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FlexibleSUSY 2.0: Extensions to investigate the phenomenology of SUSY and non-SUSY models

Peter Athron, Markus Bach, Dylan Harries, Thomas Kwasnitza, Jae-hyeon Park, Dominik Stöckinger, Alexander Voigt, Jobst Ziebell

arXiv:1710.03760v2hep-ph

TL;DR

Public software has been scarce for exploring many models beyond the familiar SUSY cases, making model-specific implementation and testing necessary. FlexibleSUSY 2.0 adds extensions for broader models, observables, and calculations. The paper reports a substantial expansion of high-precision BSM calculations, supported by correctness testing and speed tests for large-scale scans.

  • Problem

    For many models beyond the MSSM and NMSSM, few public software packages existed, so parameter-space studies required researchers to write and test private code.

    Source: §1 · p. 7

  • Method

    FlexibleSUSY 2.0 extends its generated spectrum calculators with non-SUSY and complex-parameter support, new muon and W-mass calculations, a new boundary-value solver, and a hybrid EFT/fixed-order Higgs-mass calculation.

    Source: New version program summary · p. 3, §12 · p. 87

  • Results

    The updates substantially extend high-precision BSM calculations; the system was tested against literature and other generators, and speed tests demonstrated effectiveness in large-scale scans.

    Source: §12 · pp. 87–88

  • Takeaways & Limitations

    The examples include efficient large-scale scans and low-energy effective theories of SUSY models with complicated matching-scale boundary conditions.

    Source: §12 · p. 87

    Models must remain perturbative between boundary scales, and automatic low-energy matching requires the SM gauge group as a factor.

    Source: §11 · p. 86

Abstract

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We document major new features and improvements of FlexibleSUSY, a Mathematica and C++ package with a dependency on the external package SARAH, that generates fast and precise spectrum generators. The extensions presented here significantly increase the generality and capabilities of the FlexibleSUSY package, which already works with a wide class of models, while maintaining an elegant structure and easy to use interfaces. The FlexibleBSM extension makes it possible to also create spectrum generators for non-supersymmetric extensions of the Standard Model. The FlexibleCPV extension adds the option of complex parameters to the spectrum generators, allowing the study of many interesting models with new sources of $CP$ violation. FlexibleMW computes the decay of the muon for the generated model and thereby allows FlexibleSUSY to predict the mass of the $W$ boson from the input parameters by using the more precise electroweak input of $\{ G_F, M_Z, \alpha_{\text{em}} \}$ instead of $\{ M_W, M_Z, \alpha_{\text{em}} \}$. The FlexibleAMU extension provides a calculator of the anomalous magnetic moment of the muon in any model FlexibleSUSY can generate a spectrum for. FlexibleSAS introduces a new solver for the boundary value problem which makes use of semi-analytic expressions for dimensionful parameters to find solutions in models where the classic two-scale solver will not work such as the constrained E$_6$SSM. FlexibleEFTHiggs is a hybrid calculation of the Higgs mass which combines the virtues of both effective field theory calculations and fixed-order calculations. All of these extensions are included in FlexibleSUSY 2.0, which is released simultaneously with this manual.

New version program summary

FlexibleSUSY 2.0 extends the program with new models, observables, and algorithms. Its revisions add non-supersymmetric and complex-parameter support, new muon and W-mass calculations, a semi-analytic solver, and an improved hybrid Higgs-mass calculation.

New version program summary

  • Version 2.0 supersedes the 2015 FlexibleSUSY release.Source: New version program summary · p. 3
  • FlexibleSUSY 2.0 adds support for non-supersymmetric models and complex parameters, alongside new calculations and algorithms.The listed revisions include FlexibleBSM, FlexibleCPV, FlexibleMW, FlexibleAMU, FlexibleSAS, and an improved hybrid Higgs-mass calculation.Source: New version program summary · p. 3
  • The new FlexibleMW and FlexibleAMU extensions calculate the W-boson mass from muon decay and the muon's anomalous magnetic moment, respectively.Source: New version program summary · p. 3
  • FlexibleSAS adds a semi-analytic boundary-value-problem solver, and FlexibleEFTHiggs improves the hybrid Higgs-mass calculation.Source: New version program summary · p. 3

(FlexibleEFTHiggs).

FlexibleEFTHiggs addresses the calculation of mass spectra and observables subject to constraints at multiple scales. Its solution uses nested iteration and numerical minimization, with perturbativity and parameter-support restrictions.

(FlexibleEFTHiggs).

  • The generated spectrum calculation determines masses, mixings, and observables while satisfying constraints at two or more scales linked by coupled renormalization-group equations.Source: (FlexibleEFTHiggs) · p. 3
  • A nested iterative algorithm and numerical minimization of the Higgs potential provide the solution method.Source: (FlexibleEFTHiggs) · p. 3
  • Couplings must remain perturbative between boundary scales; rank-3 tensor-like parameters are unsupported, and automatic Standard Model-like couplings require an SM gauge-group factor.The generated code can be adapted and extended to address restrictions in scope.Source: (FlexibleEFTHiggs) · p. 3

1. Introduction

The introduction motivates tools for exploring models beyond the most familiar extensions, where public software has been scarce and private implementations require substantial coding and testing. FlexibleSUSY 2.0 broadens the models and observables it supports through a set of modular extensions.

