  We report many-body calculations of the self-energy and lifetime of Shockley
and image states on the (100) and (111) surfaces of Cu that go beyond the $GW$
approximation of many-body theory. The self-energy is computed in the framework
of the GW\Gamma approximation by including short-range exchange-correlation
(XC) effects both in the screened interaction W (beyond the random-phase
approximation) and in the expansion of the self-energy in terms of W (beyond
the GW approximation). Exchange-correlation effects are described within
time-dependent density-functional theory from the knowledge of an adiabatic
nonlocal XC kernel that goes beyond the local-density approximation.
