The Minimal Supersymmetric Standard Model (MSSM) is the minimal extension to the Standard Model that realizes N=1 supersymmetry, although non-minimal extensions do exist. Supersymmetry pairs bosons with fermions; therefore every Standard Model particle has a partner that has yet to be discovered. If these supersymmetric partners exist, it is possible that they will be observed at the Large Hadron Collider, which began operations in 2009. If the superparticles are found, it may be analogous to discovering antimatter and depending on the details of what might be found, it could provide evidence for grand unification and might even, in principle, provide hints as to whether string theory describes nature.
The MSSM was originally proposed in 1981 to stabilize the weak scale, solving the hierarchy problem. The Higgs boson mass of the Standard Model is unstable to quantum corrections and the theory predicts that weak scale should be much weaker than what is observed to be. In the MSSM, the Higgs boson has a fermionic superpartner, the Higgsino, that has the same mass as it would if supersymmetry were an exact symmetry. Because fermion masses are radiatively stable, the Higgs mass inherits this stability. However, in MSSM there is a need for more than one Higgs field, as described below.
The only unambiguous way to claim discovery of supersymmetry is to produce superparticles in the laboratory. Because superparticles are expected to be 100 to 1000 times heavier than the proton, it requires a huge amount of energy to make these particles that can only be achieved at particle accelerators. The Tevatron was actively looking for evidence of the production of supersymmetric particles before it was shut down on 30 September, 2011. Most physicists believe that supersymmetry must be discovered at the LHC if it is responsible for stabilizing the weak scale. There are five classes of particle that superpartners of the Standard Model fall into: squarks, gluinos, charginos, neutralinos, and sleptons. These superparticles have their interactions and subsequent decays described by the MSSM and each has characteristic signatures.
The MSSM imposes R-parity to explain the stability of the proton. It adds supersymmetry breaking by introducing explicit soft supersymmetry breaking operators into the Lagrangian that is communicated to it by some unknown (and unspecified) dynamics. This means that there are 120 new parameters in the MSSM. Most of these parameters lead to unnacceptable phenomenology such as large flavor changing neutral currents or large electric dipole moments for the neutron and electron. To avoid these problems, the MSSM takes all of the soft supersymmetry breaking to be diagonal in flavor space and for all of the new CP violating phases to vanish.
Read more about Minimal Supersymmetric Standard Model: Theoretical Motivations, Predictions of The MSSM Regarding Hadron Colliders, MSSM Fields, The MSSM Lagrangian, Problems With The MSSM, Theories of Supersymmetry Breaking
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