Quantum electrodynamic corrections to thegfactor of heliumPstates

Abstract

The Land\'e $g$ factor describes the response of an atomic energy level to an external perturbation by a uniform and constant magnetic field. In the case of many-electron systems, the leading term is given by the interaction ${\ensuremath{\mu}}_{B}\phantom{\rule{0.16em}{0ex}}(\stackrel{P\vec}{L}+2\phantom{\rule{0.16em}{0ex}}\stackrel{P\vec}{S})\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{B}$, where $\stackrel{P\vec}{L}$ and $\stackrel{P\vec}{S}$ are the orbital and spin angular momentum operators, respectively, summed over all electrons. For helium, a long-standing experimental-theoretical discrepancy for $P$ states motivates a re-evaluation of the higher order terms which follow from relativistic quantum theory and quantum electrodynamics (QED). The tensor structure of relativistic corrections involves scalar, vector, and symmetric and antisymmetric tensor components. We perform a tensorial reduction of these operators in a Cartesian basis, using an approach which allows us to separate the internal atomic from the external degrees of freedom (magnetic field) right from the start of the calculation. The evaluation proceeds in a Cartesian basis of helium eigenstates, using a weighted sum over the magnetic projections. For the relativistic corrections, this leads to a verification of previous results obtained using the Wigner-Eckhart theorem. The same method, applied to the radiative correction (Bethe logarithm term) leads to a spin-dependent correction, which is different for singlet versus triplet $P$ states. Theoretical predictions are given for singlet and triplet $2P$ and triplet $3P$ states and compared to experimental results where available.