Threshold Effects on the Massless Neutrino in the Canonical Seesaw Mechanism

TL;DR

This paper revisits threshold effects in the canonical seesaw mechanism, deriving complete one-loop RGEs for all relevant operators, and proves that one-loop effects cannot generate mass for initially massless neutrinos in minimal models.

Contribution

It provides the first strict proof that one-loop RG running cannot induce neutrino mass in minimal seesaw models with a massless initial neutrino.

Findings

Derived complete one-loop RGEs for all three dimension-five operators in $ u$SMEFT.

Identified new contributions to the Weinberg operator's Wilson coefficient and related parameters.

Proved that one-loop RG effects cannot generate neutrino mass from zero in minimal seesaw models.

Abstract

In this work, we revisit the one-loop renormalization group equations (RGEs) among non-degenerate seesaw scales, i.e., threshold effects in the canonical seesaw mechanism, which have been obtained for more than two decades. Different from the previous work only focusing on the Weinberg operator, we derive the complete one-loop RGEs of all three dimension-five operators in the Standard Model effective field theory with right-handed neutrinos ($\nu$SMEFT) and apply them to threshold effects in the canonical seesaw mechanism. We find some contributions from the Weinberg operator to its Wilson coefficient, the neutrino Yukawa coupling matrix, and the Higgs quartic coupling absent in the previous calculations. Based on the updated one-loop RGEs, we derive the RGE of the effective neutrino mass matrix's determinant without any approximation. Then, for the first time, we provide a strict proof that the one-loop RG running effects among non-degenerate seesaw scales can not generate a non-zero mass for the initial massless neutrino in the minimal type-I seesaw mechanism or in the canonical one with a rank-degenerate neutrino Yukawa coupling matrix. One has to include two- or higher-loop corrections to achieve a non-zero mass for the massless neutrino.