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Dark matter direct detection rate in a generic model with micrOMEGAs2.2

G. Belanger, F. Boudjema, A. Pukhov, A. Semenov

arXiv:0803.2360v2hep-ph

TL;DR

Direct WIMP detection requires calculating elastic scattering with detector nuclei, while existing calculations and public codes include model-specific treatments. The paper adds automatic WIMP–nucleus direct-detection rate computation to micrOMEGAs for generic new-physics models, including tree-level cross sections and dominant QCD corrections.

  • Problem

    Direct searches for WIMP dark matter require reliable WIMP–nucleus scattering calculations, but public treatments include model-specific approaches such as neutralino calculations in supersymmetry.

  • Method

    The paper introduces a micrOMEGAs module that computes tree-level WIMP–nucleon cross sections, assembles WIMP–nucleus interactions, and includes dominant QCD corrections for generic models.

  • Results

    The direct-detection rate is computed automatically in micrOMEGAs for any implemented generic new-physics model containing a WIMP candidate.

  • Takeaways & Limitations

    The module extends micrOMEGAs direct-detection calculations beyond supersymmetric neutralino models to generic WIMP models.

  • Takeaways & Limitations

    The default gluon-contribution treatment approximates box diagrams and is justified only when mq/(M˜q −Mχ) ≪1, although users can implement a more complete calculation.

Abstract

from arXiv · show

We present a new module of the micrOMEGAs package for the calculation of WIMP-nuclei elastic scattering cross sections relevant for the direct detection of dark matter through its interaction with nuclei in a large detector. With this new module, the computation of the direct detection rate is performed automatically for a generic model of new physics which contains a WIMP candidate. This model needs to be implemented within micrOMEGAs2.1.

1 Introduction

Direct detection searches measure nuclear recoils from WIMP scattering, but interpreting them requires particle, nuclear, and astrophysical inputs. The paper addresses micrOMEGAs 2.2’s missing direct-detection module by automating rate calculations for generic WIMP models.

  • WIMPs are leading stable CDM candidates across many Standard Model extensions, with possible fermion, vector-boson, or scalar identities and masses from a few GeV to a few TeV.
  • Direct searches measure recoil energy from WIMP–nucleus scattering, whose interactions are classified as spin independent or spin dependent.Spin-independent interactions add coherently and favor heavy nuclei, whereas spin-dependent interactions mainly involve unpaired nucleons and favor light nuclei.
  • Only DAMA reported a positive signal, while Edelweiss, CDMS, and Xenon established upper limits on WIMP–nucleon cross sections.
  • Before micrOMEGAs 2.2, the package calculated relic abundance but lacked a direct-detection-rate module despite existing model-specific scattering calculations.
  • The new module computes tree-level WIMP–nucleus cross sections for generic new-physics models, includes dominant QCD corrections, and supports user-specified nuclear form factors and velocity distributions.

2 Elastic scattering of WIMPs on point-like nuclei

The paper derives zero-momentum WIMP–nucleus scattering from effective WIMP–nucleon interactions and extends the framework beyond Majorana fermions. Scalar and spin-dependent interactions are treated differently because coherent nucleon summation enhances only the former for heavy nuclei.

  • Small WIMP velocities make momentum transfer negligible, so nucleon elastic cross sections are evaluated at zero momentum and folded with nuclear form factors for finite-momentum nuclear scattering.
  • In the non-relativistic limit, WIMP–nucleon amplitudes reduce to scalar spin-independent and axial-vector spin-dependent interaction classes, with momentum-suppressed operators generally omitted.Such operators can matter when the leading scalar coefficient is strongly suppressed, but then rates are expected to be far below experimental sensitivities.
  • Spin independent interactions: Spin-independent WIMP–nucleus amplitudes sum proton and neutron contributions coherently, producing an approximately A^2 enhancement when λp ≈ λn.
  • Spin dependent interactions: Spin-dependent nuclear amplitudes require separate proton and neutron spin-current sums, with cancellations among paired nucleons and no strong nuclear enhancement expected.The relevant spin-content factors are estimated near 0.5 for nuclei with an odd proton or neutron and near zero for even nuclei.
  • Generalization to other DM candidates: The effective-Lagrangian framework is generalized from Majorana fermions to Dirac fermions, real or complex scalars, and real or complex vector fields.For scalar fields only spin-independent interactions are possible, while self-conjugate cases are recovered by setting the odd couplings to zero.
  • The detector event rate averages χ- and anti-χ–nucleus cross sections under the assumption that particles and antiparticles have equal densities.

