Physical features of light properties around the hairy black hole SVT solutions in U (1) gauge-invariant

We investigate weak-field gravitational lensing and horizon-scale optical properties of a spherically symmetric hairy black hole in the [Formula: see text] gauge-invariant scalar-vector-tensor (SVT) theory with cubic scalar-vector coupling. Within the optical geometry framework, we employ the Gibbons–Werner Gauss–Bonnet method to derive the Gaussian optical curvature and the corresponding weak deflection angle, explicitly identifying the contributions of the black-hole mass [Formula: see text], charge [Formula: see text], and cubic coupling parameter [Formula: see text]. We then extend the analysis to dispersive media by considering both a cold, non-magnetized plasma with gravitationally redshifted photon frequency and an effective phenomenological dark-matter-induced refractive index, and we determine how these environments modify the bending angle relative to vacuum propagation. In addition, using a Hamiltonian description of photon motion, we study the photon sphere and the shadow radius in vacuum and plasma backgrounds. We further estimate the frequency-dependent energy-emission rate through the geometric-optics absorption cross section and clarify its dependence on the shadow radius and Hawking temperature. In the appropriate limits, our results consistently recover the Schwarzschild and Reissner–Nordström cases. Our analysis shows that both the cubic SVT interaction and dispersive media produce systematic, model-dependent corrections to the deflection angle, photon-sphere radius, shadow size, and emission profile, which may be relevant for future lensing and horizon-scale observations.