Rapid cross over transition
C. A. S ALGADO
II.6. LHC and RHIC Data on Elliptic Flow: v 2
● Momentum space:
Reaction plane
€
E dN
d3p = 1 2π
dN
pTdpTdη
[
1+ 2v2( )pT cos 2((
φ −ψreaction plane)) ]
€
N ~ 100 ⇒1 N ~ O(v2)
€
1 N3 4 ~ 0.03 << v2
• ‘Non-flow’ effect for 2nd order cumulants
• Signal implies 2-1 asymmetry of particles production w.r.t. reaction plane.
€
v2 ≈ 0.2
Non-flow effects should disappear if we go from 2nd to 4th order cumulants.
(2.11)
2nd order cumulants do not characterize solely collectivity.
(2.12)
(2.13)
U.A.Wiedemann
b
Φ
๏ Non-central collisions
•
asymmetry in transverse planeasymmetry in momentum distribution
๏ Reaction plane is hard to determine
•
Azimuthal correlation of particles“non-flow effects”
(correlations involving few particles)
•
Correlations among more particles reduce further sensitivity to non-flow Borghini, Dinh and Ollitrault 01Heavy Ion Collisions J. Casalderrey-Solana TAE 2016
Measured Anisotropy
32
II.6. LHC and RHIC Data on Elliptic Flow: v
2● Momentum space:
Reaction plane
€
E dN
d3p = 1 2π
dN
pTdpTdη
[
1+ 2v2( )pT cos 2((
φ −ψreaction plane)) ]
€
N ~ 100 ⇒ 1 N ~ O(v2)
€
1 N3 4 ~ 0.03 << v2
• ‘Non-flow’ effect for 2nd order cumulants
• Signal implies 2-1 asymmetry of particles production w.r.t. reaction plane.
€
v2 ≈ 0.2
Non-flow effects should disappear if we go from 2nd to 4th order cumulants.
(2.11)
2nd order cumulants do not characterize solely collectivity.
(2.12)
(2.13)
U.A.Wiedemann
๏ A very large effect
v
2=0.2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.6
0.8 1.0 1.2 1.4
f 1+2v 2Cos@2fD
Heavy Ion Collisions J. Casalderrey-Solana TAE 2016
A Geometric Property
33
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14 Introduction
y
x y’ x’
Figure 1.10: An illustration showing the standard eccentricity and the participant ec- centricity (see text for details).
However even at fixed impact parameter, the number of individual nucleons partic- ipating in the collision as well as their positions in the transverse plane could fluctuate from event to event. As a consequence, the center of the overlap zone can be shifted and the orientation of the principal axes of the interaction zone can be rotated with respect to the conventional coordinate system. To correct for this the participant eccentricity is defined by
!part = !y!2 −x!2#
!y!2 +x!2# (1.8)
where the eccentricity is calculated relative to the new coordinate system defined by the major axis of the initial system (see Fig. 1.10). The eccentricity determined like this is called !part.
The average values of !std and !part are rather similar for all but the most peripheral collisions for interactions of heavy nuclei like Au+Au. For smaller systems, however, fluctuations in the nucleon positions become quite important for all centralities and the average eccentricity can vary significantly depending on how it is calculated [17].
1.4 Model description
In this section a brief description will be given of the theory and phenomenological models used in comparison with the measurements.
1.4.1 Hydrodynamics
Hydrodynamics is a macroscopic approach to describe the dynamical evolution of the ex- pansion stage of a heavy-ion collision. In the model, it is assumed that shortly after the
2 at RHIC result in more than twice as many hadrons pro-
duced in the direction of the reaction plane, than orthog- onal to it (v2(pT ∼ 2 GeV) > 0.2). The transverse mo- mentum integrated v2 reaches v2 " O(0.1). Remarkably, the experimental praxis of v2-measurements indicates that the parametric bounds on v2{2}, v2{4}, ... v2{n} given above provide realistic numerical estimates for the feasibility of v2 measurements if one uses for nmult val- ues of order of the charge multiplicity per unit rapidity dNch/dy. We summarize this information in Table I for nmult = 30, 50 and 80. These values of nmult are smaller than dNch/dy in sufficiently central heavy (Au) collisions, but they are comparable to the values in semi-peripheral collisions of lighter (Cu) nuclei.
np = 2 np = 4
v2 (nmult = 30) > 0.18 > 0.09
v2 (nmult = 50) > 0.14 > 0.05
v2 (nmult = 80) > 0.11 > 0.04
TABLE I: Estimates of the minimal signal strength v2{np}, which can be discriminated from non-flow effects in an np- cumulant analysis based on nmult particles.
