Dynamics of Quark Gluon Plasma and Interference of Thermal Photons

0.29 MB·English

by  Dinesh K. Srivastava

Dynamics of Quark Gluon Plasma and Interference of
Thermal Photons
Dinesh K. Srivastava1,∗ Rupa Chatterjee1,2, and Somnath De1
1 1Variable Energy cyclotron Centre, 1/AF,
1 Bidhan Nagar, Kolkata-700064, India and
0 2Department of Physics, University of Jyva¨skula¨, PO Box 35 (YFL), FI-40014, Finland
2
The quantum statistical interference between identical particles emitted from a com-
n
pletelychaoticsourceisexpectedtoprovidevaluableinputforthespacetimedescription
a
J of the system. Intensity interferometry of thermal photons produced in heavy ion col-
lisions is a very promising tool to explore the structure and dynamics of the collision
5
fireball. ThermalphotonshavingKT ≤ 2GeV/cget competingcontributionfrom both
hadronic and quark matter phases and this competition gives rise to a rich structure in
]
h theoutwardcorrelationfunction,owingtotheinterferencebetweenthephotonsfromthe
t two sources. The temporal separation between the two sources provides the lifetime of
-
l the system and the correlation results are found to be sensitive to quark hadron phase
c
transition temperature and the formation time of the plasma. The outward correlation
u
functionstronglydependsontheequationofstateofthestronglyinteractingmatterand
n
isseentoclearlydistinguishbetweenthelatticebasedandbagmodelequationsofstate.
[

1
v I. INTRODUCTION smaller separation in space time as well at
7 smallerrelativemomentum,providesthemost
3 directtoolto get spatio-temporalinformation
The primary motivation of the relativistic
9
of the system [7].
heavy ion collision studies is to investigate
0
. the properties of a deconfined strongly inter- In 1950s’, Robert Hanbury Brown and
1 acting matter, Quark Gluon Plasma (QGP) RichardQ.Twissmeasuredtheangularsizeof
0
produced in the collisions. The observation adistantstarbyutilizingthemethodofinten-
1
1 of elliptic flow [1, 2], jet quenching [3, 4], sity interferometry for the very first time [8].
: and the success of parton recombination [5] Theoretically the photon bunching was first
v
as a model for hadronizationat the Relativis- explained by Purcell [9], in one of the key ex-
i
X tic Heavy IonCollider (RHIC) have advanced periments of quantum optics. In 1959 Gold-
r ourunderstanding ofthis hotanddense novel haber et al. [10] observed an unexpected an-
a state of matter (Fig. 1). The evolution of the gular correlation among identical pions while
system produced in the collision is explained studyingthe ρ0 resonance,whichwasthe first
quite successfully using the powerful method observationof intensity correlationin particle
ofhydrodynamics[6]. However,numerousim- physics. Very soon it was realized that the
portant questions like; how quickly (if at all) correlations of identical particles emitted by
does the plasma thermalize, what is the life highly excited sources are sensitive not only
time of the system, if there is a phase transi- to the size of the system, but also to its life-
tion or not, are still to be addressed. time and the momentum dependent correla-
Thesmallsize(∼fm)andtransient(∼fm/c) tion function contains information about the
nature of the collision fireball make it ex- dynamics [11] of the system produced in the
tremely difficult to obtain space time infor- collisions.
mation of the system. Correlation between The heavy ion community often uses the
two final state particles which is stronger for term HBT in reference to the original work
of Hanbury Brown and Twiss for any analy-
sis related to the spatio-temporal aspects of
the particle emitting source. Several interest-
∗Electronicaddress: [email protected] ing theoretical works on correlation study in
)(pv2t000...0111.6241 200 ΛπpKG+ e+S0+ +V p :Λ πσ-/σgeo 0-80% -2dy) (GeV/c)11010-1 Thermal photDSDSJo.....AnRTdKls’ua.aES:rsm nbrAaiti vdeneuaeret+r s naieAat alt.e-u vDaTt a →l.a0.P. lT=. Te γ0T 0+r3 e0=X=0 s= 03s4 [ o755M0u080-en1 0-MV-k0 6Mo%,e0 τ.Ve 00 TcV=, 0Mτe0, 0 τn=.=e50t V0= 5rf.a0,m93 τl.030]1/c = 7Mf0m fe.m2/Vc /f,cm τ0/=c0.15 fm/c
0.08 Cascade 2π dpT10-2 PPrHoEmNpItX γ :A NuL+OAu p [Q0C-1D0 %× TcAeAn[t0ra-1l]0%]
0.06 N/(10-3
2d
0.04 10-4
0.02
Hydro curves Huovinen 10-5
0 TC = 165 MeV, Tf = 130 MeV
-0.02 10-6
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
p [GeV/c] 10-7
t 0 1 2 3 4 5 6 7 8
PPHHEENNIIXX AAuu++AAuu ((cceennttrraall ccoolllliissiioonnss)):: pT (GeV/c)
RAA10 DπηDπη00iirreecctt γγ FIG. 2: Thermal photon predictions for central
GGLLVV ppaarrttoonn eenneerrggyy lloossss ((ddNNgg//ddyy == 11110000))
Au+Aureactions at RHICcomputedwith differ-
ent hydrodynamical and dynamical fireball mod-
1 els along PHENIX experimental data [17].
