Naïve quantum gravity is one-loop renormalisable. There is a very simple way to argue that this is true: one lists all possible counter-terms that may appear at one loop, and shows that they are, up to boundary terms, identical to terms already appearing in the original Lagrangian. This requires a non-trivial cancellation that results from the fact that a certain combination of terms happens to be topological (the Euler-Poincaré characteristic, cf. this PSE post).

Question: Can a similar analysis show that ($\mathcal N=4,8$) SUGRA is two-loop renormalisable?

As far as I know, the two-loops (and three- and four-loops) renormalisability of supersymmetric quantum gravity has been established by computing certain tree-level graphs and using the optical theorem or similar techniques. I guess that enumerating all possible counter-terms to three and four-loops is very cumbersome, but to two-loops it seems feasible. I don't know whether this has been attempted and the analysis was inconclusive (not enough symmetries to rule out all possibilities), or whether the computation is just so cumbersome it is not worth it. It seems to be a very direct approach, so it would be nice if it could be done.


This does seem to be the way that 2-loop finiteness of $\mathcal{N}=1$ SUGRA was discovered. A discussion of the construction of possible counterterms is given in e.g. this reference: [arXiv1506.03757]. They show there simply isn't a supersymmetric counterterm at 2-loop order, and hence find that there will be no divergence in $\mathcal{N}=1$ SUGRA (and therefore, this means no counterterm will exist for $\mathcal{N}=4$ or $\mathcal{N}=8$ SUGRA as well, since these theories of course have $\mathcal{N}=1$ supersymmetry).

For higher loop order in the more supersymmetric theories, they appear to have better UV behavior beyond what you would expect even from counterterm arguments. There is some explanation in this reference: [arXiv:1703.08927]. There is also a conjecture that $\mathcal{N}=8$ SUGRA may be perturbatively finite at all loop orders, which, if true, does not seem to be able to follow from some symmetry principle. So in particular, the absence of an appropriately symmetric counterterm will fail at some loop order to explain the good UV behavior of SUGRA scattering amplitudes.


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