Trapping Dirac fermions in tubes generated by two scalar fields

In this work we consider (1,1)-dimensional resonant Dirac fermionic states on tubelike topological defects. The defects are formed by rings in (2,1) dimensions, constructed with two scalar fields phi and chi, and embedded in the (3,1)-dimensional Minkowski spacetime. The tubelike 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 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)(psi) over bar psi. With a convenient decomposition of the fermionic fields in left and right components, 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 Schrodinger-like equations whose eigenvalues are the masses of the fermionic states. With F(phi,chi) = phi chi the Schrodinger-like equations are numerically solved with appropriated boundary conditions. Several resonance peaks for both components are obtained, and the results are confronted with the qualitative analysis of the Schrodinger-like potentials.