Fermions on thick branes in the background of sine-Gordon kinks

Abstract

A class of thick branes in the background of sine-Gordon kinks with a scalar potential $V(\ensuremath{\phi})=p(1+\mathrm{cos}\frac{2\ensuremath{\phi}}{q})$ was constructed by R. Koley and S. Kar [Classical Quantum Gravity 22, 753 (2005)]. In this paper, in the background of the warped geometry, we investigate the issue of localization of spin-$1/2$ fermions on these branes in the presence of two types of scalar-fermion couplings: $\ensuremath{\eta}\overline{\ensuremath{\Psi}}\ensuremath{\phi}\ensuremath{\Psi}$ and $\ensuremath{\eta}\overline{\ensuremath{\Psi}}\mathrm{sin}\ensuremath{\phi}\ensuremath{\Psi}$. By presenting the mass-independent potentials in the corresponding Schr\"odinger equations, we obtain the lowest Kaluza-Klein modes and a continuous gapless spectrum of Kaluza-Klein states with ${m}^{2}>0$ for both types of couplings. For the Yukawa coupling $\ensuremath{\eta}\overline{\ensuremath{\Psi}}\ensuremath{\phi}\ensuremath{\Psi}$, the effective potential of the right chiral fermions for positive $q$ and $\ensuremath{\eta}$ 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 is discussed extensively in the literature. However, for the coupling $\ensuremath{\eta}\overline{\ensuremath{\Psi}}\mathrm{sin}\ensuremath{\phi}\ensuremath{\Psi}$, the effective potential of the right chiral fermions for positive $q$ and $\ensuremath{\eta}$ 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 following remarkable conclusion: for positive $\ensuremath{\eta}$ and $q$, the potentials of both the left and right chiral fermions could trap the corresponding zero modes under certain restrictions.