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Fluid dynamics, viscosity and flow in ultrarelativistic heavy-ioncollisions
Harri Niemi
Department of Physics, University of Jyvaskyla
Hiukkasfysiikan paiva 23.11.2018
based onHN, K. J. Eskola and R. Paatelainen, PRC 93, 024907 (2016), arXiv:1505.02677
HN, K. J. Eskola, R. Paatelainen and K. Tuominen, PRC 93, 014912 (2016), arXiv:1511.04296HN, K. J. Eskola, R. Paatelainen and K. Tuominen, PRC 93, 014912 (2016), arXiv:1511.04296
K. J. Eskola, HN, R. Paatelainen and K. Tuominen, PRC 97, no. 3, 034911 (2018), arXiv:1711.09803
withKari J. Eskola, University of Jyvaskyla
Risto Paatelainen, CERNKimmo Tuominen, University of Helsinki
Harri Niemi pQCD + saturation + hydro
EKRT model: pQCD + saturation + hydro
QCD based computation of ET production−→ Initial conditions for fluid dynamics−→ Predicted (
√s,A, centrality)-dependence
Fluid dynamical evolution
Cooper-Frye freeze-out
−→ Constraints for η/s(T ) from experimental data
Harri Niemi pQCD + saturation + hydro
Initial states comes in all sizes and shapes:
Initial temperature profiles in transverse plane (orthogonal to the beam axis)
Harri Niemi pQCD + saturation + hydro
Global shape of the initial densityprofile characterized by eccentricities:
εneinΦn = {rne inφ}/{rn}
{(· · · )} =
∫dxdye(x , y , τ0)(· · · )
εn = eccentricityΦn = “participant plane” angle
n = 2 n = 3 n = 4
Harri Niemi pQCD + saturation + hydro
Fluid dynamical evolution: spatial structure −→ momentum space
Harri Niemi pQCD + saturation + hydro
Characterize azimuthal momentum distribution by its Fourier coefficients vn:
εn, Φn −→ vn, Ψn
−→
dN
dydφ=
dN
dy(1 + 2vn cos[n(φ−Ψn)])
Conversion efficiency from εn to vn depends on the properties of QCD matter(viscosity, equation of state)
Harri Niemi pQCD + saturation + hydro
Initial energy density from the EKRT model
NLO pQCD calculation of transverse energy ET
EPS09 nuclear parton distributions (Eskola et. al. JHEP 0904, 065 (2009))with impact parameter dependence (Helenius et. al. JHEP 1207 073 (2012))
dσAB→kl··· ∼ fi/A(x1,Q2)⊗ fj/B(x2,Q
2)⊗ σ
Essential quantity σ 〈ET 〉 with pT cut-off p0
σ 〈ET 〉 (p0,∆y , β) =
∫ √s
0
dETETdσ
dETθ(yi ∈ ∆y , pT > p0,ET > βp0)
e =dET
τ0∆yd2s= TA(s− b
2)TA(s +
b
2)σ 〈ET 〉p0,∆y
τ0∆y
TA = (fluctuating) nucleon density
Harri Niemi pQCD + saturation + hydro
Local saturation condition
Lower cut-off p0 determined from a local saturation condition
dET
d2s= TA(s− b
2)TA(s +
b
2)σ 〈ET 〉p0,∆y =
Ksat
πp3
0∆y
10-1 100 101
TA TA [fm−4 ]
1
2
3
psa
t [G
eV] Ksat =0.50 β=0.8
LHC 2.76 TeV Pb +Pb
RHIC 200 GeV Au +Au
C(a+TA TA )n −ban
The full calculation can besummarized by a simpleparametrization
Event-by-event fluctuationsthrough fluctuations in TATA.
Energy density at time τ0 = 1/psat
e(s, τ0 = 1/psat) = Ksatpsat(s)4/π
Ksat free parameter that needto be fixed once.
