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SuperIso v3.0: A program for calculating flavor physics observables in 2HDM and supersymmetry

F. Mahmoudi

arXiv:0808.3144v3hep-ph

TL;DR

SuperIso v3.0 addresses the need to evaluate important indirect observables and constraints for several extensions of the Standard Model. It expands model and observable coverage, revises key calculations including b →sγ at NNLO, and uses SLHA-based inputs with external spectrum calculators or user-provided files. The resulting package supports phenomenological constraints and consistency checks before and alongside direct searches.

  • Problem

    Indirect observables provide complementary probes of physics beyond the Standard Model, but evaluating important constraints across 2HDM and supersymmetric models requires dedicated program support.

  • Method

    SuperIso reads SLHA or LHA-inspired model inputs, optionally obtains spectra from external calculators, and computes flavor observables through revised and newly added routines.

  • Results

    SuperIso v3.0 evaluates numerous flavor observables and the muon anomalous magnetic moment across SM, 2HDM, MSSM, and NMSSM scenarios.

  • Takeaways & Limitations

    Indirect constraints from SuperIso can provide information before LHC data, guide direct searches, and support consistency checks.

  • Takeaways & Limitations

    The program does not yet cover beyond-minimal-flavor-violation scenarios, whose extension remains under development.

Abstract

from arXiv · show

We describe SuperIso v3.0 which is a public program for evaluation of flavor physics observables in the Standard Model (SM), general two-Higgs-doublet model (2HDM), minimal supersymmetric Standard Model (MSSM) and next to minimal supersymmetric Standard Model (NMSSM). SuperIso v3.0, in addition to the branching ratio of B -> X_s gamma and the isospin asymmetry of B -> K* gamma, incorporates other flavor observables such as the branching ratio of B_s -> mu+ mu-, the branching ratio of B -> tau nu_tau, the branching ratio of B -> D tau nu_tau, the branching ratio of K -> mu nu_mu and the branching ratios of D_s -> tau nu_tau and D_s -> mu nu_mu. The calculation of the branching ratio of B -> X_s gamma is also improved, as it includes NNLO Standard Model contributions. The program also computes the muon anomalous magnetic moment (g-2). Nine sample models are included in the package, namely SM, 2HDM, and mSUGRA, NUHM, AMSB and GMSB for the MSSM, and CNMSSM, NGMSB and NNUHM for the NMSSM. SuperIso uses a SUSY Les Houches Accord file (SLHA1 or SLHA2) as input, which can be either generated automatically by the program via a call to external spectrum calculators, or provided by the user. The calculation of the observables is detailed in the Appendices, where a suggestion for the allowed intervals for each observable is also provided.

1 Introduction

SuperIso evaluates important indirect flavor observables and constraints in the SM, 2HDM, MSSM, and NMSSM. Version 3.0 broadens the observable and model coverage while supporting automated or user-provided input files.

  • Motivation: SuperIso evaluates indirect observables and constraints relevant to exploring physics beyond the Standard Model.Indirect searches test SM predictions, reveal indirect new-physics effects, complement direct searches, and check consistency with direct results.
  • Capabilities: The program includes flavor observables beyond isospin symmetry breaking and b →sγ, together with the muon anomalous magnetic moment.The broader set includes several leptonic and rare-meson branching ratios, as described in the paper context.
  • Scope: Version 3.0 covers the SM, general 2HDM, MSSM, and NMSSM.The program was extended to the general 2HDM in version 2.6 and to the NMSSM in version 3.0.
  • Input and models: SuperIso accepts SUSY Les Houches Accord input generated by external spectrum calculators or supplied by the user, alongside an LHA-inspired 2HDM format.Supported generators include SOFTSUSY, ISAJET, NMSSMTools, and 2HDMC.
  • Paper organization: The paper presents the package content, usage procedure, inputs and outputs, example results, observable formulas, and suggested allowed intervals.The formulas and interval suggestions are provided in the Appendices.

2 Content of the SuperIso v3.0 package

SuperIso v3.0 is a C99 package that reads model parameters, computes spectra and couplings when needed, and evaluates a broad set of observables through dedicated routines. Its revised calculations include NNLO b →sγ, improved isospin asymmetry, and newly implemented leptonic observables.

