Table Of ContentCollective non-Abelian instabilities in a melting Color Glass Condensate
Paul Romatschkea and Raju Venugopalanb
a Fakult¨at fu¨r Physik, Universit¨at Bielefeld,
33615 Bielefeld, Germany
b Physics Department, Brookhaven National Laboratory,
Upton, N.Y. 11973, U.S.A.
(Dated: February 2, 2008)
Wepresentfirstresultsfor3+1-DsimulationsofSU(2)Yang-Millsequationsformatterexpanding
into the vacuum after a heavy ion collision. Violations of boost invariance cause a non-Abelian
Weibel instability leading soft modes to grow with proper time τ as exp(Γ g2µτ), where g2µ is a
scale arising from the saturation of gluons in the nuclear wavefunction. The scale for the growth
p
6 rate Γ is set by a plasmon mass, defined as ωpl =κ0 g2τµ, generated dynamically in the collision.
0 We compare the numerical ratio Γ/κ0 to the correspqonding value predicted by the Hard Thermal
0 Loop formalism for anisotropic plasmas.
2
n
Experiments at the Relativistic Heavy Ion Collider give τ 1 . This thermalization time scale,
Ja (RHIC) at Brookhaven National Laboratory indicate at RHtIhCermen.e∼rgiαes1S3,/i5sQssignificantly larger than 1 fm/c.
9 that a thermalized Quark Gluon Plasma (QGP) may
Scattering rates in the bottom up scenario are con-
1 be formed in ultrarelativistic collisions of beams of gold
ions [1]. Phenomenologicalanalyses of RHIC data point trolled by the Debye mass, which sets the range of par-
2 ticle interactions. Interestingly, the Debye mass squared
to thermalization of matter on early time scales of or-
v can change sign for anisotropic “CGC like” initial con-
der 1 fm/c after the collision [2]. Understanding how
1
ditions, giving rise to an imaginary component in the
2 wavefunctions of high energy nuclei decohere completely
gluon dispersion relation [14]. This corresponds to the
1 on short time scales to form a thermalized QGP is an
0 outstanding theoretical puzzle of great interest. non-Abelian analog of the collective Weibel instabil-
1 ity [10, 11, 12] leading to an exponential growth of soft
5 At central rapidities, where thermalization is most modes. Thisphenomenonmayinvalidateoratleastmod-
0 likely, the wavefunctions of colliding nuclei are domi- ify [13] bottom up thermalization. Recently, several nu-
/
h nated by small x modes which, due to their large oc- merical studies have explored the behavior of instabili-
p cupation numbers, are described as classical fields [3]. tiesinthefinitetemperatureHardThermalLoop(HTL)
p- Thesefields,andtheirevolutionwithenergy,canbecom- formalism[15]extendedtoanisotropicnon-Abelianplas-
e puted in a classicaleffective field theory called the Color mas [16]. These studies find exponentially growing soft
h Glass Condensate (CGC) [4] characterized by a semi– modes. However, the most recent detailed 3+1-D simu-
v: hardscale,Qs(x)>>ΛQCD. This “saturation”scale en- lations find that the exponential modes saturate at late
i sures that the dynamics can in principle be understood times when non-linear self-interactions become impor-
X
in weak coupling; it is estimated [4] to be Qs 1.4 GeV tant. Alsointerestingisasubsequentphenomenon,anal-
r ≈
a for RHIC energies and Qs 2.2 GeV at Large Hadron ogous to Kolmogorov cascading in turbulent plasmas,
≈
Collider (LHC) energies of √s = 5.5 TeV/nucleon. The where energy is transferred back to hard modes [17].
initial conditions for the collision can be expressed self- While all these studies consider anisotropic particle dis-
consistently in terms of the computed classical fields of tributions a la the CGC, neither the expansion of the
each of the nuclei [5]. system nor the specific CGC dynamics following from
physics of the small x modes is implemented.
The classical dynamics of fields produced in the col-
lision is approximately boost invariant. Assuming it to We present in this letter, for an SU(2) Yang-Mills
be strictly so, the resulting 2+1 dimensional Yang-Mills theory, a first 3+1-dimensional numerical study of non-
equations generating the space-time dynamics of fields Abelian collective instabilities generated in the expand-
(the“meltingoftheCGC”)wasinvestigatednumerically, ing CGC. The three spatial dimensions here are the
andtheenergyandnumberdistributionsoftheclassically two transverse co-ordinates x⊥, the space time rapidity
produced gluons computed [6]. On proper time scales η = 12ln(tt+−zz). The fourth dimension is given by the
τ 1/Q , the energy density behaves as ε 1/τ, sug- proper time τ = √t2 z2. The strict boost invariance
s
∼ ∼ −
gestingfreestreamingofgluonsinthetransverseplaneat of fields (their lack of dependence on η) in Ref. [6] is
these early times. In their “bottom up” scenario, Baier relaxed [20]. Violations of boost invariance arise from
et al. suggested that thermalization is a consequence of finite energy constraints and from small x quantum cor-
subsequent re-scattering of on-shell gluons by 2 2 [7] rections. The magnitude of the violations from the for-
↔
and 2 3 processes [8, 9]. The bottom up estimates mer are small. The latter are significantlylargerandare
↔
2
0.0001
of order unity over rapidity intervals of order Y 1/α .