3 WIMP elastic scattering on quarks

The section formulates WIMP–quark interactions using a compact operator basis and describes micrOMEGAs 2.2’s numerical extraction of low-energy coefficients. It also covers nucleon matrix elements, heavy-quark/gluon effects, and validation against MSSM results.

  • Operator basis: Only a few point-like operators are needed in the q^2 → 0 limit to expand generic WIMP–quark interactions.The basis separates spin-independent and spin-dependent interactions and supports scalar, fermion, and vector WIMPs.
  • Numerical approach: micrOMEGAs 2.2 estimates operator coefficients numerically by projecting zero-momentum χq → χq matrix elements onto auxiliary operators.CalcHEP generates the relevant amplitudes from the model’s Feynman rules, including quark and antiquark interference for even and odd operators.
  • Tensor operators: Tensor operators are extracted from the momentum dependence of forward scattering and then subtracted from the rest-limit amplitude to isolate the scalar coefficient.The method uses a numerical second derivative with respect to p1·p2 and does not require explicitly implementing tensor operators in CalcHEP.
  • Tensor operators: In typical MSSM models, accurate twist-2 treatment changes the result by less than 1%, although the correction can be larger when squark–WIMP mass differences are small.The automated extraction agrees numerically with the complete Drees–Nojiri calculation at the percent level.
  • Nucleon matching: The default effective-condensate treatment is not generally justified for box diagrams and is valid only when mq/(M˜q − Mχ) ≪ 1.micrOMEGAs uses this approximation by default but allows users to implement a more complete box-diagram calculation.

4 The special case of supersymmetry

The supersymmetric treatment incorporates model-specific higher-order effects into WIMP–nucleon scattering, while adapting the MSSM implementation for light-quark couplings and squark mixing. These corrections can materially depend on parameters such as μ and tan β, although some loop and QCD effects remain small or partially cancel.

  • Higher-order corrections: MSSM and related models receive explicit corrections to Higgs-exchange and quark- or squark-exchange box diagrams, extendable to CPVMSSM and NMSSM scenarios.The treatment includes complex phases and an additional singlet field in the corresponding extensions.
  • Model implementation: Light-quark Higgs couplings and SUSY-QCD effects must be retained because direct-detection scalar scattering is dominated by Higgs exchange involving light quarks.These effects are less safely neglected than in relic-density calculations.
  • Model implementation: Light-generation squark mixing is included because both Higgsino couplings and off-diagonal squark mixing terms depend on the quark mass.The model file therefore requires additional light-quark vertex terms despite SLHA inputs commonly assuming no first- and second-generation mixing.
  • Higher-order corrections: QCD corrections for heavy quarks typically increase σSI by about 5%, whereas SUSY-QCD corrections depend on the sign of μ and are enhanced at large tan β.For μ > 0 the cross section decreases after SUSY-QCD effects, while for μ < 0 heavy-Higgs and squark contributions increase.
  • Higher-order corrections: Loop-correction discrepancies are below the percent level in the compared cases but can become large in the general MSSM when squark and neutralino masses are similar.The expected correction scale is of order mq/(M˜q − Mχ) for heavy quarks.

5 Scattering rates on nuclei: form factors and velocity distribution

The direct-detection rate combines WIMP kinematics, nuclear form factors, and the halo velocity distribution for both spin-independent and spin-dependent scattering. The implementation uses Woods–Saxon or Gaussian approximations where appropriate, with comparisons showing good agreement in the stated recoil range.