1. Introduction

  • The motivation for broader model exploration includes unresolved dark matter, unexplained gauge structure and fractional charges, and the weak-scale stability problem.The introduction identifies supersymmetry as a prominent response to the weak-scale stability problem.Source: §1 · p. 6
  • FlexibleSUSY uses SARAH-derived model expressions and translates them into C++ routines embedded in a boundary-value-problem solver.The generated expressions include renormalization-group equations, self-energies, tadpoles, mass matrices, and electroweak-symmetry-breaking conditions.Source: §1 · p. 6
  • For many models beyond the MSSM and NMSSM, public software has been scarce, leaving researchers to write and test private code to explore parameter space.The text notes that bugs are more likely in private tools than in well-tested public ones.Source: §1 · p. 7
  • The first release had limitations that restricted the models, phenomenology, and calculation precision that could be studied.The paper presents version 2.0 as an extension of the program's model and observable coverage.Source: §1 · pp. 7–8
  • FlexibleSUSY 2.0 expands the program to non-supersymmetric models, complex parameters, and constrained non-minimal SUSY models that the two-scale solver cannot handle.The introduction also lists predictions of the W mass and calculations of the muon's anomalous magnetic moment and electric dipole moments.Source: §1 · pp. 7–8
  • The modular extensions cover non-SUSY models, muon observables, CP violation, W-mass prediction, a new solver, and a hybrid EFT/fixed-order Higgs-mass calculation.The paper organizes the extensions so readers can consult particular modules independently.Source: §1 · p. 8

2. Quick start

The quick-start material lists the requirements and steps for building and using a spectrum generator, then explains how to adapt FlexibleSUSY for alternative models. It illustrates both generated outputs and model-file customization.

2.1. Requirements

  • Building a generator requires Mathematica, SARAH, C++11 and FORTRAN compilers, and supporting libraries.The listed requirements also include LoopTools, Eigen, Boost, and the GNU scientific library.Source: §2.1 · p. 8

2.2. Downloading

  • A predefined model's generator is created by generating its model directory, configuring the build, and compiling; it can then read SLHA input and output a spectrum.The quick start also shows a Mathematica interface for calling a generator and accessing the returned mass spectrum and running parameters.Source: §2.2 · pp. 9–10

2.3. Spectrum generators for alternative models

  • For an alternative model, users need SARAH model files and can modify a FlexibleSUSY model file to set separate GUT-scale Aλ and Aκ values.The example specifies those values in the SLHA input file's EXTPAR block using entries 63 and 64.Source: §2.3 · pp. 10–11

3. Mathematica interface

FlexibleSUSY’s Mathematica interface lets users configure a model’s spectrum generator, calculate spectra and observables, and use handles to manage parameter points. It also supports spectrum export and parameter scans.

3. Mathematica interface

  • Users load the model’s library and open a handle with settings, Standard Model inputs, and model-specific inputs before calculating a spectrum or observables.Unspecified model input parameters default to zero.Source: §3 · pp. 11–13
  • Handles reference a parameter point, its spectrum, and observables, and allow separate Mathematica kernels to calculate spectra for different points in parallel.The interface can also be used for parameter scans and uncertainty estimates.Source: §3 · pp. 12, 16
  • The spectrum output includes pole masses and running masses and parameters at the chosen output scale, using SARAH conventions rather than SLHA conventions.Pole masses and mixing matrices carry the Pole[] head; particle masses are non-negative and mixing matrices are generally complex.Source: §3 · pp. 13–14
  • The observables function returns replacement rules, including the anomalous magnetic moment of the muon and effective Higgs couplings to photons and gluons.The Higgs couplings are returned for all members when the relevant CP-even or CP-odd states are multiplets.Source: §3 · pp. 15–16
  • Users can print a calculated spectrum in an SLHA-compatible format with the model’s ToSLHA function.The interface documentation also points to examples for parameter scans and uncertainty estimates.Source: §3 · p. 16

4. FlexibleSUSY model file extensions

This section describes extensions to FlexibleSUSY model files, adding configurable higher-order contributions and more expressive boundary conditions, including high-scale matching to UV completions. Their use and applicability depend on model-specific requirements.

