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Model Independent Form Factors for Spin Independent Neutralino-Nucleon Scattering from Elastic Electron Scattering Data

Gintaras Duda, Ann Kemper, Paolo Gondolo

TL;DR

The paper addresses nuclear-structure uncertainties in direct dark matter searches by replacing conventional Helm form factors with model-independent analytic form factors derived from elastic electron scattering. Using Sum-of-Gaussians (SOG) and Fourier-Bessel (FB) expansions, it derives analytic $F(q)$ for nuclei such as $^{28}$Si, $^{40}$Ca, $^{70}$Ge, and $^{129}$Xe and compares them to the Helm parameterization. It finds 10–20% shifts in predicted cross sections and upper limits, especially for heavier targets, demonstrating that model-independent form factors can significantly affect interpretation of direct-detection results. The work advocates adopting FB/SOG form factors in place of, or alongside, Helm-based estimates and discusses implications for mass versus charge distributions, with implementation in the DarkSUSY toolkit.

Abstract

Theoretical calculations of neutralino-nucleon interaction rates with various nuclei are of great interest to direct dark matter searches such as CDMS, EDELWEISS, ZEPLIN, and other experiments since they are used to establish upper bounds on the WIMP-proton cross section. These interaction rates and cross sections are generally computed with standard, one or two parameter model-dependent nuclear form factors, which may not exactly mirror the actual form factor for the particular nucleus in question. As is well known, elastic electron scattering can allow for very precise determinations of nuclear form factors and hence nuclear charge densities for spherical or near-spherical nuclei. We use charge densities derived from elastic electron scattering data to calculate model independent, analytic form factors for various target nuclei important in dark matter searches, such as Si, Ge, S, Ca and others. We have found that for nuclear recoils in the range of 1-100 keV significant differences in cross sections and rates exist when the model independent form factors are used: at 30 keV nuclear recoil the form factors squared differ by a factor of 1.06 for $^{28}$Si, 1.11 for $^{40}$Ca, 1.27 for $^{70}$Ge, and 1.92 for $^{129}$Xe. We show the effect of different form factors on the upper limit on the WIMP-proton cross section obtained with a hypothetical $^{70}$Ge detector during a 100 kg-day effective exposure. Helm form factors with various parameter choices differ at most by 10--20% from the best (Fourier Bessel) form factor, and can approach it to better than 1% if the parameters are chosen to mimic the actual nuclear density.

Model Independent Form Factors for Spin Independent Neutralino-Nucleon Scattering from Elastic Electron Scattering Data

TL;DR

The paper addresses nuclear-structure uncertainties in direct dark matter searches by replacing conventional Helm form factors with model-independent analytic form factors derived from elastic electron scattering. Using Sum-of-Gaussians (SOG) and Fourier-Bessel (FB) expansions, it derives analytic for nuclei such as Si, Ca, Ge, and Xe and compares them to the Helm parameterization. It finds 10–20% shifts in predicted cross sections and upper limits, especially for heavier targets, demonstrating that model-independent form factors can significantly affect interpretation of direct-detection results. The work advocates adopting FB/SOG form factors in place of, or alongside, Helm-based estimates and discusses implications for mass versus charge distributions, with implementation in the DarkSUSY toolkit.

Abstract

Theoretical calculations of neutralino-nucleon interaction rates with various nuclei are of great interest to direct dark matter searches such as CDMS, EDELWEISS, ZEPLIN, and other experiments since they are used to establish upper bounds on the WIMP-proton cross section. These interaction rates and cross sections are generally computed with standard, one or two parameter model-dependent nuclear form factors, which may not exactly mirror the actual form factor for the particular nucleus in question. As is well known, elastic electron scattering can allow for very precise determinations of nuclear form factors and hence nuclear charge densities for spherical or near-spherical nuclei. We use charge densities derived from elastic electron scattering data to calculate model independent, analytic form factors for various target nuclei important in dark matter searches, such as Si, Ge, S, Ca and others. We have found that for nuclear recoils in the range of 1-100 keV significant differences in cross sections and rates exist when the model independent form factors are used: at 30 keV nuclear recoil the form factors squared differ by a factor of 1.06 for Si, 1.11 for Ca, 1.27 for Ge, and 1.92 for Xe. We show the effect of different form factors on the upper limit on the WIMP-proton cross section obtained with a hypothetical Ge detector during a 100 kg-day effective exposure. Helm form factors with various parameter choices differ at most by 10--20% from the best (Fourier Bessel) form factor, and can approach it to better than 1% if the parameters are chosen to mimic the actual nuclear density.

Paper Structure

This paper contains 9 sections, 32 equations, 8 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Nuclear Charge Densities and Form Factors
  3. Elastic Electron Scattering and Model Independent Form Factors
  4. Nuclear Charge Density Parameters from Muonic Atom Spectroscopy
  5. Discussion and Results
  6. Charge vs. Mass Form Factors
  7. Conclusion
  8. Acknowledgements
  9. Appendix I: Nuclear Charge Density Parameters for Nuclear Form Factors

Figures (8)

  • Figure 1: Charge density for $^{40}$Ca and $^{208}$Pb as obtained from elastic electron scattering data.
  • Figure 2: Helm, FB, and SOG form factors for $^{28}$Si versus nuclear radius as obtained from elastic electron scattering data. Lines for momentum transfers corresponding to 10 and 100 keV nuclear recoil are included.
  • Figure 3: Charge density for $^{70}$Ge in the Woods-Saxon and FB parameterizations.
  • Figure 4: Helm and FB form factors for $^{70}$Ge.
  • Figure 5: Helm and FB form factors for $^{70}$Ge with 10 and 100 keV recoil lines for small momentum transfers.
  • ...and 3 more figures