Harri Niemi pQCD + saturation + hydro
Model the space-time evolution of A+A collisions by relativistic fluid dynamics:
Neglect net-baryon number, bulk viscosity & heat flow
Conservation of energy and momentum:
∂µTµν = 0
Viscosity (Israel-Stewart theory):
Dπ〈µν〉 = − 1
τπ
(πµν − 2η∇〈µuν〉
)− 4
3πµν
(∇λuλ
)− 10
7π〈µλ σν〉λ
Longitudinal expansion is treated using boost invariance: ∂p∂ηs
= 0, vz = zt
Equation of state: Petreczky/Huovinen: NPA 837, 26-53 (2010)
Chemical freeze-out T = 175 MeV, kinetic T = 100 MeV
Harri Niemi pQCD + saturation + hydro
Transverse momentum spectrum: Cooper-Frye integral over decoupling surface
EdN
d3k=
dN
dyd2pT=
∫Σ
dΣµkµf (x , k),
where f (x , k) is the particle momentum distribution at the decopling.
Need to convert fluid quantities to momentumdistributions:Tµν −→ f (x , k)
Momentum distribution function: 14-moment approximation
fk = f0k + δfk = f0k + f0k1
2(e + p)T 2k〈µkν〉π
µν
is consistent with Tµν :
Tµν(x) =
∫dKkµkν f (x , k)
Harri Niemi pQCD + saturation + hydro
Temperature dependent η/s
100 150 200 250 300 350 400 450 500T [MeV]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
η/s
η/s=0.20
η/s=param1
η/s=param2
η/s=param3
η/s=param4
tuned to reproduce v2{2} at LHC mid-peripheral collisions.
relaxation time τπ(T ) = 5ηε+p
.
Harri Niemi pQCD + saturation + hydro
charged hadron multiplicity, LHC 2.76 TeV and RHIC 200GeV
0 10 20 30 40 50 60 70 80centrality [%]
102
103
dN
ch/dη |η|<
0.5 (a)
LHC 2.76 TeV Pb +Pb
ALICE
η/s=0.20
η/s=param1
η/s=param2
η/s=param3
η/s=param4
0 10 20 30 40 50 60 70 80centrality [%]
101
102
103
dN
ch/dη |η|<
0.5
(b)
RHIC 200 GeV Au +Au
η/s=0.20
η/s=param1
η/s=param2
η/s=param3
η/s=param4
STAR
PHENIX
Important check for the initial conditions
EKRT model: centrality,√s and nuclear mass number dependence
Harri Niemi pQCD + saturation + hydro
Flow coefficients, LHC 2.76 TeV and RHIC 200 GeV
0 10 20 30 40 50 60 70 80centrality [%]
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
v n{ 2}
(a)
pT =[0.2…5.0] GeV
LHC 2.76 TeV Pb +Pb
η/s=0.20
η/s=param1
η/s=param2
η/s=param3
η/s=param4
ALICE vn
{2}
0 10 20 30 40 50 60 70 80centrality [%]
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
v 2/3
{ 2} ,v4
{ 3}
(b)
pT =[0.15…2.0] GeV
RHIC
200 GeV
Au +Au
η/s=0.20
η/s=param1
η/s=param2
η/s=param3
η/s=param4
STAR v2
{2}
STAR v3
{2}
STAR v4
{3}
The magnitude of the flow coefficients is the most direct constrain for η/s√s dependence −→ constraints for T -dependence
Harri Niemi pQCD + saturation + hydro
Flow fluctuations: δv2 = v2−〈v2〉〈v2〉
1.0 0.5 0.0 0.5 1.0 1.5δε2 , δv2
10-2
10-1
100
P(δv 2
), P(δε 2
)
5−10 %
δv2
δε12
δε2
ATLAS
1.0 0.5 0.0 0.5 1.0 1.5δε2 , δv2
10-2
10-1
100
P(δv 2
), P(δε 2
)
35−40 %
δv2
δε12
δε2
ATLAS
1.0 0.5 0.0 0.5 1.0 1.5δε2 , δv2
10-2
10-1
100
P(δv 2
), P(δε 2
)
55−60 %
δv2
δε12
δε2
ATLAS
Relative flow fluctuation spectrainsensitive to shear viscosity η/s.
