Fourth-Order Vacuum Polarization Correction to the Positronium Hyperfine Structure

It is shown that the fourth-order vacuum polarization correction to the annihilation diagram contributes $\Delta \nu =-\frac{1}{4}{\alpha }^4{R}_{\infty }\ln {\alpha }^{-1\mathrm{}}$ to the positronium ground-state hyperfine structure. This leads to a reduction of 11.3 MHz in the previous theoretical prediction which greatly improves the agreement between theory and experiment.