Relativistic Aspects of Nuclear Physics

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Dissipative hydrodynamics in RHIC LHC
energy A. K. Chaudhuri Variable Energy
Cyclotron Centre, Kolkata
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Introduction why do we need
dissipative hydrodynamics? Relativistic fluid
dynamics brief theory, 21
dimensional Hydrodynamics with shear viscosity
Minimally viscous fluid, 1st order phase
transition AuAu_at_RHIC,
differential v2, pT-spectra,
centrality dependence of dNch/dy, ltpTgt, pT
integrated v2. PbPb_at_LHC
some predictions Viscous fluid, 2nd order phase
transition AuAu_at_RHIC
PbPb_at_LHC Summary conclusions
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QCD lattice simulations indicate that at high
temperature confined hadronic states undergoes a
phase transition to a deconfined state called
Quark Gluon Plasma.
Cheng et al. 0710.0354 simulation with two
light quark flavor heavier strange Quark, with
almost physical current quark masses (pion
mass220 MeV). Energy density, pressure, entropy
density do not attain non-interacting
Stefan-Boltzmann limit at high temperature.
Strongly interacting QGP (sQGP)! A second order
confinement-deconfinement phase transition at
Tc196(3) MeV. KarschLaermann,
hep-lat/0305025, Tc170(15) MeV.
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There are enough evidences that AuAu collisions
has produced some form of sQGP
high pT suppresion
disappearance of away side jet
elliptic flow in non-central coll. explained in
hydro models.
elliptic flow is not explained in transport
models unless partonic cross-section is large
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Hydrodynamics provide a convenient model to
analyse RHIC data. (at sufficiently high
temperature, the equilibrium state of any field
theory can be regarded as a fluid.)
freeze-out at TF
AuAu collision can be viewed as
hadronic phase TltTc
mixed phase at Tc
QGP phase
ti,e(x,y,h), v(x,y,h)
Freeze-out
Equation of State
inputs required ideal hydro
ideal hydrodynamicsti0.6 fm, ei30 GeV/fm3,
TF100 MeV, EOS (QGPHRG, 1st order phase
transtion at Tc164 MeV HeinzKolb), explains a
large volume of RHIC data, e.g. pT-spectra,
elliptic flow upto pT1.5 GeV. Not HBT.
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shortcomings of ideal fluid dynamics may be
removed in dissipative hydrodynamics. Characterise
QGP by its dissipation coefficients e.g.
viscosity, conductivity.
Viscosity
shear viscosity ability to transfer Momentum.
Opposes flow.
Bulk viscosity resistance to expansion
Arnold, Moore, YaffeJHEP05(2003) weakly coupled
QCD, h/s ? 1 Meyer PRD07 gauge theory,h/slt 1,
best estimate h/s0.134 at T1.6 Tc. Son et al
PRL87 strongly coupled ADS/CFT, h/s 1/4p
z0, in a conformal theory Kharzeev,
Tuchin0705.4280 Recent lattice dataLow energy
theorem z/s?0.8 at Tc
shear or bulk viscosity from first principle for
a strongly coupled QCD medium is yet to be
obtained. Can hydro simulations give
phenomenological limit? possibly no.
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Muronga PRL88(2002) Solves 2nd order theory in
one dimension. Generated the interest in the
community. With Rischke nucl-th/0407114 solve
2nd order theory 11 dimension. D. Teaney
PRC68(2003) 1st order theory, blast wave
model. Compared with RHIC data. Data require
viscosity less than perturbative estimate.
ChaudhuriHeinznucl-th/0504022solved 2nd order
theory in 11 dimensions. Heinz, Song and
Chaudhuri PRC73(2006)give explicit equations
for 2nd order theory in 21 dimensions.
Chaudhuri PRC74(2006) solves 1st order theory
in 21 dimensions. Baier, Romatschke, Wiedemann
nucl-th/0602249 re-derived relaxation
equations from kinetic theory. Romatschke
Romatschke nucl-th/07061522 solved 2nd order
theory in 21 dimensions, compare with
expt. Chaudhuri 0704.0134, 0708.1252,
0801.3180 solves 21 for QGP phase and later
with phase transition. compare with expt. Heinz
Song0709.0742 ,0712.3715  solves 2nd order
theory in 21 dimension. Koide, Denicol, Mota and
Kodama PRC75(2007) A new formulation of causal
dissipative hydrodynamics. More simpler than
Israel-Stewart formulation. Trying to implement
the theory in 31 dimensions. Van
Biro0704.2039 Give a new concept that
internal energy can be divided in to dissipative
and non-dissipative part. 1st order theory can be
stable. Dusling Teaney PRC08 a different
Ottinger-Grmela formulation
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Brief theory of dissipative hydrodynamics Israel
and Stewart Ann. Physics,118(1979) Simple
fluid is fully specified by primary variables Nm,
Tmn and Sm, and an unspecified number of
additional variables. Primary variables satisfy
conservation laws and the entropy law.

