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

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## Abstract

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

## Keywords

Heavy Ion Phenomenology## Notes

### **Open Access**

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## References

- [1]CMS collaboration,
*Charged particle multiplicities in pp interactions at*\( \sqrt{s}=0.9,\;2.36 \)*and*7*TeV*,*JHEP***01**(2011) 079 [arXiv:1011.5531] [INSPIRE]. - [2]CMS collaboration,
*Multiplicity and Transverse Momentum Dependence of Two- and Four-Particle Correlations in pPb and PbPb Collisions*,*Phys. Lett.***B 724**(2013) 213 [arXiv:1305.0609] [INSPIRE]. - [3]ALICE collaboration,
*Insight into particle production mechanisms via angular correlations of identified particles in pp collisions at*\( \sqrt{\mathrm{s}} = 7 \)*TeV*,*Eur. Phys. J.***C 77**(2017) 569 [arXiv:1612.08975] [INSPIRE]. - [4]ATLAS collaboration,
*Measurement of forward-backward multiplicity correlations in lead-lead, proton-lead and proton-proton collisions with the ATLAS detector*,*Phys. Rev.***C 95**(2017) 064914 [arXiv:1606.08170] [INSPIRE]. - [5]ATLAS collaboration,
*Correlated long-range mixed-harmonic fluctuations measured in pp, p+Pb and low-multiplicity Pb+Pb collisions with the ATLAS detector*,*Phys. Lett.***B 789**(2019) 444 [arXiv:1807.02012] [INSPIRE]. - [6]E. Iancu, A. Leonidov and L. McLerran,
*The color glass condensate: An introduction*, in*QCD perspectives on hot and dense matter. Proceedings, NATO Advanced Study Institute, Summer School, Cargese, France, August 6-18, 2001*, pp. 73-145, 2002, hep-ph/0202270 [INSPIRE]. - [7]
- [8]F. Gelis, E. Iancu, J. Jalilian-Marian and R. Venugopalan,
*The Color Glass Condensate*,*Ann. Rev. Nucl. Part. Sci.***60**(2010) 463 [arXiv:1002.0333] [INSPIRE]. - [9]Y.V. Kovchegov and E. Levin,
*Quantum chromodynamics at high energy*, vol. 33, Cambridge University Press, (2012).Google Scholar - [10]A. Kovner and M. Lublinsky,
*One gluon, two gluon: Multigluon production via high energy evolution*,*JHEP***11**(2006) 083 [hep-ph/0609227] [INSPIRE]. - [11]E. Iancu and D.N. Triantafyllopoulos,
*JIMWLK evolution for multi-particle production in Langevin form*,*JHEP***11**(2013) 067 [arXiv:1307.1559] [INSPIRE].ADSCrossRefGoogle Scholar - [12]A. Kovner and M. Lublinsky,
*Entanglement entropy and entropy production in the Color Glass Condensate framework*,*Phys. Rev.***D 92**(2015) 034016 [arXiv:1506.05394] [INSPIRE]. - [13]A. Kovner, M. Lublinsky and M. Serino,
*Entanglement entropy, entropy production and time evolution in high energy QCD*,*Phys. Lett.***B 792**(2019) 4 [arXiv:1806.01089] [INSPIRE]. - [14]H.-T. Elze,
*Entropy, quantum decoherence and pointer states in scalar*‘*parton*’*fields*,*Phys. Lett.***B 369**(1996) 295 [hep-th/9406085] [INSPIRE]. - [15]K. Kutak,
*Gluon saturation and entropy production in proton-proton collisions*,*Phys. Lett.***B 705**(2011) 217 [arXiv:1103.3654] [INSPIRE]. - [16]R. Peschanski,
*Dynamical entropy of dense QCD states*,*Phys. Rev.***D 87**(2013) 034042 [arXiv:1211.6911] [INSPIRE]. - [17]R. Peschanski and S. Seki,
*Entanglement Entropy of Scattering Particles*,*Phys. Lett.***B 758**(2016) 89 [arXiv:1602.00720] [INSPIRE]. - [18]J. Berges, S. Floerchinger and R. Venugopalan,
*Thermal excitation spectrum from entanglement in an expanding quantum string*,*Phys. Lett.***B 778**(2018) 442 [arXiv:1707.05338] [INSPIRE]. - [19]J. Berges, S. Floerchinger and R. Venugopalan,
