Black-Hole-like Solutions in Scalar-Torsion Gravity with Non-minimal Derivative Couplings

We study black-hole-like solutions in scalar–torsion gravity with a non-minimal derivative coupling between the scalar field and the torsion scalar. Starting from the corrected master equation, we apply the BPS Lagrangian method and obtain Schwarzschild-like and Reissner–Nordström-like geometries of the BPS-reduced system. These geometries become solutions of the full Euler–Lagrange equations in the strong derivative-coupling limit [Formula: see text]. The solutions possess horizon-like surfaces where the lapse function vanishes, the radial metric coefficient degenerates, and the null expansions show marginal or trapped behaviour. The Schwarzschild-like branch admits a horizon candidate at [Formula: see text], with trapped-surface character controlled by the sign of the coupling constant [Formula: see text]. Unlike the standard Schwarzschild geometry, the region [Formula: see text] does not exhibit the usual interchange between temporal and radial roles; instead, the metric becomes Euclidean. The Reissner–Nordström-like branch contains an additional charge-like parameter and admits two real horizon candidates when [Formula: see text]. Both branches are non-asymptotically flat and approach the cylindrical product geometry [Formula: see text] in Lorentzian signature. Thus the solutions are best interpreted as BPS-limit black-hole-like compact horizon geometries in scalar–torsion gravity, rather than as ordinary asymptotically flat black holes of general relativity.