The Migdal effect in Semiconductors for the Effective Field Theory of Dark Matter Direct Detection

Abstract

The Migdal effect in semiconductors, prompt ionization from a primary nuclear scattering event, can be described across all kinematic regimes using an effective field theory that encodes the complex vibrational and electronic degrees of freedom of the crystal in measurable structure factors. Simultaneously, general dark matter-nucleus interactions can be systematically described using non-relativistic effective field theory operators. We combine these two effective field theory frameworks to calculate the Migdal effect in semiconductors for all ten dimension-six non-relativistic operators. From the effective Hamiltonian, we find that DM-nucleus scattering factorizes from the ionization and vibrational excitation signal as it does in the free-atom case. Using data from EDELWEISS that was taken with a germanium detector, we derive new experimental bounds on each operator and compare these limits to other direct-detection constraints in the literature. We find the accessible parameter space to be disfavored by bounds on heavy mediators contained in simple UV completions that generate the effective operators.

Figures (6)

• Figure 1: DM-Migdal phonon spectrum for a germanium detector (consisting of 7.7% Ge-73) for DM spin $j_\chi = 1/2$ and DM mass (left)$m_\chi = 10$ MeV and (right)$m_\chi = 100$ MeV. Each curve is normalized to a maximum value of 1 event/(kg-yr-eV), and represents the differential rate with respect to the phonon energy for the reference cross sections given in \ref{['tab:phonon_scale']} for each operator. Note, at $10$ MeV, deviations between operators are so small that we choose to show the spectrum on a log scale. In addition, for 10 MeV, the spectrum for $\mathcal{O}_3$ is similar to the spectra in green.
• Figure 2: DM-Migdal ionization spectrum for a germanium detector containing 7.7% Ge-73, for DM with spin $j_\chi = 1/2$ and for a DM mass of $m_\chi = 10$ MeV (in black, blue, and purple ) and $m_\chi = 1$ GeV (in red). Each curve is normalized to a maximum value of 1 event/(kg-yr-eV), and represents the differential rate with respect to the ionization energy for the reference cross sections given in \ref{['tab:ionization_scale']}. For heavier DM masses, the ionization spectrum for each operator approximately shares the same characteristic shape.
• Figure 3: We report the projected sensitivity for a 1 kg-year germanium detector (with 7.7% Ge-73) as a solid dark-blue line for $\mathcal{O}_4$, $\mathcal{O}_{6}$, and $\mathcal{O}_{10}$. Additionally, re-casted EDELWEISS direct detection constraints using data from EDELWEISS2020 are shaded in dark-blue. In dot-dashed light-blue, we show the mean free path, above which DM will scatter at least once when traversing 1700 m of the Earth's crust. We include combined direct detection constraints from Kang:2018rad and TOMAR2023102851, which we have re-casted to a DM-nucleon cross section, in pink. Each figure also contains the estimated EFT validity bound in dashed gray, which are projected from the relativistic couplings (see text for details). Work from Ramani2019, which is shown in dotted gray for $\mathcal{O}_4$ and is below the scale of the plot for $\mathcal{O}_6$ and $\mathcal{O}_{10}$, suggests the parameter space shown for these three operators is disfavored due to constraints on heavy mediator couplings to the SM.
• Figure 4: We report the projected sensitivity for a 1 kg-year germanium detector (with 7.7% Ge-73) as a solid dark-blue line for $\mathcal{O}_7$, $\mathcal{O}_8$, $\mathcal{O}_9$, and $\mathcal{O}_{11}$. Additionally, re-casted EDELWEISS direct detection constraints using data from EDELWEISS2020 are shaded in dark-blue. In dot-dashed light-blue, we show the mean free path, above which DM will scatter at least once when traversing 1700 m of Earth's crust. Additionally, we include combined direct detection constraints from Kang:2018rad and TOMAR2023102851, which we have re-casted to a DM-nucleon cross section, in pink. Each figure also contains the estimated EFT validity bound, which are projected from the relativistic couplings (see text for details).
• Figure 5: We report the projected sensitivity for a 1 kg-year germanium detector (with 7.7% Ge-73) as a solid dark-blue line for $\mathcal{O}_3$, $\mathcal{O}_5$. Additionally, re-casted EDELWEISS direct detection constraints using data from EDELWEISS2020 are shaded in dark-blue. In dot-dashed light-blue, we show the mean free path, above which DM will scatter at least once when traversing 1700 m of Earth's crust. Each figure also contains the estimated EFT validity bound, which are projected from the relativistic couplings (see text for details).