The Impossible Triangle: A No-Go for Symmetry-Protected Scalar Portals in Interacting Dark Energy

Abstract

The $S_8$ tension motivates interacting dark energy (IDE), but embedding IDE in UV-complete physics faces severe naturalness challenges. We analyze four symmetry-protected DM--DE portals ( quartic ($\tfrac{1}{2}\lambda\phi^2\chi^2$), trilinear ($g\phi\chi^2$), derivative ($(c_6/\Lambda^2)(\partial_\mu\phi)^2\chi^2$), and fermionic Yukawa ($y\phi\bar{\psi}\psi$) )within a $Z_2$-symmetric Inert Doublet + Singlet Model. The trilinear portal requires $\beta \sim 0.45$ ($g \sim 10^{-16}\,\mathrm{GeV}$), overshooting the radiative bound $g \lesssim 10^{-42}\,\mathrm{GeV}$ by $\sim 26$ orders (tuning $\Delta \sim 10^{52}$). The quartic portal needs $\lambda \sim \mathcal{O}(1\text{--}10)$ versus $\lambda \lesssim 10^{-86}$ ($\Delta \sim 10^{87}$). The derivative portal saturates dynamically at $\lesssim 4\%$ suppression. The Yukawa portal yields $\Delta \sim 10^{52}$, persisting even with SUSY cancellation. No single-mediator model simultaneously satisfies technical naturalness and resolves the $S_8$ tension. Viable solutions require either multi-field tuned cancellations or explicit symmetry breaking with quantified fine-tuning.