The subtle interplay of elastic and inelastic collisions in the thermalization of the quark-gluon plasma

0.37 MB·

by  Jean-Paul Blaizot

Nuclear
Physics A
NuclearPhysicsA00(2016)1–4
www.elsevier.com/locate/procedia
The subtle interplay of elastic and inelastic collisions in the
thermalization of the quark-gluon plasma
Jean-Paul Blaizota, Jinfeng Liaob,c, Yacine Mehtar-Tanid
6
1 aTheoreticalPhysics,CEA,Saclay,France
0 bPhysicsDepartmentandCenterforExplorationofEnergyandMatter,IndianaUniversity,2401NMiloB.SampsonLane,
2 Bloomington,IN47408,USA
cRIKENBNLResearchCenter,Bldg.510A,BrookhavenNationalLaboratory,Upton,NY11973,USA
n dInstituteforNuclearTheory,UniversityofWashington,Seattle,WA98195-1550,USA
a
J

3

]
h Abstract
t
- Usingkinetictheory,weanalyzetheinterplayofelasticandinelasticcollisionsinthethermalizationofthequark-gluon
l
c plasma.Themainfocusisthedynamicsandequilibrationoflongwavelengthmodes.
u
n Keywords: quark-gluonplasma,thermalization,heavyioncollisions
[

1 1. Introduction
v
8 Understanding how the system of gluons that are created in the early stage of a heavy ion collision
0
evolvesintoalocallyequilibratedsystemremainsanimportantandchallengingproblem,withmanyinter-
3
esting facets and open issues (see Ref. [1, 2] for reviews – a more up-to-date discussion is presented by
0
0 A. Kurkela at this conference [3]). Sticking to weak coupling approaches, much of the basic physics has
. been identified over the last decade or so. This involves the plasma instabilities, and their potential role
1
in isotropizing the momentum distribution, the elastic and inelastic scatterings, and among the latter the
0
6 importanceofsoftradiation[4,5],theroleofthelongitudinalexpansion,etc. Thereexistdetailed[4],and
1 verydetailed[6],parametricanalysisatweakcouplingofthesevariousphysicalprocesses. Therearealso
: manynumericalcalculations,usingeitherstatisticalclassicalfieldtheoryorkineticequations.
v
i Thepresentcontributionreportsonworkbasedonkinetictheory. Wefeelthatthereisroom,asidefrom
X numericalcalculationsandparametricestimates,forunderstandingbasedon(differential)equationsthatwe
r cancontrol(semi)analytically. Ourgoalinthesestudiesistoidentifyrobust,genericqualitativebehaviors
a
thatemergefromsolutionsofsimplekineticequations. Wearewellawareofpotentiallimitationsofkinetic
theory in the present context, especially in the description of the very longwavelength modes. However,
the vast body of works on the thermalization of the quark-gluon plasma referring directly or indirectly to
kinetictheory,aswellastheinsightgainedbysimpleanalyticalsolutions,justifiesinourviewthepresent
exploration,andthepostponementofitsrigorousjustificationtoalaterstage.
2 J.-P.Blaizotetal./NuclearPhysicsA00(2016)1–4
2. Generalsetting
Wedescribethesystemofgluons1 thatisproducedintheinitialstageofanultra-relativisticheavyion
collisionusingkinetictheorywithaphase-spacedistributionfunction f(x,p)whichisindependentofspin
and color, uniform in space, and isotropic in momentum space2. The initial condition is inspired by the
colorglass(CGC)picture[7], f(p) = f θ(1− p/Q ),with Q thesaturationscale. Thefinalstateisgiven
0 s s
byaBosedistributionoftheform f(p) = [e(p−µeq)/Teq −1]−1,withTeq andµeq respectivelytheequilibrium
temperature and chemical potential. A non vanishing µ appears only when number changing processes
eq
can be neglected. Then µ can be either negative (underpopulation) or zero (overpopulation), with in the
eq
lattercaseformationofacondensate. Inallcaseswhereµ =0,theequilibriumdistributioniscompletely
eq
determinedbythetemperature,whichisitselffixedbytheinitialcondition,andenergyconservation.
