Coulomb scattering in the massless Nelson model II. Regularity of ground states

TL;DR

This paper analyzes the dependence of ground states on momentum and infrared cut-off in the massless Nelson model, providing bounds that are crucial for understanding scattering phenomena in quantum field theory.

Contribution

It introduces new bounds on the derivatives of ground states with respect to momentum, applicable in both infrared-regular and singular cases, advancing the analysis of infraparticle scattering.

Findings

Derived bounds on derivatives of ground states with respect to momentum.

Applicable in both infrared-regular and infrared-singular cases.

Facilitates construction of scattering states in the Nelson model.

Abstract

For the massless Nelson model we provide detailed information about the dependence of the normalized ground states $\check{\psi}_{P,\sigma}$ of the fiber single-electron Hamiltonians $H_{P,\sigma}$ on the total momentum $P$ and the infrared cut-off $\sigma$. This information is obtained with the help of the iterative analytic perturbation theory. In particular, we derive bounds of the form \[ \|\partial_{P_i}\check{\psi}_{P,\sigma}\|,\ \| \partial_{P_i} \partial_{P_j}\check{\psi}_{P,\sigma} \|\leq \frac{c}{\sigma^{\delta_{\lambda_0}}}, \] for some constant $c$ and a function of the maximal admissible coupling constant $\lambda_0\mapsto \delta_{\lambda_0}$ s.t. $\lim_{\lambda_0\to 0}\delta_{\lambda_0}=0$. These results hold both in the infrared-regular and infrared-singular case. They are exploited in part I of this series to construct the two-electron scattering states in the infrared-regular massless Nelson model (in the absence of an infrared cut-off) along the lines of Haag-Ruelle scattering theory. They should also be relevant for the problem of scattering of two infraparticles in the infrared-singular Nelson model, whose solution is the goal of this series of papers. Although a part of a larger investigation, the present work is written in a self-contained fashion.