The role of the gluonic gg<->ggg interactions in early thermalization in ultrarelativistic heavy-ion collisions

0.17 MB·English

by  Zhe Xu

EPJ manuscript No.
(will be inserted by the editor)
The role of the gluonic gg ↔ ggg interactions in early
thermalization in ultrarelativistic heavy-ion collisions
Zhe Xu1 a and Carsten Greiner1
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Institut fu¨r Theoretische Physik, Johann Wolfgang Goethe-Universit¨at Frankfurt,Germany
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Received: date/ Revised version: date
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Abstract. We “quantify” the role of elastic as well as inelastic gg ↔ ggg pQCD processes in kinetic
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equilibration within a pQCD inspired parton cascade. The contributions of different processes to kinetic
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equilibration are manifested by the transport collision rates. We find that in a central Au+Au collision
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at RHIC energy pQCD Bremstrahlung processes are much more efficient for momentum isotropization

compared to elastic scatterings. For the parameters chosen the ratio of their transport collision rates
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v amounts to 5:1.
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4 PACS. 05.60.-k Transport processes – 25.75.-q Relativistic heavy-ion collisions – 24.10.Lx Monte Carlo
1 simulations
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7 1 Introduction itstransportcrosssectionisonlythesameasthatofelas-
0 tic scatterings.Therefore one cannot understand why the
/
h It was speculated that a strongly coupled quark-gluon inclusion of the inelastic processes brings so much to ki-
p plasma(sQGP)[1]isformedinAu+AucollisionsatRHIC. netic equilibration. At first sight it seems to be an un-
- This strong coupling, or strong interaction, makes the solvableproblem.Ontheotherhand,however,thereisno
p
QGP to a fluid with very small viscosity. However, how reason to believe that momentum isotropization should
e
h strong the coupling must be in order to generate a quasi- relate to the angular distribution by means of the trans-
: ideal fluid is an open question. port cross section and not by another formula. The con-
v
Recentlywehavedevelopedanon-shellpartoncascade ceptofthetransportcrosssectionmaybemoreintuitively
i
X including elastic as well as inelastic gg ↔ggg pQCD pro- than mathematically correct. In this work we will find a
r cesses to study the issue of thermalization [2]. Although mathematically correct way to quantify the contribution
a thetotalcrosssectionofthepQCDscatteringsisafewmb, of different processes to thermal equilibration and com-
it is enough to drive the system into thermal equilibrium parethemwitheachother.Thecoreissueisthe transport
and also to generate a large elliptic flow v in noncentral collision rate. With this quantity we “quantify” the role
2
Au+Au collisions [3]. of different collision processes in kinetic equilibration.
Since with elastic pQCD scatterings alone no ther-
malization is expected to be achieved [2,4], the inelastic
gg ↔ ggg processes seem to play a leading role in early
thermalization, although for the parameters chosen in [2]
2 Parton cascade
the cross section of gg → ggg collisions is a factor of 2
smallerthanthatofelasticscatterings.Inorderto under-
stand this a transport cross section,
dσ Thebuildupofthepartoncascadeisbasedonthestochas-
σtr. = dθ sin2θ, (1) tic interpretation of the transition rate. This guarantees
dθ
Z detailedbalance,whichis,bycontrast,difficultwhenusing
wasintroducedasapertinentquantitymeasuringthecon- the geometricalconceptof the crosssection[6], especially
tributions of different collision processes to kinetic equili- formultiple scatteringslikeggg →gg.The particularfea-
bration [5], since large-angle collisions should contribute ture of the numerical implementation in the parton cas-
more to momentum isotropization. The results (see Fig. cade is the subdivision of space into small cell units. In
48 in [2]) showed that even due to the almost isotropic cells the transitionprobabilities areevaluated for random
distribution of the collision angles in inelastic collisions sampling whether a particular scattering occurs or not.
The smaller the cells, the more locally transitions will be
a [email protected] realized.
