In our previous work (arxiv:2106.05301), we computed photon and graviton scattering amplitudes with internal massive spin-J propagation. However, the amplitudes obtained were complex. In particular, expressions of four graviton scattering amplitudes take up more than an entire page. So our capacity to analyze them is limited. Since we know that supersymmetry can simplify physics and in our undying hope that supersymmetry is right around the corner in LHC, we wanted to see the constraints of supersymmetry on higher spin amplitudes in four spacetime dimensions.   

We utilized the formalism of On shell supersymmetry to compute supersymmetry constraints for various levels of supersymmetry (N =1,2 and 4) on scattering amplitudes of photons and gluons, and we integrated these constraints with results from our previous work. We could show that the basis for four-photon and four gluon amplitudes (Parity preserving sector) reduces from 7 to 2 and 1 in N=2 and N=4 supersymmetric theories, respectively, and we fixed the form factors of these amplitudes. In the work of (Alday and Maldacena), it was shown using AdS-CFT that the tensor factor of four gluon/photon amplitudes for maximal supersymmetry N=4 is unique. We showed the consistency of this result in higher spin theory, and we also showed the uniqueness of the form factor, which is the Legendre polynomials.

It is a well-known fact that Einstein's gravity is the unique effective theory of Gravitation valid at large distances (small energy scales) relative to Planck mass/length. Computing quantum observables using Einstein gravity will not be valid near and beyond Planck mass/length, and we observe a breakdown of Unitarity (Diverging amplitudes) at these energies. In addition, as are all effective field theories, Einstein's gravity is not a renormalizable theory. It is a firm belief in Physics that a Renormalizable field theory should describe the fundamental forces of nature. In addition, the Hawking-Penrose singularity theorem shows that any generic configuration in Einstein's gravity will evolve into a singularity. 

One of the popular approaches to constructing a model of Quantum gravity (Unitary at all Length scales, Renormalizable) is to deform Einstein's theory with higher derivative corrections. A seminal work (https://doi.org/10.1007/JHEP02(2016)020) showed that Higher derivative corrections violate causality unless supplemented by infinitely many higher spin particles interacting with gravity. This model strongly resembles string theory which has an infinite spectrum organized as Regge trajectories. Therefore, it is interesting to see if we consider a theory of infinitely many massive higher spin particles interacting with gravity. Do the principles of quantum field theory, namely Unitarity and Causality, constrain this theory non-trivially?

The four-point amplitudes of graviton with a massive spin-J internal leg with arbitrary positive integer J were needed to pursue this question. Only the four scalar amplitude with an arbitrary spin-J internal leg was known. This is because the expression of propagator for an arbitrary spin-J, which was known since the 1950s, the particle is complex with explicit symmetrization required. In addition to this, the three-point functions describing the interaction of two gravitons (even two photons) with an arbitrary spin-J particle are more complex than the scalar particles case. So closed-form expression of said amplitudes is difficult to obtain through the standard method of glueing two three-point functions through a propagator.

In our work, we found a way of obtaining amplitudes with photon and graviton external legs from four scalar amplitudes by a sequential action of operators. The consistency of the amplitudes was verified using polynomial identities and Angularity distributions in four spacetime dimensions.