Positronium hyperfine splitting at order $m \alpha^7$: light-by-light scattering in the two-photon-exchange channel

We have calculated the contribution to the positronium hyperfine splitting at order $m \alpha^7$ of the light-by-light scattering process in the exchange of two photons between the electron and positron. Our result is $\Delta E = -0.235355(8) m \alpha^4 \left ( \frac{\alpha}{\pi} \right )^3 = -1.034 kHz$. As a check of our approach we confirm earlier evaluations of the analogous correction for a bound system (such as muonium) with unequal masses.