Direct photon production from viscous QGP

0.19 MB·

by  A. K. Chaudhuri

Direct photon production from viscous QGP
A. K. Chaudhuri∗ and Bikash Sinha†
Variable Energy Cyclotron Centre,
1/AF, Bidhan Nagar, Kolkata 700 064, India
WesimulatedirectphotonproductioninevolutionofviscousQGPmedium. PhotonsfromComp-
tonandannihilationprocessesareconsidered. Viscouseffectonphotonproductionisverystrongand
reliablesimulationispossibleonlyinalimitedpT range. Forminimallyviscousfluidη/s=0.08),di-
rectphotonscanbereliablycomputedonlyuptopT ≤1.3GeV.Withreducedviscosity(η/s=0.04),
thelimit increases to pT ≤2GeV.
PACSnumbers: PACSnumbers(s):25.75.-q,12.38.Mh
1
1 I. INTRODUCTION 6 GeV/c), interactionjets withQGP couldbe animpor-
0 tant source of direct photons [8, 9].
2
In ideal hydrodynamic models, photon production in
Recent experiments at Relativistic Heavy Ion Col-
n Au+Au collisions at RHIC energy has been studied ex-
a lider(RHIC)haveproducedconvincingevidencesthatin tensively [10, 11]. However, it is now realized that the
J √s =200 GeV Au+Au collisions, a collective medium
NN strongly interacting medium, produced in Au+Au col-
0 is produced [1–4] . Strong elliptic flow in non-central lisions, must be treated as a viscous medium. Gravity
2 collisions is key evidence to this understanding. Hy-
dualtheoriessuggestthatspecific viscosity,i.e. viscosity
drodynamical analysis of experimental charged particles
to entropy ratio of any matter has a lower bound, the
]
data, also suggests that in Au+Au collisions, a collec-
h so called KSS bound η/s = 1/4π [12, 13]. Even though,
t tive medium, with viscosity to entropy ratio close to the photons are importantprobe ofQGP matter, viscous ef-
-
l AdS/CFT lower bound of viscosity, η/s 1/4π is pro- fects on photon production are not studied much. Only
c ≥
duced [5–7]. Whether the collective medium is a decon-
u recently, Dusling [14] studied viscous effects on photon
fined Quark-GluonPlasma (QGP) or not, is still a ques-
n production from a QGP medium. However, the model
[ tion of debate. Hopefully, the issue will be settled in appears to have some inconsistency. In viscous evo-
Pb+Pb collisions at the Large Hadron Collider (LHC). lution, photon production is affected due to (i) modi-
1
v Direct photons probe the early medium produced in a fied fluid evolution and (ii) non-equilibrium correction
3 collision better than the charged hadrons. Hadrons, be- to equilibrium distribution function. In [14], while the
2 ing strongly interacting, are emitted from the surface of non-equilibrium correction to the distribution function
8 the thermalised matter and carry information about the was included, modification of the fluid evolution, due to
3
freeze-out surface only. They are unaware of the condi- viscosity was neglected. In the present paper, the incon-
.
1 tion of the interior of the matter and can provide infor- sistency is removed.
0 mationaboutthedeepinterioronlyinanindirectway. In
1
ahydrodynamicmodel,onefixesthe initialconditionsof
1
the fluid suchthat the ”experimental”freeze-outsurface II. HYDRODYNICAL EQUATIONS, EQUATION
:
v iscorrectlyreproduced. Incontrasttohadrons,photons, OF STATE AND INITIAL CONDITIONS
Xi beingweaklyinteracting,areemittedfromwholevolume
of the matter. Throughout the evolution of the matter, In a hydrodynamicalmodel, the invariantdistribution
r
a photons are emitted. Conditions of the produced mat- of direct photons is obtained by convoluting the photon
ter, at its deep interior, are better probed by the pho- production rate with space-time evolution of the fluid.