define a time like hydrodynamic 4 velocity u
(u21) and projector
In equilibrium, an unique hydrodynamic
4-velocity u exists such that,
An equilibrium state is fully specified by
5-parameters, (n, e,u) or (am/T,bmum/T). They
span a 5-dim. hyperspace S0 (a,bm).
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in a state close to an equilibrium state,

Qm undetermined quantity of 2nd order in the
deviations
entropy production can be expressed as

dissipative flow X thermodynamic forces

for shear viscosity only
2nd order theory
1st order theory
Causality is violated. If in a fluid cell,
thermodynamic force happen to vanish,
dissipative flows stops immediately.
relaxation equation for the shear stress tensor
makes the theory causal. In a conformal fluid
there could be additional terms.
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Viscous hydrodynamics with boost-invariance
Non-zero Christoffels
Flow velocity
3 energy-momentum conservation equations



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Properties of viscous fluxes
5-independent components. Assumption of
boost-invariance reduces independent components
to 3. dependent components are obtained using
tracelessness and transversality condition. We
choose,

relaxation eqs.

3-conservation equations and 3 relaxation
equations are solved by,
AZHYDRO-KOLKATA developed at the Cyclotron
Centre, Kolkata.



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Minimally viscous fluid, h/s1/4p, in QGP and
Hadronic phase. Temp. independent. Csernai,Kapusta
McLerran nucl-th/0604032, h/s has a minima
around Tc.
EOS Bag modelHadronic resonance gas (HRG), 1st
order transition, Tc164MeV. Initial
conditions initial time ti 0.6 fm energy
density 75 participant density 25 hard
collision central entropy density in b0 AuAu
collision, Si110 fm-3 or ei 30 GeV/fm3

independent shear stress tensor components.should
be obtained by confronting expt. Assume attained
boost invariant values.

Relaxation time Boltzmann gas value,
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Israel-Stewarts relaxation equation
R
simplified IS equation.
For minimally viscous fluid (h/s0.08), Fluid
evolution marginally depend on the term R. For
larger viscosity the term has more effect and
should be included in the relaxation eq. I will
present some simulation results with simplified
IS eq. for minimally viscous fluid.
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chaudhuri/0801.3180nucl-th
TESTING AZHYDRO-KOLKATA
fluid at the centre follow 1D scaling expansion
recover ideal hydro results as viscosity
gradually reduces.
maintain symmetry, stable against change in
integration step length
AZHYDRO-KOLKATA pass all the tests.
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Temporal evolution of momentum anisotropy
anisotropy
QGP phase
AZHYDRO-KOLKATA reproduces within 10 or less the
ep simulated by Song and Heinz.
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Transverse momentum and elliptic flow
Hydrodynamics give e( or T), and vx, vy. The
information need to be converted into particle
spectra to connect with experiment. Cooper-Frye
prescription
In viscous dynamics, system is not in equilibrium
and f(x,p) can not be approximated by the
equilibrium distribution function
In a slightly off-equilibrium system
With only shear viscosity
non-equilibrium correction increases
quadratically with momentum. Large pT
particles are more affected by viscosity than low
pT particles.
Invariant distribution
Elliptic flow

