Skip to content

GitLab

  • Projects
  • Groups
  • Snippets
  • Help
    • Loading...
  • Help
    • Help
    • Support
    • Community forum
    • Submit feedback
    • Contribute to GitLab
  • Sign in / Register
SARAH SARAH
  • Project overview
    • Project overview
    • Details
    • Activity
  • Packages & Registries
    • Packages & Registries
    • Container Registry
  • Analytics
    • Analytics
    • Repository
    • Value Stream
  • Wiki
    • Wiki
  • Members
    • Members
  • Activity
Collapse sidebar
  • GOODSELL Mark
  • SARAHSARAH
  • Wiki
  • Gauge_group_constants

Last edited by Martin Gabelmann Jun 28, 2019
Page history

Gauge_group_constants

Gauge group constants

SARAH supports not only chiral superfields in the fundamental representation but in any irreducible representation of SU(N). In most cases, it is possible to fix the transformation properties of the chiral superfield by stating the corresponding dimension D. If the dimension is not unique, also the Dynkin labels are needed. For calculating kinetic terms and D-terms, it is necessary to derive from representation the corresponding generators. Furthermore, the eigenvalues C2 of the quadratic Casimir for any irreducible representation r

TaTaϕ(r)=C2(r)ϕ(r)

as well as the Dynkin index I

T**r(TaTb)ϕ(r)=I**δa**bϕ(r)

are needed for the calculation of the RGEs. All of that is derived by SARAH due to the technique of Young tableaux. The following steps are evolved:

  1. The corresponding Young tableaux fitting to the dimension D is calculated using the hook formula: $D = \Pi_i \frac{N + d_i}{h_i}$

    di is the distance of the i. box to the left upper corner and hi is the hook of that box.

  2. The vector for the highest weight Λ in Dynkin basis is extracted from the tableaux.

  3. The fundamental weights for the given gauge group are calculated.

  4. The value of C2(r) is calculated using the Weyl formula C2(r)=(Λ, Λ + ρ).

    ρ is the Weyl vector.

  5. The Dynkin index I(r) is calculated from C2(r). For this step, the value for the fundamental representation is normalized to \frac{1}{2}. $I(r) = C_2(r) \frac{D(r)}{D(G)}$

    With D(G) as dimension of the adjoint representation.

  6. The number of co- and contra-variant indices is extracted from the Young tableaux. With this information, the generators are written as tensor product.

The user can calculate this information independently from the model using the new command

CheckIrrepSUN[Dim,N]

Dim is the dimension of the irreducible representation and N is the dimension of the S**U(N) gauge group. The result is a vector containing the following information: (i) repeating the dimension of the field, (ii) number of covariant indices, (iii) number of contra-variant indices, (iv) value of the quadratic Casimir C2(r), (v) value of the Dynkin index I(r), (vi) Dynkin labels for the highest weight.

Examples
  1. Fundamental representation The properties of a particle, transforming under the fundamental representation of S**U(3) are calculated via CheckIrrepSUN[3,3]. The output is the well known result {3, 1, 0, 4/3, 1/2, {1, 0}}

  2. *'Adjoint representation The properties of a field transforming as 24' of *SU(5) are calculated by CheckIrrepSUN[24,5] . The output will be {24, 1, 1, 5, 5, {1, 0, 0, 1}}

  3. 'Different representations with the same dimension The 70' under S**U(5) is not unique. Therefore, CheckIrrepSUN[{70, {0, 0, 0, 4}}, 5] returns {70, 0, 4, 72/5, 42, {0, 0, 0, 4}}

