The SM has 19 free parameters from which 13 are related to the flavour sector. That seems unsatisfactory for a fundamental physical theory. Also the pattern of the fermion masses in the SM, see Fig. 2.2, seems to demand a deeper understanding. For example, the masses of the up-type quarks differ by a factor of approximately one thousand from each other whereas the down-type quark masses differ by approximately a factor of one hundred from each other.
This becomes even worse by including neutrino masses and mixings. This introduces three additional masses, three additional mixing angles and, depending on the nature of the neutrinos, i.e. Dirac or Majorana, one or three additional CP violating phases. The hierarchy between the masses gets even stronger. It is known that the sum of the neutrino masses cannot be larger than about 0.6 eV [29]. Even if the neutrino masses are close to this upper bound the top quark would still be about 1011 times heavier.
However, this is not the only strange pattern in the flavour sector. The mixing angles in the quark and the lepton sector are very different from each other. While the mixing angles are all small in the quark sector in the lepton sector two mixing angles are known to be large while the third one is very small. One of the leptonic mixing angles is even close to maximal.
Although the SM (respectively an extension with right-handed neutrinos) can in prin- ciple describe these patterns by imposing that the masses and mixing angles are just as
18 2. The Standard Model of Particle Physics
we observe them, maybe there is a more fundamental theory which elegantly describes the flavour patterns. For example, the PMNS matrix can be described in terms of discrete symmetries which may be interpreted as a hint towards an underlying family symmetry.
CHAPTER 3
Supersymmetry
In this chapter we want to review some fundamental concepts of supersymmetry (SUSY) based on [53–55]. We start with a short motivation for supersymmetric field theories in Sec. 3.1 and then discuss some basic formal aspects of SUSY necessary for this thesis in Sec. 3.2. We end this chapter with an introduction to the MSSM in Sec. 3.3 and a brief discussion of SUSY breaking in Sec. 3.4.
3.1
Motivation
The initial motivation for introducing supersymmetry in particle physics was the realisation that despite a no-go theorem by Coleman and Mandula [56] the S-matrix can have sym- metries beyond the internal symmetries and the Poincar´e symmetry. This no-go theorem can be circumvented by introducing symmetries whose algebras fulfil anti-commutation relations instead of commutation relations [57]. This class of symmetries is called super- symmetries. But it turned out that supersymmetric field theories have other nice properties of which we give now three prominent examples.
Probably the most prominent feature of SUSY is the solution of the hierarchy problem, see Sec. 2.3.2. In a supersymmetric version of the SM, where SUSY is unbroken, the radiative corrections to the Higgs mass are exactly cancelled by diagrams with SUSY partners of the SM particle fields in the loops. Even in a field theory with softly broken SUSY the radiative corrections to the Higgs boson mass are under control as long as the scale of the SUSY particle masses is not too far above the TeV scale [7]. Therefore the prospects of finding SUSY at the LHC is very high, see, e.g. [58–63].
The second motivation which concerns us is gauge coupling unification. The sizes of the three gauge couplings in the SM depend on the energy scale at which they are probed. While they only come close to each other in the SM at a scale of roughly 1014 GeV the
gauge couplings almost perfectly unify at a scale of roughly 1016 GeV in the MSSM [64], see also Fig. 3.1. Unification of gauge couplings is an essential ingredient for GUTs where the three SM gauge groups are unified to one simple group. From that point of view SUSY and GUTs seem to fit together quite well.
20 3. Supersymmetry 100 105 108 1011 1014 1017 ΜR@GeVD 0.02 0.04 0.06 0.08 0.10 0.12 Αi
Running Gauge Couplings in the SM
Α3 Α2 Α1 100 105 108 1011 1014 1017 ΜR@GeVD 0.02 0.04 0.06 0.08 0.10 0.12 Αi
Running Gauge Couplings in the MSSM
Α3
Α2
Α1
Figure 3.1: Running of the gauge couplings on one-loop level in the SM (left) and the
MSSM (right). The SUSY scale MSUSY was set toMZ ≈90 GeV.
The third argument often invoked in favour of SUSY is that within SUSY models in which the lightest SUSY particle (LSP) is stable on cosmological time scales this particle is a viable candidate for dark matter. This is true, for example, in R-parity conserving SUSY models like the MSSM. There, the LSP is stable and if it carries neither colour nor electric charge, like a neutralino or the gravitino, it is a good candidate for dark matter, for a recent review see [65]. Nevertheless, this third argument in favour of SUSY plays only a minor role in this thesis since our main focus is not on cosmological aspects.