From quarks to nucleons in dark matter direct detection
  We provide expressions for the nonperturbative matching of the effective
field theory describing dark matter interactions with quarks and gluons to the
effective theory of nonrelativistic dark matter interacting with
nonrelativistic nucleons. We give the leading and subleading order expressions
in chiral counting. In general, a single partonic operator already matches onto
several nonrelativistic operators at leading order in chiral counting. Thus,
keeping only one operator at the time in the nonrelativistic effective theory
does not properly describe the scattering in direct detection. Moreover, the
matching of the axial--axial partonic level operator, as well as the matching
of the operators coupling DM to the QCD anomaly term, naively include momentum
suppressed terms. However, these are still of leading chiral order due to pion
poles and can be numerically important. We illustrate the impact of these
effects with several examples.
