Body Decays, e.g. Glu_[Model].f90

  • The three-body-decay routines calculate fermion decays to three fermions, with SARAH generating the processes and diagrams and mapping them to SPheno phase-space functions.The routines calculate branching ratios and decay widths.Source: §F.1.15 · p. 110

F.2. LESHOUCHES INPUT FILE

  • LesHouches input templates can be supplied under custom filenames, and their MODSEL and SPHENOINPUT blocks provide model boundary-condition and calculation options.A separate low-scale-input template is also provided.Source: §F.2 · p. 111, MODSEL · p. 112, SPHENOINPUT · p. 112

SMINPUTS

  • SMINPUTS entries provide electroweak and strong couplings, the Fermi constant, the Z-boson mass, and fermion masses in the stated pole or running-mass schemes.The listed values include αs(mZ) in the MS scheme and running quark masses at their specified scales.Source: SMINPUTS · pp. 111–112

F.3 Decays calculated by SPheno

  • SPheno calculates SUSY and Higgs decays, running couplings to each decaying particle’s mass and including dominant QCD corrections for specified Higgs channels.Fermionic SUSY particles receive tree-level two- and three-body decays; Higgs calculations also include off-shell ZZ* and WW* final states and loop-induced decays to photons and gluons with dominant NLO corrections.Source: §F.3 · p. 114

MSSM.m

  • The MSSM example uses mSugra inputs m0, M1/2, A0, signµ, and tan β, and solves the tadpole equations for µ and Bµ.It allows a dynamically adjusted SUSY scale, with the initial guess m0^2 + 4 m12^2 and subsequent scale MSu[1] * MSu[6]; the example treats m0 and tan β as real.Source: MSSM.m · pp. 116–117
  • The MSSM example allows low-scale input of all free parameters, optional boundary conditions for dynamically calculating SUSY VEVs, and automatic generation of two- and three-body decay code for SUSY and Higgs particles.Source: MSSM.m · p. 117

NMSSM.m

  • The NMSSM adds κ, Aκ, λ, Aλ, and vS to the mSugra parameters, with κ and λ set at the SUSY scale and Aκ and Aλ at the GUT scale.Other parameters use GUT-scale boundaries except vS, which is set at the SUSY scale; Higgs and singlet soft masses are chosen to solve the tadpole equations.Source: NMSSM.m · p. 120

Seesaw1.m

  • Three generations of right-handed gauge-singlet neutrinos extend seesaw I; the heavy fields are removed at the SUSY scale, leaving low-energy masses and vertices unchanged but not the RGEs.The model also includes a neutrino Yukawa interaction, a mass term, and an effective operator generated after integrating out the right-handed neutrino.Source: Seesaw1.m · p. 122

SPheno.m

  • Seesaw I assigns each singlet generation a threshold near its mass and matches its contribution into the Weinberg operator.The matching contribution for generation i is −Yv[i, index1] Yv[index2, i] / MassOfv[i]; fermion–scalar mass splitting is neglected when heavy superfields are integrated out.Source: SPheno.m · p. 123

Seesaw2.m

  • Seesaw II adds a scalar SU(2)L triplet within an SU(5) 15-plet, implements the new superpotential interactions and mass terms, and removes the 15-plet components at low scale.Source: Seesaw2.m · pp. 123–124

SPheno.m

  • With one 15-plet, the seesaw II setup uses a single threshold, sets Yt, λ1, and λ2 there, and initializes the Weinberg operator across the threshold.Source: SPheno.m · p. 125

Seesaw3.m

  • Under SU(5)-invariant GUT boundary conditions, seesaw III uses three generations of 24-plets because one generation would not produce sufficiently large neutrino mass splitting.The 24-plet decomposes into the fields GM, WM, BM, XM, and X̄M with the listed Standard Model representations.Source: Seesaw3.m · pp. 125–126

SPheno.m

  • The seesaw III mSugra-like, SU(5)-invariant boundary conditions yield small mass splittings among 24-plet members, allowing one generation to be integrated out at each scale.Source: SPheno.m · pp. 126–127
  • At thresholds, the Weinberg-operator matching adds seesaw terms with coefficients 1/2 for Yw3Yw3/MassOfHw3 and 3/10 for Yb3Yb3/MassOfHb3.The supplied definitions show these terms for threshold indices 3, 2, and 1.Source: SPheno.m · p. 127

F.6 Implementation of a model with a gauge symmetry breaking scale in SARAH and SPheno

The section details implementing a left-right supersymmetric model in SARAH and SPheno, simplifying the example to one threshold scale. It summarizes the model’s particle content and superpotential above the SU(2)R × U(1)B−L breaking scale.

F.6 Implementation of a model with a gauge symmetry breaking scale in SARAH and SPheno

  • The implementation uses one threshold scale instead of the two discussed in; adding the second is described as straightforward but redundant for this discussion.Source: §F.6 · p. 128

F.6.1 Summary of the model

  • The model summary identifies SU(2)R × U(1)B−L breaking as occurring at a single threshold scale.Source: §F.6.1 · p. 128

From GUT scale to SU(2)R × U(1)B−L breaking scale

F.6. IMPLEMENTATION OF A MODEL WITH A GAUGE SYMMETRY BREAKING SCALE IN SARAH AND SPHENO

This section describes implementing a supersymmetric model with an SU(2)R × U(1)B−L breaking scale in SARAH and SPheno, with separate model files above and below the threshold and boundary conditions connecting the low-energy MSSM plus an effective Weinberg operator. It also records matrix conventions, including a mixing matrix diagonalized by ZA and a charged-Higgs mass matrix diagonalized by Z+.

