Black sphere model for the linewidths ofP-state protonium

The annihilation widths of $\mathrm{nP}$ $\overline{p}p$ bound states (protonium) are calculated in the Wentzel-Kramers-Brillouin approximation for a totally absorbing sphere, including the coupling to $\tilde{n}n$ states. It is shown that the annihilation occurs preferentially in one of the two possible isospin eigenstates, and that on the average it is 100 times stronger than the radiative $2P-1S$ transition. A simple formula for the $n$ dependence of the annihilation widths is also given.