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In $\mathcal{N}=1$ super electrodynamics, one has the following vector superfield

$$V(x,\theta,\bar{\theta})=\bar{\theta}\bar{\sigma}^{\mu}\theta v_{\mu}(x)+\bar{\theta}^{2}\theta\lambda+\theta^{2}\bar{\theta}\bar{\lambda}+\theta^{2}\bar{\theta}^{2}D(x)$$

in Wess-Zumino gauge. One can define the following gauge invariant spinorial superfield

$$W_{\alpha}=-\frac{1}{4}\bar{D}^{2}D_{\alpha}V,$$

where $\bar{D}_{\dot{\alpha}}=\bar{\partial}_{\dot{\alpha}}-i(\bar{\sigma}^{\mu})_{\dot{\alpha}\beta}\theta^{\beta}\partial_{\mu}$, and $D_{\alpha}=\partial_{\alpha}-i(\sigma^{\mu})_{\alpha\dot{\beta}}\bar{\theta}^{\dot{\beta}}\partial_{\mu}$ are the supercovariant derivatives.

One can find that the gauge invariant spinorial superfield has the following component expansion:

$$W_{\alpha}(x,\theta,\bar{\theta})=\lambda_{\alpha}(x)+2\theta_{\alpha}D(x)+\frac{i}{2}\theta_{\beta}(\sigma^{\mu\nu})^{\beta}_{\,\,\,\alpha}f_{\mu\nu}(x)+i\theta^{2}(\bar{\sigma}^{\mu})_{\dot{\beta}\alpha}\partial_{\mu}\bar{\lambda}^{\dot{\beta}}(x)-i\theta\sigma^{\mu}\bar{\theta}\partial_{\mu}\lambda_{\alpha}(x)-2i\theta_{\alpha}\theta\sigma^{\mu}\bar{\theta}\partial_{\mu}D(x)+\frac{1}{2}\theta\sigma^{\mu}\bar{\theta}\theta_{\beta}(\sigma^{\mu\nu})^{\beta}_{\,\,\,\alpha}\partial_{\rho}f_{\mu\nu}(x)+\frac{1}{4}\theta^{2}\bar{\theta}^{2}\Box\lambda_{\alpha}(x),$$

where $f_{\mu\nu}=\partial_{\mu}v_{\nu}-\partial_{\nu}v_{\mu}$.

I am interested in the following integral

$$\int d^{2}\theta W^{\alpha}W_{\alpha}.$$

The relevant terms in the above product are

\begin{align} &\epsilon^{\alpha\beta}W_{\beta}W_{\alpha} \\ =\epsilon^{\alpha\beta}&\left[\lambda_{\beta}+2\theta_{\beta}D+\frac{i}{2}\theta_{\gamma}(\sigma^{\mu\nu})^{\gamma}_{\,\,\,\beta}f_{\mu\nu}+i\theta^{2}(\bar{\sigma}^{\mu})_{\dot{\beta}\beta}\partial_{\mu}\bar{\lambda}^{\dot{\beta}}\right]\times \\ &\left[\lambda_{\alpha}+2\theta_{\alpha}D+\frac{i}{2}\theta_{\rho}(\sigma^{\mu\nu})^{\rho}_{\,\,\,\alpha}f_{\mu\nu}+i\theta^{2}(\bar{\sigma}^{\mu})_{\dot{\gamma}\alpha}\partial_{\mu}\bar{\lambda}^{\dot{\gamma}}\right]. \end{align}

My question comes from the following terms:

$$\epsilon^{\alpha\beta}(i\theta^{2}(\bar{\sigma}^{\mu})_{\dot{\gamma}\alpha}(\partial_{\mu}\bar{\lambda}^{\dot{\gamma}})\lambda_{\beta}+i\theta^{2}(\bar{\sigma}^{\mu})_{\dot{\beta}\beta}(\partial_{\mu}\bar{\lambda}^{\dot{\beta}})\lambda_{\alpha})$$

It seems that the above two terms cancel, which must be wrong. Where did I make the mistake?

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In the first term of your last equation $\lambda_\beta$ should be on the left side of $\partial\bar{\lambda}^{\dot{\gamma}}$. Spinors anticommute.

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  • $\begingroup$ I thought spinors $\lambda_{\beta}$ and $\bar{\lambda}_{\dot{\beta}}$ are $\mathbb{C}$-valued complex spinors in Minkowski spacetime. $\endgroup$ – The Last Knight of Silk Road Apr 11 at 16:01
  • $\begingroup$ @TheLastKnightofSilkRoad Yes, they are. Does that contradict their anticommutativity? $\endgroup$ – Kosm Apr 11 at 16:14
  • $\begingroup$ If you agree that they are $\mathbb{C}$-valued spinors, then how could they anti-commute? $\endgroup$ – The Last Knight of Silk Road Apr 11 at 16:19
  • $\begingroup$ @TheLastKnightofSilkRoad Because they are Grassmann-odd. I'm not 100% sure how to formally call components of such spinors, but my guess would be complex valued supernumbers. Regardless of classical formalism, after quantization they become operator valued. $\endgroup$ – Kosm Apr 11 at 16:38
  • $\begingroup$ You are right. I just found a detailed explanation from the book "A Walk Through Superspace". In superspace formalism, the spinor fields in the component expansion of a superfield are actually Grassmann-valued, not $\mathbb{C}$-valued. Thank you for your answer. $\endgroup$ – The Last Knight of Silk Road Apr 11 at 18:31

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