Trapping Dirac fermions in tubes generated by two scalar fields

In this work we consider $(1,1)-$dimensional resonant Dirac fermionic states on tube-like topological defects. The defects are formed by rings in $(2,1)$ dimensions, constructed with two scalar field $\phi$ and $\chi$, and embedded in the $(3,1)-$dimensional Minkowski spacetime. The tube-like defects are attained from a lagrangian density explicitly dependent with the radial distance $r$ relative to the ring axis and the radius and thickness of the its cross-section are related to the energy density. For our purposes we analyze a general Yukawa-like coupling between the topological defect and the fermionic field $\eta F(\phi,\chi)\bar\psi\psi$. With a convenient decomposition of the fermionic fields in left- and right- chiralities, we establish a coupled set of first order differential equations for the amplitudes of the left- and right- components of the Dirac field. After decoupling and decomposing the amplitudes in polar coordinates, the radial modes satisfy Schr\"odinger-like equations whose eigenvalues are the masses of the fermionic resonances. With $F(\phi,\chi)=\phi\chi$ the Schr\"odinger-like equations are numerically solved with appropriated boundary conditions. Several resonance peaks for both chiralities are obtained, and the results are confronted with the qualitative analysis of the Schr\"odinger-like potentials.