  It is known since the work of Dyachenko \& Zakharov \cite{zd} that for the
weakly nonlinear 2d infinite depth water waves, there are no 3-wave
interactions and all of the 4-wave interaction coefficients vanish on the
non-trivial resonant manifold. In this paper we study this partial
integrability from a different point of view.
  We construct, directly in the physical space, a sequence of energy
functionals $\mathfrak E_j(t)$ which are explicit in the Riemann mapping
variable and involve material derivatives of order $j$ of the solutions for the
2d water wave equation, so that $\frac d{dt} \mathfrak E_j(t)$ is quintic or
higher order. We show that if some scaling invariant norm, and a norm involving
one spatial derivative above the scaling of the initial data are of size no
more than $\varepsilon$, then the lifespan of the solution for the 2d water
wave equation is at least of order $O(\varepsilon^{-3})$, and the solution
remains as regular as the initial data during this time. If only the scaling
invariant norm of the data is of size $\varepsilon$, then the lifespan of the
solution is at least of order $O(\varepsilon^{-5/2})$. Our long time existence
results do not impose size restrictions on the slope of the initial interface
and the magnitude of the initial velocity, they allow the interface to have
arbitrary large steepnesses and initial velocities to have arbitrary large
magnitudes.
