We present high-precision calculations of isotope shifts for the $2{}^{1}S\text{−}2{}^{3}S$ and $2{}^{3}S\text{−}2{}^{3}P$ transitions in heliumlike ${}^{12,13,14}\mathrm{C}^{4+}$, with mass-shift uncertainties below 5 ppm and field-shift factors determined to $0.1–0.2\%$. Combining theory with experiment yields the mean-square charge-radius difference $\delta {\langle R^2\rangle }^{13,12}=-0.1253\left(66\right){\text{fm}}^2$, consistent with muonic-carbon results at the $1.2\sigma$ level. Using heliumlike ions as a reference, we assess the feasibility of extracting new-physics signatures from isotope shifts. We show that higher-order field-shift nonlinearities can obscure such signals and that the sensitivity to new bosons mediating electron-neutron interactions decreases with nuclear charge. For ${\mathrm{C}}^{4+}$, detecting a new-physics effect comparable to that suggested in helium would require $\delta \langle R^2\rangle$ determinations improved by 2 orders of magnitude. Our work provides a quantitative framework for extracting nuclear charge radii and for assessing potential new-physics signatures beyond the standard model in heliumlike systems.

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• Fine & hyperfine structure
• Isotope shift
• Quantum electrodynamics