10−1 II. INTENSITY INTERFEROMETRY
0 2 4 6 8 10 12 14 16 OF PHOTONS
p (GeV/c)
T
(a) π++π- (PHENIX) p+p (PHENIX) (b)
0.1 KK+S0+ (KS-T (APRH)ENIX) ΞΛ-++ΛΞ +( S(STTAARR)) theErlmecotmroemteargnoeftitcheramdieadtiiuomn,, khnaoswlonngasbethene
considered as one of the most promising and
nq efficient probes for the hot and dense state
/v20.05 ofmatterproducedinthe relativistic collision
of heavy nuclei [17]. Intensity interferometry
withidenticalhadronsorlightnucleihasbeen
animportanttooltolearnaboutthedynamics
00 0.5 1 1.5 2 0 0.5 1 1.5 2 of the subatomic or nuclear collisions. How-
pT/nq (GeV/c) KET/nq (GeV) ever,thehadronssufferfinalstateinteractions
and their correlations mainly carry informa-
FIG. 1: Elliptic flow (upper panel) [14], jet
tion aboutthe late dilute stageof the system.
quenching(middle panel) [15], and thesuccess of
parton recombination as a model for hadroniza- Photons on the other hand, decouple from
tion(lowerpanel)[16]attheRHIChaveadvanced the system immediately after their creation,
our understanding of the hot and dense medium suffering negligible re-scatterings with the
produced in relativistic heavy ion collisions. medium (α ≪ α ), and carry undistorted
s
information about the circumstances of their
production to the detector. Obtaining direct
information about the earliest hot and dense
stageofthesystembystudyinglargeK pho-
T
heavyioncollisionswerereportedin1970s’by tons is very promising. A large production
Shuryak [12], Gyulassy et al. [13] and many of photons having very high transverse mo-
others. Other important contributions in the menta is expected from the pre-equilibrium
eighties include parametrization of the source stage,andsome additionalsourcesofphotons
function and a more detailed analysis of the havealsobeenproposedrecently[18,19]. The
role of final state interactions in the system major problem of-course arises from the mea-
(see review article [7] for detail). ger emission of direct photons compared to
thehugebackgroundofdecayphotons(mostly of photons, EdN/d4xd3k from the quark and
from π0 and η mesons) produced in the colli- hadronic matter phases produced in the col-
sions. Of late, there have also been tremen- lisions and the correlation function C(q,K)
dous advancesinmethods foridentificationof is decomposed in terms of the outward, side-
single photons [20]. ward,andlongitudinalmomentumdifferences;
The theory of the intensity interferometry qo, qs and qℓ respectively. One can write the
of photons from relativistic heavy ion colli- four-momentum of the ith photon in-terms of
sions has been pursued in considerable detail transverse momentum k , rapidity y, and az-
T
byseveralauthors[21–23]inlasttwodecades. imuthal angle ψ as,
These studies further refined the early ex-
pectations of photon as powerful tool to get ki =(kiT cosψi,kiT sinψi,kiT sinh yi), (2)
spatio-temporal aspects of the fireball evolu-
tion. In a recent study by Frodermann and andthethreemomentumdifferencesqo,qsand
Heinz [24], the details of angular dependent qℓ are of the form:
intensity correlation of thermal photons from
qT·KT
non-central collision of heavy nuclei as well qo =
K
the dual nature of thermal photon emission T
resulting from the superposition of QGP and = (k12T −k22T)
hadronresonancegasphotonproductionhave pk12T +k22T +2k1Tk2T cos(ψ1−ψ2)
been highlighted.
