-
\overline{e}Y_e^\dagger P_L
\ell  -  \overline{d}Y_d^\dagger q_L +
\varepsilon \overline{q_L}Y_u u$$ where $Y_f$ are Yukawa matrices. This has been used to exchange the more usual $ (H^\dagger H)^2 \overline{\chi} \chi$, and $ (H^\dagger H) \overline{\chi} \chi$ operators for the double-derivative interactions between dark matter and the Higgs given in eqn (\[higgs\]). Notice that it is possible to use the equations of motion to replace two operators ($\overline{\chi} \chi H^\dagger H$ and $\overline{\chi} \chi (H^\dagger H)^2$) with one (involving the DM and $H^\dagger D^2 H$), because I am only interested in the $h$-$\chi$-$\bar{\chi}$ interaction induced by these operators. The linear combination of operators $[\mu^2  H^\dagger H -\lambda (H^\dagger H)^2]\overline{\chi}\chi$, which is orthogonal to the combination in the Equations of Motion, gives a vanishing $h$-$\chi$-$\bar{\chi}$ interaction, due to the minimisation condition of the Higgs potential. As in the case of the $Z$, the derivative operators of eqn (\[higgs\]) are interesting, because they give a higgs coupling to dark matter $\propto p_h^2$, which has the desirable feature of being relevant at the LHC where the Higgs is in the effective theory, but not contributing at low energy.

This note focusses on the interactions of $\chi$ with the $Z$ (eqn \[inteff\]), and with the 