Relativistic Correction to the Magnetic Moment of the Charged Lepton

TL;DR

The paper derives a gauge-consistent, general relativistic Hamiltonian by reducing the Dirac equation to a two-component Dirac-Pauli form, applicable to arbitrary, time-dependent electromagnetic fields. It then computes the $O(\alpha^2)$ relativistic correction to the Landé $g$-factor in hydrogen-like atoms, uncovering a novel $m_j^2$-dependence that generalizes the Breit result. The framework unifies relativistic corrections with kinetic and field-gradient effects, showing these corrections can rival QFT contributions in high-$Z$ or strong-field regimes and enabling direct tests in Penning traps and muonic atoms. These results provide a practical, testable pathway to quantify relativistic magnetic-moment corrections across bound and scattering systems with implications for precision spectroscopy and fundamental tests of QED.

Abstract

We derive a general relativistic Hamiltonian valid for both bound and scattering systems by reducing the four-component Dirac equation to a two-component Dirac-Pauli form. Unlike conventional approaches, our formulation includes first-order relativistic corrections in a compact, gauge-consistent expression applicable to arbitrary electromagnetic fields - including non-uniform and time-dependent configurations. As an application, we compute the O(alpha^2) relativistic correction to the Lande g-factor in hydrogen-like atoms, revealing a novel mj^2-dependent term that generalizes the Breit result. This correction is experimentally testable in Penning trap spectroscopy. We further show that relativistic effects become comparable to QFT corrections in highly charged ions where Z ~ 1/sqrt(alpha)

Figures (3)

• Figure 1: First-order relativistic correction $\Delta E_{\text{rel}}$ to the Zeeman energy level for $n = 2$ hydrogen-like states, plotted as a function of $m_j$ for each fine-structure multiplet: $2p_{3/2}$ (blue), $2p_{1/2}$ (green), and $2s_{1/2}$ (red). The energy scale is normalized by $Z^2\alpha^2 \mu_B B$, where $\mu_B = \frac{e\hbar}{2mc} \approx 5.788 \times 10^{-5}$ eV/T is the Bohr magneton. The energy shift is proportional to $\Delta E_{\text{rel}} Z^2 \mu_B B$, showing how relativistic effects scale with atomic number $Z^2$. The results are based on equations \ref{['Hzeman']}, \ref{['g+']}, and \ref{['g-']}. Spectroscopic notation $nl_j$ is used to denote the principal quantum number $n$, orbital angular momentum $l$ ($s=0$, $p=1$), and total angular momentum $j_{\pm} = l \pm 1/2$
• Figure 2: Decomposition of $g$-factor corrections. Left (blue): Leading QFT term ($\alpha/2\pi$) from electron self-energy via virtual photon exchange. Right (green): Relativistic $\mathcal{O}(\alpha^2)$ contributions from expectation values of kinetic and electromagnetic field operators, yielding the $m_j^2$-dependent term derived in this work. Red box: Dominant corrections in high-$Z$ or strong-field regimes.
• Figure 3: Magnitude of the relativistic Landé $g$-factor shift ($-\Delta g_{\text{rel}}$) for $2s_{1/2}$, $2p_{1/2}$ and $2p_{3/2}$ states in hydrogen-like atoms as a function of atomic number $Z$. The shift $\Delta g_{\text{rel}} = g_J - g_J^0$ is negative (reduction in $g$-factor), so $-\Delta g_{\text{rel}}$ is positive. The $2s_{1/2}$ state have the largest correction, followed by $2p_{3/2}$ and then $2p_{1/2}$. At $Z=20$, magnitudes align with Table \ref{['tab:gfactors']} ($-\Delta g_{\text{rel}} \approx 0.0024$ and $0.0028$).