Positronium hyperfine splitting: analytical value at m*alpha^6
  We present an analytic calculation of the m*alpha^6 recoil corrections to the hyperfine splitting (HFS) of the ground state energy levels in positronium. We find \Delta E_{\rm rec} = m alpha^6 (-1/6 ln(alpha) + 331/432 -ln(2)/4 - 17 Zeta(3)/(8 pi^2) + 5/(12 pi^2)) \approx m alpha^6 (- 1/6 ln(alpha) + 0.37632), confirming Pachucki's numerical result \cite{Ph}. We present a complete analytic formula for the m*alpha^6 HFS of the positronium ground state and, including m*alpha^7*ln^2(alpha) effects, find E(1^3S_1)-E(1^1S_0))_{theory} = 203 392(1) MHz. This differs from the experimental results by about 3 standard deviations. 
A. Czarnecki, K. Melnikov, A. Yelkhovsky