The Minimal Supersymmetric Standard Model

The Minimal Supersymmetric Standard Model (MSSM), is the supersymmetric extension of the Standard Model with the minimal particle content [1]. For each particle, there is a superpartner with the same internal quantum numbers, but with spin that differs by half a unit. The superpartners of the fermions are usually denoted with the prefix s, which is short for scalar (sfermion, slepton, squark etc.). while superpartners of the Standard Model bosons have the suffix ino (gaugino, higgsino, wino etc.). In shorthand notation the superpartners are distinguished from the ordinary particles by the presence of a tilde ($\tilde{e}$, $\tilde{\gamma}$, $\squ$ etc.).

The particle content of the MSSM is summarized in Tab. 1. If the particle spectrum of the MSSM appears unappealing, fortunately this is not the case for the gauge interactions, where a particle and its superpartner have identical interactions, e.g. a selectron couples to the $\Z$ in the same way as an electron. As previously mentioned, supersymmetry cannot be an exact symmetry of nature, but it must be broken in a way that does not destroy the cancellation of the quadratic divergences of the Higgs mass. One commonly refers to this as soft breaking of supersymmetry. In order to describe the supersymmetry breaking part of the MSSM, a number of parameters are introduced. In its most general formulation, the MSSM introduces a total of 105 new free parameters, which are obviously too many to construct a model that is able to make any predictions. Among these new parameters are the masses of the U(1)_Y, SU(2)_L and SU(3)_C gauginos (M₁, M₂ and M₃), all the sfermion masses ($\msferl$ and $\msferr$ , where $\small f$ runs over all Standard Model fermions) and the bi- and trilinear coupling parameters, (B and A_f). The B parameter controls the mixing between the two Higgs doublets, while a non-zero A_f parameter introduces mixing between superpartners of the left and right handed chiral state of f. To reduce the number of free parameters three assumptions are usually made. The first, so-called GUT assumption, is that the three gaugino masses are equal at $\mgut {}$,

$$M_1(\mgut {})=M_2(\mgut {})=M_3(\mgut {}) \equiv m_{1/2}$$ (3)

For the scalar fermions the Universality assumption requires that all sfermion masses are equal at the GUT scale

$$m_{\tilde{q}}(\mgut {})=m_{\tilde{l}}(\mgut {}) \equiv m_0$$ (4)

and finally one can also assume that there is a common trilinear coupling.

$$A_b(\mgut {}) = A_t(\mgut {}) = A_{\tau}(\mgut {}) \equiv A_0 ,$$ (5)

The notation refers to the third generation, since the resulting mixing is proportional to the mass of the fermionic partner, hence most notable in these cases. Having made these assumptions a considerable reduction of the number of parameters are reached, and one is left with a model with only five free parameters. Apart from the ones mentioned above, two additional parameters are needed to describe the Higgs sector, these are commonly chosen to be $\tan{\beta}$, the ratio of the vacuum expectation values of the two Higgs fields and $\mu$, the Higgs mass parameter.
The assumptions made above may seem quite arbritrary, but by considering SUSY models where supersymmetry is promoted from a global to a local symmetry, that automatically incorporates gravity and also take into account the mechanism of the breaking of supersymmetry, in this so-called Supergravity models, these assumptions follow quite naturally (for a review see e.g. Ref. [16]) .