Color Glass Condensate and initial stages of heavy-ion collisions

We introduce the concept of Color Glass Condensate, that describes the wave-function of a nucleon or nucleus at high energy. Then, we explain the relevance of this effective theory in the calculation of particle production in heavy ion collisions, and we show how it can be used in order to make predictions for the initial stages of these collisions – in particular in the task of providing initial conditions for hydrodynamical simulations. References 1. F. Gelis, T. Lappi, R. Venugopalan, Int. J. Mod. Phys. E16, (2007) 2595-2637. F. Gelis, T. Lappi, R. Venugopalan, Int. J. Mod. Phys. E16, (2007) 2595-2637. 2. F. Gelis, R. Venugopalan, Acta Phys. Polon. B37, (2006) 3253-3314. 3. E. Iancu, R. Venugopalan, in Quark Gluon Plasma 3, Eds. R.C. Hwa and X.N.Wang, (World Scientific, 2003). 4. E. Iancu, A. Leonidov, L. McLerran, in Cargese 2001, QCD perspectives on hot and dense matter, (2001) 74145. © Owned by the authors, published by EDP Sciences, 2010 DOI:10.1051/epjconf/20100701002 EPJ Web of Conferences 7, 01002 (2010) This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, which permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited. Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20100701002


Introduction
Initial correlations and hydrodynamics • The equations of hydrodynamics are non-linear.Therefore, solving hydro evolution for event averaged initial conditions is not the same as solving hydro event-by-event, and averaging observables at the end : To study hydrodynamics event by event, one needs an event generator for T μν (τ 0 , η, x ⊥ )

Gluon saturation
• consider a hadron or nucleus probed via gluon exchange

Gluon saturation
• when energy increases, new partons are emitted • the emission probability is α s dx x ∼ α s ln( 1 x ), with x the longitudinal momentum fraction of the gluon • at small-x (i.e.high energy), these logs need to be resummed

Gluon saturation
• as long as the density of constituents remains small, the evolution is linear: the number of partons produced at a given step is proportional to the number of partons at the previous step (BFKL)

Gluon saturation
• eventually, the partons start overlapping in phase-space • parton recombination becomes favorable • after this point, the evolution is non-linear: the number of partons created at a given step depends non-linearly on the number of partons present previously Note: At a given energy, the saturation scale is larger for a nucleus (for A = 200, A

CGC = effective theory of small x gluons
• The fast partons (large x) are frozen by time dilation described as static color sources on the light-cone : • Slow partons (small x) cannot be considered static over the time-scales of the collision process they must be treated as standard gauge Kelds Eikonal coupling to the current J μ : A μ J μ • The color sources ρ are random, and described by a distribution functional W Y [ρ], with Y the rapidity that separates "soft" and "hard" where • This evolution equation resums all the powers of α s ln(1/x) and of Q s /p ⊥ that arise in loop corrections • This equation simpliKes into the BFKL equation when the source ρ is small (one can expand η in powers of ρ) Power counting • Dilute regime : one parton in each projectile interact • Dilute regime : one parton in each projectile interact

Power counting
• In the saturated regime, the sources are of order 1/g (because ρρ ∼ occupation number ∼ 1/α s ) • Order of a connected diagram : • The single inclusive spectrum has a simple diagrammatic representation : • There are only connected graphs (AGK cancellation) • Perturbative expansion in the saturated regime :

Expression in terms of classical Aelds at LO
Gluon spectrum at LO : • A obeys the classical EOM : δS YM δA + J = 0 • The boundary conditions are very simple:

Initial classical Aelds
• The initial chromo-E and B Kelds form longitudinal "Lux tubes" extending between the projectiles: • The color correlation length in the transverse plane is What is factorization ?
• The naive perturbative expansion of dN 1 /d 3 p, assumes that the coefKcients c n are of order one • This assumption is upset by large logarithms of 1/x 1,2 : Leading Log terms • Factorizability: the logarithms must be universal and resummable into functionals that depend only on the projectiles being collided • The duration of the collision is very short: • The logarithms we want to resum arise from the radiation of soft gluons, which takes a long time it must happen (long) before the collision space-like interval • The duration of the collision is very short: • The logarithms we want to resum arise from the radiation of soft gluons, which takes a long time it must happen (long) before the collision • The projectiles are not in causal contact before the impact the logarithms are intrinsic properties of the projectiles, independent of the measured observable I : The NLO gluon spectrum can be written as a perturbation of the initial value of the classical Kelds on the light-cone : Factorization follows easily

Leading Log factorization
• By averaging over all the conKgurations of the sources in the two projectiles, we get a factorized formula for the resummation of the leading log terms to all orders : ] must be evolved up to the rapidity of the produced gluon • In the saturated regime, the inclusive n-gluon spectrum at Leading Order is the product of n 1-gluon spectra:

Introduction
• At LO, in a given conKguration of the sources ρ 1,2 , the n gluons are not correlated • Note: this is true for the bulk (p ⊥ Q s ), but not for the tail of the distribution

Multigluon spectrum at NLO
• At NLO, one has again: • Correlations appear at NLO thanks to the operator G( u, v ) u v , which can link two different gluons

Leading Log factorization
Factorization formula for the n-gluon spectrum • This formula tells us that (in the Leading Log approximation) all the correlations arise from the W [ρ]'s they pre-exist in the wave-function of the projectiles • Note: some short range correlations will also arise from splittings in the Knal state (not taken into account here, because does not come with a ln(s))

•
Immediately after the collision, the chromo-E and B Kelds are purely longitudinal :

•
No analytic solution for the Yang-Mills equations,

1 •
The duration of the collision is very short:τ coll ∼ E −1

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Procedure: (i) calculate the 1-loop corrections, (ii) disentangle the logarithms from the Knite contributions, (iii) show that the logs can be assigned to the projectiles • Problem: strong Kelds, analytic calculation not feasible Take advantage of the retarded nature of the boundary conditions in order to separate the initial state evolution (calculable analytically) from the collision itself (hopeless)

••••
• η-independent Kelds lead to long range correlations in the 2-particle spectrum :v r Particles emitted by different Lux tubes are not correlated (RQ s ) −2 sets the strength of the correlation• At early times, the correlation is Lat in ΔϕA collimation in Δϕ is produced later by radial Low The combinatorics of color source averages in a single glasma Lux tube leads to:N(N − 1) • • • (N − p + 1) − disc.terms = (p − 1)! N p Bose-Einstein distribution• If one superimposes k such Lux tubes emitting independently:N(N − 1) • • • (N − p + 1) − disc.terms = (p − 1)! N k p Negative binomial distribution with parameters ˙N¸, k • k is the number of Lux tubes: k ∼ Q 2 s R 2 ∼ # participants• Experimentally: it seems to work... Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008) 01002-p.25

Number of gluons per unit area :
ARecombination

cross-section :
Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008)

•
Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008) Particles emitted by different Lux tubes are not correlated (RQ s ) −2 sets the strength of the correlation • Long range correlation in Δη (rapidity)• Narrow correlation in Δϕ (azimuthal angle)t correlation ≤ t freeze out e − 1 2 |y A −y B |• Was there something independent of η at early times? the chromo-E and B Kelds produced in the collision• The color correlation length in the transverse plane is Q −1 s Lux tubes of diameter Q −1s , Klling up the transverse area