Serendipitous supersymmetric solution to the strong CP problem
The Minimal Supersymmetric Standard Model (MSSM) has several problems: 1. its $\mu$ term must be forbidden, then regenerated at the weak scale, 2. it allows for $R$-parity violating superpotential terms which lead to rapid proton decay, 3. it allows for dimension-5 proton decay operators. The usual imposition of $R$- or matter parity $P_M$ solves only the second of these, whereas anomaly-free discrete $\mathbb{Z}_n^R$ symmetries (consistent with grand unification) address all of them. Once the $\mu$-term is forbidden by the imposition of a discrete $\mathbb{Z}_n^R$ symmetry (which can emerge as a discrete remnant of string compactifications to 4-dimensions), the MSSM develops an accidental global $U(1)_{PQ}$ symmetry (thus providing a plausible origin for the global $U(1)_{PQ}$ needed for solving the strong CP problem). By coupling the Higgs fields to PQ-charged gauge singlet fields $X,\ Y$ (in the Kim-Nilles mechanism), and imposing SUSY breaking, one regenerates $\mu$ at the weak scale whilst breaking the discrete $\mathbb{Z}_n^R$ and the $U(1)_{PQ}$. The broken global $U(1)_{PQ}$ develops a pseudo-Goldstone boson, the DFSZ axion, thus (perhaps inadvertently) solving the strong CP problem. In this setting, SUSY develops a dark matter candidate, the SUSY DFSZ axion, and possibly, though not necessarily, a WIMP dark matter candidate as well, depending on the order of the induced $R$-parity violating operators.