  Based on the $SU(3)_{F}$ gauge family symmetry model which was proposed to
explain the observed mass and mixing pattern of neutrinos, we investigate the
symmetry breaking, the mixing pattern in quark and lepton sectors, and the
contribution of the new gauge bosons to some flavour changing neutral currents
(FCNC) processes at low energy. With the current data of the mass differences
in the neutral pseudo-scalar $P^{0}-\bar{P}^{0}$ systems, we find that the
$SU(3)_{F}$ symmetry breaking scale can be as low as 300TeV and the mass of the
lightest gauge boson be about $100$TeV. Other FCNC processes, such as the
lepton flavour number violation process $\mu^{-}\rightarrow e^{-}e^{+}e^{-}$
and the semi-leptonic rare decay $K\rightarrow \pi \bar{\nu} \nu$, contain
contributions via the new gauge bosons exchanging. With the constrains got from
$P^0-\bar{P}^0$ system, we estimate that the contribution of the new physics is
around $10^{-16}$, far below the current experimental bounds.
