On gravity localization in scalar braneworlds with a super-exponential warp factor

We show that within single brane tachyonic braneworld models, super-exponential warp factors of the form \(e^{-2f} \sim e^{-2c_1e^{c_2 |\sigma |}}\) are problematic when dealing with both the finiteness of the effective four-dimensional (4d) Planck mass and the localization of 4d gravity, which can be stated by the requirement that \(\int e^{-2f(\sigma )}d\sigma < \infty \), because this condition necessarily implies that \(c_1\) and \(c_2\) should be positive. As a consequence of this fact the tachyonic field \(T\) turns out to be complex in contradiction with the real nature of the starting action for the tachyonic braneworld. Conversely if one requires to have a real tachyon field, 4d gravity will not be localized and the effective gravitational coupling will be infinite. We present several typical examples where this problem occurs: we have analysed this situation for thin as well as thick tachyonic braneworlds with 4d Poincare symmetry, for the case when a bulk cosmological constant is present, and even for a brane with an induced spatially flat 4d cosmological background, and shown that in all cases the tachyon field \(T\) comes out to be inconsistently complex when imposing localization of 4d gravity on the brane. On the other hand, when dealing with a further reduction of the hierarchy problem on a two-brane system, one should carefully consider the sign of the constants \(c_1\) and \(c_2\) to avoid inconsistencies in the tachyonic braneworld model. We also present a similar discusion involving a canonical scalar field in the bulk where none of these problems arise and hence, the mass hierarchy and 4d gravity localization problems can be successfully addressed at once, i.e., with the same warp factor. Finally, the stability analysis of this scalar tensor braneworld model with a super-exponential warp factor is performed.