Estimation of isotropization time ($τ_{\rm iso}$) of QGP from direct photons

0.19 MB·English

by  Lusaka Bhattacharya

NuclearPhysicsA00(2011)1–4
Estimation of isotropization time (τ ) of QGP from direct photons
iso
LusakaBhattacharya
SahaInstituteofNuclearPhysics
1/AFBidhannagar,Kolkata,Pin700064,India
1
1
0
2

n Abstract
a
Wecalculatetransversemomentumdistributionofdirectphotonsfromvarioussourcesbytakingintoaccountthe
J
initialstatemomentumanisotropyofquarkgluonplasma(QGP).Thetotalphotonyieldisthencomparedwiththere-
1
3 centmeasurementofphotontransversemomentumdistributionbythePHENIXcollaboration. Itisalsodemonstrated
that the presence of such an anisotropy can describe the PHENIX photon data better than the isotropic case in the

] presentmodel. Weshowthattheisotropizationtimethusextractedlieswithintherange1.5≥τ ≥0.5fm/cforthe
h iso
p initialconditionusedhere.
-
p Keywords: anisotropy,QGP
e
h
[
1. Introduction

1
Theprimarygoalofrelativisticheavyioncollisionsistocreateanewstateofmatter,calledquarkgluonplasma
v
7 and to study its properties through various indirect probes. Out of all the properties of the QGP, the most difficult
5 problemliesinthedeterminationofisotropizationandthermalizationtimescales(τ andτ ). Studiesonelliptic
iso therm
8 flow(uptoabout p ∼ 1.5−2GeV)usingidealhydrodynamicsindicatethatthematterproducedinsuchcollisions
T
5 becomesisotropicwithτ ∼0.6fm/c(1).Ontheotherhand,usingsecondordertransportcoefficientswithconformal
. iso
1 symmetryitisfoundthattheisotropization/thermalizationtimehassizableuncertainties(2). Consequently,thereare
0 uncertainties in the initial temperature as well. Electromagnetic probes have been proposed to be one of the most
1
promisingtoolstocharacterizetheinitialstateofthecollisions(3,4). Becauseoftheverynatureoftheirinteractions
1
withtheconstituentsofthesystemtheytendtoleavethesystemwithoutmuchchangeoftheirenergyandmomentum.
:
v
Infact, photons(dileptonaswell)canbeusedtodeterminetheinitialtemperature, orequivalentlytheequilibration
i
X time.
r ItistobenotedthatwhileestimatingphotonsfromQGP(5,6,7),itisassumedthatthematterformedintherela-
a tivisticheavyioncollisionsisinthermalequilibrium. Themeasurementofellipticflowparameteranditstheoretical
explanationalsosupportthisassumption. Onthecontrary,perturbativeestimationsuggeststheslowerthermalization
ofQGP(8).However,recenthydrodynamicalstudies(2)haveshownthatduetothepoorknowledgeoftheinitialcon-
ditionsthereisasizableamountofuncertaintyintheestimateofthermalizationorisotropizationtime. Inviewofthe
absenceofatheoreticalproofbehindtherapidthermalizationandtheuncertaintiesinthehydrodynamicalfitsofex-
perimentaldata,suchanassumptionmaynotbejustified.Henceinsteadofequatingthethermalization/isotropization
timetotheQGPformationtime,inthiswork,wewillintroduceanintermediatetimescale(isotropizationtime,τ )to
iso
studytheeffectsofearlytimemomentum-spaceanisotropyonthetotalphotonyieldandcompareitwiththePHENIX
photondata(9,10,11). Recently,ithasbeenshowninRef.(12)thatforfixedinitialconditions,theintroductionof
apre-equilibriummomentum-spaceanisotropyenhanceshighenergydileptonsbyanorderofmagnitude. Incaseof
photontransversemomentumdistributionsimilarresultshavebeenreportedforvariousevolutionscenarios(13).
LusakaBhattacharyaetal./NuclearPhysicsA00(2011)1–4 2
Theplanofthepaperisthefollowing. Inthenextsectionwewilldiscussthemechanismsofphotonproduction
fromvariouspossiblesourcesandthespace-timeevolutionofthematterverybriefly. Section3isdevotedtodescribe
theresultsforvariousinitialconditionsandwesummarizeinsection4.
2. Formalism
2.1. Photonrate: AnisotropicQGP
ThelowestorderprocessesforphotonemissionfromQGParetheCompton(q(q¯)g → q(q¯)γ)andtheannihilation
(qq¯ → gγ) processes. The rate of photon production from anisotropic plasma due to these processes has been
calculatedinRef.(14). Thesoftcontributioniscalculatedbyevaluatingthephotonpolarizationtensorforanoblate
