No-Go Theorem for Phantom Crossing in Scalar-Tensor Gravity with Non-Minimal Derivative Coupling and Soft Scale-Invariance Breaking

Recent baryon acoustic oscillation measurements from the Dark Energy Spectroscopic Instrument (DESI 2024) reveal a preference for dynamical dark energy with an equation of state parameter crossing the phantom divide ($w<-1$) at low redshifts. We investigate whether this phantom crossing can be realized within scale-invariant scalar-tensor theories featuring non-minimal derivative coupling (NMDC) between the scalar field and the Einstein tensor. Employing the effective field theory of dark energy formalism, we systematically derive the ghost-free and gradient stability conditions, and examine their compatibility with the swampland conjectures from quantum gravity. We establish three rigorous no-go theorems demonstrating that: (i)~phantom crossing necessitates ${\xi }_{\text{eff}}<0$, directly violating the no-ghost stability requirement ${\xi }_{\text{eff}}>0$; (ii)~the swampland de Sitter and distance conjectures independently obstruct phantom crossing; and (iii)~these theoretical obstructions are mutually reinforcing. Our results show that scale-invariant NMDC models cannot account for the DESI preference for $w<-1$, providing robust theoretical guidance for dark energy model building beyond this class.