Relativistic and Recoil Corrections to Vacuum polarization in muonic systems: Three--photon exchange, gauge invariance and numerical values

TL;DR

This paper investigates relativistic and recoil corrections to vacuum polarization in muonic systems, emphasizing gauge invariance and providing numerical values for specific muonic ions using an adapted NRQED approach.

Contribution

It introduces an adapted NRQED framework for muonic systems and demonstrates gauge invariance of corrections including three-photon exchange diagrams.

Findings

Gauge invariance confirmed after including three-photon exchange diagrams.

Updated numerical values for corrections in muonic ions with Z=1,2,6.

Relativistic and recoil corrections are essential for accurate muonic system modeling.

Abstract

For an accurate theoretical description of muonic bound systems, it is crucial to consistently treat relativistic and recoil corrections to vacuum polarization. The one-loop vacuum-polarization effect is by far the dominant quantum electrodynamic (QED) energy correction for bound muons, being of order $\alpha (Z\alpha)^2 m_r$, where $\alpha$ is the fine-structure constant, $Z$ is the nuclear charge number, and $m_r$ is the reduced mass. Gauge invariance of the relativistic and recoil corrections to vacuum polarization of order $\alpha (Z\alpha)^4 m_r$ is investigated with respect to nonretarded and standard, renormalized variants of Coulomb gauge. The invariance is shown after including three-photon exchange diagrams. Our derivation is based on an adapted form of Nonrelativistic Quantum Electrodynamics for bound muon systems (NRQED$_\mu$), which is a version of NRQED where the hard scale is set at the muon mass instead of the electron mass. Updated values for the gauge-independent corrections for one-muon ions with nuclear charge numbers $Z = 1,2,6$ are presented.