QED calculations of energy levels of helium-like ions with $5 \leq Z \leq 30$

Abstract: A calculation of two-electron QED effects to all orders in the nuclear binding strength parameter $Z\alpha$ is presented for the ground and $n = 2$ excited states of helium-like ions. After subtracting the first terms of the $Z\alpha$ expansion from the all-order results, we identify the higher-order QED effects of order $m\alpha^7$ and higher. Combining the higher-order remainder with the results complete through order $m\alpha^6$ from [V. A. Yerokhin and K. Pachucki, Phys. Rev. A 81, 022507 (2010)], we obtain the most accurate theoretical predictions for the ground and non-mixing $n=2$ states of helium-like ions with $Z = 5-30$. For the mixing $2¹ P_1$ and $2³ P_1$ states, we extend the previous calculation by evaluating the higher-order mixing correction and show that it defines the uncertainty of theoretical calculations in the $LS$ coupling for $Z > 10$.