Systematic study of the d = 5 Weinberg operator at one-loop order

Abstract

We perform a systematic study of the d = 5 Weinberg operator at the one-loop level. We identify three different categories of neutrino mass generation: (1) finite irreducible diagrams; (2) finite extensions of the usual seesaw mechanisms at one-loop and (3) divergent loop realizations of the seesaws. All radiative one-loop neutrino mass models must fall into one of these classes. Case (1) gives the leading contribution to neutrino mass naturally and a classic example of this class is the Zee model. We demonstrate that in order to prevent that a tree level contribution dominates in case (2), Majorana fermions running in the loop and an additional Z 2 symmetry are needed for a genuinely leading one-loop contribution. In the type-II loop extensions, the Yukawa coupling will be generated at one loop, whereas the type-I/III extensions can be interpreted as loop-induced inverse or linear seesaw mechanisms. For the divergent diagrams in category (3), the tree level contribution cannot be avoided and is in fact needed as counter term to absorb the divergence.

Systematic classification of two-loop realizations of the Weinberg operator

“…The former cases, which are our "class-II" diagrams, can present interesting models of neutrino mass, even though they are not genuinely 2-loop. The characterization of class-II diagrams (and their corresponding models) is similar to the discussion given in [38] for the 1-loop order: class-II diagrams can give a theoretical motivation for the smallness of a particular vertex, generated at 1-loop order. This particular vertex then appears in one of the four genuine 1-loop neutrino mass diagrams (see figure 1), making the whole construction effectively 2-loop.…”

“…In [38] the Weinberg operator was studied systematically at the one-loop level. Two topologies (for a total of four diagrams) were identified to give neutrino masses at the 1-loop level genuinely (i.e.…”

“…Three more diagrams were found, that can be understood as 1-loop realizations of one of the known tree-level seesaws and the relation between tree-and 1-loop diagrams were discussed. In our current work, we extend this analysis [38] to the 2-loop level, following the same diagrammaticbased approach.…”

“…The systematic decomposition used in [38] allows one to identify all possible realizations of O 5 at a given loop level, in principle. However, while there are only 12 diagrams (out of which only seven turn out to be of any interest) at the 1-loop level, at the 2-loop level one can naively expect to find order O(100) diagrams, which need to be studied.…”

“…The four diagrams leading to genuine 1-loop neutrino mass models. The notation of [38] is used to classify these diagrams. Just to mention two examples: diagram T1-ii corresponds to the the classical Zee model [21], while an example for T-3 is the "scotogenic" model of [39].…”