We provide a generic framework to obtain stable dark matter along with naturally small Dirac neutrino masses generated at the loop level. This is achieved through the spontaneous breaking of the global U (1)B−L symmetry already present in Standard Model. The U (1)B−L symmetry is broken down to a residual even Zn; n ≥ 4 subgroup. The residual Zn symmetry simultaneously guarantees dark matter stability and protects the Dirac nature of neutrinos. The U (1)B−L symmetry in our setup is anomaly free and can also be gauged in a straightforward way. Finally, we present an explicit example using our framework to show the idea in action.

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...

We provide a complete systematic classification of all two-loop realizations of the dimension four operator for Dirac neutrino masses. Our classification is multilayered, starting first with a classification in terms of all possible distinct two loop topologies. Then we discuss the possible diagrams for each topology. Model-diagrams originating from each diagram are then considered. The criterion for genuineness is also defined and discussed at length. Finally, as examples, we construct two explicit models which also serve to highlight the intimate connection between the Dirac nature of neutrinos and the stability of dark matter.

We present a novel mode of neutrinoless double-β decay with emission of a light Majoronlike scalar particle φ. We assume it couples via an effective 7-dimensional operator with a (V + A) lepton current and (V ± A) quark currents leading to a long-range contribution that is unsuppressed by the light neutrino mass. We calculate the total double-β decay rate and determine the fully differential shape for this mode. We find that future double-β decay searches are sensitive to scales of the order Λ NP ≈ 1 TeV for the effective operator and a light scalar m φ < 0.2 MeV, based on ordinary double-β decay Majoron searches. The angular and energy distributions can deviate considerably from that of two-neutrino double-β decay, which is the main background. We point out possible ultraviolet completions where such an effective operator can emerge.