  The spectral-weight distribution in recent neutron scattering experiments on
the parent compound La$_2$CuO$_4$ (LCO), which are limited in energy range to
about 450\,meV, is studied in the framework of the Hubbard model on the square
lattice with effective nearest-neighbor transfer integral $t$ and on-site
repulsion $U$. Our study combines a number of numerical and theoretical
approaches, including, in addition to standard treatments, density matrix
renormalization group calculations for Hubbard cylinders and a suitable spinon
approach for the spin excitations. Our results confirm that the $U/8t$
magnitude suitable to LCO corresponds to intermediate $U$ values smaller than
the bandwidth $8t$, which we estimate to be $8t \approx 2.36$ eV for
$U/8t\approx 0.76$. This confirms the unsuitability of the conventional linear
spin-wave theory. Our theoretical studies provide evidence for the occurrence
of ground-state d-wave spinon pairing in the half-filled Hubbard model on the
square lattice. This pairing applies only to the rotated-electron spin degrees
of freedom, but it could play a role in a possible electron d-wave pairing
formation upon hole doping. We find that the higher-energy spin spectral weight
extends to about 566 meV and is located at and near the momentum $[\pi,\pi]$.
The continuum weight energy-integrated intensity vanishes or is extremely small
at momentum $[\pi,0]$. This behavior of this intensity is consistent with that
of the spin waves observed in recent high-energy neutron scattering
experiments, which are damped at the momentum $[\pi,0]$. We suggest that future
LCO neutron scattering experiments scan the energies between 450 meV and 566
meV and momenta around $[\pi,\pi]$.