(cid:12) KT(cid:12)
It is generally felt that the experimental qs = (cid:12)qT−qo (cid:12)
(cid:12) K (cid:12)
efforts for these studies have a larger likeli- (cid:12) T (cid:12)
hood of success at RHIC and LHC energies 2k1Tk2Tp1−cos2(ψ1−ψ2)
=
bpleacsamusaeaonfdlaarglaerrgeinsituipaplrteesmsiopneraotfuprieonosf dtuhee pk12T +k22T +2k1Tk2T cos(ψ1−ψ2)
to jet-quenching. qℓ = k1z −k2z
Till date only one measurement of photon = k1T sinhy1−k2T sinhy2. (3)
intensity correlation at very low transverse
The radii corresponding to these momentum
momentum K has been reported by WA98
T
differencesareobtainedbyapproximatingthe
collaboration for central collision of lead nu-
correlationfunction to Gaussianparametriza-
clei at CERN SPS [25].
tion as;
1
2 2 2 2
A. Formulation C(qo,qs,qℓ)=1+ exp(cid:2)−(cid:0)qoRo+qsRs
2
2 2
+ q R (cid:1)(cid:3).(4)
The spin averaged intensity correlation be- ℓ ℓ
tween two photons with momenta k1 and k2, The root mean square momentum difference
emitted from a completely chaotic source can hq2i and the radii are obtained as,
be expressed in terms of the space-time emis- i
sion function S(x,K) as [21, 26]: 2
hq2i = R (C−1)qi dqi, (5)
1 (cid:12)R d4xS(x,K)eix·q(cid:12)2 i R (C−1)dqi
C(q,K)=1+ (cid:12) (cid:12)
2R d4xS(x,k1) R d4xS(x,k2)
(1) 2 1
R = . (6)
wheretheobtainedcorrelationisafunctionof i 2hq2i
i
the relative q(= k1 − k2) and average mo-
menta K(= (k1 + k2)/2) of the two pho- Itistobenotedthatthe1/[2hqi2i]1/2 becomes
tons. The space-time emission function S is ausefulmeasurewhenthecorrelationfunction
often approximated as the rate of production has a more complex nature.
1.6 is assumed that a thermally and chemically
Au+Au@RHIC, γγ Intensity Correlation
Thermal Photons, k =1.95 GeV/c equilibrated plasma is formed at a very small
1.4 y1=y2=0, ψ1=ψ2=0, q1Ts=ql=0 initial time τ0 (0.2 fm/c for RHIC and 0.1
q sum (QM+HM) fm/cforLHC).Anisentropicexpansionofthe
1.2 o (only) QM; 1 + 0.5 |ρi,Q|2 plasmaisconsideredwheretheinitialtemper-
(only) HM, 1 + 0.5 |ρ |2
i,H ature T0 is obtained from the relations,
1.0
y1=y2=0, k1T=k2T, qo=ql=0 2π4 1 dN =4aT03τ0, (7)
1.4 |ρi,Q| = IQ exp[ - 0.5 (qiRi,Q)2] 45ζ(3)AT dy
C q |ρ | = I exp[ - 0.5 (qR )2]
s i,H H i i,H here A is the transverse area of the sys-
1.2 sum ~ 1 + 0.5 [|ρi,Q|2 + |ρi,H|2 + T
2 |ρ | |ρ | cos(q∆R)] tem, dN/dy is the particle rapidity density,
i,Q i,H i
and a = 42.25π2/90 for a plasma of mass-
1.0
less quarks and gluons. The initial energy
ψ=ψ=0, k =k , q=q=0
1 2 1T 2T o s
density is estimated by combining 25% con-
1.4
tributionfrombinarycollisionsand75%from
q
l number of wounded nucleons. The bag model
1.2 equation of state is considered for the sys-
tem evolution where a first order phase tran-
1.0 sition takes place at a temperature of about
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
q (GeV/c) 180 MeV and the freeeze-out temperature is
i
takenas100MeV.Therelevanthydrodynamic
FIG. 3: Outward (top panel), sideward (middle equations are solved under the assumption of
panel), and the longitudinal (lower panel) corre- boost invariant longitudinal and azimuthally
lation functions for thermal photons produced in symmetrictransverseexpansionusingthepro-
central collision of gold nuclei at RHIC. Symbols cedure discussed in Ref. [27] and integration
denote results of the correlation; curves denote
performedoverthehistoryofevolution. Fig.2
fits [26].
shows thermal photon results from from hy-
drodynamic model by different groups along
with direct photon data from PHENIX.
