QED corrections of order alpha (Zalpha)^2 E_F to the hyperfine splitting of P_1/2 and P_3/2 states in hydrogenlike ions

TL;DR

This paper calculates higher-order quantum electrodynamics corrections to the hyperfine splitting of P states in hydrogenlike ions, revealing state-dependent logarithmic contributions and identifying a nuclear-spin dependent correction.

Contribution

It provides a semi-analytic evaluation of order alpha (Zalpha)^2 self-energy corrections to hyperfine splitting, including state-dependent logarithmic terms and a new nuclear-spin correction.

Findings

Correction of order alpha (Zalpha)^2 involves a single ln(Zalpha) for P_1/2 states

No ln^2(Zalpha) term appears at this order for P states

A nuclear-spin dependent correction to the transition current is identified

Abstract

The hyperfine structure (HFS) of a bound electron is modified by the self-interaction of the electron with its own radiation field. This effect is known as the self-energy correction. In this work, we discuss the evaluation of higher-order self-energy corrections to the HFS of bound P states. These are expressed in a semi-analytic expansion involving powers of Zalpha and ln(Zalpha), where Z is the nuclear charge number and alpha is the fine-structure constant. We find that the correction of relative order alpha (Zalpha)^2 involves only a single logarithm ln(Zalpha) for P_1/2 states [but no term of order alpha (Zalpha)^2 ln^2(Zalpha), whereas for P_3/2 states, even the single logarithm vanishes. By a Foldy-Wouthuysen transformation, we identify a nuclear-spin dependent correction to the electron's transition current, which contributes to the HFS of P states. A comparison of the obtained analytic results to a numerical approach is made.