Dark matter could emerge along with the Higgs as a composite pseudo-Nambu-Goldstone boson $chi$ with decay constant $fsim mathrm{TeV}$. This type of WIMP is especially compelling because its leading interaction with the Standard Model, the derivative Higgs portal, has the correct annihilation strength for thermal freeze-out if $m_chi sim O(100)$ GeV, but is negligible in direct detection experiments due to the very small momentum transfer. The explicit breaking of the shift symmetry which radiatively generates $m_chi$, however, introduces non-derivative DM interactions. In existing realizations a marginal Higgs portal coupling $lambda$ is generated with size comparable to the Higgs quartic, and thus well within reach of XENON1T. Here, we present and analyze the interesting case where the pattern of explicit symmetry breaking naturally suppresses $lambda$ beyond the reach of current and future direct detection experiments. If the DM acquires mass from bottom quark loops, the bottom quark also mediates suppressed DM-nucleus scattering with cross sections that will be eventually probed by LZ. Alternatively, the DM can obtain mass from gauging its stabilizing $U(1)$ symmetry. No direct detection signal is expected even at future facilities, but the introduction of a dark photon $gamma_D$ has a number of phenomenological implications which we study in detail, treating $m_{gamma_D}$ as a free parameter. Complementary probes of the dark sector include indirect DM detection, DM self-interactions, and extra radiation, as well as collider experiments. We frame our discussion in an effective field theory, motivating our parameter choices with a detailed analysis of an $SO(7)/SO(6)$ composite Higgs model, which can yield either scenario at low energies.
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