Two-component superspace contraction signs ($\theta \theta ,\bar{\theta }\bar{\theta }$)

For left Grassmann derivatives, ${ϵ}^{12}=-{ϵ}_{12}=1$, $\theta \theta ={\theta }^{\alpha }{\theta }_{\alpha }$ and $\bar{\theta }\bar{\theta }={\bar{\theta }}_{\dot{\alpha }}{\bar{\theta }}^{\dot{\alpha }}$, the antisymmetric products are ${\theta }^{\alpha }{\theta }^{\beta }=-{ϵ}^{\alpha \beta }\theta \theta /2$ and ${\bar{\theta }}^{\dot{\alpha }}{\bar{\theta }}^{\dot{\beta }}={ϵ}^{\dot{\alpha }\dot{\beta }}\bar{\theta }\bar{\theta }/2$. With contractions ${\partial }^2={\partial }^{\alpha }{\partial }_{\alpha }$ and ${\bar{\partial }}^2={\bar{\partial }}_{\dot{\alpha }}{\bar{\partial }}^{\dot{\alpha }}$, both squared derivatives give $-4$ on the matching quadratic monomial. If $\mathrm{Tr}({\sigma }^{\mu }{\bar{\sigma }}^{\nu })=2{\eta }^{\mu \nu }$, the mixed bilinear product is $+(\theta \theta )(\bar{\theta }\bar{\theta }){\eta }^{\mu \nu }/2$. Reversing the metric-relative trace convention reverses this last sign.