tons. Depending on the transverse momentum, direct We assume that in √s =200 GeV, Au+Au collisions
NN
photons can probe different aspects of heavy ion colli- atRHIC, abaryonfree QGPfluidis formed. Space-time
sions. Athermalisedmediumofquarksandgluons,orof evolution of the fluid is obtained by solving 2nd order
hadrons,canproducesignificantnumberofthermalpho- Israel-Stewart’s theory,
tons. They are low p photons (p 3 GeV/c). Low
T T
≤
p photonscantestwhetherornot, QGPis producedin
T
Au+Au collisions. Hard photons (pT > 6 GeV/c) are of ∂µTµν = 0, (1)
pQCD origin and test the pQCD models. Fast partons
1
from’jet’ caninteractwiththermalpartonsofQGP and Dπµν = (πµν 2η <µuν>)
−τ − ∇
produce photons. At intermediate p range, (3 p π
T T
≤ ≤ [uµπνλ+uνπνλ]Du . (2)
λ
−
Eq.1 is the conservation equation for the energy-
∗E-mail:[email protected] momentum tensor, Tµν = (ε + p)uµuν pgµν + πµν,
−
†E-mail:[email protected] ε, p and u being the energy density, pressure and fluid
2
velocity respectively. πµν is the shear stress tensor equations are then solved till the freeze-out temperature
(we are neglecting bulk viscosity). Eq.2 is the relax- T = T =170 MeV. At the initial time τ , initial energy
F c i
ation equation for the shear stress tensor πµν. In Eq.2, density is assumed to be distributed as [5]
D = uµ∂ is the convective time derivative, <µuν> =
µ
1( µuν + νuµ) 1(∂.u)(gµν uµuν) is a∇symmetric
2 ∇ ∇ − 3 −
traceless tensor. η is the shear viscosity and τπ is the ε(b,x,y)=ε [(1 f N (b,x,y)+f N (b,x,y)],
relaxation time. It may be mentioned that in a confor- i − hard part hard coll
(3)
mally symmetric fluid relaxation equation can contain
where b is the impact parameter of the collision. N
additional terms [15]. Assuming boost-invariance, Eqs.1 part
and 2 are solved in (τ = √t2 z2,x,y,η = 1lnt+z) andNcoll arethe transverseprofileofthe averagepartic-
− s 2 t−z ipant and collision number respectively, calculated in a
coordinates, with the code ”‘AZHYDRO-KOLKATA”’,
Glauber model. f =0.13 is the hard scattering frac-
developed at the Cyclotron Centre, Kolkata. Details of hard
tion [21]. ε is the central energy density of the fluid
the code can be found in [6, 7, 16]. i
in impact parameter b = 0 collision. As it will be dis-
Eqs.1,2 are closed with an equation of state p =
cussed later, we have simulated Au+Au collisions for
p(ε). Lattice simulations [17–20] indicate that the
a range of initial (central) energy density and initial
confinement-deconfinement transition is a cross over,
time. We also assume that initial fluid velocity is zero,
ratherthana 1stor2ndorderphase transition. There is
v (x,y) = v (x,y) = 0. The shear stress tensor was ini-
no critical temperature for a cross-overtransition. How- x y
tialized with boost-invariant value, πxx = πyy = 2η/3τ ,
ever, one can define a pseudo critical temperature, the i
πxy=0. For the relaxation time, we use the Boltzmann
inflection point on the Polyakov loop. In Wuppertal-
estimate τ = 3η/4p. We also assume that shear viscos-
Budapestsimulation[19,20],pseudocriticaltemperature π
ity to entropy ratio is a constant throughout the evolu-
T 170 MeV. In the present simulation, for the QGP
c ≈ tion. In the following, we simulate Au+Au collisions for
phase,weuseanEOSbasedonWuppertal-Budapestsim-
η/s=0-0.12.
ulation.
