The Standard Model and Beyond

The Standard Model [5] is a gauge field theory based on the symmetry groups $\ssu$ and it describes the strong, weak and electromagnetic interaction as mediated by an exchange of spin-1 gauge fields, called gauge bosons. The strong interaction is described by the SU(3)_C symmetry and is mediated by eight massless gluons, while the weak and electromagnetic interactions are commonly described using the $\su$ symmetry group. The electroweak interaction is mediated by three massive ($\W$ and $\Z$) and a massless boson (the photon). The $\su$ group cannot however be an exact symmetry, since gauge invariance requires that the gauge bosons are massless. To resolve this apparent contradiction the Higgs mechanism is introduced. This mechanism spontaneously breaks the $\ssu$ symmetry into the observed  $\suem$. This breaking occurs when the Higgs field acquires a non-zero vacuum expectation value. The Higgs field is a scalar complex weak doublet, to which one additional physical particle belongs, the Higgs boson.

Matter consists of spin-1/2 fields (fermions) of two types, leptons and quarks. The fermions can be grouped into three families, where each family consists of a left-handed quark and lepton SU(2)_L doublet and a right-handed singlet for each massive fermion, as seen below.

$${e \choose \nu _e}_L , \quad e_{\small R} , \quad {u \choose d}_L , \quad u_{\small R} , \quad d_{\small R}$$

The difference in the structure for left- and right-handed fields, takes into account the experimental fact that charged weak interactions only act upon left-handed fields. The number of families are also fixed by experiment. Combined precision measurements of the cross-section around the $\Z$ -pole have pinpointed the number of light neutrino species to three, ( $N_{\nu}=~2.989\pm0.0012$ [6]). Under the assumption that each family contains one such neutrino this also limits the number of families. Another important confirmation of this picture was the recent discovery of the top quark, by the CDF and D0 experiments at the Tevatron. [7]

Despite its unquestionable success, there are some serious theoretical shortcomings of the Standard Model. One was mentioned in the introduction, the problem with the Higgs mass. This mass can be approximated by:

$$M_h^2\sim M_{h0}^2 +{\lambda\over 4 \pi^2}\Lambda^2 +\delta M_h^2 .$$ (1)

where $\Lambda$ is the cut-off scale of the Standard Model, $\lambda$ is a coupling constant of ${\cal O}(1)$ and $\delta M_h^2$ is the mass counter term that has to be introduced to cancel the quadratically divergent term, that would be of $\cal { O}$( $10^{18}\GeV ^2$), assuming no new physics between the Planck- and the electroweak scale. As the mass of the Higgs boson cannot exceed $\sim$1000 $\GeV$  [8], this counter term has to be tuned to a precision of 1 part in 10¹⁶, in order to give a Higgs mass at the electroweak scale. Such "fine tuning" is regarded by most as highly unnatural, and is often referred to as the naturalness problem of the Standard Model. To remedy this problem, different solutions have been proposed. One is to construct a model without fundamental scalars, as done in so-called technicolor models, another approach is the introduction of a new symmetry, called Supersymmetry. This is a symmetry that relates bosons and fermions, but leaves all internal quantum numbers and masses unchanged. In a supersymmetric theory all bosons get a fermionic "superpartner" and vice versa. This symmetry must, however, be broken at the electroweak scale since there is, for instance, no charged boson with the same mass and couplings as the electron. By introducing supersymmetry, the quadratically divergent terms involving a particle is cancelled by the contribution from its superpartner [9]. However this is only true as long as the mass difference between them is not too large. If this mass difference exceeds 1 $\TeV$ ( see for instance discussion in [11] and references therein), one would once again have to resort to an unnatural tuning of the parameters.
Another interesting property of supersymmetry is that it could explain the mechanism of the breaking of the electroweak symmetry, $\su$ $\to U(1)_{EM}$, via radiative corrections.

A more aesthetical or perhaps a more profound argument for the necessity to extend the Standard Model, is that it is not the ultimate theory of nature. One reason for this, is of course that it does not include gravity, or even a unification of the strong and electroweak interactions. The number of free parameters is large, there is no explanation of the number of observed families, charge quantization etc. In Grand Unified Theories (GUTs), these questions are partly answered by assuming that the $\ssu$ group of the Standard Model, is a part of a larger group $\G$ ,

$$SU(3)_C\otimes SU(2)_L\otimes U(1)_Y \subset \G {} .$$ (2)

The group $\G$ would have a single coupling constant, so in order for the Standard Model to be consistent with this picture, it is necessary that the evolution of the different coupling constant unify at some energy scale. In 1991, precision measurements of the coupling constants at LEP, showed that in a GUT with the particle content of the Standard Model this is not achieved, while supersymmetric models were consistent with unification [12] (See Fig. 5). Furthermore, in this case the unification scale, $\mgut {}$, was of $\cal { O}$(10¹⁶) $\GeV$. This is in agreement with the constraints imposed by the proton life time, $\tau_P$. The observed limit on $\tau_P$, requires that the grand unification scale must be above 10¹⁵ $\GeV$ [13]. This is so far the only experimental indication that supersymmetry indeed is the correct extension of the Standard Model.

Figure: Evolution of the gauge coupling constants in the Standard Model (top) and in the MSSM (bottom), from the measured values at the $\Z$ -pole. Figure from Ref. [14].