  In this study, we investigate out-of-time-order correlators (OTOCs) in
systems with power-law decaying interactions such as $R^{-\alpha}$, where $R$
is the distance. In such systems, the fast scrambling of quantum information or
the exponential growth of information propagation can potentially occur
according to the decay rate $\alpha$. In this regard, a crucial open challenge
is to identify the optimal condition for $\alpha$ such that fast scrambling
cannot occur. In this study, we disprove fast scrambling in generic long-range
interacting systems with $\alpha>D$ ($D$: spatial dimension), where the total
energy is extensive in terms of system size and the thermodynamic limit is
well-defined. We rigorously demonstrate that the OTOC shows a polynomial growth
over time as long as $\alpha>D$ and the necessary scrambling time over a
distance $R$ is larger than $t\gtrsim R^{\frac{2\alpha-2D}{2\alpha-D+1}}$.
