  We prove global existence and modified scattering property for the solutions
of the $2D$ gravity water waves system in the infinite depth setting for a
class of initial data, which is only required to be small above the level
$\dot{H}^{1/5}\times \dot{H}^{1/5+1/2}$. No assumption is assumed below this
level, therefore, it allows to have infinite energy. As a direct consequence,
the momentum condition assumed on the physical velocity in all previous small
energy results by Ionescu-Pusateri, Alazard-Delort and Ifrim-Tataru is removed.
