\mathcal{C}_{\bar{W}}&=& \mathcal{C}_{W}^{*} = \Bar{\Lambda}_K P_R \Lambda_K,\\
\mathcal{Z}_{\bar{W}}&=&\mathcal{Z}_{W}^C = i2 P_R (-\frac{1}{2}\gamma\cdot \hat{F}_K -i\mathcal{D}_K)\Lambda_K,\\
\mathcal{H}_{\bar{W}}&=&\mathcal{K}_{W}^{*} = 0,\\
\mathcal{K}_{\bar{W}}&=&\mathcal{H}_{W}^{*} = -2(2\Bar{\Lambda}_KP_R \cancel{\mathcal{D}}\Lambda_K + \hat{F}_K^+\cdot \hat{F}_K^+ - \mathcal{D}^2_K) \equiv -2\bar{F}^{\bar{W}},\\
\mathcal{B}^{\bar{Q}}_{\mu}&=&(\mathcal{B}^{W}_{\mu})^{*} = -i\mathcal{D}_{\mu}(\Bar{\Lambda}_K P_R \Lambda_K),\\
\Lambda_{\bar{W}}&=&\Lambda_{W}^C = 0,\\
\mathcal{D}_{\bar{W}}&=&\mathcal{D}_{W}^{*} = 0.\label{Multiplet_W2}
\end{eqnarray}

We also defined:

\begin{eqnarray}
&& \hat{F}_{ab}^K \equiv e_a^{\mu}e_b^{\nu} (2\partial_{[\mu }\mathcal{B}^K_{\nu]} + \bar{\psi}_{[\mu}\gamma_{\nu]}\Lambda_K),\\
&& \Tilde{\hat{F}}_{ab} \equiv -i\frac{1}{2} \varepsilon_{abcd}\hat{F}^{cd}, \qquad \hat{F}_{ab}^{\pm} \equiv \frac{1}{2} (\hat{F}_{ab} \pm \tilde{\hat{F}}_{ab}),\qquad   (\hat{F}_{ab}^{\pm})^{*}= \hat{F}_{ab}^{\mp}\\
&& \cancel{\mathcal{D}}\Lambda_K \equiv \gamma \cdot \mathcal{D} \Lambda_K,\\
&& \mathcal{D}_{\mu} \Lambda_K \equiv \Big(\partial_{\mu} -\frac{3}{2}b_{\mu} + \frac{1}{4}\omega_{\mu}^{ab} \gamma_{ab} -\frac{3}{2}i\gamma_* \mathcal{A}_{\mu} \Big) \Lambda_K - \Big(\frac{1}{4}\gamma^{ab} \hat{F}_{ab}^K + \frac{1}{2}i\gamma_* \mathcal{D}_K \Big)\psi_{\mu}.
\end{eqnarray}

Next, we shall consider a gauge fixing that is equivalent to the gauge condition given by $\eta =0$ in Ref. <cit.>. In fact, this can be obtained by imposing $\Lambda_K =0$. We will call this gauge the “Liberated SUGRA gauge”. Then, the only non-vanishing superconformal components of the multiplets $\mathcal{W}^2(K),\bar{\mathcal{W}}^2(K)$ in this gauge are given by  $\mathcal{H}_{W} = -2(\hat{F}_K^-\cdot\hat{F}_K^- -\mathcal{D}_K^2)$ and $\mathcal{K}_{\bar{W}} = -2(\hat{F}_K^+\cdot\hat{F}_K^+ -\mathcal{D}_K^2) = \mathcal{H}_{W}^{*}$.

Another representation of the chiral multiplet is

\begin{eqnarray}
\mathcal{W}^2(K) &=& \left(\mathcal{C}_{W}, \frac{i}{\sqrt{2}}\mathcal{Z}_{W},-\frac{1}{2}\mathcal{H}_{W}\right) \nonumber\\
&=& \left( \Bar{\Lambda}_K P_L \Lambda_K,~ \sqrt{2} P_L (-\gamma\cdot \hat{F}_K + 2i\mathcal{D}_K)\Lambda_K,~ 2\Bar{\Lambda}_KP_L \cancel{\mathcal{D}}\Lambda_K + \hat{F}_K^-\cdot \hat{F}_K^- - \mathcal{D}^2_K\right) \nonumber\\
&\equiv& \left( X^W, P_L\chi^W, F^W \right),\\
\bar{\mathcal{W}}^2(K) &=& \left(\mathcal{C}_{W}^{*}, -\frac{i}{\sqrt{2}}\mathcal{Z}_{W}^C,-\frac{1}{2}\mathcal{H}_{W}^{*}\right) \nonumber\\
&=& \left( \Bar{\Lambda}_K P_R \Lambda_K,~ \sqrt{2} P_R (-\gamma\cdot \hat{F}_K - 2i\mathcal{D}_K^{*})\Lambda_K^C,~ 2\Bar{\Lambda}_KP_R \cancel{\mathcal{D}}\Lambda_K + \hat{F}_K^+ \cdot \hat{F}_K^+ - (\mathcal{D}^{*}_K)^2\right) \nonumber\\
&\equiv& \left( \bar{X}^{\bar{W}}, P_R\chi^{\bar{W}}, \bar{F}^{\bar{W}} \right)
\end{eqnarray}

We will also need the following definitions:

\begin{eqnarray}
X^W = \bar{\Lambda}_K P_L \Lambda_K \equiv W &=&  -2K_{\bar{I}J}[\bar{\chi}^{J}(\overline{\cancel{\mathcal{D}}z^{\bar{I}}})-\bar{F}^{\bar{I}}\bar{\chi}^{J}]K_{{\bar{I}}'J'}[(\cancel{\mathcal{D}}z^{{\bar{I}}'})\chi^{J'}-F^{J'}\chi^{{\bar{I}}'}] \nonumber\\
&& - K_{{\bar{I}}J}[\bar{\chi}^{J}(\overline{\cancel{\mathcal{D}}z^{\bar{I}}})-\bar{F}^{\bar{I}}\bar{\chi}^{J}] K_{\bar{I}'\bar{J}'K'}[\chi^{K'}\bar{\chi}^{\bar{I}'}\chi^{\bar{J}'}] \nonumber\\
&& - K_{\bar{I}\bar{J}K}[\bar{\chi}^{\bar{J}}\chi^{\bar{I}}\bar{\chi}^{K}]K_{{\bar{I}}'J'}[(\cancel{\mathcal{D}}z^{{\bar{I}}'})\chi^{J'}-F^{J'}\chi^{{\bar{I}}'}] \nonumber\\
&& - \frac{1}{2}K_{\bar{I}\bar{J}K}[\bar{\chi}^{\bar{J}}\chi^{\bar{I}}\bar{\chi}^{K}]K_{\bar{I}'\bar{J}'K'}[\chi^{K'}\bar{\chi}^{\bar{I}'}\chi^{\bar{J}'}],\\
\bar{X}^{\bar{W}} =  \bar{\Lambda}_K P_R \Lambda_K \equiv \bar{W} &=& (\bar{\Lambda}_K P_L \Lambda_K)^C,
\end{eqnarray}

\begin{eqnarray}
\mathcal{D}_KP_L\Lambda_K &=& \bigg[2K_{I\bar{J}}
\bigg( -\mathcal{D}_{\mu} z^I \mathcal{D}^{\mu} \bar{z}^{\bar{J}} -\frac{1}{2} \bar{\chi}^I P_L \cancel{\mathcal{D}}\chi^{\bar{J}} -\frac{1}{2} \bar{\chi}^{\bar{J}} P_R \cancel{\mathcal{D}} \chi^I + F^I\bar{F}^{\bar{J}}\bigg) \nonumber\\
&&+ K_{IJ\bar{K}} \bigg( -\bar{\chi}^I \chi^J \bar{F}^{\bar{K}} + \bar{\chi}^I (\cancel{\mathcal{D}}z^J)\chi^{\bar{K}}         \bigg) + K_{\bar{I}\bar{J}K} \bigg( -\bar{\chi}^{\bar{I}} \chi^{\bar{J}} F^{K} + \bar{\chi}^{\bar{I}} (\cancel{\mathcal{D}}\bar{z}^{\bar{J}})\chi^{K}\bigg) \nonumber\\
&& + \frac{1}{2} K_{IJ\bar{K}\bar{L}} (\bar{\chi}^IP_L\chi^J )(\bar{\chi}^{\bar{K}} P_R \chi^{\bar{L}})\bigg]
\nonumber\\&&\times
\bigg[-\sqrt{2}iK_{\bar{I}'J'}[(\cancel{\mathcal{D}}z^{J'})\chi^{\bar{I}'}-\bar{F}^{\bar{I}'}\chi^{J'}] -\frac{i}{\sqrt{2}}K_{\bar{I}'\bar{J}'K'}\chi^{K'}\bar{\chi}^{\bar{I}'}\chi^{J'}\bigg].
\end{eqnarray}

\begin{eqnarray}
\mathcal{D}_K^2 &=& \bigg[2K_{I\bar{J}}
\bigg( -\mathcal{D}_{\mu} z^I \mathcal{D}^{\mu} \bar{z}^{\bar{J}} -\frac{1}{2} \bar{\chi}^I P_L \cancel{\mathcal{D}}\chi^{\bar{J}} -\frac{1}{2} \bar{\chi}^{\bar{J}} P_R \cancel{\mathcal{D}} \chi^I + F^I\bar{F}^{\bar{J}}\bigg) \nonumber\\
&&+ K_{IJ\bar{K}} \bigg( -\bar{\chi}^I \chi^J \bar{F}^{\bar{K}} + \bar{\chi}^I (\cancel{\mathcal{D}}z^J)\chi^{\bar{K}}         \bigg) + K_{\bar{I}\bar{J}K} \bigg( -\bar{\chi}^{\bar{I}} \chi^{\bar{J}} F^{K} + \bar{\chi}^{\bar{I}} (\cancel{\mathcal{D}}\bar{z}^{\bar{J}})\chi^{K}\bigg) \nonumber\\
&& + \frac{1}{2} K_{IJ\bar{K}\bar{L}} (\bar{\chi}^IP_L\chi^J )(\bar{\chi}^{\bar{K}} P_R \chi^{\bar{L}})\bigg]^2.
\end{eqnarray}

$w^2,\bar{w}^2$ Composite Complex Multiplets: (Weyl/Chiral) weights $=(-1,\pm 3)$

\begin{eqnarray}
&&  w^2 \equiv \dfrac{\mathcal{W}^2(K)}{\Upsilon^2}
= \{\mathcal{C}_w,\mathcal{Z}_w,\mathcal{H}_w,\mathcal{K}_w,\mathcal{B}^w_{\mu},\Lambda_w,\mathcal{D}_w\},\\
&&   \bar{w}^2 \equiv \dfrac{\bar{\mathcal{W}}^2(K)}{\Upsilon^2}
= \{\mathcal{C}_{\bar{w}},\mathcal{Z}_{\bar{w}},\mathcal{H}_{\bar{w}},\mathcal{K}_{\bar{w}},\mathcal{B}^{\bar{w}}_{\mu},\Lambda_{\bar{w}},\mathcal{D}_{\bar{w}}\}
\end{eqnarray}

\begin{eqnarray}
\mathcal{C}_{w} &=& \frac{\mathcal{C}_{W}}{\Upsilon^2},\\
\mathcal{Z}_{w} &=& \frac{1}{\Upsilon^2}\mathcal{Z}_{W} -2\frac{\mathcal{C}_{W}}{\Upsilon^3}\mathcal{Z}_{\Upsilon},\\
\mathcal{H}_{w} &=&  \frac{1}{\Upsilon^2}\mathcal{H}_{W} -2\frac{\mathcal{C}_{W}}{\Upsilon^3}\mathcal{H}_{\Upsilon} -\frac{1}{2}\bigg[ -4\frac{1}{\Upsilon^3} (\bar{\mathcal{Z}}_{W}P_L\mathcal{Z}_{\Upsilon}+\bar{\mathcal{Z}}_{\Upsilon}P_L\mathcal{Z}_{W})+6\frac{\mathcal{C}_{W}}{\Upsilon^4}\bar{\mathcal{Z}}_{\Upsilon}P_L\mathcal{Z}_{\Upsilon}\bigg],\\
\mathcal{K}_{w} &=&  \frac{1}{\Upsilon^2}\mathcal{K}_{W} -2\frac{\mathcal{C}_{W}}{\Upsilon^3}\mathcal{K}_{\Upsilon} -\frac{1}{2}\bigg[ -4\frac{1}{\Upsilon^3} (\bar{\mathcal{Z}}_{W}P_R\mathcal{Z}_{\Upsilon}+\bar{\mathcal{Z}}_{\Upsilon}P_R\mathcal{Z}_{W})+6\frac{\mathcal{C}_{W}}{\Upsilon^4}\bar{\mathcal{Z}}_{\Upsilon}P_R\mathcal{Z}_{\Upsilon}\bigg],\\
\mathcal{B}^{w}_{\mu} &=& \frac{1}{\Upsilon^2}\mathcal{B}^{W}_{\mu} -2\frac{\mathcal{C}_{W}}{\Upsilon^3}\mathcal{B}^{\Upsilon}_{\mu} + \frac{1}{2}i  \bigg[ -4\frac{1}{\Upsilon^3} (\bar{\mathcal{Z}}_{W}P_L\gamma_{\mu}\mathcal{Z}_{\Upsilon}+\bar{\mathcal{Z}}_{\Upsilon}P_L\gamma_{\mu}\mathcal{Z}_{W})+6\frac{\mathcal{C}_{W}}{\Upsilon^4}\bar{\mathcal{Z}}_{\Upsilon}P_L\gamma_{\mu}\mathcal{Z}_{\Upsilon}\bigg],\qquad\quad\\
\Lambda_{w} &=& \frac{1}{\Upsilon^2}\Lambda_{W} -2\frac{\mathcal{C}_{W}}{\Upsilon^3}\Lambda_{\Upsilon} + \frac{1}{2}
\bigg[ -\frac{2}{\Upsilon^3} (i\gamma_*\cancel{\mathcal{B}}_{W}+P_L\mathcal{K}_{W}+P_R\mathcal{H}_{W}-\cancel{\mathcal{D}}\mathcal{C}_{W})\mathcal{Z}_{\Upsilon} \nonumber\\
+ 6\frac{\mathcal{C}_{W}}{\Upsilon^4}(i\gamma_*\cancel{\mathcal{B}}_{\Upsilon}+P_L\mathcal{K}_{\Upsilon}+P_R\mathcal{H}_{\Upsilon}-\cancel{\mathcal{D}}\mathcal{C}_{\Upsilon})\mathcal{Z}_{\Upsilon}   \bigg],\nonumber\\
&& -\frac{1}{4}\bigg[ \frac{6}{\Upsilon^4}\frac{3!}{2!} \mathcal{Z}_{(W}\bar{\mathcal{Z}}_{\Upsilon}\mathcal{Z}_{\Upsilon)} -24 \frac{\mathcal{C}_{W}}{\Upsilon^5}\mathcal{Z}_{\Upsilon}\bar{\mathcal{Z}}_{\Upsilon}\mathcal{Z}_{\Upsilon}  \bigg],\\
\mathcal{D}_{w} &=& \frac{1}{\Upsilon^2}\mathcal{D}_{W} -2\frac{\mathcal{C}_{W}}{\Upsilon^3}\mathcal{D}_{\Upsilon} + \frac{1}{2}  \bigg[ -\frac{2}{\Upsilon^3} 2!(\mathcal{K}_{(W}\mathcal{H}_{\Upsilon)}-\mathcal{B}_{(W}\cdot \mathcal{B}_{\Upsilon)} -\mathcal{D}\mathcal{C}_{(W}\cdot \mathcal{D}\mathcal{C}_{\Upsilon)} - 2\bar{\Lambda}_{(W} \mathcal{Z}_{\Upsilon)} - \bar{\mathcal{Z}}_{(W} \cancel{\mathcal{D}}\mathcal{Z}_{\Upsilon)})\nonumber\\
&&+6\frac{\mathcal{C}_{W}}{\Upsilon^4}(\mathcal{K}_{\Upsilon}\mathcal{H}_{\Upsilon}-\mathcal{B}_{\Upsilon}\cdot \mathcal{B}_{\Upsilon} -\mathcal{D}\mathcal{C}_{\Upsilon}\cdot \mathcal{D}\mathcal{C}_{\Upsilon} - 2\bar{\Lambda}_{\Upsilon} \mathcal{Z}_{\Upsilon} - \bar{\mathcal{Z}}_{\Upsilon} \cancel{\mathcal{D}}\mathcal{Z}_{\Upsilon}) \bigg]\nonumber\\
&& -\frac{1}{4}\bigg[ \frac{6}{\Upsilon^4}\frac{3!}{2!} \bar{\mathcal{Z}}_{(W}(i\gamma_*\cancel{\mathcal{B}}_{\Upsilon}+P_L\mathcal{K}_{\Upsilon}+P_R\mathcal{H}_{\Upsilon})\mathcal{Z}_{\Upsilon)} -24\frac{\mathcal{C}_{W}}{\Upsilon^5}\bar{\mathcal{Z}}_{\Upsilon}(i\gamma_*\cancel{\mathcal{B}}_{\Upsilon}+P_L\mathcal{K}_{\Upsilon}+P_R\mathcal{H}_{\Upsilon})\mathcal{Z}_{\Upsilon}  \bigg],\nonumber\\
&& +\frac{1}{8} \bigg[ -\frac{24}{\Upsilon^5} \frac{4!}{3!} \bar{\mathcal{Z}}_{(W} P_L \mathcal{Z}_{\Upsilon} \bar{\mathcal{Z}}_{\Upsilon} P_R \mathcal{Z}_{\Upsilon)} +120\frac{\mathcal{C}_{W}}{\Upsilon^6}
\bar{\mathcal{Z}}_{\Upsilon} P_L \mathcal{Z}_{\Upsilon} \bar{\mathcal{Z}}_{\Upsilon} P_R \mathcal{Z}_{\Upsilon}
\bigg] .
\end{eqnarray}

Note that we have to insert $\mathcal{K}_{W}=\Lambda_{W}=\mathcal{D}_{W}=0$ and $\mathcal{H}_{\bar{W}}=\Lambda_{\bar{W}}=\mathcal{D}_{\bar{W}}=0$ as given in Eqs. (<ref>) and (<ref>). In addition, the complex conjugate multiplet $\bar{w}^2$ can be obtained by taking a replacement $Q \rightarrow \bar{Q}$ in the above expressions.

In the Liberated SUGRA gauge ($\Lambda_K=0$), the non-vanishing components are given by

\begin{eqnarray}
\mathcal{H}_{w}|_{\Lambda_K=0} &=& \frac{\mathcal{H}_{W}}{\Upsilon^2},\quad \mathcal{K}_{\bar{w}}|_{\Lambda_K=0} = \frac{\mathcal{K}_{\bar{W}}}{\Upsilon^2}\\
\Lambda_{w}|_{\Lambda_K=0} &=& -\frac{1}{\Upsilon^3} P_R \mathcal{H}_{W} \mathcal{Z}_{\Upsilon},\quad \Lambda_{\bar{w}}|_{\Lambda_K=0} = -\frac{1}{\Upsilon^3} P_L \mathcal{K}_{\bar{W}} \mathcal{Z}_{\Upsilon}\\
\mathcal{D}_{w}|_{\Lambda_K=0} &=& -\frac{1}{\Upsilon^3}\mathcal{K}_{\Upsilon}\mathcal{H}_{W} -\frac{3}{2\Upsilon^4} \bar{\mathcal{Z}}_{} P_R\mathcal{H}_{W} \mathcal{Z}_{\Upsilon},\quad \mathcal{D}_{\bar{w}}|_{\Lambda_K=0} = -\frac{1}{\Upsilon^3}\mathcal{K}_{\bar{W}}\mathcal{H}_{\Upsilon} -\frac{3}{2\Upsilon^4} \bar{\mathcal{Z}}_{} P_L\mathcal{K}_{\bar{W}} \mathcal{Z}_{\Upsilon}. \nonumber\\{}
\end{eqnarray}

Furthermore, with the superconformal gauge choice $\mathcal{Z}_{\Upsilon}=0$, the non-vanishing components are

\begin{eqnarray}
\mathcal{H}_{w}|_{\Lambda_K=0,\mathcal{Z}_{\Upsilon}=0} &=& \frac{\mathcal{H}_{W}}{\Upsilon^2},\quad \mathcal{K}_{\bar{w}}|_{\Lambda_K=0,\mathcal{Z}_{\Upsilon}=0} = \frac{\mathcal{K}_{\bar{W}}}{\Upsilon^2}\\
\mathcal{D}_{w}|_{\Lambda_K=0,\mathcal{Z}_{\Upsilon}=0} &=& -\frac{1}{\Upsilon^3}\mathcal{K}_{\Upsilon}\mathcal{H}_{W},\quad \mathcal{D}_{\bar{w}}|_{\Lambda_K=0,\mathcal{Z}_{\Upsilon}=0} = -\frac{1}{\Upsilon^3}\mathcal{K}_{\bar{W}}\mathcal{H}_{\Upsilon}. \nonumber\\{}
\end{eqnarray}

§.§  chiral projection multiplets: (Weyl/Chiral) weights

\begin{eqnarray}
T(\bar{w}^2) &=& \left( -\frac{1}{2}\mathcal{K}_{\bar{w}}, -\frac{1}{2} \sqrt{2} iP_L (\cancel{\mathcal{D}}\mathcal{Z}_{\bar{w}}+\Lambda_{\bar{w}}), \frac{1}{2}(\mathcal{D}_{\bar{w}}+\square^C \mathcal{C}_{\bar{w}} + i\mathcal{D}_a \mathcal{B}^a_{\bar{w}}) \right),\\
\bar{T}(w^2) &=& \left( -\frac{1}{2}\mathcal{K}_{\bar{w}}^{*}, \frac{1}{2} \sqrt{2} iP_R (\cancel{\mathcal{D}}\mathcal{Z}_{\bar{w}}^C+\Lambda_{\bar{w}}^C), \frac{1}{2}(\mathcal{D}_{\bar{w}}^{*}+\square^C \mathcal{C}_{\bar{w}}^{*} - i\mathcal{D}_a (\mathcal{B}^a_{\bar{w}})^{*}) \right) .
\end{eqnarray}

The superconformal setting for this and its complex conjugate are then given by

\begin{eqnarray}
T \equiv T(\bar{w}^2) &=& \{\mathcal{C}_T,\mathcal{Z}_T,\mathcal{H}_T,\mathcal{K}_T,\mathcal{B}_{\mu}^T,\Lambda_T,\mathcal{D}_T\} ,
\nonumber\\
\bar{T} \equiv \bar{T}(w^2) &=& \{\mathcal{C}_{\bar{T}},\mathcal{Z}_{\bar{T}},\mathcal{H}_{\bar{T}},\mathcal{K}_{\bar{T}},\mathcal{B}_{\mu}^{\bar{T}},\Lambda_{\bar{T}},\mathcal{D}_{\bar{T}}\},
\end{eqnarray}

\begin{eqnarray}
\mathcal{C}_T &=&  -\frac{1}{2} \mathcal{K}_{\bar{w}} = -\frac{1}{2\Upsilon^2}\mathcal{K}_{\bar{W}} +\frac{\mathcal{C}_{\bar{W}}}{\Upsilon^3}\mathcal{K}_{\Upsilon} +\frac{1}{4}\bigg[ -4\frac{1}{\Upsilon^3} (\bar{\mathcal{Z}}_{\bar{W}}P_R\mathcal{Z}_{\Upsilon}+\bar{\mathcal{Z}}_{\Upsilon}P_R\mathcal{Z}_{\bar{W}})+6\frac{\mathcal{C}_{\bar{W}}}{\Upsilon^4}\bar{\mathcal{Z}}_{\Upsilon}P_R\mathcal{Z}_{\Upsilon}\bigg],\nonumber\\{} \\
\mathcal{Z}_T &=& -P_L(\cancel{\mathcal{D}}\mathcal{Z}_{\bar{w}}+\Lambda_{\bar{w}}),\\
\mathcal{H}_T &=& -(\mathcal{D}_{\bar{w}}+\square^C \mathcal{C}_{\bar{w}} + i\mathcal{D}_a \mathcal{B}^a_{\bar{w}}),\\
\mathcal{K}_T &=& 0,\\
\mathcal{B}^T_{\mu} &=& -\frac{1}{2} i \mathcal{D}_{\mu} \mathcal{K}_{\bar{w}},\\
\Lambda_T &=& 0 ,\\
\mathcal{D}_T &=& 0.
\end{eqnarray}

\begin{eqnarray}
\mathcal{C}_{\bar{T}} &=&  -\frac{1}{2} \mathcal{K}_{\bar{w}}^{*},\\
\mathcal{Z}_{\bar{T}} &=& -P_R(\cancel{\mathcal{D}}\mathcal{Z}_{\bar{w}}^C+\Lambda_{\bar{w}}^C),\\
\mathcal{H}_{\bar{T}} &=& 0,\\
\mathcal{K}_{\bar{T}} &=& -(\mathcal{D}_{\bar{w}}^{*}+\square^C \mathcal{C}_{\bar{w}}^{*} - i\mathcal{D}_a (\mathcal{B}^a_{\bar{w}})^{*}),\\
\mathcal{B}^{\bar{T}}_{\mu} &=& \frac{1}{2} i \mathcal{D}_{\mu} \mathcal{K}_{\bar{w}}^{*},\\
\Lambda_{\bar{T}} &=& 0 ,\\
\mathcal{D}_{\bar{T}} &=& 0.
\end{eqnarray}

