Relativistic Bethe logarithm for triplet states of helium-like ions

We report a calculation of relativistic corrections of order $m{\alpha }^7$ to the Bethe logarithm for the $2^{3}S$ and $2^{3}P$ states of helium-like ions. The calculation is required for improving the accuracy of theoretical energies of helium-like ions and for checking the evaluation of the $m{\alpha }^7$ effects in helium performed in [V. Patkóš, V. A. Yerokhin, K. Pachucki, Phys. Rev. A 103, 042809 (2021)], where a significant discrepancy with experimental results was found. The large-$Z$ limit of the relativistic Bethe logarithm is determined numerically, in excellent agreement with the analytical results obtained from the hydrogen theory.