We (re)derive the propagators and Feynman rules for the massless scalar and vector multiplets in $N=2$ projective superspace (“projective hyperspace”). With these, we are able to calculate both the divergent and finite parts of 2, 3, & 4-point functions at 1-loop for $N=2$ super-Yang-Mills theory), explicitly in projective hyperspace itself. We find that effectively only the coupling constant needs to be renormalized, unlike in the $N=1$ case where an independent wavefunction renormalization is also required. This feature is similar to that of the background field gauge, even though we are using ordinary Fermi-Feynman gauge. The computation of 1-hoop beta-function is then straightforward and matches with the known result. We also show that it receives no 2-hoops contributions. All these calculations provide an alternative proof of the finiteness of $N=4$ super-Yang-Mills.

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