Possibility of the Traversable Wormholes in the Galactic Halos within $4D$ Einstein-Gauss-Bonnet Gravity

Recently, there has been significant interest regarding the regularization of a $D\rightarrow 4$ limit of Einstein-Gauss-Bonnet (EGB) gravity. This regularization involves re-scaling the Gauss-Bonnet (GB) coupling constant as $\alpha/(D-4)$, which bypasses Lovelock's theorem and avoids Ostrogradsky instability. A noteworthy observation is that the maximally or spherically symmetric solutions for all the regularized gravities coincide in the $4D$ scenario. Considering this, we investigate the wormhole solutions in the galactic halos based on three different choices of dark matter (DM) profiles, such as Universal Rotation Curve, Navarro-Frenk-White, and Scalar Field Dark Matter with the framework of $4D$ EGB gravity. Also, the Karmarkar condition was used to find the exact solutions for the shape functions under different non-constant redshift functions. We discussed the energy conditions for each DM profile and noticed the influence of GB coefficient $\alpha$ in violating energy conditions, especially null energy conditions. Further, some physical features of wormholes, viz. complexity factor, active gravitational mass, total gravitational energy, and embedding diagrams, have been explored.