  Axion or axion-like particle (ALP) has been usually considered as a CP-odd
Nambu-Goldstone boson (NGB) from the spontaneous breakdown of a global U(1)
symmetry. In this paper, we point out that the NGB behaves as a CP-even
particle coupled to the SM particles in a large class of simple (or perhaps the
simplest) renormalizable models. We provide a first study of the collider
phenomenology and cosmology of the CP-even ALP. In a natural parameter region,
the CP-even ALP can be produced from the Higgs boson decay in colliders. When
the mass is not very light, the signals will be Higgs exotic decays, Higgs
decay to displaced vertex $\times 2$, Higgs decay to displaced vertex + missing
energy. The signal can be discriminated from other models, e.g. hidden photon,
by measuring the decay length and the decay products of the light new particle.
In addition, when $ m_a\lesssim \,$MeV, in which case the Higgs boson invisible
decay may be probed in the colliders, the CP-even ALP is a nice Dark matter
(DM) candidate. The DM can be probed by 21cm line measurement, the future
measurement of the Milky way mass halo function in the Vera Rubin Observatory,
as well as X- or $\gamma$-ray observations. The DM production mechanisms are
discussed.
