  Weak scale supersymmetry (SUSY) remains a prime explanation for the radiative
stability of the Higgs field. A natural account of the Higgs boson mass,
however, strongly favors extensions of the Minimal Supersymmetric Standard
Model (MSSM). A plausible option is to introduce a new supersymmetric sector
coupled to the MSSM Higgs fields, whose associated states resolve the little
hierarchy problem between the third generation squark masses and the weak
scale. SUSY also accomodates a weakly interacting cold dark matter (DM)
candidate in the form of a stable neutralino. In minimal realizations, the
thus-far null results of direct DM searches, along with the DM relic abundance
constraint, introduce a level of fine-tuning as severe as the one due to the
SUSY little hierarchy problem. We analyse the generic implications of new SUSY
sectors parametrically heavier than the minimal SUSY spectrum, devised to
increase the Higgs boson mass, on this little neutralino DM problem. We focus
on the SUSY operator of smallest scaling dimension in an effective field theory
description, which modifies the Higgs and DM sectors in a correlated manner.
Within this framework, we show that recent null results from the LUX experiment
imply a tree-level fine-tuning for gaugino DM which is parametrically at least
a few times larger than that of the MSSM. Higgsino DM whose relic abundance is
generated through a thermal freeze-out mechanism remains also severely
fine-tuned, unless the DM lies below the weak boson pair-production threshold.
As in the MSSM, well-tempered gaugino-Higgsino DM is strongly disfavored by
present direct detection results.
