Ramond supercurrent zero-mode square ($G_0^2=L_0^R$)

With the standard string oscillator algebra and $G_0={\sum }_m{\alpha }_m\cdot {\psi }_{-m}$, pairing positive and negative modes gives $G_0^2={\alpha }_0^2/2+{\sum }_{m>0}({\alpha }_{-m}\cdot {\alpha }_m+m{\psi }_{-m}\cdot {\psi }_m)$. This is the vacuum-normal-ordered Ramond sector generator $L_0^R$, so $\{G_0,G_0\}=2L_0^R$. Rescaling the superconformal current rescales its square quadratically; its normalization cannot be ignored when comparing it with a fixed Virasoro generator.