A $U(1)_{(B-L)_3}$ model for dark matter and the $B^+\to K^+\nu\bar{\nu}$ excess reported by Belle II

Abstract

We explore the $U(1)_{(B-L)_3}$ gauge symmetry under which only third generation fermions are charged. The cancelation of the $[U(1)_{(B-L)_3}]^3$ anomaly requires a chiral fermion, $\chi_L$, singlet under the gauge group but charged under $U(1)_{(B-L)_3}$. This chiral fermion can play the role of thermal dark matter with a relic abundance set via the freeze-out scenario through the annihilation to pairs of the third generation fermions. In the quark mass basis, the new gauge boson will have off-diagonal couplings to the $b$ and $s$ quarks which can lead to a new contribution to $B^+\to K^+\nu_\tau\bar{\nu}_\tau$. We discuss the relevant bounds from the LHC searches for a new gauge boson, dark matter searches, electroweak precision data, $\Upsilon$ decay, the CKM matrix and the $B_s^0-\bar{B}_s^0$ mixing. We entertain the possibility of explaining the $B^+\to K^+\nu\bar{\nu}$ excess recently reported by Belle II within this model.