In the Liberated SUGRA gauge, the non-vanishing components are given by

\begin{eqnarray}
\mathcal{C}_T|_{\Lambda_K=0} &=&  -\frac{1}{2} \mathcal{K}_{\bar{w}}, \\
\mathcal{Z}_T|_{\Lambda_K=0} &=& -P_L\Lambda_{\bar{w}},\\
\mathcal{H}_T|_{\Lambda_K=0} &=& -\mathcal{D}_{\bar{w}},\\
\mathcal{B}^T_{\mu}|_{\Lambda_K=0} &=& -\frac{1}{2} i \mathcal{D}_{\mu} \mathcal{K}_{\bar{w}}.
\end{eqnarray}

With the superconformal gauge choice $\mathcal{Z}_{\Upsilon} =0$, we have

\begin{eqnarray}
\mathcal{C}_T|_{\Lambda_K=0,\mathcal{Z}_{\Upsilon} =0} &=&  -\frac{1}{2} \mathcal{K}_{\bar{w}},\qquad \mathcal{C}_{\bar{T}}|_{\Lambda_K=0,\mathcal{Z}_{\Upsilon} =0} =  -\frac{1}{2} \mathcal{H}_{w} = -\frac{1}{2} \mathcal{K}_{\bar{w}}^{*} \\
\mathcal{H}_T|_{\Lambda_K=0,\mathcal{Z}_{\Upsilon} =0} &=& -\mathcal{D}_{\bar{w}},\qquad \mathcal{K}_{\bar{T}}|_{\Lambda_K=0,\mathcal{Z}_{\Upsilon} =0} = -\mathcal{D}_{w}=-\mathcal{D}_{\bar{w}}^{*}\\
\mathcal{B}^T_{\mu}|_{\Lambda_K=0,\mathcal{Z}_{\Upsilon} =0} &=& -\frac{1}{2} i \mathcal{D}_{\mu} \mathcal{K}_{\bar{w}},\qquad \mathcal{B}^{\bar{T}}_{\mu}|_{\Lambda_K=0,\mathcal{Z}_{\Upsilon} =0} = \frac{1}{2} i \mathcal{D}_{\mu} \mathcal{H}_{w}= \frac{1}{2} i \mathcal{D}_{\mu} \mathcal{K}_{\bar{w}}^{*}.
\end{eqnarray}

Final form of the composite multiplet of liberated $\mathcal{N}=1$ supergravity

To construct the final composite multiplet which will give the new term in the action of liberated supergravity, we first collect the
following chiral multiplets:

\begin{eqnarray}
\Big( X^i \equiv  \{S_0,z^I,T(\bar{w}^2),\mathcal{W}^2(K)\},~\chi^i \equiv  \{\chi^0,\chi^I,\chi^T,\chi^W\},~F^i \equiv \{F^{0},F^{I},F^{T},F^{W}\} \Big),\nonumber\\{}
\end{eqnarray}

\begin{eqnarray}
S_0 &=& \left(s_0, P_L\chi^0, F^0 \right),\qquad
\bar{S}_0 = \left(s_0^{*}, P_R\chi^0, F^{0*} \right),\\
z^I &=& \left( z^I, P_L\chi^I, F^I \right),\qquad
\bar{z}^{\bar{I}} = \left( \bar{z}^{\bar{I}}, P_R\chi^{\bar{I}}, \bar{F}^{\bar{I}} \right),
\end{eqnarray}

\begin{eqnarray}
\mathcal{W}^2(K) &=& \left( \Bar{\Lambda}_K P_L \Lambda_K,~ \sqrt{2} P_L (-\gamma\cdot \hat{F}_K + 2i\mathcal{D}_K)\Lambda_K,~ 2\Bar{\Lambda}_KP_L \cancel{\mathcal{D}}\Lambda_K + \hat{F}_K^-\cdot \hat{F}_K^- - \mathcal{D}^2_K\right) \nonumber\\
&\equiv& \left( X^W, P_L\chi^W, F^W \right),\\
\bar{\mathcal{W}}^2(K) &=& \left( \Bar{\Lambda}_K P_R \Lambda_K,~ \sqrt{2} P_R (-\gamma\cdot \hat{F}_K - 2i\mathcal{D}_K^{*})\Lambda_K^C,~ 2\Bar{\Lambda}_KP_R \cancel{\mathcal{D}}\Lambda_K + \hat{F}_K^+\cdot \hat{F}_K^+ - (\mathcal{D}^{*}_K)^2\right) \nonumber\\
&\equiv& \left( \bar{X}^{\bar{W}}, P_R\chi^{\bar{W}}, \bar{F}^{\bar{W}} \right),
\end{eqnarray}

\begin{eqnarray}
T(\bar{w}^2) &=& \left( -\frac{1}{2}\mathcal{K}_{\bar{w}}, -\frac{1}{2} \sqrt{2} iP_L (\cancel{\mathcal{D}}\mathcal{Z}_{\bar{w}}+\Lambda_{\bar{w}}), \frac{1}{2}(\mathcal{D}_{\bar{w}}+\square^C \mathcal{C}_{\bar{w}} + i\mathcal{D}_a \mathcal{B}^a_{\bar{w}}) \right) \equiv \left( X^T, P_L\chi^T, F^T \right),\nonumber\\{}\\
\bar{T}(w^2) &=& \left( -\frac{1}{2}\mathcal{K}_{\bar{w}}^{*}, \frac{1}{2} \sqrt{2} iP_R (\cancel{\mathcal{D}}\mathcal{Z}_{\bar{w}}^C+\Lambda_{\bar{w}}^C), \frac{1}{2}(\mathcal{D}_{\bar{w}}^{*}+\square^C \mathcal{C}_{\bar{w}}^{*} - i\mathcal{D}_a (\mathcal{B}^a_{\bar{w}})^{*}) \right)
\equiv \left( \bar{X}^{\bar{T}}, P_R\chi^{\bar{T}}, \bar{F}^{\bar{T}} \right). \nonumber\\{}
\end{eqnarray}

§ LIBERATED TERM

Then, the lowest component of the final composite real multiplet is

\begin{eqnarray}
N &\equiv& \Upsilon^2 \frac{w^2\bar{w}^2}{T(\bar{w}^2)\bar{T}(w^2)}\mathcal{U} = \Upsilon^{-2}\dfrac{\mathcal{W}^2(K)\bar{\mathcal{W}}^2(K)}{T(w^2)\bar{T}(\bar{w}^2)} \mathcal{U} \nonumber\\
&=&  (X^0\bar{X}^0e^{-K(z^I,\bar{z}^{\bar{I}})/3})^{-2} X^W\bar{X}^{\bar{W}} (X^T)^{-1} (\bar{X}^{\bar{T}})^{-1}\mathcal{U}(z^I,\bar{z}^{\bar{I}}).
\end{eqnarray}

Then, since $N$ is now expressed as a function of superconformal chiral multiplets, we can represent the full off-shell expression of the superconformal Lagrangian of the liberated $\mathcal{N}=1$ supergravity via Eq. (17.19) of Freedman and Van Proeyen <cit.> as follows:

\begin{eqnarray}
\mathcal{L}_{NEW} &\equiv& [\mathbb{N}]_De^{-1} \nonumber\\
\frac{1}{2}\mathcal{D}_N - \frac{1}{4}\bar{\psi}\cdot \gamma i\gamma_* \Lambda_N -\frac{1}{6} \mathcal{C}_N R(\omega) + \frac{1}{12} \left(\mathcal{C}_N\bar{\psi}_{\mu} \gamma^{\mu\rho\sigma} - i \bar{\mathcal{Z}}_N  \gamma^{\rho\sigma} \gamma_*\right) R_{\rho\sigma}'(Q)\nonumber\\
&&+\frac{1}{8} \varepsilon^{abcd} \bar{\psi}_{a}\gamma_b \psi_c (\mathcal{B}_{d}^N -\frac{1}{2}\bar{\psi}_d \mathcal{Z}_N)
&=& N_{i\bar{j}} \bigg( -\mathcal{D}_{\mu}X^i\mathcal{D}^{\mu}\bar{X}^{\bar{j}} - \frac{1}{2} \bar{\chi}^i \cancel{\mathcal{D}} \chi^{\bar{j}} - \frac{1}{2} \bar{\chi}^{\bar{j}}\cancel{\mathcal{D}}\chi^i + F^i\bar{F}^{\bar{j}}\bigg)
\nonumber\\
&& +\frac{1}{2}\bigg[ N_{ij\bar{k}} \Big( -\bar{\chi}^i\chi^j \bar{F}^{\bar{k}} + \bar{\chi}^i (\cancel{\mathcal{D}}X^j)\chi^{\bar{k}} \Big) +h.c. \bigg]
+ \frac{1}{4}N_{ij\bar{k}\bar{l}} \bar{\chi}^i\chi^j \bar{\chi}^{\bar{k}}\chi^{\bar{l}}
\nonumber\\
&& +\bigg[\frac{1}{2\sqrt{2}}\bar{\psi}\cdot \gamma
\bigg( N_{i\bar{j}} F^i \chi^{\bar{j}} - N_{i\bar{j}} \cancel{\mathcal{D}}\bar{X}^{\bar{j}}\chi^i -\frac{1}{2}N_{ij\bar{k}} \chi^{\bar{k}}\bar{\chi}^i\chi^j  \bigg)  \nonumber\\
&& +\frac{1}{8} i\varepsilon^{\mu\nu\rho\sigma} \bar{\psi}_{\mu} \gamma_{\nu} \psi_{\rho} \bigg( N_i \mathcal{D}_{\sigma}X^i + \frac{1}{2} N_{i\bar{j}} \bar{\chi}^i \gamma_{\sigma} \chi^{\bar{j}} + \frac{1}{\sqrt{2}} N_i \bar{\psi}_{\sigma} \chi^i  \bigg)+h.c. \bigg]
\nonumber\\
&&+\frac{1}{6} N \left( -R(\omega) +\frac{1}{2} \bar{\psi}_{\mu} \gamma^{\mu \nu\rho} R'_{\nu\rho}(Q) \right)
-\frac{1}{6\sqrt{2}} \left( N_i\bar{\chi}^i + N_{\bar{i}}\bar{\chi}^{\bar{i}} \right)\gamma^{\mu\nu} R'_{\mu\nu}(Q),\nonumber\\{}\label{NEW_Lagrangian}
\end{eqnarray}

where $i,j = 0,I,W,T$ and $\bar{j},\bar{k},\bar{l} = \Bar{0},\Bar{I},\Bar{W},\Bar{T}$ and

\begin{eqnarray}
&& R_{\mu\nu ab}(\omega) \equiv  \partial_{\mu} \omega_{\nu ab} - \partial_{\nu} \omega_{\mu ab} + \omega_{\mu ac}\omega_{\nu ~ b}^{~c} - \omega_{\nu ac}\omega_{\mu ~ b}^{~c} ,\\
&& R_{\mu\nu}'(Q) \equiv 2 \left(\partial_{[\mu} + \frac{1}{2} b_{[\mu} -\frac{3}{2} A_{[\mu} \gamma_* + \frac{1}{4} \omega_{[\mu}^{ab}(e,b,\psi)\gamma_{ab} \right)\psi_{\nu]},\\
&& \omega_{\mu}^{ab}(e,b,\psi) = \omega_{\mu}^{ab}(e,b) + \frac{1}{2} \psi_{\mu} \gamma^{[a} \psi^{b]} + \frac{1}{4} \bar{\psi}^a \gamma_{\mu} \psi^b.
\end{eqnarray}

§.§ Bosonic lagrangians

The bosonic contribution to the scalar potential can be found from the term $\mathcal{D}_N$. Using the above results and Eq. (<ref>), we find the equivalent bosonic contribution to the scalar potential as follows:

\begin{eqnarray}
\mathcal{L}_B \supset N_{i\bar{j}}F^i\bar{F}^{\bar{j}} \sim N_{W\bar{W}}F^W\bar{F}^{\bar{W}} = \Upsilon^{-2}\frac{1}{\mathcal{C}_{T}\mathcal{C}_{\bar{T}}}\mathcal{U} F^W\bar{F}^{\bar{W}}.
\end{eqnarray}

Since $\mathcal{C}_{T} =- \frac{1}{2}\mathcal{K}_{\bar{w}} \sim -\frac{1}{2} \frac{\mathcal{K}_{\bar{W}}}{\Upsilon^2} =-\frac{1}{2} \frac{(-2\bar{F}^{\bar{W}})}{\Upsilon^2} = \frac{\bar{F}^{\bar{W}}}{\Upsilon^2}$ and $\mathcal{C}_{\bar{T}} = \frac{F^W}{\Upsilon^2}$, we have

\begin{eqnarray}
\mathcal{L}_B \supset \Upsilon^{-2}\frac{\Upsilon^2\Upsilon^2}{\bar{F}^{\bar{W}}F^W}\mathcal{U} F^W\bar{F}^{\bar{W}} = \Upsilon^2 \mathcal{U}.
\end{eqnarray}

In the superconformal gauge $\Upsilon=1$, we get

\begin{eqnarray}
V_{NEW} =  \mathcal{U}.
\end{eqnarray}

§.§ Fermionic lagrangians

In this section, we investigate fermionic terms in the liberated $\mathcal{N}=1$ supergravity Lagrangian. We focus in particular
on the matter-coupling and on the most divergent fermionic terms, in order to explore interesting interactions and check the limits
of validity of the liberated $\mathcal{N}=1$ supergravity as an effective theory.

First of all, let us recall that the final composite multiplet $N$ in terms of the superconformal chiral multiplets $\{S_0, z^I, W \equiv \mathcal{W}^2(K), T \equiv T(\bar{w}^2)\}$ are:

\begin{eqnarray}
N = \left(s_0s_0^{*}e^{-K(z^I,\bar{z}^{\bar{I}})/3}\right)^{-2}\dfrac{W\bar{W}}{T\bar{T}}\mathcal{U}(z^I,\bar{z}^{\bar{I}}).
\end{eqnarray}

We we also denote their lowest components as $W \equiv \bar{\Lambda}_KP_L\Lambda_K$, $\bar{W} \equiv \bar{\Lambda}_KP_R\Lambda_K$, $T\equiv \mathcal{C}_{T}$, and $\bar{T} \equiv \mathcal{C}_{\bar{T}}$. Note that the final composite multiplet consists of the four superconformal chiral multiplets only. Remember that $\mathcal{C}_{T} \sim \dfrac{\bar{F}^{\bar{W}}}{\Upsilon^2}$ and $F^W \propto \mathcal{D}_K^2|_{\textrm{boson}}\equiv \tilde{\mathcal{F}}^2$.

Generically, the matter couplings are found from the following contributions:

\begin{eqnarray}
\mathcal{L}_{F}^{\textrm{matter}}|_{\psi=0} &=& N_{i\bar{j}} \bigg( -\mathcal{D}_{\mu}X^i\mathcal{D}^{\mu}\bar{X}^{\bar{j}} - \frac{1}{2} \bar{\chi}^i \cancel{\mathcal{D}} \chi^{\bar{j}} - \frac{1}{2} \bar{\chi}^{\bar{j}}\cancel{\mathcal{D}}\chi^i + F^i\bar{F}^{\bar{j}}\bigg) - \frac{N}{6}R(\omega)|_{\psi=0}
\nonumber\\
&& +\frac{1}{2}\bigg[ N_{ij\bar{k}} \Big( -\bar{\chi}^i\chi^j \bar{F}^{\bar{k}} + \bar{\chi}^i (\cancel{\mathcal{D}}X^j)\chi^{\bar{k}} \Big) +h.c. \bigg]
+ \frac{1}{4}N_{ij\bar{k}\bar{l}} \bar{\chi}^i\chi^j \bar{\chi}^{\bar{k}}\chi^{\bar{l}}\bigg|_{\psi=0},\nonumber\\
&=& \mathcal{L}_{F1}+\mathcal{L}_{F2}+\bar{\mathcal{L}}_{F2}+\mathcal{L}_{F3} + (\mathcal{L}_{F4}+\mathcal{L}_{F5}+h.c.)+\mathcal{L}_{F6} + \mathcal{L}_{F7}
\end{eqnarray}

\begin{eqnarray}
\mathcal{D}_{\mu} X^i|_{\psi=0} &=& (\partial_{\mu} -w_ib_{\mu} -w_iA_{\mu})X^i,\nonumber\\
\mathcal{D}_{\mu} P_L\chi^i|_{\psi=0} &=& \left(\partial_{\mu} +\frac{1}{4}\omega_{\mu}^{ab}\gamma_{ab}-(w_i+1/2)b_{\mu} + (w_i-3/2)iA_{\mu}\right) P_L\chi^i -\sqrt{2}w_iX^i   P_L\phi_{\mu}. \nonumber
\end{eqnarray}

Note that the matter couplings of fermions can be classified into seven types:

\begin{eqnarray}
\mathcal{L}_{F1} &\equiv&  -N_{i\bar{j}}\mathcal{D}_{\mu}X^i\mathcal{D}^{\mu}\bar{X}^{\bar{j}}\Big|_{\psi=0},\\
\mathcal{L}_{F2} &\equiv&  -\frac{1}{2} N_{i\bar{j}} \bar{\chi}^i \cancel{\mathcal{D}} \chi^{\bar{j}}\Big|_{\psi=0}  ,\\
\mathcal{L}_{F3} &\equiv&   -N_{i\bar{j}} F^i\bar{F}^{\bar{j}} \Big|_{\psi=0},\\
\mathcal{L}_{F4} &\equiv& -\frac{1}{2} N_{ij\bar{k}} \bar{\chi}^i\chi^j \bar{F}^{\bar{k}}\Big|_{\psi=0},\\
\mathcal{L}_{F5} &\equiv& \frac{1}{2}  N_{ij\bar{k}} \bar{\chi}^i (\cancel{\mathcal{D}}X^j)\chi^{\bar{k}}\Big|_{\psi=0} ,\\
\mathcal{L}_{F6} &\equiv&  \frac{1}{4}N_{ij\bar{k}\bar{l}} \bar{\chi}^i\chi^j \bar{\chi}^{\bar{k}}\chi^{\bar{l}} \Big|_{\psi=0} ,\\
\mathcal{L}_{F7} &\equiv& - \frac{N}{6}R(\omega)|_{\psi=0}.
\end{eqnarray}

The derivatives of the $N$ are given in general by

\begin{eqnarray}
N^{(r=q+p+m+k)}_{q,p,m,k} &=& (\partial_{0}^{q}\partial_{W}^{p}\partial_{T}^m \partial_{I}^{k} N ) \nonumber\\
&=& \bigg[(\partial_0^q\partial_I^{(k-n)}\Upsilon^{-2})_{(N)}(\Upsilon^{2})_{(\mathcal{C}_T)}(\Upsilon^{2})_{(\mathcal{C}_{\bar{T}})}
\nonumber\\
\times (\Upsilon^{2m_1})_{(\partial_T^{m_1}\mathcal{C}_T)}(\Upsilon^{2m_2})_{(\partial_{\bar{T}}^{m_2}\mathcal{C}_{\bar{T}})}
(W)^{1-p_1}(\bar{W})^{1-p_2}\mathcal{U}^{(n)} \bigg]\nonumber\\
&&/\bigg[ (\tilde{\mathcal{F}}^{2})_{(\mathcal{C}_{T})}(\tilde{\mathcal{F}}^{2})_{(\mathcal{C}_{\bar{T}})}(\tilde{\mathcal{F}}^{2m_1})_{(\partial_T^{m_1}\mathcal{C}_T)}(\tilde{\mathcal{F}}^{2m_2})_{(\partial_{\bar{T}}^{m_2}\mathcal{C}_{\bar{T}})} \bigg]. \nonumber\\
&=& (\partial_0^q\partial_I^{(k-n)}\Upsilon^{-2})\Upsilon^{4+2m_1+2m_2}\frac{\mathcal{U}^{(n)}}{\tilde{\mathcal{F}}^{2+2+2m_1+2m_2}}W^{1-p_1}\bar{W}^{1-p_2} \nonumber\\
&=& (\partial_0^q\partial_I^{(k-n)}\Upsilon^{-2})\Upsilon^{4+2m}\frac{\mathcal{U}^{(n)}}{\tilde{\mathcal{F}}^{4+2m}}W^{1-p_1}\bar{W}^{1-p_2},
\end{eqnarray}

where $\tilde{\mathcal{F}} \equiv  2K_{I\bar{\jmath}} \left( -\partial_{\mu} z^I \partial^{\mu} \bar{z}^{\bar{\jmath}} +F^I\bar{F}^{\bar{\jmath}}\right)$;
$\mathcal{U}^{(n)}$ ($0 \leq n \leq 4$) is the $n$-th derivative of the
function $\mathcal{U}(z^I,\bar{z}^{\bar{\imath}})$ with respect to $z^I,\bar{z}^I$,
which are the lowest component of the matter chiral  multiplets; $q=q_1+q_2$ where $q_1$
($q_2$) is the order of the derivative with respect to the compensator scalar $s_0$ ($s_0^{*}$); $p=p_1+p_2$ where $p_1$ ($p_2$) is the order of the derivative with respect to the field strength multiplet scalar $W$ ($\bar{W}$); $m=m_1+m_2$ where $m_1$ ($m_2$) is the
order of the derivative with respect to the chiral projection multiplet scalar $T(\bar{w}^2)$ ($\bar{T}(w^2)$); $k$ is the order of the derivative with respect to the matter multiplet scalar $z^I$;
$n$ is the order of the derivative acting on the new term $\mathcal{U}$ with respect to the matter multiplet; $q$ is the total order of
derivative with respect to the compensator scalars $s_0$ and $s_0^{*}$. An explicit form of the derivatives of $N$ is given in
Appendix A.

The mass dimension[the mass dimensions of the multiplets' lowest components are $[s_0]=1,[z^I]=0,[W]=3,[T]=0$, which gives $[\tilde{\mathcal{F}}]=2$.] of the derivatives of the $N$ is $[N^{(r=q+p+m+k)}_{q,p,m,k}]=-3p-4m-2$. This implies that the mass dimension of the operator coupled to $N^{(r=q+p+m+k)}_{q,p,m,k}$ must be equal to $3p+4m+2$.

Now, let us focus on the case such that $q=0$ and $k=n$ which gives the most singular fermionic terms in the limit
that the D-term vanishes. The most singular terms are those that contain the highest power of the auxiliary field $D$ in the
denominator and therefore are the nonrenormalizable operators associated with the smallest UV cutoff mass scale. That is, we consider that the matter scalar derivatives act only on the new term $\mathcal{U}$ and there are no the derivatives with respect to the compensator scalar. Then, we have

\begin{eqnarray}
N^{(r=p+m+n)}_{p,m,n} = \Upsilon^{2+2m}\frac{\mathcal{U}^{(n)}}{\tilde{\mathcal{F}}^{4+2m}}W^{1-p_1}\bar{W}^{1-p_2}.
\end{eqnarray}

In particular, since $r = p+m+n$ ($0 \leq r \leq 4$), it reduces to

\begin{eqnarray}
N_{i \ldots l}^{(r)} = N^{(r)}_{p,m,n} = \Upsilon^{2(1+r-n-p)}\frac{\mathcal{U}^{(n)}}{\tilde{\mathcal{F}}^{2(2+r-n-p)}}W^{1-p_1}\bar{W}^{1-p_2}.
\end{eqnarray}

Remember that we called $r$ the total order of the derivatives acting on the $N$. Here $n$ is the number of
derivatives acting on the
matter scalars in the new term $\mathcal{U}$; they produce $\mathcal{U}^{(n)}$. Finally, $p=p_1+p_2$ is the sum of the number of derivatives w.r.t the multiplets $(W,\bar{W})$ acting on $\mathcal{U}$.

Structure of the fermionic components

Next, let us explore the detailed structure of the chiral fermions of the superconformal multiplets. First of all, the compensator and matter fermions are given by

\begin{eqnarray}
\chi^0 &=& P_L\chi^0,\qquad \chi^I = P_L\chi^I.
\end{eqnarray}

Note that $\chi^0$ and $\chi^I$ are fundamental fermions, and later in the $S$-gauge, the compensator chiral fermions will be replaced by the matter ones according to: $P_L\chi^0 = \frac{1}{3}s_0 K_I P_L\chi^I$ and $P_R\chi^0 = \frac{1}{3}s_0^{*} K_{\bar{I}} P_R\chi^{\bar{I}}$.

On the contrary, the other fermions $\chi^W$ and $\chi^T$ are composite. Hence, we need to find their specific structure.

