Minimal Majoron Dark Matter from a Discrete $Z_N$ Gauge Symmetry
We investigate majoron dark matter in a minimal setup, where the Standard Model is extended by three right-handed neutrinos and a complex scalar field. The theory is defined by an exact discrete gauge symmetry, $Z_N\subset U(1)_{B-L}$, while the global $U(1)_{B-L}$ symmetry emerges only as an accidental symmetry at low energies. For nontrivial choices of the discrete symmetry $Z_N$, such as $Z_5$, $Z_7$, $Z_{11}$, and $Z_{13}$, Planck-suppressed operators explicitly break this accidental symmetry and generate a small majoron mass, making the resulting pseudo-Nambu--Goldstone boson a well-motivated dark matter candidate. We study its production via the misalignment mechanism after inflation, considering both radiation-dominated and early matter-dominated cosmological histories, and confront the viable parameter space with isocurvature bounds, cosmological constraints, and indirect dark matter searches. We find that the $Z_5$ model is excluded by limits on the dominant dark matter decay into neutrinos, whereas the other models remain viable. In particular, the $Z_7$ scenario predicts a majoron mass in the $1$--$10\,{\rm MeV}$ range and can be sensitively probed by future MeV gamma-ray observations, especially with COSI, through the 511$\,$keV line from the majoron decay into an electron--positron pair and the monochromatic gamma-ray line from its decay into two photons.