2 ZheXu, Carsten Greiner: Therole of the gluonic gg↔ggg interactions in early thermalization
The three-body gluonic interactions are described by simulation with pure elastic processes starting with the
the matrix element [7] same initial conditions. From Fig. 1 we see that while the
gluon system is still far from kinetic equilibrium in the
|Mgg→ggg|2 = 92g4(q2⊥+s2m2D)2 k2⊥[(k⊥1−2gq2⊥q)2⊥2+m2D] × sthimeumlaotmioenntwuimthapnuisroetreolpaystricelascxaetstteorinthges v(adlauseheadt ecquurivlieb)-,
rium, 1/3, in the simulation including inelastic processes
×Θ(k⊥Λg−coshy). (2)
(solid curve).
The suppressionof the radiationof soft gluons due to the Wefitthetimeevolutionofthemomentumanisotropy
Landau-Pomeranchuk-Migdal(LPM) effect[8,9,2],which using the standard relaxation formula
is expressed via the step function in Eq. (2), is mod-
1 p2 1 t−t
eled by the consideration that the time of the emission, F(t)= + Z (t )− exp(− 0). (3)
∼ 1 coshy, should be smaller than the time interval be- 3 (cid:18)(cid:28)E2(cid:29) 0 3(cid:19) θ(t0)
k⊥
tweentwo scatterings orequivalently the gluonmeanfree Forsimplicity we labelnow the momentum anisotropyby
path Λg. This leads to a lower cutoff for k⊥ and to a de- Q:=<p2Z/E2 >. F(t) is only equal to Q(t) at t=t0. For
crease of the total cross section. fixed t the relaxation time θ is a constant with respect
0
In this work we simulate the time evolution of gluons to t. Such a fit can be done at every time point t . The
0
produced in a central Au+Au collision at RHIC energy. twothin dottedcurvesinFig.1arefits withθ =0.9fm/c
The initial gluons are taken as minijets with transverse at t = 0.3 fm/c and θ = 2.4 fm/c at t = 1.2 fm/c. We
0 0
momentum being greater than 1.4 GeV, which are pro- find that an isotropic state is achieved at about 1.0 fm/c
duced via semi-hardnucleon-nucleoncollisions.Using the in the simulation including inelastic scattering processes.
Glaubergeometrythe gluonnumberisinitiallyabout700 Moreover, we see that the relaxation time θ is generally
per momentum rapidity. These gluons take about 60% of time dependent. The two values of θ in the fits are ob-
thetotalenergyenteredinthecollision.Choosingsuchan tained by guessing. Actually θ can be calculated exactly,
initial condition and performing a simulation including since in order to make a local fit one should request that
Bremsstrahlung processes we obtain dET/dy about 640 the time derivative of F(t) and Q(t) are equal at t = t0.
GeV at midrapidity at a final time of 5 fm/c, at which This leads to
theenergydensityofgluonsdecreasestothecriticalvalue
of 1 GeV/fm3. The value of dET/dy obtained from the Q˙(t) = F˙(t) =−(Q(t0)−Qeq.) 1 , (4)
simulationiscomparablewiththatfromtheexperimental t=t0 t=t0 θ(t0)
(cid:12) (cid:12)
measurements at RHIC. (cid:12) (cid:12)
We concentrate on the central region: 0 < x < 1.5 with Qeq(cid:12). =1/3. Cha(cid:12)nging t0 to t gives
T
fm and −0.2 < η < 0.2, where η denotes the space-time Q˙(t) 1
rapidity. Results which will be shown below are obtained = . (5)
Q −Q(t) θ(t)
in this region by ensemble average. eq.
TheimportanceofincludinginelasticpQCDgg ↔ggg Equation (5) expresses the relaxationrate 1/θ of the mo-
processes to momentum isotropization is clearly demon- mentum anisotropy. In the next section we analytically
stratedinFig.1,wherethetimeevolutionoftheaveraged separate the relaxation rate into different terms corre-
momentum anisotropy, < p2Z/E2 >, is depicted. pZ and spondingparticlediffusionandvariousscatteringprocesses,
and we define the transport collision rate which quanti-
fies the contribution of a certain process to momentum

0.35 isotropization.
