Order-α4R∞contributions to positronium hyperfine structure from radiative corrections to two-photon virtual annihilation

We present the details of our calculation of radiative corrections affecting the two-photon virtual-annihilation contributions to the positronium hyperfine splitting in order $m{\alpha }^6\left({\alpha }^4{R}_{\infty }\right)$. Our result, which is obtained in analytic form, is $\Delta E_{2\gamma }^A\left({\alpha }^4{R}_{\infty }\right)=-13.13\mathrm{}$ MHz. We point out the gauge invariance of the result and demonstrate the cancellation of potential infrared contributions which appear at intermediate stages of the calculation.