*Dynamics of entanglement in expanding quantum fields*,*JHEP***04**(2018) 145 [arXiv:1712.09362] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [20]Y. Hagiwara, Y. Hatta, B.-W. Xiao and F. Yuan,
*Classical and quantum entropy of parton distributions*,*Phys. Rev.***D 97**(2018) 094029 [arXiv:1801.00087] [INSPIRE]. - [21]
- [22]Y. Liu and I. Zahed,
*Entanglement in Reggeized Scattering using AdS/CFT*, arXiv:1803.09157 [INSPIRE]. - [23]X. Feal, C. Pajares and R.A. Vazquez,
*Thermal behavior and entanglement in Pb-Pb and p-p collisions*,*Phys. Rev.***C 99**(2019) 015205 [arXiv:1805.12444] [INSPIRE]. - [24]M. Hentschinski, H. Weigert and A. Schafer,
*Extension of the color glass condensate approach to diffractive reactions*,*Phys. Rev.***D 73**(2006) 051501 [hep-ph/0509272] [INSPIRE]. - [25]A. Kovner, M. Lublinsky and H. Weigert,
*Treading on the cut: Semi inclusive observables at high energy*,*Phys. Rev.***D 74**(2006) 114023 [hep-ph/0608258] [INSPIRE]. - [26]L.D. McLerran and R. Venugopalan,
*Computing quark and gluon distribution functions for very large nuclei*,*Phys. Rev.***D 49**(1994) 2233 [hep-ph/9309289] [INSPIRE]. - [27]L.D. McLerran and R. Venugopalan,
*Green*’*s functions in the color field of a large nucleus*,*Phys. Rev.***D 50**(1994) 2225 [hep-ph/9402335] [INSPIRE]. - [28]J. Jalilian-Marian, A. Kovner and H. Weigert,
*The Wilson renormalization group for low x physics: Gluon evolution at finite parton density*,*Phys. Rev.***D 59**(1998) 014015 [hep-ph/9709432] [INSPIRE]. - [29]A. Kovner, J.G. Milhano and H. Weigert,
*Relating different approaches to nonlinear QCD evolution at finite gluon density*,*Phys. Rev.***D 62**(2000) 114005 [hep-ph/0004014] [INSPIRE]. - [30]A. Kovner and J.G. Milhano,
*Vector potential versus color charge density in low x evolution*,*Phys. Rev.***D 61**(2000) 014012 [hep-ph/9904420] [INSPIRE]. - [31]
- [32]E. Iancu, A. Leonidov and L.D. McLerran,
*Nonlinear gluon evolution in the color glass condensate. 1.*,*Nucl. Phys.***A 692**(2001) 583 [hep-ph/0011241] [INSPIRE]. - [33]E. Iancu, A. Leonidov and L.D. McLerran,
*The renormalization group equation for the color glass condensate*,*Phys. Lett.***B 510**(2001) 133 [hep-ph/0102009] [INSPIRE]. - [34]E. Ferreiro, E. Iancu, A. Leonidov and L. McLerran,
*Nonlinear gluon evolution in the color glass condensate. 2.*,*Nucl. Phys.***A 703**(2002) 489 [hep-ph/0109115] [INSPIRE]. - [35]A. Kovner and M. Lublinsky,
*In pursuit of Pomeron loops: The JIMWLK equation and the Wess-Zumino term*,*Phys. Rev.***D 71**(2005) 085004 [hep-ph/0501198] [INSPIRE]. - [36]A. Kovner and M. Lublinsky,
*Remarks on high energy evolution*,*JHEP***03**(2005) 001 [hep-ph/0502071] [INSPIRE]. - [37]A. Kossakowski,
*On quantum statistical mechanics of non-hamiltonian systems*,*Rept. Math. Phys.***3**(1972) 247.ADSMathSciNetCrossRefGoogle Scholar - [38]G. Lindblad,
*On the Generators of Quantum Dynamical Semigroups*,*Commun. Math. Phys.***48**(1976) 119 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [39]T. Banks, L. Susskind and M.E. Peskin,
*Difficulties for the Evolution of Pure States Into Mixed States*,*Nucl. Phys.***B 244**(1984) 125 [INSPIRE]. - [40]E. Iancu, K. Itakura and L. McLerran,
*A Gaussian effective theory for gluon saturation*,*Nucl. Phys.***A 724**(2003) 181 [hep-ph/0212123] [INSPIRE]. - [41]E. Iancu and D.N. Triantafyllopoulos,
