  Magnetic fluctuations is the leading candidate for pairing in cuprate,
iron-based and heavy fermion superconductors. This view is challenged by the
recent discovery of nodeless superconductivity in CeCu$_2$Si$_2$, and calls for
a detailed understanding of the corresponding magnetic fluctuations. Here, we
mapped out the magnetic excitations in \ys{superconducting (S-type)}
CeCu$_2$Si$_2$ using inelastic neutron scattering, finding a strongly
asymmetric dispersion for $E\lesssim1.5$~meV, which at higher energies evolve
into broad columnar magnetic excitations that extend to $E\gtrsim 5$ meV. While
low-energy magnetic excitations exhibit marked three-dimensional
characteristics, the high-energy magnetic excitations in CeCu$_2$Si$_2$ are
almost two-dimensional, reminiscent of paramagnons found in cuprate and
iron-based superconductors. By comparing our experimental findings with
calculations in the random-phase approximation,we find that the magnetic
excitations in CeCu$_2$Si$_2$ arise from quasiparticles associated with its
heavy electron band, which are also responsible for superconductivity. Our
results provide a basis for understanding magnetism and superconductivity in
CeCu$_2$Si$_2$, and demonstrate the utility of neutron scattering in probing
band renormalization in heavy fermion metals.
