Complete One-loop Structure of the Type-(I+II) Seesaw Effective Field Theory

TL;DR

This paper derives the complete one-loop effective field theory for the combined type-(I+II) seesaw mechanism, revealing additional contributions and modifications to operator coefficients beyond the separate type-I and type-II cases.

Contribution

It provides the first complete one-loop matching of the type-(I+II) seesaw mechanism onto its effective field theory, including entangled effects of right-handed neutrinos and triplet Higgs.

Findings

All additional contributions from N_R and Φ are calculated.

The structure of dim-6 operators matches that of the type-II SEFT.

Wilson coefficients of certain operators are modified by new contributions.

Abstract

Besides the three canonical seesaw mechanisms, the hybrid scenario, i.e., the so-called type-(I+II) seesaw mechanism containing both the right-handed neutrinos $N^{}_{\rm R}$ and the triplet Higgs $\Phi$ is also an appealing extension of the Standard Model (SM) to account for tiny neutrino masses. Recently, the seesaw effective field theories (SEFTs) of the three canonical seesaw mechanisms have already been completely constructed up to one-loop level. In this work, we carry out the one-loop matching of the type-(I+II) seesaw mechanism onto the corresponding type-(I+II) SEFT, which is by no means the trivial combination of the type-I and type-II SEFTs and contains additional contributions even though the right-handed neutrinos and the triplet Higgs have no direct interactions. Employing the Feynman diagrammatic approach, we calculate all those additional contributions from the entangled effects of $N^{}_{\rm R}$ and $\Phi$, and finally achieve the complete one-loop structure of the type-(I+II) SEFT. In the type-(I+II) SEFT, the number and content of dim-6 operators are exactly the same as those in the type-II SEFT, but the Wilson coefficients of the unique dim-5 and nine dim-6 operators as well as the quartic coupling constant of the SM Higgs gain some additional contributions, which are absent in the type-I and type-II SEFTs.