  We consider the full irrotational water waves system with surface tension and
no gravity in dimension two (the capillary waves system), and prove global
regularity and modified scattering for suitably small and localized
perturbations of a flat interface. An important point of our analysis is to
develop a sufficiently robust method, based on energy estimates and dispersive
analysis, which allows us to deal with strong singularities arising from time
resonances in the applications of the normal form method and nonlinear
scattering. As a result, we are able to consider a suitable class of
perturbations with finite energy, but no other momentum conditions. Part of our
analysis relies on a new treatment of the Dirichlet-Neumann operator in
dimension two which is of independent interest. As a consequence, the results
in this paper are self-contained.
