Deducing counter-terms for canonically quantised GR

[GogoJS]

Consider the action of a massive scalar field minimally coupled to gravity, that is,

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

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?

 

[nrqed]

Consider the action of a massive scalar field minimally coupled to gravity, that is,

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

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?

You are looking for an interaction which, at tree level, will give an amplitude be proportional to the divergent one you wrote above. Since your contribution comes from a diagram with the scalar in the loop and only gravitons in the external legs you should find that you will need a counter term that contains only the Riemann tensor. Of course at some point you will need terms like $\phi R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}$ but these will arise to cure divergences of loop diagrams with a scalar in an external leg.

 

[GogoJS]

You are looking for an interaction which, at tree level, will give an amplitude be proportional to the divergent one you wrote above. Since your contribution comes from a diagram with the scalar in the loop and only gravitons in the external legs you should find that you will need a counter term that contains only the Riemann tensor. Of course at some point you will need terms like $\phi R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}$ but these will arise to cure divergences of loop diagrams with a scalar in an external leg.

Thanks, I see you what you mean. I will expand $R_{\mu\nu\lambda\sigma}R^{\mu\nu\lambda\sigma}$ and other curvature scalars in powers of $h_{\mu\nu}$, and then find the terms that are required at tree-level for the amplitude to be reproduced.