Theoretical transition frequencies beyond 0.1 ppb accuracy in H$_2^+$, HD$^+$, and antiprotonic helium

Abstract

We present improved theoretical calculations of transition frequencies for the fundamental transitions $(L\!=\!0,v\!=\!1)\to(L'\!=\!0,v'\!=\!0)$ in the hydrogen molecular ions H$_2^+$ and HD$^+$ with a relative uncertainty $4\cdot10^{-11}$ and for the two-photon transitions in the antiprotonic helium atom with a relative uncertainty $10^{-10}$. To do that, the one-loop self-energy correction of order $α(Zα)^6$ is derived in the two Coulomb center approximation, and numerically evaluated in the case of the aforementioned transitions. The final results also include a complete set of other spin-independent corrections of order $mα^7$. The leading order corrections of $α^2\ln^3(Zα)^{-2}(Zα)^6$ are also considered that allows to estimate a magnitude of yet uncalculated contributions.

10 pages, 2 figure, to be submitted to PRA