Asymptotic Properties of Self-Energy Coefficients

Abstract

We investigate the asymptotic properties of higher-order binding corrections to the one-loop selfenergy of excited states in atomic hydrogen. We evaluate the historically problematic A60 coefficient for all P states with principal quantum numbers n ≤ 7 and D states with n ≤ 8 and find that a satisfactory representation of the n-dependence of the coefficients requires a three-parameter fit. For the high-energy contribution to A60, we find exact formulas. The results obtained are relevant for the interpretation of high-precision laser spectrocopic measurements.PACS numbers: PACS numbers 12.20. Ds, 31.30.Jv, 06.20.Jr, Bound-state quantum electrodynamics (QED) occupies a unique position in theoretical physics in that it combines all conceptual intricacies of modern quantum field theories, augmented by the peculiarities of bound states, with the experimental possibilities of ultra-high resolution laser spectroscopy. Calculations in this area have a long history, and the current status of theoretical predictions is the result of continuous effort. The purpose of this Letter is twofold: first, to present improved evaluations of higher-order binding corrections to the bound-state self-energy for a large number of atomic states, including highly excited states with a principal quantum number as high as n = 8, and second, to analyze the asymptotic dependence of the analytic results on the bound-state quantum numbers. Highly-excited states (e.g, with n = 4 to 12) are of particular importance for high-precision spectroscopy experiments in hydrogen (for a summary, see for instance [1, p. 371]). In the analytic calculations, we focus on a specific higher-order binding correction, known as the A 60 coefficient or "relativistic Bethe logarithm." We write the (real part of the) one-loop self-energy shift of an electron in the field of a nucleus of charge number Z aswhere F (nl j , Zα) is a dimensionless quantity. In this Letter, we use natural units withh = c = m = 1 and e 2 = 4πα (m is the electron mass). The notation nl j is inspired by the usual spectroscopic nomenclature: n is the level number, j is the total angular momentum and l is the orbital angular momentum. The semi-analytic expansion of F (nl j , Zα) about Zα = 0 for a general atomic state with quantum numbers n, l ≥ 1 and j gives rise to the expression, F (nl j , Zα) = A 40 (nl j ) + (Zα) 2 A 61 (nl j ) ln(Zα) −2

Nonrelativistic QED approach to the Lamb shift

“…In constitutes one of three contributions to relativistic Bethe logarithm L, being defined as in [13].…”

“…(9) of Ref. [13]. The second relativistic correction E L2 is the nonrelativistic quadrupole contribution in the conventions adopted in [8,9].…”

Toward high-precision values of the self energy of non-S states in hydrogen and hydrogen-like ions

“…where the dependence of the terms on the electron state nL j is implicit; A 40 is a coefficient which has been calculated for many states; A 61 is known analytically; G SE (Zα) is a remainder whose limit as Z → 0 has recently been calculated for all the states considered here [27,28]. In order to check our calculations of F , we made sure that the numerical values of G SE (Zα) obtained from F and Eq.…”

Two-loop Bethe logarithms for non-Slevels

“…We see that the total value of B 60 is the sum of high-energy operators given by inverse powers of the principal quantum number, linear combinations of terms proportional to ln(2) and ζ functions of various arguments, and low-energy terms β 4 and β 5 which are known from one-loop calculations [28,29] (see also Tables VII and VIII (2) (23a) …”