  • Scattering rates on nuclei: The recoil spectrum is obtained by folding zero-momentum WIMP–nucleon cross sections with momentum-dependent nuclear form factors and integrating over the WIMP velocity distribution.The rate also depends on the local dark-matter density, detector mass, and exposure time.
  • 5.1 Spin independent interactions: For SI scattering, isotropic center-of-mass scattering gives a flat recoil-energy distribution up to Emax(v) before form-factor and velocity-distribution effects are included.The nucleus form factor depends on transfer momentum q.
  • 5.1 Spin independent interactions: The SI form factor is the Fourier transform of the normalized nuclear density, implemented with a Fermi distribution that yields the Woods–Saxon form factor.The default surface thickness is a = 0.52 fm, and the radius parameters are fitted from nuclear data.
  • 5.2 Spin dependent interactions: The SD calculation uses three nuclear structure functions to encode spin magnitude, spatial spin distribution, and their isoscalar/isovector combinations.Their zero-momentum normalization recovers the corresponding cross section at rest.
  • 5.2 Spin dependent interactions: For nuclei lacking precise SD form factors, a Gaussian distribution with an effective radius is fitted to known form factors, assuming Z exchange makes S11 dominant.The approximation works well, while some residual dependence on the dark-matter model remains through the q dependence of Sij.
  • 5.2 Spin dependent interactions: In model BP, Gaussian SD form factors agree well with exact form factors for 5–50 keV recoils, usually more closely than different precise form-factor sets agree.The approximation is especially adequate for light nuclei at small momentum, but is less reliable at large recoil energies.
  • 5.3 Velocity distribution of dark matter: The finite Galactic escape speed is bounded by 498 km/s < vmax < 608 km/s at 90% confidence, with median likelihood vmax = 544 km/s.This cutoff sets the upper support of the galactic-frame velocity distribution.

6 Description of routines

The module provides routines for configuring nucleon inputs, evaluating nuclear form factors, computing recoil spectra, and extracting event counts for direct-detection calculations.

  • Nucleon inputs: Proton and neutron quark-content coefficients can be redefined with setProtonFF and setNeutronFF, while NULL preserves defaults.The routines accept three-element double arrays for scalar, pseudoscalar-vector, and sigma coefficients.
  • Auxiliary calculations: The package also supplies utilities for scalar quark-content coefficients and MSSM loop factors, including tree-level operation when qBOX is NULL.FeScLoop supports spin-1/2 WIMPs and scalar squark-like particles, while MSSMDDtest computes MSSM amplitudes.
  • Nuclear form factors: Nucleus form factors are implemented for many isotopes, with named constants and routines supporting spin-dependent form-factor inspection.PlotSS displays a selected Sxx function against recoil energy for a chosen nucleus and energy range.
  • Recoil calculations: nucleusRecoil computes recoil distributions from the local dark-matter density, velocity distribution, nuclear properties, and optional loop correction.The resulting dNdE spectrum can be plotted or integrated over an energy interval with displayRecoilPlot and cutRecoilResult.
  • Nuclear form factors: The approximate spin-dependent form-factor treatment agrees well with exact form factors and with different form-factor sets.The approximation remains reliable for light nuclei because their momentum transfer is small; it is less suitable at large recoil energies.

7 Sample results

The sample results compare SI and SD predictions across SUGRA and non-SUGRA models and benchmark them against other public codes. Agreement is generally strong for dominant contributions, while exchange-specific discrepancies remain.

