Source-linked AI summary
Automatic Calculation of supersymmetric Renormalization Group Equations and Self Energies
Florian Staub
TL;DR
Existing tools and analytic results largely focused on the MSSM, while extensions require additional model implementation. SARAH automates model calculations, renormalization-group equations, loop corrections, and outputs, with results verified against established MSSM, NMSSM, and SPheno calculations.
Problem
Existing computer tools and analytical results mainly support the MSSM, making implementation of new supersymmetric models demanding.
Method
SARAH calculates model-dependent quantities, supports field rotations into mass eigenstates, derives renormalization-group equations, computes one-loop corrections, and generates files for diagram calculators.
Results
The calculated MSSM and NMSSM self-energies, masses, and renormalization-group equations agree with established analytical results and independent numerical routines.
Takeaways & Limitations
SARAH provides a framework for extending automated supersymmetric calculations beyond the MSSM while retaining agreement with established validation results.
Takeaways & Limitations
Loop-correction calculations are performed in the DR-scheme and ’t Hooft gauge, while abelian gauge couplings may require specified GUT normalization.
Abstract
from arXiv · showhide
SARAH is a Mathematica package for studying supersymmetric models. It calculates for a given model the masses, tadpole equations and all vertices at tree-level. Those information can be used by \SARAH to write model files for CalcHep/CompHep or FeynArts/FormCalc. In addition, the second version of SARAH can derive the renormalization group equations for the gauge couplings, parameters of the superpotential and soft-breaking parameters at one and two-loop level. Furthermore, it calculates the one-loop self energies and the one-loop corrections to the tadpoles. SARAH can handle all N=1 SUSY models whose gauge sector is a direct product of SU(N) and U(1) gauge groups. The particle content of the model can be an arbitrary number of chiral superfields transforming as any irreducible representation with respect to the gauge groups. To implement a new model, the user has just to define the gauge sector, the particle, the superpotential and the field rotations to mass eigenstates.
1. Introduction
SARAH addresses the lack of broadly available analytical results and computer support for supersymmetric models beyond the MSSM by automating their calculation and export.
- SARAH extends automated supersymmetric-model analysis beyond the MSSM, whose existing tools and published expressions were more readily available.The package calculates tree-level masses, tadpole equations, interactions, and generates outputs for CalcHep/CompHep and FeynArts/FormCalc.
- A new model requires only its gauge sector, particle content, superpotential, and field rotations to mass eigenstates.
- The first SARAH version already calculated tree-level masses, tadpole equations, and all model interactions.
2. Download, installation and first evaluation
SARAH is installed by placing its package in Mathematica’s application directory, where model definitions and generated calculation outputs are organized across dedicated directories and files.
- The package is extracted into Mathematica’s application directory, with Linux and Windows locations specified.
- SARAH initially contains Package and Models directories, while calculations generate an Output directory for results and diagram-calculator files.
- The archive includes documentation, delivered-model information, a command introduction, and a usage example.
- A model is initialized by a command using the model filename, with examples for the MSSM and a CKM-basis next-to-minimal model.
- The second SARAH version is documented separately through an overview of changes from the first version.
3. Tree-level calculations
SARAH evaluates a model’s tree-level Lagrangian to derive masses, tadpole equations, and vertices, while supporting rotations, eigenstate choices, and exports to diagram calculators.
- Model initialization checks gauge anomalies and charge conservation before calculating the complete tree-level Lagrangian.
- Masses and tadpoles: Mass matrices arise from second derivatives of the Lagrangian, while tadpole equations are first derivatives of the scalar potential with respect to VEVs.
- Masses and tadpoles: MassMatrix and TadpoleEquation provide access to particle mass matrices and VEV-specific tadpole equations.
- Vertices: Vertices are obtained by differentiating the Lagrangian, applying vacuum conditions, and listing separate coefficients for different Lorentz structures.
- Output: SARAH generates model files for CalcHep/CompHep and FeynArts/FormCalc, supporting gauge choices, CP violation, colored-interaction splitting, and loop-diagram workflows.
4. Renormalization Group Equations
SARAH derives one- and two-loop renormalization-group equations for supersymmetric parameters and provides configurable, structured Mathematica outputs for these results.
- SARAH calculates one- and two-loop RGEs for superpotential parameters, soft-breaking terms, gauge couplings, and VEVs.
- Options: Two-loop calculations, variable generation numbers, and matrix-multiplication formatting can be enabled or disabled through options.
