# Thermal QED theory for bound states

## TL;DR

This paper develops a finite-temperature QED framework for bound states using the Hadamard photon propagator, enabling analysis of thermal effects on atomic systems without renormalization, and discusses implications for fundamental constants.

## Contribution

It introduces a thermal QED theory for bound states employing the Hadamard propagator, ensuring gauge invariance and analytical properties, and applies it to evaluate thermal effects on atomic parameters.

## Key findings

- Derived the thermal Coulomb potential and its asymptotics.
- Analyzed thermal one-photon exchange, self-energy, and vacuum polarization effects.
- Discussed impact on proton radius and Rydberg constant measurements.

## Abstract

In present paper the Quantum Electrodynamics theory at finite temperatures for the bound states is presented. To describe the thermal effects arising in a heat bath the Hadamard form of thermal photon propagator is employed. This form permits the simple introduction of thermal gauges in a way similar to the 'ordinary' Feynman propagator and, therefore, the gauge invariance can be proved for all the considered effects. Moreover, contrary to the 'standard' form of thermal photon propagator, the Hadamard expression has a well defined analytical properties. However, this thermal photon propagator contains the divergent contribution which requires the introduction of regularization procedure within the framework of constructed theory. The method of regularization in conjunction with the physical interpretation is given in the paper. Correctness of regularization procedure is confirmed also by the gauge invariance of final results and coincidence of the results (on the example of self-energy correction) for two different forms of photon propagator. On the basis of constructed theory the thermal Coulomb potential and its asymptotics at the large distances are found. Finally, we discuss in details the thermal effects of lowest order in the fine structure constant and temperature. Such effects are presented by the thermal one-photon exchange between bound electron and nucleus, thermal one-loop self-energy, thermal vacuum polarization, recoil corrections and correction on the finite size of the nucleus. Introduction of the regularization allows us do not apply the renormalization procedure. To confirm this we describe also the thermal vertex (with one, two and three vertices) corrections within the adiabatic $S$-matrix formalism. Finally, the influence of thermal effects on the determination of proton radius and Rydberg constant is discussed in the paper.