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 π+ and π-. Applications to the hyperfine structure and the Lamb shift in muonic hydrogen are also considered.