Incoherent Fermionic Dark Matter Absorption with Nucleon Fermi Motion

We investigate the incoherent regime of the fermionic dark matter absorption by nuclei using the relativistic Fermi gas model and nuclear form factors. With the momentum transfer being roughly equal to the dark matter mass $m_{\chi }$, the incoherent regime contributes significantly to the absorption process for $m_{\chi }\gtrsim 100$,MeV with a spin-independent operator and for even smaller mass with a spin-dependent one. We also compare the situations for various target nuclei (${}^{131}$Xe, ${}^{72}$Ge, ${}^{40}$Ar, ${}^{20}$Ne and ${}^4$He) that are typically used in the dark matter direct detection. A heavier nucleus actually has the advantage of probing the incoherent scattering of the fermionic absorption dark matter. Observing both the coherent and incoherent contributions would be an important justification of the fermionic dark matter absorption.