Collinear Laser Spectroscopy of $2{}^3\text{\hspace{0.17em}⁣}{S}_1\to 2{}^3\text{\hspace{0.17em}⁣}{P}_{\text{\hspace{0.17em}⁣}J}$ transitions in helium-like ${}^{12}{\mathrm{C}}^{4+}$

Transition frequencies and fine-structure splittings of the $2{}^3\text{\hspace{0.17em}⁣}{S}_1\to 2{}^3\text{\hspace{0.17em}⁣}{P}_{\text{\hspace{0.17em}⁣}J}$ 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.