First, the $\mathcal{W}(K)^2$-multiplet fermion, say $\chi^W$, is found to be

\begin{eqnarray}
\chi^W &=& P_L \chi^W = \sqrt{2}  P_L(-\gamma\cdot \hat{F}_K + 2i\mathcal{D}_K)\Lambda_K
\nonumber\\
&=&  -2\sqrt{2} P_L\gamma^{\mu\nu} \partial_{[\mu}\mathcal{B}_{\nu]}^K\Lambda_K-\sqrt{2}P_L\gamma^{\mu\nu} \Bar{\psi}_{[\mu} \gamma_{\nu]}\Lambda_K\Lambda_K + 2\sqrt{2}iP_L\mathcal{D}_K\Lambda_K
\nonumber\\
&=& -2\sqrt{2} P_L\gamma^{\mu\nu} \partial_{[\mu}(iK_I\mathcal{D}_{\nu]} z^I -iK_{\bar{I}}\mathcal{D}_{\nu]} \bar{z}^{\bar{I}} + i K_{I\bar{J}}\bar{\chi}^I \gamma_{\nu]} \chi^{\bar{J}})\Lambda_K+ 2\sqrt{2}iP_L\mathcal{D}_K\Lambda_K
\nonumber\\
&=& 2 i \gamma^{\mu\nu} \partial_{[\mu}(K_I \Bar{\psi}_{\nu]}\chi^I -K_{\bar{I}}\Bar{\chi}^{\bar{I}}\psi_{\nu]} - \sqrt{2} K_{I\bar{J}}\bar{\chi}^I \gamma_{\nu]} \chi^{\bar{J}})P_L\Lambda_K
\nonumber\\&&
+ 2\sqrt{2}i\bigg[2K_{I\bar{J}}
\bigg( -\mathcal{D}_{\mu} z^I \mathcal{D}^{\mu} \bar{z}^{\bar{J}} -\frac{1}{2} \bar{\chi}^I P_L \cancel{\mathcal{D}}\chi^{\bar{J}} -\frac{1}{2} \bar{\chi}^{\bar{J}} P_R \cancel{\mathcal{D}} \chi^I + F^I\bar{F}^{\bar{J}}\bigg) \nonumber\\
&&+ K_{IJ\bar{K}} \bigg( -\bar{\chi}^I \chi^J \bar{F}^{\bar{K}} + \bar{\chi}^I (\cancel{\mathcal{D}}z^J)\chi^{\bar{K}}         \bigg) + K_{\bar{I}\bar{J}K} \bigg( -\bar{\chi}^{\bar{I}} \chi^{\bar{J}} F^{K} + \bar{\chi}^{\bar{I}} (\cancel{\mathcal{D}}\bar{z}^{\bar{J}})\chi^{K}\bigg) \nonumber\\
&& + \frac{1}{2} K_{IJ\bar{K}\bar{L}} (\bar{\chi}^IP_L\chi^J )(\bar{\chi}^{\bar{K}} P_R \chi^{\bar{L}})\bigg]P_L\Lambda_K
\nonumber\\
&=& 2\sqrt{2}i\tilde{\mathcal{F}} (P_L\Lambda_K)_{1f} + \cdots + 7~\textrm{fermions}\nonumber\\
&=& 2\sqrt{2}i \tilde{\mathcal{F}} (-\sqrt{2}iK_{\bar{I}J}[(\cancel{\mathcal{D}}z^{J})_{0f}\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J]) + \cdots + 7~\textrm{fermions} \nonumber\\
&=& 4 \tilde{\mathcal{F}} K_{\bar{I}J}[(\cancel{\partial}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J] + \cdots + 7~\textrm{fermions},
\end{eqnarray}

where $P_L \Lambda_K = -\sqrt{2}iK_{\bar{I}J}[(\cancel{\mathcal{D}}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J] -\frac{i}{\sqrt{2}}K_{\bar{I}\bar{J}K}\chi^{K}\bar{\chi}^{\bar{I}}\chi^J$. Note that the composite fermion $\chi^W$ contains powers of the
matter fermions ranging from one to seven, and it is nonvanishing on-shell.

Second, the chiral projection multiplet $T(\bar{w}^2)$ fermions are found to be as follows:

\begin{eqnarray}
\chi^T &=& P_L \chi^T = -\frac{i}{\sqrt{2}} P_L(\cancel{\mathcal{D}}\mathcal{Z}_{\bar{w}}+\Lambda_{\bar{w}}) = -\frac{i}{\sqrt{2}}\bigg[ \frac{(\cancel{\mathcal{D}}\mathcal{Z}_{\bar{W}})}{\Upsilon^2} - \frac{(\cancel{\mathcal{D}}\Upsilon)\mathcal{Z}_{\bar{W}}}{\Upsilon^3} -2\frac{\mathcal{C}_{\bar{W}}}{\Upsilon^3}P_L\Lambda_{\Upsilon} -\frac{i\cancel{\mathcal{B}}_{\Upsilon}\mathcal{Z}_{\bar{W}}}{\Upsilon^3} \bigg]_{\mathcal{Z}_{\Upsilon}=0}\nonumber\\
&=& \frac{1}{\Upsilon^2}\bigg[~ \cancel{\mathcal{D}}\chi^{\bar{W}} - \frac{(\cancel{\mathcal{D}}\Upsilon)\chi^{\bar{W}}}{\Upsilon} -2\frac{\Bar{\Lambda}_KP_R\Lambda_K}{\Upsilon}P_L\Lambda_{\Upsilon} -\frac{i\cancel{\mathcal{B}}_{\Upsilon}\chi^{\bar{W}}}{\Upsilon}~ \bigg]_{\mathcal{Z}_{\Upsilon}=0}
\nonumber\\
&=& \frac{1}{\Upsilon^2}\bigg[~ (\cancel{\mathcal{D}}\chi^{\bar{W}})_{1f} - \frac{[(\cancel{\mathcal{D}}\Upsilon)_{0f}+i(\cancel{\mathcal{B}}_{\Upsilon})_{0f}]}{\Upsilon}(\chi^{\bar{W}})_{1f}~ \bigg] + \cdots + 9~\textrm{fermions} ,\\
&=& \frac{1}{\Upsilon^2}\bigg[~  4 (\cancel{\partial}\tilde{\mathcal{F}}) K_{\bar{I}J}[(\cancel{\partial}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J] \nonumber\\
&&- \Big(\frac{2}{s_0^{*}}\cancel{\partial}s_0^{*} -\frac{2}{3}K_{\bar{K}}\cancel{\partial}\bar{z}^{\bar{K}} - 2\gamma^{\mu}(b_{\mu}+iA_{\mu})\Big)4 \tilde{\mathcal{F}} K_{\bar{I}J}[(\cancel{\partial}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J] ~ \bigg]_{1f}+ \cdots + 9~\textrm{fermions} ,\nonumber\\
&=& \frac{4}{\Upsilon^2}  (\cancel{\partial}\tilde{\mathcal{F}}) K_{\bar{I}J}[(\cancel{\partial}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J] + (\chi^T)_{1f}|_{\tilde{\mathcal{F}}^1} + \cdots + 9~\textrm{fermions},
\end{eqnarray}

where $\tilde{\mathcal{F}} \equiv  2K_{I\bar{J}} \left( -\partial_{\mu} z^I \partial^{\mu} \bar{z}^{\bar{J}} +F^I\bar{F}^{\bar{J}}\right)$; “$|_{\tilde{\mathcal{F}}^1}$” denotes the terms proportional to $\tilde{\mathcal{F}}^1$, and

\begin{eqnarray}
\cancel{\partial} \tilde{\mathcal{F}}&=& 2(\cancel{\partial}K_{I\bar{J}}) \left( -\partial_{\mu} z^I \partial^{\mu} \bar{z}^{\bar{J}} +F^I\bar{F}^{\bar{J}}\right)
\nonumber\\
&&+2K_{I\bar{J}} \left( -(\cancel{\partial}\partial_{\mu} z^I) \partial^{\mu} \bar{z}^{\bar{J}}-\partial_{\mu} z^I (\cancel{\partial}\partial^{\mu} \bar{z}^{\bar{J}}) +(\cancel{\partial}F^I)\bar{F}^{\bar{J}}+F^I(\cancel{\partial}\bar{F}^{\bar{J}})\right).
\end{eqnarray}

Note that $\chi^T$ is also nonvanishing on-shell. In the above calculation we have used the formula:

\begin{eqnarray}
\mathcal{D}_{\mu} P_L\chi^W &=&  \left(\partial_{\mu} +\frac{1}{4}\omega_{\mu}^{ab}\gamma_{ab}-\frac{7}{2}b_{\mu} + \frac{3}{2}iA_{\mu}\right)P_L\chi^W -\frac{1}{\sqrt{2}}P_L(\cancel{\mathcal{D}}W + F^W)\psi_{\mu} \nonumber\\
&&-3\sqrt{2}W  P_L\phi_{\mu}.
\end{eqnarray}

to find

\begin{eqnarray}
(\cancel{\mathcal{D}}\chi^W)_{1f} &=& (\cancel{\mathcal{D}}P_L\chi^W)_{1f} \nonumber\\
&=& \gamma^{\mu}\left(\partial_{\mu} +\frac{1}{4}\omega_{\mu}^{ab}\gamma_{ab}-\frac{7}{2}b_{\mu} + \frac{3}{2}iA_{\mu}\right)(P_L\chi^W)_{1f} + \frac{1}{\sqrt{2}}P_L (F^W)_{0f} \gamma^{\mu} \psi_{\mu} \nonumber\\
&=& \cancel{\partial}(P_L\chi^W)_{1f}+\gamma^{\mu}\left(\frac{1}{4}\omega_{\mu}^{ab}\gamma_{ab}-\frac{7}{2}b_{\mu} + \frac{3}{2}iA_{\mu}\right)(P_L\chi^W)_{1f} + \frac{1}{\sqrt{2}}P_L \tilde{\mathcal{F}}^2 \gamma^{\mu} \psi_{\mu}\nonumber\\
&=& 4 (\cancel{\partial}\tilde{\mathcal{F}}) K_{\bar{I}J}[(\cancel{\partial}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J] + 4 \tilde{\mathcal{F}} (\cancel{\partial}K_{\bar{I}J})[(\cancel{\partial}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J]\nonumber\\
&&+4 \tilde{\mathcal{F}} K_{\bar{I}J}[(\square z^{J})\chi^{\bar{I}}+\cancel{\partial}z^{J}\cancel{\partial}\chi^{\bar{I}}-(\cancel{\partial}\bar{F}^{\bar{I}})\chi^J-\bar{F}^{\bar{I}}(\cancel{\partial}\chi^J)] \nonumber\\
&&+\gamma^{\mu}\left(\frac{1}{4}\omega_{\mu}^{ab}\gamma_{ab}-\frac{7}{2}b_{\mu} + \frac{3}{2}iA_{\mu}\right)(4 \tilde{\mathcal{F}} K_{\bar{I}J}[(\cancel{\partial}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J] ) + \frac{1}{\sqrt{2}}P_L \tilde{\mathcal{F}}^2 \gamma^{\mu} \psi_{\mu} \nonumber\\
&=& 4 (\cancel{\partial}\tilde{\mathcal{F}}) K_{\bar{I}J}[(\cancel{\partial}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J] + (\cancel{\mathcal{D}}\chi^W)_{1f}|_{\tilde{\mathcal{F}}^1} + (\cancel{\mathcal{D}}\chi^W)_{1f}|_{\tilde{\mathcal{F}}^2}.
\end{eqnarray}

where $F^W = 2\Bar{\Lambda}_KP_L \cancel{\mathcal{D}}\Lambda_K + \hat{F}_K^-\cdot \hat{F}_K^- - \mathcal{D}^2_K$, $(\hat{F}_K^-)_{0f}=0$, and $(F^W)_{0f} = -(\mathcal{D}_K^2)_{0f} = -\tilde{\mathcal{F}}^2$, and

\begin{eqnarray}
(\cancel{\partial} \chi^W)_{1f} &=& (\cancel{\partial} P_L\chi^W)_{1f} = 4 (\cancel{\partial}\tilde{\mathcal{F}}) K_{\bar{I}J}[(\cancel{\partial}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J] + 4 \tilde{\mathcal{F}} (\cancel{\partial}K_{\bar{I}J})[(\cancel{\partial}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J]\nonumber\\
&&+4 \tilde{\mathcal{F}} K_{\bar{I}J}[(\cancel{\partial}\cancel{\partial}z^{J})\chi^{\bar{I}}+\cancel{\partial}z^{J}\cancel{\partial}\chi^{\bar{I}}-(\cancel{\partial}\bar{F}^{\bar{I}})\chi^J-\bar{F}^{\bar{I}}(\cancel{\partial}\chi^J)]\nonumber\\
&=& 4 (\cancel{\partial}\tilde{\mathcal{F}}) K_{\bar{I}J}[(\cancel{\partial}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J] + 4 \tilde{\mathcal{F}} (\cancel{\partial}K_{\bar{I}J})[(\cancel{\partial}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J]\nonumber\\
&&+4 \tilde{\mathcal{F}} K_{\bar{I}J}[(\square z^{J})\chi^{\bar{I}}+\cancel{\partial}z^{J}\cancel{\partial}\chi^{\bar{I}}-(\cancel{\partial}\bar{F}^{\bar{I}})\chi^J-\bar{F}^{\bar{I}}(\cancel{\partial}\chi^J)]
\nonumber\\
&\approx& 4 (\cancel{\partial}\tilde{\mathcal{F}}) K_{\bar{I}J}[(\cancel{\partial}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J] + 4 \tilde{\mathcal{F}} (\cancel{\partial}K_{\bar{I}J})[(\cancel{\partial}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J]\nonumber\\
&&-4 \tilde{\mathcal{F}} K_{\bar{I}J}(\cancel{\partial}\bar{F}^{\bar{I}})\chi^J .
\end{eqnarray}

Here $\approx$ means equality up to terms proportional to the equations of motion of free massless matter fields. Such
terms produce only terms that contain additional factors of the matter field masses in the numerator and therefore give rise to either renormalizable operators or nonrenormalizable operators weighted by a mass scale higher than that associated to those
terms that do not vanish on shell. Some details of the above calculation are as follows:

\begin{eqnarray}
\mathcal{D}_{\mu} z^I &=& \partial_{\mu}z^I -\frac{1}{\sqrt{2}}\bar{\psi}_{\mu} \chi^I,\\
\mathcal{C}_{\bar{W}}  &\equiv& \bar{W} = \Bar{\Lambda}_KP_R\Lambda_K =  2~\textrm{fermions} + 4~\textrm{fermions} + 6~\textrm{fermions} ,\\
P_L \Lambda_K &=& -\sqrt{2}iK_{\bar{I}J}[(\cancel{\mathcal{D}}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J] -\frac{i}{\sqrt{2}}K_{\bar{I}\bar{J}K}\chi^{K}\bar{\chi}^{\bar{I}}\chi^J,\\
(\cancel{\mathcal{D}}\chi^{\bar{W}})_{1f} &=& 2\sqrt{2}i(\cancel{\mathcal{D}}\tilde{\mathcal{F}})_{0f}(P_L\Lambda_K)_{1f} + 2\sqrt{2}i \tilde{\mathcal{F}}  (P_L\cancel{\mathcal{D}}\Lambda_K)_{1f},\\
\cancel{\mathcal{D}}\Lambda_K &=& \gamma \cdot \mathcal{D} \Lambda_K,\\
\mathcal{D}_{\mu} \Lambda_K &=& \Big(\partial_{\mu} -\frac{3}{2}b_{\mu} + \frac{1}{4}\omega_{\mu}^{ab} \gamma_{ab} -\frac{3}{2}i\gamma_* \mathcal{A}_{\mu} \Big) \Lambda_K - \Big(\frac{1}{4}\gamma \cdot  \hat{F}^K + \frac{1}{2}i\gamma_* \mathcal{D}_K \Big)\psi_{\mu},\\
(P_L\cancel{\mathcal{D}}\Lambda_K)_{1f} &=& \gamma^{\mu}\mathcal{D}_{\mu}|_{\psi=0} (P_R\Lambda_K)_{1f} + \frac{i}{2} \tilde{\mathcal{F}} \gamma^{\mu}\psi_{\mu},\\
(\gamma \cdot \hat{F}^K)_{0f} &=& 0,\qquad \mathcal{D}_K|_{0f} = \tilde{\mathcal{F}}\\
(\hat{F}^K_{\mu\nu})_{0f} &=& (2\partial_{[\mu} \mathcal{B}_{\nu]}^K)_{0f} = 2i\partial_{[\mu}(K_I\partial_{\nu]} z^I -K_{\bar{I}}\partial_{\nu]} \bar{z}^{\bar{I}}) = 2i(\partial_{[\mu}K_I\partial_{\nu]} z^I -\partial_{[\mu}K_{\bar{I}}\partial_{\nu]} \bar{z}^{\bar{I}}) \nonumber\\
&=& 2i(K_{IJ}\partial_{[\mu}z^{(J}\partial_{\nu]} z^{I)} -K_{\bar{I}\bar{J}}\partial_{[\mu}z^{(\bar{J}}\partial_{\nu]} \bar{z}^{\bar{I})}) = 0,\\
(\cancel{\mathcal{D}}\Upsilon)_{0f} &=&  (\cancel{\partial} -2\gamma^{\mu}b_{\mu} -2i\gamma^{\mu}A_{\mu})\Upsilon \nonumber\\
&=& (\cancel{\partial}s_0)s_0^{*}e^{-K/3} + s_0(\cancel{\partial}s_0^{*})e^{-K/3} + s_0s_0^{*}e^{-K/3}(-\frac{1}{3}\cancel{\partial}K) -2\gamma^{\mu}(b_{\mu}+iA_{\mu})\Upsilon \nonumber\\
&=&  \Upsilon(\frac{1}{s_0}\cancel{\partial} s_0 + \frac{1}{s_0^{*}} \cancel{\partial} s_0^{*} - \frac{1}{3}K_{I}\cancel{\partial}z^I-\frac{1}{3}K_{\bar{I}}\cancel{\partial}\bar{z}^{\bar{I}}-2\gamma^{\mu}(b_{\mu}+iA_{\mu}))
i(\cancel{\mathcal{B}}_{\Upsilon})_{0f} &=&  \Upsilon(-\frac{1}{s_0}\cancel{\partial} s_0 + \frac{1}{s_0^{*}} \cancel{\partial} s_0^{*} + \frac{1}{3}K_{I}\cancel{\partial}z^I-\frac{1}{3}K_{\bar{I}}\cancel{\partial}\bar{z}^{\bar{I}}),\nonumber\\
\frac{(\cancel{\mathcal{D}}\Upsilon)_{0f}+i(\cancel{\mathcal{B}}_{\Upsilon})_{0f} }{\Upsilon}  &=& \frac{2}{s_0^{*}}\cancel{\partial}s_0^{*} -\frac{2}{3}K_{\bar{I}}\cancel{\partial}\bar{z}^{\bar{I}} - 2\gamma^{\mu}(b_{\mu}+iA_{\mu}).
\end{eqnarray}

Finally, we present here the chiral fermions of the superconformal multiplets up to multiple fermion terms.

\begin{eqnarray}
\chi^i =
\begin{cases}
\chi^0,\\
\chi^I,\\
\chi^W = 4 \tilde{\mathcal{F}} K_{\bar{I}J}[(\cancel{\partial}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J] + \cdots 7~\textrm{fermions},\\
\chi^T = \frac{1}{\Upsilon^2}\bigg[~  4 (\cancel{\partial}\tilde{\mathcal{F}}) K_{\bar{I}J}[(\cancel{\partial}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J]- \Big(\frac{2}{s_0^{*}}\cancel{\partial}s_0^{*} -\frac{2}{3}K_{\bar{K}}\cancel{\partial}\bar{z}^{\bar{K}} - 2\gamma^{\mu}(b_{\mu}+iA_{\mu})\Big)
\\ \qquad\qquad\quad  \times 4 \tilde{\mathcal{F}} K_{\bar{I}J}[(\cancel{\partial}z^{J})\chi^{\bar{I}}-\bar{F}^{\bar{I}}\chi^J] ~ \bigg]_{1f}+ \cdots + 9~\textrm{fermions} ,
\end{cases}
\nonumber
\end{eqnarray}

\begin{eqnarray}
\tilde{\mathcal{F}} &=&  2K_{I\bar{J}} \left( -\partial_{\mu} z^I \partial^{\mu} \bar{z}^{\bar{J}} +F^I\bar{F}^{\bar{J}}\right)
\nonumber\\
\cancel{\partial} \tilde{\mathcal{F}}&=& 2(\cancel{\partial}K_{I\bar{J}}) \left( -\partial_{\mu} z^I \partial^{\mu} \bar{z}^{\bar{J}} +F^I\bar{F}^{\bar{J}}\right)
\nonumber\\
&&+2K_{I\bar{J}} \left( -(\cancel{\partial}\partial_{\mu} z^I) \partial^{\mu} \bar{z}^{\bar{J}}-\partial_{\mu} z^I (\cancel{\partial}\partial^{\mu} \bar{z}^{\bar{J}}) +(\cancel{\partial}F^I)\bar{F}^{\bar{J}}+F^I(\cancel{\partial}\bar{F}^{\bar{J}})\right).\nonumber
\end{eqnarray}

Notice that none of the chiral fermions $\chi^i$ vanish on-shell, and that only $\chi^T$ dependens on $\tilde{\mathcal{F}}$ and includes the factor of $\Upsilon^{-2}$. All of these properties affect the mass dimension of the expansion coefficients of the nonrenormalizable Lagrangians.

We finally expand $W$ and $\Bar{W}$ as follows:

\begin{eqnarray}
W &=&  -2K_{\bar{\imath}J}[\bar{\chi}^{J}(\overline{\cancel{\mathcal{D}}\bar{z}^{\bar{\imath}}})-\bar{F}^{\bar{\imath}}\bar{\chi}^{J}]K_{{\bar{\imath}}'J'}[(\cancel{\mathcal{D}}\bar{z}^{{\bar{\imath}}'})\chi^{J'}-F^{J'}\chi^{{\bar{\imath}}'}]
\nonumber\\
&& - K_{{\bar{\imath}}J}[\bar{\chi}^{J}(\overline{\cancel{\mathcal{D}}\bar{z}^{\bar{\imath}}})-\bar{F}^{\bar{\imath}}\bar{\chi}^{J}] K_{\bar{\imath}'\bar{\jmath}'K'}[\chi^{K'}\bar{\chi}^{\bar{\imath}'}\chi^{\bar{\jmath}'}] \nonumber\\
&& - K_{\bar{\imath}\bar{\jmath}K}[\bar{\chi}^{\bar{\jmath}}\chi^{\bar{\imath}}\bar{\chi}^{K}]K_{{\bar{\imath}}'J'}[(\cancel{\mathcal{D}}\bar{z}^{{\bar{\imath}}'})\chi^{J'}-F^{J'}\chi^{{\bar{\imath}}'}]  \nonumber\\
&&- \frac{1}{2}K_{\bar{\imath}\bar{\jmath}K}[\bar{\chi}^{\bar{\jmath}}\chi^{\bar{\imath}}\bar{\chi}^{K}]K_{\bar{\imath}'\bar{\jmath}'K'}[\chi^{K'}\bar{\chi}^{\bar{\imath}'}\chi^{\bar{\jmath}'}],\\
\bar{W} &=& (W)^{*}.
\end{eqnarray}

Notice that $W$ and $\Bar{W}$ can be represented by products of two, four, and six fundamental fermions.

§.§ Additional gauge fixing for physical theory in the liberated supergravity

First of all, we consider the conventional superconformal gauge which is chosen by

\begin{eqnarray}
&&\mathcal{C}_{\Upsilon} = \Upsilon = 1 \Longleftrightarrow s_0\bar{s}_0e^{-K/3}=1,\\ &&\mathcal{Z}_{\Upsilon} = 0 \implies P_L\chi^0 - \frac{1}{3}s_0 K_I P_L\chi^I =0 \quad \& \quad P_R\chi^0 - \frac{1}{3}s_0^{*} K_{\bar{I}} P_R\chi^{\bar{I}} =0, \label{conv1}\\
&& s_0 = \bar{s}_0 \implies s_0 = \bar{s}_0 = e^{K/6}, \label{conv2} \\
&& b_{\mu} = 0. \label{conv3}
\end{eqnarray}

Note that the first condition is the $D$-gauge fixing which gets us to the Einstein frame; the second one is the improved $S$-gauge fixing; the third one is the $A$-gauge fixing; the last one is the $K$-gauge fixing.

To compare our results with the formulation of liberated supergravity in <cit.>, we choose a gauge given by

\begin{eqnarray}
\Lambda_K =0 \Longleftrightarrow \chi^{W}=\chi^{T}=0.\label{gauge-fixing-lib}
\end{eqnarray}

In both the conventional superconformal gauge (<ref>,<ref>,<ref>)  and in (<ref>), the relevant multiplets are

\begin{eqnarray}
S_0 &=& \left(e^{K/6}, \frac{1}{3}e^{K/6} K_I P_L\chi^I, F^0 \right),\qquad
\bar{S}_0 = \left(e^{K/6}, \frac{1}{3}e^{K/6} K_{\bar{I}} P_R\chi^{\bar{I}}, F^{0*} \right),\\
z^I &=& \left( z^I, P_L\chi^I, F^I \right),\qquad
\bar{z}^{\bar{I}} = \left( \bar{z}^{\bar{I}}, P_R\chi^{\bar{I}}, \bar{F}^{\bar{I}} \right),\\
\mathcal{W}^2(K) &=&  \left( W, P_L\chi^W, F^W \right)=\left( 0, 0, \hat{F}_K^-\cdot \hat{F}_K^- - \mathcal{D}^2_K\right),\\
\bar{\mathcal{W}}^2(K) &=&  \left( \bar{W}, P_R\chi^{\bar{W}}, \bar{F}^{\bar{W}} \right)= \left( 0,0, \hat{F}_K^+ \cdot \hat{F}_K^+ - (\mathcal{D}^{*}_K)^2\right),\\
T(\bar{w}^2) &=& \left( T, P_L\chi^T, F^T \right) = \left( -\frac{1}{2}\mathcal{K}_{\bar{w}}, 0, \frac{1}{2}\mathcal{D}_{\bar{w}} \right),\\
\bar{T}(w^2) &=& \left( \bar{X}^{\bar{T}}, P_R\chi^{\bar{T}}, \bar{F}^{\bar{T}} \right) = \left( -\frac{1}{2}\mathcal{K}_{\bar{w}}^{*}, 0, \frac{1}{2}\mathcal{D}_{\bar{w}}^{*} \right).
\end{eqnarray}

Let us further observe that all the terms coupled to $N$ and its derivatives with respect to $0,I,T$ vanish because $N$ contains the product of $W\bar{W}$, which is zero in the liberated gauge, i.e. $W=\bar{W}=0$. We see that all the fermionic terms coupled to the derivatives of $N$ with respect to $W$ or $T$ vanish as well, because these terms always couple to the fermions $\chi^W$ and $\chi^T$ which vanish in the gauge. The only non-vanishing contribution is given by only the bosonic term, $N_{W\bar{W}}F^W\bar{F}^{\bar{W}}$ which gives us the new term $\mathcal{U}$ to the scalar potential. Therefore, in both gauges, the liberated supergravity Lagrangian is specified by

\begin{eqnarray}
\mathcal{L}_{Lib} = \mathcal{L}_{SUGRA} + \mathcal{L}_{NEW},
\end{eqnarray}

where $\mathcal{L}_{SUGRA}$ is the standard supergravity Lagrangian which contains the auxiliary fields $F^0,F^I$ and $\mathcal{L}_{NEW} = \mathcal{U}(z^I,\bar{z}^{\bar{I}})$. Then, with this Lagrangian, after solving the equations of motion for the auxiliary fields, we can obtain the usual supergravity action in addition to the new term.

