Using the renormalization group improvement technique, we study the effective potential of a model consisting of $N$ scalar fields ${\varphi }^i$ transforming in the fundamental representation of an $O\left(N\right)$ group coupled to an additional scalar field $\sigma$ via cubic interactions, defined in a six-dimensional spacetime. We find that the model presents a metastable vacuum, which can be long-lived, where the particles become massive. The existence of attractive and repulsive interactions plays a crucial role in such phenomena.