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We present two alternative proofs for the cancellation mechanism in the U(1)symmetric pseudo-Nambu-Goldstone-Boson Dark Matter (pNGB DM) model. They helpus to have a better understanding of the mechanism from multi-angle, andinspire us to propose some interesting generalizations. In the first proof, werevisit the non-linear representation method and rephrase the argument with theinteraction eigenstates. In this picture, the phase mode (DM) can only have atrilinear interaction with a derivative-squared acting on the radial mode whenthe DM is on-shell. Thus, the DM-quark scattering generated by a mass mixingbetween the radial mode and the Higgs boson vanishes in the limit ofzero-momentum transfer. Using the same method, we can easily generalize themodel to an SO(N) model with general soft-breaking structures. In particular,we study the soft-breaking cubic terms and identify those terms which preservethe cancellation mechanism for the DM candidate. In our discussion of thesecond method, we find that the cancellation relies on the special structure ofmass terms and interactions of the mediators. This condition can bestraightforwardly generalized to the vector-portal models. We provide twoexamples of the vector-portal case where the first one is an SU(2)_L \timesU(1)_Y \times U(1)_X model and the second one is an SU(2)_L \times U(1)_Y\times U(1)_(B-L) \times U(1)_X model. In the first model, the vector mediatorsare the Z_nu boson and a new U(1)_X gauge boson X_nu, while in the second modelthe mediators are the U(1)_(B-L) and U(1)_X gauge bosons. The cancellationmechanism works in both models when there are no generic kinetic mixing termsfor the gauge bosons. Once the generic kinetic mixing terms are included, thefirst model requires a fine-tuning of the mixing parameter to avoid thestringent direct detection bound, while the second model can naturallycircumvent it.

Cancellation mechanism of dark matter direct detection in Higgs-portal and vector-portal models