  The charge-density-wave (CDW) instability in the underdoped, pseudogap part
of the cuprate phase diagram has been a major recent research focus, yet
measurements of dynamic, energy-resolved CDW correlations are still in their
infancy. We report a high-resolution resonant inelastic X-ray scattering (RIXS)
study of the underdoped cuprate superconductor HgBa$_{2}$CuO$_{4+\delta}$ ($T_c
= 70$ K). At $T=250$ K, above the CDW order temperature $T_\mathrm{CDW} \approx
200$ K, we observe significant dynamic CDW correlations at about 40 meV. This
energy scale is comparable to both the superconducting gap and the previously
reported low-energy pseudogap. At $T = T_c$, a strong elastic CDW peak appears,
but the dynamic correlations around 40 meV remain virtually unchanged. In
addition, we observe a new feature: dynamic correlations at significantly
higher energy, with a characteristic scale of about 160 meV. A similar scale
was previously identified in other experiments as a high-energy pseudogap. The
existence of three distinct features in the charge response is highly unusual
for a CDW system, and suggests that charge order in the cuprates is closely
related to the pseudogap phenomenon and more complex than previously thought.
We further observe the paramagnon dispersion along [1,0], across the
two-dimensional CDW wavevector $\boldsymbol{q}_\mathrm{CDW}$, which is
consistent with magnetic excitations measured by inelastic neutron scattering.
Unlike for some other cuprates, our results point to the absence of a
discernible coupling between CDW and magnetic excitations.
