Flipped SU(5): unification, proton decay, fermion masses and gravitational waves

We study supersymmetric (SUSY) flipped $SU(5)\times U(1)$ unification, focussing on its predictions for proton decay, fermion masses and gravitational waves. We performed a two-loop renormalisation group analysis and showed that the SUSY flipped $SU(5)$ model predicts a high GUT scale $M_{\rm GUT} > 10^{16}$ GeV. We also investigated the restrictions on the $M_{B-L}$ scale which is associated with the $U(1)_{\chi}$ breaking scale. We found that the $M_{B-L}$ scale can vary in a broad region with negligible or little effect on the value of $M_{\rm GUT}$. Proton decay in this model is induced by dimension-6 operators only. The dimension-5 operator induced by SUSY contribution is suppressed due to the missing partner mechanism. We found that the partial decay width $p \to \pi^0 e^+$ is high suppressed, being at least one order of magnitude lower than the future Hyper-K sensitivity. We also studied fermion (including neutrino) masses and mixings which can also influence proton decay. We presented two scenarios of flavour textures to check the consistency of the results with fermion masses and mixing. The $B-L$ gauge breaking leads to the generation of cosmic strings. The $B-L$ scale here is not constrained by gauge coupling unification. If this scale is very close that of GUT breaking, strings can be unstable due to the decay to monopole-antimonople pair. Such metastable strings can be used to explain the NANOGrav signals of stochastic gravitational wave background, which may be interpreted here as resulting from the decay of metastable cosmic strings.