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Qualities of axion and LSP in Pati-Salam unification with $Z^R_{4}\times Z_{N}$ symmetry

In this paper we construct supersymmetric Pati-Salam (PS) models containing the minimal supersymmetric standard model and an invisible axion. The models include two discrete symmetries, $\mathbb{Z}_4^R \times \mathbb{Z}_N$, which maintain the $quality$ of the accidental Peccei-Quinn (PQ) symmetry and thus the solution to the strong CP problem. We require that the discrete anomaly conditions are satisfied for both $\mathbb{Z}_4^R \times G_{\rm PS}^2$ and $ \mathbb{Z}_N \times G_{\rm PS}^2$. The vacuum expectation value of the PQ field spontaneously breaks all the discrete symmetries. R-parity is violated if any of the PQ field(s) has an odd charge under $\mathbb{Z}_4^R$. We present two explicit models which we refer to as a minimal model where R-parity violation is extremely suppressed, and a non-minimal model where R-parity violation is significant. In the latter model, the neutralino becomes unstable even if it is the Lightest Supersymmetric Particle (LSP), and, in addition, there are new low-energy vector-like states. In both examples, R-parity violation is sufficiently suppressed such that the proton is stable.