Stress tensor of a commuting beta-gamma system ($T=(1-\lambda ):\partial \beta \gamma :-\lambda :\beta \partial \gamma :,c=12{\lambda }^2-12\lambda +2$)

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.