Elliptic flow of thermal photons and formation time of quark gluon plasma at RHIC

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by  Rupa Chatterjee

Elliptic flow of thermal photons and formation time of quark gluon plasma at RHIC
Rupa Chatterjee1 and Dinesh K. Srivastava1
1
Variable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700 064, India
(Dated: January 23, 2009)
We calculate the elliptic flow of thermal photons from Au+Au collisions at RHIC energies for a
range of values for the formation time τ0 but a fixed entropy (or particle rapidity density). The
resultsarefoundtobequitesensitivetoτ0. Thev2 forphotonsdecreasesasτ0decreasesandadmits
a larger contribution from the QGP phase which has a smaller v2. The elliptic flow coefficient for
hadrons, on the otherhand, is only marginally dependent on τ0.
It is by now generally believed that we are witness-
9
ing the birth of quark gluon plasma in relativistic colli-
0 sion of gold nuclei at RHIC energies. This view is sup- 104
0
2 portedbythecleardisplayofjet-quenching [1,2],elliptic 103 0.2 fm/c
0.4 fm/c
Jan fltthioaewlaoonnfi-sspoeattrrotoipcfylectsoolilnaezcnitmiovnuit-tcyheanaltnardanlisccoootnlrlvoiespiroysnioosfn[t3ho,efi4ri]nmhiteoiramalledsnpitnaag-, 3-2c GeV) 110021 π 001...680 fffmmm///ccc
3 and radiation of thermal photons [5]. Now the next set dy ( 100
T
2 of questions occupy the central stage in this endeavour. 2dp 10-1
] What is the temperature of the plasma? Is it in ther- dN/ 10-2
h mal and chemical equilibrium? What is the formation
10-3 Au+Au@200 AGeV; b=6 fm
t time τ0, beyond which we could possibly use the power-
-
l ful methods of hydrodynamics to describe the evolution 10-04.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
c
u of the system? pT (GeV/c)
n Thequestionofτ0 isalsorelatedtotheprocessofther-
[
malization. In the simplest treatment one assumes that

3 the partons produced in the collision have an average 0.4
v energy E and thus τ0 1/ E . It is a different ques- Au+Au@200 AGeV; b=6 fm
8 h i ≈ h i
tionwhether these arethermalized[6]. One mayuse one
4 0.3
of the many mini-jet based models and assume that the
5
0 energy and momentum of the partons are ”some-how”
. redistributed and the plasma is formed at the deposited p)T0.2 π 0.2 fm/c
9 energydensity [7]. A further refinementmay include use v(2 0.4 fm/c
0
0.6 fm/c
8 ofa partoncascademodel [8, 9, 10], a self-screenedpar- 0.8 fm/c
0.1
0 ton cascade model [11], or partonsaturation models [12] 1.0 fm/c
v: togetatimeτ0 1/pT orαs/Qs (Ref.[13])atwhichone
≈
i assumes the plasma to be thermally equilibrated. The 0.0
X 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
chemical equilibration can also be modeled using master p (GeV/c)
r equations[7,14]. Eventhoughdiscussedinliteraturefor T
a
quite some time [15], the role of plasma instabilities in
FIG. 1: (Colour on-line) Spectrum (upper panel) and v2
thermalizing the plasma is being explored in detail only
(lowerpanel)forprimarypionsfromAu+Aucollisionsat200
now (see Ref. [16]).
GeV/A for b= 6 fm for different initial times.
In view of the severaltreatments and perhaps compli-
mentarymodels availablein the literature,a more direct
measurementof the formationtime τ0 wouldbe very de-
sirable. In the present work we argue that a experimen-
suggested by the re-combination models [19] and quark-
tal determination of the v2 of thermal photons for non-
numberscalingofhadronicv2)determinesthisindirectly.