  • Package structure: SuperIso v3.0 provides C99 programs and routines for calculating constraining flavor observables across SM, 2HDM, MSSM, and NMSSM scenarios.The package includes model-specific main programs and invites users to write their own main programs.
  • Workflow: The computation proceeds by generating or supplying an SLHA file, scanning it, and calculating the observables.External generators include ISAJET, SOFTSUSY, NMSSMTools, and 2HDMC.
  • Input handling: The package supports SLHA2 reading, validity checks, charged-LSP detection, and rejection of unsupported or invalid model files.Invalid inputs can reflect failed spectrum generation, unsupported models, non-SLHA formatting, or missing important elements.
  • Improved observables: b →sγ is calculated at NNLO from LO, NLO, and NNLO Wilson coefficients evaluated at the matching and lower scales.The calculation uses renormalization-group evolution between scales before computing the inclusive branching ratio.
  • Improved observables: The isospin-asymmetry calculation uses LO and NLO Wilson coefficients at the decay and spectator scales and has separately coded integrals.An automatic calculator accepts only the SLHA filename, while the observable routine also takes Λh.
  • New observables: New routines calculate Bu →τντ branching ratios and their SM-normalized ratios, including higher-order SUSY Yukawa-coupling corrections.These leptonic decays occur at tree level, with the stated SUSY corrections included.

3 Compilation and installation instructions

SuperIso retains a simplicity-of-use design while providing executables for SM, 2HDM, MSSM, NMSSM, and SLHA workflows. Compilation uses the supplied Makefile and requires external spectrum calculators for model-specific programs.

  • Package use: The package preserves a design centered on simplicity of use.The main structure remains unchanged since the first version.
  • Installation contents: The distribution contains source files, a Makefile, ten sample main programs, and an example SLHA input file.The Makefile also specifies paths to required external executables and directories.
  • Compilation: The library is compiled first, after which individual executables are built with make name or make name.c.The executable name can be sm, thdm, msugra, amsb, gmsb, nuhm, cnmssm, ngmsb, nnuhm, or slha.
  • Executables: slha.x computes observables from an input SLHA file, while thdm.x first obtains 2HDM masses and couplings through 2HDMC.The model-specific executables similarly generate spectra before calculating observables.
  • External calculators: The AMSB, GMSB, mSUGRA, and NUHM executables use ISAJET and/or SOFTSUSY to generate spectra and couplings.CNMSSM, NGMSB, and NNUHM use NMSSMTools for the corresponding NMSSM parameter spaces.

4 Input and output description

SuperIso programs read SLHA-based inputs or model-specific parameters, calculate flavor observables, and report multiple branching ratios, ratios, magnetic-moment deviations, and exclusion flags. Separate executables support SM, MSSM scenarios, and NMSSM scenarios, with external spectrum generators supplying masses and couplings.

  • SLHA input and outputs: SuperIso's slha.x reads an SLHA file and calculates numerous flavor observables, including branching ratios, normalized ratios, a_muon, and direct-search exclusion status.An inconsistent file can indicate an invalid generated point, an unimplemented model, or missing important parameters.
  • Standard Model: The standalone sm.x program computes the different observables in the Standard Model without requiring arguments.Its example output includes delta0, several meson-decay branching ratios, normalized ratios, and the Bs →µ+µ− branching ratio.
  • MSSM scenarios: mSUGRA inputs are m0, m1/2, A0, and tan β, with optional sign(µ), top pole mass, b-quark mass, and αs(MZ).msugra.x can use ISAJET and/or SOFTSUSY to generate the mass spectrum before calculating observables.
  • MSSM scenarios: AMSB, GMSB, and NUHM executables extend the model inputs with scenario-specific parameters and calculate observables after spectrum generation.AMSB uses m0, m3/2, and tan β; GMSB uses Λ, N5, and tan β; NUHM adds µ and mA to the mSUGRA inputs.
  • NMSSM scenarios: CNMSSM and NGMSB executables use NMSSMTools-generated parameters, with CNMSSM involving λ and NGMSB involving Λ, N5, and λ.The paper calls its CNMSSM a partially constrained NMSSM because m_S^2 and κ are determined through Higgs-potential minimization.

5 Results

SuperIso observables impose complementary constraints on supersymmetric parameter spaces. In the illustrated mSUGRA and NUHM planes, low mass parameters and high tan β are generally disfavored, while tachyonic regions are separately identified.

