Initial Conditions with Flow from a McLerran Venugopalan model with Transverse Dynamics
aa r X i v : . [ nu c l - t h ] D ec Initial Conditions with Flow from a McLerranVenugopalan model with Transverse Dynamics
Guangyao Chen and Rainer J. Fries
Cyclotron Institute and Department of Physics and Astronomy, Texas A&M University,College Station, TX 77843, USAE-mail: [email protected]
Abstract.
Using a recursive solution of the Yang-Mills equation, we calculate analyticexpressions for the gluon fields created in ultra-relativistic heavy ion collisions at small times τ . We have worked out explicit solutions for the fields and the energy momentum tensor upto 4th order in an expansion in τ . We generalize the McLerran-Venugopalan model to allowfor a systematic treatment of averaged charge densities µ that vary as a function of transversecoordinates. This allows us to calculate radial, elliptic and directed flow of gluon fields. Ourresults can serve as initial conditions for hydrodynamic simulations of nuclear collisions thatinclude initial flow.
1. Introduction
Nuclear collisions at high energies at the Relativistic Heavy Ion Collider (RHIC) and the LargeHadron Collider (LHC) can create and probe novel forms of QCD matter. One example is adeconfined phase of QCD called Quark Gluon Plasma (QGP). A thermalized QGP is created innuclear collisions at RHIC and LHC after a time of about 1 fm/ c . Relativistic hydrodynamicsis the tool of choice to describe the evolution of QGP in this stage [1, 2]. Large uncertainties inthe hydrodynamic evolution come from the incomplete understanding of the initial conditions,i.e. the evolution of the nuclear collision prior to 1 fm/ c . This is where another new phase ofQCD, Color Glass Condensates (CGC) plays an important part. The natural limit of a nucleusat very high energy is characterized by a saturated gluon density in which the gluon field canbe described by a quasi-classical field [3, 4]. The earliest phase of the collision of two nuclei canthus be described as the collision of two sheets of color glass [5]. The classical field resultingfrom the collision then decays and equilibrates to a plasma of quarks and gluons.To be more precise, in the color glass formalism partons are distinguished by momentumfraction. Sources have large Bjorken- x while the small- x gluons are treated through the classicalfield F µν . They are related by the Yang-Mills equation [ D µ , F µν ] = J ν . Before the collision,the two incoming nuclei on the positive and negative light cone are dominated by transversechromo-electric and chromo-magnetic fields which are perpendicular to each other [3, 4]. Afterthe collision, the fields in the forward light cone can be parameterized as A ± = ± x ± A and atransverse field A i , i = 1 , A and A i depend on the proper time τ = √ t − z and transversecoordinates but not on the space-time rapidity η . For small τ , a series expansion A µ = P n τ n A µ ( n ) in powers of τ can be employed. A few lowest orders in time should give an adequate and analyticdescription of the gluon fields immediately after the collision [6].he collision initially ( τ → E = ig [ A i , A i ] , B = − igǫ ij [ A i , A j ]between the nuclei, where A i and A i are the fields of nucleus 1 or 2 before the collision in lightcone gauge. The next order in τ gives us linearly growing transverse electric and magnetic fields E i (1) = −
12 (sinh η [ D i , E ] + cosh η ǫ ij [ D j , B ]) , B i (1) = 12 (cosh η ǫ ij [ D j , E ] − sinh η [ D i , B ]) . We have calculated analytic expressions for the fields up to fourth order in τ explicitly andwill discuss those results elsewhere [7]. The behavior we find is qualitatively consistent withnumerical studies of classical field QCD [8].The energy momentum tensor of the gluon field after the collision can be easily calculated.Up to order O ( τ ) — we skip higher orders here for brevity — it can be written in the form T mn f = A + C B B B A + D E B ′ /τB E A − D B ′ /τ B ′ /τ B ′ /τ ( − A + C ) /τ (1)in the τ, x, y, η coordinate system. Here A = ( E + B ) / τ = 0,and B i and B ′ i are the coefficients of the Poynting vector T i = B i cosh η + B ′ i sinh η and linearin τ ( i = 1 , C , D , E ∝ τ are coefficients quadratic in τ . All thesecoefficients have been calculated analytically [7]. One can check that this energy momentumtensor is boost invariant owing to the original boost-invariant setup of the colliding nuclei.
