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

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.