III. PHOTON RATES
Solution of partial differential equations (Eqs.1,2) re-
quires initial conditions, e.g. transverse profile of the
energy density (ε(x,y)), fluid velocity (v (x,y),v (x,y)) Asmentionedearlier,inviscousevolutionphotonrates
x y
and shear stress tensor (πµν(x,y)) at the initial time τ . are modified. The photon rate equations involve distri-
i
One also need to specify the viscosity (η) and the relax- bution functions of quarks and gluons. For example, in
ationtime(τ ). Afreeze-outtemperatureisalsoneeded. 1+2 3+4 processes (e.g. Compton and annihilation
π
→
In the following, we will consider viscous effects on pho- processes), the general form of photon production rate
tons from the QGP phase only. The hydrodynamical is,
E4dd3Rp4 =N Z (2πd)33p21E1(2πd)33p22E2f1(E1)f2(E2)(2π)δ4(p1+p2−p3−p4)|M|2[1±f3(E3)](2πd)33p23E3 (4)
where M is the matrix element for the reaction. f’s in
Eq.4aredistribution functionofquarksandgluons. Un-
dR 2 sσ(s)
lfiuknecatinonidsefalisflmuiodd,iifinedvidscuoeutsoenvoonlu-etqiounil,ibearicuhmdcisotrrriebcuttiioonn, Ed3p ≈ (2π)6f1(E)Z d3p2f2(E2)[1±f3(E2) E2 (7)
Non-equilibrium correction to distribution functions
f f (1+δf ),δf <<1. (5) insidethephasespaceintegralleadstocorrectionsinthe
neq eq neq neq
→ logarithmic term and can be neglected. For equilibrium
The non-equilibrium correctionδfneq depend on dissi- distribution function, as demonstrated by Kapusta and
pative forces as well as on particle momenta. For shear Lichard[22],thephaseintegrationcanbeexplicitlyeval-
viscosity, it can be obtained as, uatedandoneobtainforbothComptonandannihilation
processes [14],
1
δf =Cp p πµν = p p πµν (6)
neq µ ν 2(ε+p)T2 µ ν
dR 5α α 3.7388E
E e sf (E)T2ln (8)
Itisobviousthatnon-equilibriumcorrectiontophoton d3p ≈ 9 2π2 neq (cid:20) g2T (cid:21)
rates is non-trivial. For Compton and annihilation pro-
cesses, in the leading log approximation, the rate equa- The invariant photon distribution then has two parts,
tion can be simplified [14], equilibrium part (EdNeq) and a non-equilibrium part
d3p
3
101 0.05
100 20-40% Au+Au 0.04 20-40% Au+Au
t i=0.6 fm, Ti=350 MeV t i=0.6 fm, Ti=350 MeV
10-1 0.03
-2V) 10-2 0.02
e w
(GT 10-3 c flo 0.01
2ydp 10-4 ellipti 0.00
d
N/ 10-5
d
-0.01
10-6 h /s=0 h /s=0
10-7 h /s=0.08 (non-eq. corr. neglected) -0.02 h /s=0.08 (non-eq. corr. neglected)
h /s=0.08 (non-eq. corr. included) h /s=0.08 (non-eq. corr. included)
10-8 -0.03
0 1 2 3 4 5 0 1 2 3 4 5
pT (GeV) pT (GeV)
FIG. 2: (color online) same as in Fig.1 but for elliptic flow.
FIG. 1: (color online) Transverse momentum spectra for
photons, from evolution of ideal and minimally viscous
(η/s=0.08)QGP.Forviscousfluid,spectraobtainedwithand
without the non-equilibrium correction to distribution func- (δN ) to photon spectra equals the equilibrium contri-
neq
tion, are shown separately. bution N ). For minimally viscous fluid, the equality
eq
occur at p 1.7 GeV. As noted earlier, viscous hydro-
T
≈
dynamics is applicable only when δN /N <<1. Ev-
(EdNneq). Since non-equilibrium correction to distribu- neq eq
d3p idently, for η/s=0.08, in a hydrodynamic model, photon
tion function is assumed to be small, it is essential that, productionfromComptonandannihilationprocessescan
not be reliably computed beyond p =1.7 GeV. Indeed,
T
if we assume that viscous hydrodynamics remain reli-
dN dN
E neq <<E eq. (9) able only until, δNeq = 0.5, the photons from Compton
d3p d3p Neq
and annihilation processes can be computed only up to
p 1.3 GeV.
