turbo-1 said:
There has been a tremendous expenditure of resources of all types (including peoples' entire careers) thrown at this problem, but I have a question. Why are we building higher-energy colliders to look for the LSP?
Well, the LSP is just one out of plenty of reasons why we build colliders like LHC. Maybe the most interesting to come out of LHC is something we had not even thought about. But anyway, let's move on:
The concentration should be on the building of detectors, because if the the LSPs exist, they should be everywhere.
Sure, and there are a lot of experiments running and being developed for direct detection. These will probably reach enough sensitivity to scan at least some part of the LSP parameter space in a few years. Besides particle physics constraints on the cross section for interaction with the detector, the chance of succeeding also highly depends on astrophysical conditions; we're not exactly sure of what local density of DM particles to expect. If we're lucky we'll find some evidence for DM through direct detection even before LHC, but due to that the LSP probably interacts very tiny with ordinary matter I personally doubt that.
However, once (and if!) we find the LSP at CERN we of course also need to confirm that it really makes up the DM, so direct detection is of course needed for the complete confirmation.
If lightest supersymmetric particle is truly the lightest, there is no lighter supersymmetrical particle that it can decay to, meaning that if they exist, the universe should be teeming with them already. Every one ever produced still exists - they are immortal.
You're are right to a certain extent. The LSP is forbidden to decay into other particles through a multiplicative quantum number called R-parity which is conserved in supersymmetric theories. (Well, actually there are SUSY theories without conserved R-parity, but for several reasons those are not as intersesting, but let's keep this simple.) Ordninary particles have R-parity (1) while their supersymmetric partners carry (-1). That's why a single LSP cannot decay into anything less massive. However, two particles with R-parity (-1) each, that is a total of (-1)*(-1)=(1) can annihilate into a standard model pair, like two gammas. Observing these gammas (for example from the dense region in the galactic centre) in fact is a way to indirectly detect the LSP, and much work has been put down on clearing such things out too. However it's not clear wheter this signal will drown into the background from other astrophysical sources.
The fact that LSPs have not been detected already should be sobering to the guys building and equipping the collidors.
The missing signal from direct detection of course puts limits on what properties the LSP might have, but the parameter space where the LSP can be is still HUGE, and direct detection need to become much more efficient before it can rule out the LSP as a DM candidate.
Has this been discussed in the literature, EL?
Oh yeah, in hundreds, maybe more, papers over the years. A good one to start with is maybe Jungman et al:
http://www.arxiv.org/abs/hep-ph/9506380
It's certainly not up to date, but the concepts are nicely explained.