Second-order corrections to the wave function at the origin in muonic hydrogen and pionium

Nonrelativistic second-order corrections to the wave function at the origin in muonic and exotic atoms are considered. The corrections are due to the electronic vacuum polarization. Such corrections are of interest due to various effective approaches, which take into account QED and hadronic effects. The wave function at the origin plays a key role in the calculation of the pionium lifetime, various finite nuclear size effects, and the hyperfine splitting. The results are obtained for the $1s$ and $2s$ states in pionic and muonic hydrogen and deuterium and in pionium, a bound system of ${\pi }^{+}$ and ${\pi }^{-}$. Applications to the hyperfine structure and the Lamb shift in muonic hydrogen are also considered.

Figure 1

Vacuum polarization corrections to the electrostatic Coulomb potential ($V_{\mathrm{VP}}$): the Uehling potential ($V_{\mathrm{VP}}^{(1)}$), and the reducible two-loop ($V_{\mathrm{VP}}^{(1·1)}$) and irreducible two-loop ($V_{\mathrm{VP}}^{(2)}$) potentials (the Källen-Sabry potential).

Figure 2

The first-order correction to the wave function at the origin. The filled circle is for $\delta (\mathbf{r})$. The double line stands for the nonrelativistic reduced Coulomb Green function.

Figure 3

The second-order vacuum polarization correction to the wave function at the origin.

Figure 4

The second-order vacuum polarization corrections to the hyperfine splitting in muonic hydrogen. The wavy line stands for the hyperfine interaction.

Figure 5

The ${\alpha }^{5}m$ correction to the Lamb shift in muonic hydrogen: the only third-order contribution of nonrelativistic perturbation theory [see (3)].