§ SPECTROSCOPHY FOR NON-RENORMALIZABLE INTERACTIONS IN THE LIBERATED $\MATHCAL{N}=1$ SUPERGRAVITY

In this section, we investigate the suppression of the nonrenormalizable fermionic terms in the liberated $\mathcal{N}=1$ supergravity in the superconformal formalism. To do this, we need to recall the EFT expansion reviewed in Sec. <ref>, and consider the structure of EFT expansion in Eq. (<ref>) and constraint given in Eq. (<ref>). Hence, we have

\begin{eqnarray}
\mathcal{L}_{EFT} \supset \sum_{\delta \geq 0}^{\textrm{Finite N}} \frac{C^{\delta}}{M^{\delta-d}} \mathcal{O}^{(\delta)} \sim \sum_{\delta \geq 0}^{\textrm{Finite N}} \frac{1}{\Lambda_{cut}^{\delta-d}} \mathcal{O}^{(\delta)}  \implies \frac{C^{\delta}}{M^{\delta-d}} \lesssim \frac{1}{\Lambda_{cut}^{\delta-d}} \nonumber
\end{eqnarray}

where $\Lambda_{cut}$ is a cutoff scale; $M$ is a characteristic mass scale of a theory; $C^{\delta}$ is a dimensionless Wilson coefficient, and $\mathcal{O}^{(\delta)}$ is an effective field operator with the mass dimension $\delta$. In our model, the dimensionless Wilson coefficient will be dependent on the liberated term $\mathcal{U}$, and the mass scale $M$ will depend on $\mathcal{F}$. Hence, the key result we will find in this section is given by

\begin{eqnarray}
\mathcal{U}^{(n)} \lesssim \begin{cases}
\mathcal{F}^{2(4-n)} \left(\dfrac{M_{pl}}{\Lambda_{cut}}\right)^{2(4-n)}  \quad \textrm{where}\quad0\leq n \leq 2\quad\textrm{for}~N_{mat} =1,\\
\mathcal{F}^{2(6-n)}\left(\dfrac{M_{pl}}{\Lambda_{cut}}\right)^{2(6-n)}  \quad \textrm{where}\quad0\leq n \leq 4\quad\textrm{for}~N_{mat} \geq 2,
\end{cases}
\end{eqnarray}

where $n$ is the total number of derivatives with respect to the chiral matter multiplets $z^I$; $\mathcal{F} \equiv \left<K_{I\Bar{J}}F^I\Bar{F}^{\Bar{J}}\right>$ is the dimensionless vacuum expectation value of $\tilde{\mathcal{F}}$; $\mathcal{U}(z,\Bar{z})$ is called “liberated term” defined as a dimensionless, gauge-invariant, general real functon of the matter scalars $z^I$'s, and $\mathcal{U}^{(n)}$ is the $n$-th derivative of the liberated term.

§.§ Fermionic terms in liberated $\mathcal{N}=1$ supergravity

The component Lagrangians of the liberated supergravity is given by

\begin{eqnarray}
&& \mathcal{L}_{NEW} \equiv [\mathbb{N}]_De^{-1} \nonumber\\
&&= N_{i\bar{\jmath}} \bigg( -\mathcal{D}_{\mu}z^i\mathcal{D}^{\mu}\bar{z}^{\bar{\jmath}} - \frac{1}{2} \bar{\chi}^i \cancel{\mathcal{D}} \chi^{\bar{\jmath}} - \frac{1}{2} \bar{\chi}^{\bar{\jmath}}\cancel{\mathcal{D}}\chi^i + F^i\bar{F}^{\bar{\jmath}}\bigg)
+\frac{1}{2}\bigg[ N_{ij\bar{k}} \Big( -\bar{\chi}^i\chi^j \bar{F}^{\bar{k}} + \bar{\chi}^i (\cancel{\mathcal{D}}z^j)\chi^{\bar{k}} \Big) +h.c. \bigg]
\nonumber\\&&+ \frac{1}{4}N_{ij\bar{k}\bar{l}} \bar{\chi}^i\chi^j \bar{\chi}^{\bar{k}}\chi^{\bar{l}}
\nonumber\\
&& +\bigg[\frac{1}{2\sqrt{2}}\bar{\psi}\cdot \gamma
\bigg( N_{i\bar{\jmath}} F^i \chi^{\bar{\jmath}} - N_{i\bar{\jmath}} \cancel{\mathcal{D}}\bar{z}^{\bar{\jmath}}\chi^i -\frac{1}{2}N_{ij\bar{k}} \chi^{\bar{k}}\bar{\chi}^i\chi^j\bigg)   +\frac{1}{8} i\varepsilon^{\mu\nu\rho\sigma} \bar{\psi}_{\mu} \gamma_{\nu} \psi_{\rho} \bigg( N_i \mathcal{D}_{\sigma}z^i + \frac{1}{2} N_{i\bar{\jmath}} \bar{\chi}^i \gamma_{\sigma} \chi^{\bar{\jmath}} \nonumber\\
&&+ \frac{1}{\sqrt{2}} N_i \bar{\psi}_{\sigma} \chi^i  \bigg)+h.c. \bigg]
+\frac{1}{6} N \left( -R(\omega) +\frac{1}{2} \bar{\psi}_{\mu} \gamma^{\mu \nu\rho} R'_{\nu\rho}(Q) \right)
-\frac{1}{6\sqrt{2}} \left( N_i\bar{\chi}^i + N_{\bar{\imath}}\bar{\chi}^{\bar{\imath}} \right)\gamma^{\mu\nu} R'_{\mu\nu}(Q),\label{componentLagrangian}
\end{eqnarray}

First of all, focusing only on matter couplings, i.e. looking at terms independent of $\psi$, we read the following terms from Eq. (<ref>)

\begin{eqnarray}
\mathcal{L}_{F1} &\equiv&  -N_{i\bar{\jmath}}\mathcal{D}_{\mu}z^i\mathcal{D}^{\mu}\bar{z}^{\bar{\jmath}}\Big|_{\psi=0},\quad
\mathcal{L}_{F2} \equiv  -\frac{1}{2} N_{i\bar{\jmath}} \bar{\chi}^i \cancel{\mathcal{D}} \chi^{\bar{\jmath}}\Big|_{\psi=0}  ,\nonumber\\
\mathcal{L}_{F3} &\equiv&   -N_{i\bar{\jmath}} F^i\bar{F}^{\bar{\jmath}} \Big|_{\psi=0},\quad
\mathcal{L}_{F4} \equiv -\frac{1}{2} N_{ij\bar{k}} \bar{\chi}^i\chi^j \bar{F}^{\bar{k}}\Big|_{\psi=0},\nonumber\\
\mathcal{L}_{F5} &\equiv& \frac{1}{2}  N_{ij\bar{k}} \bar{\chi}^i (\cancel{\mathcal{D}}z^j)\chi^{\bar{k}}\Big|_{\psi=0} ,\nonumber\\
\mathcal{L}_{F6} &\equiv&  \frac{1}{4}N_{ij\bar{k}\bar{l}} \bar{\chi}^i\chi^j \bar{\chi}^{\bar{k}}\chi^{\bar{l}} \Big|_{\psi=0} ,\quad
\mathcal{L}_{F7} \equiv - \frac{N}{6}R(\omega)|_{\psi=0}.\label{fermi_Lagrangian}
\end{eqnarray}

Here, we observe that the fermionic terms in the  effective Lagrangian contain couplings to the function $\mathcal{U}$
and its derivatives since $N \propto \mathcal{U}$.

The general structure of the fermionic terms can be found as a power series in derivatives of the composite multiplet $N$ (i.e. $N_i,N_{i\bar{\jmath}},N_{ij\bar{k}}$ and $N_{ij\bar{k}\bar{l}}$). The $r$-th derivative of $N$, denoted with
$N_{i\ldots\bar{l}}^{(r)}$ has the following generic form

\begin{eqnarray}
N_{i\ldots\bar{l}}^{(r)} &=& N^{(r=q+p+m+k)}_{q,p,m,k}
= (\partial_0^q\partial_I^{(k-n)}\Upsilon^{-2})\Upsilon^{4+2m}\frac{\mathcal{U}^{(n)}}{\tilde{\mathcal{F}}^{4+2m}}W^{1-p_1}\bar{W}^{1-p_2}
\nonumber\\
(-1)^{q_1+q_2} (q_1+1)!(q_2+1)! s_0^{-(2+q_1)} {s_0^*}^{-(2+q_2)}(\partial_I^{(k-n)}e^{2K/3})\Big) \Upsilon^{4+2m}\frac{\mathcal{U}^{(n)}}{\tilde{\mathcal{F}}^{4+2m}}W^{1-p_1}\bar{W}^{1-p_2}.
\nonumber\\{} \label{Gen_str_N_deriv}
\end{eqnarray}

\begin{eqnarray}
W &=&  -2K_{\bar{\imath}J}[\bar{\chi}^{J}(\overline{\cancel{\mathcal{D}}\bar{z}^{\bar{\imath}}})-\bar{F}^{\bar{\imath}}\bar{\chi}^{J}]K_{{\bar{\imath}}'J'}[(\cancel{\mathcal{D}}\bar{z}^{{\bar{\imath}}'})\chi^{J'}-F^{J'}\chi^{{\bar{\imath}}'}]  - K_{{\bar{\imath}}J}[\bar{\chi}^{J}(\overline{\cancel{\mathcal{D}}\bar{z}^{\bar{\imath}}})-\bar{F}^{\bar{\imath}}\bar{\chi}^{J}] K_{\bar{\imath}'\bar{\jmath}'K'}[\chi^{K'}\bar{\chi}^{\bar{\imath}'}\chi^{\bar{\jmath}'}] \nonumber\\
&& - K_{\bar{\imath}\bar{\jmath}K}[\bar{\chi}^{\bar{\jmath}}\chi^{\bar{\imath}}\bar{\chi}^{K}]K_{{\bar{\imath}}'J'}[(\cancel{\mathcal{D}}\bar{z}^{{\bar{\imath}}'})\chi^{J'}-F^{J'}\chi^{{\bar{\imath}}'}]  - \frac{1}{2}K_{\bar{\imath}\bar{\jmath}K}[\bar{\chi}^{\bar{\jmath}}\chi^{\bar{\imath}}\bar{\chi}^{K}]K_{\bar{\imath}'\bar{\jmath}'K'}[\chi^{K'}\bar{\chi}^{\bar{\imath}'}\chi^{\bar{\jmath}'}],\\
\bar{W} &=& (W)^*,
\end{eqnarray}

$\tilde{\mathcal{F}} \equiv  2K_{I\bar{\jmath}} \left( -\partial_{\mu} z^I \partial^{\mu} \bar{z}^{\bar{\jmath}} +F^I\bar{F}^{\bar{\jmath}}\right)$;
$\mathcal{U}^{(n)}$ ($0 \leq n \leq 4$) is the $n$-th derivative of the
function $\mathcal{U}(z^I,\bar{z}^{\bar{\imath}})$ with respect to $z^I,\bar{z}^I$,
which are the lowest component of the matter chiral  multiplets; $q=q_1+q_2$ where $q_1$
($q_2$) is the order of the derivative w.r.t. the compensator scalar $s_0$ ($s_0^*$); $p=p_1+p_2$ where $p_1$ ($p_2$) is the order of the derivative w.r.t. the field strength multiplet scalar $W$ ($\bar{W}$); $m=m_1+m_2$ where $m_1$ ($m_2$) is the
order of the derivative w.r.t. the chiral projection multiplet scalar $T(\bar{w}^2)$ ($\bar{T}(w^2)$); $k$ is the order of the derivative w.r.t. the matter multiplet scalar $z^I$;
$n$ is the order of the derivative acting on the new term $\mathcal{U}$ w.r.t. the matter multiplet; $q$ is the total order of
derivative w.r.t. the compensator scalars $s_0$ and $s_0^*$.

To find restrictions on $V_{NEW}$ coming from fermionic terms, we have to identify the most singular terms in the
Lagrangian. These terms can be found using the fact that powers of $\tilde{\mathcal{F}}$ in the denominator may lead to a
singularity which gets stronger when $m$ increases by taking more derivatives with respect to the lowest component of the
multiplet $T(\bar{w}^2)$ as seen from Eq. (<ref>). Hence, we will investigate the fermionic terms containing only derivatives with respect to the chiral projection
and matter scalar indices, i.e. $T$ and $I$, in order to find the terms coupled to $\mathcal{U}^{(n)}$ that contain the
maximal inverse powers of $\tilde{\mathcal{F}}$. They are those with $q=p=0$ and $k=n$. We note in particular that
if our theory has a single chiral matter multiplet then the most singular terms are
found to be the couplings to the derivatives proportional to $N_{T\bar{T}}$, $N_{WT\bar{T}}$, $N_{W\bar{W}T\bar{T}}$
while for two or more chiral matter multiplets they are $N_{TT\bar{T}\bar{T}}$. The latter terms vanish identically for a
single multiplet because of Fermi statistics.

First of all, let us examine the single matter chiral multiplet case. Due to Fermi statistics, the possible fermionic terms are
proportional only to three tems, $\mathcal{U}^{(0)}$, $\mathcal{U}^{(1)}$, and $\mathcal{U}^{(2)}$, so that the maximal
order of the derivative with respect to the chiral projection that can appear in the Lagrangians scalar is two and
appears in the terms
proportional to $N_{T\bar{T}}$, $N_{WT\bar{T}}$, and $N_{W\bar{W}T\bar{T}}$. To show that such terms do not
vanish consider

\begin{eqnarray}
\mathcal{L}_{F2}|_{q=p=0,k=n} &\supset&   \Upsilon^{2+2m}\frac{\mathcal{U}^{(n)}}{\tilde{\mathcal{F}}^{4+2m}}W\bar{W}  \bigg[-\frac{1}{2} \Big((\bar{\chi}^I)^{n_1} (\Upsilon^{-2} 4 (\cancel{\partial}\tilde{\mathcal{F}}) K_{\bar{\imath}J}(\cancel{\partial}z^{J})\bar{\chi}^{\bar{\imath}}P_R)^{m_1}\Big)
\nonumber\\
\Big((\cancel{\mathcal{D}}\chi^I)^{n_2} (\cancel{\mathcal{D}}(\Upsilon^{-2} 4 (\cancel{\partial}\tilde{\mathcal{F}}) K_{\bar{\imath}J}(\cancel{\partial}z^{J})P_R\chi^{\bar{\imath}}))^{m_2}\Big)\bigg]_{\psi=0},
\end{eqnarray}

where $m = m_1 + m_2$, $n = n_1 + n_2$, and $2=m+n$. Restoring the mass dimensions by fixing the super-Weyl gauge[Here, we use the convention of the superconformal formalism that all physical bosonic and fermionic matter
fields have dimensions 0 and 1/2 respectively and  $\mathcal{F}$ has dimension 2 while the compensator
$s_0$ has dimension 1 <cit.>. Through dimensional analysis, we find $[\mathcal{D}_{\mu}]=1,[z^i]\equiv 0+[i],[\chi^i]\equiv \frac{1}{2}+[i],[F^i]\equiv 1+[i]$ where $i=0,I,W,T$ and
$[0]=1,[I]=0,[W]=3,[T]=0$.] (i.e. $\Upsilon = M_{pl}^2$, $s_0=s_0^*=M_{pl} e^{K/6}$, $P_L\chi^0 = \frac{1}{3}s_0 K_I P_L\chi^I= \frac{1}{3}M_{pl} e^{K/6} K_I P_L\chi^I$, and $b_{\mu}=0$), we obtain

\begin{eqnarray}
\mathcal{L}_{F2}|_{q=p=0,k=n} &\supset&   M_{pl}^{2(2+2m)}\frac{\mathcal{U}^{(n)}}{\tilde{\mathcal{F}}^{4+2m}} W\bar{W}  \bigg[-\frac{1}{2} \Big((\bar{\chi}^I)^{n_1} (M_{pl}^{-4} 4 (\cancel{\partial}\tilde{\mathcal{F}}) K_{\bar{\imath}J}(\cancel{\partial}z^{J})\bar{\chi}^{\bar{\imath}}P_R)^{m_1}\Big)
\nonumber\\
\Big((\cancel{\mathcal{D}}\chi^I)^{n_2} (M_{pl}^{-4}( 4\cancel{\mathcal{D}} (\cancel{\partial}\tilde{\mathcal{F}}) K_{\bar{\imath}J}(\cancel{\partial}z^{J})P_R\chi^{\bar{\imath}}))^{m_2}\Big)\bigg]_{\psi=0}
\approx M_{pl}^{4} \frac{\mathcal{U}^{(n)}}{\tilde{\mathcal{F}}^{4+2m}} \mathcal{O}_F^{(\delta)},\quad
\end{eqnarray}

where we require $m_a+n_a = 1$ for $a=1,2$ since we are studying the second derivative term $N_{i\bar{\jmath}}$ coupled to $\bar{\chi}^i$ and $\cancel{\mathcal{D}}\chi^{\bar{\jmath}}$.
Redefining $\tilde{\mathcal{F}}$ to be dimensionless by $\tilde{\mathcal{F}} \rightarrow M_{pl}^2 \tilde{\mathcal{F}}$, we obtain

\begin{eqnarray}
\mathcal{L}_{F2}|_{q=p=0,k=n} \supset M_{pl}^{-4-2m} \frac{\mathcal{U}^{(n)}}{\tilde{\mathcal{F}}^{4+2m}}
\mathcal{O}_F'^{(\delta)}.
\end{eqnarray}

We then find $\delta = 8+4m$ by trivial dimensional analysis because the Lagrangian has mass dimension 4.
Then, since $2 = m+ n$, we find that the most singular term is

\begin{eqnarray}
\mathcal{L}_{F2}|_{q=p=0,k=n} \supset M_{pl}^{2(n-4)} \frac{\mathcal{U}^{(n)}}{\tilde{\mathcal{F}}^{2(4-n)}} \mathcal{O}_F'^{(2(6-n))}.\label{SingleChiralMatterCase}
\end{eqnarray}

Next we consider the general case with several multiplets. We shall focus on the fourth derivative term denoted by $N_{ij\bar{k}\bar{l}}$, which gives a four-fermion term. Also, we have to consider the four-fermion product made only of
chiral fermions with $i=0,I,T$ because they do not contribute one
power of the F-term $\tilde{\mathcal{F}}$ in the numerator, which would reduce the number of  inverse power of the F-term
$\tilde{\mathcal{F}}$. This is because the overall factor of $\chi^W$ contains such linear dependence.
The effective fermionic Lagrangian (<ref>) reads then as follows:

\begin{eqnarray}
\mathcal{L}_{F6}|_{q=p=0,k=n}
&\supset& \Upsilon^{2+2m}\frac{\mathcal{U}^{(n)}}{\tilde{\mathcal{F}}^{4+2m}} W\bar{W}
\Big(\frac{1}{4}(\chi^I)^n(4 \Upsilon^{-2} (\cancel{\partial}\tilde{\mathcal{F}}) K_{\bar{\imath}J}(\cancel{\partial}z^{J})P_R\chi^{\bar{\imath}})^m\Big).
\end{eqnarray}

After the super-Weyl gauge fixing  we obtain

\begin{eqnarray}
\mathcal{L}_{F6}|_{q=p=0,k=n}
&\supset& M_{pl}^{2(2+2m)}\frac{\mathcal{U}^{(n)}}{\tilde{\mathcal{F}}^{4+2m}} W\bar{W}
\Big(\frac{1}{4}(\chi^I)^n(4 M_{pl}^{-4} (\cancel{\partial}\tilde{\mathcal{F}}) K_{\bar{\imath}J}(\cancel{\partial}z^{J})P_R\chi^{\bar{\imath}})^m\Big) \nonumber\\
&\approx& c M_{pl}^{4}\frac{\mathcal{U}^{(n)}}{\tilde{\mathcal{F}}^{4+2m}} \mathcal{O}_F^{(\delta)}. \nonumber\\{}
\end{eqnarray}

where $ \mathcal{O}(1)\lesssim c \lesssim \mathcal{O}(10^3)$. After doing the same dimensional analysis as in the
single-multiplet case, we obtain $\delta = 8+4m$ and

\begin{eqnarray}
\mathcal{L}_{F6}|_{q=p=0,k=n} \supset c M_{pl}^{-4-2m}\frac{\mathcal{U}^{(n)}}{\tilde{\mathcal{F}}^{4+2m}} \mathcal{O}_F'^{(8+2m)}.
\end{eqnarray}

Then, since $4 = m+ n$, we can write the most singular terms as

\begin{eqnarray}
\mathcal{L}_{F6}|_{q=p=0,k=n} \supset cM_{pl}^{2(n-6)}\frac{\mathcal{U}^{(n)}}{\tilde{\mathcal{F}}^{2(6-n)}} \mathcal{O}_F'^{(2(8-n))},\label{GeneralChiralMatterCases}
\end{eqnarray}

Since we are going to use liberated supergravity to describe time-dependent backgrounds such as slow-roll inflation
we need to look more closely at the structure of $\tilde{\mathcal{F}}$. From its definition
$\tilde{\mathcal{F}} \equiv  2K_{I\bar{\jmath}} \left( -\partial_{\mu} z^I \partial^{\mu} \bar{z}^{\bar{\jmath}} +F^I\bar{F}^{\bar{\jmath}}\right)$,
we find $\tilde{\mathcal{F}} \equiv  2K_{I\bar{\jmath}} \left( \dot{z}^I\dot{\bar{z}}^{\bar{\jmath}} +F^I\bar{F}^{\bar{\jmath}}\right) >0$
whenever spatial gradients can be neglected. We see that the most singular behaviors of the fermionic terms arises
when $\dot{z}^I=0$. By expanding $\tilde{F}$ around a static vacuum and
reserving the notation $\mathcal{F}$ for the expectation value $\left<K_{I\bar{\jmath}}F^I\bar{F}^{\bar{\jmath}}\right>$, the
effective Lagrangian can finally be rewritten as

* For the  single chiral matter multiplet case,

\begin{eqnarray}
\mathcal{L}_{F2}|_{q=p=0,k=n} \supset  M_{pl}^{2(n-4)} \frac{\mathcal{U}^{(n)}}{\mathcal{F}^{2(4-n)}} \mathcal{O}_F'^{(2(6-n))}.\label{SingleChiralMatterCase_Vac}
\end{eqnarray}

* For two or more chiral matter multiplets,

\begin{eqnarray}
\mathcal{L}_{F6}|_{q=p=0,k=n} \supset c'M_{pl}^{2(n-6)}\frac{\mathcal{U}^{(n)}}{\mathcal{F}^{2(6-n)}} \mathcal{O}_F'^{(2(8-n))}.\label{GeneralChiralMatterCases_Vac}
\end{eqnarray}

where $ \mathcal{O}(10^{-2}) \lesssim c' \lesssim  \mathcal{O}(1)$.

