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

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

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