Dark matter scattering on electrons: Accurate calculations of atomic excitations and implications for the DAMA signal

We revisit the WIMP-type dark matter scattering on electrons that results in atomic ionization and can manifest itself in a variety of existing direct-detection experiments. Unlike the WIMP-nucleon scattering, where current experiments probe typical interaction strengths much smaller than the Fermi constant, the scattering on electrons requires a much stronger interaction to be detectable, which in turn requires new light force carriers. We account for such new forces explicitly, by introducing a mediator particle with scalar or vector couplings to dark matter and to electrons. We then perform state-of-the-art numerical calculations of atomic ionization relevant to the existing experiments. Our goals are to consistently take into account the atomic physics aspect of the problem (e.g., the relativistic effects, which can be quite significant) and to scan the parameter space—the dark matter mass, the mediator mass, and the effective coupling strength—to see if there is any part of the parameter space that could potentially explain the DAMA modulation signal. While we find that the modulation fraction of all events with energy deposition above 2 keV in NaI can be quite significant, reaching $\sim 50\%$, the relevant parts of the parameter space are excluded by the XENON10 and XENON100 experiments.

Figure 1

Summary of results: Ratio of the expected signal for XENON100 (search of electron recoil with $\mathrm{\Delta }E>1\mathrm{keV}$ following Ref. [42, 43], left) and XENON10 (limit on ionization with arbitrary energy deposit [50], right)—assuming DAMA to be a positive WIMP detection—to the 90% confidence-level limits from those respective experiments, as a function of the DM mass, $m_{\chi }$, and the mediator mass, $m_v$. The contours denote the level of exclusion; for example, the line marked “2" contours the part of the parameter space for which the expected signal in XENON100/10 is 2 times larger than the signal that has been ruled out (90% C.L.). By combining both plots, it is seen that all regions of the parameter space are excluded.

Figure 2

Example diagram for interaction of a dark matter particle ($\chi$), with an electron via the exchange of mediator $\varphi$. Double line denotes a bound atomic electron ($E_{n\kappa }<0$), and the single electron line denotes a continuum state electron with energy $ϵ>0$.

Figure 3

Normalized distributions for the DM velocity in the earth frame [see Eq. (6)]. The solid black line (avg) corresponds to the DM velocity distribution in the solar frame, and the dotted blue (min) and dashed magenta (max) lines refer to the distributions in the earth frame around December 2 and June 2, respectively.

Figure 4

Comparison of the contribution of the $3s$ state to the atomic kernel of iodine in the relativistic and nonrelativistic approximations: (top) as a function of the energy deposition ($\mathrm{\Delta }E$) for a value of the momentum transfer of $q\simeq 9\mathrm{MeV}$, and (bottom) as a function of $q$ for $\mathrm{\Delta }E=4\mathrm{keV}$.

Figure 5

Core-state contributions to the atomic kernel [defined in Eq. (9)] for I as a function of the energy deposition, $\mathrm{\Delta }E$, at momentum transfer $q\simeq 4\mathrm{MeV}$. The $s$ states dominate the amplitude; this domination only increases further at larger $q$. The contributions from the $d$ states (not shown) are orders of magnitude smaller again.

Figure 6

Contribution of the $3s$ core state to the atomic kernel for iodine as a function of the momentum transfer, $q$, at $\mathrm{\Delta }E\simeq 2\mathrm{keV}$. Shown separately are the kernels with different values for the high-$l$ cutoff for the continuum-state electron orbital angular momentum. The higher-$l$ continuum states contribute significantly at low values of $q$, however, at the values relevant to this work ($q\gtrsim \mathrm{MeV}$), they contribute negligibly.

Figure 7

Plot of the differential cross section [defined in Eq. (8), with $m_{\chi }=10\mathrm{GeV}$, $m_v=10\mathrm{MeV}$, and for simplicity ${\alpha }_{\chi }=1$] for Na, I, Xe, Ge, and Tl as a function of the total energy deposition, $\mathrm{\Delta }E$. The kinks in the curves correspond to the opening of deeper atomic shells; see Table 1. There is a clear and significant $Z$ dependence, which is due to the low-$r$ scaling of the wave functions and the relativistic effects.

Figure 8

Plots showing the $m_{\chi }$ dependence of (top) the differential cross section, and (bottom) the oscillation fraction, for ionization of iodine as a function of the deposited energy, $\mathrm{\Delta }E$. For the plots we have taken $m_v=10\mathrm{MeV}$, and ${\alpha }_{\chi }=1$. Low values of $m_{\chi }$ lead to significantly lower cross sections, however, as $m_{\chi }$ increases the increase in the effect wanes. The energy dependence of the oscillations increases with decreasing $m_{\chi }$, since in these regions only part of the DM velocity distribution can give rise to an effect, and the event rate deviates from Eq. (17).

Figure 9

Total cross section (${\mathrm{cm}}^2$) for the ionization of NaI in the 2–6 keV interval assuming ${\alpha }_{\chi }=\alpha$ for the average (i.e. spring/fall) DM velocity distribution, including the Gaussian resolution profile (18).

Figure 10

Unmodulated event rate $R_0$ for NaI in the 2–6 keV interval assuming ${\alpha }_{\chi }=\alpha$ in units of $\mathrm{cpd}/\mathrm{kg}/\mathrm{keV}$: (top) assuming perfect detector resolution; (bottom) including a Gaussian resolution profile (18).

