# Breaking of supersymmetric QED in the O'Raifeartaigh model

The action for supersymmetric QED is

\begin{align} S_{\text{SQED}} = \int d^4x\left\{8d(x)^2- F_{\mu\nu}(x)F^{\mu\nu}(x) - 4i\lambda(x)\sigma^{\mu}\partial_{\mu}\bar{\lambda}(x)+ |F_L(x)|^2 + |F_{\bar{E}}(x)|^2 \right.\\ + iD_{\mu}^*\bar{\psi}_L(x)\bar{\sigma}^{\mu} \psi_L(x) + iD_{\mu}\bar{\psi}_{\bar{E}}(x)\bar{\sigma}^{\mu} \psi_{\bar{E}}(x) + |D_{\mu}A_L(x)|^2 + |D_{\mu}^*A_{\bar{E}}(x)|^2\\ + ed(x)\left(|A_L(x)|^2 - |A_{\bar{E}}(x)|^2\right) - \frac{e}{\sqrt{2}}[\bar{\lambda}(x)[\bar{\psi}_L(x)A_L(x) - \bar{\psi}_{\bar{E}}(x)A_{\bar{E}}(x)]\\ +\lambda(x)[\psi_L(x)A^*_L(x) - \psi_{\bar{E}}(x)A_{\bar{E}}^*(x)]] - m[A_L(x)F_{\bar{E}}(x) + A_L^*(x)F_{\bar{E}}^*(x)\\ + A_{\bar{E}}(x)F_{L}(x) + A_{\bar{E}}^*(x)F_{L}^*(x) - \left. \psi_L(x)\psi_{\bar{E}}(x) - \bar{\psi}_L(x)\bar{\psi}_{\bar{E}}(x)]\right\}, \end{align} where $A_{L,\bar{E}}$ are selectrons, $\lambda(x)$ is the photino, $\psi_{L,\bar{E}}$ are electrons and $V_{\mu}$ is the photon. The selectrons and electrons have the same mass $m$, while the photon and photino have mass $m=0$. I am breaking this symmetry in the O'Raifeartaigh model which introduces three new scalar fields given by \begin{align} A &= a + \sqrt{2}\theta\psi_A + \theta\theta F_A\\ X &= x + \sqrt{2}\theta\psi_X + \theta\theta F_X\\ Y &= y + \sqrt{2}\theta\psi_Y + \theta\theta F_Y, \end{align} and the superpotential \begin{align} W[A, X, Y] \equiv nAY + gX(A^2 - M^2). \end{align} where $n, g, M\in\mathbb{R}$. This is an example of so called $F$-term breaking of SUSY where one obtains a non-vanishing VEV using the auxiliary field $F_i$.

In the derivation of $\mathcal{L}_{SQED}$, I use three fields: \begin{align} L &= A_L(x) + \sqrt{2}\theta \psi_L(x) + \theta\theta F_L(x),\\ \bar{E} &= A_E(x) + \sqrt{2}\theta \psi_E(x) + \theta\theta F_E(x),\\ V_{WZ}(x, \theta, \bar{\theta}) &= (\theta\sigma^{\mu}\bar{\theta})V_{\mu} + \theta\theta\bar{\theta}\bar{\lambda}(x) + \bar{\theta}\bar{\theta}\theta\lambda(x) + \theta\theta\bar{\theta}\bar{\theta}d(x), \end{align} where $L$ and $\bar{E}$ are lepton superfields while $V_{WZ}$ is the vector superfield in the Wess-Zumino gauge. I only have two scalar fields, while O'Raifeartaigh breaks SUSY spontaneously with three. How can I break the symmetry of the masses in the O'R. model when I'm lacking a scalar field? I'm having a hard time finding sources on this topic as well, any suggestions?