4.1. Model-specific higher-order contributions

  • FlexibleSUSY enables higher-order running and Higgs-mass contributions through model-file switches, but their validity depends on model requirements and out-of-scope use may omit contributions.The SM extensions include 3-loop beta-function running and 3-loop QCD thresholds for extracting the MS top Yukawa and strong coupling.Source: §4.1 · p. 17, §4.1.1 · pp. 20–22

4.1.1. SM-specific higher-order contributions

  • The SM 3-loop Higgs-mass calculation is complete only at O(α_t α_s^2); O(α_t^3) and O(α_t^2 α_s) contributions are incomplete, and some momentum-dependent and subdominant terms are omitted.The missing O(α_t^2 α_s) top-Yukawa matching corrections are unavailable in FlexibleSUSY, while evaluating at Q^2 = m_t^2 makes neglected O(α_t^3) terms vanish.Source: §4.1.1 · p. 19
  • In HSSUSY, 3-loop top-Yukawa corrections lower the Higgs mass by up to 500 MeV, while 2- and 3-loop strong-coupling thresholds lower it by up to 40 MeV and around 3 MeV, respectively.The explicit 3-loop Standard Model Higgs-mass contribution instead shifts the prediction upward by around 30 MeV.Source: §4.1.1 · pp. 20, 22, 23, Figure 1 · p. 20

4.1.2. (N)MSSM-specific higher-order contributions

  • FlexibleSUSY supports dominant 2-loop (N)MSSM Higgs-mass contributions, but NMSSM terms of O((α_t + α_b)^2 + α_τ^2) are available only in the MSSM limit.Consistent O((α_t + α_b)α_s) treatment beyond that limit is unavailable because NMSSM-specific 2-loop running-VEV corrections are not implemented.Source: §4.1.2 · p. 23
  • Himalaya 3-loop corrections can be included for a real MSSM-like model with two CP-even Higgs bosons, alongside recommended 3-loop running and 2-loop SUSY-QCD top-Yukawa and strong-coupling thresholds.The corrections are available in DR or MDR schemes; the pure-DR calculation requires the 2-loop SUSY-QCD correction to the MSSM DR top Yukawa coupling.Source: §4.1.2 · pp. 24–25
  • The 2-loop SUSY-QCD α_s threshold correction raises the Higgs mass by up to 2 GeV and about +1 GeV near MS ≈ 2 TeV with maximal stop mixing; FlexibleSUSY and SOFTSUSY implementations agree exactly.The potentially multi-GeV 4-loop effect also indicates substantial fixed-order uncertainty for multi-TeV stop masses.Source: §4.1.2 · pp. 27–29, Figure 3 · p. 29
  • MSSM 3-loop beta functions are supported, with Higgs pole-mass effects of a few 100 MeV and a 4–5× increase in spectrum-generator runtime.FlexibleSUSY’s expressions were numerically found to agree exactly with SOFTSUSY 3.7.0.Source: §4.1.2 · p. 27, Figure 3 · p. 29
  • Full 2-loop SUSY-QCD corrections to MSSM DR Yukawa couplings shift the Higgs mass negatively by several GeV; with explicit 3-loop terms, the prediction falls 1–2 GeV versus classic 2-loop, or 1–3 GeV for nonzero Xt.The combined prediction is closer to the pure EFT calculation; above about 2 TeV, new logarithms from the top-Yukawa correction contribute to the shift.Source: §4.1.2 · pp. 26–28

4.1.3. Split-MSSM-specific higher-order contributions

  • FlexibleSUSY adds a known part of the split-MSSM 3-loop Higgs contribution; other models using it must contain a single Higgs boson and a gluino.For consistency, the calculation also includes the 2-loop gluino contribution to the MS top mass.Source: §4.1.3 · p. 30

4.2. New features for definition of boundary conditions

  • Boundary-condition expressions can now include more complex functions, branches, and symbols, enabling high-scale matching such as HSSUSY’s 1- and 2-loop SM–MSSM Higgs-quartic matching.The available special functions and symbols are listed in Table 3.Source: §4.2 · p. 30, Table 3 · p. 31

5. FlexibleBSM extension

This section describes how FlexibleBSM extends FlexibleSUSY spectrum generation to non-supersymmetric models, including its setup, coupling determination, generated-code structure, and applications. It also presents HSSUSY and FlexibleBSM examples and tests addressing Higgs-mass uncertainties, agreement with reference calculations and other spectrum generators, and the correctness of implemented RGEs and top-Yukawa corrections.

5. FlexibleBSM extension

  • Non-SUSY spectrum generators share the SUSY generators’ core features: automatic low-energy coupling calculation, up to three boundary conditions, 2-loop running, and full 1-loop pole masses with BSM contributions.Some non-SUSY models support additional 2- and 3-loop corrections; 3-loop running is available in the SM and MSSM.Source: §5 · p. 32

5.1. Setting up a FlexibleBSM model

  • For non-SUSY models, the interface matches the SUSY case, but parameters use the MS scheme; gauge-dependent running masses and VEVs are defined in Feynman gauge with ξ_i = 1.FlexibleSUSY automatically determines low-energy gauge and Yukawa couplings in the MS scheme.Source: §5.1 · p. 32

5.2. Determination of the MS gauge and Yukawa couplings

  • When the SM gauge group is a subgroup, FlexibleSUSY determines low-energy gauge couplings from SM inputs and Yukawa couplings from running fermion masses; otherwise, gauge couplings can be supplied as inputs.The weak mixing angle can be determined from GF and MZ with full 1-loop and leading 2-loop SM corrections, or from running W and Z masses; SARAH does not currently generate beta functions for scalar tadpole terms, so they do not run.Source: §5.2 · pp. 32–33

