I'm learning SUSY with Quevedo, Cambridge Lectures on Supersymmetry and Extra Dimensions.


The SUSY transformations of the component fields of a chiral field $\Phi$ are given by (p.41)

\begin{align*} \delta_{\epsilon,\overline{\epsilon}}\varphi &= \sqrt{2}\epsilon^{\alpha}\psi_{\alpha}, \\ \delta_{\epsilon,\overline{\epsilon}}\psi_{\alpha} &= i\sqrt{2}\sigma^{\mu}_{\alpha\dot{\alpha}}\overline{\epsilon}^{\dot{\alpha}}\partial_{\mu}\varphi + \sqrt{2}\epsilon_{\alpha}F,\\ \delta_{\epsilon,\overline{\epsilon}} F &=i\sqrt{2}\overline{\epsilon}_\dot{\alpha}(\overline{\sigma}^{\mu})^{\dot{\alpha}\alpha}\partial_{\mu}\psi_{\alpha}, \end{align*} where $\varphi$ is a complex scalar, $\psi_{\alpha}$ is a left-handed Weyl spinor and $F$ is an auxiliary field.

My questions:

  1. Let us choose the superpotential $W(\Phi)\equiv \frac{m}{2}\Phi^2 + \frac{g}{3}\Phi^3$ together with kinetic part $\Phi^{\dagger}\Phi$ and remove the auxiliary field $F$ via its algebraic equations of motion. Then, the transformation rules must change as well, correct?

  2. We can use the equations of motion of the auxiliary field $F$ to remove it from the Lagrangian. How do we account for this in the transformation rules of the component fields? The transformation rules do not know anything about the model (free/interacting/massless) we are considering, so it is us who should implement this choice into the transformation rules -- but how do we do this without messing up SUSY?

  1. When we eliminate/integrate out the auxiliary field $F$, the SUSY transformation for $F$ is rendered moot, and the appearance of $F$ on the RHSs of the other SUSY transformations is replaced with its algebraic EOM.

  2. It's not true that we do not know anything about the model -- we assume that the action $S$ is SUSY-invariant. In particular, the EOM for $F$ is derived from the action.

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