  • Constraint indicators: Nine observables define distinct constraint regions, including isospin asymmetry, b →sγ, collider limits, Bs →µ+µ−, anomalous magnetic moment, and several leptonic or semileptonic decays.The figures use different colors and hatching to distinguish exclusions, favored regions, charged-LSP points, and cosmological disfavoring.
  • Constraint indicators: Suggested allowed intervals for each observable are provided in Appendix H for constraining the implemented models.These intervals complement the exclusion regions displayed for the mSUGRA and NUHM examples.
  • mSUGRA constraints: In mSUGRA with tan β = 50, A0 = 0, and µ > 0, small m0 and m1/2 values are disfavored by the observables.The unfilled bottom-left region corresponds to points with tachyonic particles.
  • NUHM constraints: In the NUHM plane with m0 = 500 GeV, m1/2 = 500 GeV, A0 = 0, and µ = 500 GeV, most observables disfavor high tan β.The white top-right triangle corresponds to a region containing tachyonic particles.

6 Conclusion

SuperIso v3.0 expands and improves flavor-observable calculations for MSSM and NMSSM studies, while supporting indirect constraints on supersymmetric parameter spaces. Its constraints disfavor specific regions and can guide direct searches and consistency checks.

  • SuperIso v3.0 adds and improves numerous flavor observables and computes the muon anomalous magnetic moment for MSSM and NMSSM studies.These indirect constraints provide phenomenological information before LHC data.
  • Indirect constraints can guide LHC direct searches and provide valuable consistency checks.

Appendix B Evolution of quark masses

The appendix describes perturbative treatments for quark masses and the b →sγ effective Hamiltonian. It specifies flavor-counting conventions, matching-scale inputs, and assumptions used in these calculations.

  • SuperIso computes quark pole masses using a three-loop formula and relates the number of lighter flavors to nfl.The stated constants are CA = 3, T = 1/2, nfl = 4 and CF = 4/3.
  • The running quark mass uses nf active flavors and the β coefficients from Eq. (2), while the 1S bottom mass is specified separately.The scale assumption given is µb = O(mb).
  • The b →sγ effective Hamiltonian is expressed through operators Oi(µ) and Wilson coefficients Ci(µ) in the standard operator basis.The coefficients are evaluated at the scale µ and include the effective-coefficient construction described in the appendix.

C.1 Standard Model contributions

The Standard Model and new-physics contributions to b →sγ are organized through Wilson coefficients, perturbative expansions, and charged- and neutral-Higgs or chargino effects. The appendix combines these contributions within the stated mass and mixing assumptions.

  • Standard Model contributions: SuperIso expresses Standard Model contributions to the b →sγ Wilson coefficients through LO, NLO and NNLO terms and associated loop functions.The appendix provides expansions and integral representations for the required functions.
  • Charged-Higgs contributions: The charged-Higgs contributions are specified at leading and next-to-leading order for the different 2HDM Yukawa sectors.Table 1 summarizes the Yukawa couplings for 2HDM types I–IV.
  • Supersymmetric contributions: Supersymmetric contributions include chargino effects, tan β-enhanced corrections, and neutral-Higgs terms generalized from the MSSM to the NMSSM.The calculation uses stated assumptions about chargino masses and neutralino mixing matrices.
  • Combination of contributions: The complete Wilson coefficients are obtained by adding the different Standard Model, Higgs and supersymmetric contributions.The appendix also specifies Higgs mixing matrices and their parameter conventions.

D.1 RGE in the standard operator basis

The standard-basis evolution transforms Wilson coefficients from the matching scale µW to the lower scale µb using perturbative evolution matrices. The construction depends on αs-scale ratios and tabulated coefficients.

  • Wilson coefficients are evolved from the matching scale µW to the lower scale µb in the standard operator basis.The evolution is expressed through U^(n) matrices acting on coefficient vectors at successive perturbative orders.
  • The evolution uses η = αs(µW)/αs(µb) and the coefficient vector ⃗C = {C1, · · · , C8}.

D.2 RGE in the traditional operator basis

This section presents the traditional operator-basis treatment for the renormalization-group evolution, distinguishing ordinary and effective Wilson coefficients and tabulating the numerical inputs used.

