One-loop vacuum polarization at $m\alpha^7$ order for the two center problem

Abstract

We present calculations of the one-loop vacuum polarization contribution (Uehling potential) for the two-center problem in the NRQED formalism. The cases of hydrogen molecular ions ($Z_1=Z_2=1$) as well as antiprotonic helium ($Z_1=2$, $Z_2=-1$) are considered. Numerical results of the vacuum polarization contribution at $m\alpha^7$ order for the fundamental transitions $(v=0,L=0)\to(v'=1,L'=0)$ in H$_2^+$ and HD$^+$ are presented.