Two-photon exchange corrections to the $g$ factor of Li-like ions

We report calculations of QED corrections to the $g$ factor of Li-like ions induced by the exchange of two virtual photons between the electrons. The calculations are performed within QED theory to all orders in the nuclear binding strength parameter $Z\alpha$, where $Z$ is the nuclear charge number and $\alpha$ is the fine-structure constant. In the region of low nuclear charges we compare results from three different methods: QED, relativistic many-body perturbation theory, and nonrelativistic QED. All three methods are shown to yield consistent results. With our calculations we improve the accuracy of the theoretical predictions of the $g$ factor of the ground state of Li-like carbon and oxygen by about an order of magnitude. Our theoretical results agree with those from previous calculations but differ by 3-4 standard deviations from the experimental results available for silicon and calcium.