central collisions can help us determine this value rather
accurately. We start with the assumption that a chemically and
Elliptic flow of thermal photons and dileptons [17, 18] thermally equilibrated quark gluon plasma is produced
has recently been proposed as a powerful tool for deter- at time τ0 beyond which we trace its evolution using an
mining the azimuthal anisotropy of momenta of partons ideal hydrodynamics. We closely follow the procedure
soonaftertheformationofthequark-gluonplasma. The outlinedearlier[17,18]insettinguptheinitialconditions
v2 forhadronsthoughseededintheearlyQGPphase(as andassumethattheentropydensitys(τ0,x,y,b)isgiven
2
by:
s(τ0,x,y,b)=κ [αnw(x,y,b)+(1 α)nb(x,y,b)] (1) 104
−
e = 0.045 GeV/fm3
whereκisaconstant,nw isnumberofwounded-nucleons 102 eff = 0.135 GeV/fm3
(participants) at impact parameter b, and nb is the -2V) π τ0 = 0.2 fm/c
nmuomdbele.rIonfabdidniatriyonc,oαlli=sio0n.7s5esistiamcaotnedstaunsitnwghtihcehGcolnautrboelrs 3y (cGe 100 ρx0.1 b = 6 fm
the relativecontributions ofsoftandhardprocesses. We pdT10-2
c0h,obos=e 0κ)=su1ch17tfhmat−1s0fo=r ths(eτ0co=llis0io.6n offmg,olxd n=uc0l,eiy[2=0] 2dN/d
10-4
at √s = 200 A GeV. We adjust κ to get the starting
pvaalruteicsleforraps0idaittyodtehnesritvyal[u21es])oisf aτ0cosuncshtatnhta,twsh0icτh0 i(morpltihees 10-06.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
p (GeV/c)
identical entropy for all the cases. Note that the profile T
oftheentropydistributionisuniquelydeterminedbythe
above equation. We employ an impact parameter de-
pendent, boost invariant hydrodynamics [22], which has 0.4
been used extensively to explore hadron production and
e = 0.045 GeV/fm3
elliptic flow of hadronsas well as photons [17] anddilep- f
0.3 e = 0.135 GeV/fm3
tons [18]. f
) T π
p 0.2
v(2 ρ
102
0.1 τ0= 0.2 fm/c
101 0.2 fm/c b = 6 fm
0.4 fm/c
-2GeV) 100 ρ 00..68 ffmm//cc 0.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
3y (c 10-1 1.0 fm/c pT (GeV/c)
d
T
p
2N/d 10-2 FIG. 3: (Colour on-line) Effect of changing the freeze-out
d energy density on the spectra (upper panel) and v2 (lower
10-3 Au+Au@200 AGeV; b=6 fm panel) for π and ρ mesons. The plot for v2 for the ρ-mesons
is shifted by 0.5 GeV/c along the x-axis.
10-04.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
p (GeV/c)
T
matter and the hadronic matter. We use the complete
leading-order results for the production of photons from
0.4 the QGP [23] and the latest results for the radiation of
Au+Au@200 AGeV; b=6 fm photonsfroma hothadronicgas[24]. As mentionedear-
lier, the equation of state (EOS Q [22]) incorporating a
0.3
phasetransitiontoquarkgluonplasmaatT 164MeV,
≈
)T and a resonance gas for the hadronic phase below the
v(p20.2 ρ 00..24 ffmm//cc energy density of 0.45 GeV/fm−3 is used to describe the
evolution. ThemixedphaseisdescribedusingMaxwell’s
0.6 fm/c
0.8 fm/c construction. The freeze-out is assumed to take place at
0.1
1.0 fm/c e = 0.075 GeV/fm3. The spectrum of hadrons is ob-
f
tained using the Cooper-Fry formulation.
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 As a first step we show the calculated spectra of pri-
p (GeV/c)
T mary pions and rho mesons along with the differential
elliptic flow for a typical impact parameter of 6 fm for
FIG. 2: (Colour on-line) Same as Fig. 1 for rho-mesons. Au+Au collisions at RHIC energies. We see that if the
initial time is large, the inverse slope of the pion spec-
The production of thermal photons is calculated by trum (Fig. 1; upper panel) is smaller due to a late start
folding the history of the evolution of the system with offlow(alltheresultsareforzerorapidity). Howeverthe
the rate for the production of photons from the quark deviation in the spectrum up to about 1.5 GeV/c is not
3
substantial, even though the spectrum at p 4 GeV/c
T
≈
changes by about a factor of three. It is very interesting
to note that the differential elliptic flow is almost inde-
101 Thermal Photons; HM
pendent of the initial time across the range of p values
T
0.2 fm/c
crmiehoronorisseimdmneeoorsteoridenc.essa,tSrbeiolmvenegianllasyrttahhrffeoeesurcughtlhoetsdmtabheryeseotconhhbe(atsflaneiogenweeF.diingWf.ot2erh)hetbahevseipenpevgrceithmrrieafiaaervidys- 3-2dy (c GeV) 111000---531 0001....4680 ffffmmmm////cccc
