Standard HiggsA perfect guy, to the point of being boring. He does everything he is supposed to do, and perfectly matches all experimental results so far (apart from the small tension with electroweak precision tests). We know everything about him except for the mass. It is believed, however, that left alone and unprotected he would acquire a large mass. This purely theoretical argument prompts most of what follows.
Susy HiggsThe marriage of Susy and Higgs has lasted for more than 30 years. Susy provides stabilization to Higgs, keeping its mass small enough. Sadly enough, bad tongues and the LEP experiment have left deep scars on this relationship. The problem is that the minimal supersymmetric model ties the Higgs boson mass to the Z boson mass. The failure to discover Higgs LEP implies that the parameters of the minimal model must be finely-tuned in order to accommodate the higher mass, thus spoiling the naturalness of the whole construction.
Composite HiggsHiggs does not have to be that elementary - it is natural to imagine that Higgs is a bound state like many other particles we have observed. For example, it could be a meson made of new quarks glued together by new strong interactions. The problem with this idea is that a simple back-of-a-napkin estimate suggests that the Higgs mass should not be much different from the scale of the new strong interactions. Since we have seen nothing like that up to a few hundreds of GeV, the mass of the composite Higgs would have to be larger, contrary to what electroweak precision tests seem to tell us. Or there must be some more structure that keeps the mass light enough...
Pseudo-Goldstone HiggsSusy does not have exclusive rights on controlling quantum corrections to the Higgs mass. Particle's masses can also be protected by spontaneously broken global symmetries. A similar mechanism operates in real life and was awarded a Nobel prize last year: thanks to that mechanism the QCD pions remain lighter than the QCD scale. The Higgs boson could also arise as a pseudo-Goldstone boson when a new strong dynamics spontaneously breaks its own global symmetries. But at the end of the day this simple idea does not work as well as it is supposed to. First, the name is unattractive and difficult to pronounciate (worse still, around Chicago it becomes a pseudo-Nambu-Goldstone-boson-Higgs monster). Besides, the new strong interactions meddle with electroweak precision observables, and at the end of the day the fine-tuning is only slightly better than in minimal supersymmetry.
Little HiggsLittle Higgs is a variation on the theme of the pseudo-Goldstone Higgs. The new strong interactions are pushed to higher scales, around 10 TeV, while an additional structure - the so-called collective symmetry breaking - protects that scale separation. While this idea can be made completely realistic and the fine-tuning of parameters can be acceptably small, fully realistic constructions are situated somewhere between late baroque and early racoco.
Fat HiggsNot that he's very pretty, but he has a cool name. This one combines the ideas of composite Higgs and supersymmetry. A strongly coupled Susy gauge theory sits in the conformal window all the way down till the TeV scale. At that point, due to the fact that some of the flavors have TeV scale masses, the theory drops out of the window and confines. The challenge is make all the numbers work and get rid of all the excess bagagge that comes along.
Invisible HiggsCould it be that Higgs was at LEP but we missed it? Actually, in models with additional singlet fields it is common that Higgs decays into exotic particles that escape from the detector without being seen. Such a cheap trick would not fool LEP, however, and invisible Higgs is just as well constrained as the standard one. Nevertheless, one can devise more complicated models where Higgs is partly invisible and hides from LEP analyses even though his mass is below 115 GeV.
UnhiggsEvery kid has to go through a negation phase at some point. It may be that Higgs is neither a god nor a particle after all. Instead, it could be a fuzzy continuum of excitations and still perfectly fulfill its role.
HiggslessFinally one should mention that Higgs might not exist. This athehigsm has some scientific support. Electroweak symmetry can be broken by a condensate in a strongly interacting theory, much as it happens to chiral symmetries in QCD. In that case Higgs is expandable, and his role is played by new resonances whose spin is one rather than zero. That is not as bad as it seems since these new resonances must have masses within the LHC reach to make the picture consistent. Higgsless theories are disfavored by electroweak precision and flavor tests, but the ultimate answer will be given by the LHC. Unless reality is Unhiggsless.
WHO IS GOING TO WIN THE RACE? WILL IT BE HIGGS-THE-PERFECT-BORING-GUY? OR HIGGS' LOVE FOR SUSY WILL OVERCOME THE OBSTACLES? OR MAYBE SOMEONE ELSE WILL MEDDLE IN THE RACE? STAY TUNED FOR THE NEXT EPISODES. To definitely nail down the nature of Higgs we'll probably need to wait for future linear colliders, but some partial answers should be provided in two years from now, if all goes well.