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Fitting data increased parameter space makes
data fitting complex in viscous dynamics. We
assume viscous fluid also have same initial
condition as in ideal dynamics,
simplified IS
and
vary TF from 160-130 MeV and compare with the
PHENIX data on pT dependence of v2 in 16-23
AuAu collisions.
(i)eq. contribution ve, marginally changed from
TF130-160 MeV. (ii)non-eq. contribution ve.
Contribute less with lowering TF.
(iii)TF160-140 MeV, viscous dynamics produces
less v2 than in expt. (iv)TF130 MeV, data are
explained upto pT3.6 GeV. Small non-eq.
correction. (v)v2 do not show saturation
(though rate of increase slows down)
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pT spectra of p-,K and proton
chaudhuri/0801.3180nucl-th
(normalised by N1.4)
simplified IS
ADS/CFT lower bound is consistent with pT
spectra. Comparable description could not be
obtained in ideal dynamics.
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chaudhuri/0801.3180nucl-th

simplified IS
pT integrated v2 10-20 less in viscous
dynamics.
Centrality dependence mid-central collisions
reasonably explained. central collisions under
predicted. peripheral collisionsover predicted.
Min.bias v2 well reproduced (even saturation).
Moderate description of v2 with ADS/CFT lower
bound on viscosity. Note description of
differential v2 is much worse in ideal dynamics.
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centrality dependence of dN/dch and mean ltpTgt
simplified IS
simplified IS


Normalised yield reproduces multiplicity data
for Npart 100.
ltpTgt is reproduced for Npart100.
ADS/CFT lower bound on viscosity is consistent
with centrality dependence of multiplicity and
mean pT in collisions with Npart 100 fm.
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Hydrodynamic predictions for PbPb collisions at
LHC.
Initial condition at LHC
Particle multiplicity in Npart350 Collisions as
a function of cm. energy.
Extrapolated particle multiplicity At LHC 927
70. Initial temperature Ti420 MeV (ini. Temp at
RHIC350 MeV). Apparently, if multiplicity
increases logarithmically with energy,
initial temperature is only 20 haigher at LHC
than in RHIC.
Soft physics will not be altered extensively.
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Particle multiplicity
simplified IS
Particle multiplicity increase by a factor of
1.6-1.8.
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Mean PT
Pion spectra
simplified IS
simplified IS
10 higher mean pT at LHC.
Slight hardening of pT spectra
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Minimum bias elliptic flow
simplified IS
15 reduction in v2.
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Centrality dependence of v2
simplified IS
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unlike in ideal evolution, in viscous evolution
entropy is generated. Then depending on
viscosity, initial entropy may be large or small
for producing a fixed final state entropy.
Initial temperature/energy density of the fluid
may not be uniquely determined from experimental
data. Study the interrelation of viscosity and
initial energy density or temperature Complete
IS equation
For this study, we use lattice based EOS with 2nd
order phase transition. Most of the hydrodynamics
for RHIC are done with EOS with 1st order
transition, even though lattice simulations
indicate otherwise. Huovinen,
NPA76192005)296 2nd order phase transition
give poorest descirption to AuAu data (in
particular proton/anti-proton data).
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Cheng PRD77(2008)lattice simulation with
increased resolution and almost physical current
quark masses.


