  Monojet searches using Effective Field Theory (EFT) operators are usually
interpreted as a robust and model independent constraint on direct detection
(DD) scattering cross-sections. At the same time, a mediator particle must be
present to produce the dark matter (DM) at the LHC. This mediator particle may
be produced on shell, so that direct searches for the mediating particle can
constrain the effective operator being applied to monojet constraints. In this
first paper, we do a case study on t-channel models in monojet searches, where
the (Standard Model singlet) DM is pair produced via a t-channel mediating
particle, whose supersymmetric analogue is the squark. We compare monojet
constraints to direct constraints on single or pair production of the mediator
from multi-jets plus missing energy searches and we identify the regions where
the latter dominate over the former. We show that computing bounds using
supersymmetric simplified models and in the narrow width approximation, as done
in previous work in the literature, misses important quantitative effects. We
perform a full event simulation and statistical analysis, and we compute the
effects of both on- and off-shell production of the mediating particle, showing
that for both the monojet and multi-jets plus missing energy searches,
previously derived bounds provided more conservative bounds than what can be
extracted by including all relevant processes in the simulation. Monojets and
searches for supersymmetry (SUSY) provide comparable bounds on a wide range of
the parameter space, with SUSY searches usually providing stronger bounds,
except in the regions where the DM particle and the mediator are very mass
degenerate. The EFT approximation rarely is able to reproduce the actual
limits. In a second paper to follow, we consider the case of s-channel
mediators.