For the chosen initial condition, the initial energy density (cid:15) = f Q4/(8π2) matches the equilibrium
in 0 s
value,(cid:15) =T4π2/30,forT = f1/4(15/4)1/4Q /π≈0.44f1/4Q .Forrealisticvaluesof f ,thistemperature
eq eq eq 0 s 0 s 0
is less than Q . It is only for a very large overpopulation, i.e., when f (cid:38) 26, that T (cid:38) Q . The initial
s 0 eq s
particlenumberisn = f Q3/(6π2), whileinequilibriumforµ = 0, n = ζ(3)T3/π2 ∝ f3/4. Theratio
in 0 s eq eq th 0
n /n =[f1/4π3]/[6ζ(3)(15/4)3/4]isunitywhen f = f ≈0.154.Dependingonwhether f > f or f < f ,
in eq 0 0 c 0 c 0 c
thenumberdensityhastodecreaseorincrease,respectively,inorderforthesystemtoreachequilibrium.
3. Elasticscatteringalone
TheapproachtoequilibriumwithelasticscatteringaloneisdescribedbytheBoltzmannequationinthe
smallangleapproximation,i.e.,aFokker-Planckequationoftheform
∂f(τ,p) 1 ∂ (cid:16) (cid:17) (cid:104) (cid:105)
=− p2J(τ,p) , J =− I ∂ f +I f(1+ f) . (1)
∂τ p2∂p a p b
IntheexpressionofthecurrentJ(τ,p),I andI arethefollowingintegrals
a b
(cid:90) (cid:90)
d3p d3p 2f(p)
I ≡ f(p)(1+ f(p)), I ≡ . (2)
a (2π)3 b (2π)3 p
(cid:82)
Thetimeτisrelatedtothephysicaltimet byτ ≡ 4πα2N2tLQ = 4π3α¯2 tLQ ,whereL = dq/qisthe
c s S
Coulomblogarithm,treatedhereasaconstant,andallmomentaareexpressedinunitsofQ .
s
Thermalizationproceedsthenwithconservedparticlenumber. Achemicalpotentialquicklydevelops,
and, atsmallmomenta, thesolutionacquirestheformofanequilibriumdistribution, f(p) (cid:39) T∗/(p−µ∗),
with T∗ and µ∗ time dependent parameters, with T∗ = I /I and µ∗ < 0. As time passes, the chemical
a b
potentialincreasesandeventuallyvanishesattheonsetofBosecondensation[8]. Beyondtheonset3 there
exists a solution[9] that behaves at small p as Θ/p, which allows for a flux F of particles at the origin,
F(p = 0) = (4πI /T∗)(T∗−Θ)Θ. At the onset of condensation, Θ = T∗. Just after onset, Θ becomes
a
rapidlylargerthanT∗,producinganegativefluxofparticlesattheorigin. ThisisillustratedinFig.1.
The non-local character of the kinetic equation Eq. (1) is worth-emphasizing: although it looks like a
localpartialdifferentialequationforthefunction f(τ,p),thereisinfactanon-linearcouplingwiththeentire
solutionthroughtheintegralsI andI . Theseintegralsencodeinparticulartheinformationofwhetherthe
a b
system is under or over populated. Just beyond the onset of condensation, however, I and I vary only
a b
mildly, while the system is rapidly driven out of equilibrium, a behavior largely determined then by local
properties.
1Weignorequarksinthepresentdiscussion.
2Weassumethatisotropizationtakesplaceovershortertimescalesthanthoseinvolvedintheprocessesthatwediscussinthis
paper.
3Equationsthatgoverntheevolutionofthecondensateinthesmallangleapproximation[10],includingfinitemasseffects[11],
haverecentlybeenobtained.However,thesimpleprocedurethatwefollowhereissufficientforthepresentdiscussion.
