Abstract
We consider scattering of Faddeev-Kulish electrons in QED and study the entanglement between the hard and soft particles in the final state at the perturbative level. The soft photon spectrum naturally splits into two parts: i) soft photons with energies less than a characteristic infrared scale $E_d$ present in the clouds accompanying the asymptotic charged particles, and ii) sufficiently low energy photons with energies greater than $E_d$, comprising the soft part of the emitted radiation. We construct the density matrix associated with tracing over the radiative soft photons and calculate the entanglement entropy perturbatively. We find that the entanglement entropy is free of any infrared divergences order by order in perturbation theory. On the other hand infrared divergences in the perturbative expansion for the entanglement entropy appear upon tracing over the entire spectrum of soft photons, including those in the clouds. To leading order the entanglement entropy is set by the square of the Fock basis amplitude for real single soft photon emission, which leads to a logarithmic infrared divergence when integrated over the photon momentum. We argue that the infrared divergences in the entanglement entropy (per particle flux per unit time) in this latter case persist to all orders in perturbation theory in the infinite volume limit.
Highlights
Symmetry renders purely hard scattering processes in QED and gravity impossible [1,2,3,4,5,6,7,8,9]
To leading order the entanglement entropy is set by the square of the Fock basis amplitude for real single soft photon emission, which leads to a logarithmic infrared divergence when integrated over the photon momentum
We argue that the infrared divergences in the entanglement entropy in this latter case persist to all orders in perturbation theory in the infinite volume limit
Summary
Symmetry renders purely hard scattering processes in QED and gravity impossible [1,2,3,4,5,6,7,8,9]. We would like to investigate if infrared divergences in the entanglement entropy appear, and whether they cancel order by order in perturbation theory We discuss both a Fock basis computation where we take a state of two bare electrons for the initial state and a proper asymptotic state where the electrons are “dressed” with infinite clouds of soft photons, in accordance with the Faddeev-Kulish construction [17,18]. In both cases and to leading order in perturbation theory, the entanglement entropy is proportional to the conventional Fock basis rate for the two initial electrons to scatter and emit at the same time a single soft photon with frequency in the range λ < ωγ < E This rate diverges logarithmically in the continuum, λ → 0 limit at tree level. For the case of Faddeev-Kulish electrons, the divergence can be traced in the overlap of the coherent states describing the soft photon clouds dressing the final state charged particles. To all orders in the electron charge, the overlap hfβjfαi vanishes in the λ → 0 limit
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