Fermion family recurrences in the Dyson-Schwinger formalism

Abstract

We study the multiple solutions of the truncated propagator Dyson-Schwinger equation for a simple fermion theory with Yukawa coupling to a scalar field. Upon increasing the coupling constant $g$, other parameters being fixed, more than one non-perturbative solution breaking chiral symmetry becomes possible and we find these numerically. These ``recurrences'' appear as a mechanism to generate different fermion generations as quanta of the same fundamental field in an interacting field theory, without assuming any composite structure. The number of recurrences or flavors is reduced to a question about the value of the Yukawa coupling, and has no special profound significance in the Standard Model. The resulting mass function can have one or more nodes and the measurement that potentially detects them can be thought of as a collider-based test of the virtual dispersion relation $E=\sqrt{p^2+M(p^2)^2}$ for the charged lepton member of each family. This requires three independent measurements of the charged lepton's energy, three-momentum and off-shellness. We illustrate how this can be achieved for the (more difficult) case of the tau lepton.