An intriguing correspondence between ingredients in geometric function theory related to the famous Bieberbach conjecture (de Branges’ theorem) and the non-perturbative crossing symmetric representation of 2-2 scattering amplitudes of identical scalars is pointed out. Using the dispersion relation and unitarity, we are able to derive several inequalities, analogous to those which arise in the discussions of the Bieberbach conjecture. We derive new and strong bounds on the ratio of certain Wilson coefficients and demonstrate that these are obeyed in one-loop \phi^4ϕ4 theory, tree level string theory as well as in the S-matrix bootstrap. Further, we find two sided bounds on the magnitude of the scattering amplitude, which are shown to be respected in all the contexts mentioned above. Translated to the usual Mandelstam variables, for large |s||s|, fixed tt, the upper bound reads |\mathcal{M}(s,t)|\lesssim |s^2||ℳ(s,t)|≲|s2|. We discuss how Szeg"{o}’s theorem corresponds to a check of univalence in an EFT expansion, while how the Grunsky inequalities translate into nontrivial, nonlinear inequalities on the Wilson coefficients.

We consider entanglement measures in 2-2 scattering in quantum field theories, focusing on relative entropy which distinguishes two different density matrices. Relative entropy is investigated in several cases which include \phi^4ϕ4 theory, chiral perturbation theory (\chi PTχPT) describing pion scattering and dilaton scattering in type II superstring theory. We derive a high energy bound on the relative entropy using known bounds on the elastic differential cross-sections in massive QFTs. In \chi PTχPT, relative entropy close to threshold has simple expressions in terms of ratios of scattering lengths. Definite sign properties are found for the relative entropy which are over and above the usual positivity of relative entropy in certain cases. We then turn to the recent numerical investigations of the S-matrix bootstrap in the context of pion scattering. By imposing these sign constraints and the \rhoρ resonance, we find restrictions on the allowed S-matrices. By performing hypothesis testing using relative entropy, we isolate two sets of S-matrices living on the boundary which give scattering lengths comparable to experiments but one of which is far from the 1-loop \chi PTχPT Adler zeros. We perform a preliminary analysis to constrain the allowed space further, using ideas involving positivity inside the extended Mandelstam region, and other quantum information theoretic measures based on entanglement in isospin.

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.

We derive bounds analogous to the Froissart bound for the absorptive part of CFT_dd Mellin amplitudes. Invoking the AdS/CFT correspondence, these amplitudes correspond to scattering in AdS_{d+1}d+1. We can take a flat space limit of the corresponding bound. We find the standard Froissart-Martin bound, including the coefficient in front for d+1=4 being \pi/\mu^2π/μ2, \muμ being the mass of the lightest exchange. For d>4d>4, the form is different. We show that while for CFT_{d\leq 6}CFTd≤6, the number of subtractions needed to write a dispersion relation for the Mellin amplitude is equal to 2, for CFT_{d>6}CFTd>6 the number of subtractions needed is greater than 2 and goes to infinity as d goes to infinity.

We develop crossing symmetric dispersion relations for describing 2-2 scattering of identical external particles carrying spin. This enables us to import techniques from Geometric Function Theory and study two sided bounds on low energy Wilson coefficients. We consider scattering of photons, gravitons in weakly coupled effective field theories. We provide general expressions for the locality/null constraints. Consideration of the positivity of the absorptive part leads to an interesting connection with the recently conjectured weak low spin dominance. We also construct the crossing symmetric amplitudes and locality constraints for the massive neutral Majorana fermions and parity violating photon and graviton theories. The techniques developed in this paper will be useful for considering numerical S-matrix bootstrap in the future.