Abstract
We develop a formalism for describing the most general notion of tree-level scattering amplitudes in 4d conformal higher spin theory. As conformal higher spin fields obey higher-derivative equations of motion, there are many distinct on-shell external states which may contribute to their scattering, some of which grow polynomially with time, leading to ill-defined amplitudes. We characterize the set of admissible scattering states which produce finite tree amplitudes, noting that there are more such states than just standard massless higher spins obeying two-derivative equations of motion. We use conformal gravity as a prime example, where the set of scattering states includes the usual Einstein graviton and a ‘ghost’ massless spin 1 particle. An extension of the usual spinor helicity formalism allows us to encode these scattering states efficiently in terms of ‘twistor-spinors’. This leads to compact momentum space expressions for all finite tree-level 3-point amplitudes of conformal higher spin theory. While some of these 3-point amplitudes vanish (including all those with only standard two-derivative higher spin external states), there are many others which are non-vanishing. We also comment on the generalization to scattering of conformal higher spins in AdS4.
Highlights
In such a higher-derivative theory, the definition of asymptotic states and scattering amplitudes is non-trivial
We develop a formalism for describing the most general notion of tree-level scattering amplitudes in 4d conformal higher spin theory
We focus in detail on the example of conformal gravity, where in addition to the massless spin 2 Einstein graviton there is an oscillating ‘ghost’ spin 1 mode which
Summary
External states in any scattering process are given by free field solutions of the equations of motion. A free bosonic CHS gauge field in 4 dimensions is a rank s totally symmetric tensor φa1···as(x) = φa(s)(x) defined up to linearised gauge transformations δφa(s) = ∂(a1 a2···as) + η(a1a2 αa3···as) ,. Two of the d.o.f. correspond to the standard two-derivative massless higher spin fields, which sit trivially inside the space of solutions to the linearised CHS equations. These twoderivative solutions are not the only on-shell states in CHS theory which are suitable for scattering in Minkowski space. There are other on-shell states which are pure oscillatory and obey the higher-derivative equations of motion in a strict sense, satisfying the linear CHS equation of order 2s without solving some other equation which is of order 2 in derivatives
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