driven mainly by the initial eccentricityfluctuations
−→ direct constrain for the initialconditions
Harri Niemi pQCD + saturation + hydro
Flow coefficients vn and the corresponding event-plane angles Ψn are notindependent
Strong correlations due to correlations in initial eccentricities εn and dueto the non-linear fluid dynamical evolution
−→ further independent constraints to initial conditions and viscosity
Event-plane correlations:
〈cos(k1Ψ1 + · · ·+ nknΨn)〉SP ≡〈v |k1|
1 · · · v |kn|n cos(k1Ψ1 + · · ·+ nknΨn)〉ev√〈v2|k1|
1 〉ev · · · 〈v2|kn|n 〉ev
,
Harri Niemi pQCD + saturation + hydro
Event-plane correlations: 2 angles
0 10 20 30 40 50 60 70centrality [%]
0.00.10.20.30.40.50.60.70.80.9
⟨ cos(
4(Ψ
2−
Ψ4))⟩ S
P
0 10 20 30 40 50 60 70centrality [%]
0.00.10.20.30.40.50.60.70.80.9
⟨ cos(
8(Ψ
2−
Ψ4))⟩ S
P
0 10 20 30 40 50 60 70centrality [%]
0.00.10.20.30.40.50.60.70.80.9
⟨ cos(
12(Ψ
2−
Ψ4))⟩ S
P
0 10 20 30 40 50 60 70centrality [%]
0.10.00.10.20.30.40.50.60.70.8
⟨ cos(
6(Ψ
2−
Ψ3))⟩ S
P
pT =[0.5…5.0] GeV
η/s=0.20
η/s=param1
η/s=param2
η/s=param3
η/s=param4
ATLAS
0 10 20 30 40 50 60 70centrality [%]
0.10.00.10.20.30.40.50.60.70.8
⟨ cos(
6(Ψ
2−
Ψ6))⟩ S
P
0 10 20 30 40 50 60 70centrality [%]
0.10.00.10.20.30.40.50.60.70.8
⟨ cos(
6(Ψ
3−
Ψ6))⟩ S
PAlready from the LHC data more constraints to η/s(T ).
Small hadronic viscosity needed to reproduce the data.
Harri Niemi pQCD + saturation + hydro
Event-plane correlations: 3 angles
0 10 20 30 40 50 60centrality [%]
0.00.10.20.30.40.50.60.70.80.9
⟨ cos(
2Ψ2
+3Ψ
3−
5Ψ5)⟩ S
P
pT =[0.5…5.0] GeV
0 10 20 30 40 50 60centrality [%]
0.00.10.20.30.40.50.60.70.80.9
⟨ cos(
2Ψ2
+4Ψ
4−
6Ψ6)⟩ S
P
0 10 20 30 40 50 60centrality [%]
0.90.80.70.60.50.40.30.20.10.0
⟨ cos(
2Ψ2−
6Ψ3
+4Ψ
4)⟩ S
P
0 10 20 30 40 50 60centrality [%]
0.10.00.10.20.30.40.50.60.70.8
⟨ cos(−
8Ψ
2+
3Ψ3
+5Ψ
5)⟩ S
P
η/s=0.20
η/s=param1
η/s=param2
η/s=param3
η/s=param4
ATLAS
0 10 20 30 40 50 60centrality [%]
0.10.00.10.20.30.40.50.60.70.8
⟨ cos(−
10Ψ
2+
4Ψ4
+6Ψ
6)⟩ S
P
0 10 20 30 40 50 60centrality [%]
0.10.00.10.20.30.40.50.60.70.8
⟨ cos(−
10Ψ
2+
6Ψ3
+4Ψ
4)⟩ S
PEqually well described by the same parametrizations that describe 2-anglecorrelations.
