R-parity revisited!

In Minimal Supersymmetric Standard Model (MSSM), we generally assume the existence of an additional discrete symmetry called R-parity defined as

where B, L corresponds to baryon and lepton number respectively whereas s denotes spin. The advantage of this ad-hoc discrete symmetry is that it keeps all the B and L violating interactions away from the superpotential and the lightest super-particle(LSP) becomes stable which can be a good dark matter candidate provided it satisfies other criteria as well. There are some models where this discrete symmetry gets spontaneously broken and we loose some of its benefits. Although the most dangerous B violating terms (leading to proton decay) remain still away, but L violating terms come into the model as a result of spontaneous symmetry breaking. These terms are not very dangerous if parametrically put under control. We can in fact see some signatures of these in ongoing experiments like Tevatron and LHC. But the LSP will no longer be stable in this case and we loose a dark matter candidate. But it is not a very serious problem that gravition (superpartner of graviton) is always there to satisfy the criteria of a good dark matter candidate in any R-parity violating model. Since gravity is very weak, gravitino can have a long life time even if R-parity is broken in the model. In fact such spontaneously broken R-parity models are much better than R-parity conserving MSSM. Initially what I used to assume that R-parity conserving models are good enough to explain the low energy phenomena as well as dark matter. But when I was trying to arrive at non-zero neutrino masses by adding right handed singlet neutrinos, R-parity gets explicitly violated at the superpotential level. Of course if we break R-parity explicitly at the superpotantial level, we can not keep the other B and L violating terms away from the superpotential. The only way to get rid of this problem is to fine tune the B-violating terms to make proton stable enough and keep L-violating terms so as to give rise to neutrino mass. This fine-tuning does not look elegant but we need to do this to explain another hierarchy problem in case of neutrino mass (why it is so small compared to other leptons and quarks). We could have realized spontaneously broken R-parity in MSSM (without right handed singlet neutrinos) by sneutrino (superpartner of left handed neutrino) vev. But that will break lepton number (a global symmetry) spontaneoysly and will give rise to Majoron which will couple to Z-boson and hence ruled out from precision measurement of Z-decay width. Thus R-parity conserving and spontaneous R-parity violating MSSM are not good enough to explain all the low energy phenomena. We need to consider either explicit R-parity violating MSSM (which needs lots of fine tuning in the R-parity breaking sector) or go beyond MSSM by enlarging the Higgs sector or gauge symmetry.