- Conventions: GUT normalization can be defined for each U(1) gauge coupling through the parameters file.
- Results: RGE results are stored in arrays for anomalous dimensions, superpotential and soft-breaking parameters, gauge couplings, and VEVs.
- Matrix notation: SARAH normally compresses generation-index sums using MatMul and trace notation, with Adj denoting matrix adjoints.
- Variable generations: With variable generations for q, the hypercharge one-loop β-function includes a NumberGenerations[q] contribution.
5. Loop Corrections
SARAH computes analytical one-loop corrections to tadpoles and particle self-energies in the DR-scheme and ’t Hooft gauge. The calculation is initiated for a chosen eigenstate set and uses abbreviated interaction couplings.
- SARAH calculates analytical one-loop corrections to the tadpoles and self-energy of all particles.
- The calculations are performed in the DR-scheme and ’t Hooft gauge.
- CalcLoopCorrections[Eigenstates] starts the loop-correction calculation for the selected eigenstates.For MSSM models, Eigenstates can be GaugeES or EWSB.
- SARAH abbreviates non-chiral three- and four-point couplings with Cp and chiral three-point couplings with PL or PR variants.
- Self-energies can provide radiative corrections to masses and mass matrices using unrotated external fields and rotated loop fields.
- Loop results are saved as contribution lists and summed expressions, with self-energies stored in SelfEnergy1LoopLis.The summed results can subsequently be written as a PDF using SARAH’s LaTeX output.
1. One-loop tadpoles.
SARAH stores one-loop tadpole corrections both as structured contribution entries and as summed expressions. A chargino-loop example records the internal particles, coupling, diagram type, charge factor, and symmetry weight.
- The chargino contribution to the one-loop tadpoles is accessed with Tadpoles1LoopList[EWSB][[1]].
- The example entry identifies a chargino loop, an unrotated Higgs–chargino vertex, an FFS diagram, charge factor 1, and weight 1/2.
- The corresponding summed tadpole term is 4*sum[gI1,1,2, A0[Mass[bar[Cha[{gI1}]]]^2]* Cp[phid,bar[Cha[{gI1}]],Cha[{gI1}]]*Mass[Cha[{gI1}]]].
2. One-loop self-energies.
SARAH represents one-loop self-energy corrections through contribution lists and summed Passarino–Veltman expressions. Examples cover a down-squark correction from a pseudoscalar Higgs interaction and a Z-boson correction from fermion loops.
- A down-squark matrix correction from a four-point pseudoscalar-Higgs interaction is stored in SelfEnergy1LoopList[EWSB][[1, 12]].
- The down-squark entry uses an SSSS diagram and corresponds to an A0-based term in SelfEnergy1LoopSum[EWSB].
- Z-boson corrections are stored in SelfEnergy1LoopList[EWSB][[15]], with entries containing internal fermions, couplings, diagram type, color factor, and weight.
- The summed Z-boson correction combines H0 and B0 integrals with left- and right-chiral couplings and internal fermion masses.
6. Definition of models
SARAH model files define gauge structure, particle content, superpotential interactions, eigenstates, and field rotations. The framework supports repeated basis changes, effective theories, and the MSSM gauge and field definitions.
- 6. Definition of models: Model information is organized in ModelName.m, particles.m, and parameters.m under PackageDirectory/Models/ModelName/.The model file is necessary for calculating the Lagrangian, while the other files define additional particle and parameter properties.
- 6. Definition of models: The model file specifies the gauge structure, particle content, superpotential, eigenstate names, and properties through DEFINITION statements.
- 6. Definition of models: SARAH supports all SU(N) gauge groups, with each vector superfield definition specifying the group, coupling, dimension, and gauge-index treatment.
- 6. Definition of models: Chiral superfields are defined by component names, generations, superfield names, and representations under the gauge groups.
- 6. Definition of models: Generators, quadratic Casimirs, and Dynkin indices are required for representations used in the kinetic Lagrangian and D-terms.
- 6. Definition of models: The superpotential assigns coefficients, couplings, and involved superfields, while gauge and generation indices are contracted automatically or fixed explicitly.
- 6. Definition of models: SARAH supports rotations, scalar decompositions, and flavor decompositions with user-chosen coefficients and mixing matrices.
- 6. Definition of models: Repeated basis changes preserve intermediate information, enabling calculations for multiple eigenstates and treatments of theories with several symmetry breakings.
