Phenomenology of Harmless Family Gauge Bosons to $K^0$-${\bar{K}}^0$ Mixing

When we try to consider family gauge bosons with a lower energy scale, a major obstacle is constraints from the observed $P^0$-${\bar{P}}^0$ mixings ($P^0=K^0,D^0,B^0,B_s^0$). Against such a conventional view, we point out that, in a U(3) family gauge boson model, the bosons are harmless to any $P^0$-${\bar{P}}^0$ mixings independently of explicit values of the family mixings, if masses $M_{ij}$ of the gauge bosons $A_i^j$ ($i,j$ are family indexes) satisfy a relation $2\mathrm{/}{M}_{ij}^2=1\mathrm{/}{M}_{ii}^2+1\mathrm{/}{M}_{jj}^2$. If such the case can be realized together with an inverted mass hierarchy $M_{33}^2 \ll M_{22}^2 \ll M_{11}^2$, we can consider family gauge bosons with a considerably lower scale, so that we can expect rich signs for family gauge bosons in a TeV scale.