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One possible extension to the Standard Model is supersymmetry (SUSY). SUSY at its essence is an additional symmetry between fermions and bosons and can best be understood by beginning with the Coleman-Mandula theorem. The Coleman-Mandula theorem states that the most general symmetries that a quantum field theory (QFT) can possess are Lorentz invariance (special relativity) and gauge symmetries like conservation of charge, lepton number, etc. (whose generators belong to Lie Algebras [33]). In other words, the Coleman-Mandula theorem is a "no-go" theorem: a relativistic QFT can have no other symmetries. In particular, there can be no change of the spin of particles. That is, there is no way in the SM to change fermions to bosons or vice-versa.

However in the mid-1970s two groups of physicists realized that the Coleman-Mandula theorem can be evaded. Supersymmetry evades the restriction of the Coleman-Mandula theorem by generalizing and loosening the restriction on the types of symmetries of a QFT (in addition to Lie Algebras one can consider graded Lie Algebras whose operators anti-commute). This additional symmetry allows for the inter-conversion of fermions and bosons. Essentially, every fermion is now associated with a superpartner boson and every boson with a superpartner fermion; adding supersymmetry to the standard model effectively doubles the number of particles. Although doubling the number of particles may seem a hopeless complication, supersymmetry is very attractive theoretically for a number of reasons.

To begin with, supersymmetry may solve the hierarchy and the fine-tuning/naturalness problem in the Standard Model. SUSY is new physics which acts at energies beyond the SM which helps to explain why the electroweak and Planck energy scales are so different. In terms of the fine-tuning problem, SUSY can explain why the Higgs mass and the Higgs vev are so small. If SUSY were an exact symmetry of nature, then the mass of each SM bosonic particle must be equal to its superpartner fermion mass. And since boson and fermion mass corrections in QFT calculations enter with opposite signs, they can cancel each other leading to a "naturally" small Higgs mass and vev. Of course supersymmetry is a broken symmetry, meaning that the symmetry is no longer valid at the typical energies and background temperatures in the Universe today. For example, we don't see a bosonic superpartner to the electron with .511 MeV mass which would be a sign of unbroken supersymmetry. Due to this breaking (which is not well understood) all superpartners must be extremely massive (much like the W and Z particles acquire mass in electroweak symmetry breaking while the photon remains massless). In order to produce acceptable corrections to the Higgs mass, the difference between boson and fermion masses must be of the order of 1 TeV.

Furthermore, precision measurements of Standard Model parameters at the LEP collider show that using only the Standard Model particle content the strong, weak, and electromagentic forces do not seem to unify at energies of about 1016 GeV. Particle physicists have long predicted that like the weak and electromagnetic forces which unify at energies of about 103 GeV, the three quantum forces should merge to become a single Grand Unified Force. However, if one adds the minimal particle content of supersymmetry, the couplings indeed seem to converge at a unification scale of M appeq 2 × 1016 GeV. [34] Additionally, supersymmetry is inherent in string theory, which currently is the only theory which has the possibility of unifying the quantum world with gravity.

And finally, and this is perhaps the most appealing characteristic of supersymmetry, the Standard Model with SUSY does in fact offer a viable dark matter candidate which we will discuss shortly. The new particles generated by adding SUSY to the SM are shown below in Table II.

 Generation 1      Generation 2      Generation 3    
 Particle Mass (GeV)  Charge   Particle Mass (GeV)  Charge   Particle Mass (GeV)  Charge 
 up squark (u) > 379 +2/3  charm squark (c) > 379 +2/3  top squark (t) > 92.6 +2/3
 down squark (d) > 379 -1/3  strange squark (s) > 379 -1/3  bottom squark (b) > 89 -1/3
 selectron (e) > 73 -1  smuon (µ) > 94 -1  stau (tau) > 81.9 -1
 e sneutrino (ve)   > 95 0  µ sneutrino (vµ) > 94 0  tau sneutrino (vtau) > 94 0


 Particle  Mass (GeV)  Description
 Neutralinos (chi1-40) > 46 Mixture of photino (gamma), zino (Z), and neutral higgsino (H0)
 Charginos (chi1,2±) > 94 Mixture of winos (W±) and charged higgsinos (H±)
 Gluinos (g) > 308 Superpartner of the gluon

Table II: The particles predicted by a supersymmetric extension of the Standard Model. Limits on the masses of particles as last reported by the Particle Data Group. [28]

When examining the particle content of the SM with SUSY, there are several possible particles which could act as dark matter. These are the neutralino (a particle state which is a superposition of the neutral superpartners of the Higgs and gauge bosons), the sneutrino (the superpartner of the neutrino), and the gravitino (the superpartner of the graviton which would come from a quantum theory of gravity). All of these particles are electrically neutral and weakly interacting, and thus are ideal WIMP-like candidates for dark matter. However, sneutrinos annihilate very rapidly in the early universe, and sneutrino relic densities are too low to be cosmologically significant. [35] And gravitinos act as hot dark matter rather than cold dark matter, and large scale structure observations are inconsistent with a universe dominated by hot dark matter. [36] This leaves the neutralino as a viable candidate.

But how can the neutralino, an extremely massive particle, exist today in sufficient numbers to make up the bulk of the dark matter (generically, massive particles decay into lighter ones)? The answer lies in what is called R-parity. In the Standard Model symmetries guarantee baryon and lepton number conservation; for this reason, the proton, the lightest baryon, cannot decay. However, with the addition of supersymmetry, this is no longer generally true due to the presence of squarks and sleptons; recall, SUSY changes quarks and leptons into bosons and vice-versa, so baryon and lepton number are violated as a matter of course. However, we know that the amount of baryon and lepton number violation (at least at low energies) must be extremely small due to sensitive tests. An interesting property of SUSY is that if one writes down a theory without lepton and baryon number violating terms, no such terms will ever appear, even through quantum loop corrections (another advantage of SUSY is that certain types of quantities never get loop corrections). Under this assumption, a new symmetry, called R-parity, may be conserved by a SUSY version of the SM. We assign +1 R-parity for all Standard Model fields (including both Higgs fields), and -1 R-parity for all superpartners. The immediate consequence of R-parity conservation is that because there are an even number of SUSY particles in every interaction, the lightest supersymmetric partner, the LSP, is stable and will not decay. If this LSP is neutral, it is an excellent candidate for dark matter.

In most SUSY versions of the standard model, the neutralino is the LSP and seems to be the most promising dark matter candidate; the relic abundance of neutralinos can be sizeable and of cosmological significance, and detection rates are high enough to be accessible in the laboratory but not high enough to be experimentally ruled out. Thus the SM with SUSY offers a single dark matter candidate: the neutralino. Although at present not one supersymmetric particle has been detected in the laboratory, supersymmetry currently offers the best hope of modeling and understanding dark matter. One clear advantage is that the minimal extension of the Standard Model using supersymmetry is well understood, and calculations, including dark matter densities and detection rates, can be performed.

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