B. Initialization and system evolution
The complete leading order results for pro-
duction of photons from quark matter by
Theexcitingpossibilityofobservationofthe Arnoldetal.[28],andtheresultsofTurbideet
interference of thermal photons in the inter- al.[29]forradiationfromthehadronicmatter
mediate K (≤ 2 GeV/c) range has been ex- areusedforthisanalysis. Thevaluesofcharge
T
plored in detail in a recent work of Srivastava particle multiplicity at mid rapidity are taken
et al. [26]. Photons in this momentum range as 1260 [19] for 200A GeV Au+Au collisions
areuniqueastheygetcompetingcontribution at RHIC and 5625 [30] for 5.5A TeV Pb+Pb
fromthetwophases;theyhavetheiroriginei- collisions at LHC.
ther in the hot and dense quark phase of the
system or in the relatively cooler but rapidly
expandinghadronicphase,wherealargebuild C. Correlation functions for thermal
up of the radial flow boosts their transverse photons
momentum. This competition gives rise to a
rich structure specially in the outward corre- The results for the outward, sideward, and
lation function, owing to the interference be- longitudinal correlation functions for thermal
tween the photons from the two sources. photons at RHIC having K ≈ 2 GeV/c are
T
For this particular study, central collisions showninFig.3. Thefour-momentaofthetwo
of gold and lead nuclei at the top RHIC and photons are chosen so that when the outward
LHC energies respectively are considered. It correlationisstudied,q andq areidentically
s ℓ
20
Au+Au@RHIC, γγ Intensity Correlation Thermal Photons
y1=y2=0, ψ1=ψ2=0, qs=ql=0 1.0 Au+Au@200A GeV TTC == 210800 MMeeVV
16 (a) C
T = 160 MeV
∆R 0.8 IH = dNH/(dNQ+dNH) C
R (fm)o128 o R I or IQH0.6
o,H 0.4
[ 2 < q2> ]-0.5
4 Ro,Q o 0.2 I = dN /(dN +dN )
Q Q Q H
00 0.4 0.8 1.2 1.6 2 0.00 0.4 0.8 1.2 1.6 2
KT (GeV/c) KT (GeV/c)
FIG.4: Transversemomentumdependenceofthe FIG.5: Transversemomentumdependenceofthe
outward radiiandtemporaldurationobtainedby fractionofthermalphotonsfromthequarkmatter
fitting the final correlation function for thermal andhadronicmatterphasesatRHICwithvarying
photonsatRHIC.Radiideterminedfromheroot- transition temperature [26].
mean-square momentum difference for the corre-
lation functionarealso givenforcomparison [26].
K ≈ 2 GeV/c, at RHIC, the value of ∆R
T o
is 12.3 fm obtained from fitting whereas ∆R
s
zero and the dependence on q0 is clearlyseen, and ∆Rℓ are of the order of zero.
andsoon. Consideringonlythe quarkmatter These results imply that, while the spa-
or the hadronic matter only, it is found that tial separation of the two sources is negligi-
the correlation functions for the two phases ble, their temporal separation is much larger,
can be approximated as which gives the lifetime of the system. If the
mixed phase is of shorter duration or absent,
2
C(q ,α)=1 + 0.5|ρ | (8)
i i,α this will obviously decrease.
The K dependence of ∆R , and the out-
where i = o, s, or ℓ, and α denotes quark T o
ward radii for the hadronic and quark mat-
matter(Q)andhadronicmatter(H)inanob-
ter sources of photons obtained from fitting
viousnotation. Thesourcedistributions|ρ |
i,α
are shownin Fig. 4 which clearly explains the
is well described as a Gaussian function,
fact mentioned above. The outward radius
|ρi,α| = Iα exp(cid:2)− 0.5 (qi2Ri2,α)(cid:3) (9) for the quark contribution depends weekly on
the transversemomentum, whichisindicative
where I = dN /(dN + dN ) and I = of a mild development of the radial flow dur-
Q Q Q H H
dN /(dN + dN ) are the fractions of the ing the quark matter phase. The correspond-
H Q H
photonsfromquarkmatterandhadronicmat- ingradiusforthehadroniccontributionshows
ter respectively. a stronger dependence on the transverse mo-
The final correlationfunctions are then ap- mentum resulting from a more robust devel-
proximated as, opmentofradialflowduringthelatehadronic
phase.
2 2
C(qi) = 1 + 0.5(cid:2) |ρi,Q| + |ρi,H| A very interesting structure is observed for
+ 2|ρ ||ρ |cos(q ∆R )] the duration of the source, which increases
i,Q i,H i i
(10) slightly at higher KT as the momenta of the
photons emitted from the hadronic phase is
where the cosine term in the above equation blueshiftedtomuchlargervalueduetostrong
clearlybringsouttheinterferencebetweenthe radial flow. The saturation of ∆R towards
o
twosources[31]. Here∆R standsforthesep- low K has its origin in the competition
i T
aration of the two sources in space and time between radial expansion and decoupling of
and q is the four momentum difference. For hadronic matter as it cools down below the
1.6 1.6
Au+Au@RHIC, γγ Ιntensity Correlation Au+Au@RHIC, γγ Ιntensity Correlation
1.5 Ty1h=eyr2m=0a,l ψP1h=o tψo2n=s0, ;k q1sTid =e=1q.9lo5n gG=0e,V 1.5 Thyer=mya=l 0P,h ψot=o nψs,= k 01;T q==2.q0=5 0GeV/c
τ=0.2 fm/c 1 2 1 2 s l
1.4 τ0=0.6 fm/c 1.4 τ0 = 0.2 fm/c TC = 160 MeV
0 T = 180 MeV
C1.3 τ=1.0 fm/c C1.3 C
0
T = 200 MeV
C
1.2 1.2
1.1 1.1
1.0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 1.0
q (GeV) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
out q (GeV/c)
o
FIG.6: τ0 dependenceoftheoutwardcorrelation
function at RHIC[26]. FIG.7: TC dependenceoftheoutwardcorrelation
function at RHIC[26].