Harri Niemi pQCD + saturation + hydro
0 20 40 60 80
centrality [%]
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16v n{2}
(a)
pT = [0.2 . . . 5.0] GeV
LHC 2.76 TeV Pb + Pb
η/s = 0.20
η/s = param1
ALICE vn{2}
0 10 20 30 40 50 60 70 80
centrality [%]
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
v 2/3
{ 2} ,v4
{ 3}
(b)
pT = [0. 15 2. 0]GeV
RHIC
200GeV
Au +Au
η/s= 0. 20
η/s= param1
STARv2
{2}
STARv3
{2}
STARv4
{3}
100 150 200 250 300 350 400 450 500T [MeV]
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
η/s
η/s=0.20
η/s=param1
0 10 20 30 40 50 60 70centrality [%]
0.00.10.20.30.40.50.60.70.80.9
⟨ cos(
4(Ψ
2−
Ψ4))⟩ S
P
LHC 2.76 TeV
Pb +Pb
0 10 20 30 40 50 60 70centrality [%]
0.00.10.20.30.40.50.60.70.80.9
⟨ cos(
8(Ψ
2−
Ψ4))⟩ S
P
0 10 20 30 40 50 60 70centrality [%]
0.00.10.20.30.40.50.60.70.80.9
⟨ cos(
12(Ψ
2−
Ψ4))⟩ S
P
0 10 20 30 40 50 60 70centrality [%]
0.10.00.10.20.30.40.50.60.70.8
⟨ cos(
6(Ψ
2−
Ψ3))⟩ S
P
pT =[0.5 5.0] GeV
η/s=0.20
η/s=param1
ATLAS
0 10 20 30 40 50 60 70centrality [%]
0.10.00.10.20.30.40.50.60.70.8
⟨ cos(
6(Ψ
2−
Ψ6))⟩ S
P
0 10 20 30 40 50 60 70centrality [%]
0.10.00.10.20.30.40.50.60.70.8
⟨ cos(
6(Ψ
3−
Ψ6))⟩ S
P
Harri Niemi pQCD + saturation + hydro
dNch
dη and vn predictions: 5.023 TeV Pb+Pb
0 10 20 30 40 50 60 70 80
centrality [%]
102
103
dN
ch/dη|η|<
0.5
η/s = 0.20
η/s = param1
ALICE 5.023 TeV
ALICE 2.76 TeV
STAR
PHENIX
20 40
centrality [%]
0.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
v n{2}(
5.02
3T
eV)/v n{2}(
2.76
TeV
)
(a)
n = 2
η/s = 0.20
η/s = param1
ALICE
20 40
centrality [%]
pT = [0.2 . . . 5.0] GeV
LHC Pb + Pb
(b)
n = 3
20 40
centrality [%]
(c)
n = 4
The framework now essentially fixed (no retuning of parameters)
Both the charged hadron multiplicity and the sligth inreaces in vn’scorrectly predicted
Harri Niemi pQCD + saturation + hydro
dNch
dη and vn predictions: 5.44 TeV Xe+Xe
0 10 20 30 40 50 60 70 80
centrality [%]
102
103
dN
ch/dη|η|<
0.5
Xe + Xe
η/s = 0.20
η/s = param1
ALICE 5.023 TeV
ALICE 5.44 TeV Xe
ALICE 2.76 TeV
STAR
PHENIX
20 40 60
centrality [%]
0.9
1.0
1.1
1.2
1.3
1.4
r n
(a)
rn = vn{2} (Xe+Xe, 5.44 TeV)vn{2} (Pb+Pb, 5.02 TeV)
n = 2
η/s = 0.20
η/s = param1
symmetric Xe
20 40 60
centrality [%]
pT = [0.2 . . . 3.0] GeV
(b)
n = 3
ALICE
20 40 60
centrality [%]
(c)
Deformed Xe :
β2 = 0.162 β4 = −0.003
n = 4
Charged hadron multiplicity: error band from the experimental error in0-5% 2.76 TeV multiplicity measurement (where EKRT model parameterKsat fixed)
The vn ratio with and without nuclear deformation
Quite strong influence of the Xe nuclear deformation on the vn ratio. (Asnoted in G. Giacalone et. al. PRC 97, 034904 (2018))
Harri Niemi pQCD + saturation + hydro
Symmetric cumulants
