Wide Field Adaptive Optics (WFAO) is a new proposed astronomical adaptive optics mode allowing a significant improvement of the seeing limited point spread function characteristics over large fields -- several arc minutes in diameter, using only one deformable mirror optically conjugated to an optimal altitude. In this paper, we present the WFAO upper limit performances, based on the assumption that the refractive index fluctuation field above the telescope is perfectly known. Our results are based on analytical developments for the residual phase power spectrum after WFAO correction, implemented in PAOLA, an analytical AO simulation tool, developed at the Herzberg Institute of Astrophysics. Results are presented for several sites: Mauna Kea, Cerro Tololo, Cerro Paranal. For each of these locations, we give the WFAO-PSF properties as a function of the field angle, the conjugation altitude of the deformable mirror, the imaging infrared wavelength, and the cone aperture angle over which the tomographic information is averaged to drive the deformable mirror actuators.