  We study the validity of effective field theory (EFT) interpretations of
monojet searches for dark matter at the LHC for vector and axial-vector
interactions. We show that the EFT approach is valid when the mediator has mass
m_med greater than 2.5 TeV. We find that the current limits on the contact
interaction scale Lambda in the EFT apply to theories that are perturbative for
dark matter mass m_DM < 800 GeV. However, for all values of m_DM in these
theories, the mediator width is larger than the mass, so that a particle-like
interpretation of the mediator is doubtful. Furthermore, consistency with the
thermal relic density occurs only for 170 <m_DM < 520 GeV. For lighter mediator
masses, the EFT limit either under-estimates the true limit (because the
process is resonantly enhanced) or over-estimates it (because the missing
energy distribution is too soft). We give some `rules of thumb' that can be
used to estimate the limit on Lambda (to an accuracy of about 50%) for any dark
matter and mediator masses from knowledge of the EFT limit. We also compare the
relative sensitivities of monojet and dark matter direct detection searches
finding that both dominate in different regions of the m_DM-m_med plane.
Comparing only the EFT limit with direct searches is misleading and can lead to
incorrect conclusions about the relative sensitivity of the two search
approaches.
