SU(5)×U(1)X axion model with observable proton decay

We propose a $SU(5)\ifmmode\times\else\texttimes\fi{}U(1{)}_{X}\ifmmode\times\else\texttimes\fi{}U(1{)}_{\mathrm{PQ}}$ model, where $U(1{)}_{X}$ is the generalization of the $B\ensuremath{-}L$ (baryon minus lepton number) gauge symmetry and $U(1{)}_{\mathrm{PQ}}$ is the global Peccei-Quinn (PQ) symmetry. There are four fermions families in $\overline{\mathbf{5}}+\mathbf{10}$ representations of $SU(5)$, a mirror family in $\mathbf{5}+\overline{\mathbf{10}}$ representations, and three $SU(5)$ singlet Majorana fermions. The $U(1{)}_{X}$ related anomalies all cancel in the presence of the Majorana neutrinos. The $SU(5)$ symmetry is broken at ${M}_{\mathrm{GUT}}\ensuremath{\simeq}(6--9)\ifmmode\times\else\texttimes\fi{}{10}^{15}\text{ }\text{ }\mathrm{GeV}$ and the proton lifetime ${\ensuremath{\tau}}_{p}$ is estimated to be well within the expected sensitivity of the future hyper-Kamiokande experiment, ${\ensuremath{\tau}}_{p}\ensuremath{\lesssim}1.3\ifmmode\times\else\texttimes\fi{}{10}^{35}\text{ }\text{ }\mathrm{years}$. The $SU(5)$ breaking also triggers the breaking of the PQ symmetry, resulting in axion dark matter (DM), with the axion decay constant ${f}_{a}$ of order ${M}_{\mathrm{GUT}}$ or somewhat larger. The CASPEr experiment can search for such an axion DM candidate. The Hubble parameter during inflation must be low, ${H}_{\mathrm{inf}}\ensuremath{\lesssim}{10}^{9}\text{ }\text{ }\mathrm{GeV}$, in order to successfully resolve the axion domain wall, axion DM isocurvature and $SU(5)$ monopole problems. With the identification of the $U(1{)}_{X}$ breaking Higgs field with the inflaton field, we implement inflection-point inflation, which is capable of realizing the desired value for ${H}_{\mathrm{inf}}$. The vectorlike fermions in the model are essential for achieving successful unification of the SM gauge couplings as well as the phenomenological viability of both axion DM and inflation scenario.