T
that similar results are obtained for all other particles, p
included in the (nearly complete) list of hadrons in our 2N/d 10-7
d Au+Au@200 AGeV
calculations. The similarity of spectra for the primaries 10-9 b = 6 fm
ensures that the spectra of particles after the resonance
decays are accounted for, will remain similar. 10-101.0 1.0 2.0 3.0 4.0 5.0 6.0
p (GeV/c)
One may argue that the freeze-out at a given (rather T
low) energy density represents an over-simplification
of the decoupling of the hadrons from the interacting
medium. Thusonecouldconsiderhadronsdecoupling at 102
a sufficiently large temperature or energy density from Thermal Photons
0.2 fm/c
the hydrodynamic evolution and populating a hadronic 100 0.4 fm/c
coansctahdeierfrinotmerwachtiicohntchreoyssu-snedcetriognosa[2fr5e,e2ze6-]o.uWtdheipleenadfiunlgl -2GeV) 10-2 001...680 fffmmm///ccc
calculation along these lines would be welcome, we can 3y (c 10-4
get a feeling of the outcome by estimating the effect of dT
p
the freeze-out density on the spectra and the v2 of the 2N/d 10-6
hadrons using hydrodynamics. d Au+Au@200 AGeV
In Fig. 3, we give our results for the p distributions 10-8 b = 6 fm
T
and differential elliptic flow, v2(pT), for pi3ons and ρ- 10-100.0 1.0 2.0 3.0 4.0 5.0 6.0
mesons for energy-densities of 0.045 GeV/fm and 0.135
p (GeV/c)
GeV/fm3, which correspond to freeze-out temperatures T
ofabout100and140MeVrespectively. Wenotethatthe
FIG. 4: (Colour on-line) Spectrum of photons radiated from
p distribution continues to evolve as the system cools
T
hadronic matter alone (upper panel) and the sum (lower
from140 MeV to 100MeV, due to the radialflow. How-
panel) for different initial times.
ever, the elliptic flow is seen to have essentially acquired
its final value when the temperature is still large. Thus
we feel that if these hadrons are fed into a hadronic cas-
cade the results for the v2 will remain unaltered. shows a much more complex and rich structure (Fig. 5).
We conclude thus, that it would not be easy to deter- Thev2forphotonsfromthequarkmatterismuchsmaller
mine τ0 on the basis of either spectra or v2 for hadrons. than that for photons from the hadronic matter [17].
Thecalculateddifferencesinthespectraatverylargeval- This, happens as the anisotropies in momenta of the
ues of p may not be quite useful, as the hydrodynamic particles producing photons take some time to develop
T
description itself may not hold at p 2 GeV/c. and the photons from the quark matter measure these
T
≫ anisotropies for early times. Note also the peak around
Next we calculate the thermal photon production for
0.5 GeV/c due to the dominating role of the reaction
the same system (Fig. 4) using τ0 varying from 0.2 fm/c
to 1 fm/c. It is seen that the contributions from the πρ πγ [17] beyond about 0.5 GeV/c. We also see an
→
hadronic matter for different starting times are only interesting trend; the v2 for thermal photons from the
marginally different at lower p , though at 6 GeV/c it quark matter increases with the increase in the initial
T
increases by an order of magnitude, as τ0 drops from 1.0 time τ0, while those for the photons from the hadronic
fm/cto0.2fm/c,duetoanincreasedflowforlowerinitial matterdecreasewithincreaseintheinitialtimeτ0. This,
times. However,thesmallerinitialtimesalsoadmitmuch wefeel,happensduetoacompetitionbetweenlargerini-
larger initial temperatures and the radiation of high p tial(spatial)anisotropiesandtemperaturesforsmallerτ0
T
anda startofcooling andequalizationofpressuregradi-
photons from the quark matter increases substantially.
Thus, while the total production of thermal photons for ents at an earlier time.
pT <1GeV/cisnotstronglyaffected,atpT =6GeV/c, The results for the final v2 for thermal photons (solid
itincreasesbymorethanfourordersofmagnitude. This curves, Fig. 5) reveal a large sensitivity to the initial
is known for quite some time in the literature. time τ0, for pT greater than about 1.5 GeV/c. This is
The differential elliptical flow for thermal photons broughtabout by the increasing contribution of photons
4
thattheNLOpQCDcalculationsnearlyfullyaccountfor
the production of photons at p 3 GeV/c.
