Supergravity

   When supersymmetry is a local symmetry, in addition to chiral and vector multiplets, there is another multiplet with the graviton and its supersymmetry partner, the gravitino. Since the graviton has spin 2, the gravitino Y_a^m, has spin 3/2, and can be seen in some way as the gauge field of local supersymmetry. Breaking supersymmetry means giving mass to the gravitino.
   As with a gauge boson, the gravitino can gain mass when the ground state of the scalar potential breaks the symmetry of the action. In the bosonic Higgs scenario, the massless Goldstone modes of the scalar field end up as the extra longitudinal components that make the massless gauge boson massive. In the supersymmetric case, in addition to Goldstone bosons, there are massless fermionic states called Goldstinos, and they provide the longitudinal modes that give mass to the gravitino and break supersymmetry.
   With supergavity, we have the interesting possibility of breaking supersymmetry through gravitational couplings. For simple N=1 supergravity with a chiral multiplet, the Kähler potential looks like

with M_P is the Planck mass and W is the superpotential of the theory. The resulting scalar potential for this theory is

In this model, the gravitino acquires a mass by eating a massless Goldstino, but because of the minus sign in the scalar potential, the total vacuum energy can be tuned to be zero. This is important because the total vacuum energy gives the cosmological constant of the theory, and the one that has been measured is extremely small.

How to test supersymmetry

   One experimental and theoretical result that is very encouraging evidence for supersymmetry is the high energy behavior of three Standard Model coupling constants (two electroweak and one strong). As stated on a previous page, the search for a Grand Unified Theory with all Standard Model fields gathered into representations of one big Lie group was encouraged by projections that the three Standard Model coupling constants meet at a single value at some energy scale M = M_GUT.
   However, when quantum corrections are included, this agreement does not occur precisely at a single value. The three coupling constants come much closer to a single value when the model in which they are being calculated is the Minimal Supersymmetric Standard Model.
   So supersymmetry suggests unification, and unification suggests supersymmetry.
   None of this is proof, but it adds a lot of excitement to the search for proof.
   One thing that a supersymmetric theory should NOT do is violate any of the observed conservation laws of particle interactions. One important observed conservation law that is easily violated by unified theories and supersymmetric theories is the conservation of baryon number.
   The proton is the lightest baryon and hence, if baryon number is conserved, the proton should be extremely stable. The observed lifetime of the proton is currently measured to be

   Grand unified theories (GUT for short) have gauge bosons that can mediate interactions that change quarks into leptons and hence allow the proton to decay by various interactions, including

where a proton, with baryon number 1, decays into a positron, which is a lepton and has baryon number 0, and a neutral pion, which is made of a quark and an antiquark and has baryon number 0. There are three quarks on the left hand side of the equation and two quarks and a lepton on the right hand side. If baryon number is not conserved, then the stable proton becomes unstable. The estimate for the proton lifetime in a GUT without supersymmetry is

   So this is bad for unification.
   In a GUT with supersymmetry there can also be baryon and lepton number violation, but for many reasons, the rate ends up being smaller so that

which is still an experimentally viable number, and a region that is close enough to the observed rate for future measurements, for example, at at the Super-Kamiokande experiment in Japan, to be able to tell us something meaningful about supersymmetry.

Dark matter and SUSY

   Because of the way stars move inside galaxies, astronomers and astrophysicists have calculated that there is a huge amount of mass in the Universe that we can't see with telescopes or other instruments because it's not giving off light the way stars do. That's why they call it dark matter.
   The presence of this dark matter can be detected by seeing how it interacts gravitationally, but it's been hard to figure out what it could be made of. One of the leading candidates for dark matter is a supersymmetry particle called the LSP, for Lightest Supersymmetric Particle.
   The success of this idea depends on the stability of the LSP. The LSP is stable in supersymmetric theories with a symmetry called R-parity, which guarantees that supersymmetric particles are produced only in pairs. This means that a supersymmetric particle can only decay into another supersymmetric particle. Hence the lightest one is stable because it can't decay into anything.
   The LSP that could make up dark matter has to be massive and electrically neutral, therefore, it could only be the supersymmetry partner of a neutral particle. The three candidates are: a gravitino (fermionic superpartner to the graviton), a sneutrino (scalar superpartner to the neutrino) or a neutralino (fermionic superpartner to a neutral gauge boson or neutral Higgs scalar).
   So far the most promising candidate for dark matter is the neutralino, because they interact weakly. Therefore they would decouple from thermal equilibrium at some early age of the universe and produce a stable residual density that could be large enough to provide the large amount of dark matter that is believed to be out there.
   There are a lot of hints that supersymmetry could be out there, because it offers ways to solve many puzzling issues in particle physics and cosmology at once.
   This is another arrow pointing to string theory, the only theory of elementary particles that requires both supersymmetry and gravity to exist in Nature.