  In the semi-light cone gauge $g_{ab}=e^{2\phi}\delta_{ab}$,
$\bar{\gamma}^+\theta=0$, we evaluate the $\phi$-dependent effective action for
the pp-wave Green-Schwarz (GS) superstring in both harmonic and group
coordinates. When we compute the fermionic $\phi$-dependent effective action in
harmonic coordinates, we find a new triangular one-loop Feynman diagram. We
show that the bosonic $\phi$-dependent effective action cancels with the
fermionic one, indicating that the pp-wave GS superstring is a conformal field
theory. We introduce the group coordinates preserving $SO(4)\times SO(4)$ and
conformal symmetry. Group coordinates are interesting because vertex operators
take simple forms in them. The new feature in group coordinates is that there
are logarithmic divergences from n-gons, so that the divergent structure is
more complicated than in harmonic coordinates. After summing over all
contributions from n-gons, we show that in group coordinates, the GS
superstring on pp-wave RR background is still a conformal field theory.
