Revisiting general dark matter-bound-electron interactions

In this Letter we revisit general dark matter (DM)-bound-electron interactions studied previously in the influential work [R. Catena {\it et al.,} Atomic responses to general dark matter-electron interactions, Phys. Rev. Res. 2, 033195 (2020)] For the most general DM-electron nonrelativistic or relativistic interactions for DM with spin up to 1, we find the average ionization matrix element squared can be organized into three terms, each of which is a product of a DM response function ($a_{0,1,2}$) and a linear combination ($\widetilde W_{0,1,2}$) of the four atomic response functions ($W_{1,2,3,4}$) given in that work, $ \widetilde W_0 = W_1, \, \widetilde W_1 = |\pmb{v}_0^\perp|^2 W_1 - 2 {m_e\, \pmb{q}\cdot \pmb{v}_0^\perp \over \pmb{q}^2} W_2 + W_3,\, \widetilde W_2 = { (\pmb{q}\cdot \pmb{v}_0^\perp)^2 \over \pmb{q}^2} W_1 - 2 {m_e\, \pmb{q}\cdot \pmb{v}_0^\perp \over \pmb{q}^2} W_2 + {m_e^2 \over \pmb{q}^2}W_4$. Furthermore, we find a crucial minus sign was missed for the calculation of $W_2$ in that work, which has significant phenomenological consequences when explaining experimental bounds on specific DM scenarios. Due to the corrected sign, there can be significant cancellations between the $W_2$ and $W_{3,4}$ terms, so that $\widetilde W_{1,2}$ are dominated by the usual response function $W_1$ in some cases. Many DM scenarios involving DM or electron axial-vector current can yield $W_2$ and thus are potentially affected by the sign. As an example, we show that the recent XENON1T constraint on the fermionic DM anapole moment is weakened by a factor of 2 or so. We also present a complete list of NR operators for spin-1 DM and compute their contributions to the DM response functions.