First-order formalism for thick branes in $f(T,T)$ Gravity

In this paper we study the thick brane scenario constructed in the recently proposed $f(T,T)$ theories of gravity, where T is called the torsion scalar and $T$ is the trace of the energy-momentum tensor. We use the first-order formalism to find analytical solutions for models that include a scalar field as a source. In particular, we describe two interesting case in which in the first we obtain a double-kink solution, which generates a splitting in the brane. In the second case, proper management of a kink solution obtained generates a splitting in the brane intensified by the torsion parameter, evinced by the energy density components satisfying the weak and strong energy conditions. In addition, we investigate the behavior of the gravitational perturbations in this scenario. The parameters that control the torsion and the trace of the energy-momentum tensor tend to shift the massive modes to the core of the brane, keeping a gapless non-localizable and stable tower of massive modes and producing more localized massless modes.