Localization of the Elko Field on the Bloch Branes

In this work, we investigate the localization properties of Elko spinor fields on the Bloch brane, a thick brane scenario generated by two interacting scalar fields and characterized by a rich internal structure. We begin by studying the dynamics of a free (massless) Elko field in the five-dimensional Bloch brane background and derive the corresponding Schrödinger-like equation governing the Kaluza–Klein modes. Our analysis shows that, due to the specific geometric features and the non-trivial warp factor of the Bloch brane, the zero mode of a free Elko field is not normalizable and therefore cannot be localized on the brane.

To resolve this issue, we introduce a non-minimal coupling term into the five-dimensional Elko spinor field action. By considering different forms of coupling between the Elko field and the background scalar fields, we demonstrate that the inclusion of such a coupling significantly alters the effective potential of the system and allows for the localization of the Elko zero mode. We identify the explicit conditions under which a normalizable and physically acceptable zero mode can be trapped on the brane.

Furthermore, we examine how the internal structure of the Bloch brane influences the localization mechanism. Our results reveal that the brane’s internal layers play a crucial role in shaping the effective potential felt by the Elko field, thus affecting both the localization behavior and the profile of the zero mode. This study provides deeper insight into the realization of dark-matter-motivated Elko fields in thick brane models.