For the S-states of muonium and positronium, the hyperfine shifts to the order α 6 of a recently derived two-fermion equation with explicit CPT -invariance are checked against the results of a nonrelativistic reduction, and the leading α 8 shifts are calculated. An additional hyperfine operator is discovered which can milden the singularity for r → 0 of the Dirac hyperfine operator, such that the resulting extended operator can be used nonperturbatively. The binding correction to magnetic moments is mentioned.

In Eq. ͑66͒, the quantum defect ␤Ј for S states should include two logarithms of mass ratios m i ϭm i /E:Ϫln͑Z␣ ͒ 2 ͪ Ϫm 2 2 ln m 2 ϪZ 2 m 1 2 ln m 1ͬ ,where r is the electric-dipole radiation enhancement factor, rϭm 2 ϩZm 1 , and Z is the nuclear charge. From dispersion-relation arguments, one expects a third ''interference logarithm'' ϪZm 1 m 2 ln m 1 m 2 inside the square bracket; it is hidden in the residual ''Salpeter shift'' ⌬E Sal Ј Eq. ͑67͒. When it is subtracted, there remains 7 3 (a n ЈϪln m 1 m 2 )Ϫ(m 2 2 ϩm 1 2 )(m 2 2Ϫm 1 2 ) Ϫ1 ln(m 2 /m 1 ) inside the bracket of Eq. ͑67͒. In the transformation of ⌬E Sal Ј following Eq. ͑77͒, the two logarithms should read Ϫln m 1 m 2 and Ϫln m 2 /m 1 . In ␦l ortho (lϭ f ), the bracket should begin with 1ϩ1/2F 2 . Also, the second form of the transformation c in Eq. ͑20͒ should have m 1 replaced by m 1 ␤, and the mass factor in Eqs. ͑7͒ and ͑8͒ should be m ϩ .

An eight-component formalism is proposed for the relativistic two-fermion problem. In QED, it extends the applicability of the Dirac equation with hyperfine interaction to the positronium case. The use of exact relativistic two-body kinematics entails a CP-invariant spectrum which is symmetric in the total cms energy. It allows the extension of recent \alpha^6 recoil corrections to the positronium case, and implies new recoil corrections to the fine and hyperfine structures and to the Bethe logarithm.Comment: Revtex, accepted for publication in Phys. Rev.

No abstract

No abstract