  • SUGRA models: Table 5 reports spin-independent proton and spin-dependent proton and neutron cross sections for sample SUGRA models.The models include modified mSUGRA-inspired benchmarks and non-universal SUGRA scenarios.
  • SUGRA models: Figure 3 compares recoil spectra dN/dE for a 131Xe detector in the BP and KP mSUGRA models.BP is shown with a full line and KP with a dashed line.
  • Code comparisons: For a given set of quark coefficients, micrOMEGAs agrees very well with Isajet for σSI when Higgs exchange dominates.Differences in squark exchange can produce 25% differences between the codes for SUGRA models.
  • Code comparisons: For σSD, micrOMEGAs is usually below Isajet, with discrepancies reaching a factor of 4 because of a sign discrepancy between Z and squark exchange diagrams.The comparison removes QCD and SUSY-QCD corrections from micrOMEGAs because the other codes neglect them.
  • Code comparisons: Against DarkSUSY 4.1, σSI discrepancies reach 25% and σSD discrepancies can reach 50% in SUGRA test models.The reported differences arise from Higgs-exchange mass choices and a factor-of-2 difference in the squark-exchange contribution.
  • Non-SUGRA models: In sample non-SUGRA models, axial-vector coefficient choices change σSD by at most 30%, less than their effect on scalar cross sections.This reduced dependence is associated with SUGRA cases where Z exchange completely dominates over squark exchange.

8 Installation

micrOMEGAs 2.2 is distributed as a single archive containing several implemented models, while users can still add other models using the documented procedure.

  • Installation: The micrOMEGAs_2.2.tgz archive contains implementations of the MSSM, CPV-MSSM, NMSSM, little Higgs, and right-handed neutrino models.The package is available from the micrOMEGAs webpage, with installation instructions provided separately.
  • Installation: Adding new models remains supported, but model files must preserve reserved identifiers and correctly assign quark PDG codes.S0 and V5 are reserved for auxiliary fields, and the direct-detection module is not compatible with earlier implementation assumptions.

9 Summary

The new direct-detection module automatically computes WIMP–nucleus scattering for generic dark-matter candidates and incorporates important quark, nuclear, and correction inputs. The examples show substantial scalar-interaction uncertainties but generally better-controlled spin-dependent uncertainties when Z exchange dominates.

  • Summary: The module supports Majorana or Dirac fermions and scalar or vector bosons, automatically obtaining the candidate mass, spin, and interactions after model implementation.It uses effective heavy-quark vertices to include dominant QCD corrections to Higgs exchange diagrams.
  • Summary: Light-quark interactions can dominate nuclear scattering, so scalar calculations require light-quark masses to be included in the model file.In the MSSM, this requires expanding model information beyond relic-density assumptions that treat first- and second-generation fermions as massless.
  • Summary: Scalar-interaction uncertainties from quark-density coefficients are very large, whereas spin-dependent uncertainties are generally more controlled when interactions arise through Z exchange.Other spin-dependent contributions can still have large uncertainties despite often lying several orders of magnitude below current limits.

Note added

An online tool was established for computing direct detection rates with micrOMEGAs 2.2.

  • The online tool computes direct detection rates using micrOMEGAs 2.2.

Appendix A - Box diagrams

The appendix addresses cases where tree-level amplitudes overestimate s-channel contributions and describes propagator modifications that reproduce box-diagram effects in suitable models.

  • Tree-level calculations overestimate s-channel amplitudes when M˜q − Mχ < mq.The paper recommends computing complete box diagrams or using the described procedure in this regime.
  • A simple modification of propagators can reproduce the box-diagram amplitude through loop functions.The box contribution is represented by modifying tree-level propagator denominators.
  • micrOMEGAs implements denominator modifications through a small change to CalcHEP.The approach applies directly to MSSM-like models with fermionic WIMPs interacting through scalar quarks.
  • The same strategy can extend to other models when appropriate loop factors are computed for the relevant WIMP and squark spins.

Appendix B - Spin dependent nucleus form factors

The appendices document spin-dependent nuclear form-factor implementations in micrOMEGAs, including alternative choices.

  • Table 8 lists the nucleus spin-dependent form factors implemented in micrOMEGAs.
  • Table 9 lists alternative nucleus spin-dependent form factors.

Appendix C - Additional routines

Additional routines support model-independent recoil calculations from WIMP mass and nucleon-level SI and SD cross sections, including event-rate output per detector mass.

  • The routines compute nucleus recoil energy from the WIMP mass and SI and SD nucleon cross sections.
  • nucleusRecoilAux returns the number of events per day and per kg of detector material.Inputs include proton and neutron SI and SD cross sections.
  • A negative proton or neutron cross section represents destructive interference between proton and neutron contributions.
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