Relative entropy in scattering and the S-matrix bootstrap

TL;DR

This work introduces quantum information measures, especially quantum relative entropy and Rényi divergences, into 2→2 scattering to quantify and constrain entanglement and distinguish between competing S-matrix amplitudes. By deriving near-threshold and high-energy expressions and applying them to φ^4 theory, χPT, and type II string dilaton scattering, the authors connect detector-based density matrices to physical observables like scattering lengths and Adler zeros. They then couple these insights to numerical S-matrix bootstrap, identifying a new 'River' of allowed amplitudes that align with χPT signs while remaining close in observables to known theories, and show how hypothesis testing using relative entropy can differentiate boundary S-matrices. The study also expands the toolkit with isospin entanglement measures and preliminary positivity in the extended Mandelstam region, suggesting a fruitful fusion of quantum information and bootstrap techniques for constraining scattering amplitudes. Overall, the paper provides both analytic structure and numerical strategies to bound and distinguish S-matrix behavior using entropic criteria, with potential experimental and formal-theory implications.

Abstract

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 $φ^4$ theory, chiral perturbation theory ($χPT$) 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 $χPT$, 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 $ρ$ 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 $χPT$ Adler zeros. We perform a preliminary analysis to constrain the allowed space further, using ideas involving positivity inside the extended Mandelstam region, and elastic unitarity.

Figures (21)

• Figure 1: The "River". We get the river by imposing only the $\rho$ resonance and the inequalities mentioned in the main text. The "Lake" and "Peninsula" in Guerrieri:2018uew are indicated. The green regions are closest, in the sense of hypothesis testing, to the 1-loop $\chi PT$ indicated by the red cross and turn out to have comparable scattering lengths. The white region is excluded. The 1-loop point is close to the "kink" in the boundary.
• Figure 2: 2-2 scattering configuration in the centre of mass frame, with a Gaussian detector being placed at a point along the ring.
• Figure 3: Two different configurations of Gaussian Detectors for 2 to 2 scattering.
• Figure 4: Behaviour of $D_Q$ corresponding to the $\phi^4$ amplitude in eq.\ref{['phi4 amplitude']} and its comparison to the high energy bounds found in eq.\ref{['phi4 S_rel monotonic high energy']}, given by the black dashed line.
• Figure 5: Behaviour of $D_Q$ for the $\chi PT$ (similar to fig.(\ref{['fig:phi4 D_Q plots comparison']}) for the $\phi^4$ amplitude) amplitude as a function of $x_1$ for different values of $s$ (with Data-fitted values of the parameters taken from Wang:2020jxr and Colangelo:2001df). The analytic bound is the one found in eq.\ref{['chiPT high energy Dq']} which is different than the one found in Section (\ref{['high energy bounds for Dq']}).