Intergenerational gauged $B-L$ model and its implication to muon $g-2$ anomaly and thermal dark matter

We study the flavor dependent $U(1)_{B_i-L_j}$ models, where an $i$th generation of quarks and $j(\neq i)$th generation of leptons are charged. By solving the anomaly free condition for the matter sector of the SM fermions and three generations of right-handed (RH) neutrinos, we find that the $j$th generation of RH neutrino is not necessarily charged under the $U(1)_{B_i-L_j}$ gauge symmetry with the charge $-1$ and the other (neither $i$th nor $j$th) generation of RH neutrino can also be. As a general solution for the anomaly cancellation conditions, the other two RN neutrinos than the charge $-1$ RH neutrino may have non-vanishing charge and be stable due to the gauge invariance, and hence it is a candidate for dark matter (DM) in our Universe. We apply this result to a $B_3-L_2$ model and consider a light thermal DM and a solution to the muon $g-2$ anomaly. We identify the parameter region to have the DM mass range from MeV to sub-GeV and simultaneously solve the muon $g-2$ anomaly. We also derive the constraints on the gauge kinetic mixing parameter by using the latest Borexino phase-II data.