Bound-state QED calculations for antiprotonic helium

We present new theoretical results for the transition energies of the hydrogen isotope molecular ions and antiprotonic helium atoms. Our consideration includes corrections at the mα⁷ order in the nonrecoil limit such as the one-loop self-energy, one-loop vacuum polarization, Wichman-Kroll, and complete two-loop contributions. That allowed to get transition energies for the fundamental transition $(v=0,L=0)\to (1,0)$ in the hydrogen molecular ion with the relative theoretical uncertainty of ∼7⋅10⁻¹² that corresponds to a fractional precision of 1.5⋅10⁻¹¹ in determination of the electron-to-proton mass ratio, m_p/m_e. Correspondingly, for the two-photon transitions in the antiprotonic helium we have 4.7⋅10⁻¹¹ as a relative uncertainty for the $(33,32)\to (31,30)$ transition frequency and a fractional precision of 3.6⋅10⁻¹¹ for an inferred antiproton-to-electron mass ratio.