Optical Landscapes and High-Energy Collisions in Hairy Horndeski Gravity

Abstract

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 $\hat{I}º{|}_{2M}$. 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 $Q=-2M$.