Combined two-loop self-energy corrections at finite and zero temperatures

In this paper we investigate higher-order corrections to the energies of bound states in hydrogen subjected to the external blackbody radiation field. In particular, within the framework of thermal quantum electrodynamics and $S$-matrix approach we analyze combined type of two-loop self-energy corrections, including one zero-vacuum and one loop at finite temperature. By utilizing the method of dimensional regularization, we derive closed analytical expressions for the energy shifts of atomic levels. Our numerical calculations demonstrate that even at room temperature these corrections can be significant for excited states, reaching the magnitude of the thermal induced Stark contribution.