5.3. Structure of the generated code

  • Generated code separates dimensionless parameters into <model>_susy_parameters and dimensionful parameters into <model>_soft_parameters, allowing dimensionless RGEs to be integrated independently.The Beta_function interface integrates RGEs to a requested scale with an adaptive Runge-Kutta algorithm.Source: §5.3 · pp. 35–36

5.4. Application: High-scale MSSM with light Higgs sector

  • In the THDM-II EFT for a high-scale MSSM with a light Higgs sector, FlexibleSUSY reproduces the results shown in Ref. for the lightest CP-even Higgs pole mass.The setup integrates out heavy SUSY particles at MS, fixes the THDM-II quartic couplings using full 1-loop and leading 2-loop MSSM threshold corrections, and evaluates the Higgs pole mass at 1 loop.Source: §5.4 · pp. 36, 38, Figure 5 · p. 38

5.5. Application: High-scale MSSM (HSSUSY)

  • FlexibleSUSY 2.0 HSSUSY includes 2-loop Higgs-quartic threshold corrections for general SUSY spectra, including generalized O(α_t^2) terms and new bottom- and tau-Yukawa contributions.HSSUSY also uses known 3-loop SM beta functions, 3-loop corrections to the running top Yukawa and strong coupling, and up to 3-loop Higgs pole-mass corrections.Source: §5.5 · p. 39
  • The supplied passage reports that h is around or below 500 MeV, supporting this value but not the stated variation methods or classification within the full EFT-calculation uncertainty.Source: §5.5 · p. 43
  • The estimated EFT uncertainty falls below 150 MeV for MS ≳ 2 TeV and decreases as MS increases.The estimate varies ΔEFT from 0 to 1 to assess terms of O(v^2/MS^2) omitted by the pure-EFT calculation.Source: §5.5 · pp. 43, 45

5.6. Tests and comparisons with other spectrum generators

  • Analytic tests found exact agreement between SARAH 4.5.3 SplitMSSM RGEs and Ref., and numeric equality between FlexibleSUSY and SOFTSUSY 3.7.0 3-loop MSSM β functions.Source: §5.6 · pp. 45–46
  • HSSUSY agrees excellently with SusyHD 1.0.1, with differences of O(100 MeV) arising from differing yt determinations and Higgs pole-mass procedures, plus HSSUSY’s additional αb and ατ corrections.HSSUSY includes generalized 2-loop threshold corrections involving αb and ατ that SusyHD does not.Source: §5.5 · p. 40, §5.6 · p. 45
  • The effective THDM reproduces Ref. and MhEFT 1.0 results for scenarios with heavy Higgsinos and gauginos.The paper points to Figure 5 as an example.Source: §5.6 · p. 45
  • The 2-loop QCD top-Yukawa threshold correction’s scale-dependent part was checked for the absence of large logarithms when matching the SM to the MSSM.Source: §5.6 · p. 46
  • Analytic checks established renormalization-scale invariance of the SM top-quark pole mass at 3-loop QCD level, validating scale-dependent 2-loop and 3-loop QCD corrections to the MS top Yukawa coupling.Source: §5.6 · p. 46

6. FlexibleAMU extension

FlexibleAMU adds a model-general calculation of the muon’s BSM anomalous magnetic moment, combining one-loop BSM diagrams with a leading-logarithmic two-loop QED contribution. Tests against GM2Calc and SPheno find typical differences of 1–10%, with larger discrepancies in scale-sensitive scenarios.

6. FlexibleAMU extension

  • The calculation sums one-loop diagrams with at least one non-SM particle and adds a universal leading-logarithmic two-loop QED contribution.The one-loop calculation is performed in the MS/DR scheme at MLCP, the running mass of the lightest electrically charged BSM particle contributing to the one-loop result.Source: §6 · pp. 46–47

6.1. Choosing FlexibleAMU in the model file

  • Users enable the muon anomalous-moment observable in an output block; GM2Calc is an alternative for two-loop MSSM calculations without flavor violation.GM2Calc is incorporated as an addon, and its uncertainty can also be requested.Source: §6.1 · pp. 47–48

6.3. Tests and comparisons with other calculations

  • 1–10% deviation from GM2Calc is typical, while FlexibleAMU and SPheno differ by up to 10%.The codes use different schemes, renormalization scales, and two-loop corrections; for some scenarios, scale sensitivity makes predictions much less precise.Source: §6.3 · p. 49
  • Around 66% scale variation for BM4′ signals an imprecise prediction, associated with scale-sensitive smuon masses and missing two-loop corrections.GM2Calc avoids this large scale uncertainty through on-shell renormalization.Source: §6.3 · p. 49

7. FlexibleCPV extension

FlexibleCPV allows model parameters to be complex and calculates fermion EDMs at one loop. Its applications illustrate physical phase dependence and comparisons with SPheno in CP-violating CMSSM scenarios.

