$$\Sigma(0), \qquad H\bar{H}(0).
\label{eq:R}$$ When supersymmetry is broken and the cosmological constant is fine-tuned, the Kähler potential interactions of Eqs. (\[eq:K-Sigma\], \[eq:K-H\]) yield the effective superpotential through supergravity effects [@Hall:1983iz]:[^3] $$W_{\rm eff} = \frac{m_\Sigma}{2} \Sigma^2 
    + \frac{\lambda_\Sigma}{3} \Sigma^3 + \frac{m_H}{2} H \bar{H} 
    + \lambda_H H \Sigma \bar{H},
\label{eq:W_eff}$$ where $m_\Sigma = c_2 m_{3/2}^*$, $\lambda_\Sigma = c_3 m_{3/2}^*/\Lambda$, $m_H = d_2 m_{3/2}^*$, and $\lambda_H = d_3 m_{3/2}^*/\Lambda$, so that $$m_\Sigma,\, m_H \sim O(\tilde{m}),
\qquad
  \lambda_\Sigma,\, \lambda_H \sim O\Bigl(\frac{\tilde{m}}{\Lambda}\Bigr).
\label{eq:orders}$$ Here, we have used $m_{3/2} \sim \tilde{m}$.[^4] The first two terms of Eq. (\[eq:W\_eff\]) provide a non-zero VEV of $\Sigma$, $\langle \Sigma \rangle \sim m_\Sigma/\lambda_\Sigma \sim 
O(\Lambda)$, breaking $SU(5)$ to the SM gauge group. In general, supersymmetry breaking effects in the $\Sigma$ potential lead to an $O(1)$ shift of the $\Sigma$ VEV,[^5] giving a VEV for the $F$-component of $\Sigma$ of order $\tilde{m} \Lambda$: $$\langle \Sigma \rangle \sim O(\Lambda),
\qquad
  F_\Sigma \sim O(\tilde{m} \Lambda).
\label{eq:Sigma-VEVs}$$ We take parameters of the model such that $\langle \Sigma \rangle$ is parametrically, e.g. a factor of a few to an order of magnitude, smaller than $\Lambda$, to ensure that there is an energy interval below $\Lambda$ in which physics is described by the $SU(5)$ theory.

Below 