inflaton field $ S$ and the others are waterfall fields ${\Psi}$ and $\bar{\Psi}$. This model is based on $ U(1)_R$ symmetry. The superpotential is given by $$W_H = S\left(-{\mu}^2+\frac{(\bar{\Psi}\Psi)^2}{M^2}\right),$$ where $ S$ has $R$ charge $ 2$ and $(\bar{\Psi}\Psi)$ has $R$ charge $0$ and $M$ is a cut-off scale. The $R$-invariant Kähler potential is given by $$K_H = |S|^2 + |\Psi|^2 + |\bar{\Psi}|^2.$$ We also introduce the $U(1)$ gauge symmetry under which the waterfall fields are transformed as $ \Psi\rightarrow e^{i\delta}\Psi$ and $\bar{\Psi}\rightarrow e^{-i\delta}\bar{\Psi}$, and $Z_2$ symmetry under which $\left(\Psi\bar{\Psi}\right)$ has a unit charge. Using the phase rotation and D-term flat condition, we can bring the complex scalar fields $\Psi$ and $\bar{\Psi}$ on the real axis as $\psi\equiv2{\rm Re}\Psi=2{\rm Re}\bar{\Psi}$.

From the superpotential and the Kähler potential, we can write down the scalar potential. Using the $R$ symmetry, we can bring the complex scalar field $ S$ on the real axis, $ {\sigma} \equiv \sqrt{2}ReS$. Neglecting higher-order terms, we can write down the scalar potential for $\sigma<1$ $$V_H \simeq 
\left(-{ \mu}^2+\frac{{\psi}^4}{16M^2}\right)^2\left(1+\frac{{\psi}^2}{2} 
    +\frac{{\sigma}^4}{8}\right)+
\frac{{\sigma}^2{\psi}^6}{16M^4}-
{\mu}^2\frac{{\sigma}^2{\psi}^4}{4M^2},$$ where $ \sigma$ is the inflaton and $ \psi$ is the waterfall field. During inflation, $ \psi$ is located at the local minimum of the potential, 