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SARAH 4: A tool for (not only SUSY) model builders

Florian Staub

arXiv:1309.7223v2hep-ph

TL;DR

SARAH 4 addresses SARAH’s prior restriction to chiral superfields by extending comparable automation to non-SUSY models. It adds general two-loop RGEs, broader model-file and spectrum-generation interfaces, improved Lie-group handling, and validation against PyR@TE.

  • Problem

    SARAH previously supported only chiral superfields in the matter sector, limiting the study of non-SUSY component fields.

  • Method

    SARAH 4 introduces separate fermion and scalar field definitions, general two-loop quantum-field-theory beta functions, interfaces to multiple computational tools, and Susyno-based Lie-group routines.

  • Results

    The two-loop RGEs agree fully with the independent PyR@TE code, while SARAH 4 supports non-SUSY calculations, color sextets, global symmetries, and Vevacious checks across SUSY and non-SUSY models.

  • Takeaways & Limitations

    SARAH 4 brings non-SUSY models closer to the automation previously available for SUSY models and connects their outputs to spectrum, observable, decay, and Monte Carlo tools.

  • Takeaways & Limitations

    SARAH’s component-field definition was previously restricted to chiral superfields, although the new version removes this restriction while retaining the older Fields format for backward compatibility.

Abstract

from arXiv · show

We present the new version of the Mathematica package SARAH which provides the same features for a non-supersymmetric model as previous versions for supersymmetric models. This includes an easy and straightforward definition of the model, the calculation of all vertices, mass matrices, tadpole equations, and self-energies. Also the two-loop renormalization group equations for a general gauge theory are now included and have been validated with the independent Python code PyR@te. Model files for FeynArts, CalcHep/CompHep, WHIZARD and in the UFO format can be written, and source code for SPheno for the calculation of the mass spectrum, a set of precision observables, and the decay widths and branching ratios of all states can be generated. Furthermore, the new version includes routines to output model files for Vevacious for both, supersymmetric and non-supersymmetric, models. Global symmetries are also supported with this version and by linking Susyno the handling of Lie groups has been improved and extended.

Program Summary

SARAH 4 extends the package from primarily supersymmetric model building to non-supersymmetric models while broadening symmetry, Lie-group, output, and vacuum-analysis support. It automates model definitions and calculations from field-theory ingredients through spectrum, observable, Monte Carlo, and vacuum-stability interfaces.

  • Automated calculations: The package calculates vertices, mass matrices, tadpole equations, self-energies, and two-loop RGEs for general quantum field theories.The non-SUSY RGE routines use general quantum-field-theory formulae rather than SUSY-specific expressions.
  • Generated outputs: Generated outputs connect models to FeynArts, CalcHep/CompHep, WHIZARD, UFO-based tools, and SPheno calculations of spectra and observables.SPheno source generation supports mass spectra, precision observables, decay widths, and branching ratios.
  • Symmetries and groups: Global symmetries and broader Lie-group support are enabled through Susyno, including SO(N), Sp(2N), and exceptional groups.Susyno supplies representation matrices, Clebsch–Gordan coefficients, and gauge-group constants.
  • Vacuum analysis: SARAH 4 generates Vevacious model files for SUSY and non-SUSY models, enabling global-minimum checks across a wide class of models.The SARAH–SPheno–Vevacious interface can account for charged or colored scalar VEVs and support metastable-vacuum lifetime calculations with Cosmotransitions.
  • Model support: SARAH 4 supports non-supersymmetric models through separate superfield, fermion-field, and scalar-field definitions.This removes the previous matter-sector restriction to chiral superfields and brings non-SUSY model definition closer to the established SUSY workflow.

Conventions to define global charges.

SARAH 4 extends model-definition conventions to global charges, non-fundamental representations, and interaction terms. It supports component-field definitions, explicit representation dimensions, simplified potential syntax, and automated index contractions.

