Self-dual gravity is a truncation of the full theory of General Relativity, where one-half of the Weyl curvature is set to zero. This theory has many interesting properties. One of them is that of integrability. Self-dual gravity is classically integrable and this is visible transparently in its twistor description. It is also perturbatively finite and one-loop exact. At tree-level, all amplitudes are vanishing to begin with, which is reminiscent of the integrable nature of this theory. However, at one-loop integrability seems to be broken and the amplitudes are non-zero. Mimicking self-dual Yang-Mills, it is the conservation of the Berends-Giele currents in this theory which leads to all the above features at tree-level, while it has been conjectured that at one-loop, this conservation law is plagued by an anomaly. Let us explain all this in further details below.
It is convenient to pass from full gravity to self-dual gravity (SDGR) in flat space using the chiral Einstein-Cartan action as has been described previously by Krasnov et al. The same helicity amplitudes in gravity are correctly captured by the simpler SDGR Feynman rules. Thus, the flat space covariant formulation of SDGR is most relevant for amplitude computations. The same helicity tree amplitudes in SDGR vanish, analogous to SDYM. The vanishing is related to the fact that to get such amplitudes, one has to remove the final leg propagator of the currents. Thus one needs to multiply the current by a factor of k^2 and take the on-shell limit k^2 → 0. However, there is no such pole to cancel this propagator and hence the amplitudes vanish. The structure of the BG currents is however more complicated in this case. In particular, to construct the current one has to take all possible permutations of the insertion of legs to the cubic vertex, instead of just cyclic permutations. However, it is possible to write down a general form of the current using the Berends-Giele recursion. Self-dual gravity is finite, despite possessing a negative dimension coupling constant. The reason being that all possible one-loop divergences are proportional to topological invariants of the underlying manifold and thus does not contribute to the S-matrix. Higher loop diagrams do not exist as will be evident from the action and therefore the theory is one-loop finite. The one-loop amplitudes are however non-trivial. They are interesting on their own right because of the simplicity in the structure of them. In particular, all such one-loop amplitudes are rational functions of the momenta involved and are cut-free. The only singularities are those of 2-particle poles. The structure of these amplitudes is very similar to the Yang-Mills case and a possible anomaly interpretation is conjectured. It remains to be understood what kind of anomaly may give rise to the non-vanishing of such amplitudes. Further, these amplitudes have recently been linked to the 2-loop divergence in quantum gravity. It thus becomes much more interesting to investigate the reason behind the finiteness of such amplitudes. A general ansatz of these amplitudes was given by Bern and collaborators using the soft and collinear limit arguments. It is possible to compute them using supersymmetry, by replacing a graviton propagating in the loop by a scalar. However, there are no direct computations of these amplitudes from self-dual gravity Feynman rules till date.