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

Armesto, Néstor (0000000109410645, Instituto Galego de Física de Altas Enerxías IGFAE, Universidade de Santiago de Compostela, 15706, Santiago de Compostela, Galicia, Spain) ; Domínguez, Fabio (0000000109410645, Instituto Galego de Física de Altas Enerxías IGFAE, Universidade de Santiago de Compostela, 15706, Santiago de Compostela, Galicia, Spain) ; Kovner, Alex (0000 0001 0860 4915, Physics Department, University of Connecticut, 2152 Hillside Road, Storrs, CT, 06269-3046, U.S.A.) ; Lublinsky, Michael (0000 0004 1937 0511, Physics Department, Ben-Gurion University of the Negev, Beer Sheva, 84105, Israel) ; Skokov, Vladimir (0000 0001 2173 6074, Department of Physics, North Carolina State University, Raleigh, NC, 27695, U.S.A.) (0000 0001 2188 4229, RIKEN/BNL Research Center, Brookhaven National Laboratory, Upton, NY, 11973, U.S.A.)

07 May 2019

Abstract: We introduce the notion of the Color Glass Condensate (CGC) density matrix ρ ^ $\widehat{\rho}$ . 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 ρ 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 dy S e = γ $\frac{d}{dy}{S}_e=\gamma$ , where γ is the leading BFKL eigenvalue. We also discuss the evolution of ρ ^ $\widehat{\rho}$ 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.

Published in: JHEP 1905 (2019) 025