The interpretation of elliptic flow measurements in heavy ion collisions relies on the observation that v2 is correlated with the initial spatial eccentricity ! of the transverse overlap region of the two projectiles [1] ,
! =
!y!2"
− !
x!2"
$y!2% + $x!2% . (3) Here, averages are performed with respect to the mat- ter distribution right after the collision, and x! and y! denote the lengths along the main axis of an ellipsoid de- scribing this distribution. For sufficiently central heavy (Au) and lighter (Cu) ion collisions, it is found that v2 ∝ ! [10, 11]. Remarkably, this is a generic expecta- tion of fluid dynamic simulations of heavy ion collisions.
Models of ideal dissipation-free hydrodynamics, which by construction describe collision scenarios of maximal col- lective flow, can account quantitatively for the size of the elliptic flow measured at RHIC [12–14] and the expected dissipative corrections are anomalously small [15–18]. In conjuction with this interpretation, the observation of very large v2-signals [5–7] is arguably one of the most far reaching discoveries of the RHIC heavy ion program.
We now turn to the question whether elliptic flow may be measurable in p-p collisions at the LHC. In p-p col- lisions studied so far, one may explain the apparent ab- sence of an elliptic flow signal by pointing to the fact that the dNch/dy in these collisions is too low to make v2 measurable (see Table I). However, while Monte Carlo simulations for minimum bias dNch/dy distributions in
√s = 14 TeV p-p collisions peak at low values < 10 for the non-diffractive contribution, they show a pronounced
y
x
rp
r ~6 fmN y Central A−A
x’ y’
r ~ 0.56 fmp r0
x Central p−p
FIG. 1: Schematic view of region of hadron production may be located in the transverse overlap region of a central proton- proton and central nucleus-nucleus collision respectively. De- pending on the number and size of hadronically active regions, large eccentricities can result even in central collisions.
high-multiplicity tail, typically reaching values as high as dNch/dy ∼ 60. Despite their model dependence, these simulations strongly indicate that abundant samples of high-multiplicity p-p events with dNch/dy ≥ 50 will be measured at the LHC. Such a multiplicity is comparable to that reached in semi-peripheral (centrality class 40 - 60 %) Cu-Cu collisions at √sNN = 62.4 GeV at RHIC, and for these latter collisions elliptic flow has been mea- sured. Whether elliptic flow is also measurable in high- multiplicity p-p event sample at the LHC then depends on the signal strength v2 and on the relative strength of non-flow corrections in p-p collisions.
To estimate the strength of the elliptic flow signal v2, we now discuss the initial spatial eccentricity ! of hadronic collisions. In the collisions of heavy (Au or Pb) ions, this eccentricity is determined solely by the trans- verse spatial overlap. More precisely, in a nucleus-nucleus collisions dNch/dy scales approximately with the average number Npart of participant nucleons, which scales with the area of the nuclear overlap. Therefore, selecting a multiplicity class in A-A amounts to selecting on impact parameter and determines the shape of the nuclear over- lap region. For large Npart, it is reasonable to make the smoothness assumption that the interactions between the Npart nucleons result in a homogeneous density distribu- tion within the area of the nuclear overlap (for illustra- tion, see right hand side of Fig. 1). If this assumption would carry over to p-p collisions, then the highest mul- tiplicity p-p collisions would be the most central ones, their spatial eccentricity would be close to zero, and so would be the flow signal v2 ∝ !. Previous estimates of the magnitude of v2 were based on this smoothness as- sumption [19, 20] or on other methods [21] and reported small, non-measurable values.
However, sizable deviations from the smoothness as- sumption have been found in modeling lighter (Cu) ion collisions [10, 11]. In these systems, the relatively small number of nucleon-nucleon interactions results in event- by-event fluctuations of the density distribution which
๏ Relation of flow in different systems
•
Take into account event by event•
v
2 scales with eccentricity (geometrical property)•
v
2 is a function of transverse density (density interactions)๏ Scaling over a large variety of systems and collision energies.
Heavy Ion Collisions J. Casalderrey-Solana TAE 2016
Microscopic Modeling
34
๏ If we understand the microsocpic d.o.f and their interactions:
Boltzmann eq:
•
Can be derived at very high temperatures in QCD (g(T)≪ 1)
Collision rates
computed via underlying theory
•
In experiments, applicability is questionable (T<0.6 GeV)•
Leading order (2-2) computations underpredicts flow effects๏ Can we have a model independent description?
