Factoring the Strong CP Problem

We present a mechanism to solve the strong CP problem by embedding QCD in a SU(3)^N product gauge group at high energies. The simplest version of this mechanism realizes a heavy QCD axion by embedding QCD in the diagonal subgroup of a SU(3)xSU(3) gauge group at a scale M~10^14 GeV. The vacuum is naturally aligned to ensure θ=0 at low energies, and the axions in the theory have masses much larger than for the standard QCD axion. Unlike other heavy QCD axion models, there are no discrete Z2 symmetries or mirror copies of SM particles. Another realization of this mechanism can revive the massless up-quark solution to the strong CP problem with a flavorful embedding of the SM fermions in a SU(3)xSU(3)xSU(3) gauge group at a scale M~100 TeV.