= Stress tensor of a commuting beta-gamma system
{title2=$T=(1-\lambda):\partial\beta\,\gamma:-\lambda:\beta\,\partial\gamma:,\quad c=12\lambda^2-12\lambda+2$}
Using the <radial-ordering sign in commuting beta-gamma contractions>, single contractions give $T(z)\beta(w)\sim\lambda\beta(w)/(z-w)^2+\partial\beta(w)/(z-w)$ and the corresponding weight $1-\lambda$ for $\gamma$. Put $A=:\partial\beta\,\gamma:$ and $B=:\beta\,\partial\gamma:$. Their double-contraction coefficients in $AA,BB,AB,BA$ are respectively $1,1,2,2$. The fourth-order pole of $TT$ is therefore $(1-\lambda)^2+\lambda^2-4\lambda(1-\lambda)=6\lambda^2-6\lambda+1$, which equals $c/2$. There is no anticommuting Wick sign.
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