  We consider the initial value problem for a system of cubic nonlinear
Schr\"odinger equations with different masses in one space dimension. Under a
suitable structural condition on the nonlinearity, we will show that the small
amplitude solution exists globally and decays of the rate $O(t^{-1/2}(\log
t)^{-1/2})$ in $L^\infty$ as $t$ tends to infinity, if the system satisfies
certain mass relations.
