Fermions on Thick Branes in the Background of Sine-Gordon Kinks

A class of thick branes in the background of sine-Gordon kinks with a scalar potential $V(\phi)=p(1+\cos\frac{2\phi}{q})$ was constructed by R. Koley and S. Kar [Classical Quantum Gravity \textbf{22}, 753 (2005)]. In this paper, in the background of the warped geometry, we investigate the issue of localization of spin half fermions on these branes in the presence of two types of scalar-fermion couplings: $\eta\bar{\Psi}\phi\Psi$ and $\eta\bar{\Psi}\sin\phi \Psi$. By presenting the mass-independent potentials in the corresponding Schr\"{o}dinger equations, we obtain the lowest Kaluza--Klein (KK) modes and a continuous gapless spectrum of KK states with $m^2>0$ for both types of couplings. For the Yukawa coupling $\eta\bar{\Psi}\phi\Psi$, the effective potential of the right chiral fermions for positive $q$ and $\eta$ is always positive, hence only the effective potential of the left chiral fermions could trap the corresponding zero mode. This is a well-known conclusion which had been discussed extensively in the literature. However, for the coupling $\eta\bar{\Psi}\sin\phi \Psi$, the effective potential of the right chiral fermions for positive $q$ and $\eta$ is no longer always positive. Although the value of the potential at the location of the brane is still positive, it has a series of wells and barriers on each side, which ensures that the right chiral fermion zero mode could be trapped. Thus we may draw the remarkable conclusion: for positive $\eta$ and $q$, the potentials of both the left and right chiral fermions could trap the corresponding zero modes under certain restrictions.