The Higgs Boson discovery must still meet several tests to verify that it is the standard model Higgs. It’s coupling in various production channels of particles must show that its couplings are exactly proportional to the masses of the particles that it couples to. It must also be verified that it is a spinless scalar and not pseudoscalar under parity, which it is on its way to be verified.
If there is a grand unification of the strong QCD quark gluon interactions with the electroweak interactions, then the grand unification must be broken, again possibly by a Higgs mechanism. The new, very heavy Higgs’ would have a mass of around the unification scale of 10^17 GeV, where a GeV (giga or billion electron volts) is around the mass of the proton and neutron. But since the standard model Higgs couples proportional to mass, it would interact exceedingly strongly with the very heavy Higgs and its mass would be driven up to the very heavy mass. Something must intervene to save this from happening since we see the standard model Higgs at relatively low mass.
The popular theory for this is Supersymmetry, or SUSY theory. Here there is a symmetry that gives every elementary spin 1/2 particle (fermion) a spin 0 (really two of them) partner with the same couplings and coupling strength. The partners of quarks are called squarks, and of leptons are called sleptons (need to sleep on that?). Then every elementary spin 1 boson has a SUSY partner with spin 1/2 and the same couplings and coupling strength. For gluons the partners are called gluinos, for W’s they are called Winos (pronounced Weenos), for the B it is called the Bino, and for the photon, they are called photinos. The Higgs has a spin 1/2 partner called the Higgsino. (Oh, and the spin 2 graviton has a supersymmetric partner of spin 3/2 called the gravitino). All of these particles are heavier than hundreds of GeVs, since none of them has been found. The theory, however, predicts that their masses should be around a 1,000 GeV (or a TeV tera electron volt), or lower. Limits by the LHC are now near this range or exceeding it in some instances.
The Minimal SUSY model has two higgs type fields. This would reduce the couplings of the Standard Model Higgs and add a higher mass neutral Higgs field, as well as a charged Higgs field. So improving the accuracy of the probability of producing the new Higgs compared to the Standard Model expectation could hint at higher mass Higgses.
The supersymmetric particles probably need to be produced in pairs, or they would have been easier to create. This is formally called R parity, which they are odd in, while ordinary particles are even in them. Conserving R parity multiplicatively, means that the product of the minuses of each in the pair created gives a positive. In their decays then, the lightest susy particle would be stable. If this is stable and neutral and only weak interacting, then it is a candidate for the Dark Matter in the universe, in the catagory called a WIMP, a weakly interacting massive particle. So the LHC is hoped to produce these in the SUSY theory. They may be a combination of the four neutral SUSY partners the Wino, Bino, Photino and Higgsino called the Neutralino.
In cosmology, the mysterious cause of inflation in the first instants of the Big Bang, might be a Higgs type field, called the quintessence. Also, the cosmological field that enters the theory as Einstein’s cosmological constant may be due to a Higgs type field that is accelerating the present universe.