Thick brane in Palatini formalism with a non-minimally coupled bulk scalar field

We study a thick brane scenario within the Palatini formulation of gravity, where the metric and affine connection are treated as independent variables. By introducing a non-minimal coupling between a bulk scalar field and the Ricci scalar, we obtain analytic solutions under a flat, four-dimensional Poincaré-invariant metric with a kink-like scalar configuration. The warp factor exhibits a bell-shaped profile, while the scalar potential forms a symmetric volcano-like structure, characteristic of a finite-thickness brane. The corresponding energy density is regular and localized, featuring a central peak with symmetrically placed negative minima. Through the analysis of linear tensor perturbations, we derive a Schrödinger-like equation with supersymmetric factorization, ensuring the absence of tachyonic modes and thus the stability of the background configuration. The effective potential also takes a volcano-like form that supports a localized graviton zero mode, confirming the recovery of four-dimensional gravity on the brane. A numerical study of the massive Kaluza--Klein spectrum reveals the progressive delocalization of massive modes into the bulk. Our results demonstrate a stable and physically consistent thick brane configuration within the Palatini gravity framework, offering new insights into gravity localization and braneworld phenomenology.

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