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Logarithmic corrections in the two-body QED problem

The logarithmic part of the Lamb shift, the contribution of the relative order α3 log (1/α) to the atomic state energy, is related to the usual infrared divergence. This fact allows one to calculate easily such corrections in positronium, and derive the recoil and electron-electron terms in the Lambshift Hamiltonian in many-electron atoms. Logarithmic energy corrections of the next order, α4 log (1/α), are of a different, relativistic nature. Their calculation is reduced to the ordinary perturbation theory for the nonrelativistic Schrödinger equation. The perturbation operators have the Breit-type structure and are found by the calculation of on-mass-shell diagrams. For positronium, the calculated log-arithmic correction survives only in n3S1 states and constitutes (5/24)mα6 log (1/α)/n3. Logarithmic corrections of the relative order α2 log (1/α) to the positronium decay rate are also of the relativistic origin and can be easily computed within the same approach. Arguments are presented in favour of a large numerical factor in the (α/π)2 correction to the positronium decay rate.