Abstract: The Standard Model (SM) is a chiral theory, where right- and left-handed fermion fields transform differently under the gauge group. Extra fermions, if they do exist, need to be heavy otherwise they would have already been observed. With no complex mechanisms at work, such as confining interactions or extra-dimensions, this can only be achieved if every extra right-handed fermion comes paired with a left-handed one transforming in the same way under the Standard Model gauge group, otherwise the new states would only get a mass after electroweak symmetry breaking, which would necessarily be small (similar to 100 GeV). Such a simple requirement severely constrains the fermion content of Grand Unified Theories (GUTs). It is known for example that three copies of the representations (5) over bar + 10 of SU(5) or three copies of the 16 of SO(10) can reproduce the Standard Model's chirality, but how unique are these arrangements? In a systematic way, this paper looks at the possibility of having non-standard mixtures of fermion GUT representations yielding the correct Standard Model chirality. Family unification is possible with large special unitary groups for example, the 171 representation of SU(19) may decompose as 3(16) + 120 + 3(1) under SO(10).

Abstract: Assuming that neutrinos are Majorana particles, we perform a complete classification of all possible mixing matrices which are fully determined by residual symmetries in the charged-lepton and neutrino mass matrices. The classification is based on the assumption that the residual symmetries originate from a finite flavour symmetry group. The mathematical tools which allow us to accomplish this classification are theorems on sums of roots of unity. We find 17 sporadic cases plus one infinite series of mixing matrices associated with three-flavour mixing, all of which have already been discussed in the literature. Only the infinite series contains mixing matrices which are compatible with the data at the 3 sigma level.

Abstract: We study systematically the decomposition of the Weinberg operator at three-loop order. There are more than four thousand connected topologies. However, the vast majority of these are infinite corrections to lower order neutrino mass diagrams and only a very small percentage yields models for which the three-loop diagrams are the leading order contribution to the neutrino mass matrix. We identify 73 topologies that can lead to genuine three-loop models with fermions and scalars, i.e. models for which lower order diagrams are automatically absent without the need to invoke additional symmetries. The 73 genuine topologies can be divided into two sub-classes: normal genuine ones (44 cases) and special genuine topologies (29 cases). The latter are a special class of topologies, which can lead to genuine diagrams only for very specific choices of fields. The genuine topologies generate 374 diagrams in the weak basis, which can be reduced to only 30 distinct diagrams in the mass eigenstate basis. We also discuss how all the mass eigenstate diagrams can be described in terms of only five master integrals. We present some concrete models and for two of them we give numerical estimates for the typical size of neutrino masses they generate. Our results can be readily applied to construct other d = 5 neutrino mass models with three loops.