Localization of $q$-form field on squared curvature gravity domain wall brane coupling with gravity and background scalar

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, Schr\"{o}dinger-like equations, and a four-dimensional effective action. We suggest a new function $F(R,\varphi)$ to stand for the coupling of the $q$-form field with gravity and background scalar fields. Our analysis highlights the significance of the parameters $\text C_2$ and $\displaystyle t$ in the localization processes.