We summarize recent significant progress in the development of a
first-principles formalism to describe the formation and evolution of matter in
very high energy heavy ion collisions. The key role of quantum fluctuations
both before and after a collision is emphasized. Systematic computations are
now feasible to address early time dynamics essential to quantifying properties
of strongly interacting quark-gluon matter.Comment: Talk by R.V. at Quark Matter 2011, Annecy, France, May 23-28, 2011.
  LaTex, 4 pages; v2, final version to appear in J. Phys. 

Berry M V

Dusling K Gelis F Venugopalan R

François Gelis

Fukushima K Gelis F

Kevin Dusling

Polarski D

Raju Venugopalan

Son D T

Sorensen P Bolliet B Mocsy A Pandit Y Pruthi N

English

arXiv

Crossref

Journal of Physics G Nuclear and Particle Physics

Ekpyrosis and inflationary dynamics in heavy ion collisions: the role of quantum fluctuations

We summarize recent significant progress in the development of a first-principles formalism to describe the formation and evolution of matter in very high energy heavy ion collisions. The key role of quantum fluctuations both before and after a collision is emphasized. Systematic computations are now feasible to address early time isotropization, flow, parton energy loss and the Chiral Magnetic Effect

Dusling, K.

Venugopalan, R.

Gelis, F.

