Considering this post: Quantum String action the action given is of the lowest order but the effective action, for low energies, is given by:

$$ S_{ef.}= -\frac{1}{2k^2} \left( S^{(0)}+ \alpha S^{(1)} + \alpha^2 S^{(2)}+... \right)$$

where $S^{(0)}$ , $S^{(1)}$ , $S^{(2)}$ ,...etc are operators of increasing dimension that envolve the product of the massless fields ($G_{MN}$ and $B_{MN}$) and their derivatives. Considering $S^{(1)}$ and that $H_{MNP}=0$, I was told that we should expect to have $9$ terms and that the Lagrangean density for this case is of the form:

$$ \mathfrak{L}=c e^{m \phi} \left[... \right] $$

I'm having trouble figuring out what these terms are (they are supposed to appear in those parenthesis above but I dont know what they are) and why are there $9$ of them.



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