√
momentum-space anisotropy of the system where the cut-off scale is fixed at k ∼ gp . Here p is a hard-
c hard hard
momentumscalethatappearsinthedistributionfunctions. Thedifferentialphotonproductionratefor1+2→3+γ
processesinananisotropicmediumisgivenby(14):
(cid:90)
dN N d3p d3p d3p
E = 1 2 3 f (p ,p ,ξ)f (p ,p ,ξ)
d4xd3p 2(2π)3 2E (2π)32E (2π)32E (2π)3 1 1 hard 2 2 hard
1 2 3
× (2π)4δ(p +p −p −p)|M|2[1± f (p ,p ,ξ)] (1)
1 2 3 3 3 hard
where,|M|2 representsthespinaveragedmatrixelementsquaredforoneofthoseprocesseswhichcontributestothe
photonrateandN isthedegeneracyfactorofthecorrespondingprocess. ξisaparametercontrollingthestrengthof
theanisotropywithξ > −1. f , f and f aretheanisotropicdistributionfunctionsofthemediumpartons. Hereitis
1 2 3
assumedthattheinfraredsingularitiescanbeshieldedbythethermalmassesfortheparticipatingpartons. Thisisa
goodapproximationatshorttimescomparedtothetimescalewhenplasmainstabilitiesstarttoplayanimportantrole.
Theanisotropicdistributionfunctioncanbeobtained(15)bysqueezingorstretchinganarbitraryisotropicdistribution
(cid:112)
functionalongthepreferreddirectioninmomentumspace, f(k,ξ,p )= fiso( k2+ξ(k.n)2,p ),wherenisthe
i hard i hard
directionofanisotropy. Itisimportanttonoticethatξ >0correspondstoacontractionofthedistributionfunctionin
thedirectionofanisotropyand−1 < ξ < 0correspondstoastretchinginthedirectionofanisotropy. Inthecontext
of relativistic heavy ion collisions, one can identify the direction of anisotropy with the beam axis along which the
systemexpandsinitially. Thehardmomentumscale p isdirectlyrelatedtotheaveragemomentumofthepartons.
hard
InthecaseofanisotropicQGP, p canbeidentifiedwiththeplasmatemperature(T).
hard
2.2. Photonrate: Isotropiccase
As mentioned earlier the QGP evolves hydrodynamically from τ onwards. In such case the distribution func-
iso
tionsbecomeFermi-DiracorBose-Einsteindistributions. Thephotonemissionrate,inisotropiccase,fromCompton
(q(q¯)g → q(q¯)γ)andannihilation(qq¯ → gγ)processeshasbeencalculatedfromtheimaginarypartofthephoton
self-energybyKapustaetal.(16)inthe1-loopapproximation. However,ithasbeenshownbyAurancheetal.(17)
that the two loop contribution is of the same order as the one loop due to the shielding of infra-red singularities.
ThecompletecalculationuptotwoloopwasdonebyArnoldetal.(18). Inthispaperwehavecalculatedthephoton
productionratefromhothadronicmatter. WefollowthecalculationsdoneinRef.(19)whereconvenientparameteri-
zationshavebeengivenforthereactionsconsidered.Theseparameterizationswillbeusedwhiledoingthespace-time
evolutiontocalculatethephotonyieldfrommeson-mesonreactions. Thephotonemissionrate(static)fromreactions
of the type BM → Bγ (B denotes baryon) has been calculated in Ref. (20). It is shown that this contribution is
not negligible compared to that meson-meson reactions. To evaluate photon rate due to nucleon (and antinucleon)
scatteringfromπ,ρ,ω,ηanda mesonsinthethermalbathweusethephenomenologicalinteractionsdescribedin
1
Ref. (20). Besides the thermal photons from QGP and hadronic matter we also calculate photons from initial hard
scatteringfromthereactionofthetypeh h → γX usingperturbativeQCD.Weincludethetransversemomentum
A B
broadeningintheinitialstatepartons(21,22).
LusakaBhattacharyaetal./NuclearPhysicsA00(2011)1–4 3
100
τ = 0.1 fm/c
10-2 τiso = 1 fm/c
iso
τ = 2 fm/c
-2V] 10-4 iso
e
G
dy[ 10-6
T
p
2N/d 10-8
d
10-10
10-12 2 4 6 8 10
p [GeV/c]
T
Figure1: (Coloronline)Mediumphotonspectrum,dN/d2pTdy,aty = 0forthefree-streaminginterpolatingmodel(δ = 2)forthreedifferent
valuesofisotropizationtime,τiso,withinitialconditions,Ti=440Gevandτı=0.1fm/c.
2.3. Space-timeevolution
The expected total photon rate must be convoluted with the space-time evolution of the fireball. The system
evolvesanisotropicallyfromτ toτ whereoneneedstoknowthetimedependenceof p andξ. Wehaveuseda
i iso hard
phenomenologicalmodel(12,13)todescribethetimedependenceof p andξ. Intheframeworkofthismodel,
