Mass Renormalization in Non-relativistic Quantum Electrodynamics with Spin 1/2

TL;DR

This paper analyzes the effective mass in nonrelativistic QED with spin 1/2, revealing how spin interaction influences mass renormalization and providing explicit asymptotic behaviors as the ultraviolet cutoff increases.

Contribution

It provides explicit formulas and asymptotic analysis for the effective mass in nonrelativistic QED with spin 1/2, highlighting the impact of spin interaction on mass enhancement.

Findings

Spin interaction enhances the effective mass.

The coefficient a_1 grows logarithmically with cutoff.

The coefficient a_2 diverges as -( ext{cutoff})^2, not as ( ext{log cutoff})^2.

Abstract

The effective mass $\mass$ of the the Pauli-Fierz Hamiltonain with ultraviolet cutoff $\Lambda$ and the bare mass $m$ in nonrelativistic QED with spin 1/2 is investigated. Analytic properties of $\mass$ in coupling constant $e$ are shown and explicit forms of constants $a_1(\Lambda/m)$ and $a_2(\Lambda/m)$ depending on $\Lambda/m$ such that $$\mass/m =1 + a_1(\Lambda/m) e^2+ a_2(\Lambda/m) e^4+ {\mathcal O}(e^6)$$ are given. It is shown that the spin interaction enhances the effective mass and that there exist strictly positive constants $b_1,b_2, c_1$ and $c_2$ such that $$\d b_1\leq \lim_{\Lambda\to\infty} \frac{a_1(\Lamdda/m)}{\log (\Lambda/m)}\leq b_2, c_1\leq \lim_{\Lambda\to\infty} \frac{a_2(\Lambda/m)}{(\Lambda/m)^2}\leq -c_2.$$ In particular $a_2(\Lambda/m)$ does not diverges as $\pm [\log(\Lambda/m)]^2$ but $-(\Lambda/m)^2$.