Consider the action of a massive scalar field minimally coupled to gravity, that is,(adsbygoogle = window.adsbygoogle || []).push({});

$$S = \int d^4x \, \sqrt{-g} \, \left( 2\kappa^{-1} R + \partial_\mu \phi \partial^\mu \phi - m^2 \phi^2\right)$$

The theory I consider is canonically quantised gravity, with ##g_{\mu\nu} = \eta_{\mu\nu} + \kappa h_{\mu\nu}## and with the corresponding Feynman rules I evaluated the one loop contribution to graviton to graviton scattering with a single scalar loop. I found the amplitude was of the form,

$$\mathcal M \sim -\frac{\kappa^2}{32\pi^2\epsilon} \left[ \eta_{\mu\nu}\eta_{\lambda\sigma} f_1(p) + (\eta_{\mu\lambda}\eta_{\nu\sigma} + \eta_{\mu\sigma}\eta_{\nu\lambda})f_2(p)\right] + \mathrm{finite \, \, (\epsilon \to 0)}$$

in dimensional regularisation, where the functions ##f_1(p)## and ##f_2(p)## are polynomials in the external momentum ##p## and scalar mass ##m##. To eliminate the divergence I would like to add the required counter-terms to the action, as one does in effective field theory. However, I am not sure how to deduce which terms to add.

Those allowed by symmetries would include terms like ##R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}## but I'm a little iffy on how I can relate these to the amplitude I've computed to absorb the divergence. In addition, I am unsure as to where or not we would need terms like ##\phi R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}## involving the scalar. How can we reason what diagrams would require counter-terms coupling the scalar to curvature scalars?

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# A Deducing counter-terms for canonically quantised GR

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