*JIMWLK evolution in the Gaussian approximation*,*JHEP***04**(2012) 025 [arXiv:1112.1104] [INSPIRE].ADSCrossRefGoogle Scholar - [42]A. Dumitru, J. Jalilian-Marian, T. Lappi, B. Schenke and R. Venugopalan,
*Renormalization group evolution of multi-gluon correlators in high energy QCD*,*Phys. Lett.***B 706**(2011) 219 [arXiv:1108.4764] [INSPIRE]. - [43]E.A. Kuraev, L.N. Lipatov and V.S. Fadin,
*Multi-Reggeon Processes in the Yang-Mills Theory*,*Sov. Phys. JETP***44**(1976) 443 [INSPIRE].ADSGoogle Scholar - [44]V.S. Fadin, E.A. Kuraev and L.N. Lipatov,
*On the Pomeranchuk Singularity in Asymptotically Free Theories*,*Phys. Lett.***60B**(1975) 50 [INSPIRE].ADSCrossRefGoogle Scholar - [45]T. Altinoluk, C. Contreras, A. Kovner, E. Levin, M. Lublinsky and A. Shulkin,
*QCD Reggeon Calculus From KLWMIJ/JIMWLK Evolution: Vertices, Reggeization and All*,*JHEP***09**(2013) 115 [arXiv:1306.2794] [INSPIRE].ADSCrossRefGoogle Scholar - [46]I. Balitsky,
*Operator expansion for high-energy scattering*,*Nucl. Phys.***B 463**(1996) 99 [hep-ph/9509348] [INSPIRE]. - [47]Y.V. Kovchegov,
*Unitarization of the BFKL Pomeron on a nucleus*,*Phys. Rev.***D 61**(2000) 074018 [hep-ph/9905214] [INSPIRE]. - [48]A. Kovner and M. Lublinsky,
*From target to projectile and back again: Selfduality of high energy evolution*,*Phys. Rev. Lett.***94**(2005) 181603 [hep-ph/0502119] [INSPIRE]. - [49]A. Kovner and A.H. Rezaeian,
*Double parton scattering in the CGC: Double quark production and effects of quantum statistics*,*Phys. Rev.***D 96**(2017) 074018 [arXiv:1707.06985] [INSPIRE]. - [50]T. Altinoluk, N. Armesto, A. Kovner and M. Lublinsky,
*Double and triple inclusive gluon production at mid rapidity: quantum interference in p-A scattering*,*Eur. Phys. J.***C 78**(2018) 702 [arXiv:1805.07739] [INSPIRE]. - [51]A. Kovner and V.V. Skokov,
*Bose enhancement, the Liouville effective action and the high multiplicity tail in p-A collisions*,*Phys. Rev.***D 98**(2018) 014004 [arXiv:1805.09296] [INSPIRE]. - [52]E. Levin and K. Tuchin,
*Solution to the evolution equation for high parton density QCD*,*Nucl. Phys.***B 573**(2000) 833 [hep-ph/9908317] [INSPIRE]. - [53]S. Mrowczynski and B. Müller,
*Wigner functional approach to quantum field dynamics*,*Phys. Rev.***D 50**(1994) 7542 [hep-th/9405036] [INSPIRE]. - [54]S. Mrowczynski,
*Wigner functional of fermionic fields*,*Phys. Rev.***D 87**(2013) 065026 [arXiv:1212.5703] [INSPIRE]. - [55]I. Bialynicki-Birula,
*Quantum fluctuations of geometry in a hot Universe*,*Class. Quant. Grav.***32**(2015) 215015 [arXiv:1501.07405] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [56]A. Kovner, M. Lublinsky and Y. Mulian,
*Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner evolution at next to leading order*,*Phys. Rev.***D 89**(2014) 061704 [arXiv:1310.0378] [INSPIRE]. - [57]I. Balitsky and G.A. Chirilli,
*Rapidity evolution of Wilson lines at the next-to-leading order*,*Phys. Rev.***D 88**(2013) 111501 [arXiv:1309.7644] [INSPIRE]. - [58]A. Kovner, M. Lublinsky and Y. Mulian,
*NLO JIMWLK evolution unabridged*,*JHEP***08**(2014) 114 [arXiv:1405.0418] [INSPIRE].ADSCrossRefGoogle Scholar - [59]M. Lublinsky and Y. Mulian,
*High Energy QCD at NLO: from light-cone wave function to JIMWLK evolution*,*JHEP***05**(2017) 097 [arXiv:1610.03453] [INSPIRE].ADSCrossRefGoogle Scholar - [60]D.E. Kharzeev and E.M. Levin,
*Deep inelastic scattering as a probe of entanglement*,*Phys. Rev.***D 95**(2017) 114008 [arXiv:1702.03489] [INSPIRE].