J.-P.Blaizotetal./NuclearPhysicsA00(2016)1–4 3
1.0
pΘ0.9
o
sl
IR0.8
&
II/ab0.7
0.6
5 10 15 20
τ
Fig.1. (Coloronline). The(blue)lowercurverepresentstheeffectivetemperatureT∗ =Ia/Ibasafunctionoftime. The(red)upper
curverepresentsthecoefficientΘ. Beforeonset, thedistributionfunctionisregularattheoriginandΘ = 0. Atonset, Θ = T∗.
Aftertheonsetofcondensation,aflowofparticlesquicklydevelopsattheoriginofmomentumspaceandthesystemisrapidlydriven
awayfromlocalequilibrium,withΘbecomingsignificantlylargerthanthelocaleffectivetemperatureT∗=Ia/Ibwhichstaysalmost
constantacrossthetransition. Atlatertimes,ΘdecreasesandeventuallyconvergesbacktoT∗. Thecalculationisdoneformoderate
overpopulation, f0=1.As f0increasesmoreandmoreparticlestendtocondense,andasaresultthegrowthofthecoefficientΘjust
beyondonsetismoreimportant.
4. Inelasticscatteringalone
Thermalizationcanalsobeachievedbyinelasticprocessesalone,whicharedominatedbytheemission
or the absorption of soft gluons. To treat those processes, we exploit recent progress in their analytical
understandinginourworkonmediuminducedcascades[12]. Wedescribethosewithageneralizationofan
equationthathasbeenstudiedin[12],thatusesasimplifiedsplittingkernel,andthatcanbebroughttothe
form4
RT∗ (cid:40)(cid:90) ∞ (cid:90) p (cid:41) p(cid:48)3
∂ f(p,τ)= dkK(p,p+k)Φ(p,p+k)− dkK(k,p)Φ(k,p) , K(p,p(cid:48))= , (3)
τ p3 p(cid:48)−p
0 0
whereΦ(p,p(cid:48)) ≡ f(p(cid:48))+ f(p(cid:48))f(p)+ f(p(cid:48)− p)(cid:2)f(p(cid:48))− f(p)(cid:3).ThequantityRisaparameter,generically
oforderunity[13],thatwekeepasafreeparametertocontroltherelativestrengthofelasticandinelastic
scatterings. Asimpleanalysisrevealsthatinthesmall pregion,thisequationreducesapproximatelyto
∂ f(p)≈ RIa (cid:2)T∗−pf(p)(cid:3), (4)
τ p3
whosesolutioniseasilyobtained. WhenoneignoresthetimedependenceofI andT∗,itreads
a
(cid:32) (cid:33)
T∗ T∗ I
f(p,τ)= p + f0− p e−P2∗(τ)/p2, P2∗(τ)=RIaτ, T∗ = Ia. (5)
b
One sees that the small momentum behavior is controlled by an essential singularity, which implies that
thefixedpoint∼ 1/pisreachedinnotime. Astimeprogressesthefixedpointsolutionispopulatedviaa
diffusionwavecenteredatP (τ),asillustratedinFig.2.
∗
Thisisincompleteconsistencyfromwhatoneobtainsfromexactlysolvingthefullkernelnumerically.
5. Interplayofelasticandinelasticscattering
Common to both types of processes is the rapid growth of the low momentum modes, and in the case
of overpopulated systems, the disappearance of particles at low momenta where the distribution function
behaves approximately as ∼ 1/p. However the way this is achieved is different in the elastic and the
inelasticcases. Intheelasticcase,the1/pbehaviordevelopsgradually,viathebuildingofastrongcurrent
of particles towards small momenta, leading eventually to Bose condensation that eliminates the particles
4ItisalsoasimplifiedversionoftheequationusedinRef.[14].
4 J.-P.Blaizotetal./NuclearPhysicsA00(2016)1–4
1000
100
f 10
1
0.1
0.001 0.005 0.010 0.050 0.100
p
Fig.2. (Coloronline). Thesolution f(p,τ)ofEq.(5)atsmallmomentaandshorttimes. ThefixedpointsolutionT∗/pisindicated
√
bythebluedashedline. Thewavefrontofthediffusionwave(centeredatP∗ ∼ τ)hastheshapeofatransientradiationspectrum
∼1/p−3(yellowdashedline).Thehorizontallineindicatestheinitialcondition f0=1.
that cannot be accomodated in the thermal spectrum. In the inelastic cases, the population of soft modes
is driven by radiation, which leads to a rapidly growing spectrum ∼ 1/p3, forcing local equilibrium to be
reachedinstantlyatsmall p.
