Aspects of self-dual Yang-Mills and self-dual gravity - Chattopadhyay, Pratik (2022)
In this thesis, we study the all same helicity loop amplitudes in self-dual Yang-Mills and self-dual gravity. These amplitudes have long been conjectured to be interpreted as an anomaly and are recently linked to the UV divergence of two-loop quantum gravity. In the first part of the thesis, we study the loop amplitudes in self-dual Yang-Mills. We show that the four point one-loop amplitude can be reduced to a computation of shifts, which strongly suggests a case for an anomaly interpretation. We next propose a new formula for the one-loop amplitudes at all multiplicity, in terms of the Berends-Giele currents connected by an effective propagator. We prove the formula by observing that it readily implies the correct collinear properties. To demonstrate the validity of our formula, we do an explicit computation at 3, 4 and 5 points and reproduce the known results. The region momenta variables play an important role in our formula and thus it points to both the worldsheet and the momentum twistor interpretations. In the second part of the thesis, we study the one loop behaviour of chiral Einstein-Cartan gravity and the one-loop amplitudes in self-dual gravity. We develop the ghost Lagrangian in chiral Einstein-Cartan gravity for a general Einstein background using the BRST formalism and compute the ghost contribution to the one-loop effective action. We next construct the one-loop graphs contributing to the four point same helicity amplitude. The double copy property is manifest in the diagrams. We also perform a shift computation of the self-energy bubble in gravity and show that the result is the square of Yang-Mills. The bubble is interpreted as an effective propagator, in complete analogy with Yang-Mills. However, the interpretation of the shift parameters in this case is not clear and thus the computation of the four point amplitude remains incomplete. We comment on a possible way to resolve this ambiguity.