7. FlexibleCPV extension

  • Since FlexibleSUSY 1.1.0, model parameters can be complex, with real-valued treatment configurable through the model file.For compatibility with version 1.0, the RealParameters list defaults to treating all parameters as real.Source: §7 · p. 50

7.2. Application: CMSSM with CP-violation

  • The CMSSMCPV setup chooses Re Bµ, Im Bµ, and |µ| as EWSB outputs, leaving the phase of µ as a free input.The MSSM/CPV model permits a nonzero relative Higgs-doublet phase and has three independent EWSB equations.Source: §7.2 · p. 51

7.3. Application: Electric dipole moments of fermions

  • In the CP-violating MSSM example, electron-EDM contours depend on the rephasing-invariant phases arg(µM1/2) and arg(µAe), with the experimental upper bound shown.The EDM calculation sums one-loop FFS- and SSF-type diagrams in the DR scheme.Source: §7.3 · pp. 51–53

7.4. Tests and comparisons with other spectrum generators

  • The lightest Higgs mass shows the same phase-angle behavior in FlexibleSUSY and SPheno, with an approximately 0.7 GeV offset attributed mainly to different higher-order top-Yukawa treatments.For the trilinear coupling running, the programs agree over 14 orders of magnitude, with a maximum deviation around 2% near Q ≈ 10^16 GeV.Source: §7.4 · pp. 53–54

8. FlexibleMW extension

FlexibleMW uses muon decay to determine the weak mixing angle and predict the W pole mass across a broader class of BSM models. Comparisons with other implementations show agreement after identified differences and inconsistencies are corrected.

8. FlexibleMW extension

  • FlexibleMW enables a W pole-mass prediction and more accurate electroweak gauge couplings, using muon decay rather than the W mass as input.The muon-decay approach relates coupling uncertainties to those of GF and MZ, rather than the less precise MW input.Source: §8 · pp. 54–55
  • The calculation derives the weak mixing angle from electroweak inputs, iterates for a self-consistent solution, and then computes the W pole mass.The implementation includes model-specific corrections and one-loop muon-decay wave-function, vertex, and box contributions.Source: §8 · pp. 54–56

8.1. Choosing FlexibleMW in the model file

  • FlexibleSUSY 2.0 applies the muon-decay approach to BSM models containing W and Z bosons and the SM gauge group as a factor.The default Automatic setting selects this approach when its conditions are met; otherwise, it uses the W-mass method.Source: §8 · p. 55, §8.1 · p. 57

8.2. Structure of FlexibleMW code

  • The interface returns sin θW and MW from a model object and SM inputs, with the spectrum-generator iteration updating MW for consistency.The calculation is exposed through the model’s weinberg_angle class.Source: §8.2 · p. 57

8.3. Tests and comparisons with other spectrum generators

  • The implementation agrees excellently with the SOFTSUSY algorithm in SM and CMSSM tests, and its generalized tree-level ρ expression was analytically checked in several models.The checks included UMSSM, MRSSM, and E6SSM.Source: §8.3 · p. 58
  • Correcting SARAH/SPheno and SPheno implementation differences yields good agreement with FlexibleMW for CMSSM and MRSSM W-mass results.The initial discrepancies involved top-mass use in corrections and an inconsistency in the final MW calculation; Figure 10 plots the comparisons against M1/2 and Λu.Source: §8.3 · p. 58, Figure 10 · p. 59

9. FlexibleSAS extension

This section presents FlexibleSAS, a semi-analytic boundary-value solver added to FlexibleSUSY 2.0 to extend spectrum generation to models where the two-scale solver may fail, and describes its method, model-file requirements, and nested iterations. It reports close agreement with the two-scale solver on stable benchmarks and reproduction of viable CSE6SSM solution regions and parameter values from earlier code, while noting sensitivity to convergence behavior and numerical precision.

9. FlexibleSAS extension

  • The two-scale solver can miss valid BVP solutions, find at most one where multiple exist, and fail entirely for models including the CNMSSM and CE6SSM.These models can require parameters determined by electroweak symmetry breaking to satisfy high-scale constraints, making additional solvers valuable.Source: §9 · pp. 59–60
  • FlexibleSAS solves the BVP using semi-analytic RGE solutions for parameter subsets, exploiting hierarchical running so dimensionful parameters can be expressed in terms of boundary values.The solver computes the coefficients numerically after determining the analytic forms from the boundary conditions.Source: §9 · p. 60

9.1. Choosing FlexibleSAS in the model file

  • Users enable FlexibleSAS through FSBVPSolvers and must give the semi-analytically expanded parameters compatible boundary conditions; the solver automatically substitutes its solutions into EWSB conditions.In the CNMSSM, this permits maintaining universal high-scale soft scalar masses while determining m0 from the EWSB conditions.Source: §9.1 · pp. 63, 65