S
We will only consider small violations of boost∼invari- 1e-05 c+c Exp(0.427 Sqrt(g2 µ τ))
0 1
ance here; as a consequence, the gluon field configura- 1e-06 c+c Exp(0.00544 g2 µ τ)
η 0 1
tions are highly anisotropic in momentum space. Ki- 3µ L 1e-07
netic theory studies [11, 12, 14] show that anisotropic 4 g 1e-08
distributions of hard modes rather than details of their η /
ηT 1e-09
dynamics drive the non-Abelian Weibel instability. Our 2τ
initial conditions are therefore similar, in this respect, ax 1e-10
m
because the high momentum anisotropically distributed 1e-11
hard modes of the field with k⊥ Qs play the role of 1e-12
∼
“hard”particles in their coupling to the softmodes with
1e-13
k⊥ ≈ τ−1kη << Qs. The effects of large amplitude vio- 0 500 1000 150g02 µ 2τ000 2500 3000 3500
lations of boost invariance will be commented on briefly
and discussed further in a follow up to this work [18]. FIG. 1: The maximum Fourier mode amplitudes of τ2Tηη
In Aτ = 0 gauge, initial conditions that violate strict for g2µL = 67.9, N = Nη = 64, Nηaη = 1.6. Also shown
boost invariance for the dynamical fields A⊥,Aη and are best fits with ex⊥pτ and exp√τ behavior. The former is
clearly ruled out by thedata.
their canonically conjugate momenta E⊥, Eη are
A = ; A =0 ; E =δE ; E = +δE ,
i i η i i η η η
A E
The lattice parameters, in dimensionless units, are a)
where, for each configuration of the color charges of the
atwreotnhuecbleoi,osAtiin≡vaAriia(nxt⊥i)n,iAtiaηl=co0n,dEitiio=ns0s,tuEdηie≡dEinη(pxr⊥e)- dNi⊥recatniodnNs rηe,stpheectnivuemlyb;ebr)ofg2laµtati⊥ceasnidtesaηin, tthheerxe⊥spaecntdivηe
vious simulations [6]. The rapidity dependent functions lattice spacings; c) τ0/a⊥ and δτ, the time at which the
simulations are initiated and the stepping size respec-
δE and δE are constructed to satisfy Gauss’s law,
i η tively; and finally d) ∆, the initial size of the rapidity
D δE +D E = 0, at the initial time τ = τ . For each
i i η η 0 fluctuations. The continuum limit is obtained by hold-
ciδnoEgnifi,hgδruEa¯rnia(dtxoi⊥omn)sδcEao¯rnie(fiymg⊥uu)rliatit=piolinδesd(2δ)b(Ey¯xi⊥a(xn−o⊥t)yh⊥earr)e.radTnrdhaowemsne,frusaanntcditsiofoymn- ianηgNtηhe=phLyηsificaxledcowmhbiilneasteionndsingg2µδτa,⊥gN2µ⊥a=⊥ agn2dµLaηantdo
zero. For this study, we pick L = 1.6 units of rapidity.
f(η)=∂ F(η). One obtains, at τ =τ , η
η 0 VariationsofL willbecommentedonlater. Themagni-
η
δEi(x⊥,η)=f(η)δE¯i(x⊥) ; δEη = F(η)DiδE¯i(x⊥). tude of violations of boost invariance, as represented by
−
∆,isaphysicalquantity;here,westudyresultsforsmall
The fields F(η) are Gaussian white noise distributed, valuesof∆ 10−11 10−8. Theinitialtimeischosento
F(η)F(η′) = ∆2δ(η η′), with the amplitude of vi- ∼ −
h i − ensure that for ∆ = 0, we recover earlier 2+1-D results;
olations of boost invariance governed by the parameter
we set τ0 = 0.05a⊥. Our results are insensitive to vari-
∆. These initial conditions are not unique. Their virtue
ations that are a factor of 2 larger or smaller than this
isthatGauss’lawismanifestandperiodicboundarycon-
choice.