T
≈
In Fig.2, we have shown the simulation results for el-
IV. EFFECT OF VISCOSITY ON PHOTON liptic flow for photons. At very low p elliptic flow is
T
SPECTRA AND ELLIPTIC FLOW negative. At low p 0.5GeV present model is not
T
≤
reliable. The logarithmicfactor inthe photonrate equa-
Gravity dual theories [12] indicate that viscosity to tion is negative when E < 0.27g2T. For fluid (central)
entropy ratio of strongly interacting matter is bounded temperature T0=350 MeV, reliable computation is pos-
from the lower side, η/s 1/4π. We first consider ef- sible only beyond pT =0.5 GeV. In ideal fluid evolution,
≥
fect of minimal viscosity on photon production. We as- at pT >0.5 GeV, elliptic flow increases with pT, till a
sume that minimally viscous (η/s=0.08) QGP fluid is maxima is reached at pT 2 GeV. At even higher pT, it
≈
thermalised in the time scale τi=0.6 fm to initial central decreasesagain. Notethateventhehighestv2(pT)isnot
temperature T0=350 MeV. A large number of charged large, less that ∼ 2%. Compare this value to v2(pT) ∼
particle’s data e.g. identified particles spectra, elliptic 20% for charged particles. In viscous evolution, elliptic
flow etc. data are explained in hydrodynamical model flow is reduced further. Incidentally, very small photon
with similar initial time and temperature scale [5]. In elliptic flow is consistent with experiments. In experi-
Fig.1, simulated photon spectra in 20-40% Au+Au col- ments also, photons do not flow [27, 28]. In Fig.2, one
lisions, from evolution of ideal and minimally viscous observes that if non-equilibrium correction is neglected,
QGP fluid, are shown. The spectra with or without ellipticflowinviscousevolutionismorethanflowinideal
the non-equilibrium correction are shown separately. If fluid evolution. However, with non-equilibrium correc-
non-equilibriumcorrectiontodistributionfunctionisne- tion included, flow is reduced in viscous evolution. It
glected, p spectra of photons is increased by a factor shows that it is important to have a consistent model,
T
of 1.2-1.5. The increase is largely p independent. otherwise,onecanconclude wrongly. The arrowinFig.2
T
∼
Incontrast,whennon-equilibriumcorrectionisincluded, indicate that approximate pT when δNneq Neq.
≈
photon production is increasedmore at large p than at Photon spectra from Compton and annihilation pro-
T
low p , It is also expected, non-equilibrium correction cesses,forfourvaluesofviscositytoentropyratio,η/s=0
T
increaseswithp (seeEq.6). ThearrowinFig.1indicate (idealfluid),0.04,0.08(AdS/CFTlowerbound)and0.12
T
the approximate p when non-equilibrium contribution are shown in Fig.3. Initial time and temperatures are
T
4
100 101
20-40% Au+Au
T0=350 MeV, t i=0.6 fm
h /s=0 100
10-1 h /s=0.04
-2V) hh //ss==00..1028 -2V) 10-1
e e
G G
(T 10-2 (T 10-2
p p
2 2
d d 20-40% Au+Au
y y
N/d N/d 10-3 sit i=60 fm-2, h /s=0.08
d 10-3 d t i=0.2 fm
t i=0.6 fm
10-4 t i=1.0 fm
t i=1.4 fm
10-4 10-5
0 1 2 3 0 1 2 3
pT (GeV) pT (GeV)
FIG.3: (coloronline)PhotonspectrafromevolutionofQGP FIG.4: (coloronline)Initialtimedependenceofphotonspec-
fluid, with viscosity to entropy ratio, η/s=0, 0.04, 0.08 and trafromevolutionofQGPfluidwithviscosityη/s=0.08. The
0.12 are shown. Theinitial timeis τi=0.6 fm and initial cen- initialentropydensitytimestheinitialtimeisfixedatsiτi=60
tral temperature is Ti=350 MeV. fm−2.
