minimal supersymmetric $SU(5)$ GUT. The matter content of the model consists of the $\Sigma({\bf 24})$, $H({\bf 5})$, and $\bar{H}({\bf 5}^*)$ Higgs fields as well as three generations of the matter fields $T({\bf 10})$ and $\bar{F}({\bf 5}^*)$, where we have suppressed the generation indices. (Right-handed neutrino superfields $N({\bf 1})$’s will be introduced in Section \[sec:scales\] when we discuss neutrino masses.)

As in Ref. [@Hall:2013eko], we consider that the potential for the GUT-breaking field $\Sigma$ arises from the Kähler potential: $$K \supset \frac{c_2}{2} \Sigma^2 + \frac{c_3}{3\Lambda} \Sigma^3 
    + {\rm h.c.},
\label{eq:K-Sigma}$$ where $c_{2,3}$ are dimensionless couplings of order unity, while $\Lambda 
\sim M_{\rm Pl}$ is the UV cutoff of the unified theory. Similarly, here we also consider that the potential associated with the $H, \bar{H}$ Higgs fields arises from the Kähler potential terms $$K \supset d_2 H \bar{H} + \frac{d_3}{\Lambda} H \Sigma \bar{H} 
    + {\rm h.c.},
\label{eq:K-H}$$ where $d_{2,3}$ are dimensionless couplings of order unity. We assume that there is no interaction in the superpotential corresponding to the terms in Eqs. (\[eq:K-Sigma\], \[eq:K-H\]) at the fundamental level (i.e. in the supersymmetric flat-space limit). This can be achieved, e.g., if the theory possesses an $R$ symmetry under which $\Sigma$ and $H\bar{H}$ are neutral: 