On the long standing issue of an R phi(2) coupling between gravity and a massless free scalar field
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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, φ̄] = (Latin small letter esh) d4x√ - ḡ [(R̄/2k) + (1/2) ḡμν∂μφ̄∂ 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).
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Accioly, Antonio Jose et al. On the long standing issue of an R phi(2) coupling between gravity and a massless free scalar field. Progress of Theoretical Physics, v. 97, n. 1, p. 121-128, 1997. Available at: <http://hdl.handle.net/11449/130567>.