The effective operators we obtained are generically nonzero even after considering possible cancellations due to Fermi
statistics or nonlinear field redefinitions. As an example we can take terms containing $\chi^i$. They are made of two
chiral multiplets $\chi^W$ and $\chi^T$ and these produce terms that do not vanish on shell (i.e. imposing $\cancel{\partial}P_L\chi^I\approx 0$ for matter fermions).
For instance, in a theory with only one matter chiral multiplet $(z,P_L\chi,F)$, we have
$W = -2K_{\bar{z}z}[\{(\cancel{\partial}z)^2- F^*\cancel{\partial}z\}(\bar{\chi}P_R\chi)+\{{F^*}^2-F^*\cancel{\partial}z\}
(\bar{\chi}P_L\chi) ]+2K_{\bar{z}z}K_{\bar{z}\bar{z}z}(\cancel{\partial}z-F^*)(\bar{\chi}P_L\chi) (\bar{\chi}P_R\chi)$, and
$\bar{W} = (W)^*$, so that $W\bar{W} = 4K_{\bar{z}z}^4(|\cancel{\partial}z|^2+|F|^2)||\cancel{\partial}z-F^*|^2(\bar{\chi}
P_L\chi)(\bar{\chi}P_R\chi)$. Hence, looking at the possible fermionic terms from $\mathcal{L}_{F1}$, when $i=z,\bar{\jmath}
=\bar{z}$ (i.e. $q=p=m=0,k=n=2$), we get

\begin{eqnarray}
\mathcal{L}_{F1} &\supset&  \Upsilon^2\frac{\mathcal{U}^{(2)}}{\tilde{\mathcal{F}}^4}W\bar{W}(\partial_{\mu} z\partial^{\mu}\bar{z})
= \Upsilon^2\frac{\mathcal{U}^{(2)}}{\tilde{\mathcal{F}}^4}4K_{\bar{z}z}^4(|\cancel{\partial}z|^2+|F|^2)||\cancel{\partial}z-F^*|^2(\partial_{\mu} z\partial^{\mu}\bar{z})(\bar{\chi}P_L\chi)(\bar{\chi}P_R\chi) . \nonumber\\
\end{eqnarray}

It is easy to see that this operator does not vanish on the mass shell of the matter scalars, $\square z \approx 0$.
As another example, from $\mathcal{L}_{F2}$ we get terms containing up to three matter fermions when we consider $q=m=p_1=0,k=n=1,p_2=p=1$

\begin{eqnarray}
\mathcal{L}_{F2} &\supset& \Upsilon^2\frac{\mathcal{U}^{(1)}}{\tilde{\mathcal{F}}^4} \frac{1}{2} W \bar{\chi} \cancel{\partial}P_R\chi^{\bar{W}} \approx  \Upsilon^2\frac{\mathcal{U}^{(1)}}{\tilde{\mathcal{F}}^4} 4K_{\bar{z}z}^2(\cancel{\partial}\tilde{\mathcal{F}})(\cancel{\partial}z)(\bar{\chi}P_R\chi) P_L(\cancel{\partial}z-F^*)^2\chi|_{\mbox{\small 3-fermion terms}}
\end{eqnarray}

Back to the results in Eqs. (<ref>) and (<ref>), the general effective Lagrangians can be cast in the form

\begin{eqnarray}
\mathcal{L}_F = \Lambda_{cut}^{4-\delta}\mathcal{O}_F'^{(\delta)}=
\begin{cases}
M_{pl}^{2(n-4)} \dfrac{\mathcal{U}^{(n)}}{\mathcal{F}^{2(4-n)}} \mathcal{O}_F'^{(\delta  = 2(6-n))}~\qquad \textrm{for}~N_{mat} =1,\\
c'M_{pl}^{2(n-6)}\dfrac{\mathcal{U}^{(n)}}{\mathcal{F}^{2(6-n)}} \mathcal{O}_F'^{(\delta = 2(8-n))} ~\qquad\textrm{for}~N_{mat} \geq 2.
\end{cases}
\end{eqnarray}

where $ \mathcal{O}(10^{-2}) \lesssim c' \lesssim \mathcal{O}(1)$; $N_{mat}$ and $\Lambda_{cut}$ are defined to be the number of chiral multiplets of matter, and the cutoff scale of our effective theory, respectively.

If we demand that our effective theory describe physics up to the energy scale  $\Lambda_{cut}$ we obtain the following

\begin{eqnarray}
\mathcal{U}^{(n)} \lesssim \begin{cases}
\mathcal{F}^{2(4-n)} \left(\dfrac{M_{pl}}{\Lambda_{cut}}\right)^{2(4-n)}  \quad \textrm{where}\quad0\leq n \leq 2\quad\textrm{for}~N_{mat} =1,\\
\mathcal{F}^{2(6-n)}\left(\dfrac{M_{pl}}{\Lambda_{cut}}\right)^{2(6-n)}  \quad \textrm{where}\quad0\leq n \leq 4\quad\textrm{for}~N_{mat} \geq 2.
\end{cases}\label{pre_inequality}
\end{eqnarray}

A conventional definition of the supersymmetry breaking scale $M_S$ is in terms of F-term expectation value so we define
$M_S^4=M_{pl}^4\mathcal{F}$, so the constraints on $\mathcal{U}^{(n)}$ become

\begin{eqnarray}
\mathcal{U}^{(n)} \lesssim \begin{cases}
\left(\dfrac{M_S}{M_{pl}}\right)^{8(4-n)}\left(\dfrac{M_{pl}}{\Lambda_{cut}}\right)^{2(4-n)} \quad \textrm{where}\quad0\leq n \leq 2\quad\textrm{for}~N_{mat} =1,\\
\left(\dfrac{M_S}{M_{pl}}\right)^{8(6-n)}\left(\dfrac{M_{pl}}{\Lambda_{cut}}\right)^{2(6-n)} \quad\textrm{where}\quad0\leq n \leq 4\quad\textrm{for}~N_{mat} \geq 2.
\end{cases}\label{generic_constraints_here}
\end{eqnarray}

Equation (<ref>) is the crucial one in our paper, as it constrains precisely the new
function $\mathcal{U}$
introduced by liberated $\mathcal{N}=1$ supergravity. The constraint depends on the reduced Planck scale $M_{pl}$, the
supersymmetry breaking scale $M_S$, $\Lambda_{cut}$ and the number of chiral multiplets of matter in the theory.
Of course, when we push both the cutoff and supersymmetry breaking scales to the reduced Planck scale, i.e. $\Lambda_{cut} \sim M_S \sim M_{pl}$, we obtain a model-independent universal constraint

\begin{eqnarray}
\forall n:~ \mathcal{U}^{(n)} \lesssim 1.
\label{Universal_Constraint1}
\end{eqnarray}

A model where supersymmetry is broken at the Planck scale is hardly the most interesting. In the more interesting case that
$M_S\ll M_{pl}$ we need the constraints (<ref>) again, so we need to first determine how many matter
chiral multiplets we have in our theory. The constraints will then depend only on our choice  of
$\Lambda_{cut}$ and $M_S$.

In the rest of this section we will examine the constraints in two cases. The first is the true, post-inflationary vacuum of the
theory. To make a supergravity theory meaningful we want it to be valid at least
up to energies $\Lambda_{cut}\gtrsim M_S$.
The second is slow-roll inflation. In this case we must have $\Lambda_{cut}\gtrsim H$, with $H$ the Hubble constant
during inflation.

For the post-inflationary vacuum the interesting regime is when $M_S$ is relatively small, say
$M_S\sim 10\,\mathrm{TeV} \approx 10^{-15} M_{pl}$ and the effective theory is valid up to an energy scale not smaller
than $M_S$, i.e. $\Lambda_{cut}\gtrsim M_S$. If $\Lambda_{cut} < M_S$ liberated supergravity would be a useless
complication, since in its domain of validity supersymmetry would be always nonlinearly realized.
In the post-inflationary vacuum, for the single matter chiral multiplet case, the constraints (<ref>) thus give for
𝒰^(0) ≲(M_S/M_pl)^32( M_pl/M_S )^8 𝒰^(0) ∼10^-360,

𝒰^(1) ≲(M_S/M_pl)^24( M_pl/M_S )^6𝒰^(1) ∼10^-270,

𝒰^(2) ≲(M_S/M_pl)^16( M_pl/M_S )^4𝒰^(2) ∼10^-180.
Notice that $\mathcal{U}^{(3)},\mathcal{U}^{(4)}$ are not restricted. For two or more matter chiral multiplets, the
constraints  are given by
𝒰^(0) ≲(M_S/M_pl)^48( M_pl/M_S )^12 𝒰^(0) ∼10^-540,

𝒰^(1) ≲(M_S/M_pl)^40( M_pl/M_S )^10𝒰^(1) ∼10^-450,

𝒰^(2) ≲(M_S/M_pl)^32( M_pl/M_S )^8𝒰^(2) ∼10^-360.
From the constraints on
$\mathcal{U}^{(0)}$ and $\mathcal{U}^{(2)}$, we find that the liberated scalar potential contributes only a negligibly small
cosmological constant and negligibly small corrections to the mass terms
of the chiral multiplet scalars. For the single chiral multiplet case, restoring dimensions we get a vacuum energy density
𝒰^(0) ≲10^-360 M_pl^4
and scalar masses
M_z ≲M_pl √(|𝒰^(2)|) = 10^-90M_pl.
These constraints become even tighter if the theory contains more than one chiral multiplet, but the ones we obtained are
already so stringent as to rule out any observable contribution to the cosmological constant and scalar masses from the
new terms made possible by liberated supergravity. We can say that  Eqs. (<ref>) and (<ref>) already send
liberated supergravity back to prison after the end of inflation.

The constraints during inflation instead can be easily satisfied if during inflation the supersymmetry breaking scale is
very high, say $M_S = M_{pl}$. In that case, $\mathcal{U}^{(0)} \lesssim \mathcal{O}(1)$.
After inflation the “worst case scenario” constraints coming from Eq. (<ref>) with $N_{mat}\geq 2$ and
$M_S = 10^{-15}M_{pl}$ are

\begin{eqnarray}
\forall n:~\mathcal{U}^{(n)} \lesssim 10^{-120(6-n)}.\label{Option2_Const_after_inf}
\end{eqnarray}

A simple way to satisfy all these constraints is to choose a no scale structure for the supersymmetric part of the scalar potential. This ensures the vanishing of the F-term contribution to the potential independently of the magnitude of the

\begin{eqnarray}
V_F =e^G(G_IG^{I\bar{\jmath}}G_{\bar{\jmath}}-3) =0,\label{No_scale_structure}
\end{eqnarray}

The total scalar potential is then given by $V = V_D+V_{NEW}$. Thanks to the no-scale structure, we can have
both $M_S\sim M_{pl}$ and $\mathcal{U}^{(0)} \sim H^2 \sim 10^{-10}$ during inflation.

Our scenario has $M_S = M_{pl}$ during inflation and $M_S = 10^{-15}M_{pl}$ at the true vacuum in the post-inflation phase, so we see that to satisfy all constraints a transition between the two different epochs must occur, for which the scale of the composite F-term $\mathcal{F}$ changes from $\mathcal{O}(M_{pl})$ to $\mathcal{O}(10^{-15}M_{pl})$.

CHAPTER: REVISITING NEW KÄHLER-INVARIANT FAYET-ILIOPOULOS TERMS IN THE SUPERCONFORMAL FORMALISM

[1.05]This chapter is based on the author's original work in Ref. <cit.>.

§ COMPONENT ACTION OF NEW FI TERM IN SUPERCONFORMAL TENSOR CALCULUS

In this chapter, we review the component action of a new, Kähler invariant Fayet-Iliopoulos term proposed by Antoniadis, Chatrabhuti, Isono, and Knoops <cit.>, using again the superconformal tensor calculus. The full Lagrangian with the new FI terms is given by

\begin{eqnarray}
\mathcal{L} &=& -3[S_0\bar{S}_0e^{-K(z,\bar{z})}]_D + ([S_0^3W(z)]_F -\frac{1}{4} [\bar{\lambda}P_L\lambda]_F+h.c.) \nonumber\\
&&- \xi \left[(S_0\bar{S}_0e^{-K(z,\bar{z})})^{-3}\frac{(\bar{\lambda}P_L\lambda)(\bar{\lambda}P_R\lambda)}{T(\bar{w}'^2)\bar{T}(w'^2)}(V)_D\right]_D,
\end{eqnarray}

where the last term coupled to the parameter $\xi$ corresponds to the new FI terms.

We consider matter chiral multiplets $Z^i$, the chiral compensator $S_0$, a real multiplet $V$, and another real multiplet
$(V)_D$, whose lowest component is the auxiliary D term of the real multiplet $V$. Their superconformal multiplets are given as follows:

\begin{eqnarray}
&& V = \{0,0,0,0,A_{\mu},\lambda,D\} ~\textrm{in the Wess-Zumino gauge,~i.e.}~v=\zeta=\mathcal{H}=0, \\
&& Z^i = (z^i,-i\sqrt{2}P_L\chi^i,-2F^i,0,+i\mathcal{D}_{\mu}z^i,0,0) = \{ z^i, P_L\chi^i,F^i\},\\
&& \bar{Z}^{\bar{i}} = (\bar{z}^{\bar{i}},+i\sqrt{2}P_R\chi^{\bar{i}},0,-2\bar{F}^{\bar{i}},-i\mathcal{D}_{\mu}\bar{z}^{\bar{i}},0,0) = \{ \bar{z}^{\bar{i}}, P_R\chi^{\bar{i}},\bar{F}^{\bar{i}}\},\\
&& S_0 = (s_0,-i\sqrt{2}P_L\chi^0,-2F_0,0,+i\mathcal{D}_{\mu}s_0,0,0) = \{ s_0, P_L\chi^0,F_0\},\\
&& \bar{S}_0 = (\bar{s}_0,+i\sqrt{2}P_R\chi^0,0,-2\bar{F}_0,-i\mathcal{D}_{\mu}\bar{s}_0,0,0) = \{\bar{s}_0, P_R\chi^0,\bar{F}_0\},\\
&& \bar{\lambda}P_L\lambda = (\bar{\lambda}P_L\lambda,-i\sqrt{2}P_L\Lambda,2D_-^2,0,+i\mathcal{D}_{\mu}(\bar{\lambda}P_L\lambda),0,0) = \{\bar{\lambda}P_L\lambda, P_L\Lambda,-D_-^2\},\\
&& \bar{\lambda}P_R\lambda = (\bar{\lambda}P_R\lambda,+i\sqrt{2}P_R\Lambda,0,2D_+^2,-i\mathcal{D}_{\mu}(\bar{\lambda}P_R\lambda),0,0) = \{\bar{\lambda}P_R\lambda, P_R\Lambda,-D_+^2\},\\
&& (V)_D = (D,\cancel{\mathcal{D}}\lambda,0,0,\mathcal{D}^{b}\hat{F}_{ab},-\cancel{\mathcal{D}}\cancel{\mathcal{D}}\lambda,-\square^CD),
\end{eqnarray}

\begin{eqnarray}
&& P_L\Lambda \equiv \sqrt{2}P_L(-\frac{1}{2}\gamma\cdot \hat{F} + iD)\lambda,\qquad P_R\Lambda \equiv \sqrt{2}P_R(-\frac{1}{2}\gamma\cdot \hat{F} - iD)\lambda,\\
&& D_-^2 \equiv D^2 - \hat{F}^-\cdot\hat{F}^- - 2  \bar{\lambda}P_L\cancel{\mathcal{D}}\lambda,\qquad D_+^2 \equiv D^2 - \hat{F}^+\cdot\hat{F}^+ - 2  \bar{\lambda}P_R\cancel{\mathcal{D}}\lambda,\\
&& \mathcal{D}_{\mu}\lambda \equiv \bigg(\partial_{\mu}-\frac{3}{2}b_{\mu}+\frac{1}{4}w_{\mu}^{ab}\gamma_{ab}-\frac{3}{2}i\gamma_*\mathcal{A}_{\mu}\bigg)\lambda - \bigg(\frac{1}{4}\gamma^{ab}\hat{F}_{ab}+\frac{1}{2}i\gamma_* D\bigg)\psi_{\mu}
\\
&& \hat{F}_{ab} \equiv F_{ab} + e_a^{~\mu}e_b^{~\nu} \bar{\psi}_{[\mu}\gamma_{\nu]}\lambda,\qquad F_{ab} \equiv e_a^{~\mu}e_b^{~\nu} (2\partial_{[\mu}A_{\nu]}),\\
&& \hat{F}^{\pm}_{\mu\nu} \equiv \frac{1}{2}(\hat{F}_{\mu\nu}\pm \tilde{\hat{F}}_{\mu\nu}), \qquad \tilde{\hat{F}}_{\mu\nu} \equiv -\frac{1}{2} i\epsilon_{\mu\nu\rho\sigma}\hat{F}^{\rho\sigma} .
\end{eqnarray}

§.§  Composite complex multiplets: (Weyl/Chiral) weights  and

We show here the components of the first superconformal composite complex multiplets $w'^2$ and $\Bar{w}'^2$ with Weyl/chiral weights $(-1,3)$ and $(-1,-3)$ respectively. These composite multiplets are defined to be

\begin{eqnarray}
&& w'^2 \equiv \frac{\bar{\lambda}P_L\lambda}{(S_0\bar{S}_0e^{-K/3})^2} = \{\mathcal{C}_w,\mathcal{Z}_w,\mathcal{H}_w,\mathcal{K}_w,\mathcal{B}^w_{\mu},\Lambda_w,\mathcal{D}_w\} \\
&& \bar{w}'^2 \equiv \frac{\bar{\lambda}P_R\lambda}{(S_0\bar{S}_0e^{-K/3})^2}
= \{\mathcal{C}_{\bar{w}},\mathcal{Z}_{\bar{w}},\mathcal{H}_{\bar{w}},\mathcal{K}_{\bar{w}},\mathcal{B}^{\bar{w}}_{\mu},\Lambda_{\bar{w}},\mathcal{D}_{\bar{w}}\}.
\end{eqnarray}

\begin{eqnarray}
\mathcal{C}_w &=& h \equiv \frac{\bar{\lambda}P_L\lambda}{(s_0\bar{s}_0e^{-K(z,\bar{z})/3})^2},\\
\mathcal{Z}_w &=& i\sqrt{2}(-h_a\Omega^a + h_{\bar{a}}\Omega^{\bar{a}}),\\
\mathcal{H}_w &=& -2h_aF^a + h_{ab}\bar{\Omega}^a\Omega^b,\\
\mathcal{K}_w &=& -2h_{\bar{a}}F^{\bar{a}} + h_{\bar{a}\bar{b}}\bar{\Omega}^{\bar{a}}\Omega^{\bar{b}},\\
\mathcal{B}^w_{\mu} &=& ih_a\mathcal{D}_{\mu}X^a-ih_{\bar{a}}\mathcal{D}_{\mu}\bar{X}^{\bar{a}}+ih_{a\bar{b}}\bar{\Omega}^{a}\gamma_{\mu}\Omega^{\bar{b}},\\
P_L\Lambda_w &=& -\sqrt{2}ih_{\bar{a}b}[(\cancel{\mathcal{D}}X^b)\Omega^{\bar{a}}-F^{\bar{a}}\Omega^b]-\frac{i}{\sqrt{2}}h_{\bar{a}\bar{b}c}\Omega^c\bar{\Omega}^{\bar{a}}\Omega^{\bar{b}},\\
P_R\Lambda_w &=& \sqrt{2}ih_{a\bar{b}}[(\cancel{\mathcal{D}}\bar{X}^{\bar{b}})\Omega^{a}-F^{a}\Omega^{\bar{b}}]+\frac{i}{\sqrt{2}}h_{ab\bar{c}}\Omega^{\bar{c}}\bar{\Omega}^{a}\Omega^{b},\\
\mathcal{D}_w &=& 2h_{a\bar{b}}\Big(-\mathcal{D}_{\mu}X^a\mathcal{D}^{\mu}\bar{X}^{\bar{b}}-\frac{1}{2}\bar{\Omega}^aP_L\cancel{\mathcal{D}}\Omega^{\bar{b}}-\frac{1}{2}\bar{\Omega}^{\bar{b}}P_R\cancel{\mathcal{D}}\Omega^a+F^aF^{\bar{b}}\Big) \nonumber\\
&&+h_{ab\bar{c}}(-\bar{\Omega}^a\Omega^bF^{\bar{c}}+\bar{\Omega}^a(\cancel{\mathcal{D}}X^b)\Omega^{\bar{c}})+ h_{\bar{a}\bar{b}c}(-\bar{\Omega}^{\bar{a}}\Omega^{\bar{b}}F^{c}+\bar{\Omega}^{\bar{a}}(\cancel{\mathcal{D}}\bar{X}^{\bar{b}})\Omega^{c}) \nonumber\\
&&+ \frac{1}{2}h_{ab\bar{c}\bar{d}}(\bar{\Omega}^aP_L\Omega^b)(\bar{\Omega}^{\bar{c}}P_R\Omega^{\bar{d}}).
\end{eqnarray}

Notice that when finding the multiplet $\bar{w}'^2$, we can just replace $h$ by its complex conjugate $h^{*}$.

§.§  chiral projection multiplets: (Weyl/Chiral) weights

The second superconformal multiplets that we need are the composite chiral projection multiplets $T(\Bar{w}'^2)$ and $\Bar{T}(w'^2)$ with Weyl/chiral weights $(0,0)$. From their component supermultiplets defined by

\begin{eqnarray}
T(\bar{w}'^2) &=& \left( -\frac{1}{2}\mathcal{K}_{\bar{w}}, -\frac{1}{2} \sqrt{2} iP_L (\cancel{\mathcal{D}}\mathcal{Z}_{\bar{w}}+\Lambda_{\bar{w}}), \frac{1}{2}(\mathcal{D}_{\bar{w}}+\square^C \mathcal{C}_{\bar{w}} + i\mathcal{D}_a \mathcal{B}^a_{\bar{w}}) \right),\\
\bar{T}(w'^2) &=& \left( -\frac{1}{2}\mathcal{K}_{\bar{w}}^{*}, \frac{1}{2} \sqrt{2} iP_R (\cancel{\mathcal{D}}\mathcal{Z}_{\bar{w}}^C+\Lambda_{\bar{w}}^C), \frac{1}{2}(\mathcal{D}_{\bar{w}}^{*}+\square^C \mathcal{C}_{\bar{w}}^{*} - i\mathcal{D}_a (\mathcal{B}^a_{\bar{w}})^{*}) \right)
\end{eqnarray}

we find the corresponding superconformal multiplets and their complex conjugates as follows:

\begin{eqnarray}
T \equiv T(\bar{w}'^2) &=& \{\mathcal{C}_T,\mathcal{Z}_T,\mathcal{H}_T,\mathcal{K}_T,\mathcal{B}_{\mu}^T,\Lambda_T,\mathcal{D}_T\} \nonumber\\
\bar{T} \equiv \bar{T}(w'^2) &=& \{\mathcal{C}_{\bar{T}},\mathcal{Z}_{\bar{T}},\mathcal{H}_{\bar{T}},\mathcal{K}_{\bar{T}},\mathcal{B}_{\mu}^{\bar{T}},\Lambda_{\bar{T}},\mathcal{D}_{\bar{T}}\},
\end{eqnarray}

whose superconformal components are given by

\begin{eqnarray}
\mathcal{C}_T &=&  -\frac{1}{2} \mathcal{K}_{\bar{w}} = h^{*}_{\bar{a}}F^{\bar{a}} -\frac{1}{2} h^{*}_{\bar{a}\bar{b}}\bar{\Omega}^{\bar{a}}\Omega^{\bar{b}} \equiv C_T\\
\mathcal{Z}_T &=& -\sqrt{2}iP_L\bigg[\cancel{\mathcal{D}}(-h^{*}_a\Omega^a + h^{*}_{\bar{a}}\Omega^{\bar{a}})-h^{*}_{\bar{a}b}[(\cancel{\mathcal{D}}X^b)\Omega^{\bar{a}}-F^{\bar{a}}\Omega^b]-\frac{1}{2}h^{*}_{\bar{a}\bar{b}c}\Omega^c\bar{\Omega}^{\bar{a}}\Omega^{\bar{b}}\nonumber\\
&&\qquad\qquad+h^{*}_{a\bar{b}}[(\cancel{\mathcal{D}}\bar{X}^{\bar{b}})\Omega^{a}-F^{a}\Omega^{\bar{b}}]+\frac{1}{2}h^{*}_{ab\bar{c}}\Omega^{\bar{c}}\bar{\Omega}^{a}\Omega^{b}\bigg] \equiv -\sqrt{2}iP_L\Omega_T ,\\
\mathcal{H}_T &=& -2\bigg[h^{*}_{a\bar{b}}\Big(-\mathcal{D}_{\mu}X^a\mathcal{D}^{\mu}\bar{X}^{\bar{b}}-\frac{1}{2}\bar{\Omega}^aP_L\cancel{\mathcal{D}}\Omega^{\bar{b}}-\frac{1}{2}\bar{\Omega}^{\bar{b}}P_R\cancel{\mathcal{D}}\Omega^a+F^aF^{\bar{b}}\Big) \nonumber\\
&&+\frac{1}{2}h^{*}_{ab\bar{c}}(-\bar{\Omega}^a\Omega^bF^{\bar{c}}+\bar{\Omega}^a(\cancel{\mathcal{D}}X^b)\Omega^{\bar{c}})+\frac{1}{2} h^{*}_{\bar{a}\bar{b}c}(-\bar{\Omega}^{\bar{a}}\Omega^{\bar{b}}F^{c}+\bar{\Omega}^{\bar{a}}(\cancel{\mathcal{D}}\bar{X}^{\bar{b}})\Omega^{c}) \nonumber\\
&&+ \frac{1}{4}h^{*}_{ab\bar{c}\bar{d}}(\bar{\Omega}^aP_L\Omega^b)(\bar{\Omega}^{\bar{c}}P_R\Omega^{\bar{d}})+\frac{1}{2}\square^C h^{*} + \frac{1}{2}i\mathcal{D}^{\mu} (ih^{*}_a\mathcal{D}_{\mu}X^a-ih^{*}_{\bar{a}}\mathcal{D}_{\mu}\bar{X}^{\bar{a}}+ih^{*}_{a\bar{b}}\bar{\Omega}^{a}\gamma_{\mu}\Omega^{\bar{b}})\bigg] \nonumber\\
&\equiv& -2F_T,\\
\mathcal{K}_T &=& 0,\\
\mathcal{B}^T_{\mu} &=& -i\mathcal{D}_{\mu}\mathcal{C}_T,\\
\Lambda_T &=& 0 ,\\
\mathcal{D}_T &=& 0,
\end{eqnarray}

where we used $a,b,c,d = 0,i(\equiv z^i),W (\equiv \bar{\lambda}P_L\lambda)$. This gives the superfield components of the chiral projection multiplet $T$:

\begin{eqnarray}
T(\bar{w}'^2) = ( C_T, P_L\Omega_T, F_T )
\end{eqnarray}

\begin{eqnarray}
C_T &=&  h^{*}_{\bar{a}}F^{\bar{a}} -\frac{1}{2} h^{*}_{\bar{a}\bar{b}}\bar{\Omega}^{\bar{a}}\Omega^{\bar{b}},\\
P_L\Omega_T &=& \cancel{\mathcal{D}}(-h^{*}_a\Omega^a + h^{*}_{\bar{a}}\Omega^{\bar{a}})-h^{*}_{\bar{a}b}[(\cancel{\mathcal{D}}X^b)\Omega^{\bar{a}}-F^{\bar{a}}\Omega^b]-\frac{1}{2}h^{*}_{\bar{a}\bar{b}c}\Omega^c\bar{\Omega}^{\bar{a}}\Omega^{\bar{b}}\nonumber\\
F_T &=&  h^{*}_{a\bar{b}}\Big(-\mathcal{D}_{\mu}X^a\mathcal{D}^{\mu}\bar{X}^{\bar{b}}-\frac{1}{2}\bar{\Omega}^aP_L\cancel{\mathcal{D}}\Omega^{\bar{b}}-\frac{1}{2}\bar{\Omega}^{\bar{b}}P_R\cancel{\mathcal{D}}\Omega^a+F^aF^{\bar{b}}\Big) \nonumber\\
&&+\frac{1}{2}h^{*}_{ab\bar{c}}(-\bar{\Omega}^a\Omega^bF^{\bar{c}}+\bar{\Omega}^a(\cancel{\mathcal{D}}X^b)\Omega^{\bar{c}})+\frac{1}{2} h^{*}_{\bar{a}\bar{b}c}(-\bar{\Omega}^{\bar{a}}\Omega^{\bar{b}}F^{c}+\bar{\Omega}^{\bar{a}}(\cancel{\mathcal{D}}\bar{X}^{\bar{b}})\Omega^{c}) \nonumber\\
&&+ \frac{1}{4}h^{*}_{ab\bar{c}\bar{d}}(\bar{\Omega}^aP_L\Omega^b)(\bar{\Omega}^{\bar{c}}P_R\Omega^{\bar{d}})+\frac{1}{2}\square^C h^{*} - \frac{1}{2}\mathcal{D}^{\mu} (h^{*}_a\mathcal{D}_{\mu}X^a-h^{*}_{\bar{a}}\mathcal{D}_{\mu}\bar{X}^{\bar{a}}+h^{*}_{a\bar{b}}\bar{\Omega}^{a}\gamma_{\mu}\Omega^{\bar{b}}). \nonumber\\{}
\end{eqnarray}

\begin{eqnarray}
\bar{T}(w'^2) = \{ C_T^{*}, P_R\Omega_T, F_T^{*}\}
\end{eqnarray}

\begin{eqnarray}
C_T^{*} &=&  h_{a}F^{a} -\frac{1}{2} h_{ab}\bar{\Omega}^{a}\Omega^{b},\\
P_R\Omega_T &=& \cancel{\mathcal{D}}(-h_a\Omega^a + h_{\bar{a}}\Omega^{\bar{a}})-h_{\bar{a}b}[(\cancel{\mathcal{D}}X^b)\Omega^{\bar{a}}-F^{\bar{a}}\Omega^b]-\frac{1}{2}h_{\bar{a}\bar{b}c}\Omega^c\bar{\Omega}^{\bar{a}}\Omega^{\bar{b}}\nonumber\\
F_T^{*} &=&  h_{a\bar{b}}\Big(-\mathcal{D}_{\mu}X^a\mathcal{D}^{\mu}\bar{X}^{\bar{b}}-\frac{1}{2}\bar{\Omega}^aP_L\cancel{\mathcal{D}}\Omega^{\bar{b}}-\frac{1}{2}\bar{\Omega}^{\bar{b}}P_R\cancel{\mathcal{D}}\Omega^a+F^aF^{\bar{b}}\Big) \nonumber\\
&&+\frac{1}{2}h_{ab\bar{c}}(-\bar{\Omega}^a\Omega^bF^{\bar{c}}+\bar{\Omega}^a(\cancel{\mathcal{D}}X^b)\Omega^{\bar{c}})+\frac{1}{2} h_{\bar{a}\bar{b}c}(-\bar{\Omega}^{\bar{a}}\Omega^{\bar{b}}F^{c}+\bar{\Omega}^{\bar{a}}(\cancel{\mathcal{D}}\bar{X}^{\bar{b}})\Omega^{c}) \nonumber\\
&&+ \frac{1}{4}h_{ab\bar{c}\bar{d}}(\bar{\Omega}^aP_L\Omega^b)(\bar{\Omega}^{\bar{c}}P_R\Omega^{\bar{d}})+\frac{1}{2}\square^C h - \frac{1}{2}\mathcal{D}^{\mu} (h_a\mathcal{D}_{\mu}X^a-h_{\bar{a}}\mathcal{D}_{\mu}\bar{X}^{\bar{a}}+h_{a\bar{b}}\bar{\Omega}^{\bar{b}}\gamma_{\mu}\Omega^{a}). \nonumber\\{}
\end{eqnarray}

§.§ Composite real multiplet : (Weyl/Chiral) weights

We present here a superconformal composite real multiplet $\mathcal{R}$ with Weyl/chiral weights $(0,0)$. Defining some chiral multiplets $\mathcal{X}^A \equiv \{X^A,P_L\Omega^A,F^A\}$ where $A=\{ S_0,Z^i,\bar{\lambda}P_L\lambda,T(\bar{w}'^2)$} and their conjugates, we represent the composite one $\mathcal{R}$ as

\begin{eqnarray}
\mathcal{R} \equiv (S_0\bar{S}_0e^{-K/3})^{-3} \frac{(\bar{\lambda}P_L\lambda)(\bar{\lambda}P_R\lambda)}{T(\bar{w}'^2) \bar{T}(w'^2)}
\end{eqnarray}

whose lowest component is

\begin{eqnarray}
\mathcal{C}_{\mathcal{R}}  \equiv (s_0\bar{s}_0e^{-K/3})^{-3}\frac{(\bar{\lambda}P_L\lambda)(\bar{\lambda}P_R\lambda)}{C_TC_{\bar{T}}}\equiv f(X^A,\bar{X}^{\bar{A}}) \label{def-f}
\end{eqnarray}

where $C_T = -D_+^2 \Delta^{-2}$; $C_{\bar{T}} = -D_-^2\Delta^{-2}$, and $\Delta \equiv s_0\bar{s}_0e^{-K/3}$. Then, the superconformal multiplet of the new Fayet-Iliopoulos term can be written by using

\begin{eqnarray}
\mathcal{R}\cdot (V)_D = \{\tilde{\mathcal{C}},\tilde{\mathcal{Z}},\tilde{\mathcal{H}},\tilde{\mathcal{K}},\tilde{\mathcal{B}}_{\mu},\tilde{\Lambda},\tilde{\mathcal{D}}\},
\end{eqnarray}

whose superconformal multiplet components are as follows:

\begin{eqnarray}
\tilde{\mathcal{C}} &=& Df,\\
\tilde{\mathcal{Z}} &=& f\cancel{\mathcal{D}}\lambda+Di\sqrt{2}(-f_{A}\Omega^A+f_{\bar{A}}\Omega^{\bar{A}}),\\
\tilde{\mathcal{H}} &=& D(-2f_AF^A + f_{AB}\bar{\Omega}^A\Omega^B)-i\sqrt{2}(-f_{A}\bar{\Omega}^A+f_{\bar{A}}\bar{\Omega}^{\bar{A}})P_L\cancel{\mathcal{D}}\lambda,\\
\tilde{\mathcal{K}} &=& D(-2f_{\bar{A}}F^{\bar{A}} + f_{\bar{A}\bar{B}}\bar{\Omega}^{\bar{A}}\Omega^{\bar{B}})-i\sqrt{2}(-f_{A}\bar{\Omega}^A+f_{\bar{A}}\bar{\Omega}^{\bar{A}})P_R\cancel{\mathcal{D}}\lambda,\\
\tilde{\mathcal{B}} &=& (\mathcal{D}^{\nu}\hat{F}_{\mu\nu})f+D(if_A\mathcal{D}_{\mu}X^A-if_{\bar{A}}\mathcal{D}_{\mu}\bar{X}^{\bar{A}}+if_{A\bar{B}}\bar{\Omega}^A\gamma_{\mu}\Omega^{\bar{B}}),\\
\tilde{\Lambda} &=& -f\cancel{\mathcal{D}}\cancel{\mathcal{D}}\lambda + D(P_L\Lambda^f+P_R\Lambda^f)+\frac{1}{2}\Big(\gamma_*(-f_A\cancel{\mathcal{D}}X^A+f_{\bar{A}}\cancel{\mathcal{D}}\bar{X}^{\bar{A}}-f_{A\bar{B}}\bar{\Omega}^A\cancel{\gamma}\Omega^{\bar{B}})\nonumber\\
&&+P_L(-2f_{\bar{A}}F^{\bar{A}} + f_{\bar{A}\bar{B}}\bar{\Omega}^{\bar{A}}\Omega^{\bar{B}})+P_R(-2f_AF^A + f_{AB}\bar{\Omega}^A\Omega^B) -\cancel{\mathcal{D}}f\Big)\cancel{\mathcal{D}}\lambda\nonumber\\
&& +\frac{1}{2}\Big( i\gamma_*\gamma^{\mu}\mathcal{D}^{\nu}\hat{F}_{\mu\nu}   -\cancel{\mathcal{D}}D\Big)i\sqrt{2}(-f_{A}\Omega^A+f_{\bar{A}}\Omega^{\bar{A}}) ,\\
\tilde{\mathcal{D}} &=&-f\square^C D + D
\bigg\{ 2f_{A\bar{B}}(-\mathcal{D}_{\mu}X^A\mathcal{D}^{\mu}\bar{X}^{\bar{B}}-\frac{1}{2}\bar{\Omega}^AP_L\cancel{\mathcal{D}}\Omega^{\bar{B}}-\frac{1}{2}\cancel{\Omega}^{\bar{B}}P_R\cancel{\mathcal{D}}\Omega^A+F^AF^{\bar{B}}) \nonumber\\
&&+f_{AB\bar{C}}(-\bar{\Omega}^A\Omega^B F^{\bar{C}} + \bar{\Omega}^A(\cancel{\mathcal{D}}X^B)\Omega^{\bar{C}})
+f_{\bar{A}\bar{B}C}(-\bar{\Omega}^{\bar{A}}\Omega^{\bar{B}} F^C + \bar{\Omega}^{\bar{A}}(\cancel{\mathcal{D}}\bar{X}^{\bar{B}})\Omega^C) \nonumber\\
&&+\frac{1}{2}f_{AB\bar{C}\bar{D}} (\bar{\Omega}^AP_L\Omega^B)(\bar{\Omega}^{\bar{C}}P_R\Omega^{\bar{D}}) \bigg\}\nonumber\\
&& -(\mathcal{D}_{\nu}\hat{F}^{\mu\nu})(if_A\mathcal{D}_{\mu}X^A-if_{\bar{A}}\mathcal{D}_{\mu}\bar{X}^{\bar{A}}+if_{A\bar{B}}\bar{\Omega}^A\gamma_{\mu}\Omega^{\bar{B}}) \nonumber\\
&& + \bigg( \sqrt{2}if_{\bar{A}B}[(\cancel{\mathcal{D}}X^B)\Omega^{\bar{A}}-F^{\bar{A}}\Omega^B]  +\frac{i}{\sqrt{2}}f_{\bar{A}\bar{B}C} \Omega^C\bar{\Omega}^{\bar{A}}\Omega^{\bar{B}} \bigg)\cancel{\mathcal{D}}\lambda\nonumber\\
&& - \bigg( \sqrt{2}if_{A\bar{B}}[(\cancel{\mathcal{D}}\bar{X}^{\bar{B}})\Omega^{A}-F^{A}\Omega^{\bar{B}}]  +\frac{i}{\sqrt{2}}f_{AB\bar{C}} \Omega^{\bar{C}}\bar{\Omega}^{A}\Omega^{B} \bigg)\cancel{\mathcal{D}}\lambda\nonumber\\
\end{eqnarray}

where the indices $A,B,C,D$ run over $0,i,W,T$. The component action of the new FI term is then given by the D-term density formula

\begin{eqnarray}
\mathcal{L}_{NEW} \equiv -[\xi \mathcal{R}\cdot (V)_D]_D &=& -\frac{\xi}{4}\int d^4x e \bigg[  \tilde{\mathcal{D}} -\frac{1}{2}\bar{\psi}\cdot \gamma i\gamma_* \tilde{\Lambda} -\frac{1}{3}\tilde{\mathcal{C}}R(\omega)\nonumber\\
\end{eqnarray}

§.§ Bosonic term of the new FI term

The new FI term is obtained from the term $D^2f_{W\bar{W}}F^WF^{\bar{W}}$ inside the D-term $\tilde{\mathcal{D}}$. Thus, we get

\begin{eqnarray}
\mathcal{L}_{NEW} &\supset& -\xi D f_{W\bar{W}}F^WF^{\bar{W}} = -\xi D \frac{ (s_0\bar{s}_0e^{-K/3})^{-3}}{C_TC_T^{*}}F^WF^{\bar{W}}
\supset
-\xi D\frac{ (s_0\bar{s}_0e^{-K/3})^{-3}}{(h_WF^W)(h^{*}_{\bar{W}}F^{\bar{W}})}F^WF^{\bar{W}}
\nonumber\\&=& - \xi D\frac{ (s_0\bar{s}_0e^{-K/3})^{-3}}{(s_0\bar{s}_0e^{-K/3})^{-4}F^{W}F^{\bar{W}}}F^WF^{\bar{W}} = -\xi D(s_0\bar{s}_0e^{-K/3}).
\end{eqnarray}

Hence, in the superconformal gauge ($s_0\bar{s}_0e^{-K/3}=M_{pl}^2=1$), we obtain

\begin{eqnarray}
\mathcal{L}_{\textrm{new FI}}/e = -\xi D,
\end{eqnarray}

\begin{eqnarray}
\mathcal{L}_{\textrm{new FI}}/e = -M_{pl}^2\xi D.
\end{eqnarray}

§ SPECTROSCOPHY FOR NON-RENORMALIZABLE INTERACTIONS IN THE NEW FI TERMS

In this section, we carefully analyze suppression of the nonrenormalizable fermionic interactions from the new FI term from the perspective of effective field theory. To do this, as we did in the previous section of the liberated supergravity, we also need to recall the EFT expansion reviewed in Sec. <ref>, and consider the structure of EFT expansion in Eq. (<ref>) and constraint given in Eq. (<ref>). Hence, we have

\begin{eqnarray}
\mathcal{L}_{EFT} \supset \sum_{\delta \geq 0}^{\textrm{Finite N}} \frac{C^{\delta}}{M^{\delta-d}} \mathcal{O}^{(\delta)} \sim \sum_{\delta \geq 0}^{\textrm{Finite N}} \frac{1}{\Lambda_{cut}^{\delta-d}} \mathcal{O}^{(\delta)}  \implies \frac{C^{\delta}}{M^{\delta-d}} \lesssim \frac{1}{\Lambda_{cut}^{\delta-d}},
\end{eqnarray}

where $\Lambda_{cut}$ is a cutoff scale; $M$ is a characteristic mass scale of a theory; $C^{\delta}$ is a dimensionless Wilson coefficient, and $\mathcal{O}^{(\delta)}$ is an effective field operator with the mass dimension $\delta$. In the case of the new FI terms, we will see that the constraint depends on the nw FI terms $\xi$ and thus the auxiliary field $D$ of the vector multiplet. The key equation we will see in this section is given by

\begin{eqnarray}
\Lambda_{cut} \lesssim H^{\alpha},\nonumber
\end{eqnarray}

where $H$ is the Hubble scale, and $\alpha$ is a parameter that depends on the mass dimension of the effective field operator. Since $\Lambda_{cut},~ H\sim 10^{-5}M_{pl} < M_{pl}=1$, we explore whether $\alpha <1$ holds. After all, we will find that the maximum of $\alpha$ is 2/3.

In particular, for the later use, we generalize the new FI constant $\xi$ into a real gauge-invariant function $\mathcal{U}(z,\bar{z})$ of matter fields. Thus, for ACIK FI term, we can just put $\mathcal{U}(z,\bar{z}) \equiv \xi$. First, after solving the equations of motion for the auxiliary fields and fixing the superconformal gauge, the component multiplets “on-shell” are given by

\begin{eqnarray}
&& Z^i  = \{ z^i, P_L\Omega^i,F^i\},\\
&& S_0  = \{ s_0, \quad \frac{1}{3}e^{K/6}K_iP_L\Omega^i,\quad F_0\},\\
&& W \equiv \bar{\lambda}P_L\lambda = \Big\{\bar{\lambda}P_L\lambda,\quad \sqrt{2}P_L(-\frac{1}{2}\gamma\cdot \hat{F} + iD)\lambda,\quad -D^2 + \hat{F}^-\cdot\hat{F}^- \Big\},
\end{eqnarray}

\begin{eqnarray}
&& \hat{F}_{ab} \equiv F_{ab} + e_a^{~\mu}e_b^{~\nu} \bar{\psi}_{[\mu}\gamma_{\nu]}\lambda,\qquad F_{ab} \equiv e_a^{~\mu}e_b^{~\nu} (2\partial_{[\mu}A_{\nu]}),\\
&& \hat{F}^{\pm}_{\mu\nu} \equiv \frac{1}{2}(\hat{F}_{\mu\nu}\pm \tilde{\hat{F}}_{\mu\nu}), \qquad \tilde{\hat{F}}_{\mu\nu} \equiv -\frac{1}{2} i\epsilon_{\mu\nu\rho\sigma}\hat{F}^{\rho\sigma},
\end{eqnarray}

and the solutions for the auxiliary fields are identified as

\begin{eqnarray}
D&=& \mathcal{U} + \frac{1}{\mathcal{U}^2}
\bigg[
\bigg(
-F^0\mathcal{U}e^{-K/6}  + \mathcal{U}_IF^I +\frac{1}{3}\mathcal{U}K_IF^I
\bigg)(\bar{\lambda}P_L\lambda)  -i
\frac{\mathcal{U}_I}{\sqrt{2}}(\bar{\Omega}^IP_L\lambda) + h.c.
\bigg]\nonumber\\
&&+\textrm{higher order terms},\\
F^0 &=& e^{2K/3}\bar{W} - \frac{1}{3}e^{K/6}(\bar{\lambda}P_R\lambda),\\
F^{\bar{J}} &=&- 3e^{K/2}G^{I\bar{J}}\nabla_IW -G^{I\bar{J}}\Big(9\frac{\mathcal{U}_I }{\mathcal{U}}+ 3K_I\Big)(\bar{\lambda}P_L\lambda)
\end{eqnarray}

Plus, the relevant chiral projection multiplet $T(\bar{w}'^2)$ is given by

\begin{eqnarray}
T(\bar{w}'^2) &=& ( C_T, P_L\Omega_T, F_T ),\\
C_T &=&  h^{*}_{\bar{a}}F^{\bar{a}} -\frac{1}{2} h^{*}_{\bar{a}\bar{b}}\bar{\Omega}^{\bar{a}}\Omega^{\bar{b}},\\
P_L\Omega_T &=& \cancel{\mathcal{D}}(-h^{*}_a\Omega^a + h^{*}_{\bar{a}}\Omega^{\bar{a}})-h^{*}_{\bar{a}b}[(\cancel{\mathcal{D}}X^b)\Omega^{\bar{a}}-F^{\bar{a}}\Omega^b]-\frac{1}{2}h^{*}_{\bar{a}\bar{b}c}\Omega^c\bar{\Omega}^{\bar{a}}\Omega^{\bar{b}}\nonumber\\
F_T &=&  h^{*}_{a\bar{b}}\Big(-\mathcal{D}_{\mu}X^a\mathcal{D}^{\mu}\bar{X}^{\bar{b}}-\frac{1}{2}\bar{\Omega}^aP_L\cancel{\mathcal{D}}\Omega^{\bar{b}}-\frac{1}{2}\bar{\Omega}^{\bar{b}}P_R\cancel{\mathcal{D}}\Omega^a+F^aF^{\bar{b}}\Big) \nonumber\\
&&+\frac{1}{2}h^{*}_{ab\bar{c}}(-\bar{\Omega}^a\Omega^bF^{\bar{c}}+\bar{\Omega}^a(\cancel{\mathcal{D}}X^b)\Omega^{\bar{c}})+\frac{1}{2} h^{*}_{\bar{a}\bar{b}c}(-\bar{\Omega}^{\bar{a}}\Omega^{\bar{b}}F^{c}+\bar{\Omega}^{\bar{a}}(\cancel{\mathcal{D}}\bar{X}^{\bar{b}})\Omega^{c}) \nonumber\\
&&+ \frac{1}{4}h^{*}_{ab\bar{c}\bar{d}}(\bar{\Omega}^aP_L\Omega^b)(\bar{\Omega}^{\bar{c}}P_R\Omega^{\bar{d}})+\frac{1}{2}\square^C h^{*} - \frac{1}{2}\mathcal{D}^{\mu} (h^{*}_a\mathcal{D}_{\mu}X^a-h^{*}_{\bar{a}}\mathcal{D}_{\mu}\bar{X}^{\bar{a}}+h^{*}_{a\bar{b}}\bar{\Omega}^{a}\gamma_{\mu}\Omega^{\bar{b}}),\nonumber\\{}
\end{eqnarray}

where  $C_T,C_{\bar{T}} = -D^2 \Delta^{-2} + \textrm{fermions including } \lambda$; $\Delta \equiv s_0\bar{s}_0e^{-K/3}$; $a = 0,i,W(=\bar{\lambda}P_L\lambda)$, and

\begin{eqnarray}
h^* \equiv \frac{\bar{\lambda}P_R\lambda}{(s_0\bar{s}_0e^{-K(z,\bar{z})/3})^2}.
\end{eqnarray}

We note that in this chiral projection multiplet, the terms “$h^*_{\bar{W}}\bar{F}^{\bar{W}}$ of $C_T$,” “$h^*_{\bar{W}k}\bar{F}^{\bar{W}}\Omega^{k}$ of $P_L\Omega_T$,” and “$F^k\bar{F}^{\bar{W}}$ and $h^*_{jk\bar{W}}\bar{F}^{\bar{W}}\bar{\Omega}^j\Omega^k$ of $F_T$” (where $j,k=0,i$) have no $\lambda$, while the other terms must have at least one $\lambda$.