2. Flow in the McLerran Venugopalan Model
In the McLerran-Venugopalan (MV) model [3, 4], the simplest implementation of color glass,an observable O calculated from sources in the two nuclei has to be averaged over all possibleconfigurations of color source distributions ρ . Each collision samples arbitrary ρ due to the shorttime scale of the collision compared to internal time scales of the nuclear wave function. Thedistribution of charge is usually assumed to be Gaussian around zero with an average value µ of the variation. We use the definition h ρ a ( ~x ⊥ ) ρ b ( ~y ⊥ ) i = g δ ab N c − µ ( ~x ⊥ ) δ ( ~x ⊥ − ~y ⊥ ) . (2)We have suppressed the dependence on the longitudinal coordinate for brevity. Note that wehave generalized the original expression from the MV model such that µ can be a functionof transverse coordinate ~x ⊥ , instead of being an average charge density that is assumed to behomogeneous. Of course the latter is unrealistic and the generalization needs to be made if long-range transverse dynamics is to be described. We have shown explicitly that if µ is varyingslowly such that | µ ( ~x ⊥ ) | ≫ m − |∇ i µ ( ~x ⊥ ) | ≫ m − |∇ i ∇ j µ ( ~x ⊥ ) | , color glass dynamics andlong-distance dynamics which are not described by CGC can be safely separated. Here m is aninfrared regulator. We find that the generalized two gluon correlation function is [7] h A ia ( ~x ⊥ ) A jb ( ~x ⊥ ) i = δ ab g µ ( ~x ⊥ )8 π ( N c − (cid:2) δ ij ln Q (1 . m ) + ∇ k ∇ l µ ( ~x ⊥ ) m µ ( ~x ⊥ ) ( 16 δ kl δ ij − δ ik δ jl ) (cid:3) , (3)where the first term is the same as in the original MV model [9]. Q here is the saturation scale. igure 1. Left: The hydro-like flow component ~B (arrows) and the underlying energy density.Middle: The rapidity-odd flow component ~B ′ . Right: The Poynting vector T i at η = 1 togetherwith the energy density. The contour lines indicate the density profile of one of the nuclei.Let us now focus on the transverse Poynting vector T i in the energy momentum tensor whichdescribes the energy flow of the gluon field. The term even in η after averaging reads B i = − τ g N c ( N c − π ∇ i ( µ µ ) ln Q (1 . m ) . (4)This expression gives radial and elliptic flow of energy ∝ −∇ i ǫ . This is similar to what ahydrodynamic flow field would look like for the same energy density ǫ if the hydrodynamicpressure is a monotonous function of ǫ . The term odd in η is B ′ i = − τ g N c ( N c − π ln Q (1 . m ) [ µ ∇ i µ − µ ∇ i µ ] . (5)Interestingly this term can describe a directed flow of energy. Note that this term does notviolate boost-invariance of the energy momentum tensor. On the other hand boost-invarianthydrodynamics would never be able to create such a flow field. Rather one can trace this termback to the QCD analogon of Gauss’ Law. This makes this rapidity-odd flow field a curiousphenomenon. The flow vector fields ~B and ~B ′ from two gold nuclei with Woods-Saxon profiles µ colliding with impact parameter b = 8 fm are shown in Fig. 1. The net Poynting vector T i exhibits a clear dipole asymmetry in the transverse plane when η = 0. Flow in color glass atmidrapidity has recently also been studied numerically in [10].
3. Hydrodynamic Initial Conditions
Since the classical gluon field only describes the very earliest stage of a nuclear collision weneed to translate the energy momentum tensor into initial conditions for hydrodynamics. Thedynamics of equilibration is outside the purview of either hydrodynamics nor classical fieldtheory and would require a more complete description of quantum non-abelian dynamics. Onecan however derive a matching hydrodynamic energy momentum tensor simply using conservedcurrents both on the classical field side and the hydrodynamic side. For ideal hydrodynamicsthis was worked out in [11, 12] by just imposing energy and momentum conservation. The flowof energy generally translates into a flow of particles on the hydrodynamic side. The directionof the hydrodynamic transverse flow velocity is parallel to the Poynting vector and its size is amonotonous function of the magnitude of the Poynting vector. A similar correspondence can beworked out in the case of matching to viscous hydrodynamics [7]. . Directed Flow Phenomenology
Without a further hydrodynamic evolution we can nevertheless check wether our somewhatsurprising direct flow result is qualitatively consistent with experiments. The first thing tocheck is the direction of the flow. It turns out that our directed flow vector ~B ′ points away fromthe spectators for a given side η > η < v is alsoroughly consistent with a sinh y shape as a function of particle rapidity y [13, 14].Note that ~B ′ vanishes for b = 0 if the two nuclei are equal. Interestingly, if an asymmetricsystem like Cu+Au collides with b = 0 the flow vector ~B ′ becomes radial, but points outwardin the direction of the smaller nucleus and inward in the direction of the larger nucleus. Anasymmetric system with b = 0 would exhibit a striking blend of both effects. Looking for sucha strong dependence of flow on rapidity in central collisions of asymmetric systems could be asignature for the eminence of strong gluon fields early in nuclear collisions.
5. Conclusion
To summarize, if the McLerran Venugopalan model is generalized to allow for variations of theaverage charge density across the nucleus flow phenomena in early stages of nuclear collisions canbe studied. Here we use analytic results for early gluon fields and their energy momentum tensorthat can be obtained in an expansion of the Yang-Mills equations around the time of collision.We find radial and elliptic flow patterns similar to hydrodynamic behavior. In addition we alsoobtain a rapidity-odd term in the flow of gluon energy which resembles directed flow. We haveargued that this flow will translate into directed flow of particles in the further course of thecollision. The results are in qualitative agreement with results from the STAR experiment.
Acknowledgments
This work is supported by NSF CAREER grant PHY-0847538, JET Collaboration and DOEgrant DE-FG02-10ER41682. We thank Joseph Kapusta and Yang Li who contributed to thisproject.
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