  In this paper, we will prove the existence of full dimensional tori for
1-dimensional nonlinear Schr\"odinger equation with periodic boundary
conditions \begin{equation*}\label{L1}
\mathbf{i}u_t-u_{xx}+V*u+\epsilon|u|^4u=0,\hspace{12pt}x\in\mathbb{T},
\end{equation*} where $V*$ is the convolution potential. Here the radius of the
invariant torus satisfies a slower decay, i.e. \begin{equation*}\label{031601}
I_n\sim e^{- \ln^{\sigma}|n|},\qquad \mbox{as}\ |n|\rightarrow\infty,
\end{equation*} for any $\sigma>2$, which improves the result given by Bourgain
(J. Funct. Anal. 229 (2005), no.1, 62-94).
