Scattering Theory for Mathematical Models of the Weak Interaction

TL;DR

This paper develops a rigorous scattering theory framework for mathematical models of weak interactions involving vector bosons and leptons, proving asymptotic completeness in both massive and massless neutrino cases.

Contribution

It introduces a self-adjoint Hamiltonian model with cut-offs for weak decay processes and proves asymptotic completeness for both massive and massless neutrino models.

Findings

Proves asymptotic completeness for massive neutrino models.

Establishes asymptotic completeness for massless neutrino models using Mourre's theory.

Models are constructed with rigorous mathematical foundations and cut-offs.

Abstract

We consider mathematical models of the weak decay of the vector bosons $W^{\pm}$ into leptons. The free quantum field hamiltonian is perturbed by an interaction term from the standard model of particle physics. After the introduction of high energy and spatial cut-offs, the total quantum hamiltonian defines a self-adjoint operator on a tensor product of Fock spaces. We study the scattering theory for such models. First, the masses of the neutrinos are supposed to be positive: for all values of the coupling constant, we prove asymptotic completeness of the wave operators. In a second model, neutrinos are treated as massless particles and we consider a simpler interaction Hamiltonian: for small enough values of the coupling constant, we prove again asymptotic completeness, using singular Mourre's theory, suitable propagation estimates and the conservation of the difference of some number operators.