On Dyson-Schwinger equations and the number of fermion families

TL;DR

This paper investigates Dyson-Schwinger equations for fermion propagators interacting with a massive gauge boson, revealing at least two solutions that suggest the existence of multiple fermion families with realistic mass gaps.

Contribution

It demonstrates that the Dyson-Schwinger equations admit multiple solutions corresponding to different fermion families, providing a theoretical basis for the number of fermion generations.

Findings

Existence of at least two solutions indicating multiple fermion families

Mass gap between solutions aligns with observed fermion mass differences

Solutions depend on gauge boson mass, coupling, and cutoff parameters

Abstract

We study Dyson-Schwinger equations for propagators of Dirac fermions interacting with a massive gauge boson in the ladder approximation. The equations have the form of the coupled nonlinear integral Fredholm equations of the second kind in the spacelike domain. The solutions in the timelike domain are completely defined by evaluations of integrals of the spacelike domain solutions. We solve the equations and analyze the behavior of solutions on the mass of the gauge boson, the coupling constant, and the ultraviolet cutoff. We find that there are at least two solutions for the fixed gauge boson mass, coupling, and the ultraviolet cutoff, thus there are at least two fermion families. The zero-node solution represents the heaviest Dirac fermion state, while the one-node solution is the lighter one. The mass gap between the two families is of the order of magnitude observed in nature.