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Thermal one-loop self-energy correction for hydrogen-like systems: relativistic approach

M. Reiter, D. Solovyev, A. Bobylev, D. Glazov, T. Zalialiutdinov

TL;DR

The paper develops a fully relativistic finite-temperature QED framework to compute the thermal one-loop self-energy correction for bound electrons, focusing on the real part that shifts atomic energy levels. Using a partial-wave decomposition and a dual-kinetic-balance B-spline basis, it yields high-precision results for hydrogen and hydrogen-like ions across states up to $n=12$, maintaining the full frequency dependence rather than a static limit. The study demonstrates that the naive $1/Z^4$ scaling from nonrelativistic dipole theory is modified by relativistic effects at higher $Z$, and it identifies magic temperatures where thermal shifts vanish for selected transitions; hyperfine contributions to these magic points are negligible. Overall, the work reconciles QED and QM pictures of thermal shifts, refines prior results by including Lamb shift and fine-structure corrections, and provides data essential for precision spectroscopy and clock experiments.

Abstract

Within a fully relativistic framework, the one-loop self-energy correction for a bound electron is derived and extended to incorporate the effects of external thermal radiation. In a series of previous works, it was shown that in quantum electrodynamics at finite temperature (QED), the description of effects caused by blackbody radiation can be reduced to using the thermal part of the photon propagator. As a consequence of the non-relativistic approximation in the calculation of the thermal one-loop self-energy correction, well-known quantum-mechanical (QM) phenomena emerge at successive orders: the Stark effect arises at leading order in $αZ$, the Zeeman effect appears in the next-to-leading non-relativistic correction, accompanied by diamagnetic contributions and their relativistic refinements, among other perturbative corrections. The fully relativistic approach used in this work for calculating the SE contribution allows for accurate calculations of the thermal shift of atomic levels, in which all these effects are automatically taken into account. The hydrogen atom serves as the basis for testing a fully relativistic approach to such calculations. Additionally, an analysis is presented of the behavior of the thermal shift caused by the thermal one-loop correction to the self-energy of a bound electron for hydrogen-like ions with an arbitrary nuclear charge $Z$. The significance of these calculations lies in their relevance to contemporary high-precision experiments, where thermal radiation constitutes one of the major contributions to the overall uncertainty budget.

Thermal one-loop self-energy correction for hydrogen-like systems: relativistic approach

TL;DR

The paper develops a fully relativistic finite-temperature QED framework to compute the thermal one-loop self-energy correction for bound electrons, focusing on the real part that shifts atomic energy levels. Using a partial-wave decomposition and a dual-kinetic-balance B-spline basis, it yields high-precision results for hydrogen and hydrogen-like ions across states up to , maintaining the full frequency dependence rather than a static limit. The study demonstrates that the naive scaling from nonrelativistic dipole theory is modified by relativistic effects at higher , and it identifies magic temperatures where thermal shifts vanish for selected transitions; hyperfine contributions to these magic points are negligible. Overall, the work reconciles QED and QM pictures of thermal shifts, refines prior results by including Lamb shift and fine-structure corrections, and provides data essential for precision spectroscopy and clock experiments.

Abstract

Within a fully relativistic framework, the one-loop self-energy correction for a bound electron is derived and extended to incorporate the effects of external thermal radiation. In a series of previous works, it was shown that in quantum electrodynamics at finite temperature (QED), the description of effects caused by blackbody radiation can be reduced to using the thermal part of the photon propagator. As a consequence of the non-relativistic approximation in the calculation of the thermal one-loop self-energy correction, well-known quantum-mechanical (QM) phenomena emerge at successive orders: the Stark effect arises at leading order in , the Zeeman effect appears in the next-to-leading non-relativistic correction, accompanied by diamagnetic contributions and their relativistic refinements, among other perturbative corrections. The fully relativistic approach used in this work for calculating the SE contribution allows for accurate calculations of the thermal shift of atomic levels, in which all these effects are automatically taken into account. The hydrogen atom serves as the basis for testing a fully relativistic approach to such calculations. Additionally, an analysis is presented of the behavior of the thermal shift caused by the thermal one-loop correction to the self-energy of a bound electron for hydrogen-like ions with an arbitrary nuclear charge . The significance of these calculations lies in their relevance to contemporary high-precision experiments, where thermal radiation constitutes one of the major contributions to the overall uncertainty budget.

Paper Structure

This paper contains 9 sections, 17 equations, 7 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Theory
  3. QED description of thermal shifts and line broadening
  4. Partial-wave expansion
  5. Transitions between hyperfine sublevels
  6. Technical details, numerical results and discussion
  7. Conclusions
  8. Acknowledgments
  9. Reduced matrix element

Figures (7)

  • Figure 1: One-loop self-energy Feynman diagram for bound electron. The double line indicates the electron propagator in the external field of the nucleus, the wavy line denotes the photon propagator.
  • Figure 2: Thermal shift for the ground state of hydrogen-like ions as a function of the nuclear charge $Z$ on a logarithmic scale. The solid line corresponds to the asymptotic estimate $1/Z^4$, with shift values in Hz taken from Table \ref{['table:re_1s_z_300k']}.
  • Figure 3: The intersection point for the thermal shifts of the states $1s_{1/2}$ (solid line, red online) and $2s_{1/2}$ (dashed line, blue online) in hydrogen, $T_m=65.732(24)$ K. The shift values are plotted in Hz as a function of temperature (in Kelvin). The results were obtained without considering hyperfine structure.
  • Figure 4: The intersection point for the thermal shifts of the states $1s_{1/2}$ (solid line, red online) and $3s_{1/2}$ (dashed line, blue online) in hydrogen, $T_m=29.826(13)$ K. The shift values are plotted in Hz as a function of temperature (in Kelvin). The results were obtained without considering hyperfine structure.
  • Figure 5: Thermal shift of the hyperfine transition frequency, corresponding to the real part of the one-loop self-energy correction for a bound electron, for various temperatures and the $1s_{1/2}$ (solid line, red online), $2s_{1/2}$ (dashed line, blue online) states in the hydrogen atom.
  • ...and 2 more figures