Effects of the variation of mass on fermion localization and resonances on thick branes

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A few years ago, Campos investigated the critical phenomena of thick branes in warped spacetimes [Phys. Rev. Lett. 88 (2002) 141602]. Inspired by his work, we consider a toy model of thick branes generated by a real scalar field with the potential V(ϕ)=aϕ²-bϕ⁴+cϕ⁶, and investigate the variation of the mass parameter a on the branes as well as the localization and resonances of fermions. An interesting result is found: there is a critical value for the mass parameter a, and when the critical value of a is reached the solution of the background scalar field is not unique and has the shape of a double kink. This happens in both cases with and without gravity. It is also shown that the numbers of the bound Kaluza-Klein modes of fermions on the gravity-free brane and the resonant states of fermions on the brane with gravity increase with the value of a.