We study the space of all kinematically allowed four photon and four graviton S-matrices, polynomial in scattering momenta. We demonstrate that this space is the permutation invariant sector of a module over the ring of polynomials of the Mandelstam invariants s, t and u. We construct these modules for every value of the spacetime dimension D, and so explicitly count and parameterize the most general four photon and four graviton S-matrix at any given derivative order. We also explicitly list the local Lagrangians that give rise to these S-matrices. We then conjecture that the Regge growth of S-matrices in all physically acceptable classical theories is bounded by s 2 at fixed t. A four parameter subset of the polynomial photon S-matrices constructed above satisfies this Regge criterion. For gravitons, on the other hand, no polynomial addition to the Einstein S-matrix obeys this bound for D ≤ 6. For D ≥ 7 there is a single six derivative polynomial Lagrangian consistent with our conjectured Regge growth bound. Our conjecture thus implies that the Einstein four graviton S-matrix does not admit any physically acceptable polynomial modifications for D ≤ 6. A preliminary analysis also suggests that every finite sum of pole exchange contributions to four graviton scattering also violates our conjectured Regge growth bound, at least when D ≤ 6, even when the exchanged particles have low spin.

We derive constraints on three-point functions involving the stress tensor, T , and a conserved U(1) current, j, in 2+1 dimensional conformal field theories that violate parity, using conformal collider bounds introduced by Hofman and Maldacena. Conformal invariance allows parity-odd tensor-structures for the T T T and jjT correlation functions which are unique to three space-time dimensions. Let the parameters which determine the T T T correlation function be t 4 and α T , where α T is the parity-violating contribution. Similarly let the parameters which determine jjT correlation function be a 2 , and α J , where α J is the parity-violating contribution. We show that the parameters (t 4 , α T ) and (a 2 , α J ) are bounded to lie inside a disc at the origin of the t 4 -α T plane and the a 2 -α J plane respectively. We then show that large N Chern-Simons theories coupled to a fundamental fermion/boson lie on the circle which bounds these discs. The 't Hooft coupling determines the location of these theories on the boundary circles.

We study the Regge trajectories of the Mellin amplitudes of the 0−, 1− and 2− magnon correlators of the Fishnet theory. Since fishnet theory is both integrable and conformal, the correlation functions are known exactly. We find that while for 0 and 1 magnon correlators, the Regge poles can be exactly determined as a function of coupling, 2-magnon correlators can only be dealt with perturbatively. We evaluate the resulting Mellin amplitudes at weak coupling, while for strong coupling we do an order of magnitude calculation. II Superstring theory [5], the Regge limit of the scattering amplitude scales as s 2+ α t 2 which denotes graviton dominance in the high energy limit (t is negative). Similarly for QCD, one can see from [6] that the LLA (leading log approximation) contribution to the Regge limit comes from,(1.5)The same can be shown in a perturbative manner for the N = 4 SYM [7] for which in weak coupling,In contrast, for the fishnet theories under consideration, we find that for the 0, 1, 2−magnon cases, in the weak coupling, the leading Regge theory is dominated by,respectively. This is expected to be connected with the inherent non-unitarity of the theory so that the effective exchanges in the Regge limit has negative spins. In this case, the LLA contribution is expected 1 Unlike other theories, where the Regge trajectories are only known in certain limits (say the weak coupling limit), here the trajectories are exact functions of the coupling ξ.

In this paper, we classify four-point local gluon S-matrices in arbitrary dimensions. This is along the same lines as [1] where four-point local photon S-matrices and graviton S-matrices were classified. We do the classification explicitly for gauge groups SO(N) and SU(N) for all N but our method is easily generalizable to other Lie groups. The construction involves combining not-necessarily-permutation-symmetric four-point S-matrices of photons and those of adjoint scalars into permutation symmetric four-point gluon S-matrix. We explicitly list both the components of the construction, i.e permutation symmetric as well as non-symmetric four point S-matrices, for both the photons as well as the adjoint scalars for arbitrary dimensions and for gauge groups SO(N) and SU(N) for all N. In this paper, we explicitly list the local Lagrangians that generate the local gluon S-matrices for D ≥ 9 and present the relevant counting for lower dimensions. Local Lagrangians for gluon S-matrices in lower dimensions can be written down following the same method. We also express the Yang-Mills four gluon S-matrix with gluon exchange in terms of our basis structures.

We study four-point functions of scalars, conserved currents, and stress tensors in a conformal field theory, generated by a local contact term in the bulk dual description, in two different causal configurations. The first of these is the standard Regge configuration in which the chaos bound applies. The second is the ‘causally scattering configuration’ in which the correlator develops a bulk point singularity. We find an expression for the coefficient of the bulk point singularity in terms of the bulk S matrix of the bulk dual metric, gauge fields and scalars, and use it to determine the Regge scaling of the correlator on the causally scattering sheet in terms of the Regge growth of this S matrix. We then demonstrate that the Regge scaling on this sheet is governed by the same power as in the standard Regge configuration, and so is constrained by the chaos bound, which turns out to be violated unless the bulk flat space S matrix grows no faster than s2 in the Regge limit. It follows that in the context of the AdS/CFT correspondence, the chaos bound applied to the boundary field theory implies that the S matrices of the dual bulk scalars, gauge fields, and gravitons obey the Classical Regge Growth (CRG) conjecture.