ditionscanbeappliedintheη direction. Abroaderclass
To study the growth rate of instabilities due to viola-
of initial conditions will be considered in future.
tions of boost invariance, we define
Our numerical procedure is as follows [21]. The small
x classicalfields before the collision are determined from
their respective classical color source densities by solv- T˜µν(kη,k⊥ =0)= dη exp(iηkη) Tµν(x⊥,η) ⊥,ρ,
h i
ing Poisson’s equations [6]. These color charge distribu- Z
tions areGaussiandistributed witha varianceg2µ. This where Tµν denote components of the stress energy ten-
momentum scale (closely related and similar in magni- sor and ⊥,ρ denotes an average over the transverse co-
hi
tude [22] to Q ) and the size of the system L are the ordinatesandoverallcolorchargeconfigurationsrespec-
s
physicaldimensionfulscalesintheproblem. Forperiodic tively. WewilllookinparticularatthisFouriertransform
boundary conditions, L2 = πR2, where R is the radius for the longitudinal pressure τ2Tηη. The magnitude of
ofthenuclei. Foreachconfigurationofcolorcharges,the this quantity is a useful measure of isotropization; mo-
initial conditions are determined, and an adaptive leap- mentumdistributionsareisotropicwhen2τ2Tηη/(Txx+
frog algorithm evolves Hamilton’s equations for the dy- Tyy) 1. InFig.1,we plotthe maximalvalue ofτ2T˜ηη
→
namical fields and their canonically conjugate momenta at each time step, as a function of g2µτ. The data are
on a discretized space-time lattice. Physical results, ob- for a 643 lattice and correspond to g2µL = 67.9 and
tained by averaging the results over all color configura- L = 1.6. The maximal value remains nearly constant
η
tionsofthesources,areexpressedintermsofg2µandthe until g2µτ 250, beyond which it grows rapidly. A
≈
dimensionlesscombinationg2µL. Higherenergiesand/or best fit to the functional form c + c exp(c τc3) gives
0 1 2
larger nuclei correspond to larger values of g2µL. c = 0.427 0.01 for c = 0.5; the coefficients c , c are
2 3 0 1
±
3
1
120
110 0.9 322
100 0.8 6142282
90
0.7
80
70 1/2µ) 0.6
k η 5600 2ωτ ( / gPl.00..45
40
0.3
30
0.2
20
10 0.1
min kη
0
0 500 1000 1500 2000 2500 3000 0 50 100 150 200 250 300 350
g2 µ τ g2µ τ
FIG. 2: The mode number kη for which the maximum am- FIG.3: Timeevolutionofωpl.,forfixedg2µL=22.6andlat-
plitude of τ2Tηη occurs (see Fig.1); again for g2µL = 67.9, tice spacings g2µa = 0.707,0.354,0.177 (N = 32,64,128),
N = Nη = 64, Nηaη = 1.6. kmin denotes the smallest kη respectively. ⊥ ⊥
m⊥ode that could be excited in thelattice simulation.
case is closely related to a mass gap generated by the
smallnumbersproportionaltotheinitialseed. Itisclear
highly non-linear dynamics of soft modes in the expand-
fromFig. 1 thatthe formexp(Γ g2µτ) is preferredto a
ing system. As in Ref. [6], fixing the residual gauge free-
fit with an exponential growth in τ.
p dom ⊥ A⊥, the gluon dispersion is given by relation
We digressbrieflytonotethatthe relevanttimescales ∇ ·
[24]
in Fig. 1, for RHIC collisions where g2µ 1 GeV, are
≈
ttinoimvoaelrainragneecceetsostaboreybinefctoeorrmeesextt.vreTismhibeellseye.hsomTwhaeelvlepvrrioceoclairstreieosnbpseohnoadfvtibooortohosetf ω(k⊥)= τ1sTTrr[A[Ei(ik(k⊥⊥)AEii((−−kk⊥⊥))++ττ−2E2Aη(ηk(k⊥⊥)E)Aη(η−(−kk⊥⊥)])].
non-Abelian Weibel instabilities in a melting CGC are
Remarkably, a mass-gap, which we associate with the
more transparent for small initial seeds. plasmon mass ωpl ω(k⊥ = 0), is generated. After
In Fig. 2, we plot the most unstable k mode, corre- ≡
η initial transient behavior, it satisfies the relation
sponding to the maximal τ2T˜ηη in Fig. 1, as a function
of g2µτ. Our result suggests that the soft k modes are 1
η ω (τ)=κ g2µ .