temperature of freeze-out at the edges.
the temporal distribution of the source func-
As seenfromEq.(6),theinverserootmean
tion, where a large duration of the hadronic
square momentum, which is a measure of the
phase is observed which controls the parame-
correlationradius,isseentovaryrapidlywith
ter ∆R .
transversemomentumfortheoutwardcorrela- o
tionowingtorapidvariationofthecompeting
contribution of photons from the two phases.
D. Sensitivity to formation time and
Asthenatureoftheoverallcorrelationfunc-
transition temperature
tion strongly depends on the relative contri-
bution from the two phases, the fraction of
the quark (I ) and hadronic (I ) contribu- We know that a smaller value of the initial
Q H
tions are investigated as a function of trans- formation time of the plasma implies a larger
verse momentum as well as varying critical initialtemperature. Thesensitivityofthecor-
temperature(seeFig.5). Onecanseethatthe relationfunctionstotheinitialformationtime
transversemomentumatwhichthequarkand of the plasma is investigated by considering
hadronic contributions become equal is sensi- systems with identical entropies but varying
tive to the transition temperature and a de- formation time. Smaller τ0 leads to a larger
creasing T increases the fraction of photons production of photons from the quark matter
C
coming from the quark matter. phase and as a result the outward correlation
The variation of the source function as a functionchangesitsnatureasshowninFig.6.
function of transverse distance and time for It is to be noted that a larger value of for-
different values of transverse momentum has mation time (∼ 1 fm/c) would necessitate
been studied in detail (not shown here) in or- inclusion of the pre-equilibrium contribution
der to understandthe correlationsresultsand of the photons, that must surely be there at
the relativecontributionsofthe twophasesin leastatlargervaluesofthetransversemomen-
it. It is observed that the radial distribution tum. Another important contribution, pho-
for both the phases are centered at r near tons from jet conversion processes also likely
T
zero and the source function for the hadronic to contribute at higher KT. These contribu-
phase extends considerably beyond the same tionswouldincreasethefractionofquarkmat-
for the QGP phase. This is of course due ter contribution in the correlation measure-
the large transverse expansion of the system ment, which in turn could mimic an effective
and results in a good amount of thermal pho- smaller value of the formation time.
ton production from hadronic phase at larger In another potentially powerfulobservation
radii. A more valuable insight is provided by thesensitivityoftheoutwardcorrelationfunc-
103
0.3 HHL
102 HHB
-2V ) 101 0-5% PHENIX
2 Cs00..21 lhhhhaaaaatddddtirrrrcooooennnn p ++gg4HHaa;ssaa N gg(veeτ=oddl8oo. rr cnno ggraar.ss) (vol. corr.) 2dN/dpdy ( GeT1100-01 Aup+iAonu@RHIC, central collsion
T = 100 MeV
10-2 f
00 0.1 0.2 0.3 0.4 0.5 10-03.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
T [GeV] pT (GeV/c)
FIG. 8: Speed of sound for (i) hadron resonance FIG.9: PionpT spectrafortheequationofstate,
gas, (ii) volume corrected hadron resonance gas, HHB and HHL for central collisions of Au nuclei
(iii)hadron andHagedorn resonancegas, and(iv) atRHIC.Experimentaldataforpions(0-5%cen-
volumecorrectedhadronandHagedornresonance trality bin) [36] are shown for comparison [32].
gas with lattice results [32].
having M < 2 GeV and all mesons having
tiontothetransitiontemperatureT isshown
C M < 1.5 GeV, along with Hagedorn reso-
inFig.7, wherethe changeinthe interference
nances [33] in thermal and chemical equilib-
pattern at larger q0 can be easily understood
rium, matches rather smoothly with lattice
in terms of the relative life time of the two
equation of state (p4 action, N = 8) [34]
phases. τ
for zero baryon chemical potential for T up
ThecorrelationresultsforPb+Pbcollisions
to ∼ 200 MeV, when corrected for the finite
at LHC energy also show similar qualitative
volume of hadrons.
