On the Long Standing Issue of an R 2 Coupling between Gravity and a Massless Free Scalar Field

Abstract

Using 2-body trees on a flat space background, it is shown that the actions A[g, φ] = (Latin small letter esh) d4x√-g [(R/2K) + (1/2)(gμν ∂μφ∂νφ + λRφ2)] and Ā[ḡ, φ] = (Latin small letter esh) d4x√ - ḡ [(R/2k) + (1/2) ḡμν∂μφ∂ νφ] 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 λ == -1/6 (conformal coupling).