A Composite Axion from a Supersymmetric Product Group

A global $U(1{)}_{\text{PQ}}$ symmetry is protected from gravitational effects in the s-confining $SU(N{)}^k$ product group theory with $A+4Q+N\bar{Q}$ matter. If the $SU(4)$ family symmetry is gauged and an appropriate tree-level superpotential is added, then the dynamically generated superpotential spontaneously breaks $SU(4)\times U(1{)}_{\text{PQ}}\to SU(3{)}_c$ and produces a QCD axion. Small values of the $CP$-violating $\theta$ parameter are then possible without any fine-tuning, as long as the product group is suitably large. By introducing a second copy of the s-confining $SU(N)$ product group also coupled to the gauged $SU(4)$, we find that values as small as $N=7$ are consistent with $\bar{\theta }<{10}^{-10}$, even under the pessimistic assumption that the dominant contribution to the axion quality is at tree level.