Scattering of fermionic isodoublets on the sine-Gordon kink

TL;DR

This paper investigates how Dirac fermions scatter off the sine-Gordon kink, providing analytical and numerical insights into wave functions, transmission, reflection, and zero modes, revealing differences in scattering behavior based on fermionic states.

Contribution

It introduces a detailed analysis of fermionic scattering on the sine-Gordon kink using a nonlinear sigma-model framework, deriving explicit expressions and numerical results for scattering coefficients and zero modes.

Findings

Diagonal and antidiagonal fermionic states interact independently with the kink.

Explicit expressions for transmission and reflection coefficients are obtained.

Zero modes exist without fragmenting fermionic charge, but can polarize the vacuum.

Abstract

The scattering of Dirac fermions on the sine-Gordon kink is studied both analytically and numerically. To achieve invariance with respect to a discrete symmetry, the sine-Gordon model is treated as a nonlinear $\sigma$-model with a circular target space that interacts with fermionic isodublets through the Yukawa interaction. It is shown that the diagonal and antidiagonal parts of the fermionic wave function interact independently with the external field of the sine-Gordon kink. The wave functions of the fermionic scattering states are expressed in terms of the Heun functions. General expressions for the transmission and reflection coefficients are derived, and their dependences on the fermion momentum and mass are studied numerically. The existence condition is found for two fermionic zero modes, and their analytical expressions are obtained. It is shown that the zero modes do not lead to fragmentation of the fermionic charge, but can lead to polarization of the fermionic vacuum. The scattering of the diagonal and antidiagonal fermionic states is found to be significantly different; this difference is shown to be due to the different dependences of the energy levels of these bound states on the fermion mass, and is in accordance with Levinson's theorem.