Of course, these searches present all sorts of questions. Why do some searches see something at close to a four sigma level, while others rule out the same thing to similar degrees of certainty?
If we are seeing something, why are we seeing particles at masses that have been pretty definitively ruled out by precision measurements of weak force decays and neutrino searches? There shouldn't be anything, either a fundamental particle or a composite one, between bottom quark mass (between 4 and 5 GeV) and 40 GeV that interacts with the electromagnetic or weak force or is quark or includes a quark or is a Standard Model Higgs Boson. Certainly, nothing in the Standard Model fits. All known baryons and all known mesons are too light, topping out at about 5 GeV, apart from a couple of bottom-bottom quark possibilities, and the bottom-bottom quark mesons would be extremely unstable, making them poor dark matter candidates. (Top quarks at 173 GeV are too heavy in addition to being unstable.)
One could imagine, without stretching the Standard Model too far, right handed neutrinos whose masses are many orders of magnitude heavier than their left handed cousins. Glueballs aren't entirely ruled out, but don't seem like a very good fit either. Ground state glueballs should have much lower masses and even if they might be stable in a ground state, they probably wouldn't be stable at higher mass excited states. They are also predicted to get lighter as they gain momentum. Other exotic mesons are also a poor fit.
There are also good theoretical grounds to think that a graviton, if one exists, has no mass.
Even SUSY particles in most of the less ambitious versions of that theory (including the SUSY Higgs boson), are ruled out in that mass range by collider experiments, although Lubos holds out hope for a photino or bino. Gordon Kane, a strong SUSY supporter, hasn't seriously proposed any particle not yet discovered under 120 GeV, and suggests a gluino mass might be as low as 500 GeV. Even LEP put a limit on the order of 50 GeV on actual particles, although the scalar mass parameter used to determine particle masses in SUSY could be as low as 12 GeV (see also here for post-LEP limits).
There are abundant varieties of candidates for heavier WIMP-like dark matter particles, and are a few candidates for lighter WIMP-like dark matter particles, but really nothing but a right handed neutrino is well motivated in this mass range, and nothing in any well developed theory explains why some of these experiments should be seeing something with considerable significance that others are excluding with considerable significance.
In addition to the limitations on mass, astronomy also tightly limits the cross sections of interactions for dark matter based on examples like the bullet cluster interaction.
We have combined results from new X-ray, optical and lensing observations and our N-body simulations of the merging galaxy cluster 1E 0657-56 in order to derive an upper limit on the self-interaction cross-section of dark matter particles, sigma /m. We give constraints on sigma/m based on two independent methods: from the lack of offset between the total mass peak and galaxy centroid of the subcluster that would arise during the merger due to drag on the subcluster halo from DM particle collisions, and from the lack of a decreased mass-to-light ratio of the subcluster due to scattering of DM particles. From the former, we find sigma/m < 1.25 cm2 g−1, and from the latter, sigma /m < 0.7 cm2 g−1, which includes the uncertainty in the impact parameter of the merger (upper limits are from 68% confidence intervals). Our best constraint is a modest improvement of the previous best constraint from conservative analytic estimates of sigma /m < 1 cm2 g−1 (M04). Furthermore, our limit of sigma/m < 1.25 cm2 g−1 is more robust than the best analytic limit, since this method does not depend on the assumption that the subcluster and main cluster M/L ratios were equal prior to the merger. Previous studies have found that sigma /m ∼ 0.5−5 cm2 g−1 is needed produce the observational effects that self-interacting dark matter has been invoked to explain (e.g., nonpeaked galaxy mass profiles and the underabundance of small halos within larger systems). Our results rule out almost this full range of values, at least under the assumption that is velocity-independent.
In other words, the data from long range astronomy would suggest that the cross section of interaction for dark matter needs to be 0.5-1.25 cm2 g-1, which takes a bit of a unit conversion to match to the way that the dark matter parameter searches report their data in chart form.
"The cross section for a typical interaction involving a neutrino is 5*10^-44 (E/[1 MeV])^2 cm^2."
Usually, these sorts of conflicts end up being resolved in favor of the no new physics direction after more evidence accumulates or more analysis is conducted.
The contradictory evidence and ever tightening constraints imposed by efforts to directly detect dark matter pose a serious quandry for physics. Gravitationally driven behavior attributed to dark matter is common place in astronomy. It is everywhere and well quantified and described at different scale and in different kinds of systems. But, we are increasingly close to ruling out any candidates for dark matter that could fit the criteria imposed by the observations in astronomy that motivate the search.
Since dark matter candidates are often the main phenomenological predictions of leading beyond the Standard Model physics, a failure to find to directly detect the dark matter candidates that they predict also undermines those beyond the Standard Model theories.