  We perform a comprehensive analysis for the light scalar dark matter (DM)in
the Inert two Higgs doublet model (i2HDM) with compressed mass spectra, small
mass splittings among three $\mathbb{Z}_2$ odd particles---scalar $S$,
pseudo-scalar $A$, and charged Higgs $H^\pm$. In such a case, the
co-annihilation processes play a significant role to reduce DM relic density.
As long as a co-annihilation governs the total interaction rate in the early
universe, a small annihilation rate is expected to reach a correct DM relic
density and its coupling $\lambda_S$ between DM pair and Higgs boson shall be
tiny. Consequently, a negligible DM-nucleon elastic scattering cross section is
predicted at the tree-level. In this work, we include the one-loop quantum
corrections of the DM-nucleon elastic scattering cross section. We found that
the quartic self-coupling $\lambda_2$ between $\mathbb{Z}_2$ odd particles
indeed contributes to the one-loop quantum correction and behaves non-trivially
for the co-annihilation scenario. Interestingly, the parameter space, which is
allowed by the current constraints considered in this study, can predict the DM
mass and annihilation cross section at the present compatible with the AMS-02
antiproton excess. The parameter space can be further probed at the future high
luminosity LHC.