nature as observed at RHIC. As the initial
temperature likely to be attained at LHC is Furthermore, two equations of state for
much larger than that at RHIC, the study of strongly interacting matter are constructed;
the properties of QGP phase and the dynam- one, HHL, in which the above is matched to
ics of its evolution will be more interesting at the lattice equation of state at T = 165 MeV
LHC.Ahigherinitialtemperaturewouldlead and the other, HHB, where it is matched to a
toalongerdurationoftheinteractingsystem, bagmodelequationofstatewithcriticaltem-
which in turn would provide ample opportu- perature TC = 165 MeV. The lattice equa-
nity for the mechanism of expansion to de- tion of state displays a sharp cross-over for
velop. 180 < T < 190 MeV.
Theresultsfortheenergydensity,pressure,
andtheentropydensityofthehadronicmatter
III. EQUATION OF STATE AND withsuccessivelyincreasingrichnessofthede-
INTENSITY CORRELATION OF scription; viz., hadrongas, hadron+Hagedorn
THERMAL PHOTONS gas,andvolumecorrectedhadron+Hagedorn
gas have been studied in detail (see Ref. [32]
In a very recent work by De et al. [32] the for detail). Best agreement with the lattice
equation of state (EOS) of strongly interact- calculation is obtained when the hadron +
ing matter and how the change in EOS af- Hagedorngasiscorrectedforthefinitevolume
fects the results of system evolution, particle of the particles and as a result the hadronic
spectra as wellas the intensity interferometry matter for both HHL and HHB equations of
of thermal photons have been explored in de- state is constructedby combining hadronand
tail. It is shown that an equation of state for Hagedorngaswithvolumecorrection. There-
hot hadronic matter consisting of all baryons sults for the square of speed of sound for the
1.6 Au+Au@RHIC, γγIntensity Correlation 103 Thermal photon spectrum for Au+Au @RHIC
1.5 Thermal Photons, k1T=1.7 GeV/c
1.4 HHHHLB -2GeV) 101 HH0-HH20LB% PHENIX
C1.3 dy (10-1
1.2 QM+HM 2N/dk T QM+HM
d10-3
1.1
1.00 0.04 0.08 0.12 0.16 10-50 0.5 1 1.5 2 2.5 3 3.5 4
q (GeV/c) k (GeV/c)
o T
FIG. 10: The outward correlation function for FIG.11: Thermalphotonspectrafortheequation
thermal photons at RHIC[32]. of state, HHB and HHL for central collisions of
AunucleiatRHIC.Experimentaldatafor(0-20%
centralitybin)[37]areshownforcomparison[32].
four description of the hadronic matter and
their comparison with the one obtained from
forthebagmodelandlatticeEOSinRef.[38]
the lattice calculations are shown in Fig. 8.
atRHIC earlier. It is to be notedthat a com-
pletedescriptionofphotondatawouldinvolve
additionof promptcontributionscaledby ap-
A. Particle spectra for HHB and HHL propriate nuclear overlapping function.
equations of state
Corresponding results for the LHC (see
Ref. [32] for detail) show that the difference
Considering an ideal hydrodynamic (az- between the transverse momentum distribu-
imuthally symmetricandlongitudinallyboost tion of photons and the hadrons for the two
invariant) evolution of the system and the EOS is further reduced. However, a slight in-
same initial conditions as used in Ref. [26] crease in the inverse slope for the HHL EOS
for Au+Au collisions at RHIC as well as for is observedas a result of the larger lifetime of
Pb+Pb collisions at LHC, transversemomen- the system at the LHC energy.
tum spectra of various hadrons and photons One can conclude from these results that
are calculated for the two equations of state. particle spectra (even for thermal photons)
The freeze-out is assumed to take place at can not distinguish between the equations of
a temperature of about 100 MeV and the state,oneadmittingafirstorderphasetransi-
particle spectra are obtained using Cooper- tionandtheotheradmittingarapidcross-over
Frye formulation [35]. Whereas for the pho- as suggestedby lattice calculations. However,
ton production, the standard rates from the it is to be noted that the history of evolution
Refs. [28, 29] are used. ofthetwosystemsisdifferentastheyaresub-
Transversemomentumspectraofpionsand jected to different rates of expansion due to
thermalphotonsforcentralcollisionofAunu- the varying speed of sound.
