# How to calculate a real (vector) superfield in Wess-Zumino gauge

In Wess, Bagger's "SUSY and SUGRA", Eq. (6.6), they write down the real superfield in WZ-gauge as $$V = -\theta\sigma^\mu \overline\theta v_\mu +\text{i}\theta\theta\overline\theta\overline\lambda - \text{i}\overline\theta\overline\theta\theta\lambda+\frac{1}{2}\theta\theta\overline\theta\overline\theta D,$$ where the degrees of freedom are a real vector field $v_\mu$, a spinor field $\lambda$ and a real scalar field $D$. However, in Hori et al.'s "Mirror Symmetry", they write the real vector field in WZ gauge in Eq. (15.26) as \begin{align*} V &= \theta^-\overline\theta^- (v_0-v_1) + \theta^+\overline\theta^+ (v_0+v_1)-\theta^-\overline\theta^+\sigma-\theta^+\overline\theta^-\overline\sigma\\ &\qquad +\text{i}\theta^-\theta^+ (\overline\theta^-\overline\lambda_- + \overline\theta^+\overline\lambda_+) + \text{i}\overline\theta^+\overline\theta^- (\theta^-\lambda_- + \theta^+\lambda_+) + \theta^-\theta^+ \overline\theta^+\overline\theta^- D. \end{align*} Here, the degrees of freedom are again a vector field $v_\mu$, a spinor field $\lambda$ and a real scalar field $D$. But additionally, there's a complex scalar field $\sigma$.

Why are these two expressions different? Where does this additional scalar field come from?

Wess and Bagger's spacetime is four dimensional, so $v_\mu$ has 4 components. For $\sigma^\mu = (1,\vec\sigma)$, $$\sigma^\mu v_\mu = \begin{pmatrix} v_0+v_3 &v_1-\text iv_2\\ v_1+\text iv_2&v_0-v_3 \end{pmatrix},$$ which means that $$\begin{array}{ll} \text{Wess, Bagger} & \text{Hori et al.}\\ v_0+v_3 & v_0+v_1\\ v_1+\text iv_2 & \sigma \end{array}$$