![]() In the first model the vector mediators are the Z μ boson and a new U(1) X gauge boson X ν, while in the second model the mediators are the U(1) B−L and U(1) X gauge bosons. We provide two examples of the vector-portal case where the first one is an SU(2) L × U(1) Y × U(1) X model and the second one is an SU(2) L × U(1) Y × U(1) B−L × U(1) X model. This condition can be straightforwardly generalized to the vector-portal models. In our discussion of the second method, we find that the cancellation relies on the special structure of mass terms and interactions of the mediators. In particular, we study the soft-breaking cubic terms and identify those terms which preserve the cancellation mechanism for the DM candidate. ![]() Using the same method, we can easily generalize the model to an SO(N) model with general soft-breaking structures. Thus, the DM-quark scattering generated by a mass mixing between the radial mode and the Higgs boson vanishes in the limit of zero-momentum transfer. In this picture, the phase mode (DM) can only have a trilinear interaction with a derivative-squared acting on the radial mode when the DM is on-shell. In the first proof, we revisit the non-linear representation method and rephrase the argument with the interaction eigenstates. They help us to have a better understanding of the mechanism from multi-angle, and inspire us to propose some interesting generalizations. We present two alternative proofs for the cancellation mechanism in the U(1) symmetric pseudo-Nambu-Goldstone-Boson Dark Matter (pNGB DM) model.
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