clei at RHIC and at midrapidity are shown in Inthesamestudyitisobservedthatthere-
Figs. 9 and 11 along with experimental data sultsforthetemporalevolutionofaverageen-
from PHENIX [36, 37] for the two cases. It ergy density, temperature and radial flow ve-
is observed that both the equations of state locityforboththeRHICandLHCenergiesdo
give a reasonable description of the particle notshowsignificantdifferenceforthetwoEOS
distribution with a slight preference for the HHL and HHB. The most profound variation
HHLEOSandtheinverseslopeofthespectra is observed for the radial velocity for the two
for HHL EOS is larger than the same for the EOS, where the radial velocity for HHL rises
HHB EOS.Similarresultshavebeenreported continuously, stays below that for the HHB,
Thermal Photons, Au+Au@RHIC A very clear difference in the underlying
20 radii for the hadronic matter contribution
HHL
(shown in Fig. 12) for the two equations of
16 HHB
∆R state is observed; where the one for the HHB
m) o
R (fo12 Ro,H equOanteiocnaonfsseteattehbateitnhgeedffieffcetrivenelcyem∆uRcho,lwarhgiecrh.
8
corresponds to the lifetime of the system is
4 Ro,Q slightlysmallerfortheHHLequationofstate,
asitdoesnotincorporatemixedphaselikethe
0 0.4 0.8 1.2 1.6 2.0 HHB. The Ro for the hadronic matter for the
KT (GeV/c) HHLequationofstateisalsomuchsmallerfor
the same reason.
FIG. 12: Transverse momentum dependence of Similar qualitative nature for the correla-
the outward radii for hadronic and quark matter
tion functions and radii is observed at LHC
sources for thermal photons [32].
energyalso. Inviewoftheseimportantobser-
vations, it would be quite interesting to ana-
lyze this behavior using a fully 3+1D hydro-
but overshoots it once the later stalls due to
dynamics [39].
the onset of the mixed phase. As a result, fi-
nal v fortheHHL EOSisslightlylargerthan
t
that for the HHB.
IV. SUMMARY AND CONCLUSIONS
B. EOS and intensity interferometry of The interference of thermal photons from
thermal photons the quark matter and hadronic matter phases
produced in relativistic heavy ion collisions
gives rise to a unique structure, specially in
It is observed that the HHL equation state
the outward intensity correlation function for
leads to a larger production of photons at
intermediate transverse momenta. The corre-
smaller radial distances as well as at inter-
lation results are found to be quite sensitive
mediate times. We know that the photons
to the initial conditions and the equation of
fromquarkmatteroriginateatearlytimesand
state of the strongly interacting matter. This
those from hadronic matter are emitted from
can be used to distinguish between the bag
the later part of the system evolution As the
model and lattice based equations of state.
interferencebetweenthetwodepends ontheir
relativecontributions,thisdifferenceholdsout
a promise that we may see a difference in the
V. ACKNOWLEDGMENTS
intensityinterferometryofphotonsforthetwo
equations of state.
It is found that the sideward correlation RC and SD gratefully acknowledge the
is essentially identical for the two equations financial supports from Finnish Academy
of state at both RHIC and LHC energies, project’Hardprocessesinheavy-ioncollisions
whereas the longitudinal correlation function attheLargeHadronCollider’andtheDepart-
shows a slight variation when the EOS is ment of Atomic Energy respectively.
changed from HHB to HHL (see Ref. [32] for
details). The results for the outward corre-
lation (shown in Fig. 10) are more dramatic References
and show a clear difference for the two equa-
tionsofstate. ThesametreatmentofRef.[26] [1] C. Adler et al. [STAR Collaboration],
is followed to estimate the source size of the Phys. Rev. Lett. 87, 182301 (2001); ibid
hadronic and the quark matter contributions. 89,132301(2002);ibid90,032301(2003);
S.S.Adleretal.[PHENIXCollaboration], [19]R. J. Fries, B. Mu¨ller, and D. K. Srivas-
Phys. Rev. Lett. 91, 182301 (2003) tava, Phys. Rev. Lett.90,132301(2003).
[2] P.Huovinen,P.Kolb,U.Heinz,P.V.Ru- [20]T. Dahms [PHENIX Collaboration], J.
uskanen, and S. Voloshin, Phys. Lett. B Phys. G 35, 104118 (2008); A.
503, 58 (2001); D. Teaney, J. Lauret, and Adare et al. [PHENIX Collaboration],
E. Shuryak, nucl-th/0110037. arXiv:0804.4168[nucl-ex].
[3] X. N. Wang, Phys. Rev. C 63, 054902 [21]D. K. Srivastava, Phys. Rev. C 71,
(2001); M. Gyulassy, I. Vitev, and 034905(2005).
