  Assuming for Weakly Interacting Massive Particles (WIMPs) a Maxwellian
velocity distribution in the Galaxy we provide an assessment of the sensitivity
of existing Dark Matter (DM) direct detection (DD) experiments to operators up
to dimension 7 of the relativistic effective field theory describing dark
matter interactions with quarks and gluons . In particular we focus on a
systematic approach, including an extensive set of experiments and large number
of couplings, both exceeding for completeness similar analyses in the
literature. The relativistic effective theory requires to fix one coupling for
each quark flavor, so in principle for each different combination the bounds
should be recalculated starting from direct detection experimental data. To
address this problem we propose an approximate model-independent procedure that
allows to directly calculate the bounds for any combination of couplings in
terms of model-independent limits on the Wilson coefficients of the
non-relativistic theory expressed in terms of the WIMP mass and of the
neutron-to-proton coupling ratio $c^n/c^p$. We test the result of the
approximate procedure against that of a full calculation, and discuss its
possible pitfalls and limitations. We also provide a simple interpolating
interface in Python that allows to apply our method quantitatively.
