Anisotropic Dark Matter Bosonic Stars in regularized 4D Einstein$-$Gauss$-$Bonnet gravity

In this work, we have constructed anisotropic bosonic dark-matter star (DMS) solutions in the context of a regularized four-dimensional Einstein$-$Gauss$-$Bonnet (4D EGB) gravity theory. Using dimensional regularization, we solve modified Tolman$-$Oppenheimer$-$Volkoff equations for a self-interacting complex scalar field in the dilute polytropic regime, $p_r = K \rho^2$, with anisotropy parameterized as $\sigma = \beta\, p_r \left( 1 - e^{-2\lambda} \right)$. We perform a comprehensive numerical analysis across the $(\alpha,\beta)$ parameter domain, where $\alpha \in [0,8]~\mathrm{km}^2$ and $\beta \in [-2,0]$, to examine mass$-$radius relations and evaluate multiple stability indicators including static equilibrium $dM/dp_c$, sound-speed causality, the radial adiabatic index $\Gamma_r$, and energy conditions. Positive Gauss$-$Bonnet coupling enhances both the maximum mass and compactness (e.g., $M_{\rm max} \approx 1.62\, M_\odot$ at $\alpha=0$ rising to $\approx 2.09\, M_\odot$ at $\alpha = 8~\mathrm{km}^2$), while negative anisotropy reduces them (e.g., from $\approx 2.21\, M_\odot$ at $\beta=0$ to $\approx 1.73\, M_\odot$ at $\beta = -2$). The resulting configurations remain statically stable up to the mass peak and satisfy physical criteria. This work extends previous isotropic boson-star analyses by systematically incorporating anisotropy within a regularized 4D EGB framework. These findings provide observationally relevant predictions for compact dark-matter objects under modified gravity.