The transition from a weak-disorder (diffusive phase) to a strong-disorder (localized phase) for directed polymers in a random environment is a well studied phenomenon. In the most common setup, it is established that the phase transition is trivial when the transversal dimension $d$ equals $1$ or $2$ (the diffusive phase is reduced to $β=0$) while when $d\geq 3$, there is a critical temperature $β_c\in (0,\infty)$ which delimits the two phases. The proof of the existence of a diffusive regime for $d\geq 3$ is based on a second moment method, and thus relies heavily on the assumption that the variable which encodes the disorder intensity (which in most of the mathematics literature assumes the form $e^{βη_x}$), has finite second moment. The aim of this work is to investigate how the presence/absence of phase transition may depend on the dimension $d$ in the case when the disorder variable displays heavier tail. To this end we replace $e^{βη_x}$ by $(1+βω_x)$ where $ω_x$ is in the domain of attraction of a stable law with parameter $γ\in (1, 2)$.