Minimal Dark Matter in the Local $B-L$ Extension

The minimal gauge group extension to the standard model (SM) by the local $U(1)_{B-L}$ (MBLSM) is well known as the minimal model to understand neutrino mass origins via the seesaw mechanism, following the gauge principle. This "small" symmetry also has deep implication to another big thing, dark matter (DM) stability. We demonstrate it in the framework of minimal dark matter (MDM), which aims at addressing two basic questions on DM, stability and the nature of interactions. However, stability and perturbativity may only allow the fermionic quintuplet. The situation is very different in the MBLSM, which leaves the subgroup of $U(1)_{B-L}$, the matter parity $(-1)^{3(B-L)}$, unbroken; it is able to stabilize all of the weakly-interacting {MDM candidates } after assigning a proper $U(1)_{B-L}$ charge. For the candidates with nonzero hypercharge, the phenomenological challenge comes from realizing the inelastic DM scenario thus evading the very strict DM direct detention bounds. We present two approaches that can slightly split the CP-even and -odd parts of the neutral components: 1) using the dimension 5 operators, which works for the $U(1)_{B-L}$ spontaneously breaking at very high scale; 2) mixing with {other fields} having zero hypercharge, which instead works for a low $U(1)_{B-L}$ breaking scale.