In summary, the components of the multiplets can be found to be

\begin{eqnarray}
&&S_0:\quad s_0 \sim e^{K/6},\quad  \cancel{\mathcal{D}}s_0 \sim
,  \nonumber\\
&& \qquad P_L\Omega^0 \sim K_ie^{K/6}\mathcal{O}^{(3/2)}_i, \quad F^0 \sim We^{2K/3},\\
&&Z^i:\quad z^i \sim z^i ,\quad \cancel{\mathcal{D}}z^i\sim \mathcal{O}^{(2)}+\mathcal{O}^{(4)}_{\bar{\psi}}, \quad P_L\Omega^i \sim \mathcal{O}^{(3/2)}_i , \quad F^i \sim F^i,\\
&&W:\quad (\Bar{\lambda}P_L\lambda) \sim \mathcal{O}^{(3)}_{\lambda\lambda
}, \quad \cancel{\mathcal{D}}(\Bar{\lambda}P_L\lambda) \sim \mathcal{O}^{(4)}_{\Bar{\lambda}\lambda} + \mathcal{O}^{(5)}_{\bar{\psi}\lambda}+D\mathcal{O}^{(3)}_{\Bar{\psi}\lambda},
\nonumber\\
&&\qquad \Omega^W \sim \mathcal{O}^{(7/2)}_{\lambda} + D \mathcal{O}^{3/2}_{\lambda}, \quad F^W \sim D^2 + \mathcal{O}^{(4)}\nonumber\\{}\\
&&T:\quad  C_T \supset  h^*_{\bar{W}}\bar{F}^{\bar{W}}|_{\textrm{lowest}} \sim D^2 + DK_i \mathcal{O}^{(3)}_{i\lambda} + K_i \mathcal{O}^{(5)}_{i\lambda},\\
&&{}\qquad\cancel{\mathcal{D}}C_T \sim \cancel{\partial}C_T \sim DD_i \mathcal{O}^{(2)}_i + D_i K_j\mathcal{O}^{(5)}_{ij\lambda} + DK_{ij}\mathcal{O}^{(5)}_{ij\lambda} +K_{ij}\mathcal{O}^{(7)}_{ij\lambda} \\
&&{}\qquad P_L \Omega_T \supset  P_L\cancel{\mathcal{D}}\mathcal{Z}_{\Bar{w}}+ h^*_{\bar{W}k}\bar{F}^{\bar{W}}\Omega^{k}|_{\textrm{lowest}}\sim D^2K_i\mathcal{O}^{(3/2)}_i + D\mathcal{O}_{\lambda}^{(5/2)}+K_i\mathcal{O}^{(11/2)}_{i\lambda},\\
&&{}\qquad F_T \supset F^k\bar{F}^{\bar{W}}+h^*_{jk\bar{W}}\bar{F}^{\bar{W}}\bar{\Omega}^j\Omega^k |_{\textrm{lowest}} \sim
D^2 F^k + D^2(K'^2+K'')\mathcal{O}^{(3)}
\end{eqnarray}

where we used the following superconformal covariant derivatives of the lowest component scalar $\mathcal{C}$ with the weights $(w,c)$ and the first fermion $\mathcal{Z}$ in a superconformal multiplet:

\begin{eqnarray}
\mathcal{D}_{\mu}\mathcal{C} &=& (\partial_{\mu}-wb_{\mu}-icA_{\mu})\mathcal{C}-\frac{i}{2}\bar{\psi}_{\mu}\gamma_*\mathcal{Z},\\
P_L\mathcal{D}_{\mu}\mathcal{Z} &=&
\Big(\partial_{\mu}-(w+1/2)b_{\mu}-i(c-3/2)A_{\mu}+\frac{1}{4}\omega_{\mu}^{ab}\gamma_{ab}\Big)\mathcal{Z} \nonumber\\
&&-\frac{1}{2}P_L(i\mathcal{H}-\gamma^a\mathcal{B}_a-i\gamma^a\mathcal{D}_a\mathcal{C})\psi_{\mu} -i(w+c)P_L\phi_{\mu}\mathcal{C}
\end{eqnarray}

The full action of the new FI term “on-shell” (i.e. $\mathcal{D}^{\nu}\hat{F}_{\mu\nu}=0,~ \cancel{\mathcal{D}}\lambda=0$) is given by

\begin{eqnarray}
\mathcal{L}_{\textrm{new FI}} \equiv -[ \mathcal{R}\cdot (V)_D]_D &=& -\frac{1}{4}\int d^4x e \bigg[  \tilde{\mathcal{D}} -\frac{1}{2}\bar{\psi}\cdot \gamma i\gamma_* \tilde{\Lambda} -\frac{1}{3}\tilde{\mathcal{C}}R(\omega)\nonumber\\
\end{eqnarray}

where the relevant components are given, for $f \equiv  (s_0\bar{s}_0e^{-K(z^i,\bar{z}^i)/3})^{-3}\frac{(\bar{\lambda}P_L\lambda)(\bar{\lambda}P_R\lambda)}{C_TC_{\bar{T}}}\mathcal{U}(z^i,\bar{z}^i)$ (where $C_T,C_{\bar{T}} = -D^2 \Delta^{-2} + \textrm{fermions including } \lambda$; $\Delta \equiv s_0\bar{s}_0e^{-K/3}$), as follows

\begin{eqnarray}
\tilde{\mathcal{C}} &=& Df,\\
\tilde{\mathcal{Z}} &=& Di\sqrt{2}(-f_{A}\Omega^A+f_{\bar{A}}\Omega^{\bar{A}}),\\
\tilde{\mathcal{H}} &=& D(-2f_AF^A + f_{AB}\bar{\Omega}^A\Omega^B),\\
\tilde{\mathcal{K}} &=& D(-2f_{\bar{A}}F^{\bar{A}} + f_{\bar{A}\bar{B}}\bar{\Omega}^{\bar{A}}\Omega^{\bar{B}}),\\
\tilde{\mathcal{B}} &=& D(if_A\mathcal{D}_{\mu}X^A-if_{\bar{A}}\mathcal{D}_{\mu}\bar{X}^{\bar{A}}+if_{A\bar{B}}\bar{\Omega}^A\gamma_{\mu}\Omega^{\bar{B}}),\\
\tilde{\Lambda} &=&   D\bigg(  -\sqrt{2}if_{\bar{A}B}[(\cancel{\mathcal{D}}X^B)\Omega^{\bar{A}}-F^{\bar{A}}\Omega^B]-\frac{i}{\sqrt{2}}f_{\bar{A}\bar{B}C}\Omega^C\bar{\Omega}^{\bar{A}}\Omega^{\bar{B}} \nonumber\\
&&+ \sqrt{2}if_{A\bar{B}}[(\cancel{\mathcal{D}}\bar{X}^{\bar{B}})\Omega^{A}-F^{A}\Omega^{\bar{B}}]+\frac{i}{\sqrt{2}}f_{AB\bar{C}}\Omega^{\bar{C}}\bar{\Omega}^{A}\Omega^{B}\bigg) \nonumber\\
&&+\frac{1}{2}\Big( -\cancel{\mathcal{D}}D\Big)i\sqrt{2}(-f_{A}\Omega^A+f_{\bar{A}}\Omega^{\bar{A}}) ,\\
\tilde{\mathcal{D}} &=&-f\square^C D + D
\bigg\{ 2f_{A\bar{B}}(-\mathcal{D}_{\mu}X^A\mathcal{D}^{\mu}\bar{X}^{\bar{B}}-\frac{1}{2}\bar{\Omega}^AP_L\cancel{\mathcal{D}}\Omega^{\bar{B}}-\frac{1}{2}\cancel{\Omega}^{\bar{B}}P_R\cancel{\mathcal{D}}\Omega^A+F^AF^{\bar{B}}) \nonumber\\
&&+f_{AB\bar{C}}(-\bar{\Omega}^A\Omega^B F^{\bar{C}} + \bar{\Omega}^A(\cancel{\mathcal{D}}X^B)\Omega^{\bar{C}})
+f_{\bar{A}\bar{B}C}(-\bar{\Omega}^{\bar{A}}\Omega^{\bar{B}} F^C + \bar{\Omega}^{\bar{A}}(\cancel{\mathcal{D}}\bar{X}^{\bar{B}})\Omega^C) \nonumber\\
&&+\frac{1}{2}f_{AB\bar{C}\bar{D}} (\bar{\Omega}^AP_L\Omega^B)(\bar{\Omega}^{\bar{C}}P_R\Omega^{\bar{D}}) \bigg\}-(\mathcal{D}_{\mu}f)(\mathcal{D}^{\mu}D),
\end{eqnarray}

where the indices $A,B,C,D$ run over $0,i,W,T$. Plus, we have

\begin{eqnarray}
&& \square^CD= e^{a\mu}[\partial_{\mu}\mathcal{D}_aD -3b_{\mu}\mathcal{D}_aD + \omega_{\mu ab} \mathcal{D}^b D + 4 f_{\mu a}D], \nonumber\\
&& \mathcal{D}_aD|_{on-shell}=e_a^{\mu}(\partial_{\mu}-2b_{\mu})D, \quad \delta D = 2\lambda_D D,\nonumber\\
&& \mathcal{D}_aD |_{b=0}=  \mathcal{U}_i \partial_{\mu}z^i \sim \mathcal{U}_i \mathcal{O}^{(2)} \textrm{  and  } D e^{a\mu} f_{\mu a} = -D\frac{R}{12} \sim D\mathcal{O}^{(2)}.
\end{eqnarray}

Since we consider $D|_{bos}=\mathcal{U}$ as a quadratic function of matter fields, we can assume that $\mathcal{U}_i \sim 0$ at the minimum of the potential. Then, the components reduce to

\begin{eqnarray}
\tilde{\mathcal{C}} &=& Df,\\
\tilde{\mathcal{Z}} &=& Di\sqrt{2}(-f_{A}\Omega^A+f_{\bar{A}}\Omega^{\bar{A}}),\\
\tilde{\mathcal{H}} &=& D(-2f_AF^A + f_{AB}\bar{\Omega}^A\Omega^B),\\
\tilde{\mathcal{K}} &=& D(-2f_{\bar{A}}F^{\bar{A}} + f_{\bar{A}\bar{B}}\bar{\Omega}^{\bar{A}}\Omega^{\bar{B}}),\\
\tilde{\mathcal{B}} &=& D(if_A\mathcal{D}_{\mu}X^A-if_{\bar{A}}\mathcal{D}_{\mu}\bar{X}^{\bar{A}}+if_{A\bar{B}}\bar{\Omega}^A\gamma_{\mu}\Omega^{\bar{B}}),\\
\tilde{\Lambda} &=&   D\bigg(  -\sqrt{2}if_{\bar{A}B}[(\cancel{\mathcal{D}}X^B)\Omega^{\bar{A}}-F^{\bar{A}}\Omega^B]-\frac{i}{\sqrt{2}}f_{\bar{A}\bar{B}C}\Omega^C\bar{\Omega}^{\bar{A}}\Omega^{\bar{B}} \nonumber\\
&&+ \sqrt{2}if_{A\bar{B}}[(\cancel{\mathcal{D}}\bar{X}^{\bar{B}})\Omega^{A}-F^{A}\Omega^{\bar{B}}]+\frac{i}{\sqrt{2}}f_{AB\bar{C}}\Omega^{\bar{C}}\bar{\Omega}^{A}\Omega^{B}\bigg),\\
\tilde{\mathcal{D}} &=&\frac{DRf}{3}+ D
\bigg\{ 2f_{A\bar{B}}(-\mathcal{D}_{\mu}X^A\mathcal{D}^{\mu}\bar{X}^{\bar{B}}-\frac{1}{2}\bar{\Omega}^AP_L\cancel{\mathcal{D}}\Omega^{\bar{B}}-\frac{1}{2}\cancel{\Omega}^{\bar{B}}P_R\cancel{\mathcal{D}}\Omega^A+F^AF^{\bar{B}}) \nonumber\\
&&+f_{AB\bar{C}}(-\bar{\Omega}^A\Omega^B F^{\bar{C}} + \bar{\Omega}^A(\cancel{\mathcal{D}}X^B)\Omega^{\bar{C}})
+f_{\bar{A}\bar{B}C}(-\bar{\Omega}^{\bar{A}}\Omega^{\bar{B}} F^C + \bar{\Omega}^{\bar{A}}(\cancel{\mathcal{D}}\bar{X}^{\bar{B}})\Omega^C) \nonumber\\
&&+\frac{1}{2}f_{AB\bar{C}\bar{D}} (\bar{\Omega}^AP_L\Omega^B)(\bar{\Omega}^{\bar{C}}P_R\Omega^{\bar{D}}) \bigg\},
\end{eqnarray}

Now we represent the possible interaction terms using the following expression

\begin{eqnarray}
\mathcal{L}_{\textrm{new FI}} &\supset&
\partial_{i}^m \partial_0^c \partial_W^b \partial_T^p \bigg[\frac{W\Bar{W}}{T\Bar{T}}Y\bigg]\cdot D \cdot \hat{\mathcal{O}}^{(\mathcal{D}_0;\Lambda_{\Bar{\psi}3/2};\mathcal{C}_{R(\omega)2};\mathcal{C}_{\Bar{\psi}R'(Q)4}; \mathcal{Z}_{R'(Q)5/2};\mathcal{B}_{\Bar{\psi}\psi 3};\mathcal{Z}_{\Bar{\psi}\psi\Bar{\psi}9/2})} \nonumber\\
&&\times  \{ (\cancel{D}S_0)^{c_1}(\Omega^0)^{c_2}(F^0)^{c_3}\}\{ (\cancel{D}z^i)^{m_1}(\Omega^i)^{m_2}(F^i)^{m_3}\} \nonumber\\
&&\times \{ (\cancel{D}W)^{b_1}(\Omega^W)^{b_2}(F^W)^{b_3}\}
\{ (\cancel{D}T)^{p_1}(\Omega^T)^{p_2}(F^T)^{p_3}\},
\end{eqnarray}

where $\Delta \equiv s_0\bar{s}_0e^{-K/3}$; $Y\equiv \Delta^{-3}\mathcal{U}(z,\Bar{z})$, and several parameters are defined as the number of derivatives with respect to the corresponding variable, such as $c=c_1+c_2+c_3$, $m=m_1+m_2+m_3$, $b=b_1+b_2+b_3 \leq 2$ (Here, the number $b$ must be constrained by “2” because each of the bilinear terms $W$ and $\Bar{W}$ appears only once in the numerator.), and $p=p_1+p_2+p_3$. Also, $\Gamma \equiv m+c+b+p \leq 4$ since we can take the four field-derivatives at most. The notation $\Lambda_{\Bar{\psi}3/2}$ means that the operator $\hat{\mathcal{O}}$ is given by the dimension-(3/2) operator $\Bar{\psi}$ which couples to the $\tilde{\Lambda}$ of the new FI term.

Plus, the operator $\hat{\mathcal{O}}$ denotes the additional contribution to the effective operator from the couplings to gravitino $\psi_{\mu}$ or curvatures $R(\omega)$ and $R'(Q)$ in the final action of new FI term in Eq. (<ref>).

Then, taking the derivatives and inserting $T \sim D^2$, we obtain

\begin{eqnarray}
\mathcal{L}_{\textrm{new FI}}|_{\textrm{with no numerical factors}} &\supset&
D^{-3-2p}(\partial^{m+c}Y)\mathcal{O}^{(2\delta_{\Gamma,0})}_R\mathcal{O}^{(6_{4\lambda}-3_{2\lambda}b)}\cdot  \hat{\mathcal{O}} \nonumber\\
&&\times  \{ (\cancel{D}S_0)^{c_1}(\Omega^0)^{c_2}(F^0)^{c_3}\}\{ (\cancel{D}z^i)^{m_1}(\Omega^i)^{m_2}(F^i)^{m_3}\} \nonumber\\
&&\times \{ (\cancel{D}W)^{b_1}(\Omega^W)^{b_2}(F^W)^{b_3}\}
\{ (\cancel{D}T)^{p_1}(\Omega^T)^{p_2}(F^T)^{p_3}\},
\end{eqnarray}

where we define $\delta_{\Gamma,0}=1$ if $\Gamma =0$, and $W^{1-b_{2\lambda}}\Bar{W}^{1-\bar{b}_{2\lambda}} \equiv \mathcal{O}^{(3_{2\lambda}(1-b_{2\lambda})+3_{2\lambda}(1-\bar{b}_{2\lambda}))} \equiv \mathcal{O}^{(6_{4\lambda}-3_{2\lambda}b)}$; that is, $6_{4\lambda}$ means that the dimension “6” comes from the four $\lambda$-fermions. Next, we introduce particular parameters that constitute the powers of the terms like $c_1,c_2,c_3$ in detail. Since the component actions depend only on $\cancel{\mathcal{D}}z^i$, $P_L\Omega^i$, and $F^i$, we define $d_i,f_i,a_i$ as the powers of “each term” (like $\mathcal{O}^{(7/2)}_{\lambda}$ and $D \mathcal{O}^{3/2}_{\lambda}$ of $\Omega^W$) within the expression of $\cancel{\mathcal{D}}z^i$ (“d”erivative), $P_L\Omega^i$ (“f”ermion), and $F^i$ (“a”uxiliary field). Equivalently, we define the correponding parameters as $d_s,f_s,a_s$, $d_W,f_W,a_W$, and $d_T,f_T,a_T$. Then, we find

\begin{eqnarray}
\{ (\cancel{D}S_0)^{c_1}(\Omega^0)^{c_2}(F^0)^{c_3}\} \sim K'^{d_{s1}+d_{s3}^{\psi}+f_{s1}}{F^0}^{a_{s1}}\mathcal{O}^{(2d_{s1}+d_{s2}+4d_{s3}^{\psi}+\frac{3}{2}f_{s1})},
\end{eqnarray}

where $d_{s1}+d_{s2}+d_{s3}^{\psi}+f_{s1}+a_{s1} =c_1+c_2+c_3 =c \leq 4$. For example, if $d_{s3}^{\psi}=2$ in the effective operator $\mathcal{O}$, then it means that there are two gravitinos in the operator. The next one is

\begin{eqnarray}
\{ (\cancel{D}z^i)^{m_1}(\Omega^i)^{m_2}(F^i)^{m_3}\} \sim {F^i}^{a_{z1}}\mathcal{O}^{(2d_{z1}+4d_{z2}^{\psi}+\frac{3}{2}f_{z1})},
\end{eqnarray}

where $d_{z1}+d_{z2}^{\psi}+f_{z1}+a_{z1}=m_1+m_2+m_3=m \leq 4$. The $W$ case is

\begin{eqnarray}
\{ (\cancel{D}W)^{b_1}(\Omega^W)^{b_2}(F^W)^{b_3}\} \sim
D^{d_{W3}^{\psi\lambda}+f_{W2}^{\lambda}+2a_{W1}} \mathcal{O}^{(4d_{W1}^{\lambda\lambda}+5d_{W2}^{\psi\lambda}+3d_{W3}^{\psi\lambda}+\frac{7}{2}f_{W1}^{\lambda}+\frac{3}{2}f_{W2}^{\lambda}+4a_{W2})},
\end{eqnarray}

where $d_{W1}^{\lambda\lambda}+d_{W2}^{\psi\lambda}+d_{W3}^{\psi\lambda}+f_{W1}^{\lambda}+f_{W2}^{\lambda}+a_{W1}+a_{W2} =b_1+b_2 +b_3= b\leq2$. As an example, also, if $d_{W1}^{\lambda\lambda}=2$ in $\mathcal{O}$, then it means that there are four $\lambda$ fermions in the effective operator. Next, the chiral projection multiplet $T$ case is given by

\begin{eqnarray}
\{ (\cancel{D}T)^{p_1}(\Omega^T)^{p_2}(F^T)^{p_3}\} &\sim&
D_i^{d_{T1}+d_{T2}^{\lambda}} \nonumber\\
&& \times K'^{d_{T2}^{\lambda}+f_{T1}+f_{T3}^{\lambda}+2a_{T2}}
K''^{d_{T3}^{\lambda}+d_{T4}^{\lambda}+a_{T2}} {F^i}^{a_{T1}} \nonumber\\
&&\times \mathcal{O}^{(2d_{T1}+5d_{T2}^{\lambda}+5d_{T3}^{\lambda}+7d_{T4}^{\lambda}+\frac{3}{2}f_{T1}+\frac{5}{2}f_{T2}^{\lambda}+\frac{11}{2}f_{T3}^{\lambda}+3 a_{T2})},
\end{eqnarray}

where $d_{T1}+d_{T2}^{\lambda}+d_{T3}^{\lambda}+d_{T4}^{\lambda}+f_{T1}+f_{T2}^{\lambda}+f_{T3}^{\lambda}+a_{T1}+a_{T2}=p_1+p_2+p_3=p\leq 4$. In addition, we can also get

\begin{eqnarray}
\partial^{m+c}Y &=&  \partial^{m+c} \Big(\Delta^{-3} \mathcal{U}(z,\Bar{z}) \Big) = \partial^s \Big( (s_0\Bar{s})^{-3} \Big) \partial^m (e^K\mathcal{U}) \nonumber\\
&& \sim (K')^{k_1}(K'')^{k_2}(K''')^{k_3}(K'''')^{k_1} (\mathcal{U})^{u_0}(\mathcal{U}^{(1)})^{u_1}(\mathcal{U}^{(2)})^{u_2}(\mathcal{U}^{(3)})^{u_3}(\mathcal{U}^{(4)})^{u_4},
\end{eqnarray}

where $k_1+k_2+k_3+k_4 + u_1 + u_2 + u_3 + u_4 = m$ (here, we define $k_i,u_i$ to be present only when $i\leq  m$ and $0\leq k_i\leq m$, $0\leq u_i \leq 1$) and $u_0 =\delta_{m,k_{tot}}$ (which is defined as a Kronecker delta) for $k_{tot}\equiv k_1+k_2+k_3+k_4 $.  In particular, we can consider $\mathcal{U} \sim D$. Combining these results, we can rewrite the terms as

\begin{eqnarray}
\mathcal{L}_{\textrm{new FI}} &\supset& (K')^{k_1}(K'')^{k_2}(K''')^{k_3}(K'''')^{k_1} (\mathcal{U})^{u_0}(\mathcal{U}^{(1)})^{u_1}(\mathcal{U}^{(2)})^{u_2}(\mathcal{U}^{(3)})^{u_3}(\mathcal{U}^{(4)})^{u_4}\nonumber\\
&& \times
D^{-3-2p}(\partial^{m+c}Y)\mathcal{O}^{(2n_R)}_R\mathcal{O}^{(6_{4\lambda}-3_{2\lambda}b)}\cdot  \hat{\mathcal{O}} \nonumber\\
&&\times K'^{d_{s1}+d_{s3}^{\psi}+f_{s1}}{F^0}^{a_{s1}}\mathcal{O}^{(2d_{s1}+d_{s2}+4d_{s3}^{\psi}+\frac{3}{2}f_{s1})} \nonumber\\
&&\times {F^i}^{a_{z1}}\mathcal{O}^{(2d_{z1}+4d_{z2}^{\psi}+\frac{3}{2}f_{z1})} \nonumber\\
&&\times  D^{d_{W3}^{\psi\lambda}+f_{W2}^{\lambda}+2a_{W1}} \mathcal{O}^{(4d_{W1}^{\lambda\lambda}+5d_{W2}^{\psi\lambda}+3d_{W3}^{\psi\lambda}+\frac{7}{2}f_{W1}^{\lambda}+\frac{3}{2}f_{W2}^{\lambda}+4a_{W2})}\nonumber\\
&&\times D^{d_{T1}+d_{T3}^{\lambda}+2f_{T1}+f_{T2}^{\lambda}+2a_{T1}+2a_{T2}}
D_i^{d_{T1}+d_{T2}^{\lambda}} \nonumber\\
&& \times K'^{d_{T2}^{\lambda}+f_{T1}+f_{T3}^{\lambda}+2a_{T2}}
K''^{d_{T3}^{\lambda}+d_{T4}^{\lambda}+a_{T2}} {F^i}^{a_{T1}} \nonumber\\
&&\times \mathcal{O}^{(2d_{T1}+5d_{T2}^{\lambda}+5d_{T3}^{\lambda}+7d_{T4}^{\lambda}+\frac{3}{2}f_{T1}+\frac{5}{2}f_{T2}^{\lambda}+\frac{11}{2}f_{T3}^{\lambda}+3 a_{T2})}.
\end{eqnarray}

The simplified expression can be given by

\begin{eqnarray}
\mathcal{L}_{\textrm{new FI}} &\supset&
(K''')^{k_3}(K'''')^{k_1} (\mathcal{U}^{(1)})^{\alpha}(\mathcal{U}^{(2)})^{u_2}(\mathcal{U}^{(3)})^{u_3}(\mathcal{U}^{(4)})^{u_4} \nonumber\\
&&\times (D)^{-n}(F)^{r}(K')^{\beta}(K'')^{\gamma} \mathcal{O}^{(\delta_{pre})}\hat{\mathcal{O}}^{(\delta_{add})},
\end{eqnarray}

\begin{eqnarray}
n &=&-u_0+ 3+2p-(d_{W3}^{\psi\lambda}+f_{W2}^{\lambda}+2a_{W1}+d_{T1}+d_{T3}^{\lambda}+2f_{T1}+f_{T2}^{\lambda}+2a_{T1}+2a_{T2}) \nonumber\\
&=& -u_0+3 + 2(d_{T1}+d_{T2}^{\lambda}+d_{T3}^{\lambda}+d_{T4}^{\lambda}+f_{T1}+f_{T2}^{\lambda}+f_{T3}^{\lambda}+a_{T1}+a_{T2}) \nonumber\\ &&-(d_{W3}^{\psi\lambda}+f_{W2}^{\lambda}+2a_{W1}+d_{T1}+d_{T3}^{\lambda}+2f_{T1}+f_{T2}^{\lambda}+2a_{T1}+2a_{T2}) \nonumber\\
&=&-u_0+ 3 + d_{T1} + 2d_{T2}^{\lambda} + d_{T3}^{\lambda} + 2d_{T4}^{\lambda} - d_{W3}^{\psi\lambda} + f_{T2}^{\lambda} + 2f_{T3}^{\lambda} -f_{W2}^{\lambda}-2a_{W1},\\
r &=& a_{s1} +a_{z1} + a_{T1}.
\end{eqnarray}

\begin{eqnarray}
\alpha &=& u_1+d_{T1} + d_{T2}^{\lambda},\\
\beta &=& k_1+ d_{s1}+d_{s3}^{\psi}+f_{s1}+d_{T2}^{\lambda}+f_{T1}+f_{T3}^{\lambda}+2a_{T2},\\
\gamma &=&k_2 + d_{T3}^{\lambda}+d_{T4}^{\lambda}+a_{T2}
\end{eqnarray}

\begin{eqnarray}
\delta_{pre} &=& 2\delta_{\Gamma,0}+6_{4\lambda}-3_{2\lambda}b +2d_{s1}+d_{s2}+4d_{s3}^{\psi}+\frac{3}{2}f_{s1}+2d_{z1}+4d_{z2}^{\psi}+\frac{3}{2}f_{z1} \nonumber\\
&& + 4d_{W1}^{\lambda\lambda}+5d_{W2}^{\psi\lambda}+3d_{W3}^{\psi\lambda}+\frac{7}{2}f_{W1}^{\lambda}+\frac{3}{2}f_{W2}^{\lambda}+4a_{W2} \nonumber\\
&& +2d_{T1}+5d_{T2}^{\lambda}+5d_{T3}^{\lambda}+7d_{T4}^{\lambda}+\frac{3}{2}f_{T1}+\frac{5}{2}f_{T2}^{\lambda}+\frac{11}{2}f_{T3}^{\lambda}+3 a_{T2}
\end{eqnarray}

\begin{eqnarray}
\delta_{add} &=& \textrm{one of  } \mathcal{D}_0;\Lambda_{\Bar{\psi}3/2};\mathcal{C}_{R(\omega)2};\mathcal{C}_{\Bar{\psi}R'(Q)4}; \mathcal{Z}_{R'(Q)5/2};\mathcal{B}_{\Bar{\psi}\psi 3};\mathcal{Z}_{\Bar{\psi}\psi\Bar{\psi}9/2}.
\end{eqnarray}

Plus, we define a parameter $\Gamma$ as the number of derivatives among $\partial_i^m$, $\partial_0^c$, $\partial_W^b$, and $\partial_T^p$ as follows:

\begin{eqnarray}
\Gamma \equiv c+m+b+p&=& d_{s1}+d_{s2}+d_{s3}^{\psi}+f_{s1}+a_{s1}+d_{z1}+d_{z2}^{\psi}+f_{z1}+a_{z1} \nonumber\\
&& +d_{W1}^{\lambda\lambda}+d_{W2}^{\psi\lambda}+d_{W3}^{\psi\lambda}+f_{W1}^{\lambda}+f_{W2}^{\lambda}+a_{W1}+a_{W2} \nonumber\\
&& +d_{T1}+d_{T2}^{\lambda}+d_{T3}^{\lambda}+d_{T4}^{\lambda}+f_{T1}+f_{T2}^{\lambda}+f_{T3}^{\lambda}+a_{T1}+a_{T2} \leq 4.
\end{eqnarray}

Furthermore, we define selection rules for the parameter set $\{(d,f,a)\}$ such that $\Gamma=\sum_{all} (d+f+a)$:

\begin{eqnarray}
\textrm{For  } (d,f,a),
&& \Tilde{\mathcal{C}}:\quad (0,0,0) \implies \Gamma(\Tilde{\mathcal{C}})=0 \qquad \delta_{add} = 2,4,\nonumber\\
&& \Tilde{\mathcal{Z}}:\quad {\color{red}(0, 1,0)}\implies \Gamma(\Tilde{\mathcal{Z}})=1\qquad \delta_{add} =5/2,~9/2,\nonumber\\
&& \Tilde{\mathcal{B}_a}:\quad {\color{orange} (1,0,0)},{\color{blue}(0,2,0)}\implies \Gamma(\Tilde{\mathcal{B}_a})=1,2\qquad \delta_{add} =3,\nonumber\\
&& \Tilde{\Lambda}:\quad {\color{red}(1,1,0)},{\color{red}(1,1,1)},{\color{blue}(0,3,0)}\implies \Gamma(\Tilde{\Lambda})=2,3\qquad \delta_{add} =3/2,\nonumber\\
&& \Tilde{\mathcal{D}}:\quad (0,0,0),{\color{orange}(2,0,0)},{\color{blue}(0,2,0)},{\color{orange}(0,0,2)},\nonumber\\
&& \qquad \qquad {\color{blue}(0,2,1)},{\color{blue}(1,2,0)},{\color{blue}(0,4,0)}\implies \Gamma(\Tilde{\mathcal{D}})=0,2,3,4\qquad \delta_{add} =0. \nonumber\label{selection_rule}
\end{eqnarray}

For example, when we consider the case $\Gamma=1$ (which means that we look at the terms coupling to $f_A$ in the action (<ref>).), then we can only consider the following selections $(1,0,0)$ (i.e. only one $d$ and others are zero.) from $\tilde{\mathcal{B}}_a$ or $(0,1,0)$ (i.e. only one $f$ and others are zero.) from $\tilde{\mathcal{Z}}$, and the couplings to $R'_{\rho\sigma}(Q)$ and $\Bar{\psi}\psi$ in the action (<ref>).

