The authors 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 the authors' analysis is to develop a sufficiently robust method (the ``quasilinear I-method'') which allows the authors to deal with strong singularities arising from time resonances in the applications of the normal form method (the so-called ``division problem''). As a result, they are able to consider a suitable class of perturbations with finite energy, but no other momentum conditions.Part of the authors' 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.