We propose an $SU(5)$ grand unified model with an invisible axion and the unification of the three coupling constants which is in agreement with the values, at $M_Z$, of $\alpha$, ${\alpha }_s$, and ${sin}^2{\theta }_W$. A discrete, anomalous, $Z_{13}$ symmetry implies that the Peccei-Quinn symmetry is an automatic symmetry of the classical Lagrangian protecting, at the same time, the invisible axion against possible semiclassical gravity effects. Although the unification scale is of the order of the Peccei-Quinn scale the proton is stabilized by the fact that in this model the standard model fields form the $SU(5)$ multiplets completed by new exotic fields and, also, because it is protected by the $Z_{13}$ symmetry.

• Received 3 September 2007

DOI:https://doi.org/10.1103/PhysRevD.76.115010