τ =0.6 fm and T =350 MeV. As expected, high p yield be used to put stringent bound on QGP viscosity and
i i T
is increased in more viscous fluid evolution. The arrows initial (formation) time.
inFig.3indicatetheapproximatepT whenδNneq =Neq. In Fig.4, simulated photon spectra from evolution of
Photonproductioncannotbe computedreliablybeyond minimally viscous (η/s=0.08)QGP fluid, for four values
pT=1.3, 1.7 and 2.4 GeV, for QGP viscosity, η/s=0.12, ofinitialtimeτi=0.2,0.6,1.0and1.4fmareshown. The
0.08 and 0.04 respectively. Very limited pT range over initial temperature is obtained from the condition that
whichviscoushydrodynamicsremainapplicablefor pho- initialentropydensitytimestheinitialtimeisaconstant,
ton production raises certain issues, which will be dis- s τ = 60fm−3. As the initial time is reduced, photon
i i
cussedlater. We justmentionthatviscouseffectonpho- spectraarehardened. Photonyieldisalsoincreased. Itis
ton production is much stronger than in charged par- easilyunderstood,forfixeds τ ,initialtimeandtemper-
i i
ticles. Indeed, for charged particles, viscous hydrody- atureareinverselyrelated. Astheinitialtimeisreduced,
namics remain applicable over a much wider pT range high pT photon production is increased due to increased
[6, 7]. The reason is understood. Photons are emitted fluid temperature. The arrows in Fig.4, indicate the p
T
through out the evolution. At early time, shear stress when non-equilibrium correction to spectra is equal to
tensorshavefinite valuesandnon-equilibriumcorrection the equilibrium contribution.
tophotonproductionislarge. Incontrast,chargedparti- ToobtaintheinverseslopeparameterT ,wehavefit-
eff
clesareemitted fromthe freeze-outsurface. Shearstress ted the spectra in the p range 1.5 p 2.5GeV with
T T
tensorsevolvesveryfast. Comparedtoearlytimes,stress an exponential (dN/d2pT e−pT/Te≤ff ). ≤Ideal hydrody-
tensorsatfreeze-outaremuchsmallerinmagnitude. Ac- namicsimulationsforphot∝onspectrasuggestthatinthis
cordingly, non-equilibrium correction is small. We have p range, QGP photons dominate the spectra [10, 11].
T
not shown simulation results for elliptic flow as a func- InFig.5,forfluidviscosity,η/s=0,0.04and0.08,inverse
tion of viscosity. As shown earlier, v2(pT) for photons slope parameter is shown as a function of initial time.
