It's been often assumed that the R-parity is conserved. If it were so, the lightest particle with [negative R-parity] . . . i.e. the lightest superpartner (LSP) would be stable because the conservation law for energy and for R-parity prevents it from all decays: there aren't any lighter particles with the [negative R-parity] . . . and the same value is required. The light LSP could be a gravitino or neutralino (either closer to a bino or to a neutral wino).From Lubos Motl.
Such a particle (LSP) could therefore constitute the bulk of the dark matter! For the neutralino, the R-parity conservation is rather critical if it wants to be employed as a dark matter particle.
However, the R-parity is likely to be broken, at least a little bit. After all, the numbers B and L or their combinations aren't exactly conserved quantum numbers. Evaporating black holes (and probably lighter objects as well) must be able to change their values (the conservation laws can't be exact because there would have to be new long-range forces analogous to electromagnetism if the symmetries were exact but there aren't any because they would destroy the tests of the equivalence principle). So the exponential is rather likely to be un-conserved, too. Now, the question is whether the R-parity violation is strong or weak.
Again, people typically assume – or derive from a deeper starting point (but I don't understand any of these derivations myself) – that the R-parity violation must be so weak that it can't be seen by the present accelerators. For example, that's a proposition behind the models by Gordon Kane, Bobby Acharya, and others. The conservation of R-parity makes it easier to preserve the longevity of the proton. While the baryon and lepton number conservation laws are allowed to be violated, the R-parity still bans some "intermediate steps" involving virtual R-parity-odd particles and makes the decay harder.
However, if one allows R-parity to be strongly violated, one should better only allow R-parity-violating terms that also violate the baryon number but that preserve the lepton number; or only terms that also violate the lepton number but that preserve the baryon number. If both lepton number and baryon number were violated, the proton decay would almost certainly be rapid, in a flagrant contradiction with the observations.
It's my impression that the baryon-number-violating R-parity-violating (BNV RPV) models have been gaining an upper hand in recent months. A cool feature is that the stop squarks' virtual activity could explain not only the stop squark gossip but also the Tevatron forward-backward asymmetry: a squark-like particle in a t-channel has always been the favorite explanation of mine for this only recent "new physics" claim by the Tevatron that wasn't self-evidently nonsensical.
Motl fails to mention that despite some theoretical arguments for a lack of baryon number conservation and lepton number convervation (separately), there has never been an instance where either baryon number violation or lepton number violation has actually been observed, and that there are arguably other ways to solve the issues that he identified with complete B and L number conservation. Also, often the theoretical concern cited for a need for lack of B and L number conservation related to baryogenesis and leptogenesis, not undiscovered long range forces and black hole evaporation which may be more model dependent.
Also, recall that one of the main phenomenological motives for SUSY these days is the need for a dark matter candidate. Yet, the LSP not excluded by LHC data (i.e. in the hundreds of GeV) is too heavy for the astronomical data to fill a viable WIMP candidate role, and R-parity violation creates a real problem in which amount of dark matter in the universe should steadily decline over the billions of years of the existence of the universe, unless it is to create counter-experimental proton decay levels as he notes. No proton decay has ever been observed either and SUSY parameters are updated every time the predicted level of proton decay in SUSY models fails to materialize.
Motl argues that R-parity violation could make the model dependent LSP detections impossible to observe as missing energy, making collider exclusions based on a lack of missing energy irrelevant, but it comes across as special pleading without a lot of thought to collateral implications that R-parity violating LSP decays could cause. For example, if there is a new production channel via supersymmetric particles in high energy decays, it should screw up the observed branching fractions in those decays.
Finally, as I understand it, the Tevatron forward-backward asymmetry is a doubtful experimental result. As of May, 2012, the more powerful LHC's data contradicts the Tevatron data (also here) and the large error bars at Tevatron suggest that its result is not really inconsistent with the Standard Model prediction. A theory that explains an asymmetry that isn't really there is a theory in trouble.
Bottom line: the experimental data that would motivate SUSY just isn't there, once again.