Symmetry–Protected Momentum Exchange between Dark Matter and Dark Energy

We present a particle physics motivated realization of interacting dark energy in which a radiatively stable dark energy sector couples to weakly interacting massive particle dark matter through pure momentum exchange. The dark energy field arises as a pseudo–Nambu–Goldstone boson from a complex scalar singlet charged under a softly broken global U(1)ₛ, while dark matter is identified with an inert scalar doublet stabilized by a discrete Z₄ symmetry. This symmetry structure allows renormalizable dark matter–dark energy portal operators; however, requiring the dark energy field to emerge as a radiatively stable pseudo–Nambu–Goldstone boson necessitates their absence, leaving derivative interactions as the leading coupling. As a result, energy transfer between the dark sectors is absent at the background level, while momentum exchange modifies the evolution of cosmological perturbations. We implement the resulting interacting dark energy model self–consistently in the Boltzmann code CLASS and study its impact on the growth of structure. We find that, despite sizeable momentum exchange, the suppression of the clustering amplitude σ₈ saturates above the level required to fully resolve current low–redshift tensions. Our results demonstrate that symmetry protected, momentum–exchange–only dark sector interactions possess an intrinsic limit on structure suppression, providing a theoretically controlled benchmark for interacting dark energy scenarios.