UNT Digital Library

   BNL-96184-2011-CP     Ekpyrosis and inflationary dynamics in heavy ion collisions: the role of quantum fluctuations  K. Dusling(1), F. Gelis(2) and R. Venugopalan(3) 1Physics Department, North Carolina State University, Raleigh, NC 27695, USA 2Institut de Physique Théorique (URA 2306 du CNRS), CEA/DSM/Saclay, 91191, Gif-sur-Yvette Cedex, France 3Physics Department, Bldg. 510A, Brookhaven National Laboratory, Upton, NY 11973, USA  Presented at the XXII International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions (Quark Matter 2011) Annecy, France July 22 to 28, 2011   August 2011    Physics Department/Nuclear Theory Group/Office of Science  Brookhaven National Laboratory    U.S. Department of Energy DOE Program Office of Science, Nuclear Physics    Notice: This manuscript has been authored by employees of Brookhaven Science Associates, LLC under Contract No. DE-AC02-98CH10886 with the U.S. Department of Energy. The publisher by accepting the manuscript for publication acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.  This preprint is intended for publication in a journal or proceedings.  Since changes may be made before publication, it may not be cited or reproduced without the author’s permission.    DISCLAIMER  This report was prepared as an account of work sponsored by an agency of the United States Government.  Neither the United States Government nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or any third party’s use or the results of such use of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof or its contractors or subcontractors.  The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.   arXiv:1107.0247v1  [hep-ph]  1 Jul 2011Ekpyrosis and inflationary dynamics in heavy ioncollisions: the role of quantum fluctuationsKevin Dusling1, François Gelis2, and Raju Venugopalan31 Physics Department, North Carolina State University, Raleigh, NC 27695, USA2 Institut de Physique Théorique (URA 2306 du CNRS), CEA/DSM/Saclay, 91191,Gif-sur-Yvette Cedex, France3 Physics Department, Bldg. 510A, Brookhaven National Laboratory, Upton, NY11973, USAE-mail: raju@bnl.govAbstract.We summarize recent significant progress in the development of a first-principlesformalism to describe the formation and evolution of matter in very high energyheavy ion collisions. The key role of quantum fluctuations both before and after acollision is emphasized. Systematic computations are now feasible to address earlytime isotropization, flow, parton energy loss and the Chiral Magnetic Effect.PACS numbers:1. IntroductionThe evolution of matter produced in the “little bang” of heavy ion collisions, to borrowa term from a recent cosmological model [1], is a process of Ekpyrosis‡, in which highlyoccupied coherent gluon states in the incoming nuclei are excited, producing very hotand dense matter shortly after the collision. The analog of the decaying inflaton fieldin this conflagration is a highly occupied Glasma [8, 9, 10] field. Quantum fluctuationsplay an essential role: a dominant class of these are responsible for the energy evolutionof the Color Glass Condensate (CGC) nuclear states before the collision. The otherdominant fluctuations grow exponentially with the square root of the proper time afterthe collision and likely isotropize matter in the Glasma. The mechanism of isotropizationand possible thermalization bears close analogy to the inflationary dynamics of “pre-heating” in the early universe [2, 3, 4]. We will outline here recent work quantifying therole of quantum fluctuations in the early time dynamics in heavy ion collisions‡ A term from Stoic cosmogony whereby the universe is destroyed in a fiery conflagration andsubsequently reconstituted at regular intervals. Heavy ion collisions containing billions of eventsrepeatedly undergo this process and are therefore closer in spirit to the Ekpyrotic scenario than theorigin of the universe in an initial singularity.Initial Conditions 22. Factorization and resummation of quantum fluctuationsThe treatment of high energy nuclear collisions in quantum field theory is motivated bythe natural separation of scales between large x static color sources ρa1,2 in the nuclearwavefunctions and small x dynamical gauge fields Aµ,a. The evolution of this separationof scales leads to a Wilsonian renormalization group description formalized as the CGCeffective field theory [5]. At high energies, the color sources are strong (ρa1,2 ∼ 1/g, whereg ≪ 1 is the QCD coupling) and color fields become time dependent immediately afterthe collision. Inclusive quantities, computed in the presence of these time dependentcolor fields§, are expressed in terms of retarded propagators, thereby allowing real timecomputations with initial data at negative infinity [6, 7].Before the collision, the effective ground states of the nuclei are coherent statesdescribed as classical fields with occupation numbers O(1/g2). In QCD, a classicaldescription of Ekpyrosis is obtained by solving Yang-Mills equations with static lightfront color sources [8, 11, 12]. At this classical level, energy dependence is introduced“by hand” via the saturation scale QS, which is closely related to the variance of theGaussian random distribution of color charges in the MV model [13, 14] of high nuclearwavefunctions. The energy dependence arises at next-to-leading (NLO) order fromquantum fluctuations (of relative strength g) about the classical background of thenuclei. These naively sub-leading contributions are enhanced by logarithms, which giveαS ln(x1,2) ∼ 1 for small x1,2 and have to be resummed to all orders in perturbationtheory. In the strong field regime, there is an additional resummation (gρ1,2)n ateach order in the αS expansion. These respectively radiative and multiple scatteringcontributions are generated by the JIMWLK renormalization group (RG) equation,∂W [ρ1,2]∂ ln(x1,2)= H1,2 W [ρ1,2], where W [ρ1,2] are gauge invariant weight functionals for thedistributions of sources in the two nuclei, and H1,2 is the corresponding JIMWLKHamiltonian [15, 16].An important consideration is to show that quantum fluctuations from the twonuclei don’t talk to each other before the collision. Albeit required by causality, it isby no means assured in a weak coupling treatment; we are able to show formally thatfactorization of the contributions of the weight functionals W [ρ1,2] to inclusive quantitiesis obtained at leading logs in x accuracy [17, 18, 19]. These quantum modes are boost-invariant pη = 0 modes, where pη is Fourier conjugate to the space-time rapidity.In the process of Ekpyrosis, nuclear coherence is lost and pη 6= 0 modes aregenerated. These modes are generically unstable [20, 21, 22, 25], and grow in anexpanding system