Fermions embedded in a scalar-vector kink-like smooth potential

TL;DR

This paper investigates the behavior of massive fermions in scalar-vector kink-like potentials, revealing bound states, scattering properties, and localization effects, with implications for relativistic quantum systems.

Contribution

It introduces a Sturm-Liouville approach to analyze fermions in kink-like potentials and explores the conditions for bound states and their disappearance.

Findings

Bound states appear as poles in the transmission amplitude in strong coupling.

Isolated bound solutions vanish when scalar and vector couplings are equal.

High localization of fermions is consistent with the Heisenberg principle in relativistic regimes.

Abstract

The behaviour of massive fermions is analyzed with scalar and vector potentials. A continuous chiral-conjugation transformation decouples the equation for the upper component of the Dirac spinor provided the vector coupling does not exceed the scalar coupling. It is shown that a Sturm-Liouville perspective is convenient for studying scattering as well as bound states. One possible isolated solution (excluded from the Sturm-Liouville problem) corresponding to a bound state might also come into sight. For potentials with kink-like profiles, beyond the intrinsically relativistic isolated bound-state solution corresponding to the zero-mode solution of the massive Jackiw-Rebbi model in the case of no vector coupling, a finite set of bound-state solutions might appear as poles of the transmission amplitude in a strong coupling regime. It is also shown that the possible isolated bound solution disappears asymptotically as the magnitude of the scalar and vector coupling becomes the same. Furthermore, we show that due to the sizeable mass gain from the scalar background the high localization of the fermion in an extreme relativistic regime is conformable to comply with the Heisenberg uncertainty principle.