  • The traditional operator basis is used to express the operators and evolve the relevant Wilson coefficients.
  • P7 and P8 coincide with O7 and O8, so their C7 and C8 coefficients are indistinguishable in both bases.
  • The effective coefficients nevertheless differ between the traditional and alternative bases, despite coincident leading-order coefficients.
  • The numerical quantities needed for the evolution are supplied in Tables 2, 4, 5, 6, and 7.
  • The appendix provides detailed observable formulas and lists meson, lifetime, form-factor, and CKM inputs used in subsequent calculations.

E.1 Branching ratio of B →Xsγ

The B →Xsγ calculation combines Standard Model and model-dependent loop contributions with perturbative, nonperturbative, electromagnetic, and NNLO corrections. For mc < mb/2, an interpolation is used because one NNLO term is analytically known only in the opposite limit.

  • B →Xsγ proceeds through electromagnetic penguin loops involving W bosons in the SM and additional charged-Higgs, chargino, neutralino, and gluino loops in supersymmetry.Neutralino and gluino contributions are negligible in minimal flavor-violating scenarios.
  • The branching ratio combines perturbative P(E0), nonperturbative N(E0), and electromagnetic corrections, with E0 denoting the photon-energy cut.
  • The remaining NNLO correction to P(E0) is separated into β0-dependent and remainder components, while selected contributions are neglected.The text specifically states that contributions with i,j = 3,4,5,6 and remaining φ(2)β0ij functions are neglected.
  • For mc < mb/2, SuperIso interpolates P(2)rem using a linear combination whose constants are fixed by matching and evaluated under two x5 assumptions.The two resulting branching ratios are averaged, following the cited prescription.

E.2 Isospin asymmetry of B →K∗γ

This section describes calculations for radiative-decay isospin asymmetry and several leptonic or semileptonic observables across the SM, 2HDM, and supersymmetric models. The formulas incorporate spectator effects, Wilson coefficients, hadronic inputs, and model-specific Higgs contributions.

  • E.2 Isospin asymmetry of B →K∗γ: The isospin asymmetry Δ0 in B →K∗γ arises from photon emission by the spectator quark, making charged and neutral B decays different.
  • E.2 Isospin asymmetry of B →K∗γ: The B →K∗γ calculation uses spectator-dependent coefficients, hard-scattering functions, meson distribution amplitudes, and a parameterization of a logarithmically divergent integral.The parameter X = ln(mB/Λh)(1 + eiϕ) is introduced for the divergent integral.
  • E.2 Isospin asymmetry of B →K∗γ: SuperIso computes the B →K∗γ asymmetry from numerical integrals and Wilson coefficients, including charged-Higgs and chargino contributions in supersymmetry.
  • E.3 Branching ratio of Bu →τντ: Bu →τντ receives W-mediated and charged-Higgs-mediated annihilation contributions, with charged-Higgs exchange lacking the SM helicity suppression at high tan β.The decay is therefore described as sensitive to charged-Higgs effects and useful for constraints.
  • The B →Dτντ, Bs →µ+µ−, K →µνµ, and Ds →ℓνℓ observables are calculated using kinematic distributions, hadronic form factors, Wilson coefficients, and charged- or neutral-Higgs contributions.For Bs →µ+µ−, supersymmetric effects can be enhanced at large tan β, while Ds charged-Higgs effects can only suppress the branching ratio.
  • Appendix F Muon anomalous magnetic moment: The muon anomalous magnetic moment is defined by aµ = (g −2)/2 as the higher-order correction to the Dirac magnetic moment.

Appendix H Suggested limits

Appendix H provides direct-search mass limits and suggested observable limits for constraining supersymmetric parameters. The paper cautions that uncertainties in CKM elements and hadronic parameters limit how strongly these constraints should be interpreted.

  • The mass limits may depend on auxiliary conditions, which are also implemented in SuperIso and can be updated by the user.
  • Table 9 presents suggested 95% C.L. bounds for the observables implemented in SuperIso, including experimental and theoretical uncertainties.
  • Constraints derived from some observables should not be over-interpreted because CKM-matrix and hadronic-parameter determinations carry large uncertainties.
  • Table 8 lists direct-search limits on Higgs and MSSM particle masses, with a 3 GeV intrinsic uncertainty included for the h0 mass limit.
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