  The existence of coherent quasiparticles near the Fermi energy in the low
temperature state of high-temperature superconductors has been well established
by angle-resolved photoemission spectroscopy (ARPES). This technique directly
probes the momentum-resolved electronic excitation spectrum of the CuO$_2$
planes. We present a study of close to optimally doped
La$_{1.83}$Sr$_{0.17}$CuO$_4$ in the superconducting state and report an abrupt
change in the quasiparticle spectral function, as we follow the dispersion of
the ARPES signal from the Fermi energy up to 0.6 eV. The interruption in the
quasiparticle dispersion separates coherent quasiparticle peaks at low energies
from broad incoherent excitations at high energies. We find that the boundary
between these low-energy and high-energy features exhibits a cosine-shaped
momentum dependence, reminiscent of the superconducting d-wave gap. Further
intriguing similarities between characteristics of the incoherent excitations
and quasiparticle properties (lifetime, Fermi arcs) suggest a close relation
between the electronic response at high and low energies in cuprate
superconductors.
