Theta in new QCD-like sectors

New QCD-like `hypercolor' sectors can generate a broad class of new signatures at hadron colliders and furnish a variety of dark matter candidates. Paired diboson resonances are a particularly important collider signature, arising both from $CP$-conserving vector hypermeson decays of the form $\widetilde\rho \rightarrow\widetilde\pi\widetilde\pi\rightarrow 4V$ and from $CP$-violating pseudoscalar hypermeson decays of the form $\widetilde\eta\rightarrow \widetilde\pi\widetilde\pi\rightarrow 4V$. The latter are sensitive to the vacuum angle $\widetilde \theta$ in the hypercolor sector. We study single- and paired-diboson resonance signatures in final states involving gluons and photons at the LHC and a future 100 TeV $pp$ collider, illustrating the discovery potential at both colliders in simple benchmark models. We also describe some of the theoretical and cosmological consequences of $\widetilde \theta$. If $CP$-violating hypermeson decays are observable at hadron colliders, ordinary QCD must have an axion. Such scenarios also provide a natural setting for a dark pion component of dark matter, with its relic abundance set by $CP$-violating annihilations. If the new vacuum angle is relaxed to zero by a dark axion, the relic density can instead be a mixture of axions and dark axions. Overproduction of dark axions is most easily avoided if the universe underwent a period of early matter domination.