Supersymmetric derivatives in chiral coordinates ($y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$)

For left Grassmann derivatives and $D_{\alpha }={\partial }_{\alpha }+i({\sigma }^{\mu }\bar{\theta }{)}_{\alpha }{\partial }_{\mu }$, ${\bar{D}}_{\dot{\alpha }}=-{\bar{\partial }}_{\dot{\alpha }}-i(\theta {\sigma }^{\mu }{)}_{\dot{\alpha }}{\partial }_{\mu }$, the coordinate $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$ gives $D_{\alpha }={\partial }_{\alpha }{|}_y+2i({\sigma }^{\mu }\bar{\theta }{)}_{\alpha }{\partial }_{y^{\mu }}$ and ${\bar{D}}_{\dot{\alpha }}=-{\bar{\partial }}_{\dot{\alpha }}{|}_y$. The odd chain rule gives ${\bar{\partial }}_{\dot{\alpha }}(\theta {\sigma }^{\mu }\bar{\theta })=-(\theta {\sigma }^{\mu }{)}_{\dot{\alpha }}$. A chiral superfield is consequently independent of $\bar{\theta }$ at fixed $y$.