One-loop matching of the type-II seesaw model onto the Standard Model effective field theory

Abstract

A bstract In this paper, we continue to construct the low-energy effective field theories (EFTs) of the canonical seesaw models, which are natural extensions of the Standard Model (SM) to accommodate tiny but nonzero neutrino masses. Different from three right-handed neutrino singlets in the type-I seesaw model, the Higgs triplet in the type-II seesaw model participates directly in the electroweak gauge interactions, rendering the EFT construction more challenging. By integrating out the heavy Higgs triplet in the functional-integral formalism, we carry out a complete one-loop matching of the type-II seesaw model onto the so-called Standard Model Effective Field Theory (SMEFT). It turns out that 41 dimension-six operators (barring flavor structures and Hermitian conjugates) in the Warsaw basis of the SMEFT can be obtained, covering all those 31 dimension-six operators in the case of type-I seesaw model. The Wilson coefficients for 41 dimension-six operators are computed up to $$ \mathcal{O}\left({M}_{\Delta }^{-2}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mtext>∆</mml:mtext> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> </mml:mfenced> </mml:math> with M ∆ being the mass scale of the Higgs triplet. Moreover, the branching ratios of rare radiative decays of charged leptons $$ {l}_{\alpha}^{-}\to {l}_{\beta}^{-}+\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>l</mml:mi> <mml:mi>α</mml:mi> <mml:mo>−</mml:mo> </mml:msubsup> <mml:mo>→</mml:mo> <mml:msubsup> <mml:mi>l</mml:mi> <mml:mi>β</mml:mi> <mml:mo>−</mml:mo> </mml:msubsup> <mml:mo>+</mml:mo> <mml:mi>γ</mml:mi> </mml:math> are calculated in the EFT and compared with that in the full theory in order to demonstrate the practical application and the correctness of our EFT construction.