  We construct models of dark matter with suppressed spin-independent
scattering cross section utilizing the existing simplified model framework.
Even simple combinations of simplified models can exhibit interference effects
that cause the tree level contribution to the scattering cross section to
vanish, thus demonstrating that direct detection limits on simplified models
are not robust when embedded in a more complicated and realistic framework. In
general for fermionic WIMP masses >10 GeV direct detection limits on the
spin-independent scattering cross section are much stronger than those coming
from the LHC. However these model combinations, which we call less-simplified
models, represent situations where LHC searches become more competitive than
direct detection experiments even for moderate dark matter mass. We show that a
complementary use of several searches at the LHC can strongly constrain the
direct detection blind spots by setting limits on the coupling constants and
mediators' mass. We derive the strongest limits for combinations of vector +
scalar, vector + "squark", and "squark" + scalar mediator, and present the
corresponding projections for the LHC 14 TeV for a number of searches:
mono-jet, jets + missing energy, and searches for heavy vector resonances.

High Energy Physics - Phenomenology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Experiment