The constrained MSSM (CMSSM) is a SUSY breaking scenario inspired by supergravity (SUGRA) theories where SUSY breaking is mediated via gravitational interactions [88,89]. To avoid large flavour changing neutral currents (FCNC) usually the assumption is made that the couplings of the SUSY breaking sector to the visible sector is universal in flavour space. This kind of model is called minimal supergravity (mSUGRA) since a minimal choice for the K¨ahler potential is used [89]. There is a subtle difference between CMSSM and mSUGRA concerning the gravitino mass which shall not bother us here since we always assume that the gravitino is not the LSP and for the other parts it is irrelevant.
The boundary conditions for the soft SUSY breaking parameters at the GUT scale are then Ma(MGUT) = m1/2 , AY(MGUT) = A0Y , ˜ m2f˜(MGUT) = m20 , (3.61)
where Ma, a = 1,2,3, are the gaugino masses, AY are the trilinear couplings which are
assumed to be proportional to the particular Yukawa matrices Y and ˜mf˜are the sfermion
masses. All in all in the CMSSM we have therefore four parameters fully describing the MSSM spectrum. They arem1/2,A0,m0 and tanβ. The absolute value of µis determined
from the conditions for EWSB and the sign ofµis set to be positive due to the constraint from the anomalous magnetic moment of the muon.
CHAPTER 4
Grand Unified Theories
In this chapter we discuss the idea of Grand Unified Theories (GUTs) on the basis ofSU(5) gauge theory which is the smallest simple gauge group containing the SM gauge group and being compatible with the field content of the SM. This GUT is of special importance for this work since later on we construct explicit flavour models within SU(5).
The guiding principle behind GUTs is the quest for unification of forces. The SM knows three gauge interactions which can be described in terms of symmetry groups. Therefore it seems compelling to look for a larger group which contains the SM gauge group and hence describes the SM gauge interactions in terms of one simple group. Besides unification of forces, GUTs also shed some light on the questions of neutrino masses, charge quantisation and anomaly cancellation. Very recently it was even shown how to embed the inflaton into a GUT representation [90].
The first step towards a GUT was the unification of colour and lepton number in the Pati–Salam (PS) model [46] with the gauge group SU(4)C ×SU(2)L×SU(2)R. Later on
also simple groups were proposed, likeSU(5) [11],SO(10) [12] orE6 [91]. Here we want to
discuss the relevant features of GUTs via the example of SU(5). Our treatment of GUTs is based on the presentations in [92].
We start with an introduction to SU(5) and describe the field content in Sec. 4.1. Afterwards we discuss how electric charge is quantised in Sec. 4.2, the unification of gauge couplings in Sec. 4.3 and proton decay in Sec. 4.4. We end this chapter with a brief summary about symmetry breaking and fermion masses within SU(5) in Sec. 4.5.
4.1
Introduction to
SU(5) and Field Content
A general representation ofSU(5) can be written as a tensor underSU(5), for more details, see App. B. The transformation matrices U are in this case
[U]im = [exp(iαaλa/2)]im , (4.1) where the indicesi, m= 1, . . . ,5 and a= 0, . . . ,23. Theαaare the transformation param-
38 4. Grand Unified Theories
The generators of SU(5) are Hermitian, traceless 5×5 matrices with the normalisation Tr(λaλb) = 2δab, for example λ3 = 0 0 0 1 −1 and λ0 = √1 15 2 2 2 −3 −3 . (4.2)
The SU(3)C × SU(2)L decomposition of a given representation is given by identifying
the first three components of an SU(5) index i = 1,2,3 as colour indices and the last two components of an SU(5) index i = 4,5 as weak isospin indices. Below we label for convenience colour indices with Greek letters,α,β,. . ., and weak isospin indices with Latin letters, r, s, . . ..
Taking a closer look at the SM field content, cf. Tab. 2.1, we see that all the left-handed matter fields fit nicely into the two representations
F =5= (3,1) + (1,2) = dc R dcB dcG e −ν L , T =10 = (3,1) + (3,2) + (3,1) = √1 2 0 −ucG ucB −uR −dR uc G 0 −ucR −uB −dB −uc B ucR 0 −uG −dG uR uB uG 0 −ec dR dB dG ec 0 L . (4.3)
where we have given the decomposition under SU(3)C ×SU(2)L, see also [93]. The lower
indices R, B and G denote the quark colours and the index L denotes that F and T are left-handed. The hypercharge of the fields is discussed in the next section.
The gauge bosons of the SM are all unified into the twenty-four-dimensional adjoint representation of SU(5) Ai
j which can be decomposed as
Aij =24= (8,1) + (1,3) + (1,1) + (3,2) + (3,2). (4.4) Using the index-labelling convention as described above we can identify the SM gauge bosons as
• Aαβ = (8,1) are the gluons of SU(3)C.
• Ars = (1,3) are the W bosons of SU(2)L.
• p3/20Ar r−
p
1/15Aα
α = (1,1) is the U(1)Y B-field.
Then twelve of the gauge bosons in Ai
j are still unassigned. They carry both SU(3)C and
SU(2)L indices