Tension between neutrino masses and gauge coupling unification in natural grand unified theories

The natural grand unified theories (GUT) solve various problems of the supersymmetric GUT and give realistic quark and lepton mass matrices under the natural assumption that all terms allowed by the symmetry are introduced with $O\left(1\right)$ coefficients. However, because of the natural assumption, it is difficult to achieve the gauge coupling unification without tuning, while keeping neutrino masses at realistic values. In this paper, we try to avoid this tension between the neutrino masses and the gauge coupling unification, by introducing suppression factors for several terms. These suppression factors can be understood by approximate symmetries in some of the solutions. We show that one of the most important results in the natural GUT scenario, that the nucleon decay mediated by superheavy gauge fields is enhanced due to a smaller unification scale while the nucleon decay mediated by superheavy colored Higgs is suppressed, may change in those models proposed in this paper.