Figure 11

Modulation amplitude $R_m$ for NaI in the 2–6 keV interval assuming ${\alpha }_{\chi }=\alpha$ in units of $\mathrm{cpd}/\mathrm{kg}/\mathrm{keV}$ (including the Gaussian resolution profile).

Figure 12

The calculated modulation fraction ($R_m/R_0$) expected for the scintillation signal in the 2–6 keV interval for NaI (including the Gaussian resolution profile).

Figure 13

The value that ${\alpha }_{\chi }$ must take in order to reproduce the DAMA modulation signal of 0.0112 $\mathrm{cpd}/\mathrm{kg}/\mathrm{keV}$ in the 2–6 keV interval: (top) assuming perfect detector resolution; (bottom) including the Gaussian resolution profile (18).

Figure 14

Calculated modulated event rate spectrum $R_m$ for DAMA for a few specific choices of DM parameters which are able to replicate the amplitude of the observed modulation.

Figure 15

Calculated scintillation event rate for Xe (top) for 2 PE, and (bottom) for 3 PE.

Figure 16

The calculated (top) unmodulated event rate (for fixed ${\alpha }_{\chi }=\alpha$), and (bottom) modulation fraction ($R_m/R_0$), for the scintillation signal in the 3–14 PE interval (corresponding to 2–6 keV) for Xe.

Figure 17

(Top) The umodulated event rate, and (bottom) the modulation amplitude, that would be expected in the XENON100 scintillation experiment in the 3–14 PE interval (corresponding to 2–6 keV) assuming the DAMA modulation signal is a positive WIMP detection (the value of ${\alpha }_{\chi }$ for each point on the parameter plot is shown in Fig. 13).

Figure 18

Ratio of the calculated event rate for DAMA (in the 2–6 keV interval) to that expected for XENON100 (in the 3–14 PE interval). Note that the ratio is highly dependent on the detector efficiency and resolution, but is essentially independent of the DM velocity distribution.

Figure 19

Calculation of the modulated ionization event rate spectrum $R_m$ for xenon (relevant to XENON100) for a few specific choices of DM parameters which are able to replicate the amplitude of the observed DAMA modulation. The corresponding signals generated in NaI are plotted in thin solid lines (almost indistinguishable from the xenon rates on this scale).

Figure 20

Unmodulated event rate (top), and modulation fraction (bottom), for DAMA in the 2–6 keV interval, for the massless mediator case $(m_v=0)$.

Figure 21

(Top) The unmodulated event rate for XENON100 in the 3–14 PE interval (corresponding to 2–6 keV), for the massless mediator case $(m_v=0)$. (Bottom) Ratio of the event rate for XENON100 to that of DAMA (in the $2–6\mathrm{keV}/3–14\mathrm{PE}$ range) for the $m_v=0$ case.

Figure 22

Calculation of (top) the unmodulated single primary-electron ionization signal in xenon (relevant to the XENON10 experiment [50]) for a fixed coupling of ${\alpha }_{\chi }=\alpha$, and (bottom) the modulation fraction.

Figure 23

Calculation of the spectral shape for the single primary-electron ionization signal (unmodulated) in xenon for a fixed coupling of ${\alpha }_{\chi }={10}^{-2}\alpha$ and $m_v=0.3\mathrm{MeV}$, for a few DM masses.

Figure 24

Calculation of the expected unmodulated ionization-only signal in xenon (relevant to the XENON10 experiment [50]) assuming the DAMA modulation signal (19) is a positive WIMP detection.

Figure 25

Calculations for the expected ionization-only signal in xenon (relevant to the XENON10 experiment [50]) assuming the modulation signal observed in the XENON100 scintillation experiment [43] is due to WIMP–electron scattering. (Top) Value that ${\alpha }_{\chi }$ must take to explain the XENON100 modulation (midpoint); (bottom) the resulting unmodulated event rate $R_0$ for XENON10 (single-electron primary ionizations only).

Figure 26

Comparison of different Lorentz structures for the $3s$ core-state contribution to the atomic kernel for I as a function of the energy deposition, $\mathrm{\Delta }E$ (with $q\simeq 4\mathrm{MeV}$), and of the momentum transfer, $q$ (with $\mathrm{\Delta }E\simeq 2\mathrm{keV}$). The pseudoscalar case gives a large effect (at higher $q$), since in this case the radial integrals include a contribution from initial and final $s$-states with $L=1$ ($L=0$ for the $s-s$ contribution to the vector and scalar cases); the pseudovector case (temporal contribution) gives by far the smallest contribution due to very large cancellations in the radial integrals, see Appendix pp2. The pseudovector case here includes only the temporal part of the interaction; to lowest-order, the spatial components for the pseudovector case behave like the vector/scalar case.

Figure 27

Plots of the atomic kernel (9) for several dominating core states of Na, Ge, I, and Xe, as a function of momentum transfer $q$, for a fixed energy deposition $\mathrm{\Delta }E=2.0\mathrm{keV}$.

Figure 28

Plots of the atomic kernel (9) for a few dominating core states of Na, Ge, I, and Xe, as a function of energy deposition $\mathrm{\Delta }E$, for a fixed momentum transfer $q=3.73\mathrm{MeV}$.