Fermions in a mixed vector-scalar double-step potential via continuous chiral transformation

TL;DR

This paper investigates fermion behavior in a mixed scalar-vector double-step potential using continuous chiral transformation, revealing relativistic bound states, transmission oscillations, and the effects of symmetries and mass on localization.

Contribution

It introduces a method to analyze fermions in mixed potentials via chiral transformation and uncovers conditions for bound states and symmetry effects in relativistic regimes.

Findings

Bound states appear as poles in transmission amplitude.

Oscillations in transmission coefficient occur with strong coupling.

High localization does not violate the Heisenberg principle due to scalar mass.

Abstract

The behaviour of fermions in the background of a double-step potential is analyzed with a general mixing of scalar and vector couplings via continuous chiral-conjugation transformation. Provided the vector coupling does not exceed the scalar coupling, a Sturm-Liouville approaching for the double-step potential shows that the transmission coefficient exhibits oscillations and that a finite set of intrinsically relativistic bound-state solutions might appear as poles of the transmission amplitude in a strong coupling regime. An isolated bound-state solution resulting from coupled first-order equations might also come into sight. It is also shown that all those possible bound solutions disappear asymptotically as one approaches the conditions for the realization of the so-called spin and pseudospin symmetries in a four-dimensional space-time. Furthermore, we show that due to the additional mass acquired by the fermion from the scalar background the high localization of the fermion in an extreme relativistic regime does not violate the Heisenberg uncertainty principle.