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 of the nuclear binding strength parameter $Z\ensuremath{\alpha}$, where $Z$ is the nuclear charge number and $\ensuremath{\alpha}$ 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 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 three to four standard deviations from the experimental results available for silicon and calcium.