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We propose the local (B-L)_{mu tau} model, which minimally retains the local B-L extension for the sake of neutrino phenomenologies, and at the same time presents an invisible gauge boson Z' with mass O(10) MeV to account for the discrepancy of the muon anomalous magnetic moment. But such a scenario is challenged by flavor physics. To accommodate the correct pattern of Cabibbo-Kobayashi-Maskawa matrix, we have to introduce either a SU(2)_L doublet flavon or vector-like quarks plus a singlet flavon. In either case Z' induces flavor changing neutral current (FCNC) in the quark sector at tree-level. We find that the former scheme is completely ruled out by K → pi nu nu. Whereas the latter scheme, in which FCNC merely arises in the up-type quark sector, is still free of strong constraint. In particular, it opens a new window to test our scenario by searching for flavor-changing top quark decay mode t → u/c+(invisible), and the typical branching ratio O(10^{-4}).