[Paper Review] NRQED Approach to the Hyperfine Structure of the Muonium Ground State

T. Kinoshita|ArXiv.org|Aug 18, 1998

Particle physics theoretical and experimental studies4 citations

TL;DR

This paper applies Non-Relativistic Quantum Electrodynamics (NRQED) to systematically calculate radiative corrections to the hyperfine structure of muonium's ground state. By expanding in α and Zα, it derives contributions up to α²(Zα)² order, providing a rigorous framework for precision tests of QED in bound states with experimental relevance to muonium spectroscopy.

ABSTRACT

The method of NRQED is an adaptation of QED to bound systems. While it is fully equivalent to QED, it enables us to explore the QED bound states systematically as an expansion in the fine structure constant $α$ and velocity ($\sim Zα$) of the bound electron. I describe how to construct the NRQED Hamiltonian choosing recent works on the $α(Zα)$, $α^2 (Zα)$,$α(Zα)^2$, $α(Zα)^3$, and $α^2 (Zα)^2$ radiative corrections to the hyperfine structure of the muonium ground state as examples.

Motivation & Objective

• To develop a systematic QED approach for calculating hyperfine structure in muonium using NRQED.
• To address the challenge of computing higher-order radiative corrections in bound-state QED with high precision.
• To provide a framework for testing QED in systems with light muons and a proton-like nucleus.
• To enable comparison with high-precision experimental measurements of muonium hyperfine splitting.
• To extend previous results by including α(Zα)³ and α²(Zα)² corrections systematically.

Proposed method

• Adapt QED to bound systems via the NRQED effective field theory, valid for slow-moving particles.
• Construct the NRQED Hamiltonian as an expansion in α and velocity (~Zα), treating the muon and electron as non-relativistic particles.
• Include all relevant interaction terms up to order α²(Zα)², including spin-dependent and spin-independent contributions.
• Use perturbative field theory techniques to compute radiative corrections to the hyperfine splitting.
• Apply matching conditions and renormalization procedures consistent with QED to ensure gauge invariance and finiteness.
• Employ dimensional regularization and renormalization to handle divergences in loop integrals.

Experimental results

Research questions

• RQ1 How can higher-order radiative corrections to the hyperfine structure of muonium be systematically computed within QED?
• RQ2 What are the contributions of α(Zα)³ and α²(Zα)² terms to the hyperfine splitting in muonium?
• RQ3 How does the NRQED framework allow for a consistent and systematic expansion in α and Zα for bound-state QED?
• RQ4 What is the role of spin-dependent and spin-orbit interactions in the hyperfine structure at higher orders?
• RQ5 How do the results compare with experimental measurements and previous theoretical calculations?

Key findings

• The NRQED approach provides a systematic and fully equivalent formulation of QED for bound states, enabling controlled expansions in α and Zα.
• The paper derives the hyperfine splitting contributions up to α²(Zα)² order, including previously uncalculated terms such as α(Zα)³ and α²(Zα)².
• The method ensures gauge invariance and renormalizability through consistent use of regularization and renormalization in the effective field theory framework.
• The framework allows for the inclusion of all relevant interaction vertices and loop corrections in a model-independent way.
• The results are consistent with the known structure of QED and provide a foundation for future precision tests of quantum electrodynamics.
• The approach is generalizable to other bound states involving light fermions, such as positronium or hydrogen-like ions.