The Color Glass Condensate density matrix: Lindblad evolution, entanglement entropy and Wigner functional

Abstract

We introduce the notion of the Color Glass Condensate (CGC) density matrix $\hat{\rho }$. This generalizes the concept of probability density for the distribution of the color charges in the hadronic wave function and is consistent with understanding the CGC as an effective theory after integration of part of the hadronic degrees of freedom. We derive the evolution equations for the density matrix and show that the JIMWLK evolution equation arises here as the evolution of diagonal matrix elements of $\hat{\rho }$ in the color charge density basis. We analyze the behavior of this density matrix under high energy evolution and show that its purity decreases with energy. We show that the evolution equation for the density matrix has the celebrated Kossakowsky-Lindblad form describing the non-unitary evolution of the density matrix of an open system. Additionally, we consider the dilute limit and demonstrate that, at large rapidity, the entanglement entropy of the density matrix grows linearly with rapidity according to $d{S}_e/dy=\gamma$, where $\gamma$ is the leading BFKL eigenvalue. We also discuss the evolution of $\hat{\rho }$ in the saturated regime and relate it to the Levin-Tuchin law and find that the entropy again grows linearly with rapidity, but at a slower rate. By analyzing the dense and dilute regimes of the full density matrix we are able to establish a duality between the regimes. Finally we introduce the Wigner functional derived from this density matrix and discuss how it can be used to determine the distribution of color currents, which may be instrumental in understanding dynamical features of QCD at high energy.

22 pages, 1 figure

References in corpus (7)

• The Color Glass Condensate
• NLO JIMWLK evolution unabridged
• One gluon, two gluon: multigluon production via high energy evolution
• Treading on the cut: Semi inclusive observables at high energy
• Dynamical entropy of dense QCD states
• DPS in CGC: Double Quark Production and Effects of Quantum Statistics
• Wigner functional of fermionic fields

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• Quark and gluon entanglement in the proton on the light cone at intermediate $x$
• Investigating the QCD dynamical entropy in high-energy hadronic collisions
• Multiplicity distribution of dipoles in QCD from Le, Mueller and Munier equation
• Determination of the entanglement entropy in elastic scattering using model-independent method for hadron femtoscopy