# Which is the analogous of the unitarity constraint of chiral perturbation theory in superchiral perturbation theory?

I am trying to reconstruct something and I would appreciate omeone helping me filling the gaps. To motivate my question let's first consider chiral perturbation theory with up and down quarks. This is the lowest order Lagrangian

$$\begin{equation*} \mathcal{L}_2=\frac{f^2}{4}\langle{}\partial_{\mu}U^{\dagger}\partial^{\mu}U\rangle \end{equation*}$$

where $f$ is some parameter with mass dimensions and $U$ is defined via

$$U=e^{it^a\phi_a/f}$$

where $t^a$ are the Pauli matrices and $\phi_a$ are complex scalar fields. For some reason i don't know people take this matrix to be unitary. So my first question is, why does $U$ have to be unitary?

In any case, if we demand $U$ to be unitary

$$UU^{\dagger}=1$$ $$e^{it^a\phi_a/f}e^{-it^a\phi_a^*/f}=1$$ and introducing $e^{-it^a\phi_a/f}$ on both sides

$$e^{-it^a\phi_a^*/f}=e^{-it^a\phi_a/f}$$ $$t^a\phi_a^*=t^a\phi_a$$ $$\begin{pmatrix} \pi_3&\pi_1-i\pi_2\\ \pi_1+i\pi_2&-\pi_3 \end{pmatrix}= \begin{pmatrix} \pi_3^*&\pi_1^*-i\pi_2^*\\ \pi_1^*+i\pi_2^*&-\pi_3^* \end{pmatrix}$$ now, redefining $\pi_3\equiv\pi_0$, $\pi_1+i\pi_2\equiv\sqrt{2}\pi_-$ and $\pi_1-i\pi_2\equiv\sqrt{2}\pi_+$ we see that the unitarity of $U$ imposes that $\pi_0$ is a real field after all and $\pi_-^*=\pi_+$

This up till now is nothing but the usual pion theory. I want to supersymmetrize this theory and this is where most of my doubts arise.

I have been told that I can embed the Lagrangian considered so far in

$$\mathcal{L}=f^2\int{}d^4\theta\langle\mathcal{U}^{\dagger}\mathcal{U}\rangle$$

where $$\mathcal{U}=e^{it^a\Phi_a/f}$$ where $t^a$ are still the Pauli matrices and $\Phi_a$ are chiral superfields. Now my second question. In the nonsupersymmetric theory we imposed a unitarity constraint in the matrix $U$. I have been told that there is an analogous constraint on $t^a\Phi_a$ but I don't know which it is, let alone where it comes from. So which constraint must I apply and why? my third question would be about how can I relate the pion fields of the nonsupersymetric theory with the superfields I have just introduced, or better how can I get $\frac{f^2}{4}\langle{}\partial_{\mu}U^{\dagger}\partial^{\mu}U\rangle$ from $\mathcal{L}=f^2\int{}d^4\theta\langle\mathcal{U}^{\dagger}\mathcal{U}\rangle$?