Stable branches of a solution for a fermion on domain wall

TL;DR

This paper investigates the existence and stability of solutions for a fermion occupying excited states in the field of a domain wall, revealing multiple solution branches and their stability properties within a specific coupling constant range.

Contribution

It introduces the concept of multiple stable solution branches for fermions on domain walls, including excited and polarized states, within a limited coupling constant interval.

Findings

Stable solution branches exist for 1<g<g_max≈1.65.

Different branches correspond to fermions on, off, and excited in the domain wall.

Multiple stable states are identified within the coupling constant range.

Abstract

We discuss the case when a fermion occupies an excited non-zero frequency level in the field of domain wall. We demonstrate that a solution exists for the coupling constant in the limited interval $1<g<g_{max}\approx 1.65$. We show that indeed there are different branches of stable solution for $g$ in this interval. The first one corresponds to a fermion located on the domain wall ($1<g<\sqrt[4]{2\pi}$). The second branch, which belongs to the interval $\sqrt[4]{2\pi}\le g\le g_{max}$, describes a polarized fermion off the domain wall. The third branch with $1<g<g_{max}$ describes an excited antifermion in the field of the domain wall.