is very small in ideal fluid evolution. It further reduces Inverse slope parameter T decreases with increasing
eff
with η/s. initialtime,aswellaswithdecreasingviscosity. InFig.5,
Since photons are emitted throughout the evolution, the shaded region indicates the experimental slope mea-
including the earliest phase of QGP, one hope to ex- sured in the PHENIX experiment. For ideal fluid, sim-
tract QGP formation time from photon measurements. ulated T agree with experiment for τ 1 fm. For
eff i
≥
Indeed, in [25, 26] it was suggested that photon ellip- viscousfluid,inthetimescale,0.2 1.0fm,inverseslope
−
tic flow can be used to constrain QGP formation time. ofthesimulatedspectraarehigherthanthatobservedin
However, from the experiments [27, 28], it appears that experiment. However, we have neglected photons from
photons do not seem to experience flow. One can not the hadronic phase. Hadronic photons will dominantly
possibly extract QGP formation time from elliptic flow contribute at low p , reducing the inverse slope param-
T
measurements. In [14], Dusling suggested that inverse eter. In other word, in more realistic simulation, T
eff
slope of the photon spectra, if measured accurately, can could be smaller than obtained presently and even for
5
0.55 viscous fluid, depending on the initial time, applicabil-
20-40% Au+Au ity rangeis lessthan p =1.5GeV. Applicability rangeis
0.50 T
sit i=60 fm-2 even less in more viscous fluid. Present simulations ne-
0.45 h /s=0.0 glect Bremsstrahlung photons in the QGP phase. How-
h /s=0.04 ever, their inclusion will not improve the situation. One
0.40 h /s=0.08 wondersaboutthe photonsinthe p range1.5 p 3
V) T ≤ T ≤
e GeV. As noted earlier, ideal hydrodynamics simulations
(Geff 0.35 innadtiec.aHteowtheavteri,nprtehsiesnptTsimraunlgaeti,onthseirnmdaiclatpehothtoantsifdQoGmPi-
T 0.30
fluid is minimally viscous, the photons in the p range
T
0.25 1.5 pT 3GeVcannotbeconsideredthermalandcan
≤ ≤
0.20 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(c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nthoetmbeasdepsQcrCibDedphhoytdornosd?ynOarmaircealtlyh.eyDforowmeaunfldueidrstwainthd
viscositymuchlessthanthe AdS/CFTlowerbound? In-
0.15
deed, in certain gravity dual theories, KSS bound can
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
be violated by 36% [29]. Limited p range over which
T
t i (fm) photons can be∼reliably computed in viscous dynamics
alsoindicate thatwithin a samemodel, one possibly can
FIG. 5: (color online) Effective temperature from photon notexplain,chargedparticleproductionandphotonpro-
spectra for initial time τi=0.2, 0.6, 1.0 and 1.4 fm. The ini- duction.
tial entropy density times the initial time is fixed at siτi=60
fm−2. Resultsareshownforviscositytoentropyratioη/s=0,
0.04 and 0.8 respectively. The two dashed lines indicate the
experimental uncertainty in PHENIX measurements.
V. SUMMARY
viscous fluid could be in agreement with experiment. To summarize, we have studied effect of shear viscos-
To constrain initial time and viscosity from inverse ityonComptonandannihilationphotons. Inviscousdy-
slope of photon spectra will not be an easy task. As namics,photonproductionismodifieddueto(i)changed
seen in Fig.5, inverse slope parameter from evolution of space-timeevolutionofthefluidand(ii)non-equilibrium
lowviscosityfluidinitializedatsmallτ couldbeconfused correction to the equilibrium distribution function. The
i
with T from high viscosity fluid initialised at large τ . non-equilibrium correction grows with viscosity as well
eff i
AsshowninFig.5,forη/s=0.04 0.04,dependingonthe with transverse momentum. Viscous effects on photon
±
initialtime,changeininverseslopeparametersis 20-80 production are strong. Even for AdS/CFT lower bound
∼
MeV.Depending onthe realisticsituation,measurement of viscosity (η/s=0.08), strong viscous correction render
of inverse slope parameter with accuracy δT 20-80 the hydrodynamicsinapplicable beyondp 1.5GeV. If
eff T
≈ ≈
MeVcouldpossiblygivearangeofvaluesforinitialtime QGP viscosity is larger than the Ads/CFT limit, then
and viscosity. limitation will be even more. Photon production as
Before we summarise, few comments are in order. We a function of initial time, also suggest that if the in-
have shown that effect of viscosity is quite strong on verseslopeparameterofthephotonspectra,ismeasured
photon production. Indeed, viscous hydrodynamics in within an accuracy of 20 MeV, one can possibly limit
±
applicable only in a limited p range. For minimally the initial time and viscosity.
T
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