PositroniumS-state spectrum: Analytic results atO(mα6)

We present an analytic calculation of the $O\left(m{\alpha }^6\right)$ recoil and radiative recoil corrections to energy levels of positronium $\mathrm{nS}$ states and their hyperfine splitting. A complete analytic formula valid to $O\left(m{\alpha }^6\right)$ is given for the spectrum of S states. Technical aspects of the calculation are discussed in detail. Theoretical predictions are given for various energy intervals and compared with experimental results.