Infrared renormalization in non-relativistic QED and scaling criticality

Abstract

We consider a spin-$\frac12$ electron in a translation-invariant model of non-relativistic Quantum Electrodynamics (QED). Let $H(\vp,\sig)$ denote the fiber Hamiltonian corresponding to the conserved total momentum $\vp\in\R^3$ of the Pauli electron and the photon field, regularized by a fixed ultraviolet cutoff in the interaction term, and an infrared regularization parametrized by $0<\sig\ll1$ which we ultimately remove by taking $\sig\searrow0$. For $|\vp|<\puppbd$, all $\sig>0$, and all values of the finestructure constant $\gs<\gs_0$, with $\gs_0\ll1$ sufficiently small and {\em independent} of $\sig$, we prove the existence of a ground state eigenvalue of multiplicity two at the bottom of the essential spectrum. Moreover, we prove that the renormalized electron mass satisfies $1<m_{ren}(\vp,\sig)<1+c\alpha$, {\em uniformly} in $\sig\geq0$, in units where the bare mass has the value 1, and we prove the existence of the renormalized mass in the limit $\sig\searrow0$. Our analysis uses the isospectral renormalization group method of Bach-Fr\"ohlich-Sigal introduced in \cite{bfs1,bfs2} and further developed in \cite{bcfs1,bcfs2}. The limit $\sig\searrow0$ determines a scaling-critical renormalization group problem of endpoint type, in which the interaction is strictly marginal (of scale-independent size). The main achievement of this paper is the development of a method that provides rigorous control of the renormalization of a {\em strictly marginal} quantum field theory characterized by a {\em non-trivial scaling limit}.