7. Verification
SARAH's calculations were verified against established analytical tools and results for the MSSM, NMSSM, and an SU(5)-inspired seesaw model. Its Lagrangian construction covers supersymmetric, soft-breaking, gauge-fixing, ghost, and chiral-field interactions.
- Verification: Tree-level MSSM vertices were checked against FeynArts and CalcHep across more than 5000 generation and parameter combinations.The validation also included several 1 →2 and 2 →2 processes using old and SARAH-generated model files.
- Verification: SARAH's one-loop MSSM self-energies agree with published expressions, while corrected masses agree with SPheno and NMSSM checks.The NMSSM numerical comparison was in complete agreement with routines from the authors of the reference calculation.
- Verification: SARAH's MSSM one- and two-loop RGEs agree analytically with published results and numerically with SPheno.For the NMSSM, an apparent two-loop discrepancy in Aλ was independently confirmed as SARAH's result.
- Lagrangian construction: The Lagrangian calculation combines gauge structure and the superpotential to derive supersymmetric interactions, soft-breaking terms, and gauge-fixing and ghost interactions.Chiral-field interactions include two-fermion-one-scalar terms and four-scalar F-terms.
- Lagrangian construction: Gauge interactions include covariant-derivative terms, D-terms, gaugino–matter interactions, and three- and four-gauge-boson self-interactions.The gauge-boson self-interactions arise from the non-abelian field-strength structure.
- Lagrangian construction: Gauge fixing breaks general gauge invariance for quantization and introduces corresponding ghost interactions, with Rξ choices for unbroken and broken symmetries.The text identifies the unitary gauge and Feynman-'t Hooft gauge as common choices.
Appendix C. Calculation of Group Factors
SARAH derives representation data needed for kinetic terms, D-terms, and RGEs, then uses generic supersymmetric formulas to calculate loop-level quantities and corrected masses.
- Representations: SARAH supports chiral superfields in any irreducible SU(N) representation, not only the fundamental representation.When dimension alone is ambiguous, Dynkin labels distinguish representations.
- Representations: Representation analysis derives generators, quadratic Casimirs, Dynkin indices, Dynkin labels, and covariant-index counts from Young tableaux.The procedure uses the hook formula, highest weights, fundamental weights, the Weyl formula, and tensor-product generators.
- Representations: CheckIrrepSUN returns representation dimension, covariant and contravariant indices, C2(r), I(r), and highest-weight Dynkin labels.For SU(3)'s fundamental representation, the reported vector is {3, 1, 0, 4/3, 1/2, {1, 0}}.
- Renormalization-group equations: The VEV RGEs are proportional to the anomalous dimensions of the chiral superfields whose scalar components acquire VEVs.The formulas use quadratic Casimirs, the adjoint-representation dimension, and t = ln Q.
- One-loop quantities: One-loop self-energies and tadpole corrections are assembled from generic fermion, scalar, vector, and mixed loops with symmetry, charge, and reality factors.The resulting self-energies feed one-loop scalar, vector-boson, and fermion masses or mass matrices.
- One-loop quantities: The loop calculation uses DR-scheme scalar functions and basis integrals such as A0 and B0, with vector self-energies using additional defined functions.Integrals are regularized in n = 4 −2ε dimensions at renormalization scale Q.
Appendix G. Changes in comparison to version 1 of SARAH
Version 2 of SARAH adds new physical calculations, broadens model support, and introduces changes to model definitions and outputs. It also includes several new supersymmetric models and improved access to generated information.
- New physical output: SARAH 2 adds one- and two-loop renormalization group equations, one-loop self-energies, and one-loop corrected tadpoles.
- Model capabilities: The package supports all irreducible representations of chiral superfields and includes representation theory for SU(N) gauge groups.
- Model capabilities: Implicit charge indices are no longer restricted to SU(2)L, and SARAH can check charge conservation within a model.
- Changes in definition of models: Model definitions gain structured DEFINITION statements, global parameter and particle properties, rotated additional interactions, particle phases, and flavor-eigenstate decomposition for multigeneration fields.
- Changes in output: Output changes include faster CalcHep/CompHep and LATEX generation, running couplings in CalcHep files, optional four-scalar splitting suppression, and improved LATEX typesetting.
- New models: The release adds MSSM with trilinear R-parity violation, the singlet-extended MSSM, UMSSM, secluded UMSSM, and nMSSM models.