  • Global charges: Global-symmetry charges can be assigned uniformly to a superfield and its components or separately to the superfield, scalar, and fermionic components.For chiral superfields, the three entries denote superfield, scalar, and fermionic charges; vector-superfield entries distinguish gauginos and gauge bosons.
  • Matter fields: SARAH 4 supports separate definitions of superfields, fermion fields, and scalar fields, making non-SUSY matter definitions as straightforward as SUSY definitions.Scalar and fermion syntax follows the same general conventions, while superfield components receive systematic names.
  • Representations: Susyno enables non-fundamental irreducible representations to be handled as vectors with their appropriate dimensions rather than as tensor products of fundamental representations.For unbroken gauge groups, this representation is especially useful for color sextets and enables UFO model files for MadGraph 5.
  • Interactions: The superpotential syntax accepts an intuitive sum of terms instead of requiring a list of terms.The same sum-of-terms approach is used for potentials in non-SUSY models, with optional automatic addition of Hermitian-conjugate interactions.
  • Index contractions: SARAH automatically adds generation indices, contracts charges with Kronecker deltas, antisymmetric tensors, or Clebsch–Gordan coefficients, and can calculate numerical CGCs through Susyno.Users may inspect the contractions and override the default contraction when multiple index contractions are possible.

Symmetry of parameters.

SARAH 4 automates parameter-symmetry checks and extends renormalization-group calculations to general quantum field theories. The implementation includes kinetic mixing and gauge dependence of running vacuum expectation values.

  • Parameter symmetries: SARAH automatically tests parameters involving repeated fields for symmetric, antisymmetric, Hermitian, or anti-Hermitian structure.It uses these properties to simplify expressions during subsequent calculations, replacing earlier manual symmetry declarations.
  • General-theory RGEs: SARAH 4 calculates one- and two-loop RGEs for a general quantum field theory with the same precision used for SUSY models.The calculation includes the most general CP and flavor structure described for the preceding SUSY RGE calculations.
  • RGE effects: Non-SUSY RGE calculations include kinetic mixing and gauge dependence in the running vacuum expectation values.These effects are incorporated using the corresponding generic rules and results cited by the paper.
  • RGE output: The RGE output is organized into arrays containing each parameter, its one-loop beta function, and its two-loop beta function.For non-SUSY models, separate structures store anomalous dimensions, gauge-coupling beta functions, interaction beta functions, and VEV beta functions.

Examples.

SARAH 4 extends automated model-building workflows to non-SUSY theories, including general two-loop RGEs, spectrum generation, observable calculations, and interfaces to simulation tools.

  • Examples: SARAH computes non-SUSY RGEs using general quantum-field-theory formulae, including kinetic mixing and gauge-dependent running vacuum expectation values.
  • Examples: SARAH and PyR@TE showed full agreement for non-SUSY two-loop RGEs, with additional checks against available literature.The comparison also identified mistakes in some references.
  • Examples: RunRGEs numerically evaluates one- or two-loop running from supplied initial parameters and returns an InterpolationFunction from NDSolve.The example evolves gauge couplings from 1000 GeV to 10^16 GeV at one loop.
  • Examples: SARAH generates Fortran source for SPheno, enabling mass spectra, decay widths, branching ratios, and precision observables for model-specific calculations.These capabilities are inherited from the spectrum-generator workflow and are available for non-SUSY models with corresponding adjustments.
  • Examples: SPheno can be configured either as a low-scale version or with RGE running between electroweak, renormalization, and higher scales.The high-scale setup supports scale-dependent boundary conditions and a condition determining when the running stops.
  • Examples: SPheno output can be exchanged directly with CalcHep, MadGraph, or WHIZARD for Monte Carlo studies.The workflow uses the generated model files together with the SPheno spectrum information.

6. Vevacious

SARAH 4 adds Vevacious output for studying global minima of one-loop effective potentials in models with specified non-zero vacuum expectation values.

