Positronium hyperfine splitting at order mα7: Light-by-light scattering in the two-photon-exchange channel

Positronium plays a unique role in the continuing development of the techniques of bound-state QED and high-precision tests of the standard model as a Coulombic bound system that is almost free of strong and weak interaction contaminations and that exhibits large recoil and annihilation (virtual and real) effects. The present frontier is at the level of three-loop corrections to energy levels and decay rates. We have calculated the three-loop correction to the positronium hyperfine splitting due to light-by-light scattering in the exchange of two photons between the electron and positron. Our result is Delta E = -0.235 355(8)m alpha(4)(alpha/pi)(3) = -1.033 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.