0.30
3 Transport collision rate
> 0.25
2
E
2 / Z 0.20 Forevaluatingthemomentumanisotropyatacertainspace
<p 0.15 point one has to go to its co-moving frame, in which we
have
0.10
elastic+inelastic p2 1 d3p p2
0.05 pure elastic Q(t)= Z = Z f(p,x=0,t). (6)
fit E2 n (2π)3E2
0.00 (cid:28) (cid:29)(cid:12)(cid:12)x=0 Z
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Taking the deriva(cid:12)tive in time gives
(cid:12)
t [fm/c]
1 d3p p2 ∂f 1 d3p ∂f
Fig. 1. Momentum anisotropy. Q˙(t)= Z −Q(t) . (7)
n (2π)3E2 ∂t n (2π)3 ∂t
Z Z
We replace ∂f/∂t by
E are,respectively,longitudinalmomentumandenergyof
a gluon. The average is computed over all gluons in the ∂f p
central region. As comparison we have also performed a =− ∇f +C22+C23+C32 (8)
∂t E
ZheXu, Carsten Greiner: Therole of the gluonic gg↔ggg interactions in early thermalization 3
adcecnoortdei,nrgestpoectthieveBlyo,lttzhmeacnonlliseiqounatteiornm. oCf22g,gC→23gagn,dggC→32 σ˜23 := 21s31! dΓ1′dΓ2′dΓ3′ pE′12′Z2 |M12→1′2′3′|2×
gtrgigbuatniodngogfgd→iffergegntprporcoecsess.seIst tiso oQ˙bviisouasddtihtaivte.thEexccoepnt- ×(2πZ)4δ(4)(p1+p2−1p′1−p′2−p′3) (16)
fhoarsacosntatrtiibcustyisotnemtotQh˙e(td),iffwuhsiicohnwteermdeninotEeqb.y(8W) gene.rCally I32 := 21! dΓ1′dΓ2′|M123→1′2′|2×
diff. 22 Z
has no contribution to the second integral in Eq. (7) due ×(2π)4δ(4)(p +p +p −p′ −p′) (17)
to particle number conservation. The same is also for the 1 2 3 1 2
Esuqm. (o7f)Cto23 and C32 at chemical equilibrium. We rewrite I˜32 := 21! dΓ1′dΓ2′ pE′12′Z2 |M123→1′2′|2×
Z 1
×(2π)4δ(4)(p +p +p −p′ −p′) (18)
Q˙(t)=W (t)+W (t)+W (t)+W (t), (9) 1 2 3 1 2
diff. 22 23 32
with dΓ = d3p /(2π)32E for short. σ and σ denote
i i i 22 23
where W22, W23 and W32 correspond to the gg → gg, the standard pQCD cross section of the gg → gg and
gg → ggg and ggg → gg process, respectively. According gg → ggg process, respectively. v = s/2E E is the
rel 1 2
to (5) we then obtain relative velocity. <> and <> symbolize, respectively,
2 3
an ensemble average over pairs and triplets of incoming
1
=Rtr. (t)+Rtr.(t)+Rtr.(t)+Rtr.(t), (10) particles. In the parton cascade simulations f(p,x,t) ≈
θ(t) diff 22 23 32 δ(3)(p−p )δ(3)(x−x (t)), and we can approximately
i i i
evaluate the averages <> and <> in local cells which
2 3
where we define hPave small volume, but a sufficient number of (test) par-
ticles to achieve high statistics.
W (t)
Rtr.(t):= i . (11) The expression of W s in (12), (13) and (14) indi-
i Q −Q(t) i
eq. catesthe differenceofthe gainandlossinthe momentum
isotropization within one collision. The influence of the
We see that the relaxation rate of kinetic equilibration,
distributionofthecollisionangleonmomentumisotropiza-
1/θ,isseparatedintoadditivepartscorrespondingtopar-
tion is implicitly contained. However, the expression of
ticle diffusion and collision processes. Rtr., Rtr. and Rtr.