Parameterise entropy density.
thermodynamic relation
1st order
2nd order
Lattice based EOS is softer than Bag
modelHRG. In QGP and Hadronic phase we use the
lattice based EOS.
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study (i) ideal fluid and (ii) viscous fluid
with h/s0.08 and 0.16 . Initial time ti0.2
fm. small initial time suggested in photon
analysis ChatterjeeSrivastava0809.0548, J/y
analysis suggest even smaller time
chaudhuriPLB655,241. initial fluid velocity
zero. Freeze-out at temperature TF150
MeV Initial shear stress tensor
boost-invariant, relaxation time Boltzmann
estimate.
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energy density 10,15, 20 and 25 GeV/fm3
p- pT spectra(resonance production neglected,
normalise by N1.6). Depend on initial energy
density As e0 increases from 10?25 GeV/fm3,
yield increase by factor of 3-4. Data fitting
will require ((i) h/s0, e0 gt25
GeV/fm3 (ii) h/s0.08, e0 ?15 GeV/fm3 (iii)
h/s0.16, e0 lt10 GeV/fm3
p- Elliptic flow Marginal dependence on
e0. Elliptic flow is spatial origin.
Initial spatial anisotropy (independent of energy
density) is converted into momentum anisotropy.
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(i)ideal fluid, h/s0, e030
GeV/fm3, Ti360 MeV (ii)viscous fluid
h/s0.08, e015 GeV/fm3, Ti300
MeV (iii)viscous fluid h/s0.16, e0 8
GeV/fm3 ,Ti260 MeV
1st order PT ti0.6fm,Ti350MeV,TF130MeV
2nd order PT
centrality dependence ltpTgt, multiplicity and
integrated v2, do not distinguish between
appropriately initialised ideal or viscous
fluid. Lattice based EOS with 2nd order phase
transtion at Tc196 MeV, describe RHIC data on
centrality dependence of ltpTgt, multiplicity and
integrated v2, .
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minimum bias v2

upto pT2 GeV, min. bias v2 donot distinguish
between ideal and viscous fluid. pT gt 2 GEV,
V2decreases with increasing viscosity pT3
GeV, h/s 0?0.16, v2 decreases by 7. pT gt 3
GeV, more decrease, (however at large pT , pQCD
processes also contribute.)
1st order PT ti0.6fm,Ti350MeV,TF130MeV
2nd order PT
minimum bias v2 distinguish ideal or viscous
fluid only at large pTgt3 GeV
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Upto pT2 GeV, particle spectra donot
distinguish between suitably initialised ideal
or viscous fluid. Lattice based EOS reproduce,
(i) pion spectra upto 2.0 GeV, (ii) kaon
spactra upto 1.5 GeV, (iii) proton spectra not
reproduced.
Lattice EOS donot correctly characterise hadronic
phase. Effective degrees of freedom in hadronic
phase is much small (ghad?2) than in
Hadronic resonance gas (HRG). Particle content in
lattice EOS is limited. EOS is softer than in a
HRG. Present lattice simulations are done with
twice the physical current quark mases.
Simulation with physical current quarks masses
possibly give better description to the proton
data.
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ltpTgt, dN/dy, ltv2gt_at_ LHC
PbPb_at_LHC
min. bias v2 decreases at PbPb_at_LHC
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Summary and Conclusions (i)simulation studies
clearly indicate that unless shear viscosity is
known, fixing initial temperature from p_T
independent observables is not possible. Ideal
fluid with central temperature Ti360MeV,
produces nearly same pion multiplicity, mean pT
or integrated v2, as minimally viscous fluid
initialised with temperature Ti300 MeV. Intitial
temperature can reduced by 30 if fluid is more
viscous (h/s0.16). p_T dependent observables
possibly distinguish between ideal and viscous
but only at pTgt 3 GeV, where other processes
contribute. (ii) Lattice based EOS with 2nd
order phase transition can possibly explain part
of the RHIC data. However, proton or heavier
particles will be under produced. As of
now, lattice based EOS need to be complimented by
HRG for hadronic state EOS. (iii)PbPb_at_LHC
compared to AuAu_at_RHIC,
multiplicity increases moderately, by factor of
1.6-1.8, pT spectra harden
slightly, (mean pT increase by 10)
elliptic flow decreases by 15.
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(iv) Bulk viscosity should be accounted for
proper characterisation of the initial condition
of the fluid. Bulk viscosity essentially reduces
the scalar pressure. Radial flow will be reduced,
and one will require still lower initial
temperature to fit the data. (ii) freeze-out
fixed TF freeze-out is unsatisfactory. Even
without dissipative effects, one can very well
produce a different set of initial conditions,
with a different freeze-out temperature. Only
when consensus is reached on the final state, one
can try to compute the initial state.
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Csernai,KapustaMcLerran, nucl-th/0604032
Shear Viscosity



?/s has a minimum at the critical point where
there is a cusp. p lt pcrit, ?/s has a
discontinuity. p gt pcrit, ?/s show broad
minima. Model calculations need to account the
behavior of ?/s .