The impact of the -drift in the cross-field transport and its effect on the density and power scrape-off layer (SOL) width in the limit of low anomalous transport is studied with the fluid code SolEdge2D. In the first part of the work, the simulations are run with an isothermal reduced fluid model. It is found that a -drift dominated regime is reached in all geometries studied (JET-like, ASDEX-like and circular analytic geometries), and that the transition toward this regime comes along with the apparition of supersonic shocks, and a complex parallel equilibrium. The parametric dependencies of the SOL width in this regime are investigated, and the temperature and the poloidal magnetic field are found to be the principal parameters governing the evolution of the SOL width. In the second part of this paper, the impact of additional physics is studied (inclusion of the centrifugal drift, self-consistent variation of temperature and the treatment of the neutral species). The addition of centrifugal drift and neutral species are shown to play a role in the establishment of the parallel equilibrium, impacting the SOL's width, although the role of the centrifugal drift is limited to a low diffusion level. Finally, the numerical results are compared with the estimate of the Goldston's heuristic drift based model (HD-model), the starting point of our study, and which has shown good agreement with experimental scaling laws. We find that the particles SOL widths in the -drift dominated regime are at least two times smaller than the estimate of the HD-model. Moreover, in the parametric dependencies proposed by the HD-model, the dependency with is retrieved, but not the one on T.