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(5+1)-Dimensional Analytical Brane-World Models: Intersecting Thick Branes

Two co-dimensional thick brane-worlds are investigated in quite general terms for two intersecting scalar fields generating the extra dimension defect. In general, when one considers two co-dimensional thick brane-worlds, the warp factor is constructed as a string-like defect. Considering a twofold-warp factor constructed from two intersecting warp factors, an alternative bulk configuration is examined. With the brane localization thus driven by two crossing scalar fields, the possible solvable models obtained from such a two co-dimensional setup are systematically discussed. The obtained solutions are classified as five different models organized into two subsets for which some of their physical properties are evaluated. For models $I$ and $II$, in the first subset, Einstein equation solutions are rigidly defined, up to some arbitrary constant. For models $III$, $IV$ and $V$, in the second subset, an additional degree of freedom not constrained by Einstein equations is admitted. The solutions are all obtained from a departure statement of assuming a conformally flat metric for the internal space, which is concomitant to the proper choice of coordinates. Eventual singularities in the curvature are identified, however, without affecting the physical appeal of the solutions described in terms of the stress energy tensor patterns, which are shown to be free of singularities for model $IV$, besides admitting straightforward reductions to $(4+1)$-dimensions. In particular, from the framework of models $III$ and $IV$, one is able to achieve brane-world solutions over two different geometries of $\mathbb{S}^{2}$ which, as demonstrated, can be reduced to trivial and non-trivial extensions of the well-known $(4+1)$-dimensional brane-worlds.