T
0.20 ≥
We would like to add though that the fragmentation
Thermal Photons
contributiontothepromptphotons(about30%forp >
Au+Au@200 AGeV T
0.16
b = 6 fm 3 GeV) will be suppresseddue to jet-quenching. Assum-
ing a suppression by a factor of about 2 (for quark jets
0.12
p)T Q0.5MxHM 1.0 fm/c which fragment into photons) this would mean a reduc-
v(20.08 QM+HM 0.8 fm/c tsihoonwbnyh≈ere1.5H%owinevtehre,tphQisCwDouylideldbecosommpeawrehdattooffw-sheattbiys
0.6fm/c
the productionofphotonsdue topassageofjets through
0.04 0.2 fm/c 0.4 fm/c the QGP[29]. Amoredetailedandcomplete calculation
will consider all these sources [30]. We postpone that to
0.000.0 1.0 2.0 3.0 4.0 5.0 6.0 afuturepublicationasthepresentemphasisisontheaz-
p (GeV/c)
T imuthal anisotropy of thermal photons, which is rather
negligible for other direct photons. Similarly, one may
FIG. 5: (Colour on-line) Thedifferential elliptic flow of ther- consider the pre-equilibrium contribution as well [31], if
mal photons (solid curves) for different initial time τ0. The τ0 is assumed to be large. These contributions will not
results for emissions from the hadronic matter (long-dashed) show any v2 as they are not subjected to collectivity.
and quark matter (short dashed) are also shown. If we are unable to separate these other sources of
photons by, say, either by tagging or by calculations,
then the v2 given here will have to be moderated as
(v2th Nth+v2non−th Nnon−th)/(Nth+Nnon−th), where
103 ‘th’ ×stands for ‘therm×al’. However as vnon−th is quite
2
Thermal & Prompt Photons
-2eV) 1100-11 APHu+EANIuX@; 1200 -0 2 A0%G meVost central 0000....2468 ffffmmmm////cccc sctomonnasnlle,wctitlihloens,tditlilffheepreserunsccicseet,sinsboetfihnteghreleaNsrugLletOinfogprQvlC2arDfgoerinτs0ipn.rgelIdenicptthihnoigs-
G
3c 1.0 fm/c thepromptcontributioncanbeusedwithagreatadvan-
y ( 10-3 NLO pQCD tage as it can be subtracted from the direct photons to
d
pT greatly increase the sensitivity of the remainder to τ0.
2N/d 10-5 In order to get a more quantitative idea about
d
10-7 these, we draw the attention of the readers to FIG.6
of Ref. [30]. First of all, as remarked earlier, the
10-09.0 1.0 2.0 3.0 4.0 5.0 6.0 Compton+annihilationpartofthepromptphotonsdoes
p (GeV/c) not contribute to azimuthal anisotropy. The prompt-
T
fragmentationpart, againas suggested earlier,will show
FIG.6: (Colouron-line)Singlephotonsfor10–20%mostcen- a marginal positive v2, due to the reaction plane depen-
dence ofenergylosssufferedby the out-goingquarksbe-
tralcollisionsforAu+Ausystemat200GeV/A.NLOpQCD
resultsforpromptphotons(seetext)andthermalphotonsfor fore fragmenting. This accordingto Ref. [30] is less than
different formation times are also shown. about+1%. The jet conversionphotonsgiveriseto very
small (less than 1%) v2. The resulting v2 due to these
−
two processes is essentially zero [30]. This, along with
from quark-matter (with smaller v2) as the time τ0 de- the possibility of an accurate determination of prompt
creases. With increasing τ0 the relative contribution of photons using NLO pQCD as seen here holds out the
thermal photons from the hadronic matter increases. As promise of determining the sensitivity of elliptic flow of
these have larger v2, the over-allv2 becomes larger. thermal photons to τ0.
Before closing we show our results for single photon An experimental verification of these predictions will
production and compare it with the measurements by give a very useful information about the formation time
the PHENIX experiment [5] for the 10–20% most cen- and also validate our ideas about the evolution of the
tralcollisionsforAu+Ausystemat200GeV/A(Fig.6). hot and dense system in relativistic heavy ion collisions.
Thepromptphotonproductionhasbeencalculatedusing A theoretical reanalysis of single photon production for
NLO pQCD [27] and choosing the factorization, renor- Pb+Pbcollisions at the CERN SPS has revealedsimilar
malization, and fragmentation scales as equal to p /2. sensitivity to the formation time [32].
T
The effects of (impact parameter dependent) shadow- In brief, we have calculated the thermal photon pro-
ing[28]andiso-spinisexplicitlyaccountedfor. CTEQ5M duction for non-central collisions of gold nuclei at RHIC
structure functions are used. The thermal photon pro- energies. The initial time is varied from 0.2 fm/c to
duction for different initial times is also shown. We see 1 fm/c, keeping the total entropy fixed and assuming
5
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