as (αS exp(2√µτ))n, with µ ∼ QS, where n denotes the order inperturbation theory beyond the classical leading order contribution. These “leadinginstabilities” are comparable to the background field at τ ∼ 1/QS and have to beresummed to all orders, leading to qualitatively different behavior.This early time behavior has a robust analogy to “pre-heating” dynamics in§ Examples of similar dynamics include Schwinger’s mechanism for electron-positron pair productionin strong QED fields and Hawking radiation in the vicinity of the event horizon of a black hole.Initial Conditions 3inflation [2, 3, 4] where quantum fluctuations, enhanced by parametric resonance withthe background Inflaton field, are conjectured to play an important role in “turbulentthermalization” of the early universe. In particular, quantum evolution of such a systemcan be represented by the stochastic average over repeated evolution of the inflaton fieldto which quantum seeds drawn from a Gaussian distributed ensemble of fluctuationsare added [23, 24]. As we shall discuss, this formulations bears an exact analogy to thetreatment of the problem in QCD [26].3. Results and OutlookAfter factorization and resummation of leading logs in x1,2 and leading instabilitiesαS exp(2√µτ), the energy-momentum tensor is expressed as〈T µν〉LLx+LInst. =∫[Dρ1Dρ2] Wx1 [ρ1]Wx2 [ρ2]∫[Dα]F0[α] T µνLO[A[ρ1, ρ2]+α](x) .(1)The weight functionals Wx1,2 [ρ1,2] satisfy the JIMWLK RG equation discussedpreviously. The argument A ≡ (A,E) denotes collectively the components of theclassical fields and their canonically conjugate momenta on the initial proper timesurface; analytical expressions for these are available at τ = 0+ [8, 9]. The initialspectrum of fluctuations F0[α], Gaussian in the quantum fluctuations α, has a variancegiven by the small fluctuation propagator in the Glasma background field as τ → 0+.In practice, the path integral in α is determined by solving the classical Yang-Millsequations repeatedly with the initial conditions at τ = 0+ given byAinit. = Ainit. +∫dµK[cKaµK(x) + c∗Kaµ∗K(x)]. (2)Here A collectively denotes the quantum fields and their canonical conjugate momenta.The coefficients cK, with K collectively denoting the quantum numbers labeling thebasis of solutions, are random Gaussian-distributed complex numbers given by〈cKc∗K′〉=NK2δKK′,〈cKcK′〉=〈c∗Kc∗K′〉= 0 . (3)Explicit expressions for the small fluctuations and their conjugate momenta, denotedhere by aµK(x) were obtained in ref. [26]. The inner product of these solutions satisfies theorthogonality condition (aK|aK′) = NKδKK′with the measure dµK(a mix of integralsand discrete sums) that ensures∫dµKNKδKK′= 1.In ref. [26], a numerical algorithm was outlined to compute eq. (1), therebydescribing both Ekpyrosis and inflationary dynamics including essential leadingquantum fluctuations. (We emphasize the formalism holds for any inclusive quantity inheavy-ion collisions including for instance parton energy loss and sphaleron transitionsat early times.) We can now study how lumpy initial conditions for color chargesρa1,2 in nuclear wavefunctions at the energy/rapidity of interest transform into the flowof matter. The early universe analogy is becoming more robust experimentally with a“WMAP-like” spectrum of spatial anisotropies now being resolved in heavy ion data [27].We carried out an extensive study of the formalism outlined here for a scalar φ4model which, among several QCD-like features, has a spectrum of unstable quantumInitial Conditions 4modes which are amplified by resonant interactions with the background field [28]. In φ4(and other scale invariant theories), the amplitude of the field is inversely proportionalto its period of oscillation. Slightly different amplitudes, corresponding to differentquantum seeds, lead to differing oscillation periods; a stochastic average over theseleads to decoherence in the evolution. A striking consequence is hydrodynamic flowwith an ideal equation of state. Eq. (2) is a realization of Berry’s conjecture [29] whichis believed to be necessary for thermalization [30] of a quantum system. Thermalizationand onset of quasi-particle dynamics have been studied in the scalar theory; similarstudies are feasible in the QCD framework of ref. [26].AcknowledgmentsR.V. and K.D are supported by the US Department of Energy under DOE ContractDE-AC02-98CH10886 and DE-FG02-03ER41260.References[1] J. Khoury, B. A. Ovrut, P. J. Steinhardt, N. Turok, Phys. Rev. D64, 123522 (2001).[2] D. Polarski, A. A. Starobinsky, Class. Quant. Grav. 13, 377-392 (1996).[3] L. Kofman, A. D. Linde, A. A. Starobinsky, Phys. Rev. D56, 3258-3295 (1997).[4] R. Micha, I. I. Tkachev, Phys. Rev. D70, 043538 (2004).[5] F. Gelis, E. Iancu, J. Jalilian-Marian, R. Venugopalan, Ann. Rev. Nucl. Part. Sci. 60, 463 (2010).[6] F. Gelis, R. Venugopalan, Nucl. Phys. A776, 135-171 (2006).[7] F. Gelis, R. Venugopalan, Nucl. Phys. A779, 177-196 (2006).[8] A. Kovner, L. D. McLerran, H. Weigert, Phys. Rev. D52, 3809-3814 (1995).[9] A. Krasnitz, R. Venugopalan, Nucl. Phys. B557, 237 (1999).[10] T. Lappi, L. McLerran, Nucl. Phys. A772, 200-212 (2006).[11] A. Krasnitz, R. Venugopalan, Phys. Rev. Lett. 84, 4309-4312 (2000).[12] T. Lappi, Phys. Rev. C67, 054903 (2003).[13] L. D. McLerran, R. Venugopalan, Phys. Rev. D49, 2233-2241 (1994).[14] L. D. McLerran, R. Venugopalan, Phys. Rev. D49, 3352-3355 (1994).[15] J. Jalilian-Marian, A. Kovner, A. Leonidov, H. Weigert, Nucl. Phys. B504, 415-431 (1997).[16] E. Iancu, A. Leonidov, L. D. McLerran, Nucl. Phys. A692, 583-645 (2001).[17] F. Gelis, T. Lappi, R. Venugopalan, Phys. Rev. D78, 054019 (2008).[18] F. Gelis, T. Lappi, R. Venugopalan, Phys. Rev. D78, 054020 (2008).[19] F. Gelis, T. Lappi, R. Venugopalan, Phys. Rev. D79, 094017 (2009).[20] P. Romatschke, R. Venugopalan, Phys. Rev. Lett. 96, 062302 (2006).[21] P. Romatschke, R. Venugopalan, Phys. Rev. D 74, 045011 (2006).[22] P. Romatschke, R. Venugopalan, Eur. Phys. J. A29, 71-75 (2006).[23] D. T. Son, [hep-ph/9601377].[24] S. Y. .Khlebnikov, I. I. Tkachev, Phys. Rev. Lett. 77, 219-222 (1996).[25] K. Fukushima, F. Gelis, [arXiv:1106.1396 [hep-ph]].[26] K. Dusling, F. Gelis and R. Venugopalan, arXiv:1106.3927 [nucl-th].[27] P. Sorensen, B. Bolliet, A. Mocsy, Y. Pandit, N. Pruthi, [arXiv:1102.1403 [nucl-th]].[28] K. Dusling, T. Epelbaum, F. Gelis, R. Venugopalan, Nucl. Phys. A850, 69-109 (2011).[29] M.V. Berry, J. Phys. A: Math. Gen. 1012, 2083 (1977).[30] M. Srednicki, Phys. Rev. E 50, 888 (1994).

Ekpyrosis and inflationary dynamics in heavy ion collisions: the role of
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Ekpyrosis and inflationary dynamics in heavy ion collisions: the role of quantum fluctuations

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