  In this article actions for N=4 SYM and N=8 supergravity are formulated in
terms of a chiral superfield, which contains only the physical degrees of
freedom of either theory. In these new actions, which originate from the
lightcone superspace, the supergravity cubic vertex is the square of the gauge
theory one (omitting the color structures). Amplitude calculations using the
corresponding Feynman supergraph rules are tedious, but can be simplified by
choosing a preferred superframe. Recursive calculations of all MHV amplitudes
in N=4 SYM and the four-point N=8 supergravity amplitude are shown to agree
with the known results and connections to the BCFW recursion relations are
pointed out. Finally, the new path integrals are discussed in the context of
the double-copy property relating N=4 SYM theory to N=8 supergravity.