9.2. Structure of the generated code

  • The semi-analytic algorithm uses nested iterations: it first converges the SUSY-preserving or dimensionless parameters and scales, then computes dimensionful parameters, solves EWSB conditions, and calculates the spectrum.Updated dimensionful parameters feed back into the inner iteration to compute threshold corrections.Source: §9.2 · p. 67
  • The generated semi-analytic solver numerically determines solution coefficients by varying dimensionful inputs and integrating RGEs, then uses them to set soft parameters during the iterative spectrum calculation.In the described iteration, coefficients are determined at the EWSB scale, the DR/MS spectrum and EWSB conditions are recalculated, and the process repeats until convergence.Source: §9.2 · pp. 68–69

9.3. Tests and comparisons with other spectrum generators

  • Running-parameter and pole-mass results from the two solvers differ by at most 0.1% across CMSSM, CNMSSM, and CE6SSM benchmark tests.The comparison applies when the same solution is found and is a stable fixed point in both solvers; corresponding unit tests were added to the test suite.Source: §9.3 · p. 70
  • In CMSSM scans, solver differences can arise from distinct fixed-point stability and convergence; tighter MX convergence, higher-order Runge-Kutta integration, and a more precise EWSB solution reduce one-step changes below the permille level.Under default settings, changes for a small number of points reach 20%–30%; the figure reports changes of 1%–30% reduced below 0.001% with the improved settings.Source: §9.3 · pp. 70–71, Figure 12 · p. 71
  • In the CMSSM, the semi-analytic solver’s typical runtime is about three times that of the two-scale solver, largely because it performs more iterations and calculates coefficients between scales.The inner iteration typically requires a similar number of steps to a full two-scale run, while the number of steps decreases as each outer iteration converges.Source: §9.3 · pp. 71–72, Figure 13 · p. 72
  • The solvers provide complementary CMSSM coverage: the semi-analytic method finds focus-point solutions at small |µ| relative to m0, while the two-scale method is more effective at small m0.Each method can also fail where the other succeeds because of differing stability or input/output choices; where both find the same solution, their Higgs-mass results agree very well.Source: §9.3 · pp. 72–73, Figure 14 · p. 73
  • In a CMSSM comparison, the semi-analytic solver found additional solutions in good agreement with modified SOFTSUSY, which the two-scale solver found at most two for the parameter region studied.The semi-analytic solver did not find valid solutions for µ < −545 GeV because the tree-level CP-odd Higgs mass became tachyonic; multiple solutions can have substantially different phenomenology.Source: §9.3 · pp. 73–74, Figure 15 · p. 74
  • In the CNMSSM scan, the singlet-VEV condition constrains m0, while obtaining a 125 GeV Higgs favors small λ and large M1/2; squark masses are then predominantly set by the gaugino mass.The singlet potential requires approximately Aκ^2 ≳ 9mS^2; the reported squarks are lighter than the gluino, whose mass is approximately 2M1/2.Source: §9.3 · p. 76
  • In the CE6SSM scan, an exotic Yukawa coupling drives the soft singlet mass negative, enabling large m0; m0 is typically greater than M1/2, and squark masses are mostly set by m0.The scan found a wide GUT-scale range for λ and κ despite strong electroweak-scale constraints on λ. These results use full 2-loop RGEs and 1-loop pole masses, but no detailed quantitative comparison to prior work was made.Source: §9.3 · pp. 76–77
  • The CSE6SSM spectrum-generator results agree very well with Ref. and with a hand-written semi-analytic solver prototype used in Refs..The comparison was performed at the same level of precision.Source: §9.3 · p. 77

10. FlexibleEFTHiggs

This section presents FlexibleEFTHiggs for predicting the lightest Higgs pole mass across low and high new-physics scales, combining EFT and fixed-order calculations with NLO+NLL accuracy through improved matching. It also illustrates a scale-variation uncertainty estimate for the prediction.

10. FlexibleEFTHiggs

  • FlexibleEFTHiggs combines EFT and diagrammatic calculations to resum large logarithms, include non-logarithmic 1-loop and power-suppressed terms, and retain accuracy across scales.At low scales it agrees with fixed-order calculations; at large scales it agrees with pure EFT, while at intermediate scales it recovers the fixed-order result plus higher-order logarithms.Source: §10 · p. 78

10.1. Basic matching condition

  • The improved matching procedure preserves 1-loop exactness while resumming leading and next-to-leading logarithms and including all non-logarithmic 1-loop contributions.It avoids spurious higher-order logarithms by using separate tree-level- and 1-loop-matched BSM parameter sets and not performing the usual momentum iteration when calculating matching-scale Higgs pole masses.Source: §10.1 · pp. 79–81

10.3. Comparison of old and improved

  • For Xt = 0 and MS > 10 TeV, the improved version deviates from HSSUSY-2L by around 10 MeV, versus around 600 MeV for the old version.For maximal stop mixing, Xt/MS = 6, the implementations differ by up to 3 GeV. Above ≈1 TeV, the improved version reproduces the 1-loop pure EFT calculation for arbitrary stop mixing; both versions differ from the 2-loop pure EFT result by around 1–2 GeV for non-zero mixing.Source: §10.3 · p. 81, Figure 19 · p. 82

