Some O(α²) Annihilation Type Contributions to the Orthopositronium Width

We consider some radiative corrections to the lowest order annihilation diagram for the orthopositronium decay rate. The insertion of the renormalized vertex correction in the annihilation graph gives $1.6283 (\alpha/\pi)^2 \Gamma_0$. We compute also the contribution of the square of the lowest order annihilation amplitude, which turns out to be $0.1702 (\alpha/\pi)^2 \Gamma_0$. Finally, we obtain a term $\alpha^2 \ln\alpha \Gamma_0$ arising from the correction to the light-light scattering block due to the exchange of one coulombic photon, in agreement with earlier computations.