Laser spectroscopy of the $^2S_{1/2} - ^2P_{1/2}$, $^2P_{3/2}$ transitions in stored and cooled relativistic C$^{3+}$ ions

The $^2S_{1/2} - ^2P_{1/2}$ and $^2S_{1/2} - ^2P_{3/2}$ transitions in Li-like carbon ions stored and cooled at a velocity of $\beta \approx 0.47$ in the Experimental Storage Ring (ESR) at the GSI Helmholtz Centre in Darmstadt have been investigated in a laser spectroscopy experiment. Resonance wavelengths have been obtained using a new continuous-wave UV laser system and a novel extreme UV (XUV) detection system used to detect forward emitted fluorescence photons. The results obtained for the two transitions are compared to existing experimental and theoretical data. A discrepancy found in an earlier laser spectroscopy measurement at the ESR with results from plasma spectroscopy and interferometry could be resolved and agreement between experiment and theory is confirmed. Nonetheless, the experimental uncertainty is still smaller than that of state-of-the art theory.