Below SU(2)R × U(1)B−L breaking scale

F.6.2 Model files for SARAH

  • SARAH’s model files define the separate scale regimes, while SPheno files provide the boundary conditions needed to connect them and generate Fortran source code.The SARAH-file description focuses on parts needed for SPheno output and omits gauge-fixing terms and Dirac-spinor definitions above the threshold.Source: §F.6.2 · p. 129, §F.6.3 · p. 131

From GUT scale to SU(2)R × U(1)B−L breaking scale

  • At the highest scale, vector and chiral superfields define the gauge and particle content, the superpotential is specified, and right-sector omega and delta fields receive VEVs vR and vBL.The gauge sector comprises U(1)B−L, SU(2)L, SU(2)R and SU(3).Source: From GUT scale to SU(2)R × U(1)B−L… · pp. 129–130

Above SU(2)R × U(1)B−L breaking scale

  • SPheno’s high-scale setup marks the model as an intermediate regime and lists threshold fields, including fields integrated out only for the third generation and above.The next-scale gauge groups and integrated-out fields’ quantum numbers are specified to calculate finite shifts in gauge couplings and gaugino masses.Source: Above SU(2)R × U(1)B−L breaking scale · p. 131
  • The high-scale vacuum conditions solve for Mdelta and Momega using an approximate tadpole solution that neglects soft-breaking terms.The section states that no closed analytical solution is available.Source: Above SU(2)R × U(1)B−L breaking scale · p. 132

Below SU(2)R × U(1)B−L breaking scale

  • The low-scale SPheno setup specifies mSugra inputs alongside B0, a, the signs of MΩ and M∆, vR, vBL, and the threshold scale.At the GUT scale, scalar masses, gaugino masses, trilinear couplings, and bilinear couplings follow the stated mSugra boundary conditions; µ, f, and α are read from input.Source: Below SU(2)R × U(1)B−L breaking scale · p. 132
  • Threshold boundary conditions match Yukawa and soft-term parameters, Higgs-sector quantities, and gauge couplings and gaugino masses between the two regimes.The matching uses the angles Θ1 and Θ2; below the breaking scale, the Weinberg operator is formed from Yv, f, and vR.Source: Below SU(2)R × U(1)B−L breaking scale · pp. 133–134

G.1. CONVENTIONS

  • A mixing matrix is parametrized and diagonalized by ZA.The supplied fragments do not identify the matrix or provide its parametrization.Source: §G.1 · p. 140
  • The charged-Higgs mass matrix is given in a specified basis and diagonalized by Z+.The supplied fragments do not specify the basis components or matrix entries.Source: §G.1 · p. 140

G.2. IMPLEMENTATION IN SARAH

The section shows how SARAH model files encode the MSSM gauge structure, fields, interactions, symmetry breaking, and mass eigenstates. It also points to references describing validation of SARAH’s output.

G.2 Implementation in SARAH

  • Gauge declarations specify each group and coupling, while field declarations encode component names, generations, superfields, and gauge representations.The examples define the U(1) × SU(2) × SU(3) gauge sector and MSSM doublet and singlet fields.Source: §G.2 · p. 141
  • The MSSM superpotential is represented as a list of coupling and field tuples, and SARAH automatically adds soft masses for scalars.Source: §G.2 · pp. 141–142
  • After electroweak symmetry breaking, SARAH encodes Higgs VEV decompositions, particle rotations into mass eigenstates, and vector-boson and gaugino mixing.The listed rotation matrices are defined in the corresponding model’s parameter file.Source: §G.2 · p. 142

G.2. IMPLEMENTATION IN SARAH

  • The mixing declaration lists gauge eigenstates, then the new mass eigenstates and their mixing matrix; its eighth line specifies chargino mixing.Source: §G.2 · p. 143
  • The example leaves the integrate-out and particle-deletion lists empty and defines Dirac spinors for the mass eigenstates.The replacements support conversion from four-component to two-component formalism.Source: §G.2 · p. 143

Verification of Output

  • Validation checks for SARAH outputs are referred to in prior work on RGEs and self-energies, model-file formats, and UFO output.The cited checks cover FeynArts/FormCalc and CalcHep/CompHep, WHIZARD and SPheno, and UFO.Source: Verification of Output · p. 144

Evaluation Time

The section reports evaluation times for SARAH commands and catalogs models included in its public version. The timing measurements are tied to a specified Mathematica version and computer.

Evaluation Time

  • Table I.1 reports the time in seconds needed to evaluate several SARAH commands in Mathematica 5.2.Measurements used a Lenovo ThinkPad X220 with an Intel i7-2620M 2.70 GHz core.Source: Table I.1 · p. 145, Evaluation Time · p. 145

Models included in the public version

Related papers

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