Abstract
Due to precision tests of quantum electrodynamics (QED), determination of accurate values of fundamental constants, and constraints on new physics, it is important in a consistent way to evaluate a number of QED observables such as the Lamb shift in hydrogen-like atomic systems. Even in a pure leptonic case, those QED variables are in fact not pure QED ones since hadronic effects are involved through intermediate states while accounting for higher-order effects. One of them is hadronic vacuum polarization (hVP). Complex evaluations often involve a number of QED quantities, for which treatment of hVP is not consistent. The highest accuracy for a calculation of the hVP term is required for the anomalous magnetic moment of a muon. However, a standard data-driven treatment of hVP, based on a dispersion integration of experimental data on electron-positron annihilation to hadrons and some other phenomena, leads to a contradiction with the experimental value of a_mu . This experimental value can be considered as an indirect determination of the hVP contribution to a_mu and the scatter of theory and experiment allows one to obtain a conservative estimation of the related hVP contribution. In this paper, we derive exact and approximate relations between the leading-order (LO) hVP contributions to various observables. Using those relations, we obtain for them a consistent set of the results, based on the scatter of a_mu values. While calculating the LO hVP term, we have to remember that next-to-LO (NLO) hVP corrections are often comparable with the uncertainty of the LO term. Special attention is payed to hVP contribution to simple atoms. In particular, we discuss the NLO contribution to the Lamb shift in ordinary and muonic hydrogen and other two-body atoms for Zle 10. We also consider the NLO contribution of the muonic vacuum polarization to the Lamb shift in hydrogen-like atoms. With the a_mu puzzle unresolved, one may still require present-days values of the hVP contributions to various observable for comparison to experiment etc. the presence of contradicting values and a lack of consistency means an additional uncertainty for a_mu and for key contributions to it, including the LO hVP one. We present here an estimation of such a propagated uncertainty in hVP contributions to different QED observables and recommend a consistent set of the related LO hVP contributions.Graphic
Highlights
Precision low-energy tests of quantum electrodynamics are not entirely pure QED tests
A few of recent and accurate evaluations of the hadronic vacuum polarization (hVP) contribution to ae have been available in the literature, which allows us to improve the Lamb shift results. (We remind that our concern is not an accuracy of the calculations by itself, but their reliability and their overall consistency.)
To find a value of Πh(0), we use calculations of the hVP contribution to ge − 2 in (11) The latter was recently found in a few publications [2,3]
Summary
Precision low-energy tests of quantum electrodynamics are not entirely pure QED tests. Even in the case of purely leptonic systems, theory requires a certain hadronic input and, in particular, an input from hadronic vacuum polarization (hVP). While the leading-order (LO) term for Δaμ(hVP) has been recently accurately calculated in many papers, the LO contributions for Δae(hVP) ([2, 3]) and for ΔνMu(hVP) ([2]) are found in a few publications only. There are not very accurate calculations of the related contributions to Lamb shift in H [4] and μH (see, e.g., [5]). A number of results on the next-to-leadingorder (NLO) hVP contributions to aμ and LO and NLO
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