The superpotential for MSSM was shown in eqn.(2.36). This superpotential is gauge (the SM gauge group) invariant, Lorentz invariant and maintains renormalizability.
9This limit can be further relaxed to m
h0 ≤ 150 GeV, assuming all couplings in the theory remain perturbative up to the unification scale [59, 60].
However, it is natural to ask that what is preventing the following terms to appear in
WM SSM, which are also gauge and Lorentz invariant and definitely renormalizable:
Wextra = ǫab(−εiLˆaiHˆub+ 1 2λijkLˆ a iLˆbjeˆck+ λ ′ ijkLˆaiQˆbjdˆck+ 1 2λ ′′ ijkuˆcidˆcjdˆck). (2.50) Of course, all of these terms violate either lepton (L) [66, 67] or baryon (B) [66, 68] number by odd units. The second and the third terms of eqn.(2.50) violate lepton number by one unit whereas the fourth term violates baryon number by one unit.
Now it is well known that in the SM, lepton and baryon numbers are conserved at the perturbative level. In the SM, L and B are the accidental symmetry of the Lagrangian, that is to say that these are not symmetries imposed on the Lagrangian, rather they are consequence of the gauge and Lorentz invariance, renormalizability and, of course, particle content of the SM. Moreover, these numbers are no way related to any fundamental symmetries of nature, since they are known to be violated by non- perturbative electroweak effects [69]. So it is rather difficult to drop these terms from a general MSSM superpotential unless one assumes B, L conservation as a postulate for the MSSM. However, in the presence of these terms there exists new contribution to the proton decay process (p→ ℓ+π0 with ℓ+ = e+, µ+) as shown in figure 2.4. This process
p
u
d
u
λ
′λ
′′ℓ
+¯
u
π
0 ed
ckFigure 2.4: Feynman diagrams for the process p→ ℓ+π0 with ℓ+= e+, µ+.
(see figure 2.4) will yield a proton life time ≈ 10−9 sec, assuming λ′, λ′′
∼ O (10−1)
and TeV scale squark masses. However, the known experimental bound for proton lifetime is > 1032years [30,70]. So in order to explain proton stability either these new
couplings (λ, λ′, λ′′) are extremely small (which again requires explanation) or their products (appear in the decay width for the process p → ℓ+π0) are very small or these
terms are somehow forbidden from the MSSM superpotential. In fact, to avoid very fast proton decay mediated through squarks of masses of the order of the electroweak scale, simultaneous presence of λ′, λ′′ type couplings must be forbidden unless the
product λ′λ′′ is severely constrained (see figure 2.4). The λ type of operators are not
so stringently suppressed, and therefore still a lot of freedom remains (see ref. [71] and references therein).
It turns out that since these new terms (see eqn.(2.50)) violate either lepton or baryon number by odd units it is possible to restrict them from appearing in WM SSM
by imposing a discrete symmetry called R-parity (Rp),10 [66, 72–74] defined as,
Rp = (−1)3(B−L)+2s, (2.51)
where s is the spin of the particle. Since L is an integer, an alternative expression for Rp is also given by
Rp = (−1)3B+L+2s. (2.52)
It is interesting to note that since different states within a supermultiplet have different spins, they must have different Rp. It turns out that by construction all the SM
particles have Rp = +1 and for all superpartners, Rp = −1. This is a discrete Z2
symmetry and multiplicative in nature. It is important to note that Rp conservation
would require (1) even number of sparticles at each interaction vertex, and (2) the lightest supersymmetric particle (LSP) has no lighter Rp = −1 states to decay and
thus it is absolutely stable (see figure 2.5). Thus the LSP for a supersymmetric model with conserved Rp can act as a natural dark matter candidate. It must be remembered
that the soft supersymmetry breaking Lagrangian will also contain Rp violating terms
[75, 76]. Particles RP= +1 Sparticles RP=−1 Sparticles RP=−1 RP conserved Particles RP= +1 Particles RP= +1 Sparticles RP=−1 RP violated Particles RP= +1 Sparticles RP=−1 Sparticles RP=−1 RP conserved Particles RP= +1 Particles RP= +1 Sparticles RP=−1 RP violated
Figure 2.5: With Rp conservation the LSP is forced to be stable due to unavailability
of any lighter sparticle states (left), whereas for the Rp-violating scenario the LSP can
decay into SM particles (right).
