This thesis presents results pertaining to scattering amplitudes in Conformal Higher Spin (CHS) theory, most of which was published in \cite{Joung:2015eny, Beccaria:2016syk, Adamo:2018srx}.
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CHS theory contains Maxwell theory, Conformal Gravity and generalises them for higher spin. After briefly introducing the general field of Higher Spins, we therefore discuss Conformal Gravity as a warm up.
Since it is a 4-derivative theory, it contains more on-shell states than just the usual 2-derivative Einstein Gravitons.
Some of these states are found to be admissible for scattering and lead to finite expressions for amplitudes. We compute three point tree-level amplitudes scattering all possible states. We give a formula which captures these amplitudes using twistor spinors.
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We then define CHS theory and its symmetries. We descirbe how it is obtained as the logarithmically divergent part of the partition function for a free scalar coupled to general spin background sources. We characterise its scattering states and proceed to present a series of amplitude computations.
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We first compute four-point amplitudes for an external scalar interacting with the full tower of CHS fields. These amplitudes need a natural prescription for summing over that infinite tower of fields. Doing so in a way that is compatible with CHS symmetry leads to vanishing amplitudes.
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We then present similar amplitudes in pure CHS theory where the external legs are 2-derivative spin $1$ and 2 CHS modes. Once again, these amplitudes are trivial. As the theory is conformal, it has a natural description in the language of twistor-spinors and we give a formula for three-point tree level amplitudes of \emph{all} states, including those which are not associated with 2 derivative equations of motion.
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Finally we look at the theory in curved spacetime, where its quadratic sector is non-diagonal. We compute some of these terms and their contributions to the conformal anomaly $c$-coefficient.