  • 6. Vevacious: Vevacious checks the global minimum of a model’s one-loop effective potential while allowing a specified set of non-zero VEVs.Its inputs include tadpole equations, the polynomial potential, and mass matrices assuming allowed VEVs are non-zero.
  • 6. Vevacious: SARAH generates Vevacious model files with MakeVevacious[] after the model is initialized.
  • 6. Vevacious: The output supports complex-parameter selection, parameter exclusion, custom output filenames, and DR or MS renormalization schemes.The default scheme is DR for SUSY models and MS for non-SUSY models.
  • 6. Vevacious: Parameters can be treated selectively in Vevacious output so subdominant terms may be neglected to reduce evaluation time.The resulting speed-up has not yet been quantified.

7. Conclusion

SARAH 4 broadens automated model-building support for non-SUSY theories while extending RGE, spectrum, Monte Carlo, symmetry, Lie-group, and vacuum-stability workflows.

  • 7. Conclusion: SARAH 4 implements complete two-loop RGEs for general quantum field theories and supports SPheno calculations for non-SUSY spectra and observables.
  • 7. Conclusion: The SPheno information can be passed to Monte Carlo tools, while Susyno generalizes Lie-group handling and enables models with color sextets for such tools.
  • 7. Conclusion: Global symmetries and Vevacious interfaces are supported for broad classes of SUSY and non-SUSY models.Vevacious enables global-minimum checks across these models.

Appendix A. Other Improvements in SARAH 4

SARAH 4 broadens model definition and tadpole-solving capabilities, adding flexible VEV specifications and numerical solutions while warning that numerical searches may find only one, potentially non-global minimum.

  • VEV definitions: SARAH 4 supports VEV alignment, allowing only selected scalar generations to acquire vacuum expectation values.This is particularly useful for Vevacious output.
  • VEV definitions: Complex VEVs can be defined with separate real and imaginary components rather than only through a phase.The new representation introduces separate real and imaginary VEV parameters.
  • Numerical tadpole solutions: Numerical tadpole solving removes the requirement that tadpole equations have an analytical solution.The numerical solution uses a Broydn method and selects the solution closest to a supplied starting point.
  • Numerical tadpole solutions: Initialization parameters provide starting values for numerical tadpole solving, such as µ -> m0 and Bµ -> m0^2.These values are used to initialize the numerical routines when analytical solving is skipped.
  • Numerical tadpole solutions: Numerical tadpole solving can find only one minimum, which may not be the global minimum; Vevacious can check globality or metastable-vacuum lifetimes.The limitation is especially relevant when solving for VEVs entering equations with third power.

Appendix A.4. Running VEVs including Gauge Dependence

SARAH 4 calculates gauge-dependent one- and two-loop running VEV equations alongside loop-correction and model-analysis routines, extending renormalization-group output for general calculations.

  • Running VEVs: SARAH 4 includes gauge dependence in the one- and two-loop beta functions of all model VEVs.The gauge parameter Xi is defined so that Xi = 0 is Landau gauge and Xi = 1 is Feynman gauge.
  • Running VEVs: The running-VEV equations incorporate Yukawa and gauge-coupling structures at one and two loops.The displayed MSSM example for vd contains gauge couplings, Yukawa traces, and Xi-dependent terms.
  • Loop corrections: SARAH calculates one-loop corrections to one- and two-point functions for one-loop mass-spectrum calculations.These corrections are computed in ’t Hooft gauge, with the DR scheme used so far.
  • RGE outputs: The package exposes beta-function outputs for gauge couplings, scalar masses, soft-breaking parameters, and gaugino masses.Separate entries are available for bilinear and linear soft terms, scalar squared masses, Majorana masses, Dirac masses, and gauge couplings.
  • Loop corrections: Loop-correction results are stored as summed or separately listed tadpole and self-energy contributions.The outputs are provided through Tadpoles1LoopSums, SelfEnergy1LoopSum, Tadpoles1LoopList, and SelfEnergy1LoopList.

