Accurate electron-recoil ionization factors for dark matter direct detection in xenon, krypton, and argon

While most scintillation-based dark matter experiments search for weakly interacting massive particles (WIMPs), a sub-GeV WIMP-like particle may also be detectable in these experiments. While dark matter of this type and scale would not leave appreciable nuclear recoil signals, it may instead induce ionization of atomic electrons. Accurate modeling of the atomic wave functions is key to investigating this possibility, with incorrect treatment leading to a large suppression in the atomic excitation factors. We have calculated these atomic factors for argon, krypton, and xenon and present the tabulated results for use with a range of dark matter models. This is made possible by the separability of the atomic and dark matter form factor, allowing the atomic factors to be calculated for general couplings; we include tables for vector, scalar, pseudovector, and pseudoscalar electron couplings. Additionally, we calculate electron-impact total ionization cross sections for xenon using the tabulated results as a test of accuracy. Lastly, we provide an example calculation of the event rate for dark matter scattering on electrons in XENON1T and show that these calculations depend heavily on how the low-energy response of the detector is modeled.

Figure 1

Comparison of the total atomic excitation factor as a function of energy deposition for xenon for vector (solid line), scalar (dotted line), pseudovector (dashed line), and pseudoscalar (dashed-dotted line) electron couplings at a fixed momentum transfer of $q=4\mathrm{MeV}$.

Figure 2

Total atomic excitation factors as a function of momentum transfer for xenon at a fixed energy deposition of $E=2\mathrm{keV}$, with the same electron couplings as in Fig. 1.

Figure 3

Velocity-averaged differential cross section for xenon when accounting for the hole-particle interaction (solid line), and when excluding it (“frozen” Hartree-Fock, dashed line). In this case, we have set the DM mass to be $m_{\chi }=1\mathrm{GeV}$. The DM form factor is set to $F_{\chi }=1$, corresponding to a heavy mediator as per Eq. (7). We use a vector electron coupling with free-electron cross section ${\overline{\sigma }}_e={10}^{-37}{\mathrm{cm}}^2$.

Figure 4

Goldstone diagrams for the direct (left), exchange (middle), and hole-particle (right) contributions to the atomic potential. The wavy line is the Coulomb interaction, external lines are either bound atomic states or unbound continuum states, and internal lines are core states (holes). The hole-particle effect arises from the deviation of the direct potential for the ejected continuum electron from that of the bound electrons.

Figure 5

Goldstone diagrams for core polarization correction to matrix elements, showing the direct (left) and exchange (right) contributions. The diagram notation is the same as in Fig. 4, with the dotted line representing interaction with the external field (DM particle), and the internal forward lines representing virtual excited atomic states. In the all-order RPA method, each of the external field vertices is replaced with these four diagrams; this process is repeated until the matrix elements converge.

Figure 6

Calculations of total electron impact ionization cross sections for xenon with (HF) and without (frozen HF) accounting for the hole-particle interaction, and comparison to calculations by Bartlett et al. [45] and experimental results from Sorokin et al. [46] and Wetzel et al. [47].

Figure 7

Example event-rate calculations before (dotted lines) and after (solid lines) smearing with the Gaussian and correcting for detection efficiency [49] of the XENON1T detector, corresponding to Eqs. (24) and (26), respectively. The spikes seen in the calculated event rate are due to deeper shells becoming accessible as the energy deposition increases, which we can also see in the cross section. A cutoff has been applied to the observable event rate at 0.5 keV to indicate the minimum energy threshold. We take ${\overline{\sigma }}_e={10}^{-37}{\mathrm{cm}}^2$. Uncertainties in the detector response lead to large, underestimated uncertainties in observable event rates at energies close to the detector threshold.

Figure 8

Binned event rates (1-keV bin widths) for a range of DM masses with vector electron couplings and a heavy mediator ($F_{\chi }=1$), calculated according to the modeling of the XENON100 detector, using Eqs. (28) and (29). We take ${\overline{\sigma }}_e={10}^{-37}{\mathrm{cm}}^2$. Uncertainties in the detector response used here may also lead to large, underestimated uncertainties in observable event rates at energies close to the detector threshold.