10.4. Choosing

  • A FlexibleEFTHiggs spectrum generator is enabled by setting its model-file flag, and the user must specify how electroweak Higgs vacuum expectation values match to the BSM model.The matching scale defaults to the SUSY scale, and the matching conditions can be specified or overridden through MatchingScaleInput.Source: §10.4 · pp. 81–84
  • The example computes a central Higgs-mass value, varies pole and matching scales separately, and sums the largest absolute deviation from the central value for each variation as uncertainty.Source: §10.4 · p. 86

11. Current limitations and workarounds

FlexibleSUSY 2.0 supports a broad range of models but retains restrictions on perturbativity, gauge structure, and parameter types. The authors describe manual workarounds for some cases and note that the modular code is straightforward to adapt.

11. Current limitations and workarounds

  • Model couplings must remain perturbative at every scale between the highest and lowest boundary conditions.Source: §11 · p. 86
  • Models generally need the SM gauge group as a factor; otherwise, gauge couplings must be fixed manually in a boundary condition.The restriction permits unambiguous low-energy matching to the SM.Source: §11 · p. 86
  • Rank-3 tensor-like Lagrangian parameters are unsupported, but can be decomposed into rank-2 matrix-like couplings as a workaround.Source: §11 · p. 87
  • Running Yukawa extraction is unsupported when a fourth fermion generation mixes with SM fermions; users can instead fix couplings manually from running SM fermion masses.Source: §11 · p. 87
  • The authors say the modular generated code makes adapting or extending it to overcome scope restrictions straightforward.Source: §11 · p. 87

12. Conclusions

FlexibleSUSY 2.0 broadens the models and calculations supported by the package, including new solvers and observables. The authors illustrate applications and report testing for correctness and speed.

12. Conclusions

  • FlexibleSUSY 2.0 adds support for non-SUSY and complex-parameter models, FlexibleSAS, muon observables, and an updated hybrid EFT/fixed-order Higgs-mass calculation.The updates also include model-specific higher-order corrections and higher-order log resummation for the Higgs mass.Source: §12 · p. 87
  • Applications include efficient large-scale scans on multiple CPU cores and low-energy effective field theories with complicated matching-scale boundary conditions.The authors also note existing use in global fits and precision Higgs-mass studies.Source: §12 · p. 87
  • The modular spectrum generators allow model-specific higher-order corrections, including three-loop Higgs contributions from Himalaya and power-suppressed terms in HSSUSY.The paper also illustrates calculations of muon anomalous magnetic and electric dipole moments, and FlexibleSAS studies in several constrained models.Source: §12 · p. 88
  • The system was tested against literature and other spectrum-generator results, and speed tests found it effective for large-scale scans.Generated C++ components are designed to be readable and reusable for users' analysis tools.Source: §12 · p. 88

A. SM input parameters

FlexibleSUSY accepts SM inputs through SLHA blocks or its Mathematica interface. The appendix describes the block-to-symbol mapping, defaults, and CKM and PMNS input conventions.

A. SM input parameters

  • The SLHA SMINPUTS block contains electromagnetic and strong couplings, the Fermi constant, and SM particle masses; omitted entries use table defaults.Table A.12 maps SLHA block entries to Mathematica symbols.Source: A. SM input parameters · p. 89, Table A.12 · p. 91
  • CKM parameters use the Wolfenstein parametrization in VCKMIN, with missing entries set to zero; Mathematica instead takes the exact angle-and-phase parametrization.Source: A. SM input parameters · pp. 89–90
  • PMNS parameters are supplied through UPMNSIN, with missing entries set to zero, and are accessible as PMNS symbols in model files.Source: A. SM input parameters · pp. 89–90
  • FlexibleSUSYInput supplies additional parameters for applications such as the muon anomalous magnetic moment, and omitted entries take their defaults.In Mathematica, SM parameters are passed as replacement rules; unset parameters use table defaults.Source: A. SM input parameters · pp. 89–90

B. FlexibleSUSY configuration options

This section documents FlexibleSUSY configuration options for controlling numerical precision, solver choice, perturbative and threshold calculations, and model-specific outputs. It covers FlexibleEFTHiggs settings, toggling specified three-loop Higgs pole-mass contributions, and selecting the scale for outputting running model parameters.

