On the long standing issue of an R phi(2) coupling between gravity and a massless free scalar field - Accioly, Antonio Jose [UNESP]; Aldrovandi, R.; Novaes, Sérgio Ferraz [UNESP]; Spehler, D.; Mukai, H. (Kyoto Univ, 1997)
Using 2-body trees on a flat space background, it is shown that the actions A[g,phi]=integral d(4)x root-g [(R/2 kappa)+(1/2)(g(mu nu)partial derivative(mu)phi partial derivative(mu)phi+lambda R phi(2))] and (A) over bar[(g) over bar,<(phi)over bar>]=integral d(4)x root-(g) over bar[((R) over bar/2 kappa)+(1/2)(g) over bar(mu nu)partial derivative(mu)<(phi)over bar>partial derivative(nu)phi] describe the same theory at the tree-level in this case. We also demonstrate the quantum equivalence (at one-loop) of the barred and unbarred systems for lambda=-1/6 (conformal coupling).