hard
ξ =0atτ=τ anditgrowswithtime(τ)andreachesmaximumatτ=τ ,afterthatξdecreasestozeroatτ>>τ .
i iso iso
We shall follow the work of Ref. (12, 13) to evaluate the p distribution of photons from the first few Fermi of the
T
plasmaevolution. Inourcalculation,weassumeafirst-orderphasetransitionbeginningatthetimeτ (p (τ )=T )
c hard c c
andendingatτ =r τ wherer =g /g istheratioofthedegreesoffreedominthetwo(QGPphaseandhadronic
H d c d Q H
phase)phases. Therefore,thetotalthermalphotonyield,arisingfromthepresentscenarioisgivenby,
(cid:34)(cid:90) (cid:35) (cid:34)(cid:90) (cid:35)
dN dR dR
= d4xE + d4xE , (2)
d2p dy d3p d3p
T aniso hydro
where the first term denotes the contribution from the anisotropic QGP phase and the second term represents the
contributionsevaluatedinidealhydrodynamicsscenario.
3. Results
We have considered the initial condition, T = 440 MeV, τ = 0.1 fm/c and free-streaming interpolating model
i i
(δ = 2) (13, 23) for the pre-equilibrium evolution. In this initial condition the maximum value of ξ will be ∼ 70
at τ = τ . In Fig. (1) we present the photon yield due to Compton and annihilation processes in the mid rapidity
iso
(θ = π/2, θ being the angle between the photon momentum and the anisotropy direction) as a function of photon
γ γ
transverse momentum. In estimating this result, we have used α = 0.3. Different lines in Fig. 1 correspond to
s
differentisotropizationtimes,τ . Weclearlyobserveenhancementofphotonyieldwhenτ >τ. Theenhancement
iso iso i
ofphotonyieldinthetransversedirections(y = 0)isduetothefactthatmomentum-spaceanisotropyenhancesthe
density of plasma partons moving at the mid rapidity (13). To show that the presence of initial state momentum
anisotropy and the importance of the contribution from baryon-meson reactions we plot the the total photon yield
assuming hydrodynamic evolution from the very begining as well as with finite τ (right panel describes the total
iso
contribution with and without the initial state momentum space anisotropy only for τ = 1 fm/c) in Fig. (2). It is
iso
clearlyseenthatsomeamountofanisotropyisneededtoreproducethedata. Wenotethatthevalueofτ neededto
iso
describethedataalsoliesintherange1.5fm/c≥τ ≥0.5fm/cforbothvaluesofthetransitiontemperatures.
iso
LusakaBhattacharyaetal./NuclearPhysicsA00(2011)1–4 4
100 MMMMMM ++(BBpuMMre ((hτpyudre=ro 1h, yTfdmcr=o/c1, ,9T T2c =M=119e9V2 2)M MeVeV)) 100 MMMM++BBMM ((τpisuor=e1 h fymdr/oc,, TTcc==117700 M MeeVV))
-2y (GeV) 10-2 ττMiissMoo== 01(τ..55iso ff=mm1// iccsfmo/c, Tc=192c MeV) -2y (GeV) 10-2 Ti=440 MeV, τi=0.1 fm/c
2dpdT Ti=440 MeV, τi=0.1 fm/c 2dpdT10-4
dN/ 10-4 dN/
10-6
(a) (b)
10-60 1 2 3 4 5 6 7 10-80 2 4 6 8 10 12 14
p (GeV/c) p (GeV/c)
T T
Figure2:(Coloronline)PhotonpT distributionsatRHICenergieswithinitialconditionTi=440GeV,for(a)Tc=192MeVand(b)170MeV.
4. Conclusion
Tosummarize,wehavecalculatedtotalsinglephotontransversemomentumdistributionsbytakingintoaccount
the effects of the pre-equilibrium momentum space anisotropy of the QGP and late stage transverse expansion on
photonsfromhadronicmatterwithvariousinitialconditions. Todescribespace-timeevolutionintheveryearlystage
we have used the phenomenological model described in Ref. (12) for the time dependence of the hard momentum
scale(p )andplasmaanisotropyparameter(ξ).Tocalculatethehardphotoncontributionsweincludethetransverse
hard
momentumbroadeningintheinitialhardscattering.ThetotalphotonyieldisthencomparedwiththePHENIXphoton
data. Withintheambitofthepresentmodelitisshownthatthedatacanbedescribedquitewellifτ isintherange
iso
of 0.5 - 1.5 fm/c for all the combinations of initial conditions and transition temperatures considered here. It is to
be noted that the apparent hump observed in all the figures (except Fig.(5)) needs to be understood and we wish to
discussitinasubsequentpaper.
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