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

TL;DR

This paper systematically classifies one-loop neutrino mass generation mechanisms from the $d=5$ Weinberg operator, identifying three categories and analyzing their features, including the conditions for dominance and the role of symmetries.

Contribution

It provides a comprehensive classification of one-loop neutrino mass models based on the Weinberg operator, including conditions for their dominance and the necessity of symmetries.

Findings

Category (1) includes finite irreducible diagrams like the Zee model.

Category (2) involves finite loop extensions of seesaws requiring symmetries.

Category (3) contains divergent diagrams that need counter terms.

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 $\mathbb{Z}_2$ symmetry are needed for a genuinely leading one-loop contribution. In the type-II loop extensions, the lepton number violating 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.