Systematic classification of three-loop realizations of the Weinberg operator

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. vacuum stability constraints for the AKS model have been studied in [35][36][37].The same topologies and diagrams as in the AKS model appear also in [38]. However, the authors of [38] use doubly charged vector-like fermions and a scalar doublet with hypercharge Y = 3/2 (plus the singlets of the AKS model). The diagrams D M 6 and D M 7 appear also in [39]. Here, however, these diagrams descend from our topologies T 40 and T 33 . D M 6 appears also in a model based on singlets [40], again descending from T 40 . The last 3-loop model we mention is the one discussed in [41]. It contains a scalar septet and a fermionic quintuplet, generating the diagrams D M 6 (from T 37 ) and D M 7 (from T 32 ). The model contains an accidental Z 2 , but still one needs to impose an additional Z 2 by hand.Our classification scheme for the different topologies concentrates on identifying 3-loop genuine topologies, which are those associated with the dominant contributions to the neutrino mass matrix. We discuss thoroughly the concept of "genuineness" in section 2. There, we also define two classes of such topologies: Normal (or ordinary) genuine topologies (in total there are 44 of them) and special genuine topologies, which require very special fields (29 cases). The full list is given in appendix A.As we will explain later, these special genuine topologies are associated to finite loop integrals, even though they generate some particular 3-or 4-point interaction at loop level. This happens because the corresponding tree-level renormalizable vertex vanishes due to the antisymmetric nature of some SU (2) L contractions. Our 29 special genuine topologies are of this type. However, there are some 3-loop models in...

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