Looking at eqn.(2.50) it is clear that sources for Rp violation (6Rp) (see references
[77–89]) are either bilinear (ǫ) [90–102] or trilinear (λ, λ′, λ′′) [76, 81, 84, 97, 103–106] in nature. The simple most example of6Rpturns out to be bilinear. It is interesting to note
that these bilinear terms are removable from superpotential by using field redefinitions,
10
however they reappear as trilinear couplings both in superpotential and in soft SUSY breaking Lagrangian [67, 107, 108] along with the original bilinear Rp-violating terms,
that were in the soft SUSY breaking Lagrangian to start with. The effect of rotating away LiHu term from the superpotential by a redefinition of the lepton and Higgs
superfields are bound to show up via the scalar potential [92]. Also even if bilinear terms are rotated away at one energy scale, they reappear in some other energy scale as the couplings evolve radiatively [109]. The trilinear couplings can also give rise to bilinear terms in one-loops (see figure 2.6) [76]. Note that 6Rp can be either explicit
(like eqn.(2.50)) [67, 77, 107, 108] or spontaneous [77, 78, 110–116].
Li λijk Y jk e (λ′ ijk) (Ydjk) Hd Lj(Qj) ec k(dck) (a) Li λijk (λ′ ijk) Yjk e (Ydjk) Hu Hd µ Lj(Qj) ec k(dck) (b)
Figure 2.6: One loop diagrams contributing to bilinear terms like LiHu, LiHdusing the
trilinear couplings λ, λ′.
Here as a digression it should be mentioned that Rp can be embedded into a larger
continuous group (see, for example, ref. [117] and references therein) which is finally abandoned for phenomenological reasons11. However, its Z
2 subgroup could still be
retained, which is the Rp.
To summarize, it seems that Rp violation is a natural feature for supersymmet-
ric theories, since Rp-violating terms (see eqn.(2.50)) are not forbidden to appear in
the MSSM superpotential by the arguments of gauge and Lorentz invariance or renor- malizability. On the contrary, assumption of Rp-conservation to prevent proton decay
appears to be an ad hoc one. Besides, models with Rp-violation are also phenomeno-
logically very rich. Of course, it is natural to ask about the fate of the proton. But considering either lepton or baryon number violation at a time proton stability can be achieved.
It is true that with 6Rp the LSP is no longer stable and can decay into the SM
particles. The stable LSP (in case it is colour and charge neutral) can be a natural candidate for the Dark matter [118,119]. However, their exist other viable dark matter candidates even for a theory with 6Rp, namely, gravitino [120–122], axion [123,124] and
axino [125, 126] (supersymmetric partner of axion).
It is important to note that a decaying LSP has very different and enriched im- plications in a collider study. Unlike models with Rp conservation, which yield large
missing energy signature at the end of any supersymmetric process, effect of6Rp can of-
ten produce interesting visible final states detectable in a collider experiments. Models with bilinear 6Rp are especially interesting concerning collider studies [122, 127–140], as
they admit direct mixing between neutrino and neutralinos.
Finally, it remains to be mentioned the most important aspect of Rp violation,
namely, generation of the neutrino mass. It is impossible to generate neutrino masses in a supersymmetric model with Rp conservation along with minimal field content
(see eqn.(2.36)). It is rather important to clarify the importance of 6Rp in neutrino
mass generation. There are other ways to generate light neutrino masses, both in supersymmetric or non-supersymmetric models like adding extra particles or enhancing the gauge group (left-right symmetric models [141] for example) and many others. But generating massive neutrinos with 6Rp is a pure supersymmetric phenomenon without
any SM analog. More on the issue of light neutrino mass generation and 6Rp will be
addressed in the next chapter.
To complete the discussion, it is important to mention that these6Rp couplings are
highly constrained by experimental limits on different physical processes, like neutron- anti neutron scattering [142–145], neutrinoless double beta decay [103,146–150], preci- sion measurements of Z decay [151–153], proton decay [154–156], Majorana masses for neutrinos [105, 157–161] etc. Discussion on different supersymmetric models with and without Rp conservation, proposed in the literature is given in a recent review [162].