Mass Renormalization and Energy Level Shift in Non-Relativistic QED

TL;DR

This paper calculates the electron's binding energy in non-relativistic QED, demonstrating mass renormalization and the removal of ultraviolet cut-off, with results aligning with Bethe's formula for small Zα.

Contribution

It provides a rigorous analysis of mass renormalization and energy level shifts in non-relativistic QED, extending understanding beyond small Zα values.

Findings

Mass renormalization leads to finite energy shifts as cut-off is removed.

The energy shift matches Bethe's formula for small Zα.

Different behavior observed for larger Zα values.

Abstract

Using the Pauli-Fierz model of non-relativistic quantum electrodynamics, we calculate the binding energy of an electron in the field of a nucleus of charge $Z$ and in presence of the quantized radiation field. We consider the case of small coupling constant $\alpha$, but fixed $Z\alpha$ and ultraviolet cut-off $\Lambda$. We prove that after renormalizing the mass the binding energy has, to leading order in $\alpha$, a finite limit as $\Lambda$ goes to infinity; i.e., the cut-off can be removed. The expression for the ground state energy shift thus obtained agrees with Bethe's formula for small values of $Z\alpha$, but shows a different behavior for bigger values.