22 23 32 the transport collision rate Rtr. is clearly different from
stand for the transport collision rates of the respective i
n < v σtr. > by the formula (1). (The index i denotes
interactions and quantify their contributions to kinetic rel i 2
22 or 23.) Only in the special case that all particles are
equilibration.Theextensiontomorethanthree-bodypro-
movingalongtheZ−axis(irrespectiveof±sign)andhave
cesses is straightforwardbecause the collision term is ad-
the same energy E,
ditive. We note that the definition of Rtr. in Eq. (11) de-
i
pends on which momentum anisotropy we are looking at. f(p,x,t)∝δ(p )δ(p )δ(p −E)+δ(p )δ(p )δ(p +E),
X Y Z X Y Z
If one defines < |pZ|/E > as the momentum anisotropy (19)
for instance, the form Ritr. will change accordingly. the lab frame is the CM system for every colliding pair.
Putting the explicit expression of the collision term In this case p′2 /E′2 =cos2θ∗ and
1Z 1
via the matrix element of transition into (8), we obtain
explicit expressions for W , which are summarized in the Rtr. ∼n<v σtr. > . (20)
i i rel i 2
following:
This result does not depend on the chosen direction of
p2 initialmomentum.Theonlynecessaryconditionsarethat
W22(t)= n<vrelσ˜22 >2 −n vrel E1Z2 σ22 , (12) all particles move along the same direction and have the
(cid:28) 1 (cid:29)2 same energy.If we define the momentum anisotropy as <
W (t)= 3n<v σ˜ > −n v p21Z σ |pZ|/E >,Ritr. maintainsitsform(20),butσitr. ischanged
23 2 rel 23 2 rel E2 23 to
(cid:28) 1 (cid:29)2 dσ
1 σtr. = dθ∗ i(1−cosθ∗). (21)
− Q(t)n<vrelσ23 >2, (13) i dθ∗
2 Z
1 I˜ 1 p2 I Figure 2 shows the transportrates Ritr. obtained from
W (t)= n2 32 − n2 1Z 32 simulationsemployingthepartoncascade.Thesolid,dashed
32 3 8E E E 2 E2 8E E E
* 1 2 3+3 (cid:28) 1 1 2 3(cid:29)3 anddash-dottedcurvesdepict,respectively,the transport
+1Q(t)n2 I32 , (14) ctioolnliswiointhrabtoetshRe2trl2a.,stRic2tr3.ananddinRe3tlra2.sctiaclccuollaltiseidonins.tWheesrimeaulilzae-
6 8E E E
(cid:28) 1 2 3(cid:29)3 the dominance of inelastic collisions in kinetic equilibra-
tion by computing the ratio (Rtr. +Rtr.)/Rtr. ≈ 5. The
where 23 32 22
thindotted curveinFig.2 presentsRtr. in the simulation
22
σ˜22 := 21s21! dΓ1′dΓ2′ pE′12′Z2 |M12→1′2′|2× wsoiltihdpcuurreveeloansteiccapnroncoetsrseeas.liWzehmenucchomdipffaerrienngcteh.iswiththe
Z 1 The contribution of particle diffusion to kinetic equi-
×(2π)4δ(4)(p +p −p′ −p′) (15) libration, Rtr. , calculated in the simulation with both
1 2 1 2 diff.
4 ZheXu, Carsten Greiner: Therole of the gluonic gg↔ggg interactions in early thermalization

2.5 1.0
gg gg (elastic+inelastic)
gg ggg (elastic+inelastic) gg gg (elastic+inelastic)
-1 m] 2.0 gdggigffgusgioggng * (((ep-1lua)rset iecl+aisnteicla)stic) -1 m] 0.8 gggg ggggg ((epluarset iec+laisnteicla) stic)
e [f 1.5 > [f 0.6
rat tr.
port 1.0 < vrel 0.4
ns n
ra 0.5 0.2
t
0.0
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
time [fm/c] t [fm/c]
Fig. 2. Transport rate. Fig. 3. “Transport collision rate” in the concept of transport
cross section.
elastic andinelasticprocesses,is depicted inFig.2 by the
symbols multiplied by −1. Rtr. , which is expressed by acomparisonisnecessaryforunderstandingwhythe con-
diff. cept of transport cross section cannot explain the strong
effectwhenincludinginelasticscatteringprocesses.InFig.
