  The usage of Effective Field Theories (EFT) for LHC new physics searches is
receiving increasing attention. It is thus important to clarify all the aspects
related with the applicability of the EFT formalism in the LHC environment,
where the large available energy can produce reactions that overcome the
maximal range of validity, i.e. the cutoff, of the theory. We show that this
does forbid to set rigorous limits on the EFT parameter space through a
modified version of the ordinary binned likelihood hypothesis test, which we
design and validate. Our limit-setting strategy can be carried on in its
full-fledged form by the LHC experimental collaborations, or performed
externally to the collaborations, through the Simplified Likelihood approach,
by relying on certain approximations. We apply it to the recent CMS mono-jet
analysis and derive limits on a Dark Matter (DM) EFT model. DM is selected as a
case study because the limited reach on the DM production EFT Wilson
coefficient and the structure of the theory suggests that the cutoff might be
dangerously low, well within the LHC reach. However our strategy can also be
applied to EFT's parametrising the indirect effects of heavy new physics in the
Electroweak and Higgs sectors.