sensitive to the non-Abelian Weibel instability while the pl 0 g2µτ
r
hard modes are not. Further, the maximal mode num-
The time evolution of this plasmon mass in units of
berk growsasafunctionoftime. The amplitude ofthe
η ω (τ/g2µ)1/2 is shown in Fig. 3 for g2µL = 22.6, with
softmodesdominatesthespectrumforg2µτ >250. This pl
κ = 0.3 0.01; it is robust as one approaches the con-
timescale,however,is weaklydependent onourchoiceof 0 ±
tinuum limit. The dependence of ω on g2µL is shown
L = 1.6; raising L to L = 3.2 lowers the timescale pl
η η η in Fig. 4 [25].
fortheonsetofgrowthbysome20%,whiletheextracted
We now assume the growthrate in the expanding sys-
growth rate stays the same. The large volume limit will
temismodifiedasγ τ γ(τ)τ,andfurtherassume,a)
be examined further in an upcoming work [18]. stat →
We now turn to the correspondence between our re- γ(τ) = m∞(τ)/√2, and b) m2∞(τ) = 32ωp2l(τ), as in the
static case. Since the plasmon mass is determined inde-
sults and the HTL formalism for anisotropic plasmas.
pendently,theseHTLrelationspredicttheFourierampli-
In the latter, the maximal Fourier amplitudes of compo-
nentsofthestress-energytensorgrowasexp(2γτ),where tudegrowsasexp Γth. g2µτ ,withΓth. =√3κ0. This
the growth rate γ satisfies the relation γ = 1m∞ for HTL motivated va(cid:16)lue opf the g(cid:17)rowth rate is compared to
2 thegrowthrateextracteddirectlyfromourfitsintheta-
maximally anisotropic particle distributions [q12]. Here, bleforseveralvaluesofg2µL(computedforN⊥ =32,64
d3p f(p~) and 128, respectively). Remarkably, we find an agree-
m2∞ =g2Nc (2π)3 p , ment to within 4−6% accuracy. However, a consistent
Z treatmentgivesthegrowthrateintheexpandingcasebe
τ ′ ′
where f is the anisotropic single particle distribution of exp(2 dτ γ(τ )). If the HTL relation of γ to the plas-
0
the hard modes. For both isotropic as well as anistropic mon mass is unchanged, an additional factor of 2 is ob-
R
distributions(withinthemodelusedin[14], [23])m2 = tainedinΓ relativetoΓ . ItisnotobviousthatHTL
∞ th. fit
3ω2. relationsgeneralizetothecaseofCGCinitialconditions.
2 pl
Interestingly, in close analogy to the HTL case, the In particular, studies are in progress to investigate how
exp(Γ g2µτ) growth of the unstable soft modes in our our measuredω relates to the HTL plasmonfrequency.
pl
p
4
this work. PR was supported by DFG-Forschergruppe
0.5 EN164/4-4. We thank D. B¨odeker, J. Engels, F. Gelis,
D.Kharzeev,A.Krasnitz,M.Laine,T.Lappi,L.McLer-
0.45
ran, Y. Nara and M. Strickland for fruitful discussions.
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[18] P. Romatschke and R.Venugopalan, in preparation.
alytical and numerical work is in progress to test these
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RV’s researchis supportedbyDOEContractNo. DE- fieldsfromthatofsingleparticledistributions.Thelatter
AC02-98CH10886. He thanks the Alexander von Hum- can be preserved even if the former is violated.
boldt foundation for support during the early stages of [21] This procedure is similar to that of Ref. [6]. Further de-
5
tails will bepresented in Ref. [18]. [24] Theeffectofsmalllongitudinalfluctuationsontransverse
[22] Ilnn(gth2µe/cΛla),sswichaelrethΛeoirsyaonfitnhfrearCeGdCsc,aQle2sof=orgd4eµr2ΛNQc/C2Dπ. · qωu(kant)i,tiefsorshcooumldpubteatrioanthaelrcsomnvaelln.ieWnecet,hferroemforeae2x+tr1a-cDt
[23] Averaging over longitudinal and transverse modes (see simu⊥lation.
the2ndpaperbyRebhanetal.inRef.[16]),oneobtains [25] ThemassgapwasnotedinRef.[6]incontextofinfrared
ω2 = m2 arctan√ξ and m2 = m2 arctan√ξ, where ξ finitenumberdistributions.Interestingly,thebehaviorof
depnlotes3√thξe strength of the∞anisot2r√oξpy and m2 is a soft ωpl versusg2µLissimilar tothat ofthegluon liberation
scale proportional to the Debye mass. Therefore, ω2 = factor fN.
pl
2m2 .
3
∞
