Quantum fermion emission from excited kinks

TL;DR

This work analyzes the decay of an excited topological kink in a 1+1D λϕ4 theory by coupling the scalar field to a quantum Dirac fermion through a Yukawa interaction. The authors develop a semi-classical framework: they solve the time dependent Dirac equation in the kink background, perform canonical quantization, and compute fermion production via Bogoliubov transformations, contrasting it with the purely scalar decay channel. They identify resonant and non resonant regimes determined by the Yukawa coupling g relative to the shape mode frequency ω_s; in the resonant regime, fermion production dominates and drives exponential decay of the shape mode, while in the non resonant regime the fermionic channel is exponentially suppressed. The study provides a nonperturbative numerical exploration of fermion emission, clarifies the role of mode mixing, and offers a foundation for extending the analysis to full quantum field theory and to more complex solitonic defects.

Abstract

The amplitude of an excited shape mode in a kink is expected to decay with a well-known power law via scalar radiation emission due to the nonlinear self-coupling of the scalar field. In this work we propose an alternative decay mechanism via pair production of fermions in a simple extension of the $φ^4$ model in which the scalar field is coupled to a (quantum) fermionic field through a Yukawa-like interaction term. We study the power emitted through fermions as a function of the coupling constant in the semi-classical limit (without backreaction) and compare it to the case of purely scalar radiation emission.

Figures (11)

• Figure 1: Energy spectrum of the time-independent Dirac equation in the kink background as a function of $g$. Independently of the value of $g$, a zero fermion mode will always be present. Whenever $g$ surpasses an integer value, a new bound fermion mode can be found. Above the mass threshold, represented by a dashed black line, scattering fermion modes exist. The energy of the shape mode is represented by a dashed red line.
• Figure 2: Profile of the switching function $F(t)$ (left panel) and time-dependent part of the perturbation (right panel). The parameters of the switching function are taken to be ${\cal A}=0.1$, $T=25$ and $s=1$. The asymptotic times are chosen to be $\pm50$.
• Figure 3: Probabilities of scattering states $n_k$ evaluated at the asymptotic future with respect to their wave number $k$, for increasing values of the Yukawa coupling $g$. The narrower panels to the right of each of the graphs show the discrete probabilities associated to the existing bound fermion states.
• Figure 4: Energies corresponding to the maxima in the probabilities in Figure \ref{['fig: nk vs k']} (dark red).
• Figure 5: Time evolution of the total energy for different values of $g$.