Threshold and infrared singularities: time evolution, asymptotic state and entanglement entropy

TL;DR

This paper investigates how threshold and infrared singularities influence particle production, decay, and entanglement entropy in quantum field theory, revealing their similarities to decay processes and their role as efficient production mechanisms.

Contribution

It introduces a non-perturbative dynamical resummation method to analyze time evolution and asymptotic states in models with divergences, highlighting their qualitative similarities to decay.

Findings

Decay probability decreases exponentially with time.

Threshold and infrared cases exhibit different decay laws: square root and power-law.

Asymptotic states are entangled, with distribution functions satisfying unitarity.

Abstract

Threshold and infrared divergences are studied as possible mechanisms of particle production and compared to the usual decay process in a model quantum field theory from which generalizations are obtained. A spectral representation of the propagator of the decaying particle suggests that decay, threshold and infrared singularities while seemingly different phenomena are qualitatively related. We implement a non-perturbative dynamical resummation method to study the time evolution of an initial state. It is manifestly unitary and yields the asymptotic state and the distribution function of produced particles. Whereas the survival probability in a decay process falls off as $e^{-\Gamma t}$, for threshold and infrared divergent cases falls off instead as $e^{-\sqrt{t/t^*}}$ and $t^{-\Delta}$ respectively, with $\Gamma, \Delta \propto (coupling)^2$ whereas $1/t^* \propto (coupling)^4$. Despite the different decay dynamics, the asymptotic state is qualitatively similar: a kinematically entangled state of the daughter particles with a distribution function which fulfills the unitarity condition and is strongly peaked at energy conserving transitions but broadened by the "lifetime" $1/\Gamma~;~ t^*$ for usual decay and threshold singularity, whereas it scales with the anomalous dimension $\Delta$ for the infrared singular case. Threshold and infrared instabilities are production mechanisms just as efficient as particle decay. If one of the particles is in a dark sector and not observed, the loss of information yields an entanglement entropy determined by the distribution functions and increases upon unitary time evolution.