Appendix B.3. Model Files and LATEX Output

SARAH 4 exports model files and derived expressions to several computational formats, supports SPheno code generation, and includes an expanded library of supersymmetric benchmark models.

  • Model-file export: SARAH 4 writes model files for UFO, CalcHep/CompHep, WHIZARD/O’Mega, FeynArts/FormCalc, and Vevacious.The corresponding commands support format-specific options such as excluded vertex classes, counterterms, gauge choices, and coupling limits.
  • SPheno and observables: SPheno source-code generation enables calculations of loop-corrected spectra, precision observables, decay widths, and branching ratios.Generated output can also connect to HiggsBounds and HiggsSignals.
  • LaTeX output: All derived information, including mass matrices, vertices, RGEs, and loop corrections, can be exported to human-readable LaTeX.Options can add Feynman diagrams and SARAH-specific field and parameter information.
  • Supported models: The model library includes standard, singlet-, triplet-, and U(1)-extended supersymmetric models with variants for flavor, CP, R-parity, seesaw, and Dirac-gaugino structures.Listed examples include NMSSM, UMSSM, R-parity-violating models, seesaw extensions, and Dirac-gaugino models.

Appendix C.2. Non-Supersymmetric Models

SARAH 4 includes several non-supersymmetric Standard Model extensions, covering inert, B-L, scalar-octet, two-Higgs-doublet, and singlet-extended scenarios.

  • Standard and extended models: The non-SUSY model set includes the Standard Model and its CKM-basis version.These provide baseline non-supersymmetric model definitions in the library.
  • Scalar-sector extensions: SARAH 4 includes inert Higgs doublet and two-Higgs-doublet models.These models extend the scalar sector without requiring supersymmetry.
  • Gauge and neutrino extensions: The library includes B-L-extended Standard Models with and without an inverse seesaw.These entries represent both the gauge extension and its inverse-seesaw variant.
  • Scalar-sector extensions: SARAH 4 also supports Standard Model extensions by a scalar color octet and by a singlet.These entries add colored or gauge-singlet scalar degrees of freedom, respectively.

Appendix D. The Model File of the NMSSM in SARAH 4

The NMSSM model file defines global symmetries, gauge groups, superfields, a superpotential, and particle-content mappings for SARAH 4.

  • Gauge structure: It defines the U(1), SU(2), and SU(3) gauge groups with couplings g1, g2, and g3.The entries label the groups as hypercharge, left, and color, respectively.
  • Superpotential: The superpotential contains Yukawa interactions Yu, Yd, and Ye together with Lambda and Kappa singlet-Higgs terms.The displayed expression includes Yu u.q.Hu, Yd d.q.Hd, Ye e.l.Hd, Lambda Hu.Hd.s, and Kappa s.s.s/3.

Appendix E. The Model File of the SM in SARAH 4

The SM model file defines its gauge and matter content, Higgs sector, scalar potential, Yukawa interactions, and mappings to physical states.

  • Gauge structure: The file specifies the U(1), SU(2), and SU(3) gauge groups with couplings g1, g2, and g3.The groups are labeled hypercharge, left, and color.
  • Matter fields: The matter-field declarations include quark and lepton multiplets with three generations and their Standard Model gauge quantum numbers.The entries define q, l, d, u, and e, including their component fields, hypercharges, weak representations, and color representations.
  • Scalar sector: The scalar sector contains one Higgs doublet H with charged and neutral components and vacuum expectation value v.The neutral field is decomposed into v, Ah, and hh components with the displayed normalization factors.
  • Interactions: The scalar potential and Yukawa Lagrangian are specified through Mu2, Lambda, Yd, Ye, and Yu interactions.LagNoHC gives the Higgs mass and quartic terms, while LagHC contains down-type, lepton, and up-type Yukawa couplings.
  • Particle content: The particle-content mappings associate gauge bosons, Higgs states, and fermions with their physical mass eigenstates.The mappings include photon, Z, W, Higgs, pseudoscalar, and flavor-state assignments.
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