  We show that any $4$-dimensional complete self-shrinker in $\mathbb{R}^{5}$
with constant squared norm $S$ of the second fundamental form, $f_{3}=0$ and
constant $f_{4}$ is isometric to $\mathbb{R}^{4}$, where $h_{ij}$ are
components of the second fundamental form, $S=\sum h_{ij}^{2}$, $f_{3}=\sum
h_{ij}h_{jk}h_{ki}$ and $f_{4}=\sum h_{ij}h_{jk}h_{kl}h_{li}$. As an
application, we obtain a classification result.
