Little Higgs is a framework designed for breaking the electroweak symmetry of the Standard Model without running into the hierarchy problem. Roughly speaking, the idea is that some new strong interactions at 10 TeV produce bound states, a subset of which ends up being much lighter than 10 TeV. These mesons, described as pseudo-goldstone bosons, are identified with the Higgs field in the Stadard Model. All this may appear involved, but similar dynamics is observed in the real-life QCD, where relatively light pions emerge as bound states. In Little Higgs, somewhat more complicated structures (new gauge bosons and new quarks) at the TeV scale are required to keep the electroweak scale ~ 100 GeV stable.
Little Higgs models have received a lot of attention and there is a good hope that something like that may turn up at the LHC. However, generic models have a hard time to comply with electroweak precision constraints. One way out is to introduce a parity symmetry that forces the new TeV scale particles to couple only in pairs to the Standard Model. In such a case, the new particles can contribute to the electroweak observables only via loop processes and their contribution may be sufficiently suppressed. Such a symmetry, dubbed T-parity, were introduced by Cheng and Low. Little Higgs models with T parity, although somewhat more involved than the minimal ones, lead to electroweak symmetry breaking without fine-tuning. As a by-product, the lightest particle with negative T parity, usually one of the new neutral gauge bosons, is forbidden to decay. This 'heavy photon' thus constitutes a nice dark matter candidate. Perfect.
From the recent paper by Hill&Hill it follows that things are not so bright. The paper is rather technical but the message is clear. Little Higgs models are usually formulated as a low energy effective theory without bothering about the strongly interacting theory at 10 TeV that gave birth to it. Hill^2 argue that in any conceivable UV completion of the Little Higgs models the T-parity is broken. The effect is induced by anomalies and is similar in spirit to that allowing the pion to decay into two photons. At the technical level, the low energy lagrangian of Little Higgs models should be augmented with the Wess-Zumino-Witten term that reflects the structure of anomalies in the UV theory. This WZW term breaks T-parity
What are the consequences? The new T-parity breaking terms should not mess up with the precision electroweak observables as they are related to anomalies and therefore loop-suppressed. But the lightest T-odd particle is no longer stable and does not constitute a good dark matter candidate. In particular, the heavy photon may decay into two W bosons, too quickly to play a role of a dark matter particle.
In the earlier paper by the same authors you can find more technical details about the WZW terms in the context of Little Higgs.
What if the heavy photon is lighter than a pair of W bosons? Can the day be saved?