  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.