1 1 d3p p p2
Rdtri.ff.(t)= Q −Q(t)n (2π)3E ·∇f Q(t)− EZ2 , 3 we depict n < vrelσtr. >2 calculated from the cascade
eq. Z (cid:18) (cid:19) simulations. At first we look at the results in the simula-
(22)
tionwithbothelasticandinelasticcollisionsandcalculate
is not computed at a certain time as performed for the
theinelastictoelasticratio(dashedversussolidcurve).It
transport collision rates, because of the inaccurate ex-
is almost a constant around 1.7 in time, which indicates
traction of ∇f. The diffusion rate is obtained by explicit
the dominance of the gg → ggg scattering processes in
countingoftheparticleswhichcomeinaswellasgooutof
kinetic equilibration. Comparing Fig. 3 to Fig. 2 we real-
thecentralregionwithinatimeinterval.Althoughtheex-
ize that the results concerning elastic scatterings have no
tractionofRdtri.ff. hasamuchlargerstatisticalfluctuation, strong difference. On the contrary, Rtr. is a factor of 2.5
23
thesumRtr. +Rtr.+Rtr.+Rtr. givesavalueconsistent largerthann<v σtr. > .ThepQCDgg →ggg process
diff. 22 23 32 rel 23 2
with the relaxationrate 1/θ(t) as one would calculate via is much more efficient for kinetic equilibration than one
Q˙(t)/(Q −Q(t)) directly from Fig. 1. would expect via the transport cross section.
eq.
We also see that Rtr. has a negative contribution to
diff.
momentum isotropization and is quite large. We did not
plot −Rdtri.ff. from the simulation with pure elastic pro- 4 Bremsstrahlung process and the LPM effect
cesses.It is slightly smaller than Rtr. (thin dotted curve).
22
Weseethatthereisabigdifferenceinthediffusionratein
It is intuitively clear that a 2 → 3 process will bring one
bothsimulations.TounderstandthisweassumeBjorken’s
moreparticle towardsisotropythana 2→2 process.The
space-time picture of central ultrarelativistic heavy-ion
kinematic factor should be 3/2 and appears in Eq. (13)
collisions [10] and use the relation derived by Baym [11]:
when assuming the decompositions
p p ∂f p ∂f
·∇f ≈ Z =− Z . (23) p2
E E ∂Z t ∂pZ vrel E1Z2 σ ≈Q(t)<vrelσ >2 . (25)
(cid:28) 1 (cid:29)2
Inserting Eq. (23) into Eq. (22) and calculating partial
integrals give AnalogouslywecompareW23 toW32.Thesumofthelast
terminEq.(13)andEq.(14)comesfromthesecondterm
−2 p4 in Eq. (7) with C +C instead of ∂f/∂tand should be
Rtr. (t)≈ Q(t)− Z (t) . (24) 23 32
diff. (Q −Q(t))t E4 zero at chemical equilibrium: we obtain
eq. (cid:18) (cid:28) (cid:29) (cid:19)
TheformulaconfirmsthatRdtri.ff.isalwaysnegative.Using n<vrelσ23 >2= 1n2 I32 , (26)
the approximation < p4Z/E4 >≈ Q2 one can also realize 3 (cid:28)8E1E2E3(cid:29)3
that the larger is Q, the larger −Rtr. is.
diff.
We have seen that only in an extreme case the trans- orequivalentlyR23 = 32R32.Assumingfurtherthedecom-
port collision rate can be reduced to a formula directly position
proportional to the transport cross section: Rtr. ∼ n <
v σtr. > . It is interesting to know how the calculated p2 I I
rel 2 1Z 32 ≈Q(t) 32 , (27)
transport collision rates differ from n<vrelσtr. >2. Such (cid:28)E12 8E1E2E3(cid:29)3 (cid:28)8E1E2E3(cid:29)3
ZheXu, Carsten Greiner: Therole of the gluonic gg↔ggg interactions in early thermalization 5
we have at RHIC energy the old concept of the transport cross
sections would strongly underestimate the contribution
3
W23(t)≈ (n<vrelσ˜23 >2 −Q(t)n<vrelσ23 >2), of gg → ggg to kinetic equilibration. The correct results
2
showed that the inclusion of pQCD Bremsstrahlung pro-
W (t)≈ 1n2 I˜32 −Q(t)1n2 I32 . cesses increases the efficiency by a factor of 5 for ther-
32 3 8E E E 3 8E E E malization. The large efficiency stems partly from the in-
* 1 2 3+3 (cid:28) 1 2 3(cid:29)3 crease of particle number in the final state of gg → ggg
TheseapproximateexpansionstogetherwithEq.(26)lead collisions, but mainly from the almost isotropic angular
toW ≈ 3W andRtr. ≈ 3Rtr. atchemicalequilibrium. distribution in Bremsstrahlung processes due to the ef-
23 2 32 23 2 32
Aloneduetothe kinematicreasona2→3processis50% fective implementation of LPM suppression.The detailed
more efficient in kinetic equilibration than a 2 → 2 or a understanding of the latter has to be developed in future
3→2 process, when σ =σ and σ˜ =σ˜ . investigations.
