A HEFT Perspective on the Type-II Seesaw Model and the Complete Basis of Lepton-Number-Violating Operators

We perform the complete tree-level matching of the type-II seesaw model onto the Higgs effective field theory (HEFT) through $\mathcal{O}(p^4)$ in the chiral expansion. Employing a nonlinear field representation and the primary-HEFT power-counting scheme, we retain the dependence on the independent heavy-scalar masses, the neutral-scalar mixing angle, and the triplet vacuum expectation value without introducing additional expansions in these parameters. To systematically describe lepton-number violation, we construct a complete and nonredundant basis of LNV HEFT operators through $\mathcal{O}(p^4)$, including their full flavor multiplicities, using a complex dressed spurion that encodes the $B-L$ charge and custodial orientation of the LNV insertion. The basis is independently validated by Hilbert-series counting. We then compare our matching results with those for the corresponding real-triplet extension and with a previously proposed broken-phase effective field theory. These comparisons identify the effects of the additional CP-odd and doubly charged scalar states and show that the overlapping broken-phase results are recovered after the appropriate parameter expansion, while HEFT retains the unexpanded nonlinear electroweak structure. We further discuss representative implications for Higgs and electroweak precision observables, vector-boson scattering, multi-Higgs production, anomalous gauge couplings, top-quark processes, neutrinoless double-beta decay, charged-lepton flavor violation, and same-sign dilepton production.