An Area Law for Entanglement Entropy in Particle Scattering

TL;DR

This paper links entanglement entropy in particle scattering to the elastic cross section, showing it scales with the collision energy and obeys an upper bound, thus proposing a 'second law' of entanglement in high energy physics.

Contribution

It introduces a novel relation between entanglement entropy and the elastic cross section in 2-to-2 scattering using the S-matrix formalism.

Findings

Entanglement entropy is proportional to the elastic cross section divided by the wave packet size.

The entropy growth with energy is bounded by the Froissart limit.

The entropy can be interpreted as an area and a probability measure.

Abstract

The scattering cross section is the effective area of collision when two particles collide. Quantum mechanically, it is a measure of the probability for a specific process to take place. Employing wave packets to describe the scattering process, we compute the entanglement entropy in 2-to-2 scattering of particles in a general setting using the $S$-matrix formalism. Applying the optical theorem, we show that the linear entropy $\mathcal{E}_2$ is given by the elastic cross section $\sigma_{\text{el}}$ in unit of the transverse size $L^2$ of the wave packet, $\mathcal{E}_2 \sim \sigma_{\text{el}}/L^2$, when the initial states are not entangled. The result allows for dual interpretations of the entanglement entropy as an area and as a probability. Since $\sigma_{\text{el}}$ is generally believed, and observed experimentally, to grow with the collision energy $\sqrt{s}$ in the high energy regime, the result suggests a "second law" of entanglement entropy for high energy collisions. Furthermore, the Froissart bound places an upper limit on the entropy growth.