Constraint on cosmological constant in generalized Skryme-teleparallel system

The Einstein-Skyrme system is understood to defy the "no hair" conjecture by possessing black-hole solutions with fractional baryon number outside the event horizon. In this article, we extend the study of the Skyrme system to teleparallel gravity framework. We consider two scenarios, the Teleparallel Equivalent of General Relativity (TEGR) and generalized teleparallel gravity $f(T)$. In our analysis, we compute the fractional baryon number beyond the black-hole horizon and its correlation with the cosmological constant ($\Lambda$). In the TEGR context, where $f(T) = -T - 2\Lambda$, the results match with the Einstein-Skyrme model, assuming a positive $\Lambda$. More interestingly, in generalized teleparallel gravity scenario, defined by $f(T) = -T - \tau T^2 - 2\Lambda$, we show that the existence of a solution demands that not only must $\Lambda$ be positive but has to lie in a range, $\Lambda_{min} < \Lambda < \Lambda_{max}$. While the upper bound depends inversely on $\tau$, the lower bound is a linear function of it. Hence, in the limiting case with generalized teleparallel gravity converging towards TEGR ($\tau \rightarrow 0$), the constraints on the cosmological constant relax to the Einstein Skryme system ($\Lambda_{min}$ approaches zero and $\Lambda_{max}$ becomes unbounded). On the other hand, in f(T) gravity, vanishing cosmological constant solution is found only if the lower bound on the energy of the soliton is very large.