Scattering amplitudes in Quadratic Graivty in a general formalism

Abstract

In \cite{salvio}, inspired by the works \cite{pauli}-\cite{donogue}, a prescription for calculating the correlation functions in Quadratic Gravity \cite{stelle1}-\cite{stelle2} was presented and further exploited in \cite{salvio2}-\cite{salve}. A covariant formalism ensuring positive definite probabilities is worked out, which is the main drawback of Quadratic Gravity. The Gauss-Ostrogradsky method for Quadratic Gravity defines two momentum densities $P_1$ and $P_2$ and two coordinate densities $Q_1$ and $Q_2$, one pair is standard, the other ghost like. The approach in \cite{salvio} involves the continuation $P_2\to i P_2$ and $Q_2\to i Q_2$ of the ghost variables acting on kets $|>$ after taking mean values. The notable result is that, in the euclidean setting, this procedure leads to Quadratic Gravity path integral $Z(J)$, thus renormalizability is not spoiled. In view of these findings, it is natural to ask how the LSZ rules of the model have to be formulated, and this is the topic of the present work. A formalism adapted to full quartic or higher order theories is worked out, extending the results of \cite{yomismo}. The main technical point is to determine the creation annihilation algebra for the graviton modes, adapted to the present prescription, and properly dealing with gauge symmetry. This makes the problem harder than the quantization of the Pais-Uhlenbeck model. Two possible quantization schemes are discussed, they depend on whether the above prescriptions are applied at the beginning or at the end of the LSZ calculation.