The effective mass of the Pauli-Fierz electron: a cutoff-independent lower bound and the breakdown of mass renormalization

Abstract

We consider the Pauli-Fierz model of one spinless non-relativistic electron coupled to the quantized transverse electromagnetic field, without external potential, with a sharp ultraviolet cutoff Lambda and bare mass m. Let h = m_eff/m be the ratio of the effective mass, defined by the curvature of the fiber ground state energy at total momentum zero, to the bare mass; by scaling, h depends only on the fine structure constant alpha = e^2/(4 pi) and on lambda = Lambda/m. We prove two lower bounds. First, h >= 1 + 0.3441 alpha/(1 + 0.2985 sqrt(alpha) + 0.1252 alpha) for all alpha > 0 and all lambda >= 1; the bound does not depend on the cutoff, and with the band extended to the cutoff its first order coefficient in alpha is the perturbative one of Hiroshima and Spohn. The proof uses a coherent translation of the field applied to the ground state as a trial state; the term linear in the translation vanishes by minimality of the ground state. Second, for every fixed alpha, h grows faster than linearly: for every epsilon > 0, h >= lambda^(8/7 - epsilon) for all large lambda (8/7 is the limit of the method). Consequently iota_PF(alpha) := inf_{lambda > 0} h/lambda > 0, and for a fixed physical mass m_phys a bare mass reproducing it exists only if Lambda <= m_phys/iota_PF(alpha). This was expected, but not proved, by Lieb and Loss (2002). The proof of the second bound starts from the sharp self-energy asymptotics of a companion paper and uses a trial state obtained from the ground state by a phase depending on the field configuration. The new ingredients are an estimator for the position of the electron given by the local centre of a shallow level set of the photon cloud, the decoupling of a localized part of the field from the electron through a divergence-free external subspace, and a proof that shallow depressions near the edge of the localization window are rare.