•
Yes, provided we assume interactions are extremely frequent
microscopic scale variation rate of density (size)
Hydrodynamics
Heavy Ion Collisions J. Casalderrey-Solana TAE 2016
Relativistic Hydrodynamics
35
๏ The stress tensor of any (homogeneous, isotropic..) theory after all process settle
e.o.s
boosting to another frame velocity of the frame
๏ Hydro approximation: assume all variation of stress tensor are very small compared to microscopic scales gradient expansion
๏ Ideal approximation: even in dynamical situation
•
At the scale of the variation all micro-process are localwith (5) dynamical fields
๏ And 5 equations
Enough to solve dynamics
Heavy Ion Collisions J. Casalderrey-Solana TAE 2016
Viscosity
36
๏ Hydrodynamics: systematic approach gradient expansion
•
Expand to leading order in gradientsshear
viscosity bulk
viscosity
Two different symmetric structure easier in fluid rest frame
•
These are transport coefficients: Intrinsic properties of the theory๏ Same interpretation as in non-relativistic fluid Dissipative properties: shear viscosity
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η
s > 1 4π
€
1
4π 1+
135 ς(3)
8 (2λ)3 / 2 + ...
%
&
' (
) *
Strong coupling limit of N=4 SYM
Kovtun, Son, Starinets, hep-th/0309213
Arnold, Moore, Yaffe, JHEP 11 (2000) 001
€
λ ≡ g2Nc 1st lattice QCD study H. Meyer, 0805.4567
Calculable from first principles in quantum field theory (QCD)
€
Gxy,xyR (ω,0) ≡
∫
dt dx eiωt Θ( )t[
Txy(t,x),Txy(0,0)]
eq€
η ≡ −lim
ω →0
1
ω ImGxy,xyR (ω,0)
η
First rigorous example that a finite T QFT can evolve with minimal dissipation.
0 )
(
T x T (e p )uu ( p )g
u
T T
2 2 1
u
T u T
2 ) 1
(
viscous hydrodynamics
) ( p
Input: “EOS”
Shear viscosity: Bulk viscosity:
Assume zero net baryon density & heat conductivity at RHIC and LHC
Shear viscosity Bulk viscosity
Heavy Ion Collisions J. Casalderrey-Solana TAE 2016
Computations of Viscosity
37
๏ pQCD estimate via transport
stochastic process
Momentum transfer
•
Shear viscosity•
Bulk viscosity(QCD is approximately conformal)
•
Kinematic viscosity:•
Large for weakly interacting plasma•
Small for strongly interacting plasmaHeavy Ion Collisions J. Casalderrey-Solana TAE 2016
Field Theory Extractions
38
๏ Transport coefficients can be computed via correlation functions
•
Derived via linear response:equate to hydro response metric perturbation (source)
๏ General expression but... hard to use in QCD
•
small ω problematic in thermal perturbation theory (IR problems)•
Real time dynamics: hard for lattice computations๏ Can we extract this from data?
Kubo Relation
Heavy Ion Collisions J. Casalderrey-Solana TAE 2016
Hydrodynamic Modeling
39
๏ Initial value of fields
•
Most simulations assume boost invariant distributions•
Initial energy density from dynamical model or empirical•
Initiate hydrodynamics at some time after collision (model parameter)๏ Hydro solver
•
Needs e.o.s (usually taken from lattice computations)๏ Decoupling
•
At Tc~165 MeV hadrons decoupled quickly thermal abundance•
Evolved with Boltzmann equation + known cross sectionsHeavy Ion Collisions J. Casalderrey-Solana TAE 2016
Success of Hydro
40
Hydro describes spectra @ LHC
14
Text
Identified particle spectra show clear evidence of thermalization and flow.
Kinetic freeze-out is cooler
and faster flowing than @ RHIC.