Whenbothprocessesarepresent,itappearsthatthesmallpdynamicsremaindominatedbytheinelastic
ones. Thefastradiationofsoftgluons, andtheaccompanyingabsorption, forcethedistributiontobehave
as1/pinnotime. OnemightthinkthatthiscouldfacilitateBosecondensation[13],assuchabehaviorisa
prerequisite. However,theveryreasonsthattheinelasticprocessesforcelocalequilibriumnearp=0imply
a fast suppression of the excess particles, thereby hindering condensation, making it in fact unnecessary.
What is involved here is not so much the fact that the inelastic processes may be globally of comparable
magnitude as the inelastic ones (i.e. R ≈ 1). The major factor is radiation, which allows transport of
momentaoverawiderangeofmomentaandonaveryshorttimescale. Thisisaspecificfeatureofgauge
theories,thatdonotshowupinscalartheoriesforinstance.
The present note focussed on what happens near p = 0, and how the particles in excess in the initial
distributionareeliminated. Thedetailedpatternoftheevolutiontowardsequilibriumdependsonthevalue
of R. For instance if R is small, the bulk of the dynamics remains the same as in the absence of inelastic
processes, the influence of inelastic processes being significant only in the small region p < P . The
∗
descriptionofwhathappensfordifferentvaluesofRwillbepresentedinaforthcomingpublicaiton.
Acknowledgements
The research of JPB is supported by the European Research Council under the Advanced Investigator
GrantERC-AD-267258. TheresearchofJLispartlysupportedbytheNSF(GrantNo. PHY-1352368)and
bytheRIKENBNLResearchCenter
References
[1] J.Berges,J.P.BlaizotandF.Gelis,J.Phys.G39(2012)085115doi:10.1088/0954-3899/39/8/085115[arXiv:1203.2042[hep-
ph]].
[2] X.G.HuangandJ.Liao,Int.J.Mod.Phys.E23,1430003(2014)doi:10.1142/S0218301314300033[arXiv:1402.5578[nucl-th]].
[3] A.Kurkela,theseproceedings.
[4] R.Baier,A.H.Mueller,D.SchiffandD.T.Son,Phys.Lett.B502,51(2001)[arXiv:hep-ph/0009237].
[5] Z.XuandC.Greiner,Phys.Rev.C71(2005)064901doi:10.1103/PhysRevC.71.064901[hep-ph/0406278].
[6] A.KurkelaandG.D.Moore,JHEP1112(2011)044[arXiv:1107.5050[hep-ph]].
[7] J.P.Blaizot,F.Gelis,J.F.Liao,L.McLerranandR.Venugopalan,Nucl.Phys.A873(2012)68[arXiv:1107.5296[hep-ph]].
[8] J.P.Blaizot,J.LiaoandL.McLerran,Nucl.Phys.A920(2013)58[arXiv:1305.2119[hep-ph]].
[9] J.P.Blaizot,B.WuandL.Yan,Nucl.Phys.A930(2014)139[arXiv:1402.5049[hep-ph]].
[10] J.P.BlaizotandJ.Liao,arXiv:1503.07263[hep-ph].
[11] J.P.Blaizot,Y.JiangandJ.Liao,arXiv:1503.07260[hep-ph];
[12] J.P.BlaizotandY.Mehtar-Tani,arXiv:1501.03443[hep-ph].
[13] X.G.HuangandJ.Liao,Phys.Rev.D91(2015)11,116012[arXiv:1303.7214[nucl-th]].
[14] A.KurkelaandE.Lu,Phys.Rev.Lett.113(2014)18,182301[arXiv:1405.6318[hep-ph]].