Single pion contribution to the hyperfine splitting in muonic hydrogen

A detailed discussion of the long-range one-pion exchange (Yukawa potential) contribution to the 2S hyperfine splitting in muonic hydrogen which had, until recently, been disregarded is presented. We evaluate the relevant vertex amplitudes, in particular ${\pi }^0{\mu }^{+}{\mu }^{-}$, combining low energy chiral expansions together with experimental data on ${\pi }^0$ and $\eta$ decays into two leptons. A value of $\mathrm{\Delta }{E}_{HFS}^{\pi }=-(0.09\pm 0.06)\mu \mathrm{e}\mathrm{V}$ is obtained for this contribution.