Regge constraints on local four-point scattering amplitudes of massive particles with spin

In this work, we classify all the possible local four-point couplings relevant for tree-level flat space $2 \rightarrow 2$ scattering of external massive particles of spin one and spin two which do not grow faster than $s^2$ at large $s$ and fixed t. This kinematic constraint on local growth of tree-level S-matrices is known as Classical Regge Growth criteria or CRG. We first construct the spin one and spin two tree-level contact S-matrices as modules of polarisation tensors and momenta over the ring of polynomials generated by Mandelstam invariants. We then consider a general scattering process where the external scattering particles are of different masses but of same spin and constrain this space to obtain a finite number of CRG allowed local Lagrangians. Our concrete results are primarily for $D\geq 8$ but the process outlined is easily generalised to lower dimensions to include low dimensional parity violating structures. The space of CRG allowed structures reduces when we specialise to identical scattering and restrict to parity even couplings in $D=4$. We show that tree-level scattering amplitudes involving exchange diagrams and contact terms in de Rham-Gabadadze-Tolley massive gravity (dRGT) violate CRG unless the parameters of the theory take special values. The CRG allowed S-matrices, in the context of large $N$ conformal field theories (CFTs), can also be interpreted as bulk $AdS$ counterterms consistent with Chaos bound. Our classified structures therefore can be thought of as ambiguities arising in the context of conformal field theory inversion formula for four point functions of unconserved spin one and spin two operators in large $N$ CFTs.