Optical Landscapes and High-Energy Collisions in Hairy Horndeski Gravity
We investigate the null geodesic structure, photon sphere dynamics, and high-energy particle collisions within a class of static, spherically symmetric hairy Horndeski black holes. Characterized by an invariant metric root at $r = 2M$ and a scalar hair parameter $Q$, the spacetime maps onto four distinct geometric domains dictated by the surface gravity $κ|_{2M}$. We derive exact analytical expressions for the photon sphere radius $r_{\text{ph}}$ and its dynamic stability criterion, showing that external circular null orbits remain dynamically unstable across non-extremal regimes. Crucially, we prove that a stable photon sphere arises exclusively in the extremal configuration ($Q = -2M$), where it coincides precisely with the degenerate horizon ($r = 2M$). We argue that this horizon-bound stable photon orbit acts as an infinitely redshifted bound state for light and ultra-relativistic particles. Finally, we analyze the Bañados-Silk-West (BSW) effect for infalling timelike test particles, demonstrating that the center-of-mass energy $E_{\text{cm}}$ for critical collisions diverges as $E_{\text{cm}} \propto (r - r_h)^{-1/2}$ strictly at the extremal threshold $Q = -2M$.