Antiprotonic atoms with nonperturbative inclusion of vacuum polarization and finite nuclear mass

We demonstrate that energy levels of excited states in a hydrogenic system consisting of an arbitrary nucleus and an antiproton can be calculated within the framework of nonrelativistic quantum electrodynamics, even for a large nuclear charge $Z$. It is because for rotational states the expansion parameter is $Z\phantom{\rule{0.16em}{0ex}}\ensuremath{\alpha}/n$. The main advantage of this approach is the possibility of exact inclusion of the finite nuclear mass, which we achieve up to the ${(Z\phantom{\rule{0.16em}{0ex}}\ensuremath{\alpha})}^{6}$ order. In addition, we include unperturbatively the one-loop and two-loop electron vacuum polarization (EVP) potentials in the nonrelativistic Hamiltonian, as well as in the leading relativistic correction. The obtained results for $l>1$ states of antiprotonic atoms with a spinless nucleus are the most accurate to date. We make available a user-friendly mathematica code for antiprotonic atoms pbarspectr, which can be further improved by combining EVP potentials with ${(Z\phantom{\rule{0.16em}{0ex}}\ensuremath{\alpha})}^{5}$ QED effects, by adding three-loop EVP, and by extending to an arbitrary nuclear spin. Finally, we note that rotational states of antiprotonic atoms can be used to determine the mean-square nuclear charge radius much more accurately than from electronic or muonic atoms.