Tuesday, July 17, 12
Shear viscosity
19 Song, Bass, Heinz, Hirano, Shen, PRL 106 (2011) 192301
Conclusion: 1 ≤ 4πη/s ≤ 2.5
Remaining uncertainty mainly due to initial density profile
Tuesday, July 17, 12
0 0.05 0.1 0.15 0.2
0 0.5 1 1.5 2 2.5
pT [GeV]
v2
Au+Au 200 GeV 30-40% central STAR data
/s=0 /s=0.08 /s=0.16
Elliptic flow “measures” η
QGP17
η/s = 1/4π η/s = 2/4π η/s = 0
Universal strong coupling limit of non-abelian gauge theories with a gravity dual:
η/s → 1/4π
aka: the “perfect” liquid
Schenke, Jeon, Gale, PRL 106 (2011) 042301
Triangular flow
Consistency check:
0
✓
0.02 0.04 0.06 0.08
0 0.5 1 1.5 2 2.5 3
pT [GeV] v3
Au+Au 200 GeV 10-20% central STAR data
/s=0 /s=0.08 /s=0.16
Tuesday, July 17, 12
Shear viscosity
19 Song, Bass, Heinz, Hirano, Shen, PRL 106 (2011) 192301
Conclusion: 1 ≤ 4πη/s ≤ 2.5
Remaining uncertainty mainly due to initial density profile
Tuesday, July 17, 12
0 0.05 0.1 0.15 0.2
0 0.5 1 1.5 2 2.5
pT [GeV]
v2
Au+Au 200 GeV 30-40% central STAR data
/s=0 /s=0.08 /s=0.16
Elliptic flow “measures” η QGP
17
η/s = 1/4π η/s = 2/4π η/s = 0
Universal strong coupling limit of non-abelian gauge theories with a gravity dual:
η/s → 1/4π
aka: the “perfect” liquid
Schenke, Jeon, Gale, PRL 106 (2011) 042301
Triangular flow
Consistency check:
0
✓
0.02 0.04 0.06 0.08
0 0.5 1 1.5 2 2.5 3
pT [GeV] v3
Au+Au 200 GeV 10-20% central STAR data
/s=0 /s=0.08 /s=0.16
Tuesday, July 17, 12
Heavy Ion Collisions J. Casalderrey-Solana TAE 2016
Hydro from Early Times
41
๏ Hydro works but... from when
•
Typical simulations at the LHC havesize of the medium micro scale
•
It does not seem as if hydro should work...•
yet...Heavy Ion Collisions J. Casalderrey-Solana TAE 2016
Extracting QCD properties from Data
42
4
FIG. 5. (a) Fifty equations of state were generated by ran- domly choosing X0 and R in Eq. (2) from the prior distribu- tion and weighted by the posterior likelihood (b). The two upper thick lines in each figure represent the range of lat- tice equations of state shown in [4], and the lower thick line shows the equation of state of a non-interacting hadron gas.
This suggests that the matter created in heavy-ion collisions at RHIC and at the LHC has a pressure that is similar, or slightly softer, to that expected from equilibrated matter.
rameters that determine the equation of state is shown in Fig. 4. As a function of X0 and R defined in Eq.
(2), the likelihood is significant for a large band near the diagonal. Higher values of X0, which delays the ap- proach of the speed of sound to one third until higher energy densities and makes the equation of state softer, can be compensated by higher values of R, which sends the speed of sound higher just above Tc and makes the equation of state sti↵er. Fifty values of X0 and R were then taken randomly from both the prior, and weighted by the posterior likelihood. For each case the speed of sound is plotted as a function of the temperature in Fig.
2. It is clear that the experimental results significantly constrain the equation of state and we also note that the RHIC and LHC data in combination provide a bet-
ter constraint than either can alone. It appears that the speed of sound cannot fall much below the hadron gas value, ⇠ 0.15, for any extended range and that it must rise with temperature. Figure 5 also shows a range of equations of state from lattice calculations [4]. The equations of state found here show a preference for being slightly softer than those from the lattice, but the ranges overlap.
IV. CONCLUSIONS
Determining the equation of state from experiment has proven difficult due to the intertwined links between model parameters and numerous observables. The sta- tistical techniques applied here overcome these difficul- ties. The resulting constraints suggest the speed of sound gradually rises as a function of temperature from the hadron gas value. The band of equations of state from Fig. 5 is modestly softer than that of lattice calculations, but has significant overlap. This analysis strengthens the supposition that the matter created in relativistic heavy ion collisions has properties similar to that of equilibrated matter according to lattice calculations and shows that our model describes the dynamics of heavy ion collisions well enough to permit the extraction the thermodynamic and transport properties of equilibrium condensed QCD matter.
ACKNOWLEDGMENTS
This work was supported by the National Science Foundation’s Cyber-Enabled Discovery and Innovation Program through grant NSF-0941373 and by the Depart- ment of Energy Office of Science through grant number DE-FG02-03ER41259. The authors thank Ron Soltz for providing the lattice data.
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๏ Example: is data sensitive to known QCD propertie?
•
Combined hydro fits of collision data of different species and energies constraint the equation of statePratt, Sangaline, Sorensen and Wang 15
Heavy Ion Collisions J. Casalderrey-Solana TAE 2016
Extracting Viscosities
43