Massive Jackiw-Rebbi Model

In this paper we analyze a generalized Jackiw-Rebbi (J-R) model in which a massive fermion is coupled to the kink of the $\lambda\phi^4$ model as a prescribed background field. We solve this massive J-R model exactly and analytically and obtain the whole spectrum of the fermion, including the bound and continuum states. The mass term of the fermion makes the potential of the decoupled second order Schrodinger-like equations asymmetric in a way that their asymptotic values at two spatial infinities are different. Therefore, we encounter the unusual problem in which two kinds of continuum states are possible for the fermion: reflecting and scattering states. We then show the energies of all the states as a function of the parameters of the kink, i.e. its value at spatial infinity ($\theta_0$) and its slope at $x=0$ ($\mu$). The graph of the energies as a function of $\theta_0$, where the bound state energies and the two kinds of continuum states are depicted, shows peculiar features including an energy gap in the form of a triangle where no bound states exist. That is the zero mode exists only for $\theta_0$ larger than a critical value $(\theta_0^{\textrm{c}})$. This is in sharp contrast to the usual (massless) J-R model where the zero mode and hence the fermion number $\pm1/2$ for the ground state is ever present. This also makes the origin of the zero mode very clear: It is formed from the union of the two threshold bound states at $\theta_0^{\textrm{c}}$, which is zero in the massless J-R model.