Infrared renormalization in non-relativistic QED and scaling criticality

We consider a spin-12 electron in the framework of non-relativistic Quantum Electrodynamics (QED). Let H( p,σ) denote the fiber Hamiltonian corresponding to the conserved total momentum p ∈ R3 of the electron and the photon field, regularized by a fixed ultraviolet cutoff in the interaction term, and an infrared regularization parametrized by 0<σ 1. Ultimately, our goal is to remove the latter by taking σ 0. We prove that there exists a constant 0 < α0 1 independent of σ > 0 such that for all | p| < 1/3 and all values of the finestructure constant 0 < α < α0, there exists a ground state eigenvalue of multiplicity two at the bottom of the essential spectrum. Moreover, we prove that the renormalized electron mass satisfies 1