    while CheckIrrepSUN[{70, {2, 0, 0, 1}}, 5] leads to

    {70, 2, 1, 42/5, 49/2, {2, 0, 0, 1}}

See also

Clone repository

Home

Index

  • Additional terms in Lagrangian
  • Advanced usage of FlavorKit
  • Advanced usage of FlavorKit to calculate new Wilson coefficients
  • Advanced usage of FlavorKit to define new observables
  • Already defined Operators in FlavorKit
  • Already defined observables in FlavorKit
  • Auto-generated templates for particles.m and parameters.m
  • Automatic index contraction
  • Basic definitions for a non-supersymmetric model
  • Basic definitions for a supersymmetric model
  • Basic usage of FlavorKit
  • Boundary conditions in SPheno
  • CalcHep CompHep
  • Calculation of flavour and precision observables with SPheno
  • Checking the particles and parameters within Mathematica
  • Checks of implemented models
  • Conventions
  • Decay calculation with SPheno
  • Defined FlavorKit parameters
  • Definition of the properties of different eigenstates
  • Delete Particles
  • Different sets of eigenstates
  • Diphoton and digluon vertices with SPheno
  • Dirac Spinors
  • FeynArts
  • Fine-Tuning calculations with SPheno
  • Flags for SPheno Output
  • Flags in SPheno LesHouches file
  • FlavorKit
  • FlavorKit Download and Installation
  • Flavour Decomposition
  • GUT scale condition in SPheno
  • Gauge Symmetries SUSY
  • Gauge Symmetries non-SUSY
  • Gauge fixing
  • Gauge group constants
  • General information about Field Properties
  • General information about model implementations
  • Generating files with particle properties
  • Generic RGE calculation
  • Global Symmetries SUSY
  • Global Symmetries non-SUSY
  • Handling of Tadpoles with SPheno
  • Handling of non-fundamental representations
  • HiggsBounds
  • Higher dimensionsal terms in superpotential
  • Input parameters of SPheno
  • Installation
  • Installing Vevacious
  • LHCP
  • LHPC
  • LaTeX
  • Lagrangian
  • Loop Masses
  • Loop calculations
  • Loop functions
  • Low or High scale SPheno version
  • Main Commands
  • Main Model File
  • Matching to the SM in SPheno
  • MicrOmegas
  • ModelOutput
  • Model files for Monte-Carlo tools
  • Model files for other tools
  • Models with Thresholds in SPheno
  • Models with another gauge group at the SUSY scale
  • Models with several generations of Higgs doublets
  • More precise mass spectrum calculation
  • No SPheno output possible
  • Nomenclature for fields in non-supersymmetric models
  • Nomenclature for fields in supersymmetric models
  • One-Loop Self-Energies and Tadpoles
  • One-Loop Threshold Corrections in Scalar Sectors
  • Options SUSY Models
  • Options non-SUSY Models
  • Parameters.m
  • Particle Content SUSY
  • Particle Content non-SUSY
  • Particles.m
  • Phases
  • Potential
  • Presence of super-heavy particles
  • RGE Running with Mathematica
  • RGEs
  • Renormalisation procedure of SPheno
  • Rotations angles in SPheno
  • Rotations in gauge sector
  • Rotations in matter sector
  • SARAH in a Nutshell
  • SARAH wiki
  • SLHA input for Vevacious
  • SPheno
  • SPheno Higgs production
  • SPheno Output
  • SPheno and Monte-Carlo tools
  • SPheno files
  • SPheno mass calculation
  • SPheno threshold corrections
  • Setting up SPheno.m
  • Setting up Vevacious
  • Setting up the SPheno properties
  • Special fields and parameters in SARAH
  • Superpotential
  • Support of Dirac Gauginos
  • Supported Models
  • Supported gauge sectors
  • Supported global symmetries
  • Supported matter sector
  • Supported options for symmetry breaking
  • Supported particle mixing
  • Tadpole Equations
  • The renormalisation scale in SPheno
  • Tree-level calculations
  • Tree Masses
  • Two-Loop Self-Energies and Tadpoles
  • UFO
  • Usage of tadpoles equations
  • Using SPheno for two-loop masses
  • Using auxiliary parameters in SPheno
  • VEVs
  • Vertices
  • Vevacious
  • WHIZARD