Since we are interested in $\mathcal{U}$ as a quadratic function of matter fields, we have $u_3=u_4=0$, so that

\begin{eqnarray}
\mathcal{L}_{\textrm{new FI}} &\supset&
(K''')^{k_3}(K'''')^{k_1} (\mathcal{U}^{(1)})^{\alpha}(\mathcal{U}^{(2)})^{u_2}\nonumber\\
&&\times (D)^{-n}(F)^{r}(K')^{\beta}(K'')^{\gamma} \mathcal{O}^{(\delta_{pre})}\hat{\mathcal{O}}^{(\delta_{add})}.
\end{eqnarray}

Since we consider the case of $D \sim F$, it reduces to

\begin{eqnarray}
\mathcal{L}_{\textrm{new FI}} &\supset&
(K''')^{k_3}(K'''')^{k_1} (\mathcal{U}^{(1)})^{\alpha}(\mathcal{U}^{(2)})^{u_2}\nonumber\\
&&\times (D)^{-n+r}(K')^{\beta}(K'')^{\gamma} \mathcal{O}^{(\delta_{pre})}\hat{\mathcal{O}}^{(\delta_{add})}.
\end{eqnarray}

In particular, we can make these terms to be small due to $\mathcal{U}^{(1)} \sim z^i \sim 0$ along the inflationary trajectory, we can set $\alpha = u_1+d_{T1} + d_{T2}^{\lambda} =0$, which means $u_1=d_{T1}= d_{T2}^{\lambda}=0$. Moreover, the derivatives of the Kähler potential with respect to the matter fields can be small as well if it includes the quadratic function of some matter fields. Of course, the second derivative cannot be neglected like the case of $\mathcal{U}^{(2)}$. However, the moduli fields will not be small due to the logarithmic dependence in the Kähler potential. Nevertheless, it is not a problem to ignore the derivatives here because they will be suppressed for large inflaton field, while they will be of order of $\mathcal{O}(1)$ in Planck unit after inflation. In any case, we can drop the dependence of the derivatives of the Kähler potential but the second derivatives. Furthermore, the second derivatives of the Kähler potential with respect to the moduli fields can be of order of $\mathcal{O}(1)$ after inflation, so that ignoring the factor of $K''$, we get

\begin{eqnarray}
\mathcal{L}_{\textrm{new FI}} &\supset&
(\mathcal{U}^{(2)})^{u_2} (D)^{-N} \mathcal{O}^{(\delta_{pre})}\hat{\mathcal{O}}^{(\delta_{add})} \equiv (\mathcal{U}^{(2)})^{u_2} (D)^{-N} \mathcal{O}^{(\delta_{tot})},
\end{eqnarray}

where $N\equiv n-r$ and

\begin{eqnarray}
N &=& 3  + d_{T3}^{\lambda} + 2d_{T4}^{\lambda} + f_{T2}^{\lambda} + 2f_{T3}^{\lambda} \nonumber\\
&&- d_{W3}^{\psi\lambda} -f_{W2}^{\lambda}-2a_{W1} -a_{s1} -a_{z1} - a_{T1}-u_0,\\
\delta_{pre} &=& 2\delta_{\Gamma,0}+6_{4\lambda}-3_{2\lambda}b +2d_{s1}+d_{s2}+4d_{s3}^{\psi}+\frac{3}{2}f_{s1}+2d_{z1}+4d_{z2}^{\psi}+\frac{3}{2}f_{z1} \nonumber\\
&& + 4d_{W1}^{\lambda\lambda}+5d_{W2}^{\psi\lambda}+3d_{W3}^{\psi\lambda}+\frac{7}{2}f_{W1}^{\lambda}+\frac{3}{2}f_{W2}^{\lambda}+4a_{W2} \nonumber\\
&&+5d_{T3}^{\lambda}+7d_{T4}^{\lambda}+\frac{3}{2}f_{T1}+\frac{5}{2}f_{T2}^{\lambda}+\frac{11}{2}f_{T3}^{\lambda}+3 a_{T2},\\
\delta_{add} &=& \textrm{one of  } \mathcal{D}_0;\Lambda_{\Bar{\psi}3/2};\mathcal{C}_{R(\omega)2};\mathcal{C}_{\Bar{\psi}R'(Q)4}; \mathcal{Z}_{R'(Q)5/2};\mathcal{B}_{\Bar{\psi}\psi 3};\mathcal{Z}_{\Bar{\psi}\psi\Bar{\psi}9/2},\\
\delta_{tot} &\equiv& \delta_{pre} + \delta_{add}.
\end{eqnarray}

In the meantime, we had $k_1+k_2+k_3+k_4  + u_1 + u_2 + u_3 + u_4 = m$ and $d_{z1}+d_{z2}^{\psi}+f_{z1}+a_{z1}=m_1+m_2+m_3=m \leq 4$. Since we now have $u_1=u_3=u_4=0$, we get

\begin{eqnarray}
u_2 =d_{z1}+d_{z2}^{\psi}+f_{z1}+a_{z1}-k_{tot},\label{u2_value}
\end{eqnarray}

where we define $k_{tot}\equiv k_1+k_2+k_3+k_4$. Here, we point out that if the mass-squared from $\mathcal{U}^{(2)}$ is of order of the electroweak scale, then the non-renormalizable terms can be suppressed strongly. If not, we have to keep it.

Looking at the above terms, we observe that the non-renormalizable terms will have the following form

\begin{eqnarray}
\mathcal{L}_{\textrm{new FI}}  \supset  \frac{(\mathcal{U}^{(2)})^{u_2}}{D^N}  \mathcal{O}^{(\delta_{tot})} \implies  \frac{(\mathcal{U}^{(2)})^{u_2}}{D^N}  \lesssim \frac{1}{\Lambda_{cut}^{\delta_{tot}-4}} \implies \Lambda_{cut} \lesssim \left(\frac{D^N}{(\mathcal{U}^{(2)})^{u_2}}\right)^{1/(\delta_{tot}-4)}
\end{eqnarray}

where we set $M_{pl}=1$ for simplicity. Since we are interested in $D\sim \mathcal{U}^{(2)}\sim H$ where $H \sim 10^{-5}$ is the Hubble scale, the constraint can be given by

\begin{eqnarray}
\Lambda_{cut} \lesssim H^{(N-u_2)/(\delta_{tot}-4)} = H^{\alpha}.
\end{eqnarray}

We note that in order for the non-renormalizable terms to well-suppressed for $H<1$, we have to require that

\begin{eqnarray}
\alpha \equiv \frac{N-u_2}{\delta_{tot}-4} \leq 1,
\end{eqnarray}

where $\delta_{tot}-4>0$ for nonrenormalizable operators. In particular, the restrictions on the cutoff will come from the first condition only since the upperbound of the second condition exceeds $\mathcal{O}(1) \sim M_{pl} =1$. In the meantime, since we consider the non-renormalizable operators, we always have $\delta_{tot}>4$. That is, if there exists a combination of the parameters satisfying the above condition with maximal value 1, then the cutoff must be constrained as $\Lambda_{cut} \lesssim H$, which will be out of our interest.

Inserting $u_0=\delta_{m,k_{tot}}$ and the value of $u_2$ in Eq. (<ref>), the relevant parameters are as follow:

\begin{eqnarray}
N -u_2&=& 3  + d_{T3}^{\lambda} + 2d_{T4}^{\lambda} + f_{T2}^{\lambda} + 2f_{T3}^{\lambda} \nonumber\\
&&- d_{W3}^{\psi\lambda} -f_{W2}^{\lambda}-2a_{W1} -a_{s1} -a_{z1} - a_{T1}-u_0-u_2,\\
\delta_{tot}-4 &=&2\delta_{\Gamma,0}+ 6_{4\lambda}-3_{2\lambda}b +2d_{s1}+d_{s2}+4d_{s3}^{\psi}+\frac{3}{2}f_{s1}+2d_{z1}+4d_{z2}^{\psi}+\frac{3}{2}f_{z1} \nonumber\\
&& + 4d_{W1}^{\lambda\lambda}+5d_{W2}^{\psi\lambda}+3d_{W3}^{\psi\lambda}+\frac{7}{2}f_{W1}^{\lambda}+\frac{3}{2}f_{W2}^{\lambda}+4a_{W2} \nonumber\\
&&+5d_{T3}^{\lambda}+7d_{T4}^{\lambda}+\frac{3}{2}f_{T1}+\frac{5}{2}f_{T2}^{\lambda}+\frac{11}{2}f_{T3}^{\lambda}+3 a_{T2} \nonumber\\
&&+\delta_{add}-4 \label{del-4}.
\end{eqnarray}

Here is a remark. Since the derivatives of $\mathcal{U}$ but $\mathcal{U}^{(2)}$ can give us the small number from the vacuum expectation values of matter fields, we consider the case of either $m=k_{tot}=d_{z1}+f_{z1}+a_{z1}+d_{z2}^{\psi}$ (which implies that $u_i=0$ for $i=1,2,3,4$ and $u_0=\delta_{m,k_{tot}}=1$) or $u_2=1$. Thus, it reduces to

\begin{eqnarray}
N -u_2&=& 2 + d_{T3}^{\lambda} + 2d_{T4}^{\lambda} + f_{T2}^{\lambda} + 2f_{T3}^{\lambda} - d_{W3}^{\psi\lambda} -f_{W2}^{\lambda}-2a_{W1} -a_{s1} -a_{z1} - a_{T1}\label{N-u2}.
\end{eqnarray}

Specifically, we obtain the following different cases.

* Case of $b=0$ (with no derivatives):

\begin{eqnarray}
(N -u_2)_{b=0}&=& 2-a_{s1} -a_{z1} - a_{T1} ,\\
(\delta_{tot}-4)_{b=0} &=&2+ 6_{4\lambda} +\delta_{add}-4
\end{eqnarray}

* Case of $b=0$ (with derivatives):

\begin{eqnarray}
(N -u_2)_{b=0}&=& 2-a_{s1} -a_{z1} - a_{T1} ,\\
(\delta_{tot}-4)_{b=0} &=&6_{4\lambda} +2d_{s1}+d_{s2}+4d_{s3}^{\psi}+\frac{3}{2}f_{s1}+2d_{z1}
\nonumber\\
&& +4d_{z2}^{\psi}+\frac{3}{2}f_{z1} +\frac{3}{2}f_{T1}+3 a_{T2}+\delta_{add}-4
\end{eqnarray}

* Case of $b=1$:

\begin{eqnarray}
(N -u_2)_{b=1=f_{W1}^{\lambda}}&=& 2 + d_{T3}^{\lambda} + 2d_{T4}^{\lambda} + f_{T2}^{\lambda} + 2f_{T3}^{\lambda}-a_{s1} -a_{z1} - a_{T1} ,\\
(\delta_{tot}-4)_{b=1=f_{W1}^{\lambda}} &=&3_{2\lambda} +2d_{s1}+d_{s2}+4d_{s3}^{\psi}+\frac{3}{2}f_{s1}+2d_{z1}+4d_{z2}^{\psi}+\frac{3}{2}f_{z1} \nonumber\\
&&+5d_{T3}^{\lambda}+7d_{T4}^{\lambda}+\frac{3}{2}f_{T1}+\frac{5}{2}f_{T2}^{\lambda}+\frac{11}{2}f_{T3}^{\lambda}+3 a_{T2} +\delta_{add}-\frac{1}{2},
\end{eqnarray}

\begin{eqnarray}
(N -u_2)_{b=1=f_{W2}^{\lambda}}&=& 1  + d_{T3}^{\lambda} + 2d_{T4}^{\lambda} + f_{T2}^{\lambda} + 2f_{T3}^{\lambda} -a_{s1} -a_{z1} - a_{T1},\\
(\delta_{tot}-4)_{b=1=f_{W2}^{\lambda}} &=& 3_{2\lambda} +2d_{s1}+d_{s2}+4d_{s3}^{\psi}+\frac{3}{2}f_{s1}+2d_{z1}+4d_{z2}^{\psi}+\frac{3}{2}f_{z1} \nonumber\\
&&+5d_{T3}^{\lambda}+7d_{T4}^{\lambda}+\frac{3}{2}f_{T1}+\frac{5}{2}f_{T2}^{\lambda}+\frac{11}{2}f_{T3}^{\lambda}+3 a_{T2} +\delta_{add}-\frac{5}{2},
\end{eqnarray}

* Case of $b=2$:

\begin{eqnarray}
(N -u_2)_{b=2=f_{W1}^{\lambda}} &=& 2  + d_{T3}^{\lambda} + 2d_{T4}^{\lambda} + f_{T2}^{\lambda} + 2f_{T3}^{\lambda}-a_{s1} -a_{z1} - a_{T1}  ,\\
(\delta_{tot}-4)_{b=2=f_{W1}^{\lambda}} &=&
2d_{s1}+d_{s2}+4d_{s3}^{\psi}+\frac{3}{2}f_{s1}+2d_{z1}+4d_{z2}^{\psi}+\frac{3}{2}f_{z1}  \nonumber\\
&&+5d_{T3}^{\lambda}+7d_{T4}^{\lambda}+\frac{3}{2}f_{T1}+\frac{5}{2}f_{T2}^{\lambda}+\frac{11}{2}f_{T3}^{\lambda}+3 a_{T2} +\delta_{add}+3,
\end{eqnarray}

\begin{eqnarray}
(N -u_2)_{b=2=f_{W2}^{\lambda}} &=&  d_{T3}^{\lambda} + 2d_{T4}^{\lambda} + f_{T2}^{\lambda} + 2f_{T3}^{\lambda} -a_{s1} -a_{z1} - a_{T1},\\
(\delta_{tot}-4)_{b=2=f_{W2}^{\lambda}} &=&
2d_{s1}+d_{s2}+4d_{s3}^{\psi}+\frac{3}{2}f_{s1}+2d_{z1}+4d_{z2}^{\psi}+\frac{3}{2}f_{z1} \nonumber\\
&&+5d_{T3}^{\lambda}+7d_{T4}^{\lambda}+\frac{3}{2}f_{T1}+\frac{5}{2}f_{T2}^{\lambda}+\frac{11}{2}f_{T3}^{\lambda}+3 a_{T2} +\delta_{add}-1,
\end{eqnarray}

\begin{eqnarray}
(N -u_2)_{b=2,f_{W1}^{\lambda}=f_{W2}^{\lambda}=1} &=&1  + d_{T3}^{\lambda} + 2d_{T4}^{\lambda} + f_{T2}^{\lambda} + 2f_{T3}^{\lambda} -a_{s1} -a_{z1} - a_{T1},\\
(\delta_{tot}-4)_{b=2,f_{W1}^{\lambda}=f_{W2}^{\lambda}=1} &=&
2d_{s1}+d_{s2}+4d_{s3}^{\psi}+\frac{3}{2}f_{s1}+2d_{z1}+4d_{z2}^{\psi}+\frac{3}{2}f_{z1} \nonumber\\
&&+5d_{T3}^{\lambda}+7d_{T4}^{\lambda}+\frac{3}{2}f_{T1}+\frac{5}{2}f_{T2}^{\lambda}+\frac{11}{2}f_{T3}^{\lambda}+3 a_{T2} +\delta_{add}+1. \nonumber\\{}
\end{eqnarray}

The following parameter condition must be satisfied
\begin{eqnarray}
\Gamma \equiv c+m+b+p&=& d_{s1}+d_{s2}+d_{s3}^{\psi}+f_{s1}+a_{s1}+\underbrace{d_{z1}+d_{z2}^{\psi}+f_{z1}+a_{z1}}_{=k_{tot} + u_0 + u_2} \nonumber\\
&& +\underbrace{d_{W1}^{\lambda\lambda}+d_{W2}^{\psi\lambda}+d_{W3}^{\psi\lambda}+f_{W1}^{\lambda}+f_{W2}^{\lambda}+a_{W1}+a_{W2}}_{=~ b\leq 2} \nonumber\\
&& +d_{T3}^{\lambda}+d_{T4}^{\lambda}+f_{T1}+f_{T2}^{\lambda}+f_{T3}^{\lambda}+a_{T1}+a_{T2} \quad \leq 4.
\end{eqnarray}

Again, $\Gamma$ is defined as the number of derivatives among $\partial_i^m$, $\partial_0^c$, $\partial_W^b$, and $\partial_T^p$. In the appendix <ref>, the largest value of $\alpha$ is found by $\alpha =2/3$, so that

\begin{eqnarray}
\Lambda_{cut} \lesssim H^{2/3} \sim 10^{-5(2/3)} = 10^{-2.66} < M_S \sim 10^{-2.5}.
\end{eqnarray}

Hence, the theory of new FI terms is basically an effective field theory with broken supersymmetry.

CHAPTER: RELAXED SUPERGRAVITY

[1.05]This chapter is based on the author's original work in Ref. <cit.>.

In this chapter, the key result we will see is given by

\begin{eqnarray}
V = V_D + V_F - \mathcal{U}, \qquad \frac{1}{\mathcal{U}}
\lesssim \frac{1}{\Lambda_{cut}^4},
\end{eqnarray}

where $V_D,V_F$ are the supergravity D- and F-term scalar potentials and $\mathcal{U}$ is defined as a gauge-invariant generl real function of matter scalars $z$'s. In particular, we will discuss how to relax the scalar potential $V$ that can satisfy the constraint above.

§ INTRODUCTION

The application of supergravity to inflationary cosmology has recently been of great interest and studied by many authors as discussed in Ch. <ref>. However, it remains still challenging to build viable models of a certain phenomenology in the context of supergravity. For instance, it is not straightforward to realize both inflationary dynamics and minimal supersymmetric standard model (MSSM) at the same time in a unified setup. The first reason for the difficulty is due to the large hierarchy between Hubble scale $H\sim 10^{-5}M_{pl}$ for inflation and electroweak (TeV) scale of order $10^{-15}M_{pl}$ for the observable-sector dynamics of standard model (SM). The second is because standard supergravity predicts the complicated structure of the F-term scalar potential, i.e. $V_F=e^G(G_IG^{I\bar{J}}G_{\bar{J}}-3)$ where $G$ is the supergravity G-function defined by $G \equiv K + \ln W + \ln \bar{W}$ which consists of Kähler potential $K$ and superpotential $W$. This implies that one must always explore a proper choice of the supergravity G-function, which is unfortunately nontrivial in general. For these reasons, it is very demanding to construct phenomenologically-desirable scalar potentials within standard supergravity.

The $\eta$ problem <cit.> is an example of such a difficulty[This issue gives rise to a hardship for obtaining a very small slow-roll parameter $\eta$ such that $\eta \ll 1$ due to the exponentially growing behavior of the F-term potential.]. No-scale supergravity <cit.> may be a solution to the $\eta$ problem because the corresponding F-term potential can exactly vanish, i.e. $V_F =0$. This is established by a clever choice of the supergravity G-function. In fact, the gravitino mass term “$-3e^G$” plays a critical role in the no-scale cancellation. Interestingly, one can easily have such no-scale structure through a logarithmic Kähler potential of the volume modulus fields and constant superpotential in string theory <cit.>. However, certain choice of superpotential may spoil the “exact” cancellation of the F-term potential yielding a remnant as shown in Eq. (14) of Ref. <cit.>. Hence, no-scale supergravity is very sensitive to the given form of both Kähler potential and superpotential. Moreover, no-scale supergravity may cause a vast number of moduli, which correspond to degenerate vacua being along flat directions in scalar potential. This turns out that moduli stabilization, which is necessary to obtain a unique vacuum, is still required in no-scale supergravity. Thus, current no-scale supergravity is not a complete strategy for model building.

Recent developments in modification of the supergraivty scalar potential have been made, e.g. liberated supergravity recently proposed by Farakos, Kehagias, and Riotto <cit.> and various types of new Fayet-Iliopoulos (FI) terms proposed by many authors <cit.>. In particular, liberated supergravity was the first attempt to allows us to have a general scalar potential. In fact, inspired by liberated supergravity, we investigate such a general scalar potential in the other fashion in this work. However, it has recently been found that liberated supergravity is not liberated literally due to strong constraints on the general function <cit.>. On the contrary, new FI terms can modify only D-term potentials, which still have non-trivial field dependence and can give us only the non-negative-definite contribution to the scalar potential. Consequently, the recent studies do not have full generality of scalar potential.

Obviously, it has very long been thought of that a negative-definite term in scalar potential can be given only by the gravitino-mass term “$-3e^G$” in the standard $\mathcal{N}=1$ supergravity, and is not present in global supersymmetry (SUSY) <cit.>. It is thus inevitable to acquire another type of cancellation in scalar potential through a new negative term so that we have a general scalar potential being beyond no-scale supergravity and the recent works. In that sense, it remains very intriguing to answer the following open questions: How can we obtain a new negative-definite potential term in supergravity? To what extent can we reform the supergravity scalar potential in a general fashion? We affirmatively answer these questions throughout this letter.

Our work is organized as follows. In Sec. <ref>, we revisit the higher order corrections in the minimal supergravity models of inflation constructed by Ferrara, Kallosh, Linde, and Porrati (FKLP) <cit.>. We firstly identify a no-go theorem for the higher order corrections. The no-go theorem is supported by the fact that canonical kinetic term of the gauge field in the vector multiplet must be present in the supergravity lagrangian. In Sec. <ref>, we propose how to relax the strongly-constrained standard form of the scalar potential by adding a special choice of higher order correction to the standard supergravity. Here we discover a new negative-definite scalar potential as a general function. In addition, we find essential constraints on the new negative term by inspecting the suppression of nonrenormalizable lagrangians to ensure that our theory is self-consistent as an effective field theory. This leads to a cutoff which is identified with the high-scale SUSY breaking mass $M_S$ <cit.>. Next, using the new negative term, we present a relaxing procedure for generating a general scalar potential[In this sense, our proposal reserves the name “Relaxed supergravity.”]. Plus, we compare our theory with the liberated supergravity by Farakos, Kehagias, and Riotto. Next, we briefly discuss the global SUSY limit of relaxed supergravity. Finally, in Sec. <ref>, we summarize our findings and give outlook on this work.

§ NO-GO THEOREM FOR HIGHER ORDER CORRECTIONS IN FKLP MODEL

In this section, we revisit higher order corrections in the minimal supergravity models of inflation proposed by  Ferrara, Kallosh, Linde, and Porrati (FKLP) <cit.>. First, we start with considering a vector multiplet $V$, its field strength multiplet $\bar{\lambda}P_L\lambda$, and real linear multiplet $(V)_D$ whose lowest component is given by the auxiliary field $D$ of the vector multiplet $V$ as follows:

\begin{eqnarray}