Infrared Renormalization in Non-relativistic Qed for the Endpoint Case

We consider a spin 1 2 electron in a translation-invariant model of non-relativistic Quantum Electrodynamics (QED). Let H(~ p, σ) denote the fiber Hamiltonian corresponding to the conserved momentum ~ p ∈ R, regularized by a fixed ultraviolet cutoff in its interaction term, and an infrared regularization parametrized by 0 < σ ≪ 1 which we ultimately remove by taking σ ց 0. For |~ p| < 1 3 , all σ > 0, and all values of the finestructure constant α < α0, with α0 ≪ 1 sufficiently small and independent of σ, 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 < mren(~ p, σ) < 1 + cα uniformly in σ ≥ 0, in units where the bare mass has the value 1. Our analysis is based on the isospectral renormalization group method of Bach-Fröhlich-Sigal developed in [1, 2] and further extended in [3, 4]. The limit σ ց 0 determines a renormalization group problem of endpoint type, in which the interaction is strictly marginal (its size is scaleindependent). We demonstrate that the mechanism leading to its uniform boundedness is a near complete cancellation of all terms in certain oscillatory sums. We control the latter by exploiting the symmetries of the model, which we incorporate into algebraic identities satisfied by the smooth Feshbach map, and by combining the isospectral renormalization group method with the strong induction principle.