Order $\alpha^4 (m/M) R_{\infty}$ corrections to hydrogen P levels
  The order $\alpha^4 (m/M) R_{\infty}$ shift of hydrogen P levels is found. The corrections are predominantly of relativistic origin. Our approach is a straightforward extension of that developed and applied by us previously to positronium P levels. The corrections to the Lamb shift in hydrogen constitute numerically \delta E(2 P_{1/2})=0.55 kHz, \delta E(2 P_{3/2}) =0.44 kHz. 
E.A. Golosov, I.B. Khriplovich, A.I. Milstein and A.S. Yelkhovsky