Paramagnon dispersion and damping in doped Na$_x$Ca$_{2-x}$CuO$_2$Cl$_2$

11 pages, 5 figures, 2 tables. Published on Physical Review B (American Physical Society).Using Resonant Inelastic X-ray Scattering, we measure the paramagnon dispersion and damping of undoped, antiferromagnetic Ca$_2$CuO$_2$Cl$_2$ as well as doped, superconducting Na$_x$Ca$_{2-x}$CuO$_2$Cl$_2$. Our estimation of the spin-exchange parameter and width of the paramagnon peak at the zone boundary $X=(0.5,0)$ confirms that no simple relation can be drawn between these parameters and the critical temperature $T_\mathrm{c}$. Consistently with other cuprate compounds, we show that upon doping there is a slight softening at $(0.25,0)$, but not at the zone boundary $X$. In combination with these measurements we perform calculations of the dynamical spin structure factor of the one-band Hubbard model using cluster dynamical mean-field theory. The calculations are in excellent agreement with the experiment in the undoped case, both in terms of energy position and width. While the increase in width is also captured upon doping, the dynamical spin structure factor shows a sizable softening at $X$, which provides insightful information on the length-scale of the spin fluctuations in doped cuprates