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.
Highlights
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
We show that the 73 genuine topologies are associated to 374 diagrams in the weak basis, which get reduced to only 30 diagrams in the mass basis
In this paper we have discussed the complete decomposition of the Weinberg operator at 3-loop order
Summary
From the complete set of 228+146 genuine diagrams one can generate models by assigning quantum numbers to the internal fields following some basic rules. For example, a fermionic 10 corresponds to a right-handed neutrino νR As discussed above, it is possible in many cases to construct models that avoid lower order diagrams, despite the use of particles such as νR, by adding additional symmetries by hand to the model. The diagram needs only three singlets (two different scalars and one vector-like fermion) and no additional symmetry to produce a non-zero neutrino mass. All other models that we found need either (i) larger SU(2)L representations and/or (ii) a larger number of beyond SM fields and/or (iii) an additional symmetry to avoid lower order diagrams. We have chosen model 3 to show how larger SU(2)L representations can play a natural role in 3-loop neutrino mass models. Model 5 is a second example with coloured particles: it needs 5 exotic fields, but no additional symmetry. For detailed definitions of the loop integrals see the appendix C
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