First-principles upper bounds on dark matter-electron scattering rates from condensed matter sum rules
A wide variety of condensed matter systems are used or proposed as detectors to search for dark matter-electron scattering. In general, the scattering rate depends on detailed knowledge of the electronic properties of these systems. However, when dark matter couples to electron density, the dark matter-electron scattering rate can be related to the electron energy loss function, whose integrals are bounded by first-principles sum rules that rely on only a few macroscopic target properties. In this paper, we use these first-principles sum rules to derive upper bounds on the dark matter-electron scattering rate depending on only a few material properties: the plasma frequency $ω_\text{p}$, the target mass density $ρ_T$, and the static (longitudinal) dielectric function at finite momentum transfer, $\varepsilon(q, 0)$. The bulk material properties $ω_\text{p}$ and $ρ_T$ vary only over a limited range across a wide variety of materials, and to a good approximation, the generic large-$q$ dependence of $\varepsilon(q, 0)$ can be understood from a simple scaling law depending only on $ω_\text{p}$ which we verify with analytic and numerical examples. Thus, our upper bounds are largely material-agnostic, and place a fundamental limit on the sensitivity of any dark matter-electron direct detection experiment probing the coupling to electron density.