  We consider the initial value problem for cubic derivative nonlinear
Schr\"odinger equations possessing weakly dissipative structure in one space
dimension. We show that the small data solution decays like $O((\log
t)^{-1/4})$ in $L^2$ as $t\to +\infty$. Furthermore, we find that this
$L^2$-decay rate is optimal by giving a lower estimate of the same order.
