Collinear Laser Spectroscopy of $2\,{}^3\!S_1 \rightarrow 2\,{}^3\!P_{\!J}$ transitions in helium-like $^{12}\mathrm{C}^{4+}$

Transition frequencies and fine-structure splittings of the $2\,{}^3\!S_1 \rightarrow 2\,{}^3\!P_{\!J}$ transitions in helium-like $^{12}\mathrm{C}^{4+}$ were measured by collinear laser spectroscopy on a 1-ppb level. Accuracy is increased by more than three orders of magnitude with respect to previous measurements, enabling tests of recent non-relativistic QED calculations including terms up to $m\alpha^7$. Deviations between the theoretical and experimental values are within theoretical uncertainties and are ascribed to $m\alpha^8$ and higher-order contributions in the series expansion of the NR-QED calculations. Finally, prospects for an all-optical charge radius determination of light isotopes are evaluated.