Fluid dynamics, viscosity and flow in ultrarelativistic ...users.jyu.fi/~tulappi/hitupva18puheet/niemi_hitu2018.pdfFluid dynamics, viscosity and ow in ultrarelativistic heavy-ion collisions

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


Initial states comes in all sizes and shapes:

Initial temperature profiles in transverse plane (orthogonal to the beam axis)


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


Fluid dynamical evolution: spatial structure −→ momentum space


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)


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


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.


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


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)


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

.


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


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


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


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

,


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.


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.


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


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


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))


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 〉


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


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.


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


Documents

Fluid dynamics, viscosity and flow in ultrarelativistic ...users.jyu.fi/~tulappi/hitupva18puheet/niemi_hitu2018.pdfFluid dynamics, viscosity and ow in ultrarelativistic heavy-ion collisions