Bound-state QED calculations for antiprotonic helium
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Bound-state QED calculations for antiprotonic helium Vladimir I. Korobov · Laurent Hilico · Jean-Philippe Karr
© Springer International Publishing Switzerland 2015
Abstract 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 mα 7 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) → (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, mp /me . Correspondingly, for the two-photon transitions in the antiprotonic helium we have 4.7 · 10−11 as a relative uncertainty for the (33, 32) → (31, 30) transition frequency and a fractional precision of 3.6 · 10−11 for an inferred antiproton-to-electron mass ratio. Keywords Antiprotonic helium · Precision spectroscopy · Nonrelativistic QED
1 Introduction In our recent work [1] we have calculated the relativistic Bethe logarithm contribution at order mα 7 in the two Coulomb center approximation. These results then have been used for improved calculations of the transition energies for the hydrogen isotope molecular
Proceedings of the International Conference on Exotic Atoms and Related Topics (EXA 2014), Vienna, Austria, 15-19 September 2014 V. I. Korobov () Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980, Dubna, Russia e-mail: [email protected] L. Hilico · J.-P. Karr Laboratoire Kastler Brossel, Universit´e dEvry val dEssonne, UPMC-Sorbonne Universit´es, CNRS, ENS-PSL Research University, Coll`ege de France, 4 place Jussieu, 75005 Paris, France
V. I. Korobov et al. + Table 1 Fundamental vibrational transitions in H+ 2 and HD molecular ions (in MHz)
H+ 2
HD+
ΔEnr
65 687 511.0714
57 349 439.9733
ΔEα 2
1091.0400
958.1514
ΔEα 3
−276.5450
−242.1262
ΔEα 4
−1.9969
−1.7481
ΔEα 5
0.1371(1)
0.1200(1)
ΔEα 6
−0.0010(5)
−0.0009(4)
Etot
65 688 323.7055(5)
57 350 154.693(4)
Table 2 The two-photon transition: (33, 32) → (31, 30), in the antiprotonic helium. (in MHz)
ΔEnr
=
2 145 088 265.34
ΔEα 2
=
-39 349.33
ΔEα 3
=
5 857.84
ΔEα 4
=
92.97
ΔEα 5
=
-8.25(2)
ΔEα 6
=
-0.10(10)
Etotal
=
2 145 054 858.50(10)
ions and antiprotonic helium atoms [2]. The general formula for the one-loop self-energy contribution at the mα 7 order has been obtained in [2, 3]. Since that time few advances have been achieved, which we suppose to discuss below in this work. For convenience we present our ultimate numerical results here in Tables 1 and 2, where the CODATA10 recommended values of constants have been adopted in all our calculations [4]. The error bars in transition frequency (see Table 1) set a limit on the fractional precision in determination of mass ratio t
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