A very interesting paper appeared one week ago in the Arxiv. It is titled "Higgs Self-Coupling Measurements at the LHC", and it is authored by M.Dolan, C.Englert, and M.Spannovsky. The idea is that once and if a Higgs boson is found at the LHC, the next natural step of the research would be its characterization as a pure standard model object or a more complex, or just different, beast.
Of course, once a signal were established, the LHC experiments would certainly want to measure all its properties as precisely as possible: mass, angular distributions, cross section in all the production mechanisms, and decay modes.
In theoretical terms, determining angular distributions would answer the question "is this a scalar particle", or "what is its spin-parity", something technically labeled as "0+" in the standard model, much like the blood type of most of us. Definitely a very important theme; but maybe even more important would be to use the production and decay information to decrypt the higgs boson couplings to fermions and bosons, and crucially, its self couplings.
"Self coupling" means that a particle can split into two copies of themselves (in the simplest realization): definitely not something trivial to do for matter particles, and indeed, fermions cannot do it, for a number of reasons (charge conservation, or angular momentum conservation, come to mind). Bosons instead are not prevented by the conservation of angular momentum rules, so the question is whether they possess or not other quantum numbers that would change in a 1->2 or 1->3 process. Photons cannot split into two photons, for example, because of the conservation of charge conjugation.
A boson which is perfectly happy to split into copies of itself is the gluon. The presence of a self-coupling of gluons is in fact built into the theory of gluons - Quantum Chromodynamics, QCD - because QCD is based on a non-abelian gauge group, SU(3). For the Higgs, the group-theory reason of the existence of three- and four-particle vertices is to be found in the very electroweak symmetry breaking mechanism, which in the unitary gauge does the miracle of creating mass degrees of freedom to W and Z bosons. I will not bother you with mathematical expressions, but the two things really have the same source in the standard model: the Higgs has self couplings if EWSB works the way it is supposed to. So comparing measured and predicted Higgs boson self couplings is the ultimate dream of the Higgs phenomenologist !
The paper therefore examines the possibilities to measure these self couplings. What we need to do is find processes whereby two Higgs bosons are simultaneously produced: this would both be due to independent Higgs boson radiation (off a quark line, for instance), and from the production of a single Higgs boson which then materialized a pair, thanks to the self interaction; some possible leading-order diagrams are shown in the figure on the right. One would thus need to measure the production precisely in order to understand how much of the total rate is due to the self interaction diagram. And alas, double Higgs production is predicted to be quite rare at the LHC !
Theoretical calculations indicate that the rate is of the order of one thousandth of the production of a single Higgs; then of course, if self interaction were much larger (due to some new physics intervening in the double Higgs production, for instance), one would observe more than that. But can such a few-femtobarns-cross-section process be seen ? That is the question to address. One should consider the full powers of a endgame LHC, that is 14 TeV running conditions and hundreds, or even thousands, of inverse femtobarns of integrated luminosity. In that case, it seems that the thing is doable -but only marginally so.
Having read the article, which deals with details of the most promising signatures (e.g. when the two Higgs bosons are produced "boosted", recoiling against a hard jet, or when the Higgs decay into specific topologies that may beat other standard model backgrounds), I am unsure whether I should be excited by the prospect, or draw home the lesson that if the LHC finds the Higgs we need to build a different machine to study in detail its properties. The latter points to a linear collider as the natural choice: but a linear collider had been advocated for decades as "the" machine to build after the LHC "because LHC would find SUSY particles, and then we'd need to study them in detail in a linear collider". Now using the same argument with only the Higgs boson as a target appears problematic to me...
Ideas On Higgs Couplings