There are a number of ways to describe Little Higgs models. The one that I prefer starts by returning to the hierarchy problem. A natural question to ask is: why, if almost all particles are massive, is only one particular one—the Higgs—a problem? The answer is that all the other particles are in a sense naturally massless; if they were exactly massless, there is an enhanced symmetry. This symmetry would force all quantum corrections to the mass to exactly cancel. Corrections for non-zero mass must then be proportional to the symmetry-breaking parameter, the mass.

For example, the

*W*and

*Z*bosons in the Standard Model (SM) are protected by a gauge symmetry. If this gauge symmetry was exact, then they would be exactly massless due to that symmetry. Indeed, we see the same thing with the photon, and one of the standard problems in introductory field theory is to prove that quantum corrections to the photon mass vanish.

The SM fermions are protected by a different symmetry, a chiral symmetry. The origin of this is closely related to the structure of the Poincare group, the group describing Special Relativity. The result is the same; quantum corrections to, for example, the electron are proportional to the electron mass.

Now, we can use this idea to interpret some of the proposed solutions to the hierarchy problem. Supersymmetry, for example, relates the Higgs to a chiral fermion (the Higgsino). The Higgs inherits the protection of the chiral symmetry as long as supersymmetry remains unbroken.

Similarly, some theories of extra dimensions relate the Higgs to a gauge boson. (The models are imaginatively titled 'Gauge-Higgs Unification'.) The gauge symmetry protects the gauge boson mass, and consequently the Higgs mass.

However, chiral fermions and gauge bosons are not the only possible massless particles. There is one more: the Goldstone boson. What's more, chiral fermions have spin one-half and gauge bosons spin one; so they cannot

*be*the spin-zero Higgs. But Goldstone bosons have spin zero!

The idea of making the Higgs a Goldstone is an old one. It suffers from a small problem. A Goldstone boson is protected by a shift symmetry; the theory is invariant by shifting the value of the Goldstone field by a constant. All interactions are forced to involve derivatives of the Goldstone field. But the Higgs

*necessarily*has other types of couplings. This turns out to destroy the protection, leaving you no better off than in the SM.

Two things made LH models viable. The first was an experimental challenge, the so-called "Little Hierarchy Problem" where LEP's precision measurements ruled out a lot of potential models. In particular, if we demand a natural Higgs mass it seems to suggest that there should be new particles with mass less than about ten times the

*Z*; while LEP ruled out

*generic*new stuff up to scales ten times that.

The second factor was a theoretical insight. By careful construction of the theory, you can ensure that no one coupling completely breaks the symmetry protecting the Higgs. This leads to a partial protection of the Higgs mass. The first term in the perturbation series—the one loop terms—is free of the quadratic divergences that push the Higgs mass too large. You would still have quadratic divergences in the next, two-loop terms; but these are smaller by a factor of about ten. Conveniently, that was how much we needed to suppress their contribution to be consistent with LEP!

The LH family of models is very large, and unlike supersymmetry there's no clear minimal model. One candidate from the early days of this idea is the Littlest Higgs, which extended the SM with only four new particles and a handful of new parameters. Unfortunately, that idea was quickly shown to be in conflict with various precision measurements. The solution was the invention of

*T*-parity, a symmetry that prevented the problematic couplings and, conveniently, left us with a dark matter candidate. In this sense,

*T*-parity is very similar to

*R*-parity in supersymmetry. Unfortunately, it also lead to a SUSY-like proliferation of particles and couplings.

Still, despite those problems the Littlest Higgs with

*T*-parity (LHT) model gave a well-defined prediction for the dark matter, the "heavy photon". As the name suggests, this is essentially a partner of the photon which is typically two to four times the Higgs boson mass. It is thus one of the few examples of spin-one dark matter candidates, and probably the best motivated in my opinion.

The development of the LHT model lead to a number of papers looking at the dark matter phenomenology. This was obvious, low-hanging fruit; all the analysis was basically that previously done in for SUSY models. Since then not too much has been done theoretically, while there have been plenty of experiments publishing new limits. There's also been the Higgs discovery, of course. So Wang, Yang and Zhu decided to use those results to place new limits on these models.

The results, after all that, aren't too shocking. Of course, we don't have any robust signals of new physics. One interesting point relates to the Higgs sector. ATLAS, one of the two experiments at the LHC to see the Higgs, reports that the Higgs decay to two photons is enhanced; CMS, the other experiment, does not. It appears that the ATLAS enhancement is a serious problem for the LHT model, which prefers a slightly lower branching ratio.

LHT Model fits CMS Higgs data well, ATLAS data poorly. |

The main result in terms of dark matter phenomenology comes in the field of direct detection. The heavy photon couples to the Higgs, which then mediates possible scattering off atomic nuclei. The XENON100 experiment cuts into the model space, but the preferred points are still okay. However, the next generation detector, XENON1T, should rule out or discover the most convincing possibilities.

LHT Model compared to current (XENON100) and future (XENON1T) limits. |

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