  In this paper, we propose two sets of different CP-sensitive observables
inspired by the Higgs production in association with the top quark. We employ a
Dark Matter simplified model that couples a scalar particle with three
generation fermions with a mass equal to that of the top quark. The
reconstruction of the kinematic variables is presented at NLO accuracy for
events associated with this massive scalar particle, which is assumed to be
vanishing to invisible decays in a detector such as ATLAS. We build these
observables by taking advantage of the similarity between the scalar coupling
with the top quark and the factorization theorem in the total scattering
amplitude, in order to represent the basis in which the phase space is
parameterized. A twofold approach employs the direct implementation of the
four-momentum phase space measure in building CP sensitive observables such as
$b_{2}$ for the Higgs, and the spin polarization of the top-quark decays in the
narrow width approximation for the employed model. We studied the asymmetries
of these distributions to test for any improvement in increasing the exclusion
region for the $g_{u_{33}}^S-g_{u_{33}}^P$ parameters associated with this
vanishing scalar particle. We have found no significant effect in the exclusion
limits by using the forward-backward asymmetry distributions and the full
shaped ones. The best limits obtained for $g_{u_{33}}^S$ at NLO accuracy were
obtained: $b_{2}:[-0.0425, 0.0475]$ $n_{4}: [-0.0450, 0.0450] $ for an
invisible scalar with a mass of 10$^{-2}$ GeV for a luminosity of 300 fb$^{-1}$
expected for the end of Run 3 of the LHC.
