Positronium Hyperfine Splitting: Analytical Value atO(mα6)

We present an analytic calculation of the $O\left(m{\alpha }^6\right)$ recoil corrections to the hyperfine splitting (HFS) of the ground state energy levels in positronium. We find $\Delta E_{\mathrm{rec}}=m{\alpha }^6(-\frac{1}{6}\ln \alpha +\frac{331}{432}-\frac{\ln 2}{4}-\frac{17\zeta \left(3\right)}{8{\pi }^2}+\frac{5}{12{\pi }^2})\approx m{\alpha }^6(-\frac{1}{6}\ln \alpha +0.376\mathrm{}32)$, confirming Pachucki's numerical result [Phys. Rev. A 56, 297 (1997)]. We present a complete analytic formula for the $O\left(m{\alpha }^6\right)$ HFS of the positronium ground state and, including $O\left(m{\alpha }^7{ln}^2\alpha \right)$ effects, find $\left[E\right(1^3{S}_1)-E(1^1{S}_0){]}_{\mathrm{theory}}=203\mathrm{}392.01(46)\mathrm{MHz}$. This differs from the experimental results by about 3 standard deviations.