B. FlexibleSUSY configuration options

  • The precision goal controls the accuracy of running, spectrum, electroweak-symmetry-breaking, and observable calculations; 10^-4 is generally sufficient, while 10^-5 may suit models with three-loop corrections.If the iteration limit is zero, the maximum iteration count is determined by the precision goal.Source: B. FlexibleSUSY configuration options · p. 93
  • The solver option tries enabled solvers in sequence by default, or selects the two-scale or semi-analytic solver individually.Selecting a solver not enabled in the model file causes an error.Source: B. FlexibleSUSY configuration options · p. 93
  • Other switches control SM and BSM pole-mass calculations, allow output despite physical problems, and can force positive Majorana masses at the cost of generally complex mixing matrices.Positive Majorana masses violate the SLHA convention; allowing negative masses keeps the corresponding mixing matrices real.Source: B. FlexibleSUSY configuration options · pp. 93, 95, 96, 97
  • Pole-mass and electroweak-symmetry-breaking loop orders should match; separate controls set beta-function and maximum threshold-correction loop orders.The maximum threshold order limits corrections, while per-parameter threshold orders can be controlled separately.Source: B. FlexibleSUSY configuration options · pp. 95–96
  • For FlexibleEFTHiggs, users can vary effective-theory pole-mass and matching scales, choose the Higgs field used in matching, and set the quartic-coupling matching order.A zero effective-theory pole-mass scale uses Q = Mt, while a zero matching scale uses the model-file SUSYScale; one-loop matching for the quartic coupling is recommended.Source: B. FlexibleSUSY configuration options · p. 97
  • FlexibleSUSY 2.0 ignores the eftMatchingLoopOrderUp option.Source: B. FlexibleSUSY configuration options · p. 97
  • Per-parameter threshold orders are encoded as decimal digits, and changing the top-Yukawa threshold order in HSSUSY can estimate part of the Higgs-mass uncertainty.The passage describes switching between two- and three-loop QCD corrections to yt, with the global threshold order set to three.Source: B. FlexibleSUSY configuration options · pp. 98–99
  • The options higgs3loopCorrectionAtAtAs and higgs3loopCorrectionAtAtAt toggle Higgs pole-mass contributions of orders O(α2 t αs) and O(α3 t), respectively.Each option has an effect only when the corresponding model-specific contributions are enabled in the FlexibleSUSY model file.Source: B. FlexibleSUSY configuration options · p. 99
  • MODSEL[12], parameterOutputScale sets the scale in GeV at which running MS/DR model parameters are output.Setting it to 0 outputs parameters at the scale assigned to SUSYScale in the FlexibleSUSY model file.Source: B. FlexibleSUSY configuration options · p. 99

C. CMSSMCPV model file

The CMSSMCPV model file configures MSSM/CPV inputs and boundary conditions, and defines phase-aware parameters and observables for the generated spectrum.

C. CMSSMCPV model file

  • Universal soft scalar masses and trilinear terms are specified alongside complex gaugino masses and a complex phase for μ.The trilinears use a shared Azero and the gaugino masses share m12 + I Imm12.Source: C. CMSSMCPV model file · p. 100
  • The parameter setup relates the Higgs vacuum expectation values to tanβ and the electroweak vacuum expectation value.The listed low-energy inputs include vd and vu; separate entries express them using MZDRbar and gauge couplings.Source: C. CMSSMCPV model file · pp. 100–101
  • Configured observables include the muon anomalous magnetic moment, charged-lepton EDMs, and Higgs-to-photon and Higgs-to-gluon couplings.The list also includes pseudoscalar Higgs photon and gluon couplings.Source: C. CMSSMCPV model file · p. 101
  • The output mapping includes real and imaginary trilinear and soft-mass parameters, with separate imaginary-part entries for the latter.The listed mappings include ImAu, ImAd, ImAe, and ImMSOFT.Source: C. CMSSMCPV model file · pp. 101–102

D. THDM model file

This section presents model-file configuration entries for THDM matching and electroweak inputs, as well as CNMSSM parameters, boundary conditions, precision settings, and selected outputs and observables.

D. THDM model file

  • The THDM setup specifies matching and electroweak-symmetry-breaking scales, and sets v1 and v2 from MZMSbar or the low-energy vev and TanBeta.It also defines M122 using MAInput and the angle determined by TanBeta.Source: D. THDM model file · pp. 103–104
  • The THDM model file adds loop-order-dependent threshold corrections to quartic couplings, including Lambda1 and Lambda2.The listed corrections are selected using LambdaLoopOrder.Source: D. THDM model file · p. 103

E. CNMSSM model file

  • The CNMSSM model file specifies input parameters and universal soft-term boundary conditions, and defines an effective pseudoscalar mass squared.The listed soft terms use m0Sq and Azero, while the effective mass expression depends on Lambda, Kappa, vS, vu, and vd.Source: E. CNMSSM model file · pp. 109–111
  • The model file lists Chi, Sv, Su, Sd, Se, Cha, and Glu as potential LSPs, with hh, Ah, and Hpm assigned high pole-mass precision.Default pole-mass precision is medium; the medium-, low-, and high-precision sets are then explicitly set to empty, empty, and {hh, Ah, Hpm}, respectively.Source: E. CNMSSM model file · p. 111
  • The listed output configuration includes vS and m0Sq as EWSB outputs, aMuon as a low-energy observable, and Higgs-photon/gluon coupling observables.The snippet also lists NMSSMRUN entries for Kappa, Lambda- and Kappa-derivative ratios, Lambda vS/Sqrt[2], and ms2.Source: E. CNMSSM model file · p. 111

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