running the operator coefficients should be included, and the quarks must be embedded in the nucleons.

To remove the $Z$, the Greens function for two quarks and two $\chi$s in the effective theory with a $Z$, should be matching to the same Greens function in the theory without a $Z$. Since the matching is performed at zero momentum for the fermion legs, the contact interactions of eqn (\[inteff\]) do not contribute, and the coefficients of the four-fermion operators of eqn (\[qDMd6\]) remain the same after the $Z$ is “matched out”. The $Z$ vertices were taken $\propto p_Z^2$ to obtain this.

The light quark currents $\overline{q} \g^\mu P_X q$ are conserved in QCD, so do not run. Also, since $\chi$ is a SM gauge singlet and the only dark sector particle below the TeV, I suppose that the operators with vector and axial vector $\chi$ currents do not mix below the TeV. See [*e.g.*]{} [@UliSDSI] about loop effects mixing various operators involving dark matter and the SM.

Finally, the quark currents can be embedded in nucleons $N = \{p,n\}$ using identities [@BBPS] such as N| \^ Q\_i |N &=& c\_[V,i]{}\^[N]{} N| \^\_N |N where $ c_{V,u}^{p}= c_{V,d}^{n} = 2$, and $c_{V,d}^{p} 