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 $\sigma =\beta p_r(1-e^{-2\lambda })$. We perform a comprehensive numerical analysis across the $(\alpha ,\beta )$ parameter domain, where $\alpha \in [0,8]{\mathrm{k}\mathrm{m}}^2$ and $\beta \in [-2,0]$, to examine mass$-$radius relations and evaluate multiple stability indicators including static equilibrium $dM\mathrm{/}d{p}_c$, sound-speed causality, the radial adiabatic index ${\mathrm{\Gamma }}_r$, and energy conditions. Positive Gauss$-$Bonnet coupling enhances both the maximum mass and compactness (e.g., $M_{\mathrm{m}\mathrm{a}\mathrm{x}}\approx 1.62M_{\odot }$ at $\alpha =0$ rising to $\approx 2.09M_{\odot }$ at $\alpha =8{\mathrm{k}\mathrm{m}}^2$), while negative anisotropy reduces them (e.g., from $\approx 2.21M_{\odot }$ at $\beta =0$ to $\approx 1.73M_{\odot }$ 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.