Relativistic corrections to energy spectrum of hydrogen due to full one-photon-exchange interaction

The study of the energy spectrum of hydrogen-like atoms has been pivotal in the development of quantum mechanics and has remained important for over a century. However, addressing bound states within a pure quantum field theory, particularly when the non-relativistic expansion is not valid, remains challenging. Over the past fifteen years, precise experimental measurements of the Lamb shifts in hydrogen and muonic hydrogen have advanced significantly; however, they have also presented numerous challenges [1−8]. Theoretically, the bound-state perturbative theory is commonly used to estimate energy corrections beyond the Coulomb potential (see recent reviews and books [9−11] and the references therein). To reliably estimate these corrections, the effective Schrodinger-like equation [12] or effective Dirac-like equations [13], which are derived exactly from the Bethe-Salpeter (BS) equation [14,15] in quantum electrodynamics (QED) or non-relativistic quantum electrodynamics (NRQED) [16], should be employed, as there is no analytical solution for the physical BS equation.

In the bound-state perturbative theory, the interaction kernel in the effective Schrödinger-like equation or effective Dirac-like equations is expanded order by order in terms of the fine structure constant $ \alpha_e $ , velocities $ {\boldsymbol{p}}_i/m_{e,p} $ , and $ m_e/m_p $ , where $ {\boldsymbol{p}}_i $ represents the three momenta of the particles in the center-of-mass frame, and $ m_e $