Bethe logarithms for the 1 singlet S, 2 singlet S and 2 triplet S states of helium and helium-like ions

一种用于Bethe对数精密计算的简单方法

A Simple approach for precision calculation of Bethe logarithm

本研究提出一种Bethe对数精密计算的简单方法，通过特定算符取主导贡献、调$\lambda$消余项、维数正规化抵消发散，大幅降低数值计算中赝态构造复杂度，氢原子基态结果达49位有效数字，多电子体系同样简便高效。

Calculation of the relativistic Bethe logarithm in the two-center problem

We present a variational approach to evaluate relativistic corrections of order α^2 to the Bethe logarithm for the ground electronic state of the Coulomb two center problem. That allows to estimate the radiative contribution at mα^7 order in molecular-like three-body systems such as hydrogen molecular ions H_2^+ and HD^+, or antiprotonic helium atoms. While we get 10 significant digits for the nonrelativistic Bethe logarithm, calculation of the relativistic corrections is much more involved especially for small values of bond length R. We were able to achieve a level of 3-4 significant digits …

Helium atom excitations by the GW and Bethe-Salpeter many-body formalism

Helium atom is the simplest many-body electronic system provided by nature. The exact solution to the Schrödinger equation is known for helium ground and excited states, and represents a workbench for any many-body methodology. Here, we check the ab initio many-body GW approximation and Bethe-Salpeter equation (BSE) against the exact solution for helium. Starting from Hartree-Fock, we show that GW and BSE yield impressively accurate results on excitation energies and oscillator strength, systematically improving time-dependent Hartree-Fock. These findings suggest that the accuracy of BSE and G…

Relativistic corrections to the Bethe logarithm for the $2^3S$ and $2^3P$ states of He

With this work we start a project of calculating the QED contribution of order $α^7m$ to the $2^3P$-$2^3S$ transition energy in helium, aiming for an accurate determination of the nuclear charge radius $r_E$ from measurements of the corresponding transition frequency. Together with the complementary determination of $r_E$ from muonic helium, this project will provide a stringent test of universality of electromagnetic interactions of leptons in the Standard Model. We report a calculation of the relativistic corrections to the Bethe logarithm for the $2^3S$ and $2^3P$ states, which is the most …

平均场近似中的原子 Bethe 对数

Atomic Bethe logarithm in the mean-field approximation

本文开发了一种基于平均场近似和 Schwartz 方法计算多电子原子 Bethe 对数的方法，通过引入改进的指数基组解决原子核附近的基组问题，发现平均场近似误差极小，并提出了一种低成本评估轻分子系统 Lamb 移位的方案。

Two-Loop Bethe Logarithms for non-S Levels

Two-loop Bethe logarithms are calculated for excited P and D states in hydrogenlike systems, and estimates are presented for all states with higher angular momenta. These results complete our knowledge of the P and D energy levels in hydrogen at the order of alpha^8 m_e c^2, where m_e is the electron mass and c is the speed of light, and scale as Z^6, where Z is the nuclear charge number. Our analytic and numerical calculations are consistent with the complete absence of logarithmic terms of order (alpha/pi)^2 (Z alpha)^6 ln[(Z alpha)^(-2)] m_e c^2 for D states and all states with higher angul…

QED calculation of the dipole polarizability of helium atom

The QED contribution to the dipole polarizability of the $^4$He atom was computed, including the effect of finite nuclear mass. The computationally most challenging contribution of the second electric-field derivative of the Bethe logarithm was obtained using two different methods: the integral representation method of Schwartz and the sum-over-states approach of Goldman and Drake. The results of both calculations are consistent, although the former method turned out to be much more accurate. The obtained value of the electric-field derivative of the Bethe logarithm, equal to $0.048\,557\,2(14…

Relativistic and Radiative Energy Shifts for Rydberg States

We investigate relativistic and quantum electrodynamic effects for highly-excited bound states in hydrogenlike systems (Rydberg states). In particular, hydrogenic one-loop Bethe logarithms are calculated for all circular states (l = n-1) in the range 20 <= n <= 60 and successfully compared to an existing asymptotic expansion for large principal quantum number n. We provide accurate expansions of the Bethe logarithm for large values of n, for S, P and circular Rydberg states. These three expansions are expected to give any Bethe logarithms for principal quantum number n > 20 to an accuracy of f…

Application of the correlated B-spline basis functions to the leading relativistic and QED corrections of helium

B-spline functions have been widely used in computational atomic physics. Different from the traditional B-spline basis (a simple product of two B-splines), the recently developed correlated B-spline basis functions(C-BSBF), in which the interelectronic coordinate $r_{12}$ is included explicitly, have greatly improved the computational accuracy of polarizability [S. J. Yang \textit{et al}., Phys. Rev. A \textbf{95}, 062505 (2017)] and bethe logarithm [ S. J. Yang \textit{et al}., Phys. Rev. A \textbf{100}, 042509 (2019)] for singlet states of helium. Here, we report the extension of the C-BSBF…

Quantum Electrodynamic Corrections to the g Factor of Helium P States

The Lande g factor describes the response of an atomic energy level to an external perturbation by a uniform and constant magnetic field. In the case of many-electron systems, the leading term is given by the interaction mu_B*(L+2S.B), where L and S are the orbital and spin angular momentum operators, respectively, summed over all electrons. For helium, a long-standing experimental-theoretical discrepancy for P states motivates a reevaluation of the higher-order terms which follow from relativistic quantum theory and quantum electrodynamics (QED). The tensor structure of relativistic correctio…

Relativistic Reduced-Mass and Recoil Corrections to Vacuum Polarization in Muonic Hydrogen, Muonic Deuterium and Muonic Helium Ions

The reduced-mass dependence of relativistic and radiative effects in simple muonic bound systems is investigated. The spin-dependent nuclear recoil correction of order (Zalpha)^4 mu^3/m_N^2 is evaluated for muonic hydrogen and deuterium, and muonic helium ions (mu is the reduced mass and m_N is the nuclear mass). Relativistic corrections to vacuum polarization of order alpha (Zalpha)^4 mu are calculated, with a full account of the reduced-mass dependence. The results shift theoretical predictions. The radiative-recoil correction to vacuum polarization of order alpha (Zα)^5 ln^2(Zalpha) mu^2/m_…