Localization of a spin-$1/2$ fermion on the braneworld is an important and interesting problem. It is well known that a five-dimensional free massless fermion $\mathrm{\Psi }$ minimally coupled to gravity cannot be localized on the Randall-Sundrum braneworld. In order to trap such a fermion, the coupling between the fermion and bulk scalar fields should be introduced. In this paper, localization and quasilocalization of a bulk fermion on the thick braneworld generated by two scalar fields (a kink scalar $\varphi$ and a dilaton scalar $\pi$) are investigated. Two types of couplings between the fermion and two scalars are considered. One coupling is the usual Yukawa coupling $-\eta \overline{\mathrm{\Psi }}\varphi \mathrm{\Psi }$ between the fermion and kink scalar, another one is $\lambda \overline{\mathrm{\Psi }}{\mathrm{\Gamma }}^M{\partial }_M\pi {\gamma }^5\mathrm{\Psi }$ between the fermion and dilaton scalar. The left-chiral fermion zero mode can be localized on the brane, and both the left- and right-chiral fermion massive Kaluza-Klein modes may be localized or quasilocalized. Hence the four-dimensional massless left-chiral fermion and massive Dirac fermions, whose lifetime is infinite or finite, can be obtained on the brane.

In this paper, we consider a coupling mechanism between the kinetic term and the gravity, in which a coupling function F (R) is introduced into the kinetic term of the five-dimensional scalar field. Based on this scenario, we investigate the localization of scalar fields in three specific braneworld models: the Minkowski brane, the de Sitter brane, and the Anti-de Sitter brane. The brane models considered here are regular with no singularity for scalar curvature. For the Minkowshi brane case, the zero mode can always be localized on the brane, and the massive modes can be localized or quasi-localized on the brane. For the dS₄ brane case, two forms of factor F (R) is considered. The zero mode can always be localized, and the massive modes could be quasi-localized on the brane. Besides, with the second coupling factor, the scalar zero mode could be localized on the both sides of the origin of extra dimension, while the massive modes could be quasi-localized on the origin. Lastly, for the AdS₄ brane case, the localization of the scalar zero mode requires the consideration of a coupling potential V (Φ, φ), while the massive modes can still be localized on the brane with an infinite number.

In this paper, writing the most general form $f(R)={\sum }_{i=1}^n{a}_i{R}^i+\mathrm{\Lambda }$ the solutions of the pure geometric thick $f(R)$ brane are investigated. For the certain value of $n$, the analytical thick brane solution can be calculated, and when $n=3$, $n=4$, and $n=10$, the thick brane solutions are presented. The solutions are stable against linear tensor perturbations. The zero mode of gravity and scalar field can be localized on thick $f(R)$ branes naturally. The zero mode of a vector field and left-chiral fermion can be localized on thick $f(R)$ branes by introducing the coupling with scalar curvature $R$ of spacetime, and the massive resonant modes can be quasilocalized on the brane with the large coupling coefficients.

It is known that the scalar fields can be trapped on branes of different types, and the U ( 1 ) gauge vector fields can be localized on the thick de Sitter brane, or the thick Minkowski brane via coupling with gravity. The Kalb-Ramond fields can be localized on the Minkowski brane and the thick de-Sitter brane, with certain couplings. In this paper, with considering a coupling mechanism between the kinetic term of the q -form fields and the background spacetime, we investigate the localization of Kaluza-Klein modes for the q -form fields in D -dimensional spacetime. Concrete q -form fields are discussed within five-dimensional brane models with typical spacetime geometries: Minkowski, de Sitter, and anti–de Sitter. In the Minkowski brane case, the zero modes of various q -form fields can be localized on the brane. In the de Sitter brane case, the zero mode of the U ( 1 ) gauge vector fields can be localized on the brane. Lastly in the anti–de Sitter brane case, the zero mode of the Kalb-Ramond fields can be localized on the brane. For the massive Kaluza-Klein modes of these q -form fields, they could be localized or quasi-localized on the brane of different types. Besides, subtle and detailed behaviors of the Kaluza-Klein modes for q -form fields are observed: the zero modes could be localized on both sides of the brane, and the massive modes could be localized or quasilocalized at the brane position.

Unlike the duality in four-dimensional spacetime, where the q-form fields corresponds to the scalar fields or the vector fields, in higher dimensional spacetime, they denote new types of particles. In this paper, we investigate the localization of the KK modes of the q-form fields in a five dimensional brane world. We introduce the coupling between the q-form fields and both the gravity and the background scalar field. This consideration enables the localization of the q-form fields on the five-dimensional RS-2 thick brane, leading to the derivation of zero modes, Schrödinger-like equations, and a four-dimensional effective action. We suggest a new function $$F(R,\varphi )$$ F ( R , φ ) to stand for the coupling of the q-form field with gravity and background scalar fields. Our analysis highlights the significance of the parameters $$\text{ C}_2$$ C 2 and $$\displaystyle t$$ t in the localization processes.