X. N. Wang, Phys. Rev. Lett. 86, 2537 [22]D. K. Srivastava and J. Kapusta, Phys.
(2001). Lett. B 307, 1 (1993); D. K. Srivas-
[4] K. Adcox et al. [PHENIX Collaboration], tava and J. Kapusta, Phys. Rev. C 48,
Phys. Rev. Lett. 88, 022301 (2002); 1335 (1993); D. K. Srivastava, Phys.
J. Adams et al. [STAR Collaboration], Rev. D 49, 4523 (1994); D. K. Srivas-
Phys. Rev. Lett. 91, 172302 (2003). tavaandC. Gale,Phys. Lett. B319,407
[5] V. Greco, C. M. Ko and P. Levai, Phys. (1993); D. K. Srivastava and J. Kapusta,
Rev. Lett. 90 202302 (2003); R. J. Fries, Phys. Rev. C 50, 505 (1994); S. A. Bass,
B. Mu¨ller, C. Nonaka, and S. A. Bass, B. Mu¨ller, and D. K. Srivastava, Phys.
Phys. Rev. Lett. 90, 202303 (2003). Rev. Lett. 93, 162301 (2004).
[6] P.KolbandU.Heinz,“Hydrodynamicde- [23]A. Timmermann, M. Plu¨mer, L. Razu-
scriptionofultrarelativisticheavy-ioncol- mov, and R. M. Weiner, Phys. Rev. C
lisions,” arXiv:nucl-th/0305084. 50, 3060 (1994); J. Pisut, N. Pisutova,
[7] U. A. Wiedemann and U. Heinz, Phys. and B. Tomasik, Phys. Lett. B 345, 553
Rept. 319, 145 (1999). (1995);C.SlottaandU.Heinz,Phys.Lett.
[8] R. H. Brown and R. Q. Twiss, Nature B 391,469 (1997); D. Peressounko,Phys.
177, 27 (1956); ibid. 178, 1046 (1956); Rev. C 67,014905(2003);J.Alamet al.,
Proc. Roy. Soc. A242, 300 (1957); ibid. Phys. Rev. C67,054902(2003);T.Renk,
243, 291 (1957). hep-ph/0408218.
[9] E. Purcell, Nature 178, 1449 (1956). [24]E. Frodermann and U. Heinz, Phys. Rev.
[10]G. Goldhaber, S. Goldhaber, W. Lee and C 80, 044903 (2009).
A. Pais, Phys. Rev. 120, 300 (1960). [25]M. M. Aggarwal et al. [WA98 Collabora-
[11]S. Pratt, Phys. Rev. D 33, 1314 tion],Phys. Rev. Lett.93,022301(2004).
(1986), G. Bertsch, M. Gong and [26]D.K.SrivastavaandR.Chatterjee, Phys.
M. Tohyama, Phys. Rev. C 37, 1896 Rev. C 80, 054914 (2009) [Erratum-ibid.
(1988),G. F. Bertsch, Nucl. Phys. A 498, C 81, 029901 (2010)].
173C (1989). [27]H. Von Gersdorff, L. D. McLerran,
[12]E.Shuryak,Phys. Lett. B44,387(1973); M. Kataja and P. V. Ruuskanen, Phys.
Sov. J. Nucl. Phys. 18, 667 (1974). Rev. D 34 (1986) 794, J. Cleymans,
[13]M. Gyulassy, S.K. Kauffmann, and L.W. K. Redlich, and D. K. Srivastava, Phys.
Wilson, Phys. Rev. C 20, 2267 (1979). Rev. C 55, 1431 (1997).
[14]J. Adams et al. [STAR Collaboration] [28]P. Arnold, G. D. Moore, and L. G. Yaffe,
Phys. Rev. Lett. 92, 052302 (2004). JHEP 12, 009 (2001).
[15]S.S.Adleretal.[PHENIXCollaboration], [29]S. Turbide, R. Rapp, and C. Gale, Phys.
Phys. Rev. Lett. 96, 202301 (2006). Rev. C 69, 014903(2004).
[16]A. Adare et al. [PHENIX Collaboration], [30]J. Kapusta, L. McLerran, and D. K. Sri-
Phys. Rev. Lett. 98, 162301 (2007). vastava, Phys. Lett. B 283, 145 (1992).
[17]D. G. d’Enterria and D. Peressounko, [31]F. Marques, G. Martinez, G. Matulewicz,
Eur. Phys. J. C 46, 451 (2006). R.Ostendorf,andY.Schutz, Phys. Rept.
[18]S. A. Bass, B. Mu¨ller, and D. K. Srivas- 284, 91 (1997); F. Marques et al., Phys.
tava, Phys. Rev. Lett.90,082301(2003). Rev. Lett. 73, 34 (1994); Phys. Lett.