SC (n,m) = 〈cos(mφ1 + nφ2 −mφ3 − nφ4)〉− 〈cos(m(φ1 − φ2))〉〈cos(n(φ1 − φ2))〉= 〈v2
mv2n 〉 − 〈v2
m〉〈v2n 〉
NSC (n,m) =SC (n,m)
〈v2m〉〈v2
n 〉
Harri Niemi pQCD + saturation + hydro
Symmetric cumulants SC(n,m)
0 10 20 30 40 50 60 70
centrality [%]
0.0000000
0.0000005
0.0000010
0.0000015
0.0000020
0.0000025
0.0000030
〈v2 4v
2 2〉−〈v
2 4〉〈v2 2〉
η/s = 0.20η/s = param1ALICE
0 10 20 30 40 50 60 70
centrality [%]
−0.0000025
−0.0000020
−0.0000015
−0.0000010
−0.0000005
0.0000000
〈v2 3v
2 2〉−〈v
2 3〉〈v2 2〉
LHC2.76TeVPb + Pb
pT = 0.2 . . .5.0GeV
0 10 20 30 40 50 60 70
centrality [%]
−0.5
0.0
0.5
1.0
1.5
2.0
〈v2 5v
2 2〉−〈v
2 5〉〈v2 2〉
×10−7
LHC2.76TeVPb + Pb
0 10 20 30 40 50 60 70
centrality [%]
−9
−8
−7
−6
−5
−4
−3
−2
−1
0
〈v2 4v
2 3〉−〈v
2 4〉〈v2 3〉
×10−8
LHC2.76TeVPb + Pb
0 10 20 30 40 50 60 70
centrality [%]
−1
0
1
2
3
4
5
6
7
〈v2 5v
2 3〉−〈v
2 5〉〈v2 3〉
×10−8
LHC2.76TeVPb + Pb
0 10 20 30 40 50 60 70
centrality [%]
−0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
〈v2 4v
2 6〉−〈v
2 4〉〈v2 6〉
×10−9
LHC2.76TeVPb + Pb
Correlation between the magnitudes of vn (independent of the event-planeangles Ψn)
SC(n,m) also very sensitive to the absolute values of vn’s
Harri Niemi pQCD + saturation + hydro
Normalized symmetric cumulants NSC(n,m)
0 10 20 30 40 50 60 70
centrality [%]
0.0
0.2
0.4
0.6
0.8
1.0
〈v2 4v
2 2〉/〈v
2 4〉〈v2 2〉−
1
η/s = 0.20η/s = param1ALICE
0 10 20 30 40 50 60 70
centrality [%]
−0.25
−0.20
−0.15
−0.10
−0.05
0.00
〈v2 3v
2 2〉/〈v
2 3〉〈v2 2〉−
1
LHC2.76TeVPb + Pb
pT = 0.2 . . .5.0GeV
0 10 20 30 40 50 60 70
centrality [%]
0.0
0.1
0.2
0.3
0.4
0.5
〈v2 5v
2 2〉/〈v
2 5〉〈v2 2〉−
1
LHC2.76TeVPb + Pb
0 10 20 30 40 50 60 70
centrality [%]
−0.30
−0.25
−0.20
−0.15
−0.10
−0.05
0.00
0.05
〈v2 4v
2 3〉/〈v
2 4〉〈v2 3〉−
1
LHC2.76TeVPb + Pb
0 10 20 30 40 50 60 70
centrality [%]
0.00
0.25
0.50
0.75
1.00
1.25
1.50
〈v2 5v
2 3〉/〈v
2 5〉〈v2 3〉−
1
LHC2.76TeVPb + Pb
0 10 20 30 40 50 60 70
centrality [%]
0.0
0.2
0.4
0.6
0.8
1.0
1.2
〈v2 4v
2 6〉/〈v
2 4〉〈v2 6〉−
1
LHC2.76TeVPb + Pb
Correlation between the magnitudes of vn (independent of the event-planeangles Ψn)
measures the correlation, do not directly depend on the absolute values ofvn.
Harri Niemi pQCD + saturation + hydro
Summary
Presented a new EbyE framework for NLO pQCD + saturation & viscoushydro
The computed√s and centrality dependence of dNch/dη agree very well
with LHC and RHIC data: predictive power!
Most direct constraints for the IS come from the v2 fluctuations and theratio v2/v3 both are now very well reproduced!
LHC vn’s alone do not stringently constrain the T -dependence of η/s
Further constraints for η/s(T ) from the vns at RHIC and the EPcorrelations at the LHC
η/s = 0.2 (blue) and param1 with minimum at T = 150 MeV (black) andsmall hadronic η/s work best in our framework
Harri Niemi pQCD + saturation + hydro