22 23 22 23
The LPM effect stems from the interference of the ra-
diated gluons (originally photons in the QED medium)
bymultiple scatteringofapartonthoughamedium.This References
is a coherent effect which leads to suppression of radia-
tion of gluons with certain modes (w,k). w and k denote 1. Miklos Gyulassy and Larry McLerran, Nucl.Phys.A 750,
energy and momentum of a gluon respectively. Heuristi- (2005) 30-63.
cally there is no suppression for gluons with a formation 2. Zhe Xu and Carsten Greiner, Phys.Rev.C 71, (2005)
time τ = w/k2 smaller than the mean free path. This 064901.
T
is called the Bethe-Heitler limit, where the gluon radia- 3. ZheXuandCarstenGreiner,ProceedingsofQuarkMatter
tionsinducedatadifferentspace-timepointinthe course 2006, hep-ph/0509324.
of the propagation of a parton can be considered as in- 4. J.SerreauandD.Schiff,J.HighEnergyPhys.0111,(2001)
dependent events. These events within the Bethe-Heitler 039.
regime have been included in the parton cascade calcula- 5. P.DanielewiczandM.Gyulassy,Phys.Rev.D31,(1985)53.
tions. Radiation of other gluon modes with coherent sup- 6. D.Molnar,ProceedingsofQuarkMatter1999,Nucl.Phys.A
pression completely dropped out, which is indicated by 661, (1999) 205-260.
7. J.F. Gunion and G. Bertsch, Phys.Rev.D 25, (1982) 746.
the Θ-function in the matrix element (2). The inclusion
8. T.S. Biro et al, Phys.Rev.C 48, (1993) 1275.
of those radiations would speed up thermalization. How
9. S.M.H. Wong, Nucl.Phys.A 607, (1996) 442.
to implement the coherent effect into a transport model
10. J.D. Bjorken, Phys.Rev.D 27, (1983) 140.
solving the Boltzmann equation is a challenge.
11. G. Baym, Phys.Lett.B 138, (1984) 18.
The Θ-function in the matrix element (2) results in
a cut-off for k , the transverse momentum of a radiated
T
gluon, k > 1/Λ , where Λ is the mean free path of a
T g g
gluon.A highervalue ofthe cut-offwilldecreasethe total
cross section of a gg →ggg collision on the one hand and
make the collision angles large on the other hand. The
latterleadstoalargeefficiencyformomentumisotropiza-
tion.Varyingthecut-offdownwardstoasmallervalueone
would enter into the LPM suppressed regime. Although
parton cascade calculations set up with smaller cut-offs
cannot completely take the LPM effect into account, one
can roughly estimate the contribution of the coherent ef-
fecttokineticequilibration.Suchcalculationswillbedone
in a subsequent paper.
5 Conclusion
Employing the parton cascade we have investigated the
importance of including pQCD Bremsstrahlung processes
tothermalization.Thequestionaddressedishowtounder-
standthe observedfast equilibrationin theoreticalterms.
The special emphasis is put on expressing the transport
collision rate in a correct manner. The concept of trans-
port cross section only gives a qualitative understand-
ing of the dominant contribution of large-angle scatter-
ingstomomentumisotropizationbutnotacorrectwayto
manifest the various contributions. In case we are study-
ing parton thermalization in a central Au+Au collision