  We prove the global well-posedness for a $L^2$-critical defocusing cubic
higher-order Schr\"odinger equation, namely \[ i\partial_t u + \Lambda^k u =
-|u|^2 u, \] where $\Lambda=\sqrt{-\Delta}$ and $k\geq 3, k \in \mathbb{Z}$ in
$\mathbb{R}^k$ with initial data $u_0 \in H^\gamma,
\gamma>\gamma(k):=\frac{k(4k-1)}{14k-3}$.
