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A Deducing counter-terms for canonically quantised GR

  1. Dec 14, 2016 #1
    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?
  2. jcsd
  3. Dec 14, 2016 #2


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    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.
  4. Dec 15, 2016 #3
    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.
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