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

We present two alternative proofs for the cancellation mechanism in the U(1) symmetric pseudo-Nambu-Goldstone-Boson Dark Matter (pNGB DM) model. They help us to have a better understanding of the mechanism from multi-angle, and inspire us to propose some interesting generalizations. In the first proof, we revisit the non-linear representation method and rephrase the argument with the interaction eigenstates. 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. 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. Using the same method, we can easily generalize the model to an SO(N) model with general soft-breaking structures. In particular, we study the soft-breaking cubic terms and identify those terms which preserve the cancellation mechanism for the DM candidate. 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. This condition can be straightforwardly generalized to the vector-portal models. We provide two examples of the vector-portal case where the first one is an SU(2)_L \times U(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 mediators are the Z_nu boson and a new U(1)_X gauge boson X_nu, while in the second model the mediators are the U(1)_(B-L) and U(1)_X gauge bosons. The cancellation mechanism works in both models when there are no generic kinetic mixing terms for the gauge bosons. Once the generic kinetic mixing terms are included, the first model requires a fine-tuning of the mixing parameter to avoid the stringent direct detection bound, while the second model can naturally circumvent it.