  We prove a vanishing property of the normal form transformation of the 1D
cubic nonlinear Schr\"odinger (NLS) equation with periodic boundary conditions
on $[0,L]$. We apply this property to quintic resonance interactions and obtain
a description of dynamics for time up to $T=\frac{L^2}{\epsilon^4}$, if $L$ is
sufficiently large and size of initial data $\epsilon$ is small enough. Since
$T$ is the characteristic time of wave turbulence, this result implies the
absence of wave turbulence behavior of 1D cubic NLS. Our approach can be
adapted to other integrable systems without too many difficulties. In the
proof, we develop a correspondence between Feynman diagrams and terms in normal
forms, which allows us to calculate the coefficients inductively.
