  We propose a new realization of strongly interacting massive particles (SIMP)
as self-interacting dark matter, where SIMPs couple to the Standard Model
sector through an axion-like particle. Our model gets over major obstacles
accompanying the original SIMP model, such as a missing mechanism of
kinetically equilibrating SIMPs with the SM plasma as well as marginal
perturbativity of the chiral Lagrangian density. Remarkably, the parameter
region realizing $\sigma_{\rm self}/m_{\rm DM} \simeq 0.1 \textrm{--} 1 \, {\rm
cm}^{2}/{\rm g}$ is within the reach of future beam dump experiments such as
the